I 
 
- 
 
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 SAVAGE 
 
 L l D K.AK. I 
 
 OF THE 
 
 ILN I V E R.5 ITY 
 Of ILLINOIS 
 
 622. OS 
 C63c9 
 
The person charging this material is re- 
 sponsible for its return on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 are reasons for disciplinary action and may 
 result in dismissal from the University. 
 
 University of Illinois Library 
 
 L161— 0-1096 
 
' 
 
THE 
 
 T. E, SAVAGE. 
 
 COAL^METAL MINERS’ 
 POCKETBOOK 
 
 OF 
 
 . PRINCIPLES, RULES, FORMULAS, 
 
 AND TABLES. 
 
 SPECIALLY COMPILED AND PREPARED FOR THE CONVENIENT USE OF 
 MINE OFFICIALS, MINING ENGINEERS, AND STUDENTS PREPAR- 
 ING THEMSELVES FOR CERTIFICATES OF COMPETENCY 
 AS MINE INSPECTORS OR MINE FOREMEN. 
 
 NINTH EDITION: REVISED AND ENLARGED, 
 
 WITH 
 
 ORIGINAL MATTER. 
 
 “Though index learning turns no student pale, 
 It grasps the Eel ol Science by the tail.” 
 
 Pope. 
 
 SCRANTON, PA.: 
 
 INTERNATIONAL TEXTBOOK COMPANY. 
 
 1907 . 
 
Copyright, 1890 , 1893 , 1900 , 
 
 BY 
 
 Th,e Colliery Engineer Company. 
 
 Copyright, 1901 , 1905 , 
 
 BY 
 
 International Textbook Company. 
 Entered at Stationers’ Hall, London, 
 All Rights Reserved. 
 
 PRINTED BY 
 
 International textbook Company, 
 Scranton, pa., U. S. A. 
 
£2 
 
 ? T) D 
 
 e.6?c < ? 
 
 o 
 
 eg 
 
 kn ' 
 
 Xs 
 
 Preface to Ninth Edition. 
 
 In this edition, the principal changes are in the section on 
 wire rope, which has been greatly enlarged by the addition 
 of material treating on cableways, tramways, transmission of 
 \ power by wire rope, and wire-rope calculations. A glossary of 
 ~ wire-rope terms has been added which has been checked up 
 by the leading wire-rope makers of the United States, and 
 therefore represents the latest practice. 
 
 A number of changes have been made in this and previous 
 editions, based on suggestions of users of the Pocketbook. 
 We will appreciate any notice of errors or omissions so that 
 future editions may be made as complete and accurate as 
 possible. 
 
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Preface to Eighth Edition. 
 
 The eighth edition of The Coal and Metal Miners’ Pocket- 
 book is an exact reprint of the seventh edition, issued to 
 supply the demand while the ninth edition was being pre~ 
 pared, and in which a number of changes will be made. 
 
 Preface to Seventh Edition. 
 
 The speedy exhaustion of the sixth edition of The Coal and 
 Metal Miners’ Pocketbook, coupled with a steady and increas- 
 ing demand for it, has made necessary this seventh edition. 
 In it a few typographical errors which unavoidably occurred 
 (as is the case in every book of this nature and size), have 
 been corrected, all departments have been carefully revised, 
 improved, and brought up to date, and considerable new 
 matter has been added. Thus, the value of the work has been 
 materially increased. The value of suggestions as to improve- 
 ments in this edition made by users of the sixth edition is 
 acknowledged with thanks, and we request that users of this 
 new, or seventh, edition, will kindly call our attention to any 
 errors or omissions they may discover, so that future editions 
 may be kept fully abreast with the requirements of mining 
 practice as such practice continues to advance. 
 
 v 
 

 ; • 
 
 
 
Preface to Sixth Edition. 
 
 The fifth edition of The Coal and Metal Miners’ Pocketbook 
 was very kindly received, and the criticisms of it were most 
 friendly and flattering. 
 
 The sixth edition has been compiled under particularly 
 favorable circumstances and is much more complete than any 
 previous edition. 
 
 Prominent engineers and manufacturers of mining machinery 
 throughout the world have kindly criticized the previous 
 edition, have suggested wherein it could be improved, and 
 have sent to us information from their private note books 
 that has never before been published. 
 
 The staff of Mines and Minerals, the large force of Mining, 
 Mechanical, and Electrical Engineers connected with The 
 International Correspondence Schools, and many other engi- 
 neers and mine managers have contributed to it. 
 
 All this material has been carefully sifted, verified wherever 
 possible, and combined with the data in the former edition. 
 By careful selection and rewriting, or by different methods of 
 presentation, it has been possible to include essentially all that 
 was in the fifth edition, and at the same time to add from one- 
 third to one-half again as much entirely new matter, without 
 materially increasing the size of the book. 
 
 Every portion of the fifth edition has been either entirely 
 rewritten, or revised, enlarged, and brought up to date. New 
 illustrations have been drawn, and the entire book has been 
 printed from new plates. 
 
 The sections on Mathematics and Surveying have been 
 amplified by the addition of new tables and by text treating 
 of the Solar Transit and Pocky Mountain methods of sur- 
 veying. 
 
 The sections on Hydraulics; the Application of Electricity to 
 Mining; Timbering, Haulage, Blasting, Ore Dressing, and Coal 
 Washing are entirely new. 
 
 vii 
 
viii 
 
 PREFACE . 
 
 The sections on Prospecting, Ventilation, and Methods of 
 Working have been entirely rewritten, enlarged, and greatly 
 improved. 
 
 The tables of Logarithms, Trigonometric Functions, etc. have 
 been reset from the latest corrected editions of standard tables. 
 The Traverse Table has been greatly reduced in length, but 
 without affecting its efficiency, while the table of Squares, 
 Cubes, etc. has been added to by the addition of Circumfer- 
 ences and Areas of Circles. 
 
 The Glossary, which contains about 2,500 words, is believed 
 to be the most complete mining glossary ever published, as it 
 is a combination of all the mining glossaries extant of which 
 the compilers could hear. 
 
 Wherever possible, credit has been given the authorities 
 from whom data have been taken, but in such a work it is mani- 
 festly impossible to give full credit for everything that has 
 been extracted, quoted, and compiled, and we can only in this 
 very general way acknowledge our indebtedness to the large 
 number of authors and engineers whom we have failed to 
 mention by name in the text. 
 
 No one appreciates as fully as does the editor of such a pub- 
 lication the value of the suggestions and data that have been 
 so generously furnished to assist us in the compilation. We 
 shall be greatly obliged to all readers of this volume who may 
 call our attention to any errors that they may discover, or to 
 the omission of any data that they may feel the lack of, so 
 that attention may be given to these matters in future editions. 
 
TABLE OF CONTENTS. 
 
 ( For detailed Index , see back of volume. See also Glossary of Minina Terms, 
 •page 565.) 
 
 WEIGHTS AND MEASURES. 
 
 The Metric System.— l. 
 
 Weights.— Troy, 1; Apothecaries’, 1; Avoirdupois, 2; Metric, 2. 
 
 M easu res of Length.— American and British, 2; Reduction of Inches 
 to Decimals of a Foot, 2; Decimals of a Foot for Each ^ of an Inch, 3; 
 Metric, 3; Russian, 3; Prussian, Danish, and Norwegian, 3; Austrian, 3: 
 Swedish, 4; Chinese, 4. 
 
 Measures of Area.— American and British, 4; Table for Reducing 
 Square Feet to Acres, 4; Metric, 4. 
 
 Measures of Volume.— American and British, 5; Metric, 5; Liquid 
 (U. S.), 5; Dry (U. S.), 5; British Imperial (Liquid and Dry), 5; 
 Contents of Cylinders or Pipes for 1 Foot in Length, 6; Mexican, 
 Central American, and South American Weights and Measures, 7. 
 
 Conversion Tables.— Customary to Metric, 7; Metric to Customary, 9. 
 
 Money.— United States, 10; British, 10; Standard U. S. Coins, Weights and 
 Fineness, Space Required to Store, 10; Conversion of English and 
 American Money Values, 11; Value of Foreign Coins, U. S. Treasury 
 Department, 11; Carat Measures, 12. 
 
 Timber and Board Measure.— Rule for Measuring, 12; Timber 
 Measure, 12; Round Timber, Table of £ Girths, 13; Board Measure, 13. 
 
 Mathematics.— General Principles, 14; Signs and Abbreviations, 14. 
 
 Arithm etic.— To Cast the Nines Out of a Number, 15; To Prove Addition, 
 Subtraction, Multiplication, and Division, 15; Common Fractions, 15; 
 Decimals, 16; Simple and Compound Proportion, 18; Involution. 19; 
 Evolution, 19; Percentage, 20; Arithmetical Progression, 20; Geomet- 
 rical Progression, 21; Logarithms, 22. 
 
 Geometry.— Principles, 24; Practical Problems in Geometrical Construc- 
 tion, 25. 
 
 Mensuration.— Surfaces, 28; Solids, 33; Prismoidal Formula, 34. 
 
 Plane Trigonometry.— Principles of Trigonometry, 34; Practical 
 Examples in the Solution of Triangles, 35. 
 
 Surveying.— The Compass, 38; Adjustments, 38; Use of Compass, 39; 
 Magnetic Variation, 39; Isogonic Chart, 39; The Vernier, 39; The 
 Transit, 40; Adjustments, 41; The Chain, or Steel Tape and Pins, 42; 
 Plumb-Bob, 44; The Clinometer or Slope Level, 44; Field Notes for an 
 Outside Compass Survey, 44; Transit Surveying, 45; Determination of 
 Meridian by Polaris, 46; Determination of Meridian With Solar Attach- 
 ment, 47; Use of Solar Attachment, 47; General Remarks, 49; Plotting 
 49; Coordinates, 51; Contents of Coal Seam, 52; Leveling, 53; Adjust- 
 ments, 53; Use of Level, 54; Field Work 54; Notes, 55; Trigonometric 
 Leveling, 56; Underground Surveying, 56; Establishment and Marking 
 of Stations, 57; Centers, 59; Notes, 60; Stope Books, 62; Mine Corps, 66; 
 Surveying Methods, 67; Outside Surveys, 67; Inside Surveys, 67; Closing 
 Surveys, 68; Connecting Outside and Inside Works Through Shafts 
 and Slopes, 68; Notes on Mapping, 74; Locating Errors, 76; Locating 
 Special Work, 77; Calculation of Areas, 77; Railroad Curves, 78; Hints 
 to Beginners, 80; Theory of Stadia Measurements, 81; Tables, 88. 
 
X 
 
 TABLE OF CONTENTS. 
 
 Fi ruirNTS OF M EC han i CS.— Levers, 91; Wheel and Axle, 92; Inclined 
 E E Plane S 93; Sere w, 93; ^ Wedge, 93; Pulleys, 94; Composition of Forces, 95. 
 
 Friction— Coefficients, 95; Shafting, 96; Friction of Mine Cars, 96. 
 Lubrication— Best Lubricants for Different Purposes, 102. 
 
 APPLIED MECHANICS. 
 
 n__ _ . . _ TlJ Aun WnrwT OF Materials. — Wooden Beams, 102; Iron 
 S and Ste«£ Beamsf lo3? Stractural-Steel I Beams 104; Pillars or Props, 
 105; Cast-Iron Columns, 106; Specific Gravity, Weight, and Properties 
 of Materials, 107; Line Shafting, 110; Weight of Castings, Sheets, and 
 Plates, 111; Weight of Cast-Iron Pipe per Foot m Pounds, 113 ’ Wood 
 Screws, 113; Weight of Wrought Iron, 114; Iron Required for 1 Mile ot 
 Track, 117; Rails, Splices, and Bolts for 1 Mile of Track, 117; Wire and 
 Sheet-Metal Gauges, 122d. 
 
 ... « p e»-n pm l Remarks 118- Flat Ropes, 119; Standard Hoisting 
 
 lUssfep.llES 
 
 127; Ordinary Long Splice, 129; Chains, 129. 
 
 H fD P?e!™Is of ifiqu e ffis r ?n 
 
 W R^ht A*’ ffi| ht V A Nltch V 138 0t &igink Wefrt fssT CoefficffiSt g o h f 
 
 isit 
 
 SSffi water, H3; 
 
 mMMxmm 
 
 mmmm 
 
 Impulse Wheels, 158; Turbines, 158. 
 
 PuMPMaCH,^ 
 
 K^ A S g TM r ^gt|V W^and the H = wer 
 wX r Kaised by' a 
 
 Delivered'^r ^M^utef^fo^V^rhra^Sized ^mps, 16^ Mis^llanTO^w 
 
 KSSSS P A u^KpSpb for Acid Water, 165. 
 
TABLE OF CONTENTS. 
 
 xi 
 
 FORMS OF POWER. 
 
 Fuels.— Table of Combustibles, 166; Analyses and Heating Values of 
 American Coals, 168; Thermal Unit, 168; Composition of Fuels, 169; 
 Classification, Composition, and Properties of Coals, 169; Weights and 
 Measurements of Coal, 170; Coke, 172; Analysis of Coal, 173. 
 
 Steam.— High-Pressure Steam, 175; Expansion of Steam, 176: Condens- 
 ers, 176. 
 
 Bo l leRS.- Lancashire Boiler, 177; Horsepower of Boilers, 177; Heating 
 Surface, 177; Choice of a Boiler, 179; Explosions, 179; Questions to Be 
 Asked Concerning New Boilers, 180; Incrustation and Scale, 182; 
 Covering for Boilers, Steam Pipes, Etc., 183; Data for Proportioning an 
 Economizer, 185; Care of Boilers, 185; Thickness of Boiler Iron, 187; 
 Pressure of Steam at Different Temperatures, 188; Maximum Economy 
 of Plain Cylinder Boilers, 188; Scheme for Boiler Test, 188; Chim- 
 neys, 189. 
 
 Steam Engines.— What Is a Good Engine? 190; Determination of 
 M. E. P., 190; Buies for Engine Drivers, 191; Belting and Velocity of 
 Pulleys, 193. 
 
 Compressed Air.— Classification of Compressors, 194; Construction of 
 Compressors, 194; Theory of Air Compression, 194; Rating of Com- 
 pressors, 195; Cooling, 195; Dry Versus Wet Compressors, 196; Trans- 
 mission of Air in Pipes, 196; Losses in the Transmission of Compressed 
 Air, 198; Friction of Air in Pipes, 201; Loss of Pressure in Pounds per 
 Square Inch, by Flow of Air in Pipes, 202; Loss by Friction in Elbows, 
 203. 
 
 Electricity.— Practical Units, 203; Strength of Current, 203; Electric 
 Pressure or Electromotive Force, 203; Resistance, 203; Power, 204; Cir- 
 cuits, 205; Series Circuits, 205; Parallel Circuits, 206; Resistance in 
 Series and Multiple, 206; Shunt, 207; Electric Wiring, 207; Materials, 
 207; Forms of Conductors, 207; Wire Gauge, 207; Estimation of Cross- 
 Section of Wires, 207; Properties of Copper Wire, 208; Properties of 
 Aluminum and Copper, 209; Estimation of Resistance, 209; Calculation 
 of Wires for Electric Transmission, 210; Current Estimates, 212; Incan- 
 descent Lamps, 213; Arc Lamps, 214; Motors, 214; Conductors for Electric- 
 Haulage Plants, 214; Dynamos and Motors, 215; Direct-Current Dynamos, 
 215; Factors Determining E. M. F. Generated, 218; Field Excitation of 
 Dynamos, 218; Series-Wound Dynamos, 219, Shunt- Wound Dynamos, 
 219; Compound-Wound Dynamos, 219; Direct-Current Motors, 220; 
 Principles of Operation, 220; Speed Regulation of Motors, 222; Connec- 
 tions for Continuous-Current Motors, 223; Alternating-Current Dyna- 
 mos, 224; Single-Phase Alternators, 225; Multiphase Alternators, 225; 
 Uses of Multiphase Alternators, 226; Alternating-Current Motors, 226; 
 Synchronous Motors, 226; Induction. Motors, 227, Transformers, 228; 
 Electric Signaling, 229; Batteries, 229; Bell Wiring, 230; Special Mine 
 Applications, 233; Telephones, 233. 
 
 MINING. 
 
 Prospecting.— Outfit Necessary, 235; Plan of Operations, 236; Geological 
 Table, 237; Coal or Bedded Materials, 238; Formations Likely to Con- 
 tain Coal, 238; Ore Deposits, 238; Position of Veins and Ore Deposits, 239; 
 Underground Prospecting, 239; Prospecting for Placer Deposits, 240; 
 Value of Free Gold per Ton of Quartz, 241; Gems and Precious Stones, 
 241; Exploration by Drilling or Bore Holes, 242; Earth Augers, 242; 
 Percussion or Churn Drills, 242; The Diamond Drill, 243; Selecting the 
 Machine, 243; Size of Tools, 243; Drift of Diamond-Drill Holes, 243; 
 The Surveying of Diamond-Drill Holes, 243; The Value of the Record 
 Furnished by the Diamond Drill, 244; The Arrangement of Holes, 244; 
 The Cost and Speed of Drilling, 244; Records of Cost per Foot in 
 Diamond Drilling, 246; Cost of Operation per Month of Bed-Rock 
 Exploration, 247; Magnetic Prospecting, 248; Prospecting for Petro- 
 leum, Natural Gas, and Bitumen, 249; Construction of Geological Maps 
 and Cross-Sections, 249; To Obtain Dip and Strike From Bore-Hole 
 Records, 250; Sampling and Estimating the Amount of Mineral Avail- 
 able, 251; Diagram for Reporting on Mineral Lands, 252. 
 
Xll 
 
 TABLE OF CONTENTS. 
 
 Opening a Mine —Opening a Gold Mine, 258; Form of Shafts, 259; Com- 
 partments, 259; Shaft Sinking, 259; Size of Shaft, 259; Forepoling, 260; 
 Metal Linings Forced Down, 260; Pneumatic Method of Shaft Sinking, 
 260; Freezing Process, 260; Table of Well-Known Shafts, 261; Kind- 
 Chaudron Method, 262; Long-Hole Process, 262; Comparison of Methods 
 of Shaft Sinking, 262; Sinking Head-Frames, 262; Sinking Engines, 263; 
 Tools, 263; Speed and Cost of Sinking, 263; Slope Sinking, 263; The 
 Sump, 264; Driving the Gangway, 264; Levels in Metal Mines, 264; 
 Mining Tunnels, 265. 
 
 Mine Timber and Timbering. — Choice of Timber, 265, Preserva- 
 tion of Timber, 265; Placing of Timber, 266; Size of Timber, 267; Joints 
 in Mine Timbering, 267; Undersetting of Props, 267; Forms of Mine 
 Timbering and Underground Supports, 267; Gangway or Level Tim- 
 bers, 268; Shaft Timbering, 270; Square Sets, 270; Landing, Plats, or 
 Stations, 272; Special Forms of Supports, 272; Iron and Steel Supports, 
 272; Trestles, 274; Timber Head-Frames or Head-Gears, 275; Steel Shaft 
 Bottoms, 276. 
 
 Methods of Working — Open Work, 277; Steam-Shovel Mines, 278, 
 ET MUhng,°278; ^ Cableways, 278; Placers, 278; Hydraulicking, 278; Dredg- 
 ing, 279; Closed Work, 279; Bedded Deposits, 279; Coal Mining, 279; 
 Roof Pressure, 280; Character of Floor, 280; Systems of Working Coal, 
 280; Room-and-Pillar System, 280; Longwall Method, 281; Panel System, 
 283; Control of Roof Pressure, 2S4; Number of Entries, 284; Direction 
 of Face, 284; Size of Pillars, 285; Room Pillars, 286; Barrier Pillars, 28/; 
 Weight on Pillars in Pounds per Square Inch, 287; Drawing Pillars, 289; 
 Compressive Strength of Anthracite, 290; Gob Fires, 291; Spontaneous 
 Combustion, 291; Coal Storage, 291; Modifications of Room-and-Pillar 
 Methods, 291; Buggy Breasts, 291; Chute Breasts, 292; Pillar-and-Stall, 
 292; Connellsville Region, 293; Pittsburg Region, 295; Clearfield Region, 
 295; Reynoldsville Region, 295; West Virginia, 296; Alabama Methods, 
 297; George’s Creek, 297; Blossburg Region, Pa., 298; Indiana Mining, 298; 
 Iowa Mining, 299; Tesla, California, Method, 300; New Castle, Colorado 
 Method, 302; Modifications of Longwall Methods, 302; Overhand 
 Stoping, 304; Methods of Mining Anthracite, 305; Brown’s Method, 
 306; Battery Working, 307; Single-Chute Battery, 309; Double-Chute 
 Battery, 309; Rock-Chute Mining, 310; Williams’ Method, 312; Running 
 of Coal, 312; Hin ts for Working Thin Seams, 313; Flushing of Culm, 314; 
 Methods of Mining Mineral Deposits, 316; Winzes, 316; Raises, 316; 
 Stoping, 316; Flat Deposits, 318; Large Deposits, Over 8 Feet Thick 318; 
 Square Work, 318; Filling, 319; Slicing, 319; Transverse Rooming With 
 Filling 1 , 319; Caving, 320; Square-Set System, 321; Irregular Deposits, 
 321; Coyoting, 321; Special Methods, 322: Frozen Ground, 322; Leaching, 
 
 Wyoming Region (Pa. ), 325^ Prices of Coal, 326; Cost of Coking Coal, 328. 
 
 Explosives — Low Explosives, 329; High Explosives, 329; Thawing 
 Dynamite, 329; Common Blasting Explosives, 330; Drilling, 330; Diam- 
 eter of Hole, 330; Amount of Charge, 331; Tamping, 331; Firing, 331; 
 Detonators; 332; Electric Firing, 332; Power of an Explosive, 334; 
 Arrangement of Drill Holes, 335. 
 
 Machine Mining.— Pick Machines, 336; Chain-Cutter Machines, 337; 
 Shearing, 337; Capacity, 337. 
 
 Ventilation of M i n es.— Atmosphere, 337; Atmospheric Pressure, 339; 
 
 N Barometric Variations, 339; Barometers, 339; Water Column Corre- 
 sponding to Anv Mercury Column, 340; Barometric Elevations, 340; 
 Chemistry of Gases. 341; Absolute Temperature, 344; . Absolute Pressure, 
 345; Diffusion and Transpiration, 346; Gases Found in Mines, 348; Con- 
 stants for Mine Gases, 349; Gas Feeders, 352; Pressure of Occluded 
 Gases, 352; Amount of Gas, 352; Outbursts of Gas, 3o2; Testing for Gas 
 by Lamp Flame, 354; Safety Lamps, 355; Lamps tor Testing, 355; Detec- 
 tion of Small Percentages of Gas, 356; Oils for Safety Lamps, 3o6, Types 
 of Safety Lamps, 356; Locking Lamps, 358; Cleaning Safety Lamps, 3o8* 
 Relighting Stations, 359; Illuminating Power of Safety Lamps, 359 
 Explosive Conditions in Mines, 359; Derangement of Ventnating Cu.r 
 rent, 359; Sudden Increase of Gas, 360; Effect of Coal Dust in Mine 
 
TABLE OF CONTENTS. 
 
 xiii 
 
 Workings, 360; Pressure as Affecting Explosive Conditions, 361; Rapid 
 Succession of Shots in Close Workings, 361; Mine Explosions, 361; 
 Recoil of an Explosion, 361; Exploring Workings After a Serious Explo- 
 sion, 361; Quantity of Air Required for Ventilation, 362; Elements in Ven- 
 tilation, 363; Power of the Current, 363; Mine Resistance, 364; Velocity 
 of the Air-Current, 364; Measurement of Ventilating Currents, 364; 
 Water Gauge, 365; Thermometer^, 366; Calculation of Mine Resistance, 
 366; Calculation of Power or Units of Work per Minute, 367; Equivalent 
 Orifice, 367; Potential Factor of a Mine, 367; Formulas, 370; Variation 
 of the Elements, 372; Practical Splitting of the Air-Current, 373; 
 Primary Splits, 374; Secondary Splits, 374; Tertiary Splits, 374; Equal 
 Splits of Air, 374; Unequal Splits of Air, 374; Natural Division of the 
 Air-Current, 374; Calculation of Natural Splitting, 374; Proportional 
 Division of the Air-Current, 375; Box Regulator, 375; Door Regulator, 
 375; Splitting Formulas, 378; Methods and Appliances in the Ventila- 
 tion of Mines, 381; Ascensional Ventilation, 381; General Arrangement 
 of Mine Plan, 381; Natural Ventilation, 381; Ventilation of Rise and 
 Dip Workings, 382; Influence of Seasons, 382; Construction of a Mine 
 Furnacp, 383; Air Columns in Furnace Ventilation, 384; Inclined Air 
 Columns, 384; Calculation of Ventilating Pressure in Furnace Ventila- 
 tion, 384; Calculation of Motive Column or Air Column, 384; Influence 
 of Furnace Stack, 385; Mechanical Ventilators, 385; Vacuum System of 
 Ventilation, 386; Plenum System of Ventilation, 386; Comparison of 
 Vacuum and Plenum Systems, 386; Types of Centrifugal Fans, 387; 
 Manometrical Efficiency*, 390; Mechanical Efticiency, 390; Fan Con- 
 struction, 391; High-Speed and Low-Speed Motors, 392; Fan Tests, 392; 
 Conducting Air-Currents, 393; Doors, 393; Stoppings, 393; Bridges, 393; 
 Air Brattice, 394; Curtains, 394. 
 
 Hoisting.— Double Cylindrical Drums, 394; Single Cylindrical Drums, 394; 
 Koepe System, 395; Whiting System, 395; Problems in Hoisting, 396; 
 Balancing a Conical Drum, 396; Horsepower of an Engine for Hoisting, 
 396; Load That a Given Pair of Engines Will Start, 396; Approximate 
 Period of Winding on a Cylindrical Drum, 397; Head-Frames, 397; 
 Head-Sheaves, 397; Guides and Conductors, 398; Safety Catches, 398; 
 Detaching Hooks, 398. 
 
 Hau lag E.— 1 Gravity Planes, 398; Number of Cars in a Trip on a Self-Acting 
 Incline, 399; Engine Planes, 399; Size of Engines Required for Engine- 
 Plane Haulage, 399; Horsepower of an Engine to Hoist a Given Load 
 Up an Incline, 400; Rope Haulage, 400; Tail-Rope System, 400; Tension 
 of Hauling Rope, 401; Endless-Rope System, 401; Friction Pull on an 
 Endless-Rope Haulage, 402; Inclined Roads, 402; Motor Haulage, 402; 
 Locomotive Haulage, 402; Compressed-Air Haulage, 403; Tractive 
 Efforts of Compressed-Air Locomotives, 404; Electric Haulage, 406; 
 Speed of Haulage, 408; Cost of Haujage, 409; Mine Roads and Tracks, 
 410; Grade, 410; Rails, 411; Gauge, 411; Curves, 411; To Bend Rails to 
 Proper Arc for Any Radius, 412; Rail Elevation, 412; Rollers, 412; 
 Switches, 413; Turnouts, 413; Slope Bottoms, 413; Tracks for Bottom of 
 Shafts, 416; Surface Tracks for Slopes and Shafts, 417. 
 
 Ore Dressing and the Preparation of Coal. — Crushing 
 Machinery, 418; Object of Crushing, 418; Selection of a Crusher, 418; 
 Jaw Crushers, 419; Blake Crusher, 419; Dodge Crusher, 419; Roll-Jaw 
 Crushers, 420; Gyratory Crushers, 420; Rolls, 421; Cracking Rolls, 421; 
 Corrugated Rolls, 422; Disintegrating Rolls and Pulverizers, 423; Ham- 
 mers, 423; Crushing Rolls, 423; Amount Crushed, 424; Speeds, 424; Speed 
 of Rolls, 425; Crushing Mills, 426; Roller Mills, 426; Ball Mills, 427; Gravity 
 Stamps, 427; Order of Drop, 428; Speed of Stamps, 429; Shoes and Dies, 
 429; Life of the Shoes and Dies, 429; Cams, Stamp Heads, and Stems, 
 429; Tappets, 429; Battery Water, 430; Duty of Stamps, 430; Horsepower 
 of Stamps, 430; Cost of Stamping, 430; Pneumatic Stamps, 430; Power 
 Stamps, 431; Steam Stamp, 431; Miscellaneous Forms of Crushers, 431. 
 
 Sizing and Classifying Apparatus.— Stationary, Screens, Griz- 
 zlies, Head-Bars, or Platform Bars, 431; Adjustable Bars. 432; Shaking 
 Screens, 432; Revolving Screens, or Trommels, 433; Speed, 434; Duty of 
 Anthracite Screens, 434; Revolving Screen Mesh for Anthracite, 434; 
 Hydraulic Classifiers, 434; Spitzkasten, 435; Spitzlutten, 4$?; Cftluni^t 
 
xjy TABLE OF CONTENTS. 
 
 Classifier, 435; Settling Boxes, 4 35 ; Jeffrey-Robinron 
 t op- Washer 436; Scaife Trough Washer, 437; Jig&, 437, fetationary 
 Screen Jigs 437' Theory of Jigging, 439; Equal Settling Particles, 439, 
 In Acclferation, 440; Suction 440; Removal of 
 Sulphur From Coal, 441; Preparation of Anthracite, 442. 
 
 M a Kin i i mo op Material — Anthracite Coal, 413; Bituminous Coal, 443; 
 H Ore R^ck Etc 44 ^ Flumes and Launders, 443; Horizontal Pressure 
 Exerted Against Vertical Retaining, Walls 
 
 roovpvers 445' Weights and Capacities of Standard Steel BucKets, wo, 
 Malleable Iron Buckets 446 ;C<»nve^ngCapaci- 
 
 Wounds, 451; Treatment of Persons Overcome by Gas, 451. 
 
 Tables— Goal Dealers’ Table, Giving Cost of Any .N^ber ^ 
 i ABLEb. p - ce p r rp on 452 - Natural Sines and Cosines, 453, Natural lan 
 
 gents and Cotangents, 464; Logarithms of Numbers, 473; L° eanthi ms of 
 Trigonometric Functions, Slims, Cosines, Tangents, Cotangents^!, 
 Latitudes and Departures (Traverse Table), J >3 7, SquMes^ OuDes, 
 Square and Cube Roots, Circumferences, Areas, and Reciprocal^ 
 Horn 1 to 1,000, 545; Diameters, Circumferences, and Areas, ** w 
 100, 561. 
 
 Glossary of Mining Terms.—565* 
 
COAL AND METAL MINERS’ 
 
 POCKETBOOK. 
 
 WEIGHTS AND MEASURES. 
 
 THE METRIC SYSTEM. 
 
 Since the metric system is the adopted system in many countries, and as 
 it is almost universally used in connection with scientific work, a brief 
 description of it is here given as preliminary to the following tables of weights 
 and measures. 
 
 The metric system has three principal units: 
 
 1. The meter , or unit of length , supposed to be the one ten-millionth part 
 of the distance from the equator to the pole on the meridian of longitude 
 passing through the city of Paris. Its actual value is 39.370432 in., the stand- 
 ard authorized by the United States Government being 39.37 in. According 
 
 to this standard, 1 yd. = meter. 
 
 OjUOl 
 
 2. The gram , or unit of weight , is the weight of a cubic centimeter of dis- 
 tilled water at 4° centigrade and 776 millimeters of atmospheric measure. 
 The kilogram (Kg.) = 1,000 grams = 2.2046 lb., is the ordinary unit of weight 
 corresponding to the English pound. According to the United States Gov- 
 
 ernment regulations, 1 lb. avoirdupois = 
 
 1 
 
 2.2046 
 
 kilogram. 
 
 3. The liter , or unit of liquid volume , is the volume of 1,000 cubic centi- 
 meters of distilled water at 4° centigrade and 776 millimeters pressure. 
 
 Multiples of these units are obtained by prefixing to the names of the 
 printed units the Greek words deka (10), hekto (100), kilo (1,000). The sub- 
 multiples or divisions are obtained by prefixing the Latin words deci ( T V), 
 centi ( T fo), and milli (^W)- The abbreviations of these several units as given 
 in the following tables are those commonly used. The kilogram-meter is the 
 work done in raising 1 kg. through a height of 1 m., and equals 7.233 ft.-lb. 
 One metric horsepower (force de cheval or cheval vapeur) equals .98633 English 
 horsepower. 
 
 TROY WEIGHT. 
 
 24 grains = 1 pennyweight. 
 
 20 pennyweights = 1 ounce = 480 grains. 
 
 12 ounces = 1 pound = 5,760 grains = 240 pennyweights. 
 
 In troy, apothecaries’, and avoirdupois weights, the grains are the same. 
 
 APOTHECARIES’ WEIGHT. 
 
 20 grains = 1 scruple. 
 
 3 scruples = 1 dram = 60 grains. 
 
 8 drams = 1 ounce = 480 grains = 24 scruples. 
 
 12 ounces = 1 pound = 5,760 grains =,288 scruples = 96 drams. 
 
2 
 
 WEIGHTS AND MEASURES. 
 
 AVOIRDUPOIS WEIGHT. 
 
 27.34375 grains 
 16 drams 
 16 ounces 
 28 pounds 
 4 quarters 
 
 = 1 dram. 
 
 = 1 ounce 
 = 1 pound 
 = 1 quarter 
 = 1 hundredweight 
 
 20 hundredweight = 1 ton 
 1 stone 
 1 quintal 
 1 “ short ton ” 
 
 1 “long ton ” 
 
 1 ounce troy or apothecaries’ = 1.09714 
 1 pound troy or apothecaries’ = .82286 
 1 ounce avoirdupois = .911458 
 
 1 pound avoirdupois = 1.21528 
 
 = 437i grains. 
 
 = 7,000 grains = 256 drams. 
 = 448 ounces. 
 
 = 112 pounds. 
 
 = 2,240 pounds. 
 
 = 14 pounds. 
 
 = 100 pounds. 
 
 = 2,000 pounds. 
 
 = 2,240 pounds, 
 avoirdupois ounces, 
 avoirdupois pound, 
 troy or apothecaries’ ounce, 
 troy or apothecaries’ pounds. 
 
 10 milligrams (mg.) 
 10 centigrams 
 10 decigrams 
 10 grams 
 10 decagrams 
 10 hectograms 
 10 kilograms 
 10 myriagrams 
 10 quintals 
 
 METRIC WEIGHT. 
 
 = 1 centigram (eg.) 
 
 = 1 decigram (dg.) 
 
 = 1 gram (g.) 
 
 = 1 decagram (Dg.) 
 
 = 1 hectogram (Hg.) 
 
 = 1 kilogram (Kg.) 
 
 = 1 myriagram (Mg.) 
 
 = 1 quintal (Q.) 
 
 = 1 tonneau, millier, or tc 
 
 = .15432 grain. 
 
 = 1.5432 grains. 
 
 = 15.432 grains. 
 
 = .0220461b. avoir. 
 
 = .22046 lb. avoir. 
 
 == 2.2046 lb. avoir. 
 
 = 22.046 lb. avoir. 
 
 = 220.46 lb. avoir. 
 
 = 2,204 lb. avoir. 
 
 MEASURES OF LENGTH. 
 
 AMERICAN AND BRITISH. 
 
 12 inches = 1 foot. 
 
 3 feet = 1 yard = 36 in. 
 
 6 feet = 1 fathom = 2 yd. = 72 in. 
 
 66 feet = 1 chain *= 11 fath. = 22 yd. = 792 in. 
 
 10 chains = 1 furlong = 110 fath. = 220 yd. = 660 ft. = 7,620 in. 
 
 8 furlongs = 1 mile = 80 chains = 880 fath. = 1,760 yd. = 5,280 ft. = 
 
 A nautical mile, or knot = 1.15136 statute miles. [63,360 in. 
 A league = 3 nautical miles. 
 
 *The chain of 66 ft. is practically obsolete. Chains of 50 or 100 ft. are now used exclusively by 
 American surveyors. 
 
 To Reduce Inches to Decimals of a Foot.— Divide the number of inches by 
 12. Thus, 7 in. = 7 -f- 12, or .58333 ft. To reduce fractions of inches to deci- 
 mals of a foot, divide the fraction hv 12, and then divide the numerator of 
 the quotient by the denominator. Thus, f in. = | -s- 12 = g 3 g . g 3 g — .0313 ft. 
 
 The annexed scale shows on one side, proportionately reduced, a scale of 
 tenths. On the other, a scale of twelfths, corresponding to inches. To 
 reduce inches to decimal parts of a foot, find the number of inches and 
 
 TENTHS OF A FOOT 
 
 O 1 
 
 
 
 3 
 
 I 
 
 4 
 
 5 6 7 
 
 8 9 10 
 
 1.1.1-U yi, i.i.i .i. i,i.i ,1 ,i ,i ,i ,i J J uJ 
 
 
 
 rnr 
 
 5 t 
 
 j , 'i i jr r i f f r ' ! 1 ‘ 1 [ 
 
 7 S S 
 
 H 1 l • i ' ' » I | 1 i • | ' • ■ i ■ ' 1 | 
 
 ) 10 11 12 
 
 LSTCHUS 
 
 fractional parts thereof on the side marked “inches.” Opposite, on the 
 scale of tenths, will be found the decimal part of a foot. Thus, if we want 
 to find the decimal part of a foot represented by 7£ inches, we find the mark 
 corresponding to 7£ inches on the side marked “inches.” Opposite this 
 mark we read 6 tenths, 2 hundredths, and 5 thousandths; or, expressed 
 decimally, .625. ♦ 
 
MEASURES OF LENGTH. 
 
 3 
 
 DECIMALS OF A FOOT FOR EACH 1-32 OF AN INCH. 
 
 Inch. 
 
 0" 
 
 1" 
 
 2" 
 
 3" 
 
 4" 
 
 5" 
 
 6" 
 
 7" 
 
 8" 
 
 9" 
 
 10" 
 
 11" 
 
 0 
 
 0 
 
 .0833 
 
 .1667 
 
 .2500 
 
 .3333 
 
 .4167 
 
 .5000 
 
 .5833 
 
 .6667 
 
 .7500 
 
 .8333 
 
 .9167 
 
 32 
 
 .0026 
 
 .0859 
 
 .1693 
 
 .2526 
 
 .3359 
 
 .4193 
 
 .5026 
 
 .5859 
 
 .6693 
 
 .7526 
 
 .8359 
 
 .9193 
 
 1^5 
 
 .0052 
 
 .0885 
 
 .1719 
 
 .2552 
 
 .3385 
 
 .4219 
 
 .5052 
 
 .5885 
 
 .6719 
 
 .7552 
 
 .8385 
 
 .9219 
 
 3% 
 
 .0078 
 
 .0911 
 
 .1745 
 
 .2578 
 
 .3411 
 
 .4245 
 
 .5078 
 
 .5911 
 
 .6745 
 
 .7578 
 
 .8411 
 
 .9245 
 
 i 
 
 .0104 
 
 .0937 
 
 .1771 
 
 .2604 
 
 .3437 
 
 .4271 
 
 .5104 
 
 .5937 
 
 .6771 
 
 .7604 
 
 .8437 
 
 .9271 
 
 3*2 
 
 .0130 
 
 .0964 
 
 .1797 
 
 .2630 
 
 .3464 
 
 .4297 
 
 .5130 
 
 .5964 
 
 .6797 
 
 .7630 
 
 .8464 
 
 .9297 
 
 t 3 s 
 
 .0156 
 
 .0990 
 
 .1823 
 
 .2656 
 
 .3490 
 
 .4323 
 
 .5156 
 
 .5990 
 
 .6823 
 
 .7656 
 
 .8490 
 
 .9323 
 
 32 
 
 .0182 
 
 .1016 
 
 .1849 
 
 .2682 
 
 .3516 
 
 .4349 
 
 .5182 
 
 .6016 
 
 .6849 
 
 .7682 
 
 .8516 
 
 .9349 
 
 5 
 
 .0208 
 
 .1042 
 
 .1875 
 
 .2708 
 
 .3542 
 
 .4375 
 
 .5208 
 
 .6042 
 
 .6875 
 
 .7708 
 
 .8542 
 
 .9375 
 
 35 
 
 .0234 
 
 .1068 
 
 .1901 
 
 .2734 
 
 .3568 
 
 .4401 
 
 .5234 
 
 .6068 
 
 .6901 
 
 .7734 
 
 .8568 
 
 .9401 
 
 t 6 s 
 
 .0260 
 
 .1094 
 
 .1927 
 
 .2760 
 
 .3594 
 
 .4427 
 
 .5260 
 
 .6094 
 
 .6927 
 
 .7760 
 
 .8594 
 
 .9427 
 
 33 
 
 .0286 
 
 .1120 
 
 .1953 
 
 .2786 
 
 .3620 
 
 .4453 
 
 .5286 
 
 .6120 
 
 .6953 
 
 .7786 
 
 .8620 
 
 .9453 
 
 i 
 
 .0312 
 
 .1146 
 
 .1979 
 
 .2812 
 
 .3646 
 
 .4479 
 
 .5312 
 
 .6146 
 
 .6979 
 
 .7812 
 
 .8646 
 
 .9479 
 
 M 
 
 .0339 
 
 .1172 
 
 .2005 
 
 .2839 
 
 .3672 
 
 .4505 
 
 .5339 
 
 .6172 
 
 .7005 
 
 .7839 
 
 .8672 
 
 .9505 
 
 I 
 
 .0365 
 
 .1198 
 
 .2031 
 
 .2865 
 
 .3698 
 
 .4531 
 
 .5365 
 
 .6198 
 
 .7031 
 
 .7865 
 
 .8698 
 
 .9531 
 
 33 
 
 .0391 
 
 .1224 
 
 .2057 
 
 .2891 
 
 .3724 
 
 .4557 
 
 .5391 
 
 .6224 
 
 .7057 
 
 .7891 
 
 .8724 
 
 .9557 
 
 h 
 
 .0417 
 
 .1250 
 
 .2083 
 
 .2917 
 
 .3750 
 
 .4583 
 
 .5417 
 
 .6250 
 
 .7083 
 
 .7917 
 
 .8750 
 
 .9583 
 
 e. 
 
 .0443 
 
 .1276 
 
 .2109 
 
 .2943 
 
 .3776 
 
 .4609 
 
 .5443 
 
 .6276 
 
 .7109 
 
 .7943 
 
 .8776 
 
 .9609 
 
 
 .0469 
 
 .1302 
 
 .2135 
 
 .2969 
 
 .3802 
 
 .4635 
 
 .5469 
 
 .6302 
 
 .7135 
 
 .7969 
 
 .8802 
 
 .9635 
 
 if 
 
 .0495 
 
 .1328 
 
 .2161 
 
 .2995 
 
 .3828 
 
 .4661 
 
 .5495 
 
 .6328 
 
 .7161 
 
 .7995 
 
 .8828 
 
 .9661 
 
 | 
 
 .0521 
 
 .1354 
 
 .2188 
 
 .3021 
 
 .3854 
 
 .4688 
 
 .5521 
 
 .6354 
 
 .7188 
 
 .8021 
 
 .8854 
 
 .9688 
 
 flf 
 
 .0547 
 
 .1380 
 
 .2214 
 
 .3047 
 
 .3880 
 
 .4714 
 
 .5547 
 
 .6380 
 
 .7214 
 
 .8047 
 
 -.8880 
 
 .9714 
 
 tt 
 
 .0573 
 
 .1406 
 
 .2240 
 
 .3073 
 
 .3906 
 
 .4740 
 
 .5573 
 
 .6406 
 
 .7240 
 
 .8073 
 
 .8906 
 
 .9740 
 
 if 
 
 .0599 
 
 .1432 
 
 .2266 
 
 .3099 
 
 .3932 
 
 .4766 
 
 .5599 
 
 .6432 
 
 .7266 
 
 .8099 
 
 .8932 
 
 .9766 
 
 £ 
 
 .0625 
 
 .1458 
 
 .2292 
 
 .3125 
 
 .3958 
 
 .4792 
 
 .5625 
 
 .6458 
 
 .7292 
 
 .8125 
 
 .8958 
 
 .9792 
 
 If 
 
 .0651 
 
 .1484 
 
 .2318 
 
 .3151 
 
 .3984 
 
 .4818 
 
 .5651 
 
 .6484 
 
 .7318 
 
 .8151 
 
 .8984 
 
 .9818 
 
 if 
 
 .0677 
 
 .1510 
 
 .2344 
 
 .3177 
 
 .4010 
 
 .4844 
 
 .5677 
 
 .6510 
 
 .7344 
 
 .8177 
 
 .9010 
 
 .9844 
 
 if 
 
 .0703 
 
 .1536 
 
 .2370 
 
 .3203 
 
 .4036 
 
 .4870 
 
 .5703 
 
 .6536 
 
 .7370 
 
 .8203 
 
 .9036 
 
 .9870 
 
 7 
 
 .0729 
 
 .1562 
 
 .2396 
 
 .3229 
 
 .4062 
 
 .4896 
 
 .5729 
 
 .6562 
 
 .7396 
 
 .8229 
 
 .9062 
 
 .9896 
 
 If 
 
 .0755 
 
 .1589 
 
 .2422 
 
 .3255 
 
 .4089 
 
 .4922 
 
 .5755 
 
 .6589 
 
 .7422 
 
 .8255 
 
 .9089 
 
 .9922 
 
 
 .0781 
 
 .1615 
 
 .2448 
 
 .3281 
 
 .4115 
 
 .4948 
 
 .5781 
 
 .6615 
 
 .7448 
 
 .8281 
 
 .9115 
 
 .9948 
 
 33 
 
 .0807 
 
 .1641 
 
 .2474 
 
 .3307 
 
 .4141 
 
 .4974 
 
 .5807 
 
 .6641 
 
 .7474 
 
 .8307 
 
 .9141 
 
 .9974 
 
 METRIC SYSTEM. 
 
 10 millimeters (mm.) 
 10 centimeters 
 10 decimeters 
 10 meters 
 10 decameters 
 10 hectometers 
 10 kilometers 
 
 = 1 centimeter (cm.) = 
 
 = 1 decimeter (dm.) = 
 
 = 1 meter (m.) = 
 
 = 1 decameter (Dm.) = 
 = 1 hectometer (flm.) = 
 = 1 kilometer (Km.) = 
 
 — 1 myriameter(Mm.) = 
 
 .3937079 inch. 
 3.937079 inches. 
 3.2808992 feet. 
 10.9363 yards. 
 
 109.363 yards. 
 
 .6213824 mile. 
 6.213824 miles. 
 
 RUSSIAN. 
 
 12 inches = 1 foot = 1 American foot. 
 7 feet = 1 sachine, or sagene. 
 
 500 sachine = 1 verst = 3, 500 'feet. 
 
 PRUSSIAN, DANISH, AND NORWEGIAN. 
 
 12 inches = 1 foot = 1.02972 American feet. 
 12 feet = 1 ruth = 12.35664 American feet. 
 2,000 ruths = 1 mile — 4.68+ American miles. 
 
 AUSTRIAN. 
 
 12 inches = 1 foot = 1.03713 American feet. 
 
 6 feet = 1 klafter. 
 
 4,000 klafters = 1 mile = 4.71+ American miles, 
 
4 
 
 WEIGHTS AND MEASURES. 
 
 SWEDISH. 
 
 12 inches = 1 foot = .97410 American foot. 
 
 6 feet = 1 fathom. 
 
 6,000 fathoms = 1 mile = 6.64+ American miles. 
 
 CHINESE. 
 
 1 chih = 1-054 American feet. 
 
 10 chih = 1 chang = 10.54 American feet. 
 
 180 chang = 1 li = 1,897 American feet. 
 
 MEASURES OF AREA. 
 
 144 sq. inches 
 9 sq. feet 
 30i sq. yards 
 40 perches 
 4 roods 
 640 acres 
 
 AMERICAN AND BRITISH. 
 
 3 square foot. 
 
 1 square yard = 
 1 perch = 
 
 1 rood = 
 
 1 acre = 
 
 1 square mile. 
 
 1,296 sq. in. 
 
 272i sq. ft. 
 
 1,210 sq. vd. = 10,890 sq. ft. 
 
 160 perches = 4,640 sq. yd. = 43,560 sq. ft. 
 
 TABLE FOR REDUCING SQUARE FEET TO ACRES. 
 
 Square 
 
 Feet. 
 
 Acres. 
 
 Square 
 
 Feet. 
 
 Acres. 1 
 
 Square 
 Feet. 1 
 
 Acres. 
 
 Square 
 
 Feet. 
 
 Acres. 
 
 100,000,000 
 
 90.000. 000 
 
 80.000. 000 
 
 70.000. 000 
 
 60.000. 000 
 
 50.000. 000 
 
 40.000. 000 
 
 30.000. 000 
 
 20.000. 000 
 10,000,000 
 
 2,295.684 
 
 2,066.116 
 
 1,836.547 
 
 1,606.979 
 
 1,377.410 
 
 1,147.842 
 
 918.274 
 
 688.705 
 
 459.137 
 
 229.568 
 
 900,000 
 
 800,000 
 
 700.000 
 
 600.000 
 500,000 
 
 400.000 
 
 300.000 
 200.000 
 100,000 
 
 20.661 
 
 18.365 
 
 16.070 
 
 13.774 
 
 11.478 
 
 9.183 
 
 6.887 
 
 4.591 
 
 2.296 
 
 9.000 
 
 8.000 
 
 7.000 
 
 6.000 
 
 5.000 
 
 4.000 
 
 3.000 
 2,000 
 1,000 
 
 .207 
 
 .184 
 
 .161 
 
 .138 
 
 .115 
 
 .092 
 
 .069 
 
 .046 
 
 .023 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 .0021 
 
 .0018 
 
 .0016 
 
 .0014 
 
 .0011 
 
 .0009 
 
 .0007 
 
 .0005 
 
 .0002 
 
 9.000. 000 
 
 8.000. 000 
 
 7.000. 000 
 
 6.000. 000 
 5.000,000 
 
 4.000. 000 
 
 3.000. 000 
 2,000,000 
 1,000,000 
 
 206.612 
 
 183.655 
 
 160.698 
 
 137.741 
 
 114.784 
 
 91.827 
 
 68.870 
 
 45.914 
 
 22.957 
 
 90.000 
 
 80.000 
 70.000 
 60,000 
 50.000 
 
 40.000 
 
 30.000 
 20.000 
 10,000 
 
 2.066 
 
 1.836 
 
 1.607 
 
 1.377 
 
 1.148 
 
 .918 
 
 .689 
 
 .459 
 
 .230 
 
 900 
 
 800 
 
 700 
 
 600 
 
 500 
 
 400 
 
 300 
 
 200 
 
 100 
 
 .021 
 
 .018 
 
 .016 
 
 .014 
 
 .011 
 
 .009 
 
 .007 
 
 .005 
 
 .0023 
 
 9 
 
 8 
 
 7 
 
 6 , 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 .00021 
 
 .00018 
 
 .00016 
 
 .00014 
 
 .00011 
 
 .00009 
 
 .00007 
 
 .00005 
 
 .00002 
 
 METRIC SYSTEM. 
 
 1 square millimeter (sq. mm.) = 
 
 1 square centimeter (sq. cm.) 
 
 1 square decimeter (sq. dm.) -~ 
 
 1 square meter, or centare (m. 1 2 or sq. m.) — 
 1 square decameter, or are (sq. Dm.) 
 
 1 hectare (ha. ) 
 
 1 square kilometer (sq. Km.) 
 
 1 square myriameter (sq. Mm.) 
 
 .001550 sq. in. 
 .155003 sq. in. 
 15.5003 sq. in. 
 10.764101 sq. ft. 
 
 .024711 acre. 
 2.47110 acres. 
 247.110 acres. 
 38.61090 sq. mi. 
 
MEASURES OF VOLUME. 
 
 5 
 
 MEASURES OF VOLUME. 
 
 AMERICAN AND BRITISH. 
 
 1,728 cubic inches = 1 cubic foot. 
 
 27 cubic feet = 1 cubic yard. 
 
 A cord of wood = 128 cu. ft., or a pile of wood 8 ft. long, 4 ft. wide, and 
 4 ft. high = 1 cord. A perch of masonry contains 24f cu. ft.; but in practice 
 it is taken as 25 cu. ft. 
 
 A ton (2,240 lb.) of Pennsylvania anthracite, when broken for domestic 
 use, occupies about 42 cu. ft. of space; bituminous coal, about 46 cu. ft.; and 
 coke, about 88 cu. ft. A bushel of coal is 80 lb. in Kentucky, Illinois, and 
 Missouri, 76 lb. in Pennsylvania, 70 fb. in Indiana, and 76 lb. in Montana. 
 
 METRIC SYSTEM. 
 
 1 milliliter, or cu. centimeter (cc. or cm. 3 ) 
 
 1 centiliter (cl.) 
 
 1 deciliter (dl. or dl. 3 ) 
 
 1 liter, or cu. decimeter (1.) 
 
 1 decaliter, or centistere (Dl. or dal.) 
 
 1 hectoliter, or decistere (HI.) 
 
 1 kiloliter, or cu. meter, or stere (Kl. or cm. 3 ) 
 1 myrialiter, or decastere (Ml.) 
 
 .0610254 cu. in. 
 .610254 cu. in. 
 6.10254 cu. in. 
 61.0254 cu. in. 
 .353156 cu. ft. 
 3.53156 cu. ft. 
 35.3156 cu. ft. 
 353.156 cu. ft. 
 
 4 gills 
 2 pints 
 4 quarts 
 3H gallons 
 63 gallons 
 2 hogsheads 
 2 pipes 
 
 LIQUID MEASURE (u. S.). 
 
 1 pint 
 1 quart 
 1 gallon 
 1. barrel 
 1 hogshead. 
 1 pipe. 
 
 1 tun. 
 
 16 liquid oz. 
 8 gills 
 
 = 28.876 cu. in. 
 = 57.75 cu. in. 
 
 32 gills = 8 pints = 231 
 7,276i cu. in. = 4.21 
 
 cu. in. 
 cu. ft. 
 
 A box 19f in. on each side contains 1 barrel. 
 1 cu. ft. — 7.48 gallons. 
 
 DRY MEASURE (u. S.). 
 
 2 pints = 1 quart = 67.2006 cu. in. = 1.16365 liquid qt. 
 
 4 quarts = 1 gallon = 268.8025 cu. in. = 1.16365 liquid gal. 
 
 2 gallons = 1 peck = 8 quarts * = 537.6050 cu. in. 
 
 4 pecks = 1 bushel = 64 pints = 32 quarts = 8 gal. = 2,150.42 cu. in. 
 
 BRITISH IMPERIAL MEASURE, BOTH LIQUID AND DRY. 
 
 4 gills = 1 pint = 34.6592 cu. in. 
 
 2 pints = 1 quart = 69.3185 cu. in. 
 
 4 quarts = 1 gallon = 277.274 cu. in. 
 
 8 quarts = 1 peck = 554.548 cu. in. 
 
 4 pecks = 1 bushel — 2,218.192 cu. in. 
 
 The standard U. S. bushel is the Winchester bushel, which is in cylinder 
 form, 18£ inches diameter and 8 inches deep, and contains 2,150.42 cubic 
 inches. 
 
 The British Imperial bushel is based on the Imperial gallon and con- 
 tains 8 such gallons, or 2,218.192 cubic inches = 1.2837 cubic feet. 
 
 Capacity of a cylinder in U. S. gallons = square of diameter in inches 
 X height in inches X .0034 (accurate within 1 part in 100,000). 
 
 Capacity of a cylinder in U. S. bushels = square of diameter in inches 
 >< height in inches X .0003652. 
 
6 
 
 WEIGHTS AND MEASURES. 
 
 CONTENTS OF CYLINDERS OR PIPES FOR 1 FOOT IN LENGTH, 
 
 The contents of pipes or cylinders in gallons or pounds are to each other 
 as the squares of their diameters. Thus, a pipe 9 ft. in diameter will contain 
 9 times as much as a 3' pipe, or 4 times as much as a 4k' pipe. 
 
 Diameters in Inches. 
 
 1 
 
 Diam. 
 
 in 
 
 Inches. 
 
 Diameter 
 in Decimals 
 of a Foot. 
 
 Gallons of 
 231 Cu.In. 
 (U. S. Stand- 
 ard.) 
 
 Weight of 
 Water in Lb. 
 in 1 Ft. of 
 Length. 
 
 Diam. j 
 in | 
 Inches. 1 
 
 Diameter 
 in Decimals ' 
 of a Foot. | 
 
 1 
 
 - ! 
 
 Gallons of 
 231 Cu. In. 
 (U. S. Stand- 
 ard.) 
 
 Weight of 
 Water in Lb. 
 in 1 Ft. of 
 Length. 
 
 •1_ 
 
 .0208 
 
 .0025 
 
 .02122 
 
 5 
 
 .4167 
 
 1.020 
 
 8.488 
 
 l 
 
 .0417 
 
 .0102 
 
 .08488 
 
 5i 
 
 .4583 
 
 1.234 
 
 10.270 
 
 | 
 
 .0625 
 
 .0230 
 
 .19098 
 
 6 
 
 .5000 
 
 1.469 
 
 12.223 
 
 1 
 
 .0833 
 
 .0408 
 
 .33952 
 
 6£ 
 
 .5417 
 
 1.724 
 
 14.345 
 
 li 
 
 .1042 
 
 .0638 
 
 .53050 
 
 7 
 
 .5833 
 
 1.999 
 
 16.636 
 
 H 
 
 .1250 
 
 .0918 
 
 .76392 
 
 n 
 
 .6250 
 
 2.295 
 
 19.098 
 
 H 
 
 .1458 
 
 .1249 
 
 1.0398 
 
 8 
 
 .6667 
 
 2.611 
 
 21.729 
 
 2 
 
 .1667 
 
 .1632 
 
 1.3581 
 
 8i 
 
 .7083 
 
 2.948 
 
 24.530 
 
 2i 
 
 .1875 
 
 .2066 
 
 1.7188 
 
 9 
 
 .7500 
 
 3.305 
 
 27.501 
 
 2i 
 
 .2083 
 
 .2550 
 
 2.1220 
 
 9i 
 
 .7917 
 
 3.682 
 
 30.641 
 
 2f 
 
 .2292 
 
 .3085 
 
 2.5676 
 
 10 
 
 .8333 
 
 4.080 
 
 33.952 
 
 3 
 
 .2500 
 
 .3672 
 
 3.0557 
 
 10i 
 
 .8750 
 
 4.498 
 
 37.432 
 
 3i 
 
 .2917 
 
 .4998 
 
 4.1591 
 
 11 
 
 .9167 
 
 4.937 
 
 41.082 
 
 4 
 
 .3333 
 
 .6528 
 
 5.4323 
 
 1H 
 
 .9583 
 
 5.396 
 
 44.901 
 
 4i 
 
 .3750 
 
 .8263 
 
 6.8750 
 
 12 
 
 1.0000 
 
 5.875 
 
 48.891 
 
 Diameters in Feet. 
 
 li 
 
 1.25 
 
 9.18 
 
 76.392 
 
 10 
 
 10.00 
 
 587.6 
 
 4,889.12 
 
 li 
 
 1.50 
 
 13.22 
 
 110.00 
 
 10i 
 
 10.50 
 
 647.7 
 
 5,404.24 
 
 If 
 
 1.75 
 
 17.99 
 
 149.73 
 
 11 
 
 11.00 
 
 710.9 
 
 5,915.84 
 
 2 
 
 2.00 
 
 23.50 
 
 195.56 
 
 1H 
 
 11.50 
 
 777.0 
 
 6,485.72 
 
 2t 
 
 2.25 
 
 29.74 
 
 247.51 
 
 12 
 
 
 846.1 
 
 7,040.00 
 
 2k 
 
 2.50 
 
 36.72 
 
 305.57 
 
 13 
 
 
 992.8 
 
 8,710.00 
 
 2f 
 
 2.75 
 
 44.43 
 
 369.74 
 
 14 
 
 
 1,152.0 
 
 10,096.00 
 
 3 
 
 3.00 
 
 52.88 
 
 440.00 
 
 15 
 
 
 1,322.0 
 
 11,000.50 
 
 31 
 
 3.25 
 
 65.28 
 
 544.37 
 
 16 
 
 
 1,504.0 
 
 12,516.00 
 
 °4 
 
 3k 
 
 3.50 
 
 71.97 
 
 631.00 
 
 17 
 
 
 1,698.0 
 
 14,166.00 
 
 St 
 
 3.75 
 
 82.62 
 
 687.53 
 
 18 
 
 
 1,904.0 
 
 15,841.00 
 
 4 
 
 4.00 
 
 94.0 
 
 782.24 
 
 19 
 
 
 2,121.0 
 
 17,691.00 
 
 4i 
 
 4.25 
 
 106.1 
 
 885.40 
 
 20 
 
 
 2,350.0 
 
 19,556.50 
 
 4 i 
 
 4.50 
 
 119.0 
 
 990.04 
 
 21 
 
 
 2,591.0 
 
 21,617.00 
 
 4f 
 
 4.75 
 
 132.5 
 
 1,105.71 
 
 22 
 
 
 2,844.0 
 
 23,663.00 
 
 5 
 
 5.00 
 
 146.9 
 
 1,222.28 
 
 23 
 
 
 3,108.0 
 
 25,943.00 
 
 5£ 
 
 5.25 
 
 161.9 
 
 1,351.06 
 
 24 
 
 
 3,384.0 
 
 28,160.00 
 
 5k 
 
 5.50 
 
 177.7 
 
 1,478.96 
 
 25 
 
 
 3.672.0 
 
 30.557.00 
 
 5£ 
 
 5.75 
 
 194.3 
 
 1.621.43 
 
 26 
 
 
 3,971.0 
 
 34,840.00 
 
 6 
 
 6.00 
 
 211.5 
 
 1,760.00 
 
 27 
 
 t 
 
 4,283.0 
 
 35,641.00 
 
 6i 
 
 6.25 
 
 229.5 
 
 1,915.18 
 
 28 
 
 
 4,606.0 
 
 40,384.00 
 
 6| 
 
 6.50 
 
 248.2 
 
 2,177.48 
 
 29 
 
 
 4,941.0 
 
 41,117.00 
 
 6| 
 
 6.75 
 
 267.7 
 
 2,233.96 
 
 30 
 
 
 5,288.0 
 
 44.002.00 
 
 7 
 
 7.00 
 
 287.9 
 
 2,524.00 
 
 31 
 
 
 5,646.0 
 
 46,984.00 
 
 7k 
 
 7.50 
 
 330.5 
 
 2,750.12 
 
 32 
 
 
 6,017.0 
 
 50,064.00 
 
 8 
 
 8.00 
 
 376.0 
 
 3,128.96 
 
 33 
 
 
 6,398.0 
 
 53,242.00 
 
 8£ 
 
 8.50 
 
 424.5 
 
 3,541.60 
 
 34 
 
 
 6,792.0 
 
 56,664.00 
 
 9 
 
 9.00 
 
 475.9 
 
 3,960.16 
 
 35 
 
 
 7.197.0 
 
 59.891.50 
 
 9i 
 
 9.50 
 
 530.2 
 
 1 
 
 1,122.84 
 
 36 
 
 
 | 7.61 1.0 
 
 1 
 
 63.36^.00 
 
WEIGHTS AND MEASURES. 
 
 7 
 
 MEXICAN, CENTRAL AMERICAN, AND SOUTH AMERICAN 
 WEIGHTS AND MEASURES. 
 
 The following table gives weights and measures in commercial use in Mex- 
 ico and the republics of Central and South America, and their equivalents 
 in the United States. Published by the Bureau of the American Republics. 
 
 Denomination. 
 
 Where Used. 
 
 U. S. Equivalents. 
 
 Arobe ..." 
 
 Paraguay 
 
 25 pounds. 
 
 Arroba (dry) 
 
 Argentine Republic 
 
 25.3175 pounds. 
 
 Arroba (dry) 
 
 Brazil 
 
 32.38 pounds. 
 
 Arroba (dry) 
 
 Cuba 
 
 25.3664 pounds. 
 
 Arroba (dry) 
 
 Venezuela 
 
 25.4024 pounds. 
 
 Arroba (liquid) ... 
 
 Cuba and Venezuela 
 
 4.263 gallons. 
 
 Barril > 
 
 Argentine Republic and Mexico 
 
 20.0787 gallons. 
 
 Carga 
 
 Mexico and Salvador 
 
 300 pounds. 
 
 Centavo 
 
 Central America 
 
 4.2631 gallons. 
 
 Cuadra 
 
 Argentine Republic 
 
 4.2 acres. 
 
 Cuadra 
 
 Paraguay 
 
 78.9 yards. 
 
 Cuadra (square)... 
 
 Paraguav 
 
 8.077 square feet. 
 
 Cuadra 
 
 Uruguay 
 
 *2 6 A OTPS ( 71 PS 
 
 Fanega (dry) 
 
 Cen tral" America 
 
 Clvi. v/O \ IlvCli 1 V J « 
 
 1.5745 bushels. 
 
 Fanega (dry) 
 
 Chile 
 
 2.575 bushels. 
 
 Fanega (dry) 
 
 Cuba 
 
 1.599 bushels. 
 
 Fanega (dry) 
 
 Mexico 
 
 1.54728 bushels. 
 
 Fanega (dry) 
 
 Uruguay (double) 
 
 7.776 bushels. 
 
 Fanega (dry) 
 
 Uruguay (single) 
 
 3.888 bushels. 
 
 Fanega (dry) 
 
 Venezuela 
 
 1.599 bushels. 
 
 Frasco 
 
 Argentine Republic 
 
 2.5096 quarts. 
 
 Frasco 
 
 Mexico 
 
 2.5 quarts. 
 
 League (land)...'.... 
 
 Paraguay 
 
 4,633 acres. 
 
 Libra 
 
 Argentine Republic 
 
 1.0127 pounds. 
 
 Libra 
 
 Central America 
 
 1.043 pounds. 
 
 Libra 
 
 Chile 
 
 1.014 pounds. 
 
 Libra 
 
 Cuba 
 
 1.0161 pounds. 
 
 Libra 
 
 Mexico 
 
 1.01465 pounds. 
 
 Libra 
 
 Peru 
 
 1.0143 pounds. 
 
 Libra 
 
 Uruguay 
 
 1.0143 pounds. 
 
 Libra 
 
 Venezuela 
 
 1.0161 pounds. 
 
 Livre 
 
 Guiana 
 
 1.0791 pounds. 
 
 Manzana 
 
 Costa Rica 
 
 If acres. 
 
 Marc 
 
 Bolivia „ 
 
 .507 pound. 
 
 Pie 
 
 Argentine Republic 
 
 .9478 foot. 
 
 Quintal 
 
 Argentine Republic 
 
 101.42 pounds. 
 
 Quintal 
 
 Brazil 
 
 130.06 pounds. 
 
 Quintal 
 
 Chile, Mexico, and Peru 
 
 101.61 pounds. 
 
 Quintal 
 
 Paraguay 
 
 100 pounds. 
 
 Vara 
 
 Argentine Republic 
 
 34.1208 inches. 
 
 Vara 
 
 Central America 
 
 38.874 inches. 
 
 Vara 
 
 Chile and Peru 
 
 33.367 inches. 
 
 Vara 
 
 Cuba 
 
 33.384 inches. 
 
 Vara 
 
 Mexico 
 
 33 inches. 
 
 Vara 
 
 Paraguay 
 
 34 inches. 
 
 Vara 
 
 Venezuela 
 
 33.384 inches. 
 
 CONVERSION TABLES. 
 
 ( United States Coast and Geodetic Survey. ) 
 
 The method of using the following tables for converting United States 
 weights and measures into metric weights and measures will be understood 
 by the following example: 
 
 Find the number of kilometers in 125 miles. 
 
 From column “ Miles to Kilometers,” 1 mile = 1.60935 kilometers, or 100 
 miles = 160.935 kilometers; 2 miles = 3.21869 kilometers, or 20 miles — 32.1869 
 kilometers; and 5 miles •= 8.04674 kilometers. Hence, 125 miles = 160.935 -+• 
 32.1869 + 8.04674 = 201.16864 kilometers, 
 
8 WEIGHTS AND MEASURES. 
 
 CUSTOMARY TO METRIC. 
 
 Linear. 
 
 Capacity. 
 
 i 
 
 to © 
 © a 
 
 to 
 
 © 
 
 © 
 
 S 
 
 3 
 
 © 
 
 © 
 
 l 
 
 1 
 
 to 
 Vh © 
 
 to 
 
 s © 
 
 03 © 
 
 SB 
 
 aa 
 
 W 
 
 is . « 
 
 03 tO 
 
 a s 
 
 csj © 
 
 Ss S3 S 
 
 fee 
 2 o 
 
 . l 
 sg 
 3 8a 
 
 3 © r* 
 
 sga 
 
 OS 
 
 3 . 
 “ 8 
 15 © 
 5.-S 
 
 O’ 
 
 o 
 
 •M . 
 
 to to 
 
 *3 ^ 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 25.4 
 
 50.8 
 
 76.2 
 
 101.6 
 
 127.0 
 
 152.4 
 
 177.8 
 
 203.2 
 
 228.6 
 
 0.304801 
 
 0.609601 
 
 0.914402 
 
 1.219202 
 
 1.524003 
 
 1.828804 
 
 2.133604 
 
 2.438405 
 
 2.743205 
 
 ).914402 
 L. 828804 
 >.743205 
 3.657607 
 4.572009 
 3.486411 
 5.400813 
 7.315215 
 8.229616 
 
 1.60935 1 
 3.21869 2 
 4.82804 3 
 6.43739 4 
 8.04674 5 
 9.65608 6 
 11.26543 7 
 12.87478 8 
 14.48412 9 
 
 3.70 
 7.39 
 11.09 
 14.79 
 18.48 
 22.18 • 
 25.88 
 29.57 
 33.27 
 
 29.57 
 
 59.15 
 
 88.72 
 
 118.29 
 
 147.87 
 
 177.44 
 
 207.02 
 
 236.59 
 
 266.16 
 
 0.94636 
 
 1.89272 
 
 2.83908 
 
 3.78543 
 
 4.73179 
 
 5.67815 
 
 6.62451 
 
 7.57087 
 
 8.51723 
 
 3.78543 
 . 7.57087 
 v ll. 35630 
 15.14174 
 18.92717 
 22.71261 
 26.49804 
 30.28348 
 34.06891 
 
 \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 Square. 
 
 Weight. 
 
 to 
 
 © to 
 
 « £<2 
 C®5) 
 I— I P m 
 
 © O’.S 
 s- CO +3 
 
 =3 Q s 
 
 C O 
 cc 
 
 g © B ' 
 
 sfs 
 
 Cj CC-rH 
 
 co H 
 
 to 
 
 rrt 
 
 1 £ • 
 ^ H 05 
 
 t- S S3 
 
 So® 
 
 00 
 
 si 
 
 Sg 08 
 
 Sg 
 
 SB 
 
 to g3 
 
 II 
 
 to 
 
 *2 OB S 
 
 B 4 © d 
 £ © § 
 c S 
 
 co« 
 
 5 2 
 
 to 
 
 a~i 
 
 S3 “ c3 
 
 |ga 
 
 to 
 
 © . 
 © cc 
 
 s s 
 SS 
 
 ss 
 
 6.452 
 12.903 
 1 9.355 
 25.807 
 32.258 
 38.710 
 45.161 
 51.613 
 58.065 
 
 9.290 
 
 18.581 
 
 27.871 
 
 37.161 
 
 46.452 
 
 55.742 
 
 65.032 
 
 74.323 
 
 83.613 
 
 0.836 
 
 1.672 
 
 2.508 
 
 3.344 
 
 4.181 
 
 5.017 
 
 5.853 
 
 6.689 
 
 7.525 
 
 0.4047 : 
 
 0.8094 : 
 
 1.2141 : 
 
 1.6187 - 
 
 2.0234 
 2.4281 
 2.8328 
 3.2375 
 3.6422 
 
 1 64.7989 
 
 2 129.5978 
 
 3 194.3968 
 
 4 259.1957 
 
 5 323.9946 
 
 6 388.7935 
 
 7 453.5924 
 
 8 518.3914 
 
 9 583.1903 
 
 28.3495 
 
 56.6991 
 
 85.0486 
 
 113.3981 
 
 141.7476 
 
 170.0972 
 
 198.4467 
 
 226.7962 
 
 255.1457 
 
 0.45359 
 
 0.90719 
 
 1.36078 
 
 1.81437 
 
 2.26796 
 
 2.72156 
 
 3.17515 
 
 3.62874 
 
 4.08233 
 
 31.10348 
 
 62.20696 
 
 93.31044 
 
 124.41392 
 
 155.51740 
 
 186.62088 
 
 217.72437 
 
 248.82785 
 
 279.93133 
 
 
 Cubic. 
 
 Miscellaneous. 
 
 1 
 
 1 © to 
 
 §*•§1 
 t -1 S d 
 
 n O (3 
 
 5 o 
 
 ©3 to 
 2o% 
 
 P* 
 
 to 
 
 ?o . 
 © 
 
 ©o © 
 o 
 
 o£ 
 ** © 
 OQ 
 
 l's 
 
 CC © 
 
 PJ © 
 
 WW 
 
 1 Gunter’s chain = 20.1168 meters. 
 
 1 sq. statute mile = 259.000 hectares. 
 
 1 fathom = 1-829 meters. 
 
 1 nautical mile = 1,853.25 meters. 
 
 1 ft. = .304801 meter 9.4840158 log. 
 
 1 avoir, pound = 453.5924277 gram . 
 
 15,432.35639 grains = 1 kilogram. 
 
 1 
 
 2 
 
 1 
 
 ( 
 
 i 
 
 ij 
 
 . 16.387 
 
 5 32.774 
 
 i 49.161 
 [ 65.54< 
 
 j 81.93( 
 5 98.321 
 
 1 114.711 
 3 131.09' 
 d 147.48 
 
 1 .02832 
 
 t .05663 
 L .08495 
 ) .11327 
 
 3 .14158 
 
 3 .16990 
 
 3 .19822 
 
 7 .22654 
 
 4 .25485 
 
 0.765 
 
 1.529 
 
 2.294 
 
 3.058 
 
 3.823 
 
 4.587 
 
 5.352 
 
 6.116 
 
 6.881 
 
 0.35239 
 
 0.70479 
 
 1.05718 
 
 1.40957 
 
 1.76196 
 
 2.11436 
 
 2.46675 
 
 2.81914 
 
 3.17154 
 
CONVERSION TABLES. 
 
 9 
 
 The method of using the following tables for converting metric weights 
 and measures into United States weights and measures may he understood 
 by the following example: . 
 
 Find the number of yards in 86 meters. 
 
 From column “ Meters to Yards,” 8 meters = 8.748889 yards, or 80 meters 
 _ 87.48889 yards; and 6 meters — 6.561667 yards. Hence, 86 meters 
 87.48889 + 6.561667 = 94.050557 yards. 
 
 METRIC TO CUSTOMARY. 
 
 Capacity. 
 
 
 fj 
 
 -iS O 
 <D rj 
 
 o 
 
 OS -M 
 
 as as 
 a>£ 
 
 eters to 
 Yards. 
 
 Kilometers 
 to Miles. | 
 
 
 
 £ 
 
 
 1 
 
 39.37 
 
 3.28083 
 
 1.093611 
 
 0.62137 
 
 2 
 
 78.74 
 
 6.56167 
 
 2.187222 
 
 1.24274 
 
 3 
 
 118.11 
 
 9.84250 
 
 3.280833 
 
 1.86411 
 
 4 
 
 157.48 
 
 13.12333 
 
 4.374444 
 
 2.48548 
 
 5 
 
 196.85 
 
 16.40417 
 
 5.468056 
 
 3.10685 
 
 6 
 
 236.22 
 
 19.68500 
 
 6.561667 
 
 3.72822 
 
 7 
 
 275.59 
 
 22.96583 
 
 7.655278 
 
 4.34959 
 
 8 
 
 314.96 
 
 26.24667 
 
 8.748889 
 
 4.97096 
 
 9 
 
 354.33 
 
 29.52750 
 
 9.842500 
 
 5.59233 
 
 Square. 
 
 
 i »s 
 
 •iH <D 
 
 OS’S 
 
 ■A 
 O ® 
 
 os 
 
 oS 
 
 b ass., 
 
 2 
 
 
 ^ f-i l— I 
 
 as 
 
 H OS^ 
 o3 fi 
 
 li 
 
 rs i— i 
 
 
 S|-g 
 
 c3 g 3 
 a* 
 
 CO OQ 
 
 rj 0^ ^ 
 
 § 
 
 m 
 
 2 £ a> 
 a<as b 
 
 “si 
 
 GO 
 
 c3 as 
 
 ■4—* ~ 
 
 m 
 
 1 
 
 0.155 
 
 10.764 
 
 1.196 
 
 2.471 
 
 2 
 
 0.310 
 
 21.528 
 
 2.392 
 
 4.942 
 
 3 
 
 0.465 
 
 32.292 
 
 3.588 
 
 7.413 
 
 4 
 
 0.620 
 
 43.055 
 
 4.784 
 
 9.884 
 
 5 
 
 0.775 
 
 53.819 
 
 5.980 
 
 12.355 
 
 6 
 
 0.930 
 
 64.583 
 
 7.176 
 
 14.826 
 
 7 
 
 1.085 
 
 75.347 
 
 8.372 
 
 17.297 
 
 8 
 
 1.240 
 
 86.111 
 
 9.568 
 
 19.768 
 
 9 
 
 1.395 
 
 96.875 
 
 10.764 
 
 22.239 
 
 Cubic 
 
 Cubic Centi- 
 meters to 
 Cubic Inches. 
 
 Cubic Deci- 
 meters to 
 Cubic Inches. 
 
 Cubic 
 Meters to 
 Cubic Feet. 
 
 Cubic Meters 
 to 
 
 Cubic Yards. 
 
 
 .0610 
 
 61.023 
 
 35.314 
 
 1.308 
 
 1 
 
 .1220 
 
 122.047 
 
 70.629 
 
 2.616 
 
 2 
 
 .1831 
 
 183.070 
 
 105.943 
 
 3.924 
 
 3 
 
 .2441 
 
 244.094 
 
 141.258 
 
 5.232 
 
 4 
 
 .3051 
 
 305.117 
 
 176.572 
 
 6.540 
 
 5 
 
 .3661 
 
 366.140 
 
 211.887 
 
 7.848 
 
 6 
 
 .4272 
 
 427.164 
 
 247.201 
 
 9.156 
 
 7 
 
 .4882 
 
 488.187 
 
 282.516 
 
 10.464 
 
 8 
 
 .5492 
 
 549.210 
 
 317.830 I 11.771 
 
 9 
 
 Milliliters, or 
 Cubic Centi- 
 liters to 
 Fluid Drams. 
 
 Centiliters to 
 Fluid Ounces. 
 
 Liters to 
 Quarts. 
 
 Decaliters 
 
 to 
 
 Gallons. 
 
 Hectoliters 
 
 to 
 
 Bushels. 
 
 0.27 
 
 0.338 
 
 1.0567 
 
 2.6417 
 
 2.8377 
 
 0.54 
 
 0.676 
 
 2.1134 
 
 5.2834 
 
 5.6755 
 
 0.81 
 
 1.014 
 
 3.1700 
 
 7.9251 
 
 8.5132 
 
 1.08 
 
 1.353 
 
 4.2267 
 
 10.5668 
 
 11.3510 
 
 1.35 
 
 1.691 
 
 5.2834 
 
 13.2085 
 
 14.1887 
 
 1.62 
 
 2.029 
 
 6.3401 
 
 15.8502 
 
 17.0265 
 
 1.89 
 
 2.367 
 
 7.3968 
 
 18.4919 
 
 19.8642 
 
 2.16 
 
 2.705 
 
 8.4535 
 
 21.1336 
 
 22.7019 
 
 2.43 
 
 3.043 
 
 9.5101 
 
 23.7753 
 
 25.5397 
 
 Weight. 
 
 Milligrams to 
 Grains. 
 
 Kilograms to 
 Grains. 
 
 Hectograms 
 to Ounces 
 Avoir. 
 
 Kilograms 
 to Pounds 
 Avoir. 
 
 .01543 
 
 15,432.36 
 
 3.5274 
 
 2.20462 
 
 .03086 
 
 30,864.71 
 
 7.0548 
 
 4.40924 
 
 .01630 
 
 46,297.07 
 
 10.5822 
 
 6.61387 
 
 .06173 
 
 . 61,729.43 
 
 14.1096 
 
 8.81849 
 
 .07716 
 
 77,161.78 
 
 17.6370 
 
 11.02311 
 
 .09259 
 
 92,594.14 
 
 21.1644 
 
 13.22773 
 
 .10803 
 
 108,026.49 
 
 24.6918 
 
 15.43236 
 
 .12346 
 
 123,458.85 
 
 28.2192 
 
 17.63698 
 
 .13889 
 
 138,891.21 | 31.7466 
 
 19.84160 
 
 Weight— ( Continued). 
 
 ® 02 . 
 
 'S 'g.ft 
 
 I an 
 § 
 
 220.46 
 
 440.92 
 
 661.39 
 
 881.85 
 
 1,102.31 
 
 1,322.77 
 
 1,543.24 
 
 1,763.70 
 
 1,984.16 
 
 os OS 
 
 Sggf 
 
 $ 
 
 2.204.6 
 4,409.2 
 6,613.9 
 8,818.5 
 
 11,023.1 
 
 13,227.7 
 
 15,432.4 
 
 17,637.0 
 
 19.841.6 
 
 32.1507 
 
 64.3015 
 
 96.4522 
 
 128.6030 
 
 160.7537 
 
 192.9044 
 
 225.0552 
 
 257.2059 
 
 289.3567 
 
10 
 
 WEIGHTS AND MEASURES. 
 
 METRIC CONVERSION TABLE. 
 
 ( Arranged by C. W. Hunt, New York . ) 
 
 Millimeters X .03937 = in. 
 
 Millimeters -4- 25.4 = in. 
 
 Centimeters X .3937 = in. 
 
 Centimeters -4- 2.54 = in. 
 
 Meters X 39.37 = in. (Act Congress). 
 Meters X 3.281 = ft. 
 
 Meters X 1.094 = yd. 
 
 Kilometers X .621 = miles. 
 Kilometers -4- 1.6093 = miles. 
 Kilometers X 3,280.7 = ft. 
 
 Square millimeters X .0155 = sq. in. 
 Square millimeters -4- 645.1 = sq. in. 
 Square centimeters X .155 = sq. in. 
 Square centimeters -4- 6.451 = sq. in. 
 Square meters X 10.764 = sq. ft. 
 Square kilometers X 247.1 = acres. 
 Hectare X 2.471 = acres. ' ! 
 Cubic centimeters -4- 16.383 - cu. in. 
 Cubic centimeters -4- 3.69 = fluid 
 drams (U. S. P.). 
 
 Cubic centimeters -4- 29.57 = fluid oz. 
 (U. S. P.). 
 
 Cubic meters X 35.315 = cu. ft. 
 
 Cubic meters X 1.308 = cu. yd. 
 
 Cubic meters X 264.2 = gal. (231 
 cu. in.). 
 
 Liters X 61.022 = cu. in. (Act Con- 
 gress). 
 
 Liters X 33.84 = fluid oz. (U. S. Phar.). 
 Liters X .2612 = gal. (231 cu. in.). 
 Liters -4- 3.78 = gal. (231 cu. in.). 
 Liters -4- 28.316 = cu. ft. 
 
 Tonnes X 1.102 = short tons. 
 
 Tonnes “X .9839 = long tons. 
 
 Hectoliter^ X 3.531 = cu. ft. 
 
 Hectoliters X 2.84 = bu. (2,150.42 
 cu. in.). 
 
 Hectoliters X .131 = cu. yd. 
 
 Hectoliters -4- 26.42 = gal. (231 
 cu. in.). 
 
 Grams X 15.432 = gr. (Act Con- 
 gress) . 
 
 Grams -4- 981 = dynes. 
 
 Grams (water) -4- 29.57 = fluid oz. 
 
 Grams -4- 28.35 = oz. avoir. 
 
 Grams per cu. cent, -4- 27.7 = lb. per 
 cu. in. 
 
 Joule X .7373 = ft. -lb. 
 
 Kilograms X 2.2046 = lb. 
 
 Kilograms X 35.3 = oz. avoir. 
 
 Kilograms -4- 1,102.3 = ton (2,000 lb.). 
 
 Kilogr. per sq. cent. X 14.223 = lb. 
 per sq. in. 
 
 Kilogram-meters X 7.233 = ft.-lb. 
 
 Kilo per meter X .672 = lb. per ft. 
 
 Kilo per cu. meter X .026 = lb. per 
 cu. ft. 
 
 Kilo per cheval X 2.235 = lb. per 
 H. P. 
 
 Kilowatts X 1.34 = H. P. 
 
 Watts -4- 746 = H. P. 
 
 Watts -4- .7373 = ft.-lb. per sec. 
 
 Calorie X 3.968 = B. T. U. 
 
 Cheval vapeur X .9863 = H. P. 
 
 (Centigrade X 1.8) +32 = degree F. 
 
 Franc X .193 = dollars. 
 
 Gravity Paris = 980.94 centimeters 
 per sec. 
 
 United States Currency. 
 10 mills = 1 cent. 
 
 10 cents = 1 dime. 
 
 10 dimes = 1 dollar. 
 
 10 dollars = 1 eagle. 
 
 MONEY. 
 
 British Money. 
 
 4farthings = 1 penny. 
 
 12 pence 1 = 1 shilling. 
 
 20 shillings = 1 pound sterling. 
 
 21 shillings = 1 guinea. 
 
 Standard United States Coins. 
 
 Gold. 
 
 Silver. 
 
 Denomination. 
 
 Value. 
 
 Weight. 
 
 Denomination. 
 
 Value. 
 
 Weight. 
 
 * Dollar 
 
 Quarter eagle 
 
 * Three-doll ar piece 
 
 Half eagle 
 
 Eagle 
 
 Double eagle 
 
 $1.00 
 
 2.50 
 
 3.00 
 
 5.00 
 10.00 
 20.00 
 
 25.8 gr. 
 64.5 gr. 
 77.4 gr. 
 
 129.0 gr. 
 
 258.0 gr. 
 
 516.0 gr. 
 
 * Trade dollar... 
 Standard silver 
 
 dollar 
 
 Half dollar 
 
 Quarter dollar... 
 Dime 
 
 $1.00 
 
 1.00 
 
 .50 
 
 .25 
 
 .10 
 
 420.0 gr. 
 
 412.5 gr. 
 192.9 gr. 
 96.45 gr. 
 38.58 gr. 
 
 “Fineness” expresses the proportion of pure metal in 1,000 parts; thus, 
 “ 900 fine ” means that 900 of every 1,000 parts are pure metal. Fineness of 
 
 * No longer coined. 
 
MONETARY VALUES. 
 
 11 
 
 U. S. coins = 900 pure metal, 100 alloy; alloy of gold coin is copper or copper 
 and silver, but in no case shall silver exceed & of total alloy. Alloy of silver 
 coin is copper. 
 
 Piece. Weight. Contents. 
 
 5-cent( nickel) 77.16 grains 75$ copper, 25$ nickel. 
 
 *3-cent .... 30 grains 75$ copper, 25$ nickel. 
 
 *2-cent 66 grains 95$ copper, 5$tin and zinc. 
 
 1-cent (copper) 48 grains 95$ copper, 5$ tin and zinc. 
 
 *No longer coined. 
 
 Space Required to Store U. S. Coins. 
 
 Description. 
 
 Amount. 
 
 How Put Up. 
 
 Space. 
 
 Gold coins 
 
 $1,000,000 
 
 $5,000 in 8-oz. duck bags 
 
 Nearly 17 cu. ft. 
 
 Silver dollars 
 
 1,000,000 
 
 1,000 in 8-oz. duck bags 
 
 250 cu. ft. 
 
 Subsidiary silver 
 
 1,000,000 
 
 1,000 in 8-oz. duck bags 
 
 150 cu. ft. 
 
 A bag of standard silver dollars occupies a space 12 in. X 9 in. X 4 in. 
 
 To Convert Value of U. S. Coins Into English Values and Vice Versa. 
 Rule. — Cents ( U. S.) 2.02771, or X -179312 = English pence. 
 
 Example.— 100 cents X .49312 = 49.312 pence — 4s. 1.312d. 
 
 Rule. — English pence X 2.02771 — cents (U. S.). 
 
 Example.— lOOd. X 2.02771 = 202.771 cents = $2.0277. 
 
 „ , Dollars , ... 
 
 Rule.— -jgjjQfi = Rounds sterling. 
 
 Example.— = £20.548. £.548 X 240 = 131.5d. = 10s. 11.5d. 
 
 4.ooo5 
 
 Rule. — Pounds X 7.8665 = dollars ( U. S.). Shillings X 2U.332+ = cents ( U.S.). 
 
 VALUES OF FOREIGN COINS, U. S. TREASURY DEPT., JAN. 1, 1899. 
 
 Argentine, Argentine Re- 
 public $ 4.824 
 
 Bolivar, Venezuela .193 
 
 Boliviano, Bolivia 439 
 
 Centen, Cuba 5.017 
 
 Colon, Costa Rica 465 
 
 Condor, Chile 7.300 
 
 Condor, U. S. of Colombia and 
 
 Ecuador 9.647 
 
 Copeck, Russia .0075 
 
 Crown, Austria-Hungary .203 
 
 Crown, Denmark, Norway, 
 
 and Sweden 268 
 
 Crown, Germany 1.06 
 
 Crown, Great Britain 1.13 
 
 Crown, Sicily 96 
 
 Crown, Spain (half pistole) 1.95 
 
 Dollar, Bolivia .96 
 
 f Dollar, British Honduras, 
 
 British Possessions, N. A. 
 
 ( except N e wfoundland ) , 
 
 and Liberia 1.000 
 
 Dollar, Chile, Peru, and 
 
 Ecuador 93 
 
 Dollar, Mexican (gold) 983 
 
 Dollar, Mexican (silver) 477 
 
 Dollar, Newfoundland 1.014 
 
 Dollar, U. S. of Colombia 935 
 
 Doubloon, Central America $14.50 
 
 Doubloon, Chile , 3.650 
 
 Doubloon, New Granada 15.34 
 
 Doubloon, Spain and Mexico 15.65 
 
 Drachma, Greece 193 
 
 Ducat, Austria, Bohemia, 
 
 Hamburg, Hanover 2.28 
 
 Ducat, Denmark 1.11 
 
 Ducat, Sweden 2.20 
 
 Escudo, Chile 1.825 
 
 Florin, Austria-Hungary 1.929 
 
 Florin, Hanover (gold) 1.66 
 
 Florin, Hanover (silver) 56 
 
 Florin, Holland, South Ger- . 
 
 many .38 
 
 Florin, Netherlands 402 
 
 Florin, Prussia 55 
 
 Florin, Silesia 48 
 
 Franc, Belgium, Bulgaria, 
 
 France, Italy, Roumania, 
 
 Switzerland 193 
 
 Gourde, Hayti 965 
 
 Groschen, Prussian Poland .024 
 
 Guinea, Great Britain 5.11 
 
 Gulden, Baden 40 
 
 Imperial. Russia 7.92 
 
 Kran, Persia 081 
 
 Kreutzer, Bavaria .0067 
 
 tThe British dollar has the same legal value as the Mexican dollar in Hongkong, the Straits 
 Settlements, and Labuan. 
 
12 
 
 WEIGHTS AND MEASURES. 
 
 VALUES OF FOREIGN co I ns. {Continued.) 
 
 Lira, Italy - $ -193 
 
 Mark, Finland 193 
 
 Mark, German Empire 238 
 
 Maximilian, Bavaria 3.30 
 
 Milreis, Brazil 546 
 
 Milreis, Portugal 1.080 
 
 Mohur, India 7.105 
 
 Napoleon, France 3.84 
 
 Peseta, Spain 193 
 
 Peso, Argentine Republic ... .965 
 
 Peso, Chile 365 
 
 Peso, U. S. of Colombia 439 
 
 Peso, Cuba 926 
 
 Peso, Guatemala, Honduras, 
 
 Nicaragua, Salvador 365 
 
 Peso, Uruguay 1.034 
 
 Piaster, Egypt 049 
 
 Piaster, Turkey .044 
 
 Piastre, Spain .1.04 
 
 Pistole, Rome '. 3.37 
 
 Pistole, Spain 3.90 
 
 Pound, Egypt 4.943 
 
 Pound Sterling, Great Britain 4.8665 
 Ruble, Russia .515 
 
 Rupee, India S .208 
 
 Shilling, Great Britain 243 
 
 Sol, Peru 439 
 
 Sou, France 01 
 
 Sovereign, Great Britain 4.8665 
 
 Sucre, Ecuador .439 
 
 ' Amoy .710 
 
 Canton 708 
 
 Chefoo 679 
 
 Chin Kiang .693 
 
 Fuehau 656 
 
 Haikwan (Cus- 
 toms) .722 
 
 Tael, China i Hankow 664 
 
 ' Hongkong f 
 
 Niuchwang .665 
 
 Ningpo 682 
 
 Shanghai 648 
 
 Swatow 655 
 
 Takau 714 
 
 L Tientsin 688 
 
 Toman, Persia 3.409 
 
 Yen, Japan - 498 
 
 t The British dollar has the same legal value as the Mexican dollar in Hongkong, the Straits 
 Settlements, and Labuan. 
 
 The carat (a 24th part) is used to express the proportion of gold in an 
 alloy; thus, gold 18 carats tine is pure. The carat is also a unit of weight 
 for precious stones. Its value vanes according to different authorities, but 
 the international carat is 3.168 grains, or 206 milligrams. 
 
 Diamond Weight (Nystrom). 
 
 Carats. Grains. Parts, 
 
 1 = 4 = 64 = 3.2 (3.168) 
 
 .25 =1 = 16 = .8 
 
 .015625 = .0625 = 1 = .05 
 .3125 = 1.25 — 20 —1 
 
 151.5 = 1 ounce 
 
 TIMBER AND BOARD MEASURE. 
 
 TIMBER MEASURE. 
 
 Volume of Round Timber.— The volume in cubic feet equals the length 
 multiplied by one-fourth the product of mean girth and diameter, all dimen- 
 sions being in feet. If length is given in feet and girth and diameter in 
 inches, divide by 144; if all dimensions are in inches, divide by 1,728. 
 
 Volume of Square Timber.— When all dimensions are in feet; 
 
 R u | e< — Multiply the breadth by the depth and that product by the length , and the 
 product will give the volume in cubic feet. 
 
 When either of the dimensions is in inches: 
 
 Rule . — Multiply as above and divide by 12. 
 
 When any two of the dimensions are in inches: 
 
 Rule . — Multiply as before and divide by 144. 
 
TIMBER AND BOARD MEASURE. 
 Round Timber —Table of 4 Girths. 
 
 13 
 
 4 Girths. 
 Inches. 
 
 Area in 
 Feet. 
 
 1 Girths. 
 Inches. 
 
 Area in 
 Feet. 
 
 £ Girths. 
 Inches. 
 
 Area in 
 Feet. 
 
 6 
 
 .250 
 
 121 
 
 1.04 
 
 19 
 
 2.50 
 
 6 f 
 
 .272 
 
 121 
 
 1.08 
 
 194 
 
 2.64 
 
 
 .294 
 
 124 
 
 1.12 
 
 20 
 
 2.77 
 
 
 .317 
 
 13 
 
 1.17 
 
 204 
 
 2.91 
 
 7 
 
 .340 
 
 131 
 
 1.21 
 
 21 
 
 3.06 
 
 7r 
 
 .364 
 
 131 
 
 1.26 
 
 214 
 
 3.20 
 
 7£ 
 
 .390 
 
 13f 
 
 1.31 
 
 22 
 
 3.36 
 
 7| 
 
 .417 
 
 14 
 
 1.36 
 
 224 
 
 3.51 
 
 8 
 
 .444 
 
 141 
 
 1.41 
 
 23 
 
 3.67 
 
 81 
 
 .472 
 
 141 
 
 1.46 1 
 
 234 
 
 3.83 
 
 w 4 
 
 .501 
 
 14f 
 
 1.51 
 
 24 
 
 4.00 
 
 si 
 
 .531 
 
 15 
 
 1.56 
 
 m 
 
 4.16 
 
 9 
 
 .562 
 
 151 
 
 1.61 
 
 25 
 
 4.34 
 
 
 .594 
 
 151 
 
 1.66 
 
 254 
 
 4.51 
 
 94 
 
 .626 
 
 151 
 
 1.72 
 
 26 
 
 4.69. 
 
 9| 
 
 .659 
 
 16 
 
 1.77 
 
 26| 
 
 4.87 
 
 10 
 
 .694 
 
 161 
 
 1.83 
 
 27 
 
 5.06 
 
 10 r 
 
 .730 
 
 161 
 
 1.89 
 
 274 
 
 5.25 
 
 xv 4 
 
 104 
 
 .766 
 
 161 
 
 1.94 
 
 28 
 
 5.44 
 
 104 
 
 .803 
 
 17 
 
 2.00 
 
 284 
 
 5.64 
 
 11 
 
 .840 
 
 171 
 
 2.09 
 
 29 
 
 5.84 
 
 1H 
 
 .878 
 
 171 
 
 2.12 
 
 294 
 
 6.04 
 
 x-1-4 
 
 111 
 
 .918 
 
 171 
 
 2.18 
 
 30 
 
 6.25 
 
 Ilf 
 
 . .959 
 
 18 
 
 2.25 
 
 
 
 12 
 
 1.000 
 
 181 
 
 2.37 
 
 
 
 Area corresponding to 4 girth (mean) in inches multiplied by length 
 in feet equal solidity in feet and decimal parts. 
 
 BOARD MEASURE. 
 
 In measuring boards, they are assumed to be 1 inch in thickness. The 
 number of feet, board measure (B. M.),in a given board or stick of timber, 
 equals the length in feet multiplied by the breadth m feet multiplied by the 
 thickness in inches. 
 
 Breadth. 
 
 Inches. 
 
 Area of a 
 Lineal Foot. 
 
 Breadth. 
 
 Inches. 
 
 Area of a 
 Lineal Foot. 
 
 Breadth. 
 
 Inches. 
 
 Area of a 
 Lineal Foot. 
 
 A 
 
 .021 
 
 44 
 
 .354 
 
 84 
 
 .688 
 
 A 
 
 A 
 
 .042 
 
 44 
 
 .375 
 
 84 
 
 .708 
 
 a 
 
 i 
 
 .063 
 
 44 
 
 .396 
 
 84 
 
 .729 
 
 * 
 
 1 
 
 .083 
 
 5 
 
 .417 
 
 9 
 
 .750 
 
 14 
 
 .104 
 
 54 
 
 .438 
 
 94 
 
 .771 
 
 14 
 
 .125 
 
 54 
 
 .458 
 
 94 
 
 .792 
 
 ■*-3 
 
 If 
 
 .146 
 
 54 
 
 .479 
 
 94 
 
 .813 
 
 -*-4 
 
 2 
 
 .167 
 
 6 
 
 .500 
 
 10 
 
 .833 
 
 24 
 
 .188 
 
 64 
 
 .521 
 
 104 
 
 .854 
 
 "4 
 
 24 
 
 .208 
 
 64 
 
 .542 
 
 104 
 
 .875 
 
 4i a 
 
 24 
 
 .229 
 
 64 
 
 .563 
 
 104 
 
 .896 
 
 3 
 
 250 
 
 7 
 
 .583 
 
 11 
 
 .917 
 
 34 
 
 .271 
 
 74 
 
 .604 
 
 114 
 
 .938 
 
 '-’A 
 
 34 
 
 .292 
 
 74 
 
 .625 
 
 Hi 
 
 .958 
 
 3f 
 
 .313 
 
 74 
 
 .646 
 
 114 
 
 .979 
 
 4 
 
 .333 
 
 8 
 
 .667 
 
 12 
 
 1.000 
 
 Area oi a lineai i 
 tents in square feet. 
 
14 
 
 MATHEMATICS. 
 
 MATHEMATICS. 
 
 By Edward H. Williams, Jr., E. M. 
 
 Professor of Mining Engineering and Geology at the Lehigh University. 
 
 GENERAL PRINCIPLES. 
 
 Quantity or magnitude is anything that can he increased or decreased, or 
 that is capable of any sort of measurement or calculation, such as numbers, 
 lines, space, time, motion, weight, force, power, heat, light, electricity, etc. 
 We can measure a quantity by applying to it a portion of the same quantity, 
 called a unit. If the quantities are of different kinds, we cannot measure 
 them by one another, but we can compare them or institute a calculation 
 between them. 
 
 Mathematics treats of all kinds of quantity that can be numbered or meas- 
 ured. Arithmetic is that part that treats of numbering, and is called the 
 science of numbers. Geometry is the science of measuring. These two are 
 the foundation of all other parts of mathematics, and are called pure mathe- 
 matics. We can also reason about numbers by substituting letters for num- 
 bers, and represent their relations by signs. This is called algebra , and it 
 may be likened to a shorthand arithmetic. An extension of arithmetic to 
 geometry, by which angles and triangles are subjected to numerical compu- 
 tation, is called trigonometry, and plane trigonometry treats of methods of 
 computing plane angles and triangles, and embraces the investigations of the 
 relations of angles in general, which is called angular analysis. Another 
 extension of arithmetic to geometry, by which lines, areas, and volumes are 
 computed, is called mensuration. Mensuration of large portions of the earth’s 
 surface, where the curvature of the same is taken into calculation, is called 
 geodesy. If the portions are smaller and curvature is neglected, the science 
 is called surveying, and mine surveying if confined to underground work. 
 
 COMMONLY USED MATHEMATICAL SIGNS AND ABBREVIATIONS. 
 
 -f means plus, or addition. 
 
 — means minus, or subtraction. 
 
 X means multiplication. 
 
 ± means plus or minus. 
 
 + means minus or plus. 
 
 -r- means division. 
 
 means ratio. 
 
 : :: : means proportion. 
 
 2 : 3 : : 4 : 6 shows that 2 is to 3 
 as h is to 6. 
 
 = means equality. 
 
 means equivalency, 
 y means square root, 
 
 f' means cube root , etc. 
 
 l/3 square root of 3. 
 
 ^ 5 cube root of 5. 
 
 T 1 7 squared. 
 
 8 3 8 cubed. 
 
 5 = a/b, a + b. 15 -h 16 = 
 
 0 Id 
 
 . * . therefore. 
 
 > greater than. 
 
 < less than. 
 
 □ square. 
 
 □' square feet. 
 
 □" square inches. 
 
 O round. 
 
 ( ) [ 1 | | , vincula , denoting that 
 
 the numbers enclosed are 
 to be taken together; as, 
 
 (a + 6)c = 4 + 3X5 = 35. 
 degrees, arc or thermometer, 
 minutes or feet, 
 seconds or inches. 
 
 30° 40' 4" is 30 degrees 40 
 minutes 4 seconds. 
 is 4 feet 6 inches, 
 accents to distinguish letters , 
 as a', a", a'", 
 ai, « 2 , a b , a c , read a sub 1, a sub b, etc. 
 a 2 , a 3 a squared, a cubed. 
 
 as = i Yofi, at = > / a 3 . 
 
 sin a = the sine of a. 
 
 log = logarithm, 
 
 angle. 
 
 L right angle. 
 
 _|_ perpendicular to. 
 
 sin sine. 
 
 cos cosine. 
 
 4' 6" 
 
 / // nt 
 
ARITHMETIC. 
 
 15 
 
 MATHEMATICAL SIGNS AND ABBREVIATIONS— {Continued). 
 
 tan, or tang, tangent. 
 sec secant. 
 
 versin versed sine. 
 
 cot cotangent. 
 
 cosec cosecant. 
 
 covers coversed sine. 
 
 tt' pi, ratio of circumference of 
 
 circle to diameter 3.14159. 
 g acceleration due to gravity 
 
 = (32.16 ft. per sec.). 
 
 R, r radius. 
 
 W, w weight. 
 
 H. P. horsepower. 
 
 I. H. P. indicated horsepower. 
 
 B. H. P. brake horsepower. 
 
 A. W. G. American wire gauge 
 
 (Brown & Sharpe). 
 
 B. W. G. Birmingham wire gauge. 
 
 r. p. m., or rev.’ per min., revolutions 
 per minute. 
 
 A decimal point is a period (.) pre- 
 fixed to a number to show 
 that the number is less than 
 unity (1); thus, .2 = •&; 
 .35 = y 3 (fo> 5.75 = 5 yo%> or 5|. 
 
 ARITHMETIC. 
 
 To Cast the Nines Out of a Number.— Add together the digits, and find how 
 many nines are contained in their sum. Reject these nines and set down the 
 remainder to the right of the number. 
 
 Example —Cast the nines out of 18.304. 18,304. /. Ans. 
 
 To Prove Addition.— Cast the nines out of each row of figures added, and 
 out of their sum. Add together the remainder and cast the nines from its 
 sum If the remainder from this last process is equal to the remainder 
 obtained from the sum of the numbers, the addition is correct. 
 
 Example— Prove this addition: 2,1 4 3,5 6 8 2 
 
 8,560,391 5 
 
 1 0,7 0 3,9 5 9 7. Ans. 
 
 To Prove Subtraction.— Add the remainder to the lesser number; their sum 
 should equal the larger number. .. , , ... 
 
 To Prove Multiplication. —Cast the nines out of multiplicand and multi- 
 plier and multiplv the remainders together. Cast the nines out of the 
 product, and the remainder should equal the remainder obtained by cast- 
 ing the nines from the original product. 
 
 Example.— Prove this multiplication: 3,542 X 6,196 = 21,946,232. 
 
 3,5 4 2 5 
 
 6,1 9 6 4 
 
 2 1,9 4 6,2 3 2 2. Ans. 
 
 To Prove Division.— Subtract the remainder, if there be any, from the 
 dividend, and divide what remains by the quotient. If the new quotient 
 equals the old divisor, the work is right. 
 
 Example.— Divide 31,046,835 by 56. 554,407p. Ans. . 
 
 Proof.— Take 43 from 31,046,835, and divide the remainder, 31,046, /92, by 
 
 Rule.— To square any number containing the fraction multiply the whole 
 number by the next higher whole number , and add |. 
 
 Example.— (8£)* 2 — 8 X 9 + i = 72£. 
 
 COMMON FRACTIONS. 
 
 A fraction is a part of a whole, as f, etc. „ 
 
 The numerator of a fraction is the number that tells how many parts of a 
 whole are taken. Thus, 2 is the numerator of § , as it shows that two of the 
 three parts into which the whole is divided are taken. 
 
 The denominator of a fraction is the number that shows into how. many 
 parts the whole is divided. Thus, in the fraction §, the 3 is the denominator. 
 
 A common denominator is a denominator common to two or more fractions. 
 Thus 4 and f have common denominators; and again, 12 is a common de- 
 nominator for I-, and £, as they each are respectively equal to T 2 5 , T5 , 
 
 T<T Add^Common Fractions.— If of the same denominator, add together the 
 numerators only. Thus & + ts + ts = iV 
 
16 
 
 FRACTIONS. 
 
 If they have different denominators, change them to fractions with com- 
 mon denominators, and proceed as before. 
 
 Example.— What is the sum of § + i + i ? 
 
 I = hi i = ift, and i = ||. 
 
 If + If + if — If* Ans. 
 
 To Multiply Common Fractions.— Multiply the numerators together for the 
 numerator, and the denominators for the denominator. Thus, £ X & X I 
 == 9%> or tV 
 
 To Divide Common Fractions.— Invert the divisor, and multiply. 
 
 Example. — Divide & by §. 
 
 AXf = T25* Ans. 
 
 To Reduce Compound Fractions to Simple Fractions.— Multiply the integer 
 by the denominator of the fraction, add the numerator for the new numera- 
 tor, and place it over the denominator. 
 
 Example.— Reduce 5§ to a simple fraction. 
 
 5X3 + 2 = 17, or the numerator, and the fraction is therefore 
 
 To Reduce Simple Fractions to Compound Fractions.— Divide the numerator 
 by the denominator, and use the remainder as the numerator of the remain- 
 ing fraction. 
 
 Example.— Reduce - 6 # to a compound fraction. 
 
 9)64(7 
 
 6 3 Compound fraction = 7+ Ans. 
 
 “I 
 
 To Reduce Common Fractions to Decimal Fractions.— Annex ciphers to the 
 numerator, and divide by the denominator, and point off as many decimal 
 places in the quotient as there are ciphers annexed. 
 
 Example.— Reduce A to a decimal fraction. 
 
 16 ) 9.0 000 (.5 6 25 Ans. 
 
 Note.— Ciphers annexed to a deci- 
 mal do not increase its value. 1.13 is 
 the same as 1.1300. Every cipher 
 placed between the first figure of a 
 decimal and the decimal point divides 
 the decimal by 10. Thus, 
 
 .13 -r- 10 = .013, 
 
 80 
 100 
 96 
 40 
 32 
 80 
 80 
 
 Table of Fractions Reduced to Decimals. 
 
 Si 
 
 .015625 
 
 If 
 
 .265625 
 
 if 
 
 .515625 
 
 11 
 
 l 
 
 .765625 
 
 3*2 
 
 .03125 
 
 52 
 
 .28125 
 
 If 
 
 .53125 
 
 hi 
 
 
 .78125 
 
 
 .046875 
 
 19 
 
 64 
 
 .296875 
 
 if 
 
 .546875 
 
 6] 
 
 63 
 
 
 .796875 
 
 T 5 
 
 .0625 
 
 T6 
 
 .3125 
 
 is 
 
 .5625 
 
 1 c 
 
 Tt 
 
 
 .8125 
 
 Si 
 
 .078125 
 
 n 
 
 .328125 
 
 ii 
 
 .578125 
 
 6 1 
 63 
 
 
 .828125 
 
 
 .09375 
 
 ii 
 
 .34375 
 
 if 
 
 .59375 ! 
 
 hi 
 
 f 
 
 .84375 
 
 £ 
 
 .109375 
 
 if 
 
 .359375 
 
 if 
 
 .609375 
 
 11 
 
 
 .859375 
 
 
 .125 
 
 f 
 
 .375 
 
 i 
 
 .625 
 
 i 
 
 
 .875 
 
 Si 
 
 .140625 
 
 if 
 
 .390625 
 
 u 
 
 .640625 
 
 64 
 
 .890625 
 
 S2 
 
 .15625 
 
 if 
 
 .40625 
 
 hi 
 
 .65625 
 
 If 
 
 .90625 
 
 ii 
 
 .171875 
 
 if 
 
 .421875 
 
 If 
 
 .671875 
 
 IS 
 
 .921875 
 
 T6 
 
 .1875 
 
 TS 
 
 .4375 
 
 ii 
 
 .6875 
 
 ft 
 
 .9375 
 
 £f 
 
 .203125 
 
 if 
 
 .453125 
 
 If 
 
 .703125 
 
 
 fc 
 
 .953125 
 
 "52 
 
 .21875 
 
 it 
 
 .46875 
 
 it 
 
 .71875 
 
 hi 
 
 .96875 
 
 if 
 
 .234375 
 
 if 
 
 .484375 
 
 if 
 
 .734375 
 
 ii 
 
 .984375 
 
 i 
 
 .25 
 
 1 
 
 .5 
 
 f 
 
 .75 
 
 l 
 
 
 1.0000 
 
 DECIMALS. 
 
 Decimal fractions have for their denominators 10 or a power of 10, 
 but the denominator is usually omitted. Thus, .1 = .01 = T |^; .001 
 
 = Tt ha} e ^c. 
 
 To Add Decimals.— Place whole numbers under 
 whole numbers, tenths under tenths, hundredths 
 under hundredths, etc., and add, placing the deci- 
 mal point in the sum directly under the points 
 above. Thus, 1 9.6 3 1 7 
 
 .00 7 5 
 .6 3 
 1.0 6 
 
 1 7.9 3 4 2 
 
DECIMALS. 
 
 17 
 
 To Subtract Decimals.— Arrange the figures as in 5.9 69 7 8 
 
 addition, and proceed as in simple subtraction. 3. 2 8 6 9 4 
 
 Thus 2.6 8 2 8 4 
 
 To Multiply Decimals.— Proceed as in ' £ ££ 
 
 simple multiplication, pointing oft* as 
 
 many decimal places in the result as 14 0 2 5 9 3 
 
 there are decimal places in both mul- 2 3 3 7 6 5 5 
 tiplicand and multiplier. Thus 0.2 4 7 7 9 1 4 3 
 
 To Divide Decimals.— Proceed as in simple division, and 
 decimal places in the quotient as the number of decimal 
 dend exceeds those in the divisor. 
 
 (5 decimal places) 
 (3 decimal places) 
 
 (8 decimal places) 
 point off as many 
 places in the divi- 
 
 Example 1 .— Divide 4.756 by 3.3. 
 
 . 3.3 ) 4.7 5 6 0 0 ( 1.4 4 1 2 Ans. 
 
 33 
 
 145 
 
 1 ^ Example 2.— Divide .006 by 20 
 
 Too 2 0 ).0 0 60 (.0 0 0 3. Ans. 
 
 —40 — $-° 
 
 33 
 
 70 
 
 66 
 
 4 
 
 U 0TE it has been said before that algebra is a shorthand arithmetic. 
 
 Before proceeding further with the various methods of arithmetic, the 
 principles of algebra will be stated, and, after the subsequent examples are 
 worked out by arithmetical rules, an example will be given of the algebraic 
 method of doing the same. In every example, we have known quantities 
 from which we seek to find certain unknown ones. While there is no way of 
 indicating these in arithmetic , we can readily do so in algebra , by placing the 
 first letters of the alphabet as representatives of the known quantities 
 (as a, b, c), and the last letters ( x , y, z) of the unknown ones. The signs m 
 algebra are those just given for arithmetic. In addition to them, we can 
 indicate multiplication by placing a period (.) between the quantities, as a.b 
 (read a multiplied by b), or simply by placing the two letters together, as ab. 
 
 We can indicate division as in common fractions, ^ being read a divided by b. 
 
 To illustrate algebraic symbols, let l denote the length, b the breadth, and 
 h the height of a mine car. If it be desired to divide the height into the 
 product of the length and breadth, it is expressed as follows: 
 
 lb 
 
 h‘ 
 
 When two or more letters are placed together, without anything between 
 them, it is understood that the quantities represented by those letters should 
 be multiplied together. If l represents 8 and b represents 4, then 4 and 8 are 
 multiplied together; thus, 4 X 8 = 32. , ,, , ,,, 
 
 If it be desired to divide the height into the sum of the length and breadth, 
 it is expressed thus: 
 
 l -}- b 
 h '' 
 
 The square of the length multiplied by the cube of the breadth, thus: 
 
 l*b\ 
 
 The square root of the length divided by the cube root of the breadth, thus: 
 
 VT 
 
 The square root of the difference of the length and breadth divided by the 
 height, thus: 
 
 V l — b 
 
18 
 
 PROPORTION. 
 
 SIMPLE PROPORTION, OR SINGLE RULE OF THREE. 
 
 .A PJ 0 P° rti ? n is , expression of equality between equal ratios; thus, the 
 ratio of 10 to 5 = the ratio of 4 to 2, and is expressed thus: 
 
 10 : 5 : : 4 : 2. 
 
 There are four terms in proportion. The first and last are the extremes 
 and the second and third are the means. 
 
 Quantities are in proportion by alternation when antecedent is compared 
 with antecedent and consequent with consequent. Thus, if 10 : 5 : : 4 : 2, 
 then 10 ! 4 i ! 5 * 2. 
 
 Quantities are in proportion by inversion when the antecedents are made 
 consequents and the consequents antecedents. Thus, if 10 : 5 • • 4 • 2 then 
 5 : 10 : : 2 : 4. ’ 
 
 In any proportion, the product of the means will equal the product of the 
 extremes. Thus, if 10 : 5 : : 4 : 2, then 5X4-10X2. 
 
 A mean proportional between two quantities equals the square root of their 
 product. Thus, a mean proportional between 12 and 3 = the square root of 
 12 x 3, or o. 
 
 If the two means and one extreme of a proportion are given, we find the 
 other extreme by dividing the product of the means by the given extreme. 
 
 1° : : : 4 : ( )• then (4 X 5) -4- 10 = 2, and the proportion is 10 : 5 : : 4 : 2 
 
 ,. tt the two extremes and one mean are given, we find the other mean bv 
 dividing the product of the extremes bv the given mean. Thus. 10 * M • • 4 • 2 
 then (10 X 2) 4- 4 = 5, and the proportion is 10 : 5 : : 4 : 2. ’ ' " ’ 
 
 .Example.— I f6 men load 30 wagons of coal in a day, how many wagons 
 will 10 men load? (They will evidently load more, so the second term of the 
 
 proportion must be greater than the first.) 
 
 6 : 10 : : 30 : ( ); then, (10 X 30) 4- 6 = 50. 
 
 Ans. 
 
 COMPOUND PROPORTION, OR DOUBLE RULE OF THREE. 
 
 Principles. 
 
 1. The product of the simple ratios of the first couplet equals the product 
 of the simple ratios of the second couplet. T-hus, 
 
 /4 : 12) . (5 : 10) = ± 7_ _ _5 6 
 
 (7 : 14 J (6 : 18/ 12 X 14 10 X 18‘ 
 
 2. The product of all the terms in the extremes equals the product of all 
 the terms m the means. Thus, in 
 
 J 4 : 121 
 
 
 (5 : 10) 
 
 17 : 14/ 
 
 * ' 
 
 16 : 18/ 
 
 we have 4 X 7 X 10 X 18 - 12 X 14 X 5 X 6. 
 
 3. Any term in either extreme equals the product of the means divided 
 by the product of the other terms in the extremes. Thus, in the same 
 proportion, we have 
 
 4 = 5 X 6 X 12 X 14 
 7X10X18 * 
 
 4. Any term in either mean equals the product of the extremes divided 
 by the product of the other terms in the means. Thus, in 
 
 J 4 : 12) . 
 
 . (5 : 101 
 
 17 : 14/ ‘ 
 
 * 16:18/ 
 
 we have 5 = (4 X 7 X 10 X 18) 4- (6 X 12 X 14). 
 
 Ru J?-— >• Put the required quantity for the first term and the similar known 
 quantity for the second term , and form ratios with each pair of similar quantities 
 for the second couplet , as if the result depended on each pair and the second term. 
 
 II. the required term by dividing the product of the means by the product 
 
 oj inejourtn terms. 
 
 . S X / MP ^ E 1 * — If 4 men can earn $ 24 in 7 days, how much can 14 men earn 
 in 12 days? 
 
 = $144. Ans. 
 
 The sum : $24 : : : 4 j . or> the sum = 
 
 Example 2.— If 12 men in 35 days build a wall 140 rd. long, 6 ft. high, how 
 
 24 X 14 X 12 
 4X7 
 
EVOLUTION. 
 
 19 
 
 many men can, in 40 days, build a wall of the same thickness 144 rd. long, 
 5 ft. high? 
 
 12:()=??4^^ = 9. Ans. 
 
 f 40 : 35 ) 
 
 4 140 : 144 y : : 
 
 l 6:5 J 
 
 40 X 140 X 6 
 
 INVOLUTION. 
 
 To Square a Number.-— Multiply the number by itself. Thus, the square of 
 4 = 4 X 4, or 16. 
 
 To Cube a Number.— Multiply the square of the humber by the number. 
 Thus, the cube of 4 = 16 X 4 = 64. 
 
 To Find the. Fourth Power of a Number.— Multiply the cube by the number. 
 Thus,, the fourth power of 4 = 64 X 4 = 256. 
 
 To Raise a Number to the Sixth Power. — Square its cube. 
 
 To Raise a Number to the Twelfth Power.— Square its sixth power. 
 
 (See logarithms for shorter method.) 
 
 EVOLUTION. 
 
 To Find the Square Root of a Number: 
 
 Rule. — I. Separate the given number into periods of two figures each , beginning 
 at the units place. 
 
 II. Find the greatest number whose square is" contained in the period on the 
 left; this will be the first figure in the root. Subtract the square of this figure from 
 the period on the left , and to the remainder annex the next period to form a 
 dividend. 
 
 III. Divide this dividend , omitting the figure on the right , by double the part of 
 the root already found, and annex the quotient to that part , and also to the divisor; 
 then , multiply the divisor thus completed ' by the figure of the root last obtained , 
 and subtract the product from the dividend. 
 
 IV. If there are more periods to be brought down , continue the operation as 
 before. 
 
 Example.— Find the square root of 8 7'4 2'2 5 ( 9 3 5 Ans. 
 
 874,225. 8 1 
 
 18 31 
 
 TT41T 
 
 549 
 
 186 5 | 9325 
 9 3 25 
 
 OPERATION. 
 
 9 X 2 = 18. 18 into 64 goes 3 times, hence 
 new divisor = 183. 93 X 2 = 186. 186 into 
 932 goes 5 times, hence new divisor = 1,865. 
 
 (See logarithms for shorter method.) 
 
 The square root of a fraction is found by extracting the square root of the 
 numerator and denominator separately. Thus, the square root of ^ f . 
 
 When decimals occur, the number is pointed off into periods both right 
 and left from the decimal point, and there will be as many decimal places in 
 the root as there are periods to the right of the decimal point in the number. 
 Example 1.— Find the square 
 
 root of 874.225. 
 
 8'7 4.2 2'5 ( 2 9.5 6+ 
 
 4 9i 4 7 4 
 441 
 
 Example 2.— Find the square 
 root of .00874225. 
 
 .0 0'8 7'4 2'2 5 (.0 9 3 5 
 81 
 
 18 3| " 
 
 590 
 
 29 25 
 
 7] 39 75W 
 
 35442 
 
 642 
 
 549 
 
 18 6 5| 93 25 
 9325 
 
 4 3 08+ 
 
 To Find the Cube Root of a Number: 
 
 Rule. — I. Separate the given number into periods of three figures each , beginning 
 at the units place. 
 
 II. Find the greatest number whose cube is contained in the period, on the left; 
 this will be the first figure in the root. Subtract the cube of this figure from the 
 period on the left , and to the remainder annex the next period to form a dividend • 
 
20 
 
 PERCENTAGE. 
 
 Ill Divide this dividend by the partial divisor , which is 3 times the square of 
 the root aS found? considered as tens; the quotient is the second figure of the 
 
 r °°\\l To the vartial divisor add 3 times the product of the second figure of the 
 root by the firsf considered as tens , also the square of the second figure, the result 
 
 willbe com piete divisor by the second figure of the root , and subtract 
 
 the^oduct^J^dMdend. dolOTl , proceed™ before, using the 
 
 part of the root already found, the same as the JM . figure t» thepremous process. 
 Example.— Find the cube root of 12,812,904. 
 
 operation. • j ^ . . 
 
 1 2,8 1 2,9 0 4 ( 2 3 4 Ans. 
 
 
 2 } = 
 
 = 8 
 
 1st partial divisor, 3 X 20 2 = 
 3 X 20 X 3 = 
 
 1,2 0 0 
 180 
 
 4,812 
 
 3 2 = 
 
 9 
 
 4,1 6 7 
 
 1st complete divisor, 
 
 1,3 8 9 
 
 6 4 5,9 0 4 
 
 2d partial divisor, 3 X 230 2 == 
 3 X 230 X 4 = 
 
 1 5 8,7 0 0 
 2,7 6 0 
 
 6 4 5,9 0 4 
 
 42 = 
 
 16 
 
 2d complete divisor, 1 6 1,4 7 6 , 
 
 The cube root of a fraction is found by extracting the cube root of the 
 numerator and denominator separately. Thus, the cube root of 64 
 (See logarithms for shorter method.) 
 
 PERCENTAGE. 
 
 Percentage means by or on the hundred. Thus, 1 J> = xh = - 01 > 
 
 plue“theratl e SSjS ft SSZSTgSSXZXA 
 
 agebythe rate 68 ' Thus?!" ' 'the' rateVt^and thepwclnkg^sll^! thebase 
 
 ™ ^dtbe D ^S« 
 equals 115.80 -e- 1,930 = .06, or 6£ 
 
 ARITHMETICAL PROGRESSION. 
 
 sum of the terms 49. 
 
 In any arithmetical progression, 
 
 T pt f = first term; 
 
 l = last, or nth term; 
 d = common difference; 
 n = number of terms; and 
 
 s = their sum. , fA ^ 
 
 The second term =/ + ( S-D* =/+ *. the fourth term -/+(4-l)d. 
 and the nth term = /+( „_ 1)d . (1) 
 
 From equation (1) we obtain , 
 
 f=l±{n-l)d. (2) d== ^ITi 
 
 „ ^ f 4-1 
 
 (3) 
 
 
 (4) 
 
 (5) 
 
GEOMETRICAL PROGRESSION, 
 
 21 
 
 Substituting the value of l from (1), 
 
 «= 2 [2/+(w “" 1)<q * 
 
 ( 6 ) 
 
 Example 1.— A company contracts to put down a bore hole at one dollar 
 if 1 ) Per foot tor the first 100 ft.; three dollars ($3) per foot for the second 100 
 ft.; and two dollars ($2) per foot additional for each successive 100 ft. The 
 hole was 800 ft. deep. What was the cost ? 
 
 n = 8; / = 100; and d = 2. Substitute these values in formula (6). 
 
 „ „ s = f [2 X 100 + (8 - 1) 200] = $6,400. Ans. 
 
 Example 2.— If water flowing 10.12 gal. per min. be struck in a shaft 30 ft 
 SSjow the ^ rface ’ and the increase in flow be .02 gal. per ft. till the depth be 
 200 ft., and thence the flow decreases .02 gal. per ft. till the rock be dry how 
 deep was the shaft at the last point, and what was the total amount of water 
 flowing into it per minute ? 
 
 During the increase of flow, n = 170; / = 10.12; and d = .02. I Tbv 
 formula (1)] — 10.12 -f (170 — 1).02 = 13150, or 13.50 gal. flow at a depth of 200 
 20*6 ft^in^de tli ^ 2X10 ' 12 (170 — 1). 02] = 2,007.7 gal. flowing in along the first 
 
 During the decrease in flow/ = 13.50; d = .02, and l = .02. 
 
 n [formula (3)] = — + 1, for a decreasing progression 
 
 n = 1 . 
 
 Then. 
 
 13.50 -.02 
 
 .02 
 
 d 
 
 + 1 
 
 675, 
 
 the depth at which the rock will run dry, 
 
 ai ?£ a = t 2 X - 02 + (675 — 1) .02], or 4,563 gal., the amount of water that 
 will flow in per minute along the last 675 ft. The total amount of water 
 flowing in along the total depth of 875 ft. is 2,007.7 + 4,563, or 6,570,7 gal. Ans. 
 
 GEOMETRICAL PROGRESSION. 
 
 A series of quantities, in which each is derived from that which precedes 
 W by multiplication by a constant quantity, is called a geometrical progression. 
 It the multiplier be a whole number, the progression is styled increasing; if 
 it be a fraction, the progression is styled decreasing. The series 
 , 1, 2, 4, 8, 16, 32 
 
 has 2 for a multiplier, and is an increasing progression. The series 
 
 32,^16, 8, 4, 2, 1, 
 
 have \ for a multiplier, and are decreasing progressions. 
 
 The common multiplier in a geometrical progression is called the common 
 ratio; or, briefly, the ratio. 
 
 Let / = first term; 
 
 l = last term, whose number from /is n\ 
 n = number of terms; 
 r = ratio; 
 s = sum of terms. 
 
 1 = 
 
 J rpn — 1 ^ 
 
 (1) 
 
 J rpn — 1 
 
 (4) 
 
 s = 
 
 /(»•» - 1 ) 
 r — 1 
 
 (2) 
 
 / = s — r (s — l). 
 
 (5) 
 
 s = 
 
 lr -/ 
 r — 1 * 
 
 (3) 
 
 r = s — f ' 
 s — l 
 
 (6) 
 
 Example.— -If a man should contract to sink a shaft to the base of the coal 
 measures at the rate of cent for the first 50 ft.; | cent for the second 50 ft ; 
 i cent lor the third 50 ft.; and so on at the same rate, how much would be 
 due if the shaft were 1,500 ft. deep ? 
 
 0 . ... „ / = tsI n = 30; and r = 2. 
 
 Substituting m formula (1), 
 
 1 = A X (229) 33,554,432, and s [formula (3)] = 3 . 3 > 554 > 432 j< 2 — T y 
 
 = $671,088.63i|. Ans. 
 
22 
 
 LOGARITHMS. 
 
 LOGARITHMS. 
 
 USE OF LOGARITHMS. 
 
 I nparithms are designed to diminish the labor of multiplication and divi- 
 sion bv substituting in their stead addition and subtraction. A logarithm us 
 the exponent of the power to which a fixed number, called the base, must be 
 raised to produce a given number. The base of the common systein is 10, 
 and as a logarithm is the exponent of the power to which the base must be 
 raised in order to be equal to a given number, all numbers are to be regarded 
 as powers of 10; hence, 
 
 100 = 1, we have logarithm of 1 = 0. 
 
 10 1 = io, we have logarithm of 10 = 1. 
 
 102 = ioo, we have logarithm of 100 = 2. 
 
 103 = i ooo, we have logarithm of 1,000 = 3. 
 
 104 = 10,000, we have logarithm of 10,000 = 4. 
 
 The logarithms of numbers between 1 and 10 are less than unity, and are 
 pxnressed as decimals. The logarithm of any number between 10 and 100 is 
 moreXn 1 anTless than 2, hence it is equal to X plus a decimal. Between 
 
 ^Thf^ in^ral'parfof a* logarithmls'tts ^clwraoteristio, the decimal part is 
 
 ltS Example.— The log of 67.7 is 1.83059, the characteristic of this logarithm 
 
 lS 1 Th^ charter illic of a logarithm is always 1 less than the number of whole 
 ^es expressing that lumber and may ^ fher neg^i veorposvUve 
 y <rv.fi nhflTpeteristic of the logarithm of 7 is 0; of 1/ is 1, oi tu is z, cio. 
 
 Tte mantissa is the decimal portion of a logarithm, and is always considered 
 
 opposite it will be found the logarithm with its ch ^ a £^” st cf ' -proceed 
 
 of 327 is 2 51455 *As the first two figures of the decimal are the same lo 
 
 lillsSisiiss 
 
 number into an integer and a decimal.. If the decimal part contains i 
 
 ssswsss iffijssst « »«<. 
 
 for the first four figures of the given number 
 
 Example.— What is the logarithm of '234,56/? we have 
 
 and in the column headed 19, under “ P. P.,” oppos, ite t » e fl nd l^whmh 
 added to the portion of the mantissa already found, .3/014, gnes .3/U2,. 
 characteristic is 5, hence the iogBntto is 5.3 accor ding to previous 
 
LOGARITHMS. 
 
 23 
 
 of a whole number and a decimal, the characteristic is 1 less than the whole 
 number. Where it is a simple decimal, or when there are no ciphers between 
 the decimal point and the first numerator, the characteristic is negative, 
 and is expressed by 1, with a minus sign over it. Where there is one cipher 
 between the decimal point and first numerator, the characteristic is 2, with 
 a minus sign over it. Where there are 2 ciphers, the characteristic is 3, with 
 a minus sign over it. Thus: 
 
 The logarithm of 67.7 is 1.83059. 
 
 The logarithm of 6.77 is 0.83059. 
 
 The logarithm of .677 is 1.83059. 
 
 The logarithm of .0677 is 2.83059. 
 
 The logarithm of .00677 is 3.83059. 
 
 The characteristic only is negative. The decimal part is positive. 
 
 To Find the Logarithm of a Vulgar Fraction.— Subtract the logarithm of the 
 denominator from the logarithm of the numerator. The difference is the 
 logarithm of the fraction. 
 
 Example.— F ind logarithm of 
 
 Log 4 = 0.60206 
 Log 10 = L 
 
 1.60206 
 
 1.60206 is the logarithm of .4. 
 
 To Find the Natural Number Corresponding to Any Logarithm.— Look in the 
 column headed “ 0 ” for the first two figures of the decimal part; the other four 
 figures are to be looked for in the same or in one of the nine following col- 
 umns. If they are exactly found, the number must be made to correspond 
 with the characteristic by pointing off decimals or annexing ciphers. 
 
 If the decimal portion cannot be found exactly, find the next lower loga 
 rifhm, subtract it from the given logarithm, divide the difference by the 
 difference between the next lower and the next higher logarithm, and annex 
 the quotient to the natural number found opposite the lower logarithm. 
 
 To Multiply by the Use of Logarithms. — Add the logarithms of the factors 
 together; the sum will be the logarithm of their product. 
 
 Example.— 67.7 X .677. 
 
 Log 67.7 = 1.83059 
 
 Log .677 = 1.83059 
 
 1.66118 
 
 1.66118 is the logarithm of 45.833. 
 
 To Divide by the Use of Logarithms.— Subtract the logarithm of the divisor 
 from the logarithm of the dividend; the difference will be the logarithm of 
 the quotient. 
 
 Example.— Divide 67.7 by .0677. 
 
 Log 67.7 = 1.83059 
 Log .0677 = 2.83059 
 3.00000 
 
 3 is the logarithm of 1,000. 
 
 To Square a Number by the Use of Logarithms.— Multiply the logarithm of 
 the number by 2. The product will be the logarithm of the square of the 
 number. 
 
 Example.— Square .677. 
 
 Log .677 = L 83059 
 2 
 
 1.66118 
 
 1.66118 is the logarithm of .45833. 
 
 To Cube a Number.— Multiply the logarithm of the number by 3. The 
 product will be the logarithm of the cube of the number. 
 
 To Raise a Number to Any Power, as 4th, 5th, 6th, or 7th, multiply the loga- 
 rithm of the number by 4, 5, 6, or 7, and the results will be the logarithms of 
 the 4th, 5th, 6th, or 7th powers, respectively. Thus, a number can readily be 
 raised to any power required. 
 
24 
 
 GEOMETRY. 
 
 7 
 
 To Extract the Square, Cube, Fourth, Fifth, or Any Root of a Number. — Divide 
 
 the logarithm of the number by the index of the root required, and the 
 quotient will be the logarithm of the required root. 
 
 Thus, to find the square root of 625: 
 
 Logarithm of 625 = 2.79588. 
 
 2.79588 -T- 2 = 1.39794 
 
 1.39794 = logarithm of 25. 
 
 Therefore, the square root of 625 is 25. 
 
 To Find the Cube, Fourth, or Any Root.— Proceed in the same way, using 
 the index of the required root as a divisor. 
 
 To Divide a Logarithm Having a Negative Characteristic.— If the characteristic 
 is evenly divisible by the divisor, divide in the usual manner and retain the 
 negative sign for the characteristic in the quotient. But if the negative 
 characteristic is less than, or not evenly divisible by, the divisor, add such a 
 negative number to it as will make it evenly divisible, and prefix an equal 
 positive number to the decimal part of the logarithm; then divide the 
 increased negative characteristic by the divisor, to obtain the characteristic 
 of the quotient desired. To obtain the decimal part of the quotient, divide 
 the decimal part of the logarithm, with the positive number prefixed, in the 
 usual manner. To this quotient prefix the negative characteristic already 
 found, and this will be the quotient desired. 
 
 Example 1.— 
 
 6.3246846 
 
 3 
 
 2.1082282. 
 
 Example 2.— _ 
 
 14.3268 472 = (]q + 4 = i8) + (4 + .3268472) 18 +> 32 G M 72 J 2.4807608. 
 
 9 y 
 
 Example 3.— 
 
 5 /-- log .677 1.830589 5 4.830589 
 
 v - 677 = 5 = 5 ' 5 + 5 
 
 1.9661178 m .9249+. 
 
 GEOMETRY. 
 
 PRINCIPLES OF GEOMETRY. 
 
 1. The sum of all the angles formed on one side of a straight line equals 
 two right angles, or 180°. 
 
 2. The sum of all the angles formed around a point equals four right 
 
 angies, or 360°. ,, ., .. . 
 
 3. When two straight lines intersect each other, the opposite or vertical 
 
 angies are equal. . „ „ _ , 
 
 4. If two angles have their sides parallel, they are equal. 
 
 5. If two triangles have two sides, and the included angle of the one 
 equal to two sides and the included angle of the other, they are equal in all 
 
 their parts^ ^ r j an g^ es h ave two angles, and the included side of the one equal 
 to two angles and the included side of the other, they are equal in all their 
 Darts 
 
 7. ’ In any triangle, the greater side is opposite the greater angle, and the 
 greater angle is opposite the greater side. . . , 
 
 » 8. The sum of the lengths of any two sides of a triangle is greater than 
 
 the length of the third side. . 
 
 9 In an isosceles triangle, the angles opposite the equal sides are equal 
 10. In any triangle, the sum of the three angles is equal to two right 
 sniffles or 180^ 
 
 ll/’lf two angles of a triangle are given, the third may be found by 
 subtracting their sum from two right angles, or 180°. 
 
 12. A triangle must have at least two acute angles, and can have but one 
 
 obtuse triangle! a perpendicular let fall from the apex to the base is 
 
 shorter than either of the two other sides. 
 
GEOMETRY. 
 
 25 
 
 14. In any parallelogram, the opposite sides and angles are equal each to 
 each. 
 
 15. The diagonals divide any paralellogram into two equal triangles. 
 
 16. The diagonals of a parallelogram bisect each other; that is, they 
 divide each other into equal parts. 
 
 17. If the sides of a polygon be produced in the same direction, the sum 
 of the exterior angles will equal four right angles. 
 
 18. The sum of the interior angles of a polygon is equal to twice as many 
 right angles as the polygon has sides, less four right angles. 
 
 Example.— The sum of the interior angles of a quadrilateral = (2X4) 
 — 4 = 4 right angles. The sum of the interior angles of a pentagon = 
 (2 X 5) — 4 = 6 right angles. The sum of the interior angles of a hexagon 
 = (2 X 6) — 4 = 8 right angles. 
 
 19. In equiangular polygons, each interior angle equals the sum divided 
 by the number of sides. 
 
 20. The square described on the hypotenuse of a right-angled triangle is 
 equal to the sum of the squares described on the other two sides. Thus, in a 
 right-angled triangle whose base is 20 ft. and altitude 10 ft., the square of 
 the hypotenuse equals the square of 20 + the square of 10 , or 500. Then the 
 hypotenuse equals the square root of 500, or 22.3607 ft. 
 
 21. Having the hypotenuse and one side of a right-angled triangle, the 
 other side may be found by subtracting from the square of the hypotenuse 
 the square of the other known side. The remainder will be the square of 
 the required side. 
 
 22. Triangles that have an angle in each equal, are to each other as the 
 product of the sides including those equal angles. 
 
 23. Similar triangles are to each other as the squares of their correspond- 
 ing sides. 
 
 24. The perimeters of similar polygons are to each other as any two 
 corresponding sides, and their areas are to each other as the squares of 
 those sides. 
 
 25. The diameter of a circle is greater than any chord. 
 
 26. Any radius that is perpendicular to a chord, bisects the chord and the 
 arc subtended by the chord. 
 
 27. Through three points not in the same line, a circumference may be 
 made to pass. 
 
 Directions— Draw two lines connecting the three points. Erect perpen- 
 diculars from the centers of each of these two lines, and the point of inter- 
 section of the perpendiculars will be the center of the circle. 
 
 28. The circumferences of circles are to each other as their radii, and 
 their areas are to each other as the squares of their radii. 
 
 Example 1 .— If the circumference of a circle is 62.83 in. and its radius is 
 10 in., what is the circumference of a circle whose radius is 15 in. ? 
 
 10 : 15 : : 62.83 : 94.245 in. Ans. 
 
 Example 2— If a circle 6 in. in diameter has an area of 28.274 sq. in., what 
 is the area of a circle 12 in. in diameter ? 
 
 32 : 6 2 : : 28.274 : 113.096 sq. in. Ans. 
 
 PRACTICAL PROBLEMS IN GEOMETRICAL CONSTRUCTION. 
 
 \E To Bisect a Given Straight Line AB . — From 
 
 A and B as centers, with a radius greater than 
 l one-half of A B, describe arcs intersecting at E 
 
 A k B and F. Draw E F. It will bisect A B. C will 
 
 1° be the middle point, and ^Fwill be perpendic- 
 
 ular to AB. The points ^and i^will be equi- 
 distant from A, B, or C. 
 
 From a Given Point C, Without a Straight Line AB, 
 to Draw a Perpendicular to the Line.— From C as a 
 center, with a radius sufficiently great, describe 
 an arc cutting AB in points A and B; then from 
 A and B as centers, with a radius greater than 
 one-half of A B , describe two arcs cutting each 
 other at Z>, and draw CD. 
 
 C 
 
 _E 
 
26 
 
 GEOMETRY. 
 
 A 
 
 XD 
 
 A+ 
 
 4 B 
 
 At a Given Point C in a Straight Line AB, to 
 Erect a Perpendicular to That Line.— Take the points 
 A and B equally distant from C, and, with A and 
 B as centers, and a radius greater than one-half 
 of A B , describe two arcs cutting each other at 
 D, and draw the line D C. 
 
 At a Point i on a Given Straight Line A B, 
 to Make an Angle Equal to a Given Angle Ei G. 
 
 From F as a center, with any radius ± G, 
 describe the arc EG. From A as a center, 
 with the same radius, describe the arc CB\ 
 then with a radius equal to the chord EG, 
 describe an arc from £ as a center, cutting 
 Oil at D, and draw A D 
 
 To Bisect a Given Arc ACB.-With the same 
 radii and the extremities A B as centers, descnbe 
 arcs intersecting at D and E. The line D E 
 bisects the arc at C. 
 
 To Bisect an Angle A B C.-With any radius and 
 R as a center, describe an arc cutting the sides at 
 A and C With these points as centers, describe 
 arcs of equal radius intersecting at D. The line 
 B D is the bisector, and the LAB D = LB B C. & 
 
 R 
 
 Through a Given Point A to Draw a Straight 
 Line Parallel to a Given Straight L |ne CD .— From 
 A as a center, with a radius greater than 
 the shortest distance from A to CD, describe 
 an indefinite arc D E. From D as a center, 
 with the same radius, describe the arc A 
 Take D E equal to A F, and draw A B. 
 
 To Bisect an Open Angle ( Method by L. L. 
 Logan).— Let AB and CD be the sides of 
 an open angle. With any point 0 as a cen- 
 ter, describe a circle cutting the sides at 
 e f a and h, and with e and/, and g and h 
 as centers and any radius, describe arcs 
 intersecting at k and l, respectively. Draw 
 Ok and 01 and mn. With p and gas cen- 
 ters, and any radius, describe arcs intersect- 
 ing at R and S. The line drawn through 
 is the required bisector. 
 
 C ft 1 
 
 /' 
 
 \ 
 
 \ B 
 
 To Find the Center of a Given Circumference or Arc. 
 
 Take any three i>oints ,4, B.and C on the circumfer 
 ence, and unite them by the lines A 5andJ C. Bisect 
 these chords by the perpendiculars D 0 and E 0, their 
 intersection is the center of the circle. 
 
GEOMETRY. 
 
 27 
 
 Through a Given Point P, to Draw a Tangent to a Given 
 Circle.— 1 . If P be in the circumference: Find C the 
 center of the circle, draw the radius CP , and draw 
 D E perpendicular to C P. 
 
 2. If P be without the circle: Join P and 
 the center of the circle. Bisect P C in D; with 
 D as a center, and a radius D 0 , describe the cir- 
 cumference intersecting the given circumference 
 at A and B. From the intersections A and B, 
 draw B P and A P. 
 
 An acute angle having its vertex in the circumference and 
 
 subtended by an arc is equal to one-half the central 
 angle subtended by the same arc. Thus, the L A B C 
 — w L. A O C. 
 
 An acute angle included between a chord and a tangent is 
 
 equal to one-half the central angle subtended by the 
 chord. Thus, [_ ABC = £ l_ COB. 
 
 If, from a point, two tangents be drawn to a circle, they 
 will be equal, and their angle of intersection will be 
 equal to the central angle subtended by the chord 
 joining the two points of tangency. Thus, A B = AC. 
 and c DAC = L BOC. 
 
 To Divide a Straight Line Into Any Number 
 of Equal Parts.— To divide the line AB 
 into, say, 6 parts, draw the line A C from 
 A, making any angle with A B; measure 
 off 6 equal spaces on A C; draw 6 B , and 
 from 1, 2, 3, U, 5 on A Cdraw parallels to 
 6 B. These divide A B as required into 
 6 equal parts. 
 
 By a similar process a line may be 
 divided into a number of unequal parts. 
 Set off on A C divisions proportional to 
 the required divisions, and draw the 
 parallel lines as explained above. 
 
23 
 
 MENSURATION. 
 
 MENSURATION. 
 
 MENSURATION OF SURFACES. 
 
 PARALLELOGRAMS. 
 
 A parallelogram is a four-sided figure whose opposite sides are parallel. 
 
 'sJtZista* 
 
 and four right an- and opposite sides and oblique angles.) opposite sides equal.) 
 gtes.) equal.) 
 
 Tn Find the Area of Anv Parallelogram.— Multiply the length of any side 
 by the length of a perpendicular line from that side to the oppositeone. 
 Thus, 6 in ^he foregoing^ figures, the areas of the square are 
 
 the length^B by the 
 
 «o« 
 
 To Find the Area of the Largest Square That May be Inscribed in a Circle. 
 
 Snuare the radius of the circle, and multiply by 2. , . «■ «_. 
 
 ^Tn Find the Side of the Largest Square That May be Inscribed in a Circle. 
 
 Dhdde the Lmeter of the circle 4 1.41421, or multiply it by .707107. 
 
 TRIANGLES. 
 
 A triangle is a figure having three straight sides. 
 
 Equilateral. 
 
 ( Three equal 
 sides.) 
 
 Isosceles. 
 
 ( Two equal 
 sides.) 
 
 Scalene. 
 
 ( Three un- 
 equal sides.) 
 
 To Find the Area of. a Triangle.-Multiply its base by one-half the perpen- 
 
 ^Tonndf^er^ndfcu^Height of an Equilateral Triangle.-Multiply the length 
 
 *Nr&s 
 
 tipiy the length of one of its sides by .658037. 
 
TRIANGLES. 
 
 29 
 
 To Find the Diameter of a Circle of Same Area as an Equilateral Triangle. 
 
 Divide the length of one of its sides by 1.34677. 
 
 The following rules apply to any triangle: 
 
 Having Two Sides and the Included Angle, to Find the Area. — Multiply together 
 the two sides and the natural sine of the included angle, and divide the 
 product by 2. Or, by logarithms, add together the logarithms of the two 
 
 sides and the logarithmic sine of the included angle, and from the sum 
 subtract the logarithm of 2, and the result will be the logarithm of the area. 
 
 Having Three Sides of a Triangle, to Find the Area.— Add the three sides 
 together, divide the sum by 2; from the half sum, subtract each side sep- 
 arately; multiply the half sum and the three remainders continuously 
 together, and extract the square root of the product. Thus, if the triangle 
 
 has three sides whose lengths are 30, 40, and 50 ft., then = 60. 
 
 Then, subtracting from this 60 each side separately, we have: 60 — 30 = 30; 
 60 — 40 - 20; 60 - 50 = 10. Then, 60 X 30 X 20 X 10 = 360,000. The square 
 root of 360,000 = 600 sq. ft., or area. 
 
 Having the Three Sides of a Triangle, to Find Its Angles.— In the triangle 
 A B C, let A B = 21 ft., B C = 17.25 ft., and A C = 32 ft. Draw B D per- 
 pendicular to AC’, then, 
 
 32 : 21 + 17.25 = 21 — 17.25 AD - DC’, 
 
 But 
 
 Adding, 
 
 Subtracting, 
 
 AD — DC = 4.48 
 AD + DC = 32 
 2 AD — 36.48 
 AD = 18.24 
 2 DC = 27.52 
 DC = 13.76 
 
 cos A = = .86857, or A = 29° 42' 25.7". 
 
 13 76 
 
 cos 0 = Wm = * 79768 ’ or c = 370 5 ' 26.7". 
 
 B = 180° — (A + C) = 1.80 — (29° 42' 25.7"+ 37° 5' 26.7") = 113° 12' 7.6". 
 Having Two Sides and Included Angle, to Find Third Side and the Other Angles. 
 
 In the triangle ABC, let A B = 19 ft., A C = 2P ft., and A = 36° 3' 29". 
 
 Draw B D perpendicular to AC. BD = 19 X sin A = 19 X .58861 = 11.18 ft. 
 
 A D = 19 X cos A = 19 X .80842 = 15.36 ft. D C = 23 — 15.36 = 7.64 ft. 
 
 11 18 
 
 Tan C = = 1.46335, or C = 55° 39' 10". B = 180 — (A + C) = 180 
 
 - (36° 3' 29" + 55° 39' 10") = 88° 17' 21". B C = B ? = - 11 ’ 18 = 13 54 ft 
 „ . , - sin C .82562 
 
 ^naving One Side and the Two Adjacent Angles, to Find the Other Two Sides. 
 
 The third angle equals 180° minus the sum of the other two angles. This 
 third angle will be the one opposite the given side. Then the sine of the 
 
 angle opposite the given side is to the given side as the sine of either of the 
 
 other angles is to its opposite side. Thus, in the triangle ABC, let A = 60°, 
 
 B 50° 700 ’ and the Side A B = 200 ft * Then the angle C = 180 °” ( 60 ° + 70 °) 
 Then, sin 50° : 200 : : sin 60° : B C, 
 
 and sin 50° : 200 : : sin 70° : A C. 
 
 To Find the Area.— Either find the three sides as above, and follow rule 
 already given, or multiply the natural sines of the two given $pgles togethgj •. 
 
30 
 
 MENSURATION. 
 
 Thpn ns the natural sine of the single angle is to the product of the sines of 
 to riven ; ang “is the°squar^ 0 ffhe given side to twice the reqmred area, 
 rrvmc «in r • sin A y sin B •: A B 1 : to twice the area of the triangle. 
 
 The area of tny triangle is equal to half the area of a parallelogram 
 having the same base and perpendicular height. 
 
 trapezoids. 
 
 A trapezoid has four straight sides, only two of which are parallel. 
 
 F — ,Z> 
 
 To Find the Area of a Trapezoid.-Add together the two parallel sides, and 
 dl vide r by 2. Multiply the quotient by the perpendicular height. 
 
 Thus, 
 
 ABA CD 
 
 X E F = area. 
 
 TRAPEZIUMS. 
 
 A trapezium has four sides, no two of which are parallel. 
 
 To Find the Area of a Trapezium.— Divide the trape- 
 zium into two triangles, and find the area ol each 
 according to the rules given under the head ot 
 “ Triangles.” Add together the areas of the two 
 triangles, and the sum will equal the area of the 
 trapezium. The sides and angles can be found m 
 
 th mhemagon e ais a'nd the perpendiculars from them to the opposite angles 
 are given, add together the two perpendiculars, multiply the sum by the 
 
 ^The^m o^^^four 2 angles included in a trapezium always equals four 
 right angles. 
 
 POLYGONS. 
 
 All figures bounded by more than four straight lines are called polygons. 
 
 If all the sides and angles are equal, it is a regular polygon. If not, it is 
 an The^sum o? the interior angles of any polyg<m is equal to twice as many 
 
 right angles as the polygon has sides less four right angles. • 
 
 To Find thp Arpa of Anv Regular Polygon. — Square one ot its siaes anu 
 multiply by the number given in the column of areas in 
 table Or multiply the length of one of the sides by one-half the length 
 of a perpendicular drawn to the center of the figure, and this product by 
 
 Side of a Regular Polygon, to Find the Radius of a Circumscribing 
 
 Ci rG |e —Multiply the side by the corresponding number m following 
 column of outer radii. If the radius of the circumscribing circle be given 
 
CIRCLES. 
 
 31 
 
 divide it by the number in column of outer radii, and the quotient will be 
 the side of the polygon. A A „ , ,, 
 
 To Find the Area of an Irregular Polygon. — Divide it into triangles, find the 
 
 Table of Regular Polygons Whose Sides Are Unity. 
 
 Number 
 
 of 
 
 Sides. 
 
 Name of Polygon. 
 
 Areas. 
 
 Outer 
 
 Radii. 
 
 Angles Contained 
 Between 
 Two Sides. 
 
 Angle at Center 
 of 
 
 Circle. 
 
 3 
 
 Equilateral triangle 
 
 .4330 
 
 .5774 
 
 60° 
 
 120° 
 
 4 
 
 Square 
 
 1.0000 
 
 .7071 
 
 90° 
 
 90° 
 
 5 
 
 Pentagon 
 
 1.7205 
 
 .8507 
 
 108° 
 
 72° 
 
 6 
 
 Hexagon 
 
 2.5981 
 
 1.0000 
 
 120° 
 
 60° 
 
 7 
 
 Heptagon 
 
 3.6339 
 
 1.1524 
 
 128° 34' 17"+ 
 
 51° 25' 43"— 
 
 8 
 
 Octagon 
 
 4.8284 
 
 1.3066 
 
 135° 
 
 45° 
 
 9 
 
 Nonagon 
 
 6.1818 
 
 1.4619 
 
 140° 
 
 40° 
 
 10 
 
 Decagon 
 
 7.6942 
 
 1.6180 
 
 144° 
 
 36° 
 
 11 
 
 Undecagon 
 
 9.3656 
 
 1.7747 
 
 147° 16' 22"— 
 
 32° 43' 38" + 
 
 12 
 
 Dodecagon 
 
 11.1962 
 
 1.9319 
 
 150° 
 
 30° 
 
 The sum will be the area 
 
 area of each triangle, and add them together, 
 of the polygon. ... , ^ 
 
 To Find the Area of a Figure Whose Outlines Are Very Irregular.— Draw straight 
 lines around it that will enclose within them (as nearly as can be judged) 
 as much space not belonging to the figure as they exclude space belong- 
 ing to it. The area of the figure thus formed may be easily found by 
 dividing into triangles. 
 
 CIRCLES. 
 
 {See Table of Areas of Circles , Etc.) 
 
 A circle is a figure bounded by a curved line, every point of which is equi- 
 distant from the center. Or, a circle is a regular poly- 
 gon of an infinite number of sides. 
 
 The circumference of a circle equals the diameter 
 multiplied by 3.1416, or the square root of the product 
 of the area multiplied by 12.566. 
 
 To Find the Diameter.— Divide the circumference by 
 3.1416, or multiply it by .31831. 
 
 To Find the Area of a Circle.— Multiply the circumfer- 
 ence by one-fourth of the diameter, or the square of the 
 radius by 3.1416. Multiply the square of the diameter 
 by .7854, or the square of the circumference by .07958. lx . . 
 
 To Find the Diameter of a Circle Equal in Area to a Given Square.— Multiply 
 one side of the square by 1.12838. 
 
 To Find the Radius of a Circle to Circumscribe a Given Square.— Multiply 
 one side by .7071; or take one-half the diagonal. 
 
 To Find the Side of a Square Equal in Area to a Given Circle. — Multiply the 
 diameter by .88623. _ . 
 
 To Find the Side of the Greatest Square in a Given Circle.— Multiply the 
 diameter by .7071. ^ ^ . , . 
 
 To Find the Area of the Greatest Square in a Given Circle.— Square the radius 
 and multiply by 2. . . . _. . 
 
 To Find the Side of an Equilateral Triangle Equal in Area to a Given Circle. 
 Multiply the diameter by 1.3468. _ . . 
 
 Having the Chord and Rise of an Arc, to Find the Radius. — Square half the 
 chord, and divide by the rise. To the quotient add the rise, and divide by 2. 
 Or, radius = the square of the chord of half the arc divided by twice the 
 rise of the whole arc. , 
 
 Having the Chord and Radius, to Find the Rise.— Square the radius, also 
 square half the chord. Take the last square from the first. Extract square 
 root of the remainder, and subtract it from the radius if the radius is greater; 
 if not, add it to the radius. 
 
 Having the Radius and Rise,’ to Find the Chord.— From the radius subtract 
 the rise (or from the rise subtract the radius, if rise is the greater), square 
 
32 
 
 MENSURATION, 
 
 the remainder, and subtract it from the square of the radius. Extract the 
 square root of the remainder., and multiply by 2. 
 
 Having the Rise of the Arc and Diameter of Circle, to Find the- Chord.— Sub- 
 tract the rise from the diameter, and multiply the remainder by the rise. 
 Extract the square root of the product, and multiply by 2. 
 
 To Find the Breadth of a Circular Ring, Having Its Area and the Diameter of 
 the Outer Circle. — Find the area of the whole circle, and from 
 it take the area of the ring. Multiply the remainder by 
 1.2732, and the square root of the product will be the diam- 
 eter of the inner circle. Take it from the diameter of the 
 outer one, and the remainder will be twice the breadth. 
 
 To Find the Area of a Circular Ring.— Take the difference of 
 the squares of the radii, and multiply it by 3.1416. 
 
 To Find the Length of an Arc When Its Degrees and Radius Are 
 Given. —Multiply the number of degrees by .01745, and the product by the 
 radius. ... , , ,, ,. 
 
 To Find the Area of a Sector.— Multiply the arc by one-half the radius. 
 The area of the sector is to the area of the circle as the number of degrees 
 in the sector is to 360°. . x . . . . 
 
 To Find the Area of a Segment.— Find the area of the sector having the 
 same arc, and also the area of the triangie formed by the chord of the segment 
 and the radii of the sector. If the segment is greater than a semicircle, add 
 the two areas; if less, subtract them. 
 
 THE ELLJPSE. 
 
 To Find the Area of an Ellipse.— Multiply one-half of the two axes AB and 
 
 CD together, and multiply the product by 3.1416. 
 
 To Find the Perimeter of an Ellipse.— Multiply one-half 
 
 the sum of the two axes by 3.1416. 
 
 To Draw an Approximate Ellipse {Method, by Three Squares). 
 Let a be the center, b c the major, and a e half the minor 
 axis of an ellipse. Draw the rectangle bfgc, and the 
 diagonal line be; at a right angle to b e, draw line/ h cut- 
 ting B B at i . With radius a e, and from a as a center, 
 draw the dotted arc ej, giving the point j on the line B B. From k, which is 
 central between b and j, draw the semicircle b mj, cutting A A at l. Draw 
 the radius of the semicircle b mj , cutting fg at n. With radius m n, mark 
 on A A, at and from a as a center, the point o. With radius ho, and from 
 center h, draw the arc p o q. With radius a l, 
 and from b and c as centers, draw arcs cut- 
 ting p o q at the points p and q. Draw the 
 lines hpr and hqs, and also the lines pit 
 and qvw. From h as a center, draw that 
 part of the ellipse lying between r and s 
 with radius h r. From p as a center draw 
 that part of the ellipse lying between r and 
 t with the radius p r. From q, draw the 
 ellipse from s to w. With radius i t, from i as 
 a center, draw the ellipse from t to 5 with 
 radius i t, and from v as a center, draw the 
 ellipse from w to c, and one-half the ellipse 
 will be drawn. It will be seen that the whole 
 construction has been performed to find the 
 centers h, p, q, i, and v, and that while v and i 
 may be used to carry the curve around the other side or half of the ellipse, 
 new centers must be provided for h, p, and q ; these new centers correspond 
 in position to h,p, q. 
 
 Straightedge Method.— On a straightedge, lay off A B 
 equal to one-half the short diameter and A C equal to 
 one-half the long diameter. Determine points on the 
 circumference of the ellipse by marking positions of 
 A, as the point B is moved along the major axis and, at 
 the same time, the point C along the minor axis. 
 
MENSURATION OF SOLIDS. 
 
 33 
 
 MENSURATION OF SOLIDS 
 
 THE CUBE AND THE PAR ALLE LO PI PE D 
 
 To Find the Surface of a Cube.— Multiply the area of one side by 6. 
 
 To Find the Surface of a Parallelopiped.— Add together twice the area of 
 the base, twice the area of the side, and twice the area of the end. 
 
 To Find the Cubical Contents of a Cube or Parallelopiped.— Multiply the area 
 of the base by the perpendicular height. 
 
 THE PRISM. 
 
 To Find the Convex Surface of a Right Prism.— Multiply the 
 perimeter of the base by the altitude. 
 
 To find the entire surface, add the areas of the bases. 
 
 To Find the Contents of a Prism.— Multiply the area of the 
 base by the altitude of the prism. 
 
 THE CYLINDER. 
 
 To Find the Convex Surface of a Cylinder. — Multiply the circum- 
 ference of the base by the altitude. 
 
 To find the entire surface, add the areas of the ends. 
 
 To Find the Contents of a Cylinder. — Multiply the area of the 
 base by the altitude. 
 
 THE SPHERE. 
 
 To Find the Surface of a Sphere. — Multiply the diameter by the circum- 
 ference; or, square the radius and multiply it by 4 and 3.1416. 
 
 To Find the Contents of a Sphere.— Multiply the surface by 
 one-third of the radius; or, multiply the cube of the diam- 
 eter by .5236. 
 
 To Find the Surface of a Zone. — Multiply the height of the 
 zone by the circumference of a great circle of the sphere. 
 To Find the Contents of a Spherical Segment of One Base. 
 
 Add the square of the height to three times the square of 
 the radius of the base; multiply this sum by the height, 
 and the product by .5236. 
 
 The curved surface on a hemisphere is equal to twice its plane surface, and 
 the curved surface on a quarter of a sphere is equal to its plane surface. 
 
 THE PYRAMID. 
 
 To Find the Convex Surface of a Pyramid. — Multiply the per- 
 imeter of the base by one-half the slant height. 
 
 To find the entire surface, add the area of the base. 
 
 To Find the Contents of a Pyramid. — Multiply the area of the 
 base by one-third of the altitude. 
 
 THE CONE. 
 
 To Find the Convex Surface of a Cone.— Multiply 
 the circumference of the base by one-half the 
 slant height. 
 
 To find the entire surface, add the area of the 
 base. 
 
 To Find the Contents of a Cone. — Multiply the area 
 of the base by one-third of the altitude. 
 
34 
 
 MENSURATION. 
 
 THE FRUSTUM OF A PYRAMID OR CONE. 
 
 To Find the Convex Surface— Multiply one-half of the sum of the perim- 
 eters or circumferences of the two bases 
 by the slant height 
 
 ’ The entire surface is found by adding 
 the areas of the two bases. 
 
 To Find the Contents of a Frustum. — Add 
 together the sum of the two bases and 
 the square root of their product, and mul- 
 tiply the sum by one-third of the altitude of the frustum. 
 
 CYLINDRICAL RINGS. 
 
 A cylindrical ring is formed by bending a cylinder or pipe until its two 
 ends meet. , , . , 
 
 To Find the Surface of a Cylindrical Ring— To the thickness 
 of the ring, add the inner diameter, multiply this sum by the 
 thickness of the ring, and the product by 9.8696. 
 
 To Find the Contents of a Cylindrical Ring.— To the thickness 
 of the ring add the inner diameter, multiply this sum by 
 the square of one-half the thickness. 
 
 To Find the Volume of an Irregular Body.— Fill a vessel of 
 known dimensions with water, and immerse the body. The contents will 
 equal the volume of water displaced. 
 
 THE PRISMOI DAL FORMULA. 
 
 This formula is the invention of Mr. Elwood Morris, C. E., of Philadelphia, 
 and is extensively used in calculating the cubical contents of cuttings, 
 embankments, etc. . . _ , 
 
 It embraces all parallelopipeds, pnsms, pyramids, cones, wedges, etc., 
 whether regular or irregular, right or oblique, with their frustums when cut 
 parallel to their bases. In fact, it embraces all solids having two parallel 
 faces or sides, provided these two faces are united by surfaces, whether plane 
 or curved, on which, and through every point of which, a straight line may 
 be drawn from one of the parallel faces to the other. 
 
 To Find the Contents of Any Prismoid— Add together the areas of the two 
 parallel surfaces, and four times the area of the section taken half way 
 between them, and parallel to them: multiply the sum by the perpendicular 
 distance between the two parallel sides, and divide the product by 6. 
 
 PLANE TRIGONOMETRY. 
 
 Plane trigonometry treats of the solution of plane triangles. 
 
 In everv triangle, there are six parts— three sides and three angles. These 
 parts are so related that when three of the parts are given, one being a side, 
 the other parts may be found. ._ , . . 
 
 An angle is measured by the arc included between its sides, the center of 
 the circumference being at the vertex of the angle. 
 
 For measuring angles, the circumference is divided into 360 equal parts. 
 
 called degrees; each degree into 60 equal parts called 
 
 minutes. 
 
 A quadrant is one-fourth the circumference of a cir- 
 cle, or 90°. _ _ _ 
 
 The complement of an arc is 90° minus the arc; D C 
 is the complement of B C, and the angle D O C is the 
 complement of BO C. 
 
 The supolement of an arc is 180° minus the arc: A E 
 is the supplement of the arc B D E. and the angle BO E. 
 
 In trigonometrv. instead of comparing the angles of 
 triangles or the arcs that measure them, we compare 
 the trigonometric functions known as the sine, cosine , tangent, cotangent, 
 secant , and cosecant. . ... 
 
 The sine of an arc is the perpendicular let fall from one extremity of the 
 
PLANE TRIGONOMETRY. 
 
 35 
 
 arc on the diameter that passes through the other extremity. Thus, CD is 
 the sine of the arc A C. 
 
 The cosine of an arc is the sine of its complement; or it is the distance 
 from the foot of the sine to 
 the center of the circle. 
 
 Thus, C E or 0 D equals the 
 cosine of arc A C. 
 
 The tangent of an arc is a 
 line that is perpendicular to 
 the radius at one extremity 
 of an arc and limited by a 
 line passing through the cen- 
 ter of the circle and the other 
 extremity. Thus, A T is the 
 tangent of A C. 
 
 The cotangent of an arc is 
 equal to the tangent of the 
 complement of the arc. 
 
 Thus, B T' is the cotangent 
 of AC. 
 
 The secant of an arc is a 
 line drawn from the center 
 of the circle through one extremity of the arc, and limited by a tangent at 
 the other extremity. Thus, O T is the secant of AC. 
 
 The cosecant of an arc is the secant of the complement of the arc. Thus. 
 O T' is the cosecant of A C. 
 
 The versed sine of an arc is that part of the diameter included between the 
 extremity of the arc and the foot of the sine. D A is the versed sine of A C. 
 
 The coversed sine is the versed sine of the complement of the arc. Thus, 
 B E is the coversed sine of A C. 
 
 From the above definitions, we derive the following simple principles: 
 
 1. The sine of an arc equals the sine of its supplement , and the cosine of an 
 arc equals the cosine of its supplement. 
 
 2. The tangent of an arc equals the tangent of its supplement , and the cotan - 
 gent of an arc equals the cotangent of its supplement. 
 
 3. The secant of an arc equals the secant of its supplement, and the cosecant 
 equals the cosecant of its supplement. 
 
 Thus, 
 
 sine of 70° = sine of 110°. cosine of 70° = cosine of 110°. 
 
 tangent of 70° = tangent of 110°. cotangent of 70° = cotangent of 110°. 
 
 secant of 70° = secant of 110°. cosecant of 70° = cosecant of 110°. 
 
 Thus, if you want to find the sine of an angle of 120° 30', look for the sine 
 of 180 — 120° 30', or 59° 30', etc. 
 
 In the rt. [\xyz, the following relations hold: 
 
 Functions of the sum and difference of two angles: 
 
 sin {A + B) = sin A cos B + cos A sin B. 
 
 cos (A 4 - B) = cos A cos B — sin A sin B. 
 
 sin (A — B) = sin A cos B — cos A sin B. 
 
 cos ( A — B) -=■ cos A cos B 4- sin A sin B. 
 
 Natural sines, tangents, etc. are calculated for a circle whose radius is 
 unity, and logarithmic sines, tangents, etc. are calculated for a circle whose 
 radius is 10,000,000,000. 
 
 PRACTICAL EXAMPLES IN THE SOLUTION OF TRIANGLES. 
 
 CaseI. To Determine the Height of a Vertical Object Standing on a Horizon- 
 tal Plane. — Measure from the foot of the object any convenient horizontal 
 
86 
 
 PLANE TRIGONOMETRY. 
 
 distance A B; at the point A, take the angle of elevation B A C. Then, as ^ 
 is known to be a right angle, we have two angles and the included side of a 
 
 naming that the line AB is_30« ft and theangle BAC^ 40°, the angle 
 
 C sin C : A B : : sin A : B C, 
 
 or .766044 : 300 : 
 Or, by logarithms: 
 Log 300 == 
 Log sin 40° = 
 
 .642788 : ( ), or 251.73+ ft. 
 
 2.477121 
 
 9.808067 
 
 12.285188 
 Log sin 50° = 9.884254 
 
 2.400934 or log of 251.73+ ft. 
 ft. 
 
 Hence, B C = 251.73 
 
 Tase 2 To Find the Distance of a Vertical Object Whose Height is Known.-Ata 
 
 noint A take the angle of elevation to the top of the object. Knowing that 
 the^ngie B is Vrigttt angle, we have the angles B and A and the side B C. 
 burning that the side BC= .200 ft. and the angle 
 
 'i!0°, we have a triangle as follows: Angle A - 
 B = ’ 900^ c = 60°, and the side B C = 200 ft. 
 
 A 
 
 ° ^ Then * sin A: B C:: sin C : A B, 
 
 or ’ .5 : 200 : : .866025 : ( ), or 346.41 ft. 
 
 By logarithms: 
 
 Log 200 = 
 Log sin 60° = 
 
 2.301030 
 
 9.937531 
 
 Log sin 30° — 
 
 12.238561 
 9.698970 
 
 2.539591 or log of 346.41 ft. . 
 
 Case 3 To Find the Distance of an Inaccessible Object.— ^ Ie . a ^ ur ® a 
 7ontal base line A B , and take the angles formed by the lines B AC and ARC. 
 zontai ease n e , ^y e then have two angles and the included 
 
 side. Assuming the angle A to be 60°, the 
 angle B 50°, and the side A B = 500 ft., we 
 have the angle C = 180° - (60° + 50°) = 70°. 
 Then, 
 
 sin 70° : A B : : sin A : B C, 
 and sin 70° : AB : : sin B : A C; 
 or, .939693 : 500 : : .866025 : B C, or 460.8 +, 
 and .939693 : 500 : : .766044 : A C, or 407.6 +. 
 By logarithms: 
 
 Log 500 = 2.698970 
 Log sin 60° = 9.937531 
 
 Log 500 = 
 
 Log sin 50° = 
 
 12.583224 
 Logsin70°= 9.972986 
 
 12.636501 
 Log sin 70°= 9.972986 
 
 2.663515 = log of 460.8+, 
 
 2.698970 
 
 9.884254 
 
 2.610238 = log of 407.6+. 
 
 Zi.u ±\JAjOO 7 • 
 
 Case 4 To Find the Distance Between Two Objects Separated by an impassa- 
 ble Barrier— Select any convenient station, as 
 C, measure the linesCA aoAC i^and l*e angle 
 
 included between these sides, 
 two sides and the included angle. . , n . 
 
 Assuming the angle C to be 60 , the side C A, 
 600 ft., and the side CB, 500 ft., we have the 
 following formula: A A B B — A 
 
 CA + CB: CA- CB:: tan — : tan — . 
 
 + B 180° -60° 
 
 2 2 
 1,100 : 100 : : tan 60° : tan 
 
 or 60°. 
 B-A 
 
 (*) 
 
 Then, 
 
 Then, 
 
PLANE TRIGONOMETRY. 
 
 29 
 
 or, 
 
 1,100 : 100 : : 1.732050 : .157459, or tangent of 
 
 B-A 
 2 ’ 
 
 or 8° 57'. 
 
 Then, 60° + 8° 57' = 68° 57', or angle B , 
 
 and 60° — 8° 57' = 51° 03', or angle A. 
 
 Having found the angles, find the third side by the same method as Case 1. 
 The above formula, worked out by logarithms, is as follows: 
 
 ' Log 100 = 2.000000 
 Log tan 60° = 10.238561 
 
 Log 1,100 = 
 
 12.238561 
 
 3.041 393 
 
 9.197168 = log tari of 
 
 B 
 
 -, or 8° 57'. 
 
 Then, 
 
 and 
 
 M 
 
 60° + 8° 57' = 68° 57', or angle B, 
 
 60° _ 8 o br = 51 o 03', or angle A. 
 
 Note.— T he greater angle is always oppo- 
 site the greater side. 
 
 Case 5. To Find the Height of a Vertical 
 Object Standing Upon an Inclined Plane.— Meas- 
 ure any convenient distance D C on a line 
 from the foot of the object, and, at the point 
 D, measure the angles of elevation EDA 
 and ED B to foot and top of tower. We 
 then have two triangles, both of which may 
 be solved by Case 1, and the height above 
 D of both the foot and top will be known. 
 
 The difference between them is the height 
 of the tower. 
 
 Case 6. To Find the Height of an Inaccessible Object Above a Horizontal Plane. 
 
 Measure any convenient horizontal line A B directly toward the object, and 
 take the angles of elevation at A and B. We will then have sufficient data 
 
 to work .with. Assuming the line A B to be 
 1,200 ft. long, the angle A , 25°, and the angle 
 D B C, 40°, we have the following: As the 
 angle D B C is 40°, the angle A B C — 90° — 
 40°, or 50°. 
 
 Then, having the side B C , and the angle 
 D B C = 40°, and the angle B D C — 90°, we 
 find the side CD by the same method as in 
 Case 1. 
 
 Second Method . — If it is not convenient to 
 measure a horizontal base line toward the 
 object, measure any line A B, Fig. (6a), and 
 also measure the horizontal angles BAD , 
 ABD , and the angle of elevation DBG 
 Then, by means of the two triangles ABD 
 and CBD, the height CD can be found. 
 
 Then, with the line A B and the angles 
 BAD and ABD known, we have two 
 angles and the included side known. The 
 third angle is readily found, and the side 
 BD can be found. Then, in the triangle 
 B DC, we have the angle B; by measure- 
 ment, D = 90°, and we have the side B D. 
 
 Then, the side CD, or the vertical height, 
 can be found by Case 1. 
 
 Case 7. To Find the Distance Between Two 
 Inaccessible Objects When Points Can Be Found 
 From Which Both Objects Can Be Seen. — Wish- 
 ing to know the horizontal distance between 
 a tree and a house on the opposite side of a river, measure the line A B, and, 
 at point A, take the angles D AC, and DAB, and, 
 at the point B, take the angles CB A and CBD . 
 Assume the length of A B — 400 ft. 
 
 Angle DAC= 56° 30'. 
 
 Angle DAB = 42° 24'. 
 
 Angle C B A — 44° 36'. 
 
 Angle CBD = 68° 50'. 
 
 __.JP 
 
 ■0 jjjS, 
 
 03 
 
38 
 
 SURVEYING. 
 
 th 111 riple U°3ff+ GPfiO'f = U3P2S'?SiS C th?S«le 
 
 + 113° V ) = °4° HX. Then, according to Case 1. hnd the side Z> B. 
 W^hen havtT ttaL angli and two sides of the triangle A DB. * e find 
 
 the third side r we haye the angles iBC and B A C. and the 
 
 side C2> by Case 4. . 
 
 SURVEYING. 
 
 employ^ iXth are the 
 
 S aAd me^ring pins, and Sometimes eert^n aeces- 
 ^ Smenis. as clinometers or slope levels, dipping needles, etc., as » dl 
 
 deviations from the practice of the form er. 
 
 THE COMPASS. 
 
 e=^SsSESS 
 
 gs^^safiasssss 
 
 should never be relied on. 
 
 TO ADJUST THE COMPASS. 
 
 _, , . Fir ^ r \yYins the bubbles into the center by the pressure of the 
 
 The Levels.— First and theri turn the compass halfway 
 
 hand on ddfarent JK^lPrunto' the ends of the tubes, it would Indicate 
 around: should the ouDOies lower them, bv tightening the screws 
 
 that^o^ en^ we^he^h^ * hose linder t ‘ he lower ends until by 
 immediately tinder. i Tel ttie P i ate again, and repeat the 
 
 &^mtion e tSmthe^nbbles will remain in the center during an entire 
 
 revolution of Aecomi^. observing through the slits a fine hair or 
 
 The sights may next betretea y , umb Sh , lllLld the hair appear on one 
 
 S'Slt ”he dght^ mmi bfe adjusted by filing off its under surface on 
 
 theade that following manner: Having the eye nearly in 
 
 The needle ls^jnsted mine n - of the compass circle, with a small 
 
 the same plane mth the ^adi l * ^ wire brin^ cue end of the needle in line 
 
 splinter of d f^5onof the circle as the zero or 90° mark, and notice 
 
 with any prominent divmon oi tne^ c ' h opposite side: if it does, 
 
 if the other end fV^^?^^ e th d e e ^^if "ot. ben¥ the center pin by 
 
 a? s s?.bSa ris ««■ .= 
 
MAGNETIC VARIATION . 
 
 39 
 
 circle; if any error is there manifested, the correction must be made in 
 the center pin only, the needle being already straightened by the previous 
 operation. „ , 
 
 When again made to cut, it should be tried on the other quarters of the 
 circle, and corrections made in the same manner until the error is entirely 
 removed, and the needle will reverse in every point of the divided circle. 
 
 TO USE THE COMPASS. 
 
 In using the compass, the surveyor should keep the south end toward his 
 person, and read the bearings from the north end of the needle. In the sur- 
 veyor’s compass, he will observe that the position of the E and W letters on 
 the face of the compass are reversed from their natural position, in order 
 that the direction of the sight may be correctly read. 
 
 The compass circle being graduated to half degrees, a little practice will 
 enable the surveyor to read the bearings to quarters— estimating with his 
 eye the space bisected by the point of the needle. 
 
 The compass is usually divided into quadrants, and zero is placed at the 
 north and south ends. 90° is placed at the E and W marks, and the gradua- 
 tions run right and left from the zero marks to 90°. In reading the bearing, 
 the surveyor will notice that if the sights are pointed in a N W direction, the 
 north end of the needle, which always points approximately north, is to the 
 right of the front sight or front end of the telescope, and, as the number of 
 degrees is read from it, the letters marking the cardinal points of the compass 
 read correctly. If the E, or east, mark were on the right side of the circle, a 
 N W course would read N E. This same remark applies to all four quadrants. 
 The compass should always be in a level position. 
 
 MAGNETIC VARIATION. 
 
 Magnetic declination or variation of the needle is the angle made by the 
 magnetic meridian with the, true meridian or true north and south line. It 
 is east or west according as the north end of the needle lies east or west of 
 the true meridian. It is not constant, but changes from year to year, and, 
 for this reason, in rerunning the lines of a tract of land, from field notes of 
 some years’ standing, the surveyor makes an allowance in the bearing of 
 every line by means of a ver- 
 nier that is so graduated that 
 30 spaces on it equal 31 on the 
 limb of the instrument, as 
 shown in the figure. 
 
 To Read the Vernier.— As the 
 compass vernier is usually so 
 made that there . are but 15 
 spaces on each side of the zero 
 mark, it is read as follows: 
 
 Note the degrees and half 
 degrees on the limb of the instrument. If the space passed beyond the 
 degree or half-degree mark by the zero mark on the vernier is less than 
 one-half the space of half a degree on the limb, the number of minutes is, 
 of course, less than 15, and must be read from the lower row of figures. If the 
 space passed is greater than one-half the spacing on the limb, read the upper 
 row of figures. The line on the vernier that exactly coincides with a line on 
 the limb is the mark that denotes the number of minutes. If the index is 
 moved to the right, read the minutes from the left half of the vernier; if moved 
 to the left, read the right side of the vernier. 
 
 To Turn Off the Variation.— Moving the vernier to either side, and with it, of 
 course, the compass circle attached, set the compass to any variation by 
 placing the instrument on some well-defined line of the old survey, and by 
 turning the tangent screw (slow-motion screw) until the needle of the com- 
 pass indicates the same bearing as that given in the old field notes of the 
 original survey. Then screw up the clamping nut underneath the vernier 
 and run all the other lines from the old field notes without further alteration. 
 
 The reading of the vernier on the limb gives the amount of variation since 
 the original survey was made. 
 
 The accompanying man shows the general course and direction ofisogonic 
 lines (those passing through points where the magnetic needle has the same 
 
4C 
 
 SURVEYING . 
 
 declination), in all parts of the United States and Mexico for the year 1900. 
 These lines are drawn full when compiled from reliable records, but dotted 
 in other places. The declination is marked in degrees at each end of every 
 alternate line, the sign + indicating a west declination, and the sign — an 
 east declination. The yearly variation, or change of declination, for the 
 period 1895-1900 is marked in numerous places on the map. The annual 
 change in declination is given in minutes; a -l- sign signifies increasing west 
 or decreasing east declination, a — sign the reverse motion. Stations to the 
 right of the agonic curve, or curve of no declination, have west declination, 
 and those to the left, east declination. The large black circles or dots 
 indicate the capitals of the several states. 
 
 The use of this chart, is quite simple. The declination for any place 
 within its borders is either found by inspection or by simple interpolation 
 between the two adjacent curves; the value found ‘is for 1900. For any 
 other year (and fraction), a reduction for secular change between the epoch 
 and given date must be applied. The annual change of the declination 
 during the period 1895-1900, expressed in minutes of arc, is indicated in 
 the chart ( + for increasing west or decreasing east declination and — for 
 the reverse motion). The amount varies in time, but not sufficiently during 
 a brief interval of years to cause any serious inaccuracy, and the values given 
 on the chart can be used for a number of years to come for all practical 
 purposes; its variation with geographical position must be estimated from 
 the map. 
 
 THE TRANSIT. 
 
 The transit is the only instrument that should he used for measuring angles 
 in any survey where great accuracy is desired. The advantages of a transit 
 over a vernier compass are mainly due to the use of a telescope. By its use 
 angles can be measured either vertically or horizontally, and, as the vernier 
 is used throughout, extreme accuracy is secured. 
 
 The illustration shows the interior construction of the sockets of a transit 
 having two verniers to the limb, the manner in which it is detached from its 
 
 spindle, and how it 
 can be taken apart 
 when desired. The 
 limb b is attached to 
 the main socket c. 
 which is carefully 
 fitted to the conical 
 spindle h, and held 
 in place by the spring 
 catch s. 
 
 The upper plate a, 
 carrying the compass 
 circle, standards, 
 etc., is fastened to the 
 flanges of the socket 
 k , which is fitted to 
 the upper conical 
 surface of the main 
 socket c. The weight 
 of all the parts is sup- 
 ported on the small 
 bearings of the end 
 of the socket, as 
 shown, so as to make as little friction as possible where such parts are 
 being turned as a whole. 
 
 A small conical center, in which a strong screw is inserted from below, is 
 brought down firmly on the upper end of the main socket c, thus holding the 
 two plates of the instrument securely together, and. at the same time, allow- 
 ing them to move freely around each other. The steel center pin on which 
 the needle rests is held by the small disk fastened to the upper plate by 
 two small screws above the conical center. The clamp to limb d /, with 
 clamp screw, is attached to the main socket. The instrument is leveled by 
 means of the leveling screws l and placed exactly over a point by means of 
 the shifting center. The plummet is attached to the loop p. 
 
 The verniers on a transit differ from those on a compass in detail only. 
 
90 
 
 80 
 
 1 
 
 70 ° 
 
ADJUSTING TUN TRANSIT. 
 
 41 
 
 The principle is the same. The transit vernier is so divided that 30 spaces on 
 it equal in length 29 on the limb of the instrument. The method of reading 
 it is practically the same as reading a compass vernier, except that of 
 the transit the vernier is made with all of the 30 divisions on one side of the 
 zero mark. 
 
 Each division of the vernier is therefore or, in other words, 1 minute 
 shorter than the half-degree graduations on the limb. 
 
 In the figure the reading is 20° 10'. If the zero on the vernier should be 
 beyond 20i° on the limb of 
 the transit, and the line 
 marked 10 should coincide 
 with a line on the limb, the 
 reading would be 20° 40'. 
 
 In case the 12th line from 
 zero should coincide with a 
 line on the limb, the read- 
 ing would be 20° 42', etc. 
 
 In some transits, the 
 graduated limb has two 
 sets of concentric gradua- 
 tions, the zero in both being 
 the same, and, while the outside set is marked from 0° each way to 90°, and 
 thence to 0° on the opposite side of the circle, the other set is marked from 
 0° to 360° to the right, as a clock face. The inside set has the N, S, E, and W 
 points marked, the 0° of the inside set being taken as north. 
 
 The interior of the telescope is fitted up with a diaphragm or cross- wire 
 ring to which cross-wires are attached. These cross-wires are either of 
 latinum or are strands of spider web. For inside work, platinum should 
 e used, as spider web is translucent and cannot readily be seen. They are 
 set at right angles to each other and are so arranged that one can be 
 adjusted so as to be vertical and the other horizontal. This diaphragm is 
 suspended in the telescope by four capstan-headed screws, and can be moved 
 in either direction by working the screws with an ordinary adjusting pin. 
 The transit should not be subjected to sudden changes in temperature that 
 may break the cross-hairs. In case of a break, remove the cross-hair 
 diaphragm and replace the broken wire. 
 
 The intersection of the wires forms a very minute point, which, when 
 they are adjusted, determines the optical axis of the telescope, and enables 
 the'surveyor to fix it upon an object with the greatest precision. 
 
 The imaginarv line passing through the optical axis of the telescope is 
 termed the line of collimation, and the operation of bringing the intersection of 
 the wires into the optical axis is called the adjustment of the line of collimation. 
 
 All screws and movable parts should be covered, so as to keep out acid 
 water or dust. If this is not done, the mine work will soon use up a transit. 
 The vertical circle on the transit may be a full circle or a segment. The 
 former is to be preferred, as it is always ready without intermediate clamp 
 screws. If the dip of a sight is to be taken, the tape must be held at the 
 transit head, and stretched in the line of sight. If the pitch of the ground is 
 to be taken, the point of foresight must be at the same height as the axis of 
 the transit, and the sight will then be parallel to the surface. The angle of 
 dip is read “ plus” or “minus,” as it is above or below the horizontal plane. 
 If we have the dip of a sight, and the distance between the transit head and 
 the point of sight, we can get the vertical and horizontal components of that 
 distance from the table of sines and cosines. 
 
 ADJUSTMENTS OF THE TRANSIT. 
 
 The use of a transit tends to disarrange some of its parts, which detracts 
 from the accuracy of its work, but in no way injures the instrument itself. 
 Correcting this disarrangement of parts is called adjusting the transit. 
 
 First Adjustment.— To make the level tubes parallel to the vernier plate. 
 
 Plant the feet of the tripod firmly in the ground. Turn the instrument 
 until one of the levels is parallel to a pair of opposite leveling screws; the 
 other level will be parallel to the other pair. Bring the bubble in each tube 
 to the middle with the pair of leveling screws to which the tube is parallel. 
 Next turn the vernier plate half way around; that is, revolve it through an 
 angle of 180°. If the bubbles have remained in the middle of the tubes, the 
 levels are in proper adjustment. If they have not remained so, but have 
 
42 
 
 SURVEYING. 
 
 tAwarrl pither end bring them half way back to the middle of the 
 tfibes bv means of the capstan-headed screws attached to the tubes, and the 
 tu + t? hflpt hv the leveling screws. Again turn the vernier plate 
 
 through 180° and if the bubbles do not remain at the middle of the tubes, 
 the correction. Sometimes the adjustment is made by one trial, but 
 usually it is necessary to repeat the operation. Each level must be adjusted 
 
 SeP SeMnd y Adjustment —To make the line of collimation perpendicular to the 
 
 and carefully leveled, sight to a pin 
 or *. about 400 ft. distant, ^nearly level 
 
 G 
 
 B- 
 
 & 00 ' 
 
 scope; that is, tarn it over on 
 its axis until it points in the 
 opposite direction, and set a 
 point at about the same dis- 
 tance, which will be at D. 
 
 ^mestrallhtlinlwith A K It maybe necessary to repeat the operation 
 t0 horizontal axis of the telescope parallel to the 
 
 sSf S.-';Fs sas .W”lSsx 
 
 plamn turn over the telescope, and turn the plate around suffi 
 Sfvtoto take an approximately accurate sight upon the 
 r»m' n t ^4 Then clamp the instrument and again take an exact 
 Sght to 'the po?nt I P Next depress the telescope, and set another 
 P fha PTonud which will come at G. The distance is l is 
 dcmbfe the S e rror of adjustment. Correct the error by raising 
 O? lowering one end of the telescope axis by means of a small 
 
 '5A -d have 
 
 [hfstakes driven down until l the . rod jading ^ the^ame on both stakes. 
 
 gives equal rod readings on both s takes. 
 
 THE CHAIN OR STEEL TAPE AND PINS. 
 
 The chain, which is gene^l^50 or 10^ft. lon|, should temah^oOhneel^ 
 
CHAIN AND PINS. 
 
 43 
 
 provided. These handles should be attached to short links at each end, and 
 the combined length of each-of these short links and one handle should be 
 exactly 1 ft. The handles should be attached to the short link in such a 
 manner that the chain may be slightly lengthened or shortened by screwing 
 up a nut at the handle. It should be divided every 10 ft. with a brass tag, on 
 which either the number of points represents the number of tens from the 
 front end, or the number of tens may be designated by figures stamped on 
 the tags. 
 
 When a chain is purchased, one that has been warranted as *' Correct, U. S. 
 Standard,” should be selected, and, before using it, it should be stretched on 
 a level surface, care being taken that it is straight, and no kinks in it, and 
 the extremities marked by some permanent mark. These marks can be 
 used in the future to test the chain. It should be tested frequently, and the 
 length kept to the standard as marked when it was new. 
 
 In chaining, the chainmen should always remember the axiom that a 
 straight line is the shortest distance between two points. Ordinarily, the chain 
 should be held horizontally, and if either end is held above the ground, a 
 plumb-bob and line should be used to mark the end of the chain on the 
 ground. If used on a regular incline, the chain may be stretched along the 
 incline, and, by having the amount of declination, the horizontal and 
 vertical distances may either be calculated or found in the Traverse Table. 
 
 For accuracy, steel tapes are now almost exclusively used by the leading 
 mining engineers, on account of their greater accuracy as compared with 
 
 Ch The steel tape is simply a ribbon of steel, on which are marked, by etching, 
 or other means, the different graduations, which may be down to inches or 
 tenths of a foot, or may be only every foot. It is wound on a reel, and may 
 be any desired length up to 500 ft. . . , _ 
 
 A well-made tape should not vary ft. in 100 ft., at any given standard 
 of temperature. The steel of the tape should not be too high in carbon, or it 
 will be brittle and liable to snap on a short bend, nor should it be of too soft 
 steel, or it will stretch when strongly pulled. . , 
 
 Careful gunsmiths can make and repair steel tapes with a high degree of 
 accuracy and fully as reasonably as the instrument makers. For outside 
 work tapes 1,000 ft. long have been made, but 500 ft. will be found as long as 
 can well be used in a mine, owing to the lack of long sights, and to the 
 increased weight of so long a tape. The average length is 300 ft. The 300 ft. 
 are divided into 10-ft., 5-ft., 2-ft., or 1-ft. lengths, as desired, and the tenths 
 and hundredths of a foot are read by means of a pocket tape or measuring 
 pin Sometimes there is an extra division before the zero mark, which is 
 divided into feet and the first foot into tenths. With such a tape, a distance 
 can be accurately measured to tenths, or even quite approximately to 
 
 hundredths of a foot. ...... .... , 
 
 The ends are fitted with eyes on swivel-joints, to prevent straining by 
 twisting. Handles of various forms have been devised to enable the tape to 
 be stretched, or to clamp a broken end. Some parties use ordinary springs 
 to prevent overstraining, and, in certain cases, spring scales are used, and 
 the same degree of tension can be readily produced, and, m this way, the 
 exact amount of sag can be calculated for any length, and the necessary 
 correction made. To keep a mark on the tape for frequent reference, si clip 
 (made by bending sharply on itself a piece of steel £ in. X 3 in.) is slipped 
 upon the tape, where it will remain unless subjected to considerable force. 
 Reels for winding the tape are made of iron of wood, and vary greatly in 
 
 size and shape. „ ,, „ , 
 
 When distances do not come at even feet, the fractional part of the foot 
 should always be noted in tenths. Thus, 53 ft. and 6 in. should always 
 
 be noted as 53.5 ft. . , _ , , , , 
 
 Pins —pins should be from 15 to 18 m. long, made of tempered-steel wire, 
 and should be'pointed at one end, and turned with a ring for a handle. 
 When using a 50-ft. chain, a set of pins should consist of eleven, one of which 
 should be distinguished bv some peculiar mark. This should be the last pm 
 stuck by the front chainman. When all eleven pins have been stuck, the 
 front chainman calls “Out!” and the back chainman comes forward and 
 delivers him the ten pins that he has picked up, and he notes the “out.” 
 When giving the distance to the transitman, he counts his “outs,” each of 
 which consists of 500 ft., and adds to their sum the number of fifties as denoted 
 by the pins in his possession, and the odd number of feet and fractional parts 
 of a foot from the last pin to the front end of the chain. 
 
44 
 
 SURVEYING. 
 
 The accuracy and value of a survey depend as much on the careful work 
 of the chainmen as on anything else, and no one should be allowed to 
 either drag or read the chain that is not intelligent enough to appreciate the 
 importance of extreme accuracy. , , . 
 
 Pins are generally used in outside work, where they can be easily stuck 
 into the ground, readily seen, and avoided, and the chances of their being 
 disturbed are slight. Inside work generally contains S9 many chances of 
 error in their use that they are usually abandoned m favor of other 
 methods. If the sight be longer than the length of the tape, it is usua* to 
 drive a tack in a sill or a collar at a point intermediate between the stations, 
 and take a measurement to the tack from each station, with the dip 01 the 
 sights; or a tripod is set up in the line of sight, and the horizontal distance 
 is m easured from each station to the string oi the plumb-bob under the 
 trinod The first method is the more accurate. 
 
 Plumb-Bob.— The plumb-bob takes the place of the transit rod m under- 
 ground work, as the stations are usually in the roof, and strings are hung from 
 them to furnish foresights and backsights. Plumb-bobs vary m weight and 
 shape. At various times and in various countries where mine surveys have 
 been made, the idea ot sighting at a flame has been considered, and, from 
 rough methods of setting a lamp on the floor on foresight and backsight, 
 there have arisen various forms of plummet lamps. The idea is to continue 
 the practice of sighting to a flame, but to make that flame exactly under the 
 station, and to avoid the difficulty in sighting to the string of the plummet 
 The idea is good, but there has never been, devised a plummet lamp that 
 would be as free from error under all circumstances as the old-fashioned, 
 plummet, so that the majority of the best engineers have gone back to the 
 plummet. The best plummet is the one that combines the least surface with 
 the greatest weight, and the ordinary shapes used for outside work are the 
 best for inside also. In a “ windy ” place, a hole can be dug m the ballast 
 of the track and the “ bob ” let into this shelter where it will be unaffected 
 bv the air. The cord is best illuminated by placing a white paper or card- 
 board behind it and holding the lamp in front and to one side. Kie string 
 shows as a dark line against a white ground, and there is less difficulty m 
 finding it than when the light is placed exactly behind it, and in this way a 
 careless man cannot burn the string by poking the flame against it. The 
 white background will also illuminate the cross-hairs of the transit. The 
 backsight “ bob ” can be made of lead, as there are no “ centers to be set by 
 this man A number of varieties have been made for the foresight, to aid him 
 in “ center setting ” ; but all get out of order easily. A quick man will do as 
 good work with the old-style bob, and have none of the accidents common 
 to the others. In general, it may be said that the instruments used for outside 
 work will be sufficient for mine work also. 
 
 The clinometer, or slope level, is a valuable instrument for side-note work; 
 but it is not accurate enough for a survey, and its place is taken by the 
 vertical- circle on the transit. There are two styles of clinometer, with a 
 bubble and with a pendulum. The latter is the old-fashioned and more 
 accurate German “Gradbogen” that is found on some old corps. The bubble 
 variety is much more easily rendered worthless by the breaking of the bubble 
 tube and in general is not so accurate as the other style, which consists of a 
 semicircular protractor cut out of thin brass and furnished with - books i at 
 each end, that it can be hung on a stretched string so that the string v ill 
 pass through the 0° and 180° points. The dip is read by a pendulum swung 
 from the center of the circle. If made sufficiently large it will readily read 
 to Quarter degrees. By inclining the string parallel to the surface and 
 taiSngflieclinometer, the. dip will be obtained. A pocket instrument 
 combining a compass and clinometer can be obtained from any dealer in 
 surveying instruments. . 
 
 FIELD NOTES FOR AN OUTSIDE COMPASS SURVEY. 
 
 Call place of beginning Station 1. . 
 
 Stations. Bearings. Distances . 
 
 1_2 N 35° E 270.0 
 
 At 1 + 37 ft. crossed small stream 3 ft. wide. 
 
 At 1 + 116 ft. = first side of road. 
 
 At 1 + 131 ft. = second side of road. , ^ . 
 
 At 1 I 137 ft. = blazed and painted pine tree, 3 ft. left, marked for a 
 “go by.” 
 
TRANSIT S UR VE YING. 
 
 45 
 
 Station 2 is a stake at foot of white-oak tree, 
 sides for corner. 
 
 2-3 N 83i° E 
 
 blazed and painted on four 
 129.0 
 
 Station 3 is a stake-and-stones corner. 
 
 3- 4 S 57° E 222.0 
 
 3 + 64 ft. = center of small stream 2 ft. wide. 
 
 3 + 196 ft. = white oak “ go by,” 2 ft. right. 
 
 Station 4 = cut stone corner. 
 
 4- 5 S 34i° W 355.0 
 
 4 _ j- 174 ft. = ledge of sandstone 10 ft. thick, dipping 27° south. 
 
 5- 1 N56i°W 323.0 
 
 5 + 274 ft. = ledge of sandstone 10 ft. thick, dipping 25° south (evidently 
 
 continuation of same ledge as at 4 + 174). 
 
 Station 1 = place of beginning. 
 
 TRANSIT SURVEYING. 
 
 To Read an Angle.— The angle read may be included or deflected. If we 
 set up at 0 with backsight at B and foresight at C, we shall find that there 
 are two angles made by the line C O with the line BOA, namely the included 
 angle B 0 C, and the deflected angle CO A. 
 
 To Read the Included Angle.— Set the zeros of vernier and graduated limb 
 together accurately, and clamp the plates. Turn the 
 telescope on the backsight, with the level bubble down, 
 and, when set, fasten lower clamp so as to fix both 
 clamped plates to the tripod head. Loosen the upper 
 clamp and turn the telescope to C and set accurately. 
 
 The vernier will read, for example, ‘‘45° left angle.” 
 
 To Read the Deflected Angle.— Arrange verniers as above, 
 and be sure and turn the telescope over on its axis till 
 the bubble tube is up, and then take the backsight and 
 fix lower clamp. Turn the telescope back (this is 
 called “ plunging” the telescope) and sight to foresight 
 and fix as before. The vernier will read a ‘‘right angle of 135°.” The sum 
 of included and deflected angles must always be 180°. 
 
 Note.— In making a survey by included angles, we must add or subtract 
 180° at each reading to have the vernier and compass agree; by deflected 
 angles, they will agree without the above addition or subtraction, and the 
 latter method is generally used. 
 
 TO MAKE A SURVEY WITH A TRANSIT. 
 
 By Individual Angles.— Set vernier at zero of limb, plunge telescope, and, 
 when set on backsight, loosen needle and read bearing of the line from back- 
 sight to set-up. Plunge telescope back and set on foresight and read both 
 needle and vernier. The difference in needle readings should agree with the 
 vernier reading within 15', as local attraction will affect the needle equally 
 on both sights. , , , . 
 
 Note.— Any mass of iron or steel that may and will be moved during the 
 readings of the needle, will affect the same and destroy the value of the 
 needle as a check. The tape and other iron materials should not be moved 
 during the taking of angles. 
 
 By Continuous Vernier.— Set vernier at zero, unclamp compass needle, and, 
 when stationary, turn the north point of compass limb so as to coincide with 
 the north point of the needle. Fix lower clamp, plunge telescope, and take 
 backsight by loosening upper clamp. The vernier and needle should agree 
 in giving the magnetic bearing of the line from backsight to set-up. Record 
 this in note book; plunge telescope, and take foresight. Needle and vernier 
 should agree as before. After making record, set up over foresight and take 
 sight to station just left with telescope plunged, having first seen that the 
 vernier reads exactly as it did on the last foresight, as a slip in carrying the 
 transit from one station to another, which is not detected at the time, can 
 never be checked afterwards when the final work is found to be in error. 
 The foresight is taken as before. On every sight the needle and vernier 
 should agree if there is no local attraction of the needle. 
 
 If we can see all the corners of a field that is to be surveyed from a central 
 point, we can make the survey by setting up at that point, and, with one 
 
46 
 
 TR ANSI T SURVEYING. 
 
 porner as a backsight, take all the other corners as foresights with but one 
 set-up and by measuring from this point to all of the corners; or we can 
 set up’ at any corner and run a line of survey around the field. This latter 
 method is called meandering. Both methods will give the same result when 
 plotted- but the former is much quicker, as the boundaries of a tract are 
 freemen tlv overgrown with bushes that must be cleared to allow a sight, 
 3 a SLtaapSnt can frequently be found that will allow a free sight to 
 all the corners, and the distance can be measured by tape, or stadia. As the 
 central point is nearer the corners than they are to one another it follows 
 that a shorter distance must be chained or cut in the case of a central set-up. 
 ^^Outsid^'sm'veys^ be made for many purposes It matters not what 
 the purpose is, the work should be fully and accurately done, and the map 
 should contain everything that will throw light upon the subject. If the 
 outside work is to be connected with inside surveys, there are a number of 
 points to be observed, and they will be given under the head of underground 
 
 W °Meridians or Base Lines.— The surveys must be based on some meridian;, 
 and started ’from some fixed point. There are four kinds of meridians, o 
 
 bl First™ t line already on the ground, as one of the sides of the tract is 
 taken as a base. The subsequent work is ends of thl& 
 
 line and all angles measured are taken as deviations from it. 
 
 Second.— A stone post is sunk in the ground, or, better, a u iron J® PJjJ 
 into rock “in place that is, not loose rock, even if a large bolder— at such 
 a distance from the works as to be beyond the influence of moving ^.machinery 
 and a line of sight is taken to some permanent natural object, as far distant as 
 can be clearly seen under adverse circumstances, as cloudy or dark weather. 
 This line of sight is the base line, and the plug is the origin. N 9 measurements 
 of distance are needed. If no natural objects exist, a station is se ^.^P a 
 a distance so as to be as permanent as possible, and angles are turned from 
 this to other points, so as to check any movement m it. Generally there are 
 „ number of tall chimneys, church spires, etc. to be found. While this is 
 preamble to the first, it Jives no method of check in underground work, 
 
 an Tiy/-The magnetic meridian is taken as the base line. The transit is 
 setup over a plug, as just noted, and the subsequent work is as described 
 unde? running continuous vernier. As the needle is subject to constant 
 Nation this%ase line will afford a check underground only for a short 
 time after the meridian is established, and all subsequent work can be 
 checked only by applying the difference between the variation at the time 
 nf P^blishment and at the time of making the survey If the time of 
 establishing the survey should be lost, the base line would become no better 
 
 tha 2WA -The trae a me?idian is taken as a base. The true north and south 
 bnp ruav be 'determined by observing the North Star, Polaris, or by observing 
 Ihe sun y The N™Star loes not lie exactly at the North Pole, but revolves 
 it in a small circle. There are two times m a day when it is exactly 
 
 our ^transffe "do ^o^hlvetheS JmduatS hmte 
 
 S? « e d St s a tar in 
 
 crosses the meridian. . . 
 
 The true meridian will give us an invariable base 
 line At any date after the establishment of the same, 
 we can check the work above or below ground by 
 applying the variation of the needle. 
 
 To Find the True North by an Observation of the North 
 Star, Polaris, at Elongation.-This star has a motion 
 around a small circle, the azimuth angle of winch 
 from the north is known for different latitudes. The 
 star may be readily found by following the line of the 
 so-called pointers in the Big Bear, or Dipper. The 
 time of the greatest eastern or western elongation is 
 found from a table. Some 10 minutes before this time 
 the transit is carefully set up and leveled over a peg. 
 The cross-wires are made to bisect the star: they are 
 
 jpo laris 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 l 
 
 I 
 
 1 
 
 1 
 
 I 
 
 1 l 
 \ 1 
 \ ! 
 
 \l 
 
 \l 
 
 V Observer 
 
THE SO LAX ATTACHMENT. 
 
 47 
 
 illuminated by a light held under the reflector fastened on the object 
 end of the telescope. The star is followed with the cross-wires until its 
 motion toward the point of its greatest elongation ceases. The telescope is 
 lowered vertically, care being taken, of course, not to move it horizontally, 
 and a peg is set up on the line, say 300 ft. or 400 ft. distant. The next morn- 
 ing the correction is made for the star’s azimuth. These corrections are 
 different for different latitudes and different years. They are to be found in 
 
 the nautical almanac. , ... . , , , 
 
 The method “by equal shadows” may be used with considerable accu- 
 racv if we take a sufficiently long staff, or can obtain the shadows of a 
 tall spire on a level surface. A vertical staff casts equal shadpws at the 
 same time before and after noon. If we drive a stake at any time before 
 noon, in the extremity of the shadow cast by such a staff, and measure 
 its distance from the staff, we have 
 one leg of an angle. After noon we 
 wait till the shadow becomes exactly 
 as long as the distance measured, and 
 drive a stake at the extremity of the 
 shadow. A line bisecting the angle 
 made by lines drawn from these two 
 stakes to the staff will be in the 
 meridian. 
 
 Establishing a Meridian Line With the 
 Solar Attachment— The angle from 
 the equator to the horizon of a place 
 is its latitude; consequently, from the 
 zenith to the pole is the colatitude , 
 or 90 o _ latitude. The angular dis- 
 tance from the equator to the sun is 
 the decimation; consequently, from 
 the sun to the pole is the polar dis- 
 tance. The angular distance from 
 the horizon to the sun is the sun’s . , , 
 
 altitude; consequently, the zenith distance is the angular distance between 
 the sun and the zenith. . „ . A . , , 
 
 Adjustments of Burt’s Solar Attachment.— After the instrument has been 
 carefully leveled, the zero of the vernier of the solar is placed opposite the 
 zero of the arc. The horizontal plates of the instrument are clamped, and 
 the sun’s image brought between the horizontal lines of either silver plate 
 bv any manipulation of the instrument and attachment possible, keeping 
 the plates horizontal and the zero of the vernier opposite the zero on the arc. 
 When the image is accurately between the horizontal lines, the arc is 
 revolved so that the image falls on the other plate; this must be done 
 rapidly as the sun’s image moves. If it does not fall between the lines, half 
 the error is corrected by the tangent screw of the solar and half by the 
 tangent screw of the telescope. The operation is repeated until the sun s 
 image falls between the lines of the second plate, after a revolution of the 
 arc it having been made to fall between the lines of the first, as described. 
 Near noon is a good time to make this adjustment, as the sun’s apparent 
 motion is not so rapid. The zero of the vernier is now brought opposite to 
 the zero of the arc by loosening the screws that fasten the vernier, and 
 sliding it as may be necessary. It is often difficult to make the zeros come 
 exactly opposite each other, as the vernier plate is apt to move slightly 
 when the screws are tightened again. The second adjustment is to make 
 the tons of the rectangular blocks of the solar attachment level, when the 
 telescone is level and the arc of the solar is set at zero. Level the transit 
 carefully as before described, set the solar at zero and plac^ the level, 
 furnished with the solar, across the tops of the blocks. If the bubble comes 
 to the center of the tube, no correction is needed; if it does not, correct the 
 error bv turning the screws under the hour circle, care being taken m this 
 as in ali other movements of these adjusting screws, to leave them tight after 
 the correction. Revolve 180° and correct again if necessary. Placing the 
 blocks 90° horizontally from their first position, go through the same 
 operation as described until in all positions the bubble remains centered. 
 
 To Use the Solar.— Before this instrument can be used at any given place, 
 it is necessary to set off upon its arcs both the declination of the sun, as 
 affected by its refraction for the given day and hour, and the latitude of the 
 place where the observation is made. 
 
48 
 
 TRANSIT S UR VE Y1NG. 
 
 The declination of the sun as given in the ephemeris of the nautical 
 almanac from year to year, is calculated for apparent noon at Greenwich, 
 England. To determine it for any other hour at a place in the United 
 States, reference must be had. not only to the difference of time arising 
 from the difference of longitude, hut also to the change of declination 
 during that time. 
 
 The longitude of the place, and therefore its difference m time, if not given 
 directly in the tables of the almanac, can be ascertained very nearly by 
 reference to that of other places given which are situated on, or very nearly 
 on, the same meridian. 
 
 It is the practice of survevors in states east of the Mississippi to allow a 
 difference of 6 hours for the difference in longitude, calling the declination 
 given in the almanac for 12 M. that of 6 a.m. at the place of observation. 
 Beyond the meridian of Santa Fe, the allowance would be about 7 hours; 
 and in California. Oregon, and Washington, about 8 hours. Having thus 
 the difference of time, we verv readily obtain the declination for a certain 
 hour in the morning, which would be earlier or later as the longitude was 
 greater or less, and the same as that of apparent noon at Greenwich on the 
 given day. Thus, suppose the observation made at a place 5 hours later than 
 Greenwich, then the declination given in the almanac for the given day at 
 noon, affected by the refraction, would be the declination at the place of 
 observation for 7 a.m. This give us the starting point. 
 
 To obtain the declination for the other hours of the day, take from the 
 almanac the declination for apparent noon of the given day, and, as the 
 declination is increasing or decreasing, add to, or subtract from, the decli- 
 nation of the first hour the difference of one hour as given in the ephemeris, 
 that will give, when affected by the refraction, the declination of the 
 succeeding hour. Proceed in like manner to make a table of the declinations 
 for every hour of the day. 
 
 To Find the True North With the Burt Solar.— Find from an ephemeris or 
 nautical almanac the sun declination for noon of the day of observation at 
 Greenwich. Find the declination for the hour of observation at the place 
 of observation bv first figuringwhat time it is at the place of observation 
 when it is noon at Greenwich. If the place of observation is west of Green- 
 wich it will be earlier there; if east, later, and in either case the difference 
 will be one hour for everv 15° of longitude. If the place is west, subtract the 
 hour just found as described from the hour of the observation, and multiply 
 the hourly difference, also taken from the ephemeris, by the remainder. If 
 the declination is increasing from the equator either north or south, add 
 this product to it; if decreasing, subtract it. A table of refractions is given 
 in the ephemeris for the different latitudes and the different hours of the 
 dav. This refraction is to be added if the declination is north, and 
 subtracted if the declination is south. Having thus ascertained the declina- 
 tion, lay it off on the declination arc. Set the colatitude of the place off on 
 the vertical arc after having leveled the instrument carefully with clamped 
 horizontal plates at zero. Alwavs in solar observations it is well to level by 
 means of the upper telescope bubble. Now, revolve the horizontal plates still 
 clamped and also the declination arc. around its polar axis until the sun’s 
 image is exactlv between the horizontal lines of the silver plates. When the 
 sun’s image is between these lines, the object end of the telescope will be 
 pointing north. , _ 
 
 To Take the Latitude With Burt’s Solar.— A few minutes before apparent 
 noon clamp the plates at zero, level the instrument carefully, and set the 
 zero of the vernier opposite the zero of the vertical arc. Lay off the declina- 
 tion, corrected for noon at the place of observation and for refraction, on 
 the declination arc, and set the time mark on the declination arc opposite 
 XII on the hour dial. Bring the sun’s image between the horizontal lines 
 of the silver plate bv moving the plates horizontally and the telescope 
 vertically, clamp both plates and telescope and follow with the tangent 
 movements the rising sun. Be careful to stop when the sun ceases to mount. 
 For a moment before apparent noon there is no perceptible motion of the 
 image. The reading on the vertical arc is the colatitude of the place. 
 
 The colatitude should never be taken this way for direct-sight calcula- 
 tions for while it satisfies the automatic solution of the true north, it may 
 not be accurate, and the latitude needed for direct-sight calculation should 
 be true to within a minute. With the Burt solar there is at times what is 
 called a false image to guard against, an image that comes between the lines 
 of the silver plates when the object end of the telescope is not pointing 
 
TRANSIT S UR VE YINQ. 
 
 49 
 
 north If the time be observed on the hour dial, or the magnetic north be 
 noticed, no error need ever occur on this score, for with the false image the 
 time will be out considerably and also the magnetic variation. 
 
 General Remarks.— With the base line located and the survey made, we see, 
 by coming back to the point from which we started or “ closing” the work, 
 whether it be correct in distance or angle. If it be in error, see if the error 
 can be located (as will be shown under plotting), and if it can be tound, run 
 those parts over again; if not, repeat the whole survey. Shoving the work, 
 as it is called, or “doctoring” it so that it will close, is the poorest practice 
 that an engineer can engage in, as all subsequent work that depends on a 
 doctored survey must be doctored to tit the faulty work — even it it be right 
 in itself. Every engineer should be able to swear— not that he “ thinks the 
 survey is accurate,” but “that he knows it to be so,” if he should be called 
 as a witness in court. One of the causes of inaccuracy is haste. 
 
 To make a complete map, the engineer should first make a survey around 
 the tract to be worked, locating all the prominent physical features and 
 improvements. If he can do so, he should make a topographical map of the 
 tract at once; but, if time is limited, by running the vertical as well as the 
 horizontal angle, he can carry the tidal elevation or the elevation above 
 some assumed datum, to every station, and murk it on the map at that point. 
 Then as he makes subsequent surveys, he can gradually get data enough to 
 make a fairly complete topographical map in course of time. Every ledge 
 of rock in place should be located, and the amount and direction of its dip, 
 as well as the character of the rock, should be marked neatly on the map. 
 The streams of water on the tract should be regarded as of primary impor- 
 tance, and should be located with exactness. 
 
 With a true meridian base line we can connect maps made at different 
 places with little trouble. This is especially useful with adjoining mines 
 connected at but one point. Having made the survey and come home, we 
 must examine all the apparatus and see if the instruments are out of adjust- 
 ment, as such a fact will prevent our bothering over work that will not 
 close It will assure us, also, that we can start out at a moment’s notice 
 with no thought of the adjustment of our tools. A famous wit said that the 
 proper time to strop a razor was just after you had used it, as you then knew 
 how much it needed it. The same will apply to surveying instruments and 
 tools— especially for underground work. Here the lamp smoke, powder 
 gases, mine dust, paint smears, acid water from “droppers,” and the other 
 abominations incident to underground surveying, especially in a coal mine, 
 will so cover the tools that they would be useless if left uncleaned half a 
 dozqn times. As soon as the corps comes back from the mine, and before 
 the clothes are changed, the tape must be stretched, tested, wiped, and oiled. 
 It can be inspected to see if marks are too much worn, or it stands in need of 
 mending, the marking pot is cleared of “muck,” and fresh white paint is 
 mixed, if the corps is going out in 24 hours; the plummets will have their 
 strings overhauled and freed from knots; hatchets will be sharpened, and 
 axes ground, pouches overhauled, and a supply of tacks or “spads” taken. 
 Then the transitman changes his clothes and sets up the transit, wipes it 
 with a cloth wet with alcohol, so as to remove dirt, oil, and paint. If water 
 has gotten between the graduated limb and compass box, the verniers must 
 be uncovered and the whole wiped dry. If the sulphureted hydrogen from 
 the powder smoke has tarnished the silver surfaces of any of the graduated 
 circles, it must be removed with whiting. Alcohol should be always used 
 instead of water, as it will quickly evaporate and leave the parts dry. The 
 telescope glasses are then wiped with soft chamois leather, and the instru- 
 ment is tested for want of adjustment before putting it away in its box. 
 When going to and from work, the transit should not be carried on the 
 transit head, or the spindle will become sprung. Nor should it be carried 
 with the arm crooked under the telescope, as the weight comes on the axis, 
 and that soon gets sprung so that all the adjusting in the world will not 
 make it work right. When carried in the hand, it should be reversed and 
 the hand slipped under the compass plate and brought over so as to clamp 
 both plates. In this way there will be no strain on any part. In case of a 
 “fall” in the mine, remember that the transit is the baby to be protected, 
 and stand a few bumps to save a strained or broken instrument, that will 
 end the work for some time. 
 
 ‘ Plotting.— A “plot” is not only a piece of ground with bodies of water, 
 roads, vegetation, etc. upon it, but refers also to the map of the same drawn 
 to a given scale, and showing all of the above natural features. Plotting is 
 
50 
 
 TRANSIT S UR VE YING . 
 
 the making of such a map from notes of a survey, and may or may not 
 require the permanent placing of the stations on the map, by which the 
 survey is made. In- underground work, the exact location and the retention 
 of those stations is a matter of the first importance, and is secondary only to 
 the exact ploUing of the side notes. The scale of the plot is generally 
 as large as will show the points of interest m the 
 Pennsylvania, the maps for coal mines must be drawn to a scale of 100 ft. to 
 an inch There are two methods of plotting: by protractor, and by 
 coordinates. When the scale is sufficiently large, it is a matter of little choice 
 which method is used, if the work be carefully done with exact instruments, 
 but with small scales-100 . ft. or above, to the inch— we should use the 
 method bv coordinates. With the latter scale, the prick of a pm on the 
 paper will^ represent a foot square, or a circle slightly larger than a foot 
 m diameter. If the next station is to be located from the pin prick of the 
 first, and that is exactly located, we may not hit the exact cenr-CT 9*^^ 
 small indentation. In fact, the chances are greatly against our doing so, 
 and the location of the second station will probably be in error. If we have 
 a bad habit of placing the protractor or the straightedge against one side of 
 the pin pricks, or pencil marks, when the scale is large we 
 be introducing a “ personal error, as it is called; and the sum of all the 
 errors made at each of 100 stations will bring our final point very much _out 
 of the way. On this account, and from the fact that no protractor that is 
 movable can be used without the chances of shppmg while the angle is read 
 or marked, has led all careful engineers to abandon its use m favor of the 
 method by coordinates. When the scale is from 1 to 2o ft. to an inch the 
 errors are small enough to make little chances of variation in a cl ?s e often 
 or twelve stations; when the survey is of short sights from a mam line to 
 points where no further work is to be done, the protractor will afford a quick 
 
 me Th°ere°i a ?han?'e of error in both methods that must he noted here, where 
 the survey is not completed at one time. If the map be made m a day or 
 two, and will never be extended by subsequent work, there will be no 
 chance of error from a change in the paper on which it is made, due to 
 moisture or dryness; but if the map be made on a series of very damp days, 
 or a series of very dry ones, a change in the weather to the other extreme 
 will swell or shrink the map. The general tendency in a large mine map 
 that is frequently used, and is rolled and unrolled every day for ^ sg 
 years, is to stretching, so that there will be a variation of from l to 5 ft m 
 1,000. If we extend a recent survey on such a map, we are plotting it to a 
 
 different scale to that assumed by the map under , ^wri 
 
 noted. The paper on which the map is to be drawn should be tacked down 
 to the table or board, and should be covered with squares each exactly 10 in. 
 square. The sides of these squares should be the meridians, or north or south 
 lines, and the tops and bottoms should run due east and west. Mark the first 
 station on the paper, set your parallel ruler or T square on the meridian nearest 
 it and with the protractor produce the course to the next station. Measure the 
 distance with a scale, and proceed in this manner to plot all the courses using 
 each time the meridian nearest the station the course is taken from. After all 
 the stations have been plotted, fill in the side notes, marking 
 the map with great care and neatness. Always use the horizontal distances. 
 All surveys should be traversed, and all plotting should be eithei checked 
 bv the traversing, or the principal stations should be plotted by use of the 
 traverse. For a large mine map that will be in use many years, muslin- 
 backed egg-shell paper must be used. It comes in a long roll, and any 
 reasonable length, and a width up to 6 ft. can be obtained. 
 
 To Calculate the Vertical Distances.— When making the survey, read the 
 vertical angles to all stations. If the angle is one of depression, note it vuth 
 a minus sign (— ) preceding it. If it is an angle of elevation, precede it with 
 a plus sign ( + ). These will show whether the vertical distance is to be 
 added to, or subtracted from, the height of the preceding station. 
 
 Having the horizontal distance and the vertical angle: 
 
 Distance X tangent of vertical angle = vertical distance. 
 
 Having the pitch distance and vertical angle: 
 
 Distance X sine of vertical angle = vertical distance. 
 
 To Calculate the Horizontal Distance, or Latitude.— Pitch distance X cosine 
 of vertical angle = horizontal distance. . r 
 
 Vertical height, or departure h- tang.of vertical angle = horizontal distance. 
 
TRANSIT S UR VE YINQ. 
 
 51 
 
 Tv Calculate the Pitch Distance.— Horizontal distance — cosine of bearing, 
 or multiplied by secant of bearing = pitch distance. 
 
 Vertical distance -i- sine of vertical angle, or multiplied by cosecant of 
 bearing = pitch distance. 
 
 To Calculate the Vertical Angle.— The horizontal distance -- the pitch 
 distance = cosine of vertical angle. 
 
 Vertical distance -5- pitch distance = sine of vertical angle. 
 
 Vertical distance -4- horizontal distance = tangent of vertical angle. 
 
 Note.— W henever sines, cosines, tangents, etc. are here named, they mean 
 the natural sines, etc. of the angle. . 
 
 Plotting by Coordinates.— In describing the establishment of a meridian 
 and a fixed point, we made the latter a stone post, or iron plug sunk in. solid 
 rock. This point is called the origin of coordinates. We have the principal 
 meridian passing through this point in an exact north and south direction, 
 and a secondary meridian or base line passing through this point at right 
 angles to the first, or in an exact east and west line. Any point we may 
 select on the map will be a certain distance north or south, and east or west of 
 the origin. The lines drawn from this point at right angles to the two base 
 lines just given are called the coordinates of that jpoint, and we can plot the 
 point when they are given. For example, the coordinates of Station 24 are 
 North 845.67, and East 890.12. We measure 890.12 ft. east of the origin on the 
 secondary meridian and, from this point, measure 345:67 ft. north to the point 
 desired; or we can measure first on the primary meridian to the north and 
 then turn off a right angle to the east and reach the same point. In any 
 event we plot the position ol' each station independently of all the others, 
 and any error in locating one is not carried to the next. When two stations 
 are plotted, the distance between them on the map should be exactly what 
 we found for their horizontal distance on the ground. This check shows 
 whether our plotting is correct. This is also called traversing a survey if the 
 meridian be north and south, and in books on surveying there are printed 
 traverse tables , which are accurate within certain limits, but not so accurate 
 as the tables of coordinates published separately, as the latter are carried to a 
 greater number of decimals. Gordon’s Traverse Tables will enable you to 
 find, without calculation, the coordinates for a distance of 12 miles with a 
 chance of error of only half an inch, which is much more accurate than the 
 graduation of the instruments with which the work was done. 
 
 With a north and south meridian, the point from which we begin to 
 measure angles— the zero point— is the north point, and the angles are read 
 for continuous vernier in the direction of the hands of a watch. The sines 
 of angles are eastings and westings, and the cosines are northings and 
 southings. 
 
 To Traverse a Survey.— To traverse a survey, means to determine by calcu- 
 lation how far north or south and east or west any station may be from 
 another, the location of which is fixed. To do this, all distances must be 
 either measured horizontally, or calculated to horizontal distances. The 
 horizontal angles, or courses, must be either read as quadrant courses, or 
 reduced from azimuth to quadrant courses. An azimuth course is one that 
 is read on the transit which is graduated from 0° to 360°. A quadrant course 
 is one read in the quadrant of the circle, as S 67° W, N 43° E, etc. 
 
 Latitude means distance north or south, and is determined by the first 
 initial of the recorded course. Thus, if a course is S 67° W, the latitude is 
 south; if N 43° E, the latitude is north. 
 
 Departure means distance east or west, and is determined by the last 
 initial of the recorded course. Thus, if a course is S 67° W, the departure is 
 west; if N 43° E, the departure is east. 
 
 The latitude = distance X cosine of bearing. 
 
 The departure = distance X sine of bearing. 
 
 If the survey is a continuous one around a tract, and ending at the 
 place of beginning, the sum of the northings should equal the sum of 
 the southings, and the sum of the eastings should bqual the sum of the 
 westings. Or, in other words, the sum of all the latitudes north, should 
 equal the sum of all the latitudes south; and the sum of all the departures 
 east, should equal the sum of all the departures west. It is evident that by 
 coming back to the place of beginning the surveyor has traveled the same 
 distance north as he has south, and the same distance east as he has west. 
 
 The most accurate way to construct a map is to traverse the survey and 
 place all stations on it by the traversed distances, or to at least put a 
 number of the principal stations on the map by the traversed distances, and 
 
52 
 
 transit sur veying. 
 
 use the protractor to 
 
 the survey is at the fixed f ^ of this point from the 
 
 pencil sketch to find the a PPJP^i^ enera f trend of the property itself, 
 boundaries of the property , ai t t f 1 ? e g or “|j„ upon the paper so that all of 
 This will show us the place to 0 f d P about the same amount of 
 
 the property can be placed on tne map, anu north and souih 
 
 margin on all sides. It will also ^° h Ts^ settled mXthe origin by a needle 
 line must take on the paper When this f tueQ^ mar s 
 
 point, and lay the straightedge across it «) a quite hard pencil 
 
 principal meridian and draw Then lav off on both sides of the origin 
 brought to a very hue P 0 *™ • t h needle points. These must he so 
 
 distances of 5 in., and mark the an err() P of one hundredth of an 
 
 accurately located that there' ^^^dred At the point where we can get 
 inch in them, or °ne ^ot m f angles to the principal meridian, lay 
 
 the longest line on the paper at right Qf the meridian, and 
 off points for a right angle ac £ ara y f the straightedge, a lme parallel 
 draw through the three th^ accurately into 5" distances as 
 
 to the secondary meridian an^ lines at right angles 
 
 before. Through each ot the points thus manw , ,1 squares 
 
 to the lines already drawn, until the paper ^ accurately ^ flvedlulldred th. 
 5 in. on a side, and n0 .n e o^hem extremities ofthe lines passing through 
 
 Beginning with the origin mark the side 0 f the north and 
 
 it zero. All distances to the east o P tQ that p ne; those to the left are 
 south zero line are marked + P ii ne are marked + 
 
 marked - . A1 distances above : the east anawes^ _ If the coordinates of 
 with respect to that line, and a d t measur e the whole east 
 
 Stations. 
 
 , Quadrant 
 Courses. 
 
 1 
 
 Distances. 
 
 Latitudes. 
 
 Departures. 
 
 
 S 
 
 E 
 
 W 
 
 1-2 
 
 2-3 
 
 a-4 
 
 4-5 
 
 N35°E 
 N 83° 30' E 
 S 57° E 
 S 34° 15' W 
 
 270 
 
 129 
 
 222 
 
 355 
 
 221 
 
 15 
 
 121 
 
 293 
 
 155 
 
 128 
 
 186 
 
 200 
 
 1 236 
 
 414 
 
 469 
 
 200 
 
 Totals. 
 
 N 
 
 S 
 
 E 
 
 W 
 
 221 
 
 
 155 
 
 
 236 
 
 
 283 
 
 
 115 
 
 
 469 
 
 
 
 178 
 
 269 
 
 
 for latitudes and 
 
 ine iorcgumg is 2 21 ft north and 155 it. easx oi oiauuu a, 
 
 by the rule given under tPe heacl tie diyideit int0 triangles and 
 same number of stdes. If a^ nregi these area s will be the area of 
 
 ‘ K 
 
 wittfthe greatest ^arej owing 0 to^iaWUty to error through very slight mac- 
 
 curacies of measurement. Tract.— If the seam lies flat, 
 
 To Find the Contents of a Seam ofCoal Under* ^ c thickness of th e seam in 
 multiply the area of the ^act “ square leer oy i seam in feet . if the 
 
 slam is -- «nd Us area® by measuring the width of the tract on 
 
 its line 
 the hor 
 This w: 
 
 byft^hiAnessTvill give thb contente.^ . p feet x gp Gr- x 62 .5 
 
 feet. The product will ° e “f I 1 ' me^uring the width of the tract on 
 
 seam is an inclined one, hnd ^its 1 area .by mea ^ of the seam by dividing 
 
 its line of pitch and find the distance on tne p ^ the angle of inclination, 
 the horizontal distance measured by e ^°sine the p i tc h distance by the 
 strict Mid you h whl have the area’o? the seam. This multiplied 
 
 icufii'u • 4-v» a nAtitonrs . ^ 
 
 Tons of coal = 
 
 2,240 
 
LEVELING. 
 
 63 
 
 LEVELING. 
 
 Instruments.— But two instruments are used— the level and a leveling rod. 
 The level consists of a telescope to which is fitted, on the under side, a long 
 level tube. The telescope rests in a Y at each end of a revolving bar, which is 
 attached to a tripod head very similar to that used tor a transit. The 
 telescope is similar to the telescope of a transit. The leveling rod is 
 merely a straight bar of wood, 6 ft. or more in length, divided into feet and 
 tenths of a foot. A target divided into four equal parts by two lines, one 
 parallel with the staff, and the other at right angles to it, and painted red 
 and white, so as to make it prominent at a distance, slides on the rod and is 
 provided with a clamp screw. The center of the target is cut out and a 
 vernier, graduated decimally, is set in, which enables the rodman to read as 
 close as of a foot. 
 
 If a long rod is required, it is made of two sliding bars, which, when 
 closed, are similar to a single rod, as described above. When used at points 
 where it is necessary to shove the target to a greater height than 6 or 6£ ft., 
 the target is clamped at the highest graduation on the front of the rod, and 
 the rod is extended by pushing up the back part, which carries the target 
 with it. The readings, in this case, are made either from the vernier on a 
 graduated side, or a vernier on the back. The rodman must always hold his 
 rod perfectly plumb or perpendicular. 
 
 To Adjust the Level.— The proper care and adjustment of the level is of great 
 importance. A very slight error in adjustment will completely destroy the 
 utility of any work done. 
 
 To Adjust the Line of Collimation. — Set the tripod firmly, remove the Y pins 
 from the clips, so as to allow the telescope to turn freely, clamp the instru- 
 ment to the tripod head, and, by the leveling and tangent screws, bring 
 either of the wires upon a clearly marked edge of some object, distant from 
 100 ft. to 500 ft. 
 
 Then with the hand, carefully turn the telescope half way around, so that 
 the same wire is compared with the object assumed. . 
 
 Should it be found above or below, bring it half way back by moving the 
 capstan-headed screws at right angles to it, remembering always the invert- 
 ing property of the eyepiece; now bring the wire again upon the object, and 
 repeat the first pperation until it will reverse correctly. Proceed in the 
 same manner with the other wire until the adjustment is completed. 
 Should both wires be much out, it will be well to bring them nearly correct 
 before either is entirely adjusted. 
 
 To Adjust the Level Bubble.— Clamp the instrument over either pair of 
 leveling screws, and bring the bubble into the center of the tube. 
 
 Now turn the telescope in the wyes, so as to bring the level tube on either 
 side of the center of the bar. Should the bubble run to the end, it would 
 show that the vertical plane, passing through the center of the bubble, was 
 not parallel to that drawn through the axis of the telescope rings. To 
 rectify the error, bring it by estimation half way back, with the capstan- 
 headed screws, which are set in either side of the level holder, placed 
 usually at the object end of the tube. Again bring the level tube over the 
 center of the bar, and adjust the bubble in the center, turn the level to 
 either side, and, if necessary, repeat the correction until the bubble will 
 keep its position, when the tube is turned half an inch or more to either side 
 of the center of the bar. The necessity for this operation, arises from the 
 fact that when the telescope is reversed, end for end, in the wyes in the 
 other and principal adjustment of the bubble, we are not certain of placing 
 the level tube in the same vertical plane, and, therefore, it would be almost 
 impossible to effect the adjustment without a lateral correction. 
 
 Having now, in a great measure, removed the preparatory difficulties, we 
 proceed to make the level tube parallel with the bearings of the Y rings. To 
 do this, bring the bubble into the center with the leveling screws, and then, 
 without jarring the instrument, take the telescope out of the wyes and 
 reverse it end for end. Should the bubble run to either end, lower that end, 
 or, what is equivalent, raise the other by turning the small adjusting nuts, 
 on one end of the level, until, by estimation, half the correction is made; 
 again bring the bubble into the center and repeat the whole operation, until 
 the reversion can be made without causing any change in the bubble. It 
 would be well to test the lateral adjustment, and make such correction as 
 may be necessary in that, before the horizontal adjustment is entirely 
 completed. 
 
54 
 
 transit s ur verm 
 
 To Adjust the «... ,-Ha^ng - ef 
 
 now to descnbe that of . t0 thevertical axis, so that the bubble 
 
 level into a ^ entire revolution ’of the instrument, 
 
 will remain m the ^"‘rt.iihe hrectly over the center of the bar, and clamp 
 To do this, bring the level tube directly o v before over two of the leveling 
 
 the telescope hrm t 1 >7? n eket leve?^ the fubbie, and turn the instrument halt 
 screws, unclamp .the wcket level^ne dudd ' tbe ^ me position with respect 
 
 way around, so that the ievei ba > the bubble run to either end, bring 
 to the leveling beneath^ Should^the wome ^ bar; now move the 
 
 ill^cop^vefthe^ther sit of level! ^^^-ews, bring t^hbteagin into 
 
 If^crewf ’succes’Sv!?® until thl adjustment is very nearly perfected, when 
 
 brought precisely to its i ormer atuat • made and the soc kets well 
 
 completely adjusted, ! f the 11 ^men P P bubble wlll reverse over each 
 
 fitted to each other and tne tnpou . . b unable to make it 
 
 pair of screws m auy^m- f^ond the engmee c y t e 
 
 S" “ Sm prob.il, b.' 1. Ur. imperfecttuu «, rh. 
 
 interior spindle. __ tQ of the i eve l have been effected, and the bubble 
 After the adjustments oi -tne socket, the engineer should 
 
 remains in the center m an P n( j sighting upon the end of the 
 
 carefully tmn the telescope he w> ^ Jong each side of the wye, 
 
 level, which has the Q ^ r ri v z ®^ { ! 1 ^possible Whin this has been secured, 
 
 make the tube as n^lye^c^asposb vertieal e(Jge of a building, 
 
 he may observe, through the teiesc p , he should loosen two of 
 
 and the adjustments of the level will 
 
 be complete. ^otrnmpnt the legs must be set firmly 
 
 To Use the Levf .-^en usmg the ms^ment, me ^gs^ ^ be 
 
 into the ground, and neither r the > na ^ ^ brought over each pair of 
 
 g££ sting the w d il“c^y to tJ*%& the object distinctly in view, 
 
 SO that all errors of P a ^^. x t ?i ay 1 ^^fln^bserver is moved to either side of 
 This error £*&**%* telescope in which the foci of the object and 
 the center of 1£e <7^®°® Brlciselfupon the cross-wires and object; m such 
 eyeglasses are not brought precisely up surface, and the observation 
 
 a case the wires will appear to n » Te 5™ Les the wires and object should be 
 will be liable to inaccuracy. spider lines will appear to be fastened 
 
 brought toto view so i^rfe^y that the £ P to™ hoW ever the eye is moved 
 to the surface, and will lernain m i n fi ly set in the tripod head as 
 If the socket of the 11 s rument ■becomes so nrmij e shou f d place the 
 to be difficult .of remove tor l the ^ binary way toe e „ aud a 
 
 ^eS u"^ ^ to the blr, Scire also to hold his hands so as to 
 
 grasp it the moment it is free. carefully made and vertical angles 
 
 Field Work. If the survey has heen camuiiy ^ ^ where extreme 
 taken at every sight, leveling will oe nece * In mos t cases of 
 
 accuracy in regard to “eta^ng thickness of strata, 
 practical work at ^ ide r p ad etc., the elevations calculated by the 
 
 general rise or fall of j an made roa . , - Q ugh, but there are frequently 
 use of the vertical angle will be close e g g ^ ccegs in certA in work. In 
 instances when leveling ; must ^donet transit telescope is supplied 
 
 ihs note book ruled, the levelman » 
 
 ready to proceed with the work. 
 
FIELD WORK. 
 
 55 
 
 The rodman holds the rod on the starting point, the elevation of which is 
 either known or assumed. The levelman sets up his instrument somewhere 
 in the direction in which he is going, but not necessarily, or usually, in the 
 precise line. He then sights to the rod and notes the reading as a backsight 
 or +■ (plus) sight, entering it in the proper column of his note book, and 
 adding it to the elevation of the starting point as the “ height of instrument.” 
 The rodman then goes ahead about the same distance, sets his rod on some 
 well defined and solid point, and the levelman sights again to the target, 
 which the rodman moves up or down the rod till it is exactly bisected by the 
 horizontal cross-hair in the telescope, as he did when giving the backsight. 
 This reading is noted as a foresight or — (minus) sight. The foresight 
 subtracted from the height of instrument gives the elevation of the second 
 station. The rodman holds this latter point, and the levelman goes ahead 
 any convenient distance, backsights to the rod, and proceeds as before. In 
 this case we have assumed that levels are only being taken between regular 
 stations or two extreme points. 
 
 If a number of points in close proximity to each other are to be taken, 
 the rodman, after giving the backsight, hoids his rod at each point desired. 
 The readings of any number in convenient sighting distance are taken and 
 recorded as foresights, and any descriptive notes are made in the column of 
 remarks. These are each subtracted from the height of instrument, and the 
 elevation found is noted in column headed elevation. After all the inter- 
 mediate points are taken, the rodman goes ahead to some well-defined point, 
 which is called a “turning point” (T. P.) in the notes. The elevation of 
 this is found and recorded. The rodman remains at this point until the 
 levelman goes ahead, sets up and takes a. backsight. This backsight reading, 
 added to the elevation of the turning point, gives a new height of instrument 
 from which to subtract new foresights, and thus obtain the elevation of the 
 next set of points sighted to. 
 
 When running levels over a long line, the levelman should set frequent 
 “bench marks.” These are any permanent well-defined marks that can be 
 readily found and identified at any future time. By leveling to them he has 
 secured the elevation of points from which to start any subsequent levels 
 that may be necessary. A good bench mark can always be made on the side 
 or root of a large tree or stump by chopping it away so as to leave a wedge- 
 shaped projection with the point up. Drive a nail in the highest point of 
 this, to mark where the rod was held, and blaze the tree or stump above the 
 bench mark. In this blaze, either cut or paint the number of the bench 
 mark, which should, of course, correspond with the number in the note 
 book. In the mines, prominent frogs or castings in the main roads, if 
 permanent, make good bench marks. 
 
 Level Notes. 
 
 Station. 
 
 B. S. 
 
 i 
 
 F. S. 
 
 H. Inst. 
 
 Elev. 
 
 Remarks. 
 
 1 
 
 
 
 
 100. 
 
 Assumed elevation of Station 1. 
 
 
 3.412 
 
 
 103.412 
 
 
 
 2 
 
 
 4.082 
 
 
 99.33 
 
 Station 2 of survev. See page 
 
 
 
 
 
 
 Vol 
 
 
 
 6.791 
 
 
 96.621 
 
 Sight taken to ground at N. E. cor. 
 
 
 
 
 
 
 John Smith’s house. 
 
 3 = T. P. 
 
 
 4.862 
 
 
 98.55 
 
 Station 3 of survey noted above. 
 
 
 11.698 
 
 
 110.248 
 
 
 
 4 
 
 
 9.817 
 
 
 100.431 
 
 Station 4 of survey noted above. 
 
 B. M.l 
 
 
 6.311 
 
 
 103.937 
 
 B. M. 1 is on north side of large 
 
 
 
 
 
 
 white oak. 
 
 5 
 
 
 6.427 
 
 
 103.821 
 
 Station 5 of survey noted above. 
 
 In underground leveling, extreme care must be observed to record the 
 algebraic signs of the readings, which show whether the level rod was held 
 in its usual position, indicated by a + sign or the absence of any sign, or 
 upside down, indicated by the — sign. 
 
 Proof of Calculations.— The calculations are proven by adding together 
 the backsights and also the foresights taken to turning points and last 
 
56 
 
 UNDERGROUND SURVEYING. 
 
 station. Their difference equals the difference of level between the starting 
 point and last station. Thus: 
 
 Foresights. Backsights. 
 
 4.862 3.412 
 
 61427 11-698 
 
 11.289 
 
 15.110 
 
 11.289 
 
 a 821 = 108.821 — 100.0 or 3.821. 
 
 TRIGONOMETRIC LEVELING. 
 
 for the leveling of mine slopes and 
 pitching rooms where the Y level can- 
 not be used with any advantage or 
 accuracy. By reading the angles and 
 by checking the measurements a very 
 high degree of accuracy can be ob- 
 tained in trigonometric leveling. 
 
 Case 1. Assume the elevation of A 
 to be 100 ft. A. T. With the transit set 
 up over A and properly leveled, sight 
 to a point C on a rod so that B C 
 equals A D. Measure the vertical angle 
 
 ■I AQ I Q 
 
 Case 2. Assume the elevation of 
 station A in the -roof of a mine to be 
 100 ft A. T. Then with the transit 
 set up directly under A and properly 
 leveled sight to a point C upon the 
 plumb-line suspended from the station 
 B, measure the vertical angle X, m- 
 dined distance D C, and roof distance 
 B C From this, the distance C F — 
 n CY sin X. The elevation of B is 
 then found as follows: The elevation 
 of B = the elevation of A AJJ-t 
 
 diagrams the most complex modifi- 
 
 cations can be worked out. 
 
 Trigonometric Level Notes. 
 
 Station. 
 
 Vertical 1 
 Angle. 
 
 Inclined 
 
 Distance. 
 
 Vertical 
 
 Distance. 
 
 Height of 
 Instrument. 
 
 Roof 
 
 Distance. 
 
 Eleva- 
 
 tion. 
 
 A 
 
 A- B 
 B - C 
 C -D 
 D— E 
 
 H — h 1 1 
 
 100 
 
 100 
 
 100 
 
 100 
 
 +8.72 
 
 +3.49 
 
 —5.23 
 
 -6.98 
 
 2' 
 
 3' 
 
 4' 
 
 2' 
 
 3' 
 
 2' 
 
 3' 
 
 V 
 
 +100 
 
 109.72 
 
 112/21 
 
 105.98 
 
 98.00 
 
 UNDERGROUND SURVEYING. 
 
 been grouped together for convenience of reference. 
 
STATIONS. 
 
 57 
 
 The Establishment of Stations.— As this is the most important duty of an 
 engineer in surface work, so it takes the first place in work underground, as 
 the accuracy of the work depends on the location of the stations, while its 
 rapidity depends on using the fewest number consistent with completeness. 
 It also stands to reason that the fewer the numbe’r of stations, the fewer the 
 chances of error. In underground work, stations should be located under 
 the conditions of permanence, freedom from destroying agencies, and ease of 
 access. Temporary stations for a single sight need not fill all these require- 
 ments. We establish them generally in the roof of the mine— less frequently 
 in the floor. In the former case we must establish a “ center” before each 
 set-up of the transit, and thus underground differs from surface work. The 
 first surveys were made with lamps set on the floor, sighted to, and then set 
 over. Permanence was secured by driving iron nails or tacks in the sills of 
 the track or sets of timber. As acid water soon destroyed these, they were 
 followed by copper tacks or brads, and all were witnessed by notches cut on 
 both sides of the sill, as in outside work, and by a vertical paint mark on the 
 solid wall, with the number of the station. This method is faulty, as the 
 tracks in crooked gangways are seldom placed where one can get the longest 
 sight, and, as they are the traveling ways, the stations run the chance of 
 being knocked out by passing men or mules, and the whole track, on a curved 
 incline, is generally sprung by every loaded trip. As the sights must be as 
 long as carefulness of work will allow, we put them generally in the roof, as 
 that offers the greatest area for a choice, and is not under foot. Any settling 
 of the roof so as to affect the accuracy of the station would be equally effect- 
 ive in destroying the accuracy of a station in the floor. We therefore choose 
 places that will be least affected by subsequent work, and put the stations in 
 collars, lids or wedges of props, in the props themselves, when they have 
 incline sufficient to allow the transit to be set under them, or in the roof 
 itself. Wherever set, they should not project far from the surface, and thus 
 be liable to be brushed away in a low gangway by cars with topping higher 
 than usual, or knocked away by flying fragments from a shot, if near the 
 working faces. Top stations have a mark about them to call attention to 
 their location. It is generally a circle, unless there are other corps at work 
 in the same mine that use the circle, and the stations of the two surveys 
 would be confused if marked alike. In this case a corps selects some easily 
 made figure, as a triangle, square, etc. If two surveys use the same station, 
 the mark of the second survey is placed around that of the first, and the 
 “ Remarks ” give “ Station No. 234 of L. & S. corps,” etc. 
 
 Kinds of Stations. — The simplest top station is a shallow conical hole, made 
 with the point of the foresight man’s hatchet, which is dug into the top rock 
 and rotated, and is called by some a jigger station. Corps using these entirely 
 have a jigger consisting of a steel-pointed extension rod, with an offset hold- 
 ing a paint brush. The rod is long enough to allow the point to be driven 
 into the roof at any height, and its rotation marks a circle with the brush, 
 which is also used to mark the number beside it. Centers are set under such 
 stations and sights are given by another tool— also called a jigger. This is an 
 extension rod, beyond the upper end of which projects a piece of sheet iron 
 shaped like an isosceles triangle, with the upper and smaller angle cut off so 
 as to form an end one-quarter of an inch broad, and in this end is cut a 
 U-shaped groove. 
 
 The sights are given and the “ centers ” set by putting the plummet cord 
 in this groove, and placing the end in the “jigger hole” in the roof. The 
 cord must be more than twice the length of the section of the place, as it 
 must be held in the hand, run over the jigger notch, and hang vertically to 
 the plummet, which must come to the floor when the stations are set. The 
 rod and cord are held in the left hand, and the right is free to steady the 
 “bob,” give sight, or set the center. The advantage of this method lies in 
 the quickness with which -the centers are set and the sights given, and the 
 ease with which the highest stations are reached. The disadvantages are 
 the impossibility of making the jigger hole perfectly conical, so that the jig- 
 ger can be set in the same place on two successive sights, and the plummet 
 cord will hang exactly in the same place. 
 
 Second.— Common shingle nails are driven into collars, or cracks in the 
 roof. The end of the plummet line is noosed and put over the head. This 
 causes an eccentric hanging of the plummet that may cause an error in back- 
 sight and foresight of the width of the nail head, which will be quite appreci- 
 able in a short sight. To do away with this error, a variety of nails ( called spads , 
 spuds , etc.) are made of iron or copper. Iron will not corrode in dry mines, 
 
58 
 
 UNDERGRO UND S UR VE YING. 
 
 k rrmoli cheaper The simplest is made by hammering out the head of a 
 horsesSe or mule^shoe nail, punching a hole in the flattened head for insert- 
 ing the cord and cutting off the point, so as to make the finished spad an 
 inch Ion 0- This will bring, the head near the surface without having to drill 
 toodeena hole and will make them unfit for lamp picks as they are very 
 handv for such purposes, and thousands have been pulled out to this end. 
 Anv blacksmith can furnish them for less than 1 cent each. They are 
 driven broadside to the line of sight, or they will be liable to the same objec- 
 tion as the shingle nail. To remove all chance of eccentricity, a form is 
 made with a shoulder in which a hole is drilled parallel to the length of the 
 nail The practice of using staples for stations is antiquated though gi\ en 
 iii the last editions of soml modern textbooks-and should never be used 
 
 wh ^, a d C ^In : SSta Of spads are driven into a crack of the roof; but 
 such stations cannot be called permanent , as the same force that made the 
 crack will tend to open it and let the nail drop. Even if this does not hap- 
 pen we shall have the water in a wet mine coming in by these cracks, and 
 rotting 6 the nails, or the rock at the sides of the crack, and m a month after 
 tVip nlnoixiff of tli6 st&tion. it will b6 unfit for usg* . , 
 
 Fourth -Into a hole drilled in the roof, a wooden plug is driven, and into 
 thiswe drive the spad. The swelling wood clamps the same and prevents 
 i ccnfong out as readily as it was put in. The plugs are made of well-dried 
 and are carried bv the man that sets the stations. The first 
 holes were made by a jumper, and the plugs were 2 in. square and 6 m. long. 
 The modern holes are usually made by a f n 
 
 will do the work without bendin 
 slates or clays; but an ordinary 
 
 atthe shank. Such drills can be used in 
 „rill and hammer must be used in harder 
 
 U1 w^ fhe roof is too poOT to hJld them. Such stations should be checked in 
 they are used, if we wish to swear to the accuracy 
 
 in the qu^ndary^of tl^man^vhose^^mpanyi worked ac^ossthSr liSe beSuse 
 
 wood shoepeg is ci . d A cor q w m S oon rot, and, if m the 
 
 plummet is tied to the lower for whip lashes , while the wire is 
 
 S5r!Te™anSi! bu?fven this will be pulled out by catching in the topping 
 
 0f spads is dispensed with, and all the stations put in rock 
 
 Lastly. a twist drill makes a vertical hole 1 m. deep, 
 
 roof where poss b . ♦ ta ^ en the foresight man puts a steel clip with 
 
 Int ° h ^?es Thf is made by bending upon itself a thin piece of steel 
 serrated edges. This is mane d to | ether it w m go into the hole, and 
 
 & in. wide. -When the senas are ^esse^ -js hold the clip in the hole so that 
 the spring of the sides the ® through a hole in the center of the 
 
 it is hard to pul] out oord^passes tnmi *gu a _ no matter how the clip 
 
 bend and is, ^ d pressing together the ends of the clip. This is 
 
 is inserted. ]t remov ea uy p | in | as the re is no eyehole to be freed 
 
 the easiest and Q nc ] e . s V a L d ld un ti e d The hangingof the plummet 
 from dirt and no knot to be tied and manea. f n a | long as F the roof 
 
 takes a fraction of !a ^^ n a Vthe pXng of the holes inclined .to the 
 
 vcTc Jbyl careleS mamand the many roofs that are unfit for pierc.ng 
 
 with a twist drill ho , l]d hav e some regular way pf witnessing our 
 
 Marking St • vertical line on the rib calls attention to a station m 
 stations. I" general, ^vert o f station has the mark around it. as has 
 
 the floor newthe sirte marKea. * jc fl e If three regular corps are 
 been described a ^it ^s som g the same mine s, a s the company 
 
 engaged m the ^“.tgi^ividual operator, and the private corps that is 
 corps, the : corps i of : the nm p of the land owners, they must use 
 
 The most common are the circle, square, and 
 
STATIONS. 
 
 59 
 
 triangle. If the “ circle ” corps puts in the station, it has a circle about it. 
 The next corps uses it and puts a square about it and notes “ Sta. 472 = to 
 Sta. 742 of ( ) Corps.” The third corps uses it and puts a triangle about 
 the square, and notes “ Sta. 617 = to Sta. 472 of ( ) Corps, and Sta. 742 of ( ) 
 
 Corps*.” If the first corps uses the station again, it notes the numbers given 
 by the two other corps, and these three numbers will aid in identifying it if 
 one or two of the numbers are lost. 
 
 Distinguishing Stations.— Each station must be lettered or numbered so that 
 it can be readily recognized when the subsequent surveys are made. When 
 set it may have been^at the end of a gangway, while six months later 
 the gangway has been "driven hundreds of feet from that place, chambers 
 have been turned off in what was solid, and the place be so utterly unlike 
 its former state that nothing but a fixed mark belonging to that station 
 alone will enable us to recognize it. The methods of distinguishing stations 
 vary widely. In one place the writer found that each gangway and room 
 had a Station 1 at its beginning, and the various stations numbered 1 were 
 designated “ Grog Run 1, 2, 3, etc.”; “ Pat James Gangway 1, 2, etc.”; and so 
 on through the map, that showed between fifty and one hundred stations 
 numbered 1, so that a new engineer would have had to learn the mine 
 thoroughly before he could extend a survey. Another way is to use Al, 
 A2, etc., up to A100, and so through the alphabet to avoid running up too high 
 in numbers. A third was lettering the various sections of the mine A, B, 
 C, etc., and the numbers begin with 1 in each and run up indefinitely. 
 
 All of the above have disadvantages, as powder or lamp smoke, mud, 
 mold, or the misplaced ingenuity of small boys may so obliterate or obscure 
 a mark that it will be recognized only by association with its immediate 
 neighbors, and these may have shared the same fate. You may have only a 
 part of the mine map with you, and because the system of marking strives to 
 get along with as few symbols as possible, you have to go to the office, when 
 there would have been a chance of deciphering the mark if there had been 
 a number of figures to it. The best practice, therefore, courts large numbers, 
 begins with Station 0 at the mouth of the slope or drift, or the foot of the 
 shaft, and numbers consecutively in each bed. In a short time three figures 
 are reached, while in old mines the numbers require four digits. The 
 chances of obscuring such a mark are lessened, while the chances of our 
 deciphering it are increased. 
 
 Centers.— When the station is in the roof, there must be something for the 
 transit to set over, as it is easier to do so than to set under a station, and much 
 more accurate as instruments are now made. The set-up is made over a 
 “ center.” At first, a cross scratched on the floor or on a loose piece of slate, 
 a daub of white lead on the same with a small piece of coal placed under 
 the point of the plummet — when that had been steadied— or finally, a nail 
 driven into a block and afterwards pointed, were used. All of these, except 
 where the mark was on the solid floor — if they were large, enough co be 
 stable— were in the way of the observer’s feet, while, if small, they were so 
 light as to be readily displaced. It must be noted here that it is not so much 
 the errors that we can foresee and detect that influence the accuracy of the 
 work in our own eyes, but the chances of error from accidents that we 
 cannot control and that cannot be readily detected. To avoid the above 
 chances, we make the centers as small and as heavy as we can— in other 
 words, we make them of lead. A hole If in. in diameter and i in. deep is 
 bored in a thick plank, a brad is set in its center with the head down; the 
 hole is filled with melted lead and the brad is slightly raised to surround the 
 head with lead, and held with pincers in a vertical position till the lead has 
 set. The brad is cut off I in. above the lead and pointed. This ’‘center” 
 combines weight and small size, and is generally used. 
 
 Paint.— White lead, or Dutch white, thinned with linseed oil, is ordinarily 
 used. It is carried in a covered tin pail holding a pint. The cover nas a 
 hole large enough to admit the brush. The pail generally has to be cleaned 
 out after each day’s work, as the brush gathers dirt every time it is used. In 
 case the paint is to be kept for a number of days, it must be covered with 
 water, which can be poured off before using. If the ordinary paint brush 
 has too long bristles, it can be shortened and kept from wearing by winding 
 with fine wire to the proper length. The top should be wiped clean and 
 dry with a piece of cotton waste before the paint is applied, or the white 
 will be so discolored as to be scarcely visible, or if the top is dirty it will 
 flake off, and the numbers be lost. 
 
60 
 
 SURVEY NOTES. 
 
 KEEPING NOTES. 
 
 Taking Note,— Complete no^e^shoul<^be^teten^a^d^recor^dneatly^i^ 
 
 Every ledge of rock should ibe noted , its “ a separate book for 
 transit note s^an'd for ^denotes, andwhere many collieries are operated, a 
 separate set of books should be f or ■ Jgc: ^Sterile' following things: The 
 However the to check the vernfer, the 
 
 numbers of the stations, the MJ 1 distance measured, either flat or on 
 
 vernier reading, the dip ot tne si gnu f the ground; the height of 
 
 the dip; the height ot the axis ot _thetramnt Necessary remarks to 
 
 l^eight'of Srumelt), H. R- (hci^hl of tod, or poml to which 
 
 sight was taken), .^^^^starting a survey, there should he entered the 
 At the top of . g tQ be done; the na mes of 
 
 name of the mme work and those that were taken from the 
 
 the regular corps employed f O“he wotk, ai used . the date of the work, 
 
 mine to point out work or assist; thyn^mmenlsuseo u e where such 
 
 and, in case it he the ° S uch hooks are^complete records, and 
 
 work was noted must , oe set -down Such boo«a e g P fg of the kind of 
 
 Xkdon1in a 4seTlawsuit “quires Inch testimony by showing the number 
 of men, the instruments "^“^hetam«eE^toy of keeping these 
 Transit and Side Not . es -~P e ^ a ^^ P thods arrange themselves into groups, 
 an^spec\mens of foS^oups will be shown, as the most common in use m 
 
 the ^j s T-The side notes of each sight follow the transit notes of that sight, 
 V»v the side notes in the same hook. 
 
 Y Fourth. — Each set of notes has a separate hook se ts o£ 
 
 The last method is *ke best-^en “ t X same time, such a method is 
 notes, and where two men .do • the ^^^g^ate^et of hooks for ordinary 
 imperative. Each mine should have a f t P ara "f y ;f the engineer reference 
 work and special work, and |^c a pract^c a | d if the party ma kes surveys 
 notes m a portable form. Unless tms ’ succee dmg surveys m any 
 
 in twenty different n^^es^ the no f t bQth must be carr i e d. This 
 
 ssars?a ssus.ss'." 
 
 looked up, and the -notes found. outside survey is secondary to the 
 
 The taking of side nnderiTund work, of ordinary character, the 
 
 instrumental work; while m unde , are built the s j de no tes. The 
 
 lines of the survey are r skeletons upo hich ^ the forms for taking them 
 side notes are therefore of the^ hi ig tbe underground work, so that 
 
 the'mapper* can 6 reproduce themkith fu 11 y even if he may never have been 
 
 inside. Qnrmose we are setting up at 6; with backsight at 
 
 „• &ht = ffiP 27' left; and that the drstance b e 
 
 is 421.76 ft. measured on a pitch ol + 4 50 . 
 
 First Form: 
 
 Sta. 
 
 Needle. 
 B. S. 
 
 S 25° 30' W 
 
 Vernier. 
 
 Needle. 
 F. S. 
 
 Pitch. 
 
 Dist. 
 
 Sta. 
 
 A 
 
 B 
 
 L 85° 27' 
 
 L 85° 26' 
 
 S 60° 0' E 
 
 + 4° 35' 
 
 421.76 
 
 c 
 
FORMS FOR NOTES . 
 
 61 
 
 Fig. 1. 
 
 differences of level as measured 
 filled in at the office, to make the 
 
 This is read: “ Set-up at 6; backsight at a; foresight to c; first reading of 
 vernier under A; second (check) reading under B; check reading by needle 
 computed from foresight and backsight needle, etc.” Some note the 
 readings of one vernier at A, and the opposite one at B, and take only 
 one sight. The last column for stations is sometimes omitted, and the first 
 widened, so that the three can be entered as fol- 
 lows: “o-6-c.” 
 
 Second Form .—' This differs from the first in 
 having but one column for the venier reading, 
 which is not noted until two readings agree, 
 and also in omitting the last column for sta- 
 tions, as noted above. In some cases the line 
 is indicated by but two stations, the one set up 
 at, and the foresight, as in the angle given 
 above, b-c To note the backsight, the previous 
 line is always taken, and under “Sta.” we put 
 “a-5,” and the needle reading. 
 
 Third Form . — In this case a continuous ver- 
 nier is carried and the readings are put in 
 the second column, with the needle course on 
 foresights as a check in the third. In the column 
 for stationsonly the station at which the set-up 
 is made is noted on the line with the readings 
 for that set-up — the backsight going on the pre- 
 vious, and the foresight on the following, lines. 
 
 Fourth Form . — This is also a form for record- 
 ing a continuous vernier, as well as the deflected 
 angle. The right-hand page is for noting 
 by level or transit. Certain columns are 
 
 book complete as a reference in mapping, or in the mine. This form is 
 advocated “/or its compactness but there is such a thing as too much of 
 that article, as there is no room on either page for remarks, while in all the 
 other systems, the right-hand page is set aside for this purpose. 
 
 Fifth Form .— Where the leveling is performed by the transit, and each 
 sight is taken with that end in view, the level notes are added to any form 
 of transit notes chosen and they are recorded as shown in table on page 56. 
 These figures are used to calculate the differences in elevation, as shown by 
 the pitch of the sight. The minus signs show that the points noted are 
 below the stations. If the station were in the 
 +5 * floor, the sight would have a plus sign. A contin- 
 
 uous vernier can be used with this form. 
 
 Forms for Side Notes.— In every case the notes 
 should convey to the man that plots some idea of 
 the form of the place surveyed. An accurate sketch 
 cannot be made unless the whole locality can be 
 seen at a glance— which is seldom, if ev6r, the case— 
 and yet we must not go to the other extreme and 
 write down the notes without a sketch; yet that is 
 what is frequently done, and may be simply noted 
 + $/ — 4 6 as the first form, and put aside as a faulty method 
 
 with no good points. 
 
 ♦ 72 — 5 7 Second Form — In this, see Fig. 1, as in a sketch 
 
 made as a person advances with no definite idea of 
 the arrangement of the work, there is too frequently 
 is a running of the sketch off the page— on one side or 
 the other, and a cramping of certain parts. Insert- 
 ing the figures on the line of survey confuses the 
 ;» one that plots if the sketch is distorted or cramped. 
 As the hands of the note man are dirty from rub- 
 bing along the tape, to find the numbers, it generally 
 happens that the sketches are smeared and blurred 
 ' so that they are hard to decipher when the notes 
 
 Fig. 2. are most clearly kept, and a method that encourages 
 
 cramping, confusion, or obscurity must be rejected. 
 
 Third Form . — There is no attempt made at sketching in this form, Fig. 2, 
 but the red line in the center of the page of the note book is taken as the line 
 of survey, and the next parallel lines on either side are taken as the boundaries 
 of the solid on either side. The only figures on each side of the red line are 
 
 O 238 [167. 
 + 157 — 
 + /S? 
 +Mo 
 4 I/O 
 
 + 30 
 + 20 
 
 + 16 — » 5 
 + 14 
 0,23 r 
 
 L 
 
 LJ 
 
 :n 
 
62 
 
 SURVEY NOTES. 
 
 thP distances from the line to the solid, while the “pluses” at which they 
 were taken are noted at the side of the page, and the exact distance betw eei q 
 ^ two stations is enclosed in the parallelogram. This method at the 
 pluses 155 and 157 calls attention to a point where ^ a ^ c ^ e v ? r ^ e 
 
 ' cans & tlmpUla^e^s to Sve|e 
 
 32SJ5J& .5 MX'SK 
 
 corner as ttefirst ortart ^*X>f the 
 
 ;^«id tm thi 
 
 whethe^^at a ei^ e he < 10^or 100* ftf dSant.^ onVcan 1 %™^d”°n™tes^ W 
 
 accurately taken, and two persons accurately plotting such notes will reach 
 
 th \«^ Notes —These are kept as in outside work, as has been before stated 
 
 iilssasssssa 
 
 §gE| gfekSSHSSSs^SE 
 
 proper degree. For rough measurements, the length of the winding rope 
 bet By on^of these ^ mShodsw^lS'cate a bench mark below, that is connected 
 
 ImUon^W^ must 1 bear In mind that roof stations are almost certain to 
 
 kn Whenever we begin a level survey, we must measure the distance between 
 roofand fkmr'and^e' it if agrees w& the notes . U» differs, we must note 
 the fact under the original notes, as a check for future work. 
 
 STOPE BOOKS. 
 
 By Joseph Barrell.* 
 
 Tn lare-e metal mines, where the veins are more or less vertical and great 
 volumes of SI are Extracted from between the levels, it becomes important 
 to fldoDt such a system for recording the shape and location of the stopes that 
 at any P future time the engineers may he. able to give precise information 
 
 coimeming them,^withou^ent e ^ng^tt^mine for^^m purjms^.^^ means for 
 
 sketches Udth the transit work of the drifts and also the various Boors with 
 one another othetical map of a portion of two levels Figs. 2, 3, and 4 
 ^Ksed^X^okl rilled %°t\e ^er^nically 
 
 or 
 
 #S ee “Mines and Minerals,” OctoDer, 1899 , page 97 , 
 
STOPE BOOKS. 
 
 63 
 
 fifth line to be heavier. Each square will represent a square set, giving an 
 actual scale of about 20 ft. to the inch. A smaller scale does not show enough 
 detail, and a larger one is not necessary for this class of work. 
 
 The most convenient size for the bound books is 11 in. long by 5i in. wide. 
 Only the right-hand leaves are 
 numbered, so that when open 
 a page extends entirely across 
 the book, 20 in., showing 400 ft. 
 of the length of the vein and 
 wide enough for two floors on 
 one page. The floors imme- 
 diately above each other must 
 follow on consecutive pages; 
 thus, on the first double page 
 will be 400 ft. of the sill floor 
 and first floor, on the second 
 page the second and third 
 floors, and on the seventh page 
 the twelfth and thirteenth 
 floors. The eighth page is 
 reserved for cross-sections of 
 the vein, and the ninth for the 
 long upright section, these two 
 being shown by Figs. 3 and 4. 
 
 The next 400 ft. of the same 
 drift will be shown on page 10, 
 
 so that it joins on to page 1 on one side and to page 19 on the other, and 
 in this way the work of one level is kept together. For convenience, the 
 book should be indexed by placing a projecting tag with the number of the 
 level on each page of the drift floor. 
 
 Having now a general idea of the arrangement of the work in the book, it 
 remains for us to determine, precisely, how to 
 place it and how to show the relation to the 
 transit surveys. First, it is necessary to have 
 some reference line on the vein. For this 
 purpose draw on the map, through the center 
 of the shaft, a line perpendicular to the strike 
 of the vein, as shown in Fig. 1. Scale off from 
 the map the distance at which it cuts the 
 transit course from the nearest station. Then, 
 
 by adding 
 
 the known distances between sta- 
 tions to this, we obtain the surveyed distance 
 of apy point on the drift from our reference 
 line. Select the middle or end of a page in 
 the stope book for the zero or reference line, 
 such that the drift will go most conveniently 
 in the book, and on the upper line locate the 
 survey points at their proper distances from it, 
 as in Fig. 2. The vertical location of any part 
 can be told when the dimensions of the tim- 
 bering are known. In this instance the drift 
 is 7 ft. 10 in. high, and each of the following 
 12 floors are 7 ft. 2 in. If the levels are exactly 
 100 ft. apart, the thirteenth floor will conse- 
 quently be 6 ft. 2 in. in height. 
 
 The Stope Book in the Mine.— Having indi- 
 cated in the office, by a regular system, the 
 place for everything that will be found in the 
 mine, the next step is to proceed to the details 
 of sketching. The location of the stations on 
 the top line of the drift-floor pages shows on 
 what vertical line they should lie in the 
 sketch; but the fact that each square must 
 be kept as representing a square set, and that 
 any or all of them may not be exactly to our 
 scale, will cause the location made in the mine to vary slightly from that 
 made in the office. Any such discrepancy must be taken upon the edge 
 ot the page. Therefore, proceed to the station nearest the center pf the pag© 
 
64 
 
 UNDERGROUND SURVEYING. 
 
 and locate it as nearly as possible under its position at the top, remembering 
 tha/t the sets of timber, no matter how irregular, must be represented as 
 Snares Now walk along the drift, watching the character of one side at the 
 lino of the floor sketching while walking, counting the sets, and indicating 
 the Posts bfdots of the plncil. Check up the number of sets on each station 
 and Continue to the end of the drift. Sketch the other side in coming back, 
 
 . , Aniinfing the sets a second time, from station to station, a further 
 check y is"d 8 n^n the wort°On the ’correctness of the sketches of the 
 
 dri Ha^inVhnihed h thedrilh'clfmb to theSr^floor and locate the set climbed 
 through the same distance east or west of the reference line as °n the drift 
 below g see Fig 2. Since the chutes and manways will ordinarily not step off 
 
 PTonnd floor against tke hanging wall. It is well to start with 10, since then, 
 
 t^cl^e'^^O'^is^^roi^lllan^toat^rtramines^e'Mmbering^nthe first 
 SOOT On the thfrteen* floor, in this instance, as shown in Fig. 3, the man- 
 wav is in row 19 In such a manner each floor is completely located with 
 reference to the drift below and ultimately with the transit survey. 
 
 The ease and rapidity of the work will depend on the character of the 
 mine. Much is gained by practice, the work ™t being 
 
 Knt q ruinsp for sketching being made ever> tittn or tenm sei, 01 wuwevei 
 Sere a Change in the character of the wall. In drifts such as are usually 
 ~ ® .ppr.™ q nfin t n q nm ft of sketching is a good day’s work. The sketch 
 Should be tekenat^some^ definite horizon, and that of the floor level is best. 
 Features of constant recurrence must be represented by conventional signs. 
 Features ot constant ^ in Fi % c enclosed by a square indicates a 
 
 chute passing through the floor; up means a ladder 
 up; M, a manway down. A full line indicates a 
 rock wall, a broken line, lagging; cross-hatching 
 represents filling; a dashed-and-dotted line, the 
 presumed limit of filled workings, etc. It is ot 
 importance to indicate irregularities m the tim- 
 bers. If it is a short set, write S within the square; 
 if a long set, L, giving the length if necessary. If 
 there should be an angle in the timbering so that 
 there may be a set more on one side of the dntt 
 than on the other, represent it by a wedge-shaped 
 opening, as shown, being the appearance ot the 
 drift if it were straightened out. In sketching, it is 
 essential to accuracy that the attention be held 
 to a few things at a time. On reaching the top ot 
 the raise the plan views are completed, each floor 
 having been sketched on the way up. On descend- 
 ing - turn to page 8 and sketch the cross-section of 
 +v»o rnflmvflv as shown in Fig. 3. Each set is still represented by a square, 
 althOTgh the sets are higher than wide, but since that fact is known it can 
 
 lea The stonebook is taken through the mine and brought up to date at the 
 
 monih the tot &p Should have a notch cut in it and the same indicated in 
 the Tki K i™„ Section -The view of the vein that will he of most general use, 
 
 Fig. 3. 
 
THE LONG SECTION. 
 
 65 
 
 mine or the office, but it is a little better to determine the limiting points in 
 the mine. The transit stations, chutes, and raises should, of course, be 
 located upon it as common points with the map, connecting the two. The 
 final long section is drawn by merely piecing together the several parts 
 from the stope book and drawing them horizontally and vertically to the 
 same scale. 
 
 Since a vein is quite an irregular surface, the question arises as to what 
 modification of a vertical projection will give the most accurate and con- 
 venient representation of it. That devised by Mr. A. A. Abbott, of which 
 the essential feature is the horizontal adjustment space left between the 
 levels, is shown in Fig. 4. At our reference line, in this case the projection 
 of the shaft upon the vein, the zero points of the levels are placed over each 
 other. But owing to the warpings in the vein, a raise, such as No. 201, 
 started, in this case, 40 ft. east, will not break through on the similar point 
 of the level above. The simplest way in which to allow for these discrep- 
 ancies is to draw the 
 levels more than their 
 true distance apart by 
 the width of an adjust- 
 ment space, lay out 
 each level at its true 
 length, and draw the 
 raises perpendicular to 
 them provided, of 
 course, that the raises 
 only step off along the 
 dip and not along the 
 strike. All these fea- 
 tures can be appreci- 
 ated by studying Figs. 
 
 1, 2, 3, and 4 in con- 
 nection with each 
 other. Lines corre- 
 sponding to the height 
 are ruled on the sides 
 of the drawing, and 
 the floor on which the 
 work is being done is 
 ascertained by means 
 of a parallel ruler. 
 
 Such a view approxi- 
 mates to a development 
 of the vein horizon- 
 tally, but vertically to 
 a projection , since the 
 vein is projected on a 
 series of vertical planes passing through the transit courses. The drifts are 
 shown at their true lengths, but the heights are the vertical distances and 
 not the lengths up the dip. 
 
 The long section will be brought up to date every 3 or 6 months, and the 
 portions of the veins extracted during the interval indicated by cross- 
 hatching or tinting. In the illustration, the ore bodies are cross-hatched to 
 bring them out more clearly, but to do this on the regular map would 
 involve erasures with every extension in the workings. In 1 the mine, a 
 hard and sharp drafting pencil will be used, but pencil markings in a book 
 constantly in use soon become faint and blurred. It is necessary to go 
 through the book at intervals and ink in everything with waterproof draw- 
 ing ink. Two colors can be used with advantage, red for all transit lines 
 and survey figures, and black for the stopes and those symbols relating to 
 them. If there are several splits to a vein, sometimes worked together and 
 sometimes not, the work on the different splits can be readily distinguished 
 on the long section by using red for the hanging-wall split, and blue for that 
 of the foot-wall. If old filling is taken out, such as ore, as frequently hap- 
 pens, the parts extracted can be cross-hatched in red, and thus a record of 
 both the first and second extractions preserved. 
 
 The value of these methods over mere sketches made without system lies 
 in their accuracy. Where the timbering is irregular, the accuracy of the 
 results depends largely upon the time and care taken in the work. 
 
66 
 
 UNDERGROUND SURVEYING. 
 
 MINE CORPS. 
 
 The method of dividing the work in an underground survey depends on 
 the size of the corps. We will therefore consider the work of each man, 
 in order to get the right number for the corps. The chief of the party 
 should be where he can do the most good, and where he can plan the 
 work for his subordinates. The principal point ot the survey is the 
 setting 1 of the stations so as to do the work thoroughly with the fewest 
 set-ups and thus diminish the chances of error in instrumental work. 
 The chief should locate the stations and add all the necessary signs 
 to show how the work is to be done. The transitman should not 
 have his attention distracted from his particular work by questions as to 
 procedure He should work untrammeled. The chief, therefore, should not 
 run transit. Upon this basis the ideal mine corps works, and such a corps 
 consists of at least four, and better five, men from the office .and three from 
 the mine. It is divided into two sections. The chief takes the men supplied 
 bv the mine— one or more of whom are acquainted with the work done since 
 the last survey— and locates the stations; the transitman follows with the 
 second section, to measure angles and distances. By this time the stations 
 are set and the chief takes his men after the transit party and gets the side 
 notes with a check measurement of the distances between stations. 
 
 Such a corps goes to the end of the former survey and identifies the last 
 two stations. The transitman prepares to set up at the last, while the chief 
 and partv goes as far as he can see the light from the last station, or to some 
 intermediate point from which one or more sights are to be taken. He then 
 stops and sends a man along each place where a sight must be taken, as long 
 as their lights are plainly seen from top to bottom where he is standing, and 
 over this place he marks a point for a station to be inserted, and generally 
 inserts it himself unless he be pushed by work, and must leave it for another 
 to do when he places a circle about the dot, places the number at the side, 
 and as many arrows as there are new stations, the longer arrow generally 
 pointing to the sight to be last taken and where the transit is to 
 y s be set up next. Leaving a backsight at the point just set, 
 
 “ he sets, successively, stations at the points where the 
 
 foresight men have stood, in the manner just described, 
 until he has covered the new work— the mine boss or some 
 intelligent miner going with him to give him an idea of 
 the “lay of the ground,” so that the work can be covered 
 with the fewest number of stations. Sometimes the chief 
 takes the side notes and measures the distances between 
 stations as fast as they are set. In a pitching place, a circu- 
 lar brass protractor with small plummet is hung at the 
 center of a stretched tape, to give the angle at which the 
 tape is held; this serves as a check to the measurements 
 of the transit party, which are taken as the basis of the 
 work, and the other measurements are solely as checks. 
 
 In flatwork, both measurements should coincide. 
 
 In a small corps, and where time is of little importance, the foresight 
 man puts in the stations ahead of the transit, and while he is so doing the 
 transitman takes the side notes. Sometimes the side notes are taken by the 
 Se man whill one of the party is taking the transit to the next station 
 and setting it up for the next sight. There are about as many variations 
 from these two methods as there are corps. . r . 
 
 The foresiaht man should be intelligent and active, as the amount of work 
 done in a day depends on his ability to keep ahead of the transitman. Some of 
 the latter are fast enough to keep two foresight men on the jump. His duty 
 is to set the center for the next set-up under the station, and also place the 
 tripod if three are used in the work, to give the sight and, in some corps, to 
 carry the front end of the tape and assist in taking the distance. In some 
 corps he also carries the bag with tools for setting stahons, so that he gen- 
 erally has a load that makes rapidity of movement difficult and anything 
 that will diminish the weight carried will tend to quicken the work. 
 
 The raDiditv with which good work is done varies considerably, but it 
 depends on the activity of transitman and foresightman, and a good corps 
 should have no trouble in making twelve set-ups an hour and taking 
 two or three sights from each set-up It vanes also with the distances 
 between stations. The saving of time should never be sought at the expense 
 of accuracy in the work; it is to be gained by rapidity of moving about, in 
 
SURVEYING METHODS. 
 
 67 
 
 setting transit, center, etc., and in hanging plummets to give sight. The 
 foresight man and backsight man should be in position to give sight before 
 the transitman is ready, so that he can turn his instrument on one or the 
 other and find them in position. The slowest parties were those that carried 
 empty powder kegs (in the days when loose powder was allowed inside) 
 for seats, and spent the greater part of the time sitting on them. 
 
 The backsight man has little to do inside, and to compensate for this, he 
 is the one that cleans and oils the tape, gets out new plummet strings, and 
 sees that the tools are ready for the next work, as soon as the corps gets to 
 the office. 
 
 The transitman cleans the transit, unless the corps has subordinates that 
 can be trusted with so delicate an instrument. The blackening from 
 sulphureted hydrogen is rubbed from the silvered surfaces with whiting, 
 and the oil or paint smears are removed with alcohol. Alcohol should be 
 used instead of water for cleaning the instruments, and especially the lenses, 
 which are wiped with jewelers’ cotton or soft chamois skin. 
 
 SURVEYING METHODS. 
 
 Outside Surveys.— We have spoken of the points necessary to include in 
 the survey outside, and how the base line is established. It remains to call 
 attention to several points that must be known before the surface plant can 
 be protected from settling, from the removal of the deposit below. The 
 exact location of all buildings, lakes, ponds, rivers, railroads, etc. is not 
 only necessary for the making of a correct map; it is necessary for the 
 determination of the amounts and location of the beds that must be left 
 untouched by the subsequent mining. Here must be mentioned an error 
 that generally governs the location of the retaining pillars to support the 
 above and prevent damages to themselves or to the mine. The settling of 
 the ground would make all bodies of- water leak into the mine, and also 
 destroy to a greater or less degree all surface plant, as well as throw out of 
 plumb all shafts or other openings for hoisting, if it did not close them 
 entirely. The usual custom is to extend vertical planes through the bound- 
 ary lines of such objects, and leave untouched all parts of the superincum- 
 bent beds embraced by those planes. This is accurate only when the strata 
 are horizontal or vertical, as beds settle normally to the planes of the strata 
 and not in a vertical line in case the open spaces are stowed. If the spaces 
 are left open, they are first filled by falls, and then the settling goes on 
 according to the above rule. No cut is necessary to show the method of 
 settling, and the place where the bed is to be left untouched may be found 
 as follows: Draw a vertical section through the point to be supported , and also 
 the underlying bed on the line of the dip of the bed— the section being accurately 
 drawn to any scale. Draw through the extremities of the object to be supported, 
 lines to the bed, which will make right angles with it. The space included will 
 give the dimension of the pillar measured along the dip of the bed, and the 
 dimensions of the object taken at right angles to the first plane will give the other 
 dimension of the pillar. 
 
 Inside Surveys.— As the beds of anthracite lie at all angles with the hori- 
 zontal plane, the methods of surveying them vary accordingly, and can be 
 divided into flat and pitching work. Flat work is where the beds have so 
 slight a dip that the cars can be drawn to the face of the room, and where 
 there is nothing to prevent a sight to that face from the gangway. The 
 variations in the methods of work in this case depend on the accuracy with 
 which the work must be performed, as, in some cases, the workings are 
 approaching the boundary line of the property, and the sides of the rooms 
 must be located accurately. In general, the rooms are driven at right angles 
 to the gangway, unless the dip is too great to haul a car to the face on that 
 line, when they are inclined to the gangway at an acute angle. The width 
 of the rooms in fiat work is generally uniform where the roof is good, but 
 where the roof is poor the entrance is harrowed for a short distance (to better 
 support the gangway) and then widened to the full width, or the whole is 
 driven to the limit narrow, and the side is robbed when the top is drawn, 
 and the whole room caves in. This last must be surveyed before the robbing 
 begins. 
 
 The most accurate method of survey is to run a line along the gangway 
 and put a station at the entrance of each room, whence a sight is taken 
 to the face. This may be varied by putting the stations at alternate rooms 
 
68 
 
 SURVEYING 
 
 and measuring through the cross-cuts to get the thickness of the pillars of 
 the intermediate rooms, or placing stations at every third room and measur- 
 ing the thickness of pillars and width of rooms that intervene; or, finally, by 
 running out the gangway with as few sights as possible and paying no atten- 
 tion to the positions of the rooms in setting stations, thence up to the last 
 room to the face, and back through the cross-cuts nearest the face to the 
 former work, where a tie is made. When opportunity offers, sights are made 
 from the face of the rooms to the stations in the gangway for immediate ties. 
 In case a gangwav and airway have been driven considerably ahead of the 
 rooms, it is always necessary to run lines out each and tie at the last cross- 
 cut. This must be done in every case where the gangway is approaching the 
 boundary line, or old workings that have been abandoned and are full 
 of water! In addition to this check the miners must keep bore holes 20 ft. 
 ahead in the line of the gangway, and every 20 ft. must drive others from 
 the corners of the heading at an angle of 30° with the line of the gangway. 
 In this way there will be no danger of running into “a house of water, as 
 
 the Cornish miners call it, if the survey be inaccurate. 
 
 Pitching Work.— When the bed pitches so that a car cannot be run to the 
 face and when there is a good deal of firedamp in the mine, it is generally 
 difficult to see from the gangway to the face, where the roof is good and the 
 room straight, as a buggy track or chute, or both— when the pitch is slight 
 fill up the room, and, where the pitch is great, the gangway pillars generally 
 run across the face, or there is a “battery” shutting off the bottom of the 
 room so that the face can be reached only by several sights. Where the 
 roof is poor, the obstructions are increased, as the rooms are driven narrower 
 or if wide, have center props and stowing m the center. If the coal is full of 
 slate or if the partings are thick, a large part of the room is taken up with 
 piles’ of “ gob,” and with a very poor roof the body of the room that has been 
 worked out is filled with the fallen roof, and the coal sent out through the 
 triangular man wavs, where it is almost impossible to take a sight. ^ 
 
 Work of this kind is surveyed by lines out gangway and back through the 
 faces of the rooms which are generally clear, even if the bodies of the rooms 
 Se filled with the fallen top. Where chance favors sights are taken to the 
 aangwav; but this very seldom happens, as the two lines are as effectually 
 seoarated as if in different mines. From the stations in the faces, lines are 
 run down the rooms as far as possible to get their direction and to locate 
 the cross-cuts The very worst case of all is where two l^eds are separated 
 hv a thin nartina of rock and the gangway is driven in the lower one alone, 
 roomr in ^ the upper one being worked by rock chutes into the rooms 
 below, or into the chutes from those rooms. This °f work is hard o 
 ventilate and to survev where the rooms above are ventilated by the air 
 lystem ohhe lower beds; but is readily mapped where there is an air system 
 
 f ° r niosL b Iurvev.s,-To diminish the chance of error and to furnish a ready 
 check the survey must be closed upon itself or some part of a former sur ye> 
 withe^ryuvllvenew stations. With good work the error close 
 Should not exceed 1', and the error m position shouMbe ^ectffied bv 
 Errors must not be “balanced they must be detect® d a id rectified Dy 
 running the line again, if they are not readily sejen *0™ must be 
 
 given. If an incorrect survey be balanced, each su ^doeot one m 
 altered to fit this ‘incorrect work, though it may aI ^ he cf 
 
 never know where our work really stands. It well, tbeT ^^ 
 the work in arc as soon as we make a close and before the party leaves me 
 place, as it is easy to rerun the work then. 
 
 CONNECTING OUTSIDE AND 
 
 INSIDE WORK THROUGH SHAFTS AND 
 
 SLOPES. 
 
 As the dip of the bed increases, it is less easy to make % connection and 
 the chances of accuracy diminish. In a survey, B. ^ 0 ^-^”f t fc aneu)ar 
 
 locates the station with regard to former work The greatest angular 
 
 accuracy for a given value of R is where T er^. ffn^ — O and ^ cosine 
 Angle. As the pitch or vertical angle of sight increases, the above cosine 
 
 diminishes until, at a vertical sight, R. Cos. vert.. Anf lko~+ [ u n mi difficulty 
 In the case of an adit level, or a slope of less than 45 , there ^ no diffic y 
 bevond the want of absolute rigidity m setting up the transit, and the 
 danger of moving it in going about it. The difficulty increases mo e p > 
 
SHAFTS AND SLOPES. 
 
 69 
 
 than does the pitch, and as R. Cos. Vert. Ang. diminishes, though R be fixed, 
 the chances of error increase. When the slope reaches 60° there is an 
 impracticability in running a line down a slone, as the lme of collimation of 
 the telescope strikes, the graduated limb of the instrument. We can use 
 a prismatic eyepiece and see up the slope; but cannot look down. As we 
 have assumed that it is unnecessary to use an additional telescope, we shall 
 have to run the line by intermediates. Set up at the bottom of the slope 
 where the longest sight up the same can be secured and backsight on a 
 station of the underground work; or set a backsight for the occasion (both 
 stations will afterward be connected with the work below). With the 
 prismatic eyepiece, sight up the slope on a line that will give the longest 
 sight and, at the same time, afford a good intermediate place to set up 
 the transit, as, on a pitch of 60° or more, it is absolutely necessary that the 
 legs of the transit should be set solidly (in holes in the floor, or between the 
 sills of the track) so that they will not be moved by subsequent walking 
 about it. By this method, all the sights will be taken from one side alone, 
 and the tripod legs can be shortened to make the sight possible without 
 building a standing place— if the man be short-legged. 
 
 Call this station A; at the foot of the slope locate B, where the transit can 
 be readily set up, and as far up the slope as we can see (this distance must be 
 at least 100 ft. ), and in a continuation of A B, locate C. Set up at B and take 
 foresight to C\ locate D under the same conditions that governed the placing 
 of B, and, in a continuation of the line B D, place E. Set up at with 
 foresight at E, and locate F and G as before. The survey is carried by the 
 intermediates B, D, F, etc., to the top, by a series of foresights to C, E, G, etc. 
 
 Shafts.— The term shaft in American coal-mining practice is applied only 
 to vertical openings, though in metal mining, both in this country and 
 abroad, it is also applied to highly inclined slopes. For such shafts, most of 
 the methods given in the textbooks are worthless, as they are for transit 
 work and R. Cos. Vert. Ang. in rare cases may be as great as 20 ft., while R 
 varies from 100 to 1,500 ft. Again, to sight down a shaft necessitates the 
 erection of a temporary (and therefore more or less unsteady) support for 
 the tripod of the transit, and the chances of variation in its position as we 
 stand on different sides of it are so great that we cannot feel sure that a 
 movement has not taken place that will vitiate the work. 
 
 In sighting up a shaft of greater depth than 100 ft., there is annoyance— if 
 not danger — from dripping water or the fall of more solid substances. In a 
 wet shaft the object glass is instantly covered with water, and a sight is 
 impossible. We must also have a platform to stand upon, and we cannot 
 feel sure that this will be perfectly rigid. From all these considerations the 
 methods with a transit are never used by engineers in the anthracite regions, 
 and the connections are made as follows: 
 
 When the bottom of the shaft can be reached by an adit or slope in a 
 roundabout route of such length as to render errors in measurement of dis- 
 tance of great importance, the angles are carried by a transit with as long 
 sights as possible, and no distances are measured, from a point on the surface 
 in the shaft to a point vertically below it in the mine. Sometimes the guide 
 of the cage is taken when it has been recently set, as the guides are plumbed 
 into position; but the better way is to suspend an iron plummet by a copper 
 wire: sink the former in a barrel of water so as to lessen the tendency to 
 swing on account of the pull upon the “ bob ” and wires from the air-cur- 
 rents, or falling drops in a wet shaft. The top of the barrel is covered with 
 two pieces of plank with a semicircular groove of 3 in. radius cut out of the 
 middle for the passage of the wire, to catch the substances whose fall upon 
 the water would cause waves. The heavier the plummet and the lighter 
 the wire, the less the tendency to swing. This wire can be sighted at by 
 parties above and below at the same time, and the swing can be bisected 
 to get the position of the wire. A number of sights that agree can be taken 
 as accurate. 
 
 When the shaft is the only way to get below from above, it must be 
 plumbed with two or more wires suspended as just described. With two 
 wires, they are so hung that an instrument can be set up below in a line 
 passing through them produced, and at a sufficient distance from them to 
 insure an accurate sight; with more wires, the station below can be located 
 at any point whence all the wires can be seen. 
 
 Case 1. — Two wires are used, which are located as far apart as possible. 
 Two pieces of scantling c d and ef, Fig. 1, are spiked across the opposite corners 
 of two compartments of a shaft to allow the cages to pass up and down 
 
70 
 
 SURVEYING. 
 
 e<cc 
 
 Fig. 1. 
 
 without interference. The station X is (roughly) located in a line through 
 the corners x, x and is connected with the outside survey. From this station 
 locate in the line Jm two spads for holding the wires of the plumh-bobs 
 These are driven up to the head in the scantlings m such a way that the line 
 of sight passes through the center of the holes m their heads. M.easure the 
 distance^ Xa and a b This completes the work of the survey aboveground. 
 
 The light copper wire is rolled upon a 
 reel, and one end is fastened to a light 
 plumb-bob to keep it free from coils or 
 kinks in descending. It can thus be 
 readily lowered without accident. 
 When at the bottom, the upper end 
 is fastened in the spad and the heavy 
 “bob” applied to the bottom and 
 placed in the empty barrel. The cages 
 are then run slowly up and down, 
 with an observer on each to see that the wires hang free from top to bot- 
 tom By this time the wire will have stretched so that it will be straight, 
 and if there be any slack, it is taken up, the barrel is filled with water, and 
 +he top boards put in piace. As a last check, measure the distance between 
 the wires below and see if it agrees with the distance above. 
 
 Lining inbelow a point Fon the line a &, make, a hole in the roof two 
 inches in diameter, and drive in a broad plug. Setting up transit under 
 Y we sight at the wires a and b alternately.. A number of methods for illumi- 
 nating the wires have been used, and are given in textbooks. The writer has 
 always found those depending on a sight of the wire across the flame of a 
 lamp the hardest to obtain, and concludes from experience that the method 
 of illuminating the wires for mine surveying is the best for this also. A large 
 white target is placed behind both wires and illuminated by a large lamp 
 with a reflector behind it. The wire stands out black against it and can be 
 followed across the target. As there is considerable distance between the 
 wires and as the transit is comparatively near them, there will be small 
 chauce of getting a sight of one, when the telescope is focused upon the 
 other aid fo the focus has to be set between them. This gives a hazy sight 
 at each but both are shown against the white background. m strong relief. 
 After the transit head is shifted so that the lipe of sight coincides. a PPr°-p- 
 mately with both, focus upon them alternately and see if the line the 
 
 swing of each. If so, the work is done; if not, the shifting of the transit 
 head g mu st follow till the end is attained. It frequently requires two or more 
 hours of steady observation to complete the work, and, when it seems as if 
 the proper point were secured, one of the wires will show by its swaying that 
 it has been deflected from the vertical by a peculiar slant of wind, and the 
 result obtained must be checked again. When you are through there is no 
 absolute certainty that the point you have marked is m the accurate 
 extension of the line ab at the surface. Having decided on the proper place, 
 vou drive a spad into the plug overhead; hang a plumb-bob to it, and see if 
 it be over the axis of the transit, as shown by the screw on the telescope. If 
 not drive the spad so that the point of the bob does so hang, and the station 
 F is said to be in the line a b. Measure Ya and the angles to any station of 
 the underground survey; the line a b is connected with the surveys at day- 
 light and below, and the plumb-bobs may be removed. . 
 
 Criticism of the Above Method.-l. As has been stated there is no absolute 
 certainty that the point Y is in the line a b prolonged, and this want of 
 eertaintv should not exist in so important a measurement. 
 
 2 The work must be performed by daylight, and the length of time 
 neeessarv to complete it makes it impossible to work the shaft for at least 
 half a day, and may cause annoyance to the operators or if you working 
 for a lessee, lead them to refuse to let you have the use of the shaft at tne 
 
 “TTma^beha^d ^obtafnhong sight below on any .line running 
 through the larger axis of the shaft. Any shorter line would give too shorta 
 base line and would increase the chances of error. To avoid these, another 
 
 me CA°SE S 2 — Fi g 11 ™ sho ws* the top set of timbers in a shaft of two hoisting 
 
 compartments down which it is desired to carry a known course or mend- 
 fa™ on the surface to the entry below. First find out which side of the 
 shaft is best adapted for setting up the transit, as the point to be marked m 
 the mines will be vertically under the point on the surface; consequently. 
 
SHAFTS AND SLOPES, 
 
 71 
 
 NORTH 
 
 the side with the widest opening leading from the foot of the shaft should 
 1)6 selected 
 
 Having carried the meridian to a convenient point near the top of the 
 shaft, and having found that the south side of the shaft is the most accessi- 
 ble, determine, with an ordinary string, the location of the point A, from 
 which the hangers for the plumb-lines wil be exactly located by means ot 
 the transit. Now mark with chalk on 
 the timbers where the strings cross. 
 
 These marks, though not accurate, 
 serve as guides in setting the hangers. 
 
 Make a permanent station at the point 
 A and carry the meridian to it. 
 
 The hangers can be made of strap 
 iron, \ in. thick by 2 in. wide, and at 
 least 16 in. long. In one end of the 
 iron, have a jaw with a fine cut at the 
 apex, or a drill hole just large enough 
 to contain the wire to be used for 
 plumbing. There should be two or 
 three countersunk holes in the hanger, 
 through which to fasten it to the tim 
 bers by means of heavy wire nails A 
 top view of the hanger is shown m 
 Fig. 2. 
 
 In most shafts there is a space from 
 2 to 4 in. wide between the ends of 
 the cage and the sides of the timbers; i A . 
 
 and in order to hoist and lower the cage to see that the wires are hanging 
 freely, it is best to set the hangers in such a position on the timbers that 
 the wires will hang in the middle of the space. 
 
 Fasten the hangers permanently over the chalk marks previously maae 
 on the north side of the shaft, wuth the jaws pointing toward A, and on the 
 south side of the shaft the outer end of the hanger may be fastened 
 temporarily. 
 
 Now set the transit over the station at A, take the backsight, foresight on 
 the wire hole of the hanger C and set the wire hole of the hanger B on the 
 same line. Record this course and foresight on the wire hole of tne hanger 
 E fixing as before the wire hole of the hanger D in the same line. Record 
 this course, and then the meridian to be carried into the workings below is 
 established. Measure carefully and record the distances A to B, A to G, 
 and B to C, the distances A to D A to E, and D to E , and finally the dis- 
 tances B to D and C to E. The necessity of taking all these measurements is 
 for the purpose of establishing a point at the bottom of the shaft vertically 
 below A, and checking the work in the office. , , . ... 
 
 The transit party can now descend to the bottom of the shaft, taking with 
 them four buckets of oh, the weights or plumb-bobs to be attaches to the 
 wire, and all the surveying instruments, leaving a responsible party on the 
 surface to handle the wires. Having arrived at the bottom ci the shatt, 
 have the cage hoisted 3 ft. above the landing, throw several planks across 
 the timbers on which to set the buckets of oil, signal to the man on top to 
 lower a wire and to fasten it securely, passing it through the wire hole ot 
 the hanger; now attach the plumb-bob and adjust the wire to such length 
 that when sustaining the full weight of the plumb-bob. the latter will not 
 touch the bottom of the bucket. Insert the weight in the oil, using care 
 not to leave the full weight on the wire with a jerk,. but let tne weight 
 down slowly, so that the wire receives the full strain gradually. Set the 
 three remaining wires in a similar way. 
 
 After the wires have been hanging a few minutes with the weights 
 attached, the latter may move from one side to the other ot the buckets. 
 Watch this carefully and keep moving the buckets until all the weights 
 hang perfectly free, then leave everything alone until the wires become 
 steady. The cages can now be hoisted and lowered for the purpose of 
 examining the wires to see that they hang free and plumb, care being 
 that the cages are not brought so close to the landings as to disturb the 
 hangers at the top, or the buckets at the bottom. . 
 
 To find a point vertically below A, stretch a string along the wires B, C , 
 being careful not to touch them; stretch another along the wires D, E\ 
 thent with a plumb-line, determine a point on the bottom vertically below 
 
72 
 
 SURVEYING . 
 
 the intersection of the strings. Measure the distances A B, A D, B D y and 
 C E and compare them with the corresponding distances at the top of the 
 shaft. If these distances compare favorably, the wires are, in all probability, 
 steady, and the work of determining the desired course with the transit 
 may now be begun. _ . , . 
 
 Set the transit up over the point of intersection just found; backsight on 
 the wires B, C; foresight on the wires D, E, and compare the included angle 
 and the distances with the corresponding angle and distances at the 
 surface. If these do not correspond, move the transit in the direction 
 
 necessary to increase or decrease the angle 
 or distances, as the case may be. Repeat 
 this operation until the exact point verti- 
 cally below A is determined. 
 
 A simple device that is of great advan- 
 tage is to have three links from an ordinary 
 
 —B trace chain placed in the wires on the side 
 toward the transit, and a few feet above 
 the buckets. This not only enables the 
 wires to turn freely, but also enables the 
 transitman to sight through one of the links 
 to the wire beyond, whereby he can place 
 
 I I the transit in exact line with the wires more 
 
 O ' I easily than if the links were not there. 
 
 i I Case 3.— Two, three, or four wires are 
 
 / \ I employed. They are secured and hung as 
 
 / \\ I before, and are located in the angles of the 
 
 / 1\\ J compartments x, x f x, x, Fig. 3. These are 
 
 / \ \ connected with four stations A, B, C , D , the 
 
 / \ \ lines A B and C D being at right angles to 
 
 \ Tll\ one another for convenience in the subse- 
 
 \ icw aftl quent calculation, and are connected with 
 
 \ I II /I I the outside survey. From A and B, taking 
 
 \U/ AB as a base line, the points x, x , x, x are 
 
 \ I / located. The same is repeated from C and 
 
 \/ D, taking CD as a base line. We thus have 
 
 , four locations of each wire. These are tab- 
 
 7 ' * 1 * * * * * 7 ulated, and any variations in a reading 
 
 must be followed by a repetition of the 
 same. The mean of the readings gives the 
 location. (Subsequently, the subject of cab 
 culating work will be taken up.) It can be 
 briefly stated here that the bearings of each 
 wire to each of the others, as referred to the 
 base line of the survey, are then calculated 
 and the distance between the wires accu- 
 rately measured. This finishes the work at 
 daylight. , x - 
 
 There may be two general types of 
 arrangements ot the bottom of the shaft, 
 and both arrangements have been sketched 
 and lettered similarly. The first is a case 
 when the shaft is arranged across the dip 
 of the bed, and the second is parallel to the 
 same. In both cases O and 7 are taken as 
 far apart as possible, and all the wires 
 x ' x , x , x located from each station with 
 reference to the other. The distances between the wires above and below 
 are also accurately measured as a check. There wih be 
 and 7 from the four wires, and the mean of these is taken as the co rrect one. 
 In every case of angle measurement mentioned, a senes of readmgsofeach 
 angle is taken upon different parts of the graduated limb to avoid instru- 
 mental errors, and the mean of these taken as the true reading. te 
 
 locations of 0 and 7, the course between them as referred to the mean base 
 line, is calculated, and 07 is the base line for the underground work The 
 angle readings above and below can be made at the same time with differ- 
 enf instruments, and, in taking the readings below, it is not ^essa^to 
 wait for absolute quiet in the wires, as that is seldom foun ^; f i^th a 
 can be bisected by the cross-hair, and the readings are duplicated until a 
 
 Fig 
 
SHAFTS AND SLOPES 
 
 73 
 
 
 constant result is secured. By this method a greater accuracy and speed is 
 obtained, and the angles below can be accurately measured, no matter how 
 the shaft may be arranged. 
 
 The T-Square Method.*— This ingenious method of taking the line under- 
 ground is especially valuable in shafts with several small compartments or 
 in cramped places where one cannot line in 
 with the wires. 
 
 The wires are placed in separate com- 
 partments and as far apart as possible. The 
 apparatus is made by the carpenter, and 
 consists of a straightedge and T squares. The 
 former is merely a planed pine board about 
 8 in. X$ in. and a'foot longer than the distance 
 between the wires. It rests approximately 
 horizontally on slats tacked across the shaft 
 for supports. It is brought to about an } or 
 
 in. from each wire and then nailed to the 
 slats sufficiently to prevent slipping. One 
 man should be at each wire. The T squares 
 are most serviceable if made with a mov- 
 able head clamped by a thumbscrew, and 
 of planed pine, about *2£ in. X i in. Except 
 in cramped quarters, the T squares will be 
 set at right angles, and should be placed together in clamping to insure 
 that each is set at precisely the same angle. 
 
 Fig. 4 shows a cramped position such as sometimes arises and in which 
 the movable head gives more latitude in working. After clamping, the 
 T squares are slid along the straightedge until close to the wire, but not 
 touching it, and are there clamped by a “G” clamp, both "men working at one 
 T square. The ends of the T squares, C and D, must be supported on 
 blocks so that the T squares lie approximately in the same horizontal plane ; 
 Everything up to the next step need be only approximately and quickly 
 placed, but now the greatest care must be exercised in measuring out equal 
 distances, A C and B D , from the wires. If the wire vibrates, determine the 
 middle of its swing by a pencil or pin. Hold a footmark (not the end of the 
 tape) opposite the wire on the T square, measure out an even number of 
 feet, and mark the point with a sharp pencil, and insert a pin. This done 
 for both wires gives us the parallelogram A B D C\ in which the only essential 
 is that C D should be exactly parallel to A B, the line of the wires. Now set 
 the transit over the most convenient of the two points, as D. To get the 
 azimuth of D C , and, consequently, the line of the wires, sight on a known 
 point E for a backsight, and measure the angle EDC. Establish another 
 point as F , on the line of the backsight, in order that its course may be 
 preserved after the instrument and T square are removed. 
 
 By this means the closing angle at E may be read after the wires are 
 removed from the shaft. Underground, the method 
 is the same, except that D C is the known course, 
 and is used as the backsight. To give the coordinates 
 of the instrument at D with the greatest precision, 
 the angle ADC and the distance A D should be 
 HIGS-Siircd 
 
 Surveying Slopes or Inclined Shafts. — Where a single 
 sight reaches from top to bottom of the shaft, the 
 problem is simple enough. A station can be estab- 
 lished on the inside of the foot-wall plate at the collar 
 and others in similar positions at each level. The 
 instrument set up over any station can command the 
 whole shaft and the level opposite. 
 
 Fig. 5. Where the shaft is sunk on several dips, the survey 
 
 is a much more difficult matter. Fig. 5 illustrates 
 cases of common occurrence. The shaft may be divided into sections like 
 A DEB, which are convex downwards, and others such as E B C, which 
 are concave downwards. As a rule a set-up can be avoided at the convex 
 knuckles if desired, and need only be made at those that are concave. 
 
 Bent Plumb-Line Method.— A may be invisible from B, but the survey may be 
 
 • #See “Mines and Minerals,’’ January, 1899, page 242. 
 
 
74 
 
 SURVEYING. 
 
 carried from one point to the other by the ingenious method of the bent 
 
 ^ The most complicated example which can arise is shown in Fig. 5. 
 Establish a station at A, the foot- wall side of the collar, the center point 
 being a small nail head projecting horizontally. Attach a long plumb-lme 
 to this and carry the other end to B. Here it will probably be necessary to 
 use a small screw-eye, with its head turned into the vertica 
 shaft, for the. center point. Pass the plumb-line through this and draw it 
 fairly tight. Now attach a plumb-bob at an intermediate point and regulate 
 the tautness so that the line is clear at all points. 
 
 The curves in the shaft may be such that two plumb-bobs may have to be 
 hung, as at D and E. and even a third may become necessary 
 
 The plumb-line, perhaps 100 ft. long, is apt to be disturbed by the air- 
 currents, and it is often better to mark a point on a convenient timber near 
 1), and another near E, so close to the string that there is no doubt of the 
 points lying in exactly the same vertical plane as the plumb-lme. If these 
 points be once established, the string and weights can be taken out of the 
 shaft, leaving us four points in the same vertical plane, and whose horizontal 
 projections lie in the same course. , . ... „ 
 
 Now set up at A and measure the azimuth angle from the backsight to D, 
 thereby Riving the bearing from A to B. If D should be Invisible from B, 
 depress the telescope after sighting on D and locate the point N m the same 
 vertical plane, and so situated that it is visi ble from both l and B. Measure 
 the vertical angle and distance to N. Now set up at B, use the course B E for 
 a backsight, and foresight to C. Measure the vertical angle and distance 
 B N It is seen that B N might have been used as a backsight, and E only 
 serves as an additional check. N is really an intermediate station, but since 
 it lies in the course A B, a set-up ttnw is unnecessary In simple cases, it is a 
 very convenient method of car: ying' a survey from the surface to the first 
 level and a longer horizontal projection of the sight Ab can be secured 
 than if a set-up were made in the shaft at D\ but in complicated cases, such 
 ‘as the one shown, it may often be quicker to make the extra set-up than 
 to use the plumb-line. In all sights for determining azimuth keep the 
 vertical angles as low as possible, and the horizontal projection of the 
 
 C ° U Method'by a Single Wire in the Shaft -Stretch a rather fine wire, free from 
 kinks down the shaft, as shown in Fig. 6. being careful that it touches 
 nowhere in the shaft. Take two plumb-bobs provided with fine round 
 strings. Suspend one from A and the other from B so that they nearly touch 
 the same side of the wire MN. In order to have the plumb-lines as far apart 
 as possible, the line at B must be quite long and a can of water should be 
 provided to keep the bob from swinging. The plumb-line is fastened to a 
 nail B nearly in the proper position. Have a bar of wood with a block 
 fastened' to it placed to one side of B and a little below it. 
 The block must have a hole so that a small screw bolt can 
 easily screw through it. A spool is run on the bolt having a 
 small groove turned in it and being sandpapered and greased 
 so that the string will slip easily as the bolt is turned. Now, 
 place the transit in line with the two plumb-bobs as m an 
 ordinary case of shaft plumbing. Repeat this operation below. 
 The plumb-bobs in both cases hang in the same vertical plane 
 and thus the true bearings are found underground. Even the 
 plumb-lines could be dispensed with, but the method would 
 not then be so accurate. The instrument would be set nearly 
 in the vertical plane passing through the wire, leveled and 
 sighted at M. Dip the telescope until the lowest point on the 
 wire is visible, note the amount by which the cross-hair and 
 wire fail to coincide and shift the instrument accordingly. But 
 if this method were tried, the two points sighted at would not be nearly so 
 far apart horizontally as the plumb-lines, and any error in leveling wou .d 
 also vitiate the result. This method of the single wire, however, provides 
 no way of obtaining the coordinates. 
 
 Fig. 6. 
 
 NOTES ON MAPPING. 
 
 There are no general rules governing the minutiae of map making, so 
 that it may be well to note some of the variations in practice. In some 
 offices the area excavated is shown by a light wash of India ink in addition 
 
NOTES ON MAPPING. 
 
 75 
 
 to the ink line bounding the solid area. This makes a striking map, and 
 the workings stand out prominently. If the survey were never to be 
 extended and the map were simply made to show a particular state of the 
 workings, there would be no objection to the practice; but as such extern 
 sions have to be made, and old pillars removed or cut up, it requires con-' 
 siderable skill to tint the extensions, and especially the surfaces when 
 erasures have been made, so as to produce an effect uniform with the old 
 tinted surface, especially as that tint has been deepened by frequent han- 
 dling. For these reasons many offices omit the tint on the map, but some tint 
 the back of the tracing used by the corps, or sent to the mine inspector. 
 
 It is an open question whether the ends of the chambers (breasts, rooms) 
 and gangways (entries, levels) should be closed with ink. It is well to be 
 able to show on your map where the faces of the workings were at any given 
 time. Some place the date of each survey at the ends of the gangways at 
 each posting, and of every fourth or fifth chamber, and thus note the rapid 
 ity with which the mine is worked. Others use various colors to denote th.e 
 successive postings of the survey, and place across the ends of gangways ana 
 chambers.the color appropriate to the survey that located them. 
 
 Wnere there are a number of beds worked from the same shaft, slope, or 
 adit, the workings are frequently vertically above one another, and their 
 location on the same map causes confusion unless care be taken. One ot the 
 methods used to distinguish between each bed is to line in the areas worked 
 with a color appropriate to that bed. In this way, three or four beds have 
 been plotted on the same map. A better way is to make a map for each bed 
 and to combine the various beds on the tracings for the officers and the mine 
 inspector. Each bed on this is lined with its color, and tinted with a wash 
 of the same color on the back of the tracing on the parts excavated. 
 
 The lines of survey are lightly drawn bet ween stations with red ink, and 
 the stations denoted by minute circles of the same color as that used for that 
 particular seam. The survey lines should never cut the circumference of the 
 circle, as such a procedure might cause doubt as to the exact location of the 
 station i n case measurements were made on the map. Stations are numbered 
 as in the mine, and beneath each is placed the elevation above or below tide 
 of the roof of the mine. If the stations are numerous and the elevations fre- 
 quently taken, such a map would furnish the means of ascertaining the 
 shape of the bed by running contours through points of equal elevation. 
 This plan was used by a number of engineers and was adopted by the Second 
 Geological Survey of Pennsylvania in making their mine sheets. The adop- 
 tion of the scale of 100 ft. to the inch, under the ventilation law, also fur- 
 nished that survey with a means of tracing and connecting adjoining 
 properties and their workings, as had been frequently done by large com- 
 panies having adjacent collieries, and the maps of the anthracite regions ol 
 the Second Geological Survey of Pennsylvania have been thus compiled from 
 tracings of office maps, with little or no inside work by the corps of the survey. 
 
 The ground areas of all buildings are tinted a uniform red. 
 
 All railroad tracks are represented by red lines. 
 
 All bridges, etc., and, in fact, any improvements built by man, are to be 
 colored red. 
 
 In the mine, the stations are represented by a small circle (o) with the 
 number in black beside it. 
 
 The lines of survey are drawn between the circumferences of tne circles 
 
 marking the station 'o o, so as to leave the centers uncolored. The 
 
 elevations above tide are marked. 
 
 All small bodies of water are tinted with Prussian blue; all large bodies 
 with indigo — as Prussian blue is too vivid for large areas of tint. 
 
 It is the custom to allot a color to each bed, and make each bed on the 
 general tracing in its color. In this way all the beds may be mapped on the 
 same tracing, and can be distinguished though the workings may all under- 
 lie the same area. Various colors are sometimes used to denote the extent 
 of the workings at the given postings. 
 
 The paper on which the map is made should be of the best quality, as 
 frequent changes in the workings, and the removal of portions of old pillars, 
 necessitate many erasures. Ordinary paper will not work well after erasure 
 and subsequent handling, and the best practice is to use cloth-backed egg- 
 shell paper. 
 
 It remains to note that the temperature and humidity of the office, and of 
 the place where the maps are stored, should vary as little as possible. As the 
 scale is small, any variation in the paper by contraction or expansion will 
 
76 
 
 SURVEYING. 
 
 affect the scale on which the map was originally laid out, and will affect it 
 unequally, as the paper is not homogeneous. This can be seen by making 
 measurements on old maps to check work done in former times. In almost 
 every case the 500' squares are slightly in error. This must be guarded 
 against in case measurements are taken from the map. It is best to calculate 
 the distances wanted from the coordinates of the ends of the lines, and 
 insure absolute accuracy. _ , ,. 
 
 A good map is the sign of a good draftsman. The title should be subordi- 
 nate to the map. It is common to see the fifty-cent map of a small area 
 smothered under a gorgeous ten-dollar title. The lettering should be neat 
 and appropriate, and a style should be adopted that can be readily thrown 
 off — as time is quite an item in the money made by mapping. 
 
 A neat title, the judicious use of tints over the area excavated, and good 
 lettering of minor objects will add dollars to the 
 & value of a map. A small outlay of taste and care 
 
 will make a beautiful tracing out of a ragged one, 
 and double its value. 
 
 Locating Errors. — Errors in arc or distance are 
 easily located by the method of coordinates. The 
 transitman, at the completion of a closed survey, 
 should sum the angles and see if it be 360°. Thus, 
 before leaving the mine, a check will be established 
 on the transit work. In case of a variation from 
 360°, the notes are examined to see if the needle 
 readings show a similar variation with the vernier. 
 In case the notes are incomplete, or, if a continuous 
 vernier has been carried, we table the work before 
 leaving the mine and locate the error as follows: Fig. 7 represents a close of 
 four stations and an angular error at c. Starting from a we table as follows: 
 
 Sta. 
 
 Course. 
 
 Dist. 
 
 N 
 
 s 
 
 E 
 
 w 
 
 Sums. 
 
 N 
 
 S 
 
 E 
 
 W 
 
 a 
 
 b 
 
 d (d’) 
 a ( a ") 
 b 
 
 N 
 
 E 
 
 S 30 E 
 S 60 W 
 N 30 W 
 
 100 
 
 100 
 
 100 
 
 100 
 
 100 
 
 86.58 
 
 50 
 
 100 
 
 50 
 
 86.58 
 
 100 
 
 100 
 
 13.42 
 
 36.58 
 
 100 
 
 150 
 
 63.12 
 
 
 There is an error 
 course a 6 by 30°. 
 Reversing the order 
 
 in close of 30°, as the course a" b differs from the 
 of tabling, and correcting each course by 30°, we have: 
 
 Unon comparing the sums of the northings and southings and eastings 
 and westings in both, we find that c is the only station for which they agree. 
 Here the error occurred. The location of c from either direction was cor- 
 rect and what followed incorrect. From this we can deduce the rule: 
 
 To Find an Error in Arc.— Table the close from any station m both directions 
 back to the initial station. The station which has a similar sum of eastings 
 
NOTES ON MAPPING. 
 
 77 
 
 and westings and northings and southings in both tablings is the one at 
 which the error was made. 
 
 With two or more errors nothing can he done. 
 
 Errors in Distance— These maybe found in a close by tabling. Suppose 
 we have a square abed so placed that the magnetic meridian passes 
 through b d. Let the distance c d be incorrect. 
 
 The first location of a is southings 0, westings 0; the second location is 
 southings 35.35, westings 35.35. The westings are sines and the southings 
 cosines, as stated before. As sin -f- cos = tang of course referred to the base 
 line, we divide 35.35 by 35.35 and obtain 1 as the natural tangent for 45°, and, 
 as the error was in ‘southings and westings, the course on which the error 
 was made is S 45° W. The amount of the error is found by dividing the error 
 in eastings or westings as tabled by the sine of the course just found, or the 
 error in northings or southings by the cosine of the same. Both results 
 will agree, 35.35 -r- 70.71 = .50. Reducing the measured distance by this 
 amount, we find the tabulation shows an accurate close. From this we 
 deduce the rule: _ 
 
 To Find an Error in Measurement— Divide the difference between the east- 
 ings or westings of the two locations by the difference between the northings 
 or southings of the same. The quotient will be the tangent of the course on 
 which the error was made. The extent of the error is found by dividing the 
 error in eastings or westings (as tabled) by the sine of the above course,, or 
 the error in northings or southings by the cosine of the same. 
 
 Locating Special Work.— This last principle may be used for finding the 
 proper course and distance to drive a tunnel between two stations connected 
 bv a survey. In outside tunneling, the survey is generally carried over the 
 surface in a straight line. In underground work this is impossible, so that it 
 is a much more difficult task to ascertain the distance and direction to drive, 
 from the number of measurements to be made in connection with the two 
 stations, but if the work is accurately done it is much more a feat than in 
 outside work. To include all the elements that enter into such a calculation, 
 we will suppose that an underground slope is to be run between two beds of 
 coal. The distance between the two ends must be accurately obtained, as 
 well as the relation of the two stations found. Having tabled the work, we 
 get the difference between the sums of sines and cosines, as just described, 
 for the two points; the quotient from dividing the first by the second gives 
 us the course, and from the last rule the horizontal distance is found. The 
 levels give us the difference in elevation, and from these data we get the 
 slope per hundred, and the distance measured on that slope. All of the work 
 is done in the mine and while the transit is setting up at one of the end 
 stations, so that before leaving the mine, we can give the course, pitch, and 
 distance of the tunnel and set the first station for lining in the center. The 
 more important the work, the greater need of accuracy. In one case a 
 1,000' chain was constructed to measure the distance between two shafts 
 that were to be connected by work driven from both ends, and much 
 of the outside work was done on the ice of the Susquehanna River, and a 
 transit reading to 5" was used. The work closed vertically and laterally 
 within an inch. 
 
 Calculation of Areas.— In connection with the mapping of the part newly 
 worked, the engineer of the company sometimes calculates the area 
 excavated since the last posting, and estimates the royalties accruing to the 
 various parties whose lands are leased. 
 
 This method, at best, is liable to grave errors, and requires a number of 
 accurate cross-sections of the bed to determine its composition, as well as 
 
78 
 
 SURVEYING . 
 
 numerous determinations of the specific gravity to determine its weight. 
 The old method of estimating workable coal in a property allowed 1,000 tons 
 per acre per foot of thickness of the bed. To obtain this amount, the roof 
 must be good, the pillars of medium size, and the bed near the surface; 
 or the surface must be so valueless that the pillars can be -‘drawn” or 
 “robbed.” As beds increase in depth, if the surface be valuable, the ratio 
 of pillars to stall must increase or the greater pressure will cause the mine to 
 cave in It has been found that, when the surface is to be kept up and the 
 workings are to be carefully driven, but 850 tons per acre per foot of thick- 
 ness can be used in calculation, under the present system of mining. Efforts 
 are constantly being made to increase this amount, with a possible chance 
 of success , „ ,, . 
 
 To estimate the amount of coal excavated from any property, the best 
 method is to institute an account of all cars of coal taken out from the work- 
 ings under that property. As soon as the measurements on the mine tracing 
 show that a gangway or room has crossed the property line, the oflice is 
 notified, and all cars coming from those places are credited to that property. 
 This is the only absolutely accurate method of computation. The total 
 number of cars of coal run through the breaker is known, with the total 
 weight of prepared coal. The ratio of the cars coming from a given property 
 to the total number of cars is taken as the ratio between the total prepared 
 coal and the coal sold from that property. 
 
 RAILROAD CURVES. 
 
 These are arcs of circles, and are divided into simple, compound, and 
 reverse curves. A simple curve has but one radius, a compound one is con- 
 tinuous and has two or more radii, and a 
 reverse one is also continuous but com- 
 posed of arcs described in opposite direc- 
 tions. 
 
 Curves are designated by the number 
 of degrees in the central angle, which is 
 subtended by an arc whose chord is 100 ft. 
 long. Thus, if the angle BOG , Fig. 1, is 
 10° and B G is 100 ft. long, B G H C is a 10° 
 curve. 
 
 The angle FE C, formed by the pro- 
 longation of two adjacent straight por- 
 tions of a railroad, or tangents , as they are 
 technically called, is termed an intersec - 
 tion angle. 
 
 The deflection angle of a curve is the 
 angle formed at any point of the curve 
 between a tangent and a chord of 100 ft., 
 and is therefore one-half the size of the 
 y degree of the curve. If the chord B G is 
 
 100 ft., the\angle EBG is the deflection angle of the curve BGHC, and is 
 one-half the angle B O G. . n . 
 
 When the deflection angle D is given, the radius of the curve, R, is 
 found by the formula 
 
 sin D 
 
 The curve used to connect two tangents is determined mainly by the form 
 of the country. When this is decided, the point of beginning, called the 
 P. C. (point of curve), and the point where the curve ends, called the P. T. 
 (point of tangent), must be located. Both these points are the same distance 
 from the point of intersection of the tangents, called the P. I. ( point of inter- 
 section). This distance is called the tangent distance of a curve, and is found 
 by the formula 
 
 7= R tan h l 
 
 in which 7= tangent distance; 
 
 R = radius of curve; 
 
 T = intersection angle. 
 
 Having set the tangent points B and C\ Fig. 1, in order to locate points on 
 the curve, set up the transit at B , the P. C. Set the vernier at zero, and sight 
 to E, the P. I. Suppose B to be a full station on the tangent, and that it has 
 
RAILROAD CURVES. 
 
 79 
 
 been decided to set stakes at each 100 ft. Let the central angle BOG , meas- 
 ured by the 100-ft. chord B G, be 10°; then, the deflection angle E B G, having 
 its vertex B in the circumference, and being subtended by the chord B G, 
 will equal £ B G G, or 5°. Turn an angle of 5° from B, which in this case will 
 be to the right, measure 100 ft. from B, aud drive a stake at G. Turn off an 
 additional angle of 5°, making 10° from zero, and at another 100 ft. measured 
 from 6r, and drive a stake at H. Continue this process until 20°, or one-half 
 the intersection angle, has been turned off. This last deflection will bring 
 the lbrechainman to the point of tangency C, or the P. T. 
 
 When the P. C. comes between two stations it is called a substation, and 
 the chord between it and the next station on the curve is called a subchord. 
 Had the P. C. been a substation, say 32 ft. beyond a regular station, the 
 deflection angle for the measuring distance of 100 — 32 = 68 ft. would be 
 found in this manner: The deflection for 100 ft. is 5° — 300'; hence, for 1 ft. 
 
 Fig. 2. 
 
 it is = 3', and for 68 ft. it is 3 X 68 = 204' = 3° 24'. This is turned off 
 100 
 
 and a stake set in line 68 ft. from the transit. Other stations are then located 
 as above, by turning off an additional 5° each time. 
 
 Rules for'Measuring the Radius of a Curve.— Stretch a string, say 20 ft. long, or 
 longer if the curve is not a sharp one, across the 
 curve corresponding to the line from A to C, in 
 Fig. 2. Then measure from B the center of the 
 line A C, and at right angles with it, to the rail at 
 D. Multiply the distance A to B, or one-half the 
 length of the string, in inches, by itself; measure 
 the distance D to B in inches, and multiply it by 
 itself. Add these two products, and divide the sum 
 by twice the distance from B to D, measured exactly in inches and fractional 
 parts of inches. This will give the radius of the curve in inches. 
 
 It may be more convenient to use a straightedge instead of a string. Care 
 must be taken to have the ends of the string or straightedge touch the same 
 part of the rail as is taken in measuring the distance from the center. If the 
 string touches the bottom of the rail flange at each end, and the center 
 measurement is made to the rail head, the result will not be correct. 
 
 In practice, it will be found best to make trials on different parts of the 
 curve, to allow for irregularities. 
 
 Example. — Let AC be a 20-ft. string; half the distance, or A B , is then 
 10 ft., or 120 in. Suppose B D is found on measurement to be 3 in. Then 120 
 multiplied by 120 is 14,400, and 3 multiplied by 3 is 9; 14,400 added to 9 is 
 14,409, which, divided by twice 3, or 6, equals 2,401i in., or 200 ft. 1£ in., which 
 is the radius of the curve. 
 
 The formula is thus stated, 
 
 A B* + B D* 
 
 2 B D = R - 
 
 Or, applied to the above example, 
 
 1002 Q 
 
 o\7q = 2 > 401 * in - = 200 ft * in - 
 
 2 X o 
 
 To Find the Radius of a Circular Railroad Curve, the Straight Portions of a Road 
 Being Given.— If Q I and PD, Fig. 3, are the straight portions that are to be 
 connected, the radius of the curve ID may be found as follows: 
 
 Produce Q I and P D until they meet and form the angle T. Bisect the 
 angle Q TP by the line T C. From the point on either line from which the 
 curve is to begin, in this instance making the point I the point of curve, erect 
 the line I C perpendicular to Q T, and the point where this joins the line T C, 
 or C , is the center of the curve, and the line I C is the radius. To find the 
 end of the curve, or point of tangent, as D, draw a line from C, perpendicular 
 to TP. The line CD will also be a radius of the circle of which I D is the 
 arc, and the point D will be the point of tangent. 
 
 To Find the Radii of Compound Curves to Join Two Straight Portions or Road. 
 This kind of curve is adopted where the railroad is required to pass through 
 given points, as C . , D, P, P, Fig. 3 ( b ), or to avoid obstructions. 
 
 Compound railroad curves are composed of straight lines and circular 
 arcs, and have common normals, O H, O P, P I. Q J , K R. and therefore com- 
 mon tangents where the arcs are joined. The normals are perpendicular to 
 the straight portions of the road also; OH is perpendicular to A B E F is 
 perpendicular to QJ and K R. 
 
80 
 
 SURVEYING. 
 
 To find the radii 0 B, C Q, Fig. 3 (c), to connect two straight lines of rail- 
 road, AB, D E, the road has to pass from the point B, through the point C , 
 and to touch the straight road E F at any point D. 
 
 Join B and C, make the angle B CO = 0 B C, which is supposed to he 
 given, equal 90° — TB C. Draw B 0 perpendicular to AB, then OB = CO , 
 and is the radius of the arc B C. 
 
 With OB as radius, describe the arc B C; draw C F perpendicular to C Q, 
 and produce DE to meet it in F; make DF = CF\ and draw D Q perpen- 
 dicular to E F , to meet C Q in Q. Then CQ -= Q D, and the radii 0 B and 
 0 D are determined. 
 
 Practical Method of Laying Out Sharp Curves in a Mine.— Curves m a mine are 
 
 usually so sharp that they are designated as curves of so many feet radius, 
 
 instead of as curves of so many degrees. 
 
 Suppose that it is required to connect the two headings A and B, Fig. 4(a), 
 which are perpendicular to each other, with a curve of 60 ft. radius. Pre- 
 pare the device shown in Fig. 4 (6), by taking three small wires or inelastic 
 strings f g, gh, and g k, each 10 ft. long, and connecting one end of each to 
 a small ring, and the other end of two to the ends of a piece y. 00 '? ^ 3 . 
 long. Form a neat loop at the end / of the string#/. To use this device, 
 lay off on the center line of the heading B, c d and d e equal to 60 ft. and 
 3 10 ft., respectively. Place the loop / of the 
 
 device described over a small wire peg driven 
 in at e, and the ring g over a similar peg at d. 
 Take hold of the stick h k, pull the strings g h 
 and g k taut, and place the center mark on h k 
 on the center line of the heading B. Drive a 
 small peg in at m, located by the point k, which 
 is on the curve. Move the device forward, 
 place the loop / over the peg at d, the ring g 
 over the peg at m, and take hold of the stick 
 h k and pull until the strings g h and g k are 
 
 Fig. 4. 
 
 taut, and the strings f g and g h are in a straight line. The point A: will fall 
 on the curve at n, which mark by driving in a peg. To locate other 
 points, proceed exactly as in the last step. The distance c d in any case 
 is found by the formula c d = R tan £ J, in which R is the radius of the 
 curve, and I the intersection angle of the center lines of the headings. 
 
 HINTS TO BEGINNERS. 
 
 Abuse of Instruments.— Surveying instruments of value and precision are 
 not made of cast iron, as one would think from the way they are frequently 
 handled. Underground work is transacted in places dark, dirty, and con- 
 fined, so that extra care must be observed to prevent accidental knocks that 
 damage the instrument even if they do not destroy its accuracy. 
 
STADIA MEASUREMENTS. 
 
 81 
 
 As it frequently happens that long distances must be traversed under- 
 ground in going between the shaft or slope and the workings to be surveyed, 
 the transit and level should be carried so as to obviate all accidents. They 
 should never be attached to the tripod and carried on the shoulder, and, if 
 the route to be passed over is up or down a slope or working place, the 
 person carrying the instrument should be the last to descend and the first to 
 ascend, so that loose stones or dirt that may be dislodged will not affect or 
 endanger the instrument or trip the carrier. 
 
 Be sure that the tripod head is tightly screwed on to the tripod. The 
 writer remembers a case where the transitman and himself, when new to 
 the work, spent over an hour in endeavoring to obtain two readings of an 
 angle that would agree. The variations— from 8' to 2°— were caused by the 
 slight movement of an old instrument with too much “ lost motion,” and a 
 loose tripod head. 
 
 A great many engineers prefer kerosene to fish oil for their lamps. Kero- 
 sene never drops upon your book to make an unsightly smear, and perhaps 
 obliterate part of your notes. A kerosene lamp is hotter and, with the 
 glazed mine hat, is more apt to produce headaches. The writer, during the 
 latter part of his underground work, wore a straw hat, had a piece of thin 
 sheet brass riveted to its front with a hole in the top for the lamp hook. To 
 the lamp was brazed a narrow cross strip of the same metal, and the strip 
 ends, bent back upon themselves, were slid down the sides of the plate on 
 the hat and kept the lamp from swaying. With such an arrangement it is 
 not necessary to remove the lamp to read the vernier, and when the lamp 
 is used, for other purposes, the hat can be removed with the lamp fastened tcb 
 it. This arrangement keeps the hands free from lamp smoke or oil, and a 
 cleaner note book is the result. 
 
 When there is an antipathy to a lamp upon the head, and when, with a 
 long, wooden handle, one or both hands are free in going about the work, 
 a larger lamp is used of “torch” pattern, employed by wheel testers or 
 engineers in railroad practice. Kerosene can be burned in this. The handle 
 can be tucked under the left arm while taking side notes. Such a lamp is 
 convenient in finding old stations in a high place, when there is no firedamp. 
 
 For plumbing wet shafts, kerosene resists the extinguishing power of 
 water better than fish oil, and is less readily blown out by a strong ventilating 
 current. It makes more smoke, and, in tight headings, or mines with poor 
 ventilation, with a large party, fouls the air much more readily than fish oil. 
 Sometimes a mixture of the two is burnt in very drafty places, where it is 
 ham to maintain a light. Kerosene is burned in the plummet lamp unless 
 it is used with the “ safety ” attachment. Sweet oil, or any oil burning 
 without smoke, must then be used. Smoke clogs the openings in the gauze, 
 restricts the entry and escape of gases, and, especially if the gauze be damp 
 with oil, may ignite and communicate the flame from within to the outside 
 body of gas. 
 
 White lead or Dutch white (white lead and sulphate of baryta in equal 
 parts) is best for painting stations. Zinc white has been tried with less 
 success. The mixture should not contain too much linseed oil— especially 
 in wet places— or it will run and destroy the witness. 
 
 THEORY OF STADIA MEASUREMENTS. 
 
 By Arthur Winslow.* 
 
 Late Assistant Geologist, Second Geological Survey of Pennsylvania, State 
 Geologist of Missouri. 
 
 The fundamental principle on which stadia measurements are based is 
 the geometrical one that the lengths of parallel lines subtending an angle are 
 proportional to their distances from its apex. Thus if, in Fig. 1 (a), a represents 
 the length of a line subtending an angle at a distance d from its apex, and a' 
 the length of a line, parallel to, and twice the length of, a subtending the 
 same angle at a distance d' from its apex, then d' will equal 2d. 
 
 * Mr. Winslow’s calculations and tables have been proved practically correct by the several 
 corps of the Second Geological Survey of Pennsylvania. The corps in the anthracite regions, under 
 directions of Mr. Frank A. Hill, geologist in charge, took over 30,000 stadia sights, and better 
 results were obtained when tie surveys were made than in previous work in which distances 
 
 were chained. 
 
82 
 
 SUR VEYINO. 
 
 This is, in a general way, the underlying principle of stadia work; the 
 nature of the instruments used, however, introduces several modifications, 
 and these will be best understood by a consideration of the conditions under 
 which such measurements are generally made. 
 
 There are placed in the telescopes of most instruments fitted for stadia 
 work, either two horizontal wires (usually adjustable), or a glass with two 
 etched horizontal lines at the position of the cross-wires and equidistant 
 from the center wire. A self-reading stadia rod is further provided, gradu- 
 ated according to the units of measurements used. In a horizontal sight 
 
 with such a telescope and rod , 
 the positions of the stadia wires 
 are projected upon the rod, and 
 intercept a distance which, in 
 Fig. 1 (b), is represented by a. 
 
 In point of fact, there is 
 formed, at the position of the 
 stadia wires, a small conjugate 
 image of the rod that the wires 
 intersect at points b and c, 
 which are, respectively, the 
 foci of the points B and C on 
 the rod. If, for the sake of 
 simplicity, the object glass be 
 considered a simple biconvex 
 iens, then, -by a principle of 
 optics, the rays from any point 
 of an object converge to a focus at such a position that a straight line, called 
 a secondary axis, connecting the point with its image, passes through the 
 center of 'the lens. This point of intersection of the secondary axes is 
 called the optical center. Hence, it follows that lines such as c C and b B 
 in Fig 1(b) drawn from the stadia wires through the center of the object 
 glass will intersect the rod at points corresponding to those that the wires 
 cut on the image of the rod. From this follows the proportion: 
 
 d _ a 
 
 ~v ”T- 
 
 d = j- a, (1) 
 
 where d = distance of rod from center of objective; . 
 
 p == distance of stadia wires from center of objective; 
 a = distance intercepted on rod by stadia wires; 
 
 I = distance of stadia wires apart. 
 
 If p remained the same for all lengths of sight, then y- could be made a 
 
 Fig. 1. 
 
 desirable constant and d would be directly proportional to a. Unfortunately, 
 however for the simplicity of such measurements, p (the focal length) varies 
 with the length of the sight, increasing as the distance diminishes and vice 
 versa. Thus, the proportionality between d and a is variable. I he object, 
 then, is to determine exactly what function a is of d and to express the rela- 
 tion in some convenient formula. 
 
 The following is the general formula for biconvex lenses: 
 
 i + ? = 7 : (2) 
 
 f is the principal focal length of the lens, and p and p' are the focal distances 
 of image and object, and are, approximately , the same as p and d, respectively, 
 in equation (1): 
 
 Therefore, j ^ = y, approximately, 
 
 d d 
 
 and 
 
 p f L 
 
 a a 
 
 From (1), 
 
 Whence, 
 
 •** I ~ f L 
 d = -j- a + /• (2) 
 
STADIA MEASUREMENTS. 
 
 83 
 
 In this formula, it will be noticed that as / and I remain constant for 
 sights of all lengths, the factor by which a is to be multiplied is a constant, 
 and that d is thus equal to a constant times the length of a, plus/. This for- 
 mula would seem, then, to express the relation desired, and it is generally 
 considered as the fundamental one for stadia measurements. As above 
 
 stated, however, the equation ~ = — is only approximately true, and 
 
 the conjunction of this formula with (2) being, therefore, not rigidly admissi- 
 ble, equation (3) does not express the exact relation.* The equation express- 
 ing the true relation, though differing from (3) in value, agrees with it in 
 
 form, and also in that the expression corresponding to -- is a constant, .and 
 that the amount to be added remains, practically, /. The constant corre- 
 sponding to -j may be called fcf, and thus the distance of the rod from the 
 
 objective of the telescope is seen to be equal to a constant times the reading 
 on the rod, plus the principal focal length of the objective. To obtain the 
 exact distance to the center of the instrument, it is further necessary to add 
 the distance of the objective from that center to/; which sum may be called c. 
 The final expression for the distance, with a horizontal sight, is then 
 
 d = k a -f c. (4) 
 
 The necessity of adding c is somewhat of an encumbrance. In the stadia 
 work of the U. S. Government surveys, an approximate method is adopted in 
 which the total distance is read directlv from the rod. For this method the 
 rod is arbitrarily graduated, so that, at the distance of an average sight, the 
 same number of units of the graduation are intercepted, between the stadia 
 wires on the rod, as units of length are contained in the distance. For any 
 other distance, however, this proportionality does not remain the same; for, 
 according to the preceding demonstration, the reading on the rod is propor- 
 tional to its distance, not 
 from the center of the in- 
 strument, but from a point 
 at a distance “ c ” in front 
 of that center, so that, 
 when the rod is moved 
 from the position where 
 the reading expresses the 
 exact distance, to a point 
 say half that distance from 
 the instrument center, the 
 reading expresses a dis- 
 tance less than half; and, 
 at a point double that dis- 
 tance from instrument cen- 
 ter, the distance expressed 
 by the reading is more 
 than twice the distance. 
 
 The error for all distances 
 less than the average is 
 minus, and for greater dis- 
 tances, plus. The method 
 is, however, a close approx- 
 imation, and excellent re- 
 sults are obtained by its use. 
 
 Another method of get- 
 ting rid of the necessity of 
 adding the constant was 
 devised by ' Mr. Porro, a 
 Piedmontese, who constructed an instrument in which there was such a 
 combination of lenses m the objective that the readings on the rod, for all 
 lengths of sight, were exactly proportional to the distances.:}: The instrument 
 
 * This is demonstrated later on. 
 
 t k is dependent on /, and can therefore be made a convenient value in any instrument fitted 
 with adjustable stadia wires. It is generally made equal to 100, so that a reading on the rod of 1' 
 corresponds to a distance of 100' -f /. 
 
 } A notice of this instrument will be found in an article by Mr. Benjamin Smith Lyman, 
 entitled “ Telescope Measurements in Surveying,” in “Journal Franklin Institute,” May and 
 June, 1868, and a fuller description is contained in “ Annales des Mines,” Vol. XVI, fourth series. 
 
84 
 
 SURVEYING. 
 
 was however, bulky and difficult to construct, and never came into 
 
 SflS^SlSIiS 
 
 mmmm. 
 
 the rod. In Fig. 2 (&)., 
 
 A B = a = reading on rod; _ . t. r TT . 
 
 up = d == inclined distance = c + GF = c + k. CH, 
 jfP — j) = horizontal distance = d cos n; 
 pp = Q = vertical distance = D tan 
 n = vertical angle; 
 
 AG B = 2m. 
 
 and the 
 
 sines of the opposite angles, 
 
 AF sinm . 
 
 GUF = sin [90° + \n — m)Y 
 
 or, 
 
 AF = G-Fsinm 
 
 cos ( n — m ) 9 
 
 and 
 
 B F 
 GF 
 
 sin m 
 
 iET[90° — (w+m)]’ 
 
 1 
 
 or, 
 
 p p = (? p sin m 
 
 or 
 
 cos (n + m)’ 
 
 iP-(- JSP = <? F 7 sin 
 
 cos ( n — m ) ' cos {n 4- m). 
 
 a „ T , CH 1 
 
 .4 F + i? -F — a > an< ^ G F — y tan m 
 
 C II cos m 
 
 2 sin m 
 
 By substituting and reducing to a common denominator, 
 
 C TT r».os m f cos ( n_ + m ) + cos ( n — rrrf J 
 a T “i" cos (»+"»») cos (n — m) 
 
 Reducing this according to trigonometrical formulas, 
 cos 2 cos 2 m — sin 2 n sin 2 m 
 
 CH = cl 
 
 as 
 
 d = c + ka 
 
 cos n cos 2 m 
 
 d = MF=c + k. CH. 
 
 cos 2 n cos 2 m — sin 2 n sin - m 
 
 cos n cos 2 m 
 
 The horizontal distance, D = d cos n. . 2 
 
 F) = c cos n + k a cos 2 n — kci sin- n tan m. 
 
 IOST4 1- fcUWD-i* — . , , .. 
 
 The third 
 
 very small even for long g s ^^t0 07'V TlSeforeiThe final formula for 
 ffiSTX “?adir Aeld%%°cauf, and with Wires equidistant from 
 the center wire, is the following: cosB + O * cos , n , ( 5 ) 
 
STADIA MEASUREMENTS. 85 
 
 The vertical distance Q is easily obtained from the relation: Q = Dtann. 
 
 .*. Q = c sin n + a k cos n sin n\ 
 
 or, Q = c sin n + a Jc . (6)* 
 
 With the aid of formulas (5) and (6), the horizontal and vertical distances 
 can be immediately calculated when the reading from a vertical rod and 
 the angle of elevation of any sight are given. From these formulas the 
 stadia reduction tables following have been calculated. The values of a A; cos'" n 
 sin 2 n 
 
 and a k — - — were separately calculated for each 2 minutes up to 30° 
 
 of elevation; but, as the value of c sin n and c cos n has quite an inappre- 
 ciable variation for 1°, it was thought sufficient to determine these values 
 only for each degree. As c varies with different instruments, these last two 
 expressions were calculated for three different values of c, thus furnishing 
 a ratio from which values of c sin n and c cos n can be easily determined 
 for an instrument having any constant (c). 
 
 Similar tables have been computed by J. A. Ockerson and Jared Teeple, 
 of the United States Lake Survey. Their use is, however, limited, from the 
 fact that the meter is the unit of horizontal measurement, while the eleva- 
 tions are in feet. The bulk of the tables furnishes differences of level for 
 stadia readings up to 400 meters, but only up to 10° of elevation. Supple- 
 mentary tables give the elevations up to 30° for a distance of 1 meter. For 
 obtaining horizontal distances, reference has to be made to another table 
 which is somewhat an objectionable feature, and a multiplication and a 
 subtraction has to be made in order to obtain the result. Last, but not least, 
 these tables are apparently only accurate when used with an instrument 
 whose constant is .43 meter. 
 
 The many advantages of stadia measurements in surveying need not be 
 dwelt on here, both because attention has been repeatedly called to them 
 and because they are self-evident to every engineer. Neither will it be 
 within the compass of this article to describe the various forms of rods and 
 instruments, or the conventionalities of stadia work. 
 
 It is seen that, in the deduced formula, the factor by which the reading 
 on the rod is multiplied is a constant for each instrument. The question 
 now arises, Does this remain the case with a compound objective ? 
 
 In view of the difficulty of demonstrating this mathematically, it was 
 decided to make a practical test of this point with a carefully adjusted 
 instrument. The readings were taken from two targets set so that the sight 
 should be horizontal, thus preventing any personal error or prejudice from 
 affecting the reading. A distance of 500 ft. was first measured off on a level 
 stretch of ground, and each 50-ft. point accurately located. From one end 
 of this line, three successive series of stadia readings were then taken from 
 the first 50-ft. and each succeeding 100-ft. mark. The following table con- 
 tains the results: 
 
 Distances. 
 
 Feet. 
 
 Spaces Intercepted on the Rod. 
 
 1st Series. 
 Feet. 
 
 2d Series. 
 Feet. 
 
 3d Series. 
 Feet. 
 
 Mean. 
 
 Feet. 
 
 50 
 
 .485 
 
 .4860 
 
 .4855 
 
 .4855 
 
 100 
 
 .985 
 
 .9870 
 
 .9830 
 
 .9850 
 
 200 
 
 1.985 
 
 1.9860 
 
 1.9840 
 
 1.9850 
 
 300 
 
 2.989 
 
 2.9875 
 
 2.9870 
 
 2.9878 
 
 400 
 
 3.983 
 
 3.9800 
 
 3.9890 
 
 3.9840 
 
 500 | 
 
 4.985 
 
 4.9850 
 
 4.9900 
 
 4.9867 
 
 Multiplying the mean of these readings by 100, and subtracting the result 
 from the corresponding distance, we obtain the following table: 
 
 *The above demonstration is substantially that given by Mr. George J. Specht in an article 
 on Topographical Surveying in “ Van Nostrand’s Engineering Magazine,” February 1880 
 though enlarged and corrected. ’ 
 
86 
 
 SURVEYING. 
 
 Distances. 
 
 Feet. 
 
 Mean of 
 Stadia Readings 
 Times 100. 
 Feet. 
 
 1 
 
 Differences. 
 
 Feet. 
 
 Variations 
 From Mean. 
 Feet. 
 
 50 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 48.55 
 
 98.50 
 
 198.50 
 
 298.78 
 
 398.40 
 
 498.67 
 
 1.45 
 
 1.50 
 
 1.50 
 
 1.22 
 
 1.60 
 
 1.33 
 
 | 
 
 + .02 
 + .07 
 + .07 
 — .21 
 + .17 
 — .10 
 
 
 r> nr «• 1 AQ 
 
 Stun of differences = 8.60; mean of difference - 1.43. 
 
 The variations between the 4^ft Ting oSy^Tft. & A 
 
 3SS ^SbteSsaaSBraeffleas* 
 
 is a constant value t P wone having a compound, plano-convex 
 
 taken that m * settii ig th< 3 s0 set that the reading, at any dis- 
 
 ment constant and mat tne ires am0 unt of this constant. f 
 
 tance, is less tton tiie ■ true d: ish a uy* h distances and elevations 
 
 onlfaVS^^ 
 
 -eragl height of the 
 
 telescope ^^.^S^o^r^ere this method will be impracticable, and 
 
 then^hemolieif procedure must be left to the^ud^^^^^^^ur^eyon If 
 
 rod, and the difference between this as the case may be. 
 
 used in sighting to the rod eit small that in a great deal of stadia 
 
 This difference will ordniarily be so ^ma 11 | rod for the angle of 
 
 work no reduction will he nece^ry. In sghtog TO t gt always be use d. B y 
 
 depression or elevation, the center h measured For theoretical exact- 
 this means an exactly continuous line is e “easure idistant fr0 m the 
 
 ness it is necessary that the stadia wires shoui^ne d ^ tance read is for an 
 
 angle°of elevatioVdifferin^ from the true one by an amount proportional to 
 
 tte SSn«eir g hd S 
 
 measurements. . The common errors o c f 1 ‘ff I JXng S a difference of a whole 
 Sff, ”5 of .noth,,, on. .ho»M „ „ 
 
 #This applies to an in S trument witli m 0 vable S ta<na ™ es h ' ^ 
 rlass. In the latter ease, the graUnation of “• ar J e not affected by variations of 
 
 o aimed as an advantage for etched lines “ ,f ey A serles 0 f tests made with one of 
 
 srrSigi^s tS” stzlk rrfS^TSmVr^ 
 
 between^hat'produced^n' theTnXmen. by the snn of a hot summer's day and that produce y 
 •”?&& wEER evMenflJ proportional to the length of sight, with a 1,000' sight it would 
 amount to 22.5', etc. 
 
stadia measurements. 
 
 87 
 
 even footmark, and in the check, reading the other one. This is assuming 
 the measurements to be made by the ordinary method, and not by the 
 
 approximate one of the United States Engineers. y 
 
 HORIZONTAL DISTANCES AND DIFFERENCES OF LEVEL FOR 
 STADIA MEASUREMENTS. 
 
 The formulas used in the computation of the following tables were those 
 given by Mr George J . Specht in an article on Topographical Surveying 
 published m Van Nostrand’s Engineering Magazine” for February *]880 
 These formulas furnish expressions for horizontal distances and differences 
 h *° y r . st 7 adla measurements, with the conditions that the stadia rod 
 
 They ^ areas follows” ^ Stadia wires be e( l uidistan t from the center wire. 
 
 = c cos n + a k cos 2 n\ 
 = D tan n — c sin n 
 
 ak sin2« 
 
 glass, plus focal 
 
 = horizontal distance; 
 
 = difference of level; 
 
 = distance from center of instrument to center of obie< 
 length of object glass; J 
 
 a = rea^ngSi^tadia rwi;^ aSS diVided by distanCf of stadia wires apart; 
 n = vertical angle; 
 
 a k = (genirfhy WO) multiplied by k ' which is a constant for each instrument 
 
 table ?*.the vertical columns consist of two series of numbers for 
 each degree, which series represent, respectively, the different values of 
 
 a k cos 2 n and for every 2 minutes, when ak — 100. To obtain the 
 
 horizontal distance or the difference of level in any case, the corresponding 
 value of c cos n or c sin n must further be added; and the mean of each of 
 these expressions, for each degree, with three of the most common values of 
 c, is given under each column. va±ucc ui 
 
 i et itbe re( l u i red to find the horizontal distance and the 
 
 contra nt ° f 7 ^ Vei T W +n n = + 6° 18', a = 570, and the instrument 
 constant c = J5- In the > column headed 6° opposite 18' in the series for 
 
 thfrefore, when a^i 9 570° &S the expressi0lJ for ak cos 2 n when ak = 100; 
 
 m ^ ^akco&n = 98.80 X 5.70 = 563.16. 
 
 column to^e S 75 be a(Med ° C ° S ^ which> in this case > is found in the subjoined 
 In a similar manner, the required difference of level is 
 ~ 4 . (+10.91X5.70) +.08 - +62.27. 
 
 One multiplication and one addition must be made in each case. 
 
 It is to be noticed that, with the smaller angles, cos n in the expressions 
 c cos n and c sin n may be entirely neglected without appreciable error 
 For values of c, which differ from those given, an approSSate^^tlon 
 ?wo I expre4o t ns+ amount of dlffer enee, may very easily be made in these 
 
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ELEMENTS OF MECHANICS. 
 
 91 
 
 ELEMENTS OF MECHANICS. 
 
 Only the elements of machines are here treated, as all machinery how- 
 ever complicated, is merely a combination of the six elementary forms viz • 
 the lever , the wheel- and axle , the pulley , the inclined plane, the wedqe and 
 the screw; and these six can be still further reduced to the lever and the 
 inclined plane. They are termed mechanical powers, but they do not 
 produce force; they are only methods of applying and directing it 
 
 The law of all mechanics is: 
 
 The power multiplied by the distance through which it moves is equal to the 
 weight multiplied by the distance through which it moves. 
 
 Thus, 20 ilb. of power moving through 5 ft. = 100 lb. of weight moving 
 through 1 ft. In the following discussion friction is not considered the 
 idea being to give an elementary knowledge of the principles of the 
 elements of mechanics. * 
 
 Levers.— There are three classes of levers. They are: (1) power at one 
 end, weight at the other, and fulcrum between; (2) power at one end 
 fuicrum at the other, and weight between; (3) weight at one end, fulcrum 
 at the other, and power between. 
 
 The handle of a blacksmith’s bellows is a lever of the first class. The 
 hand is the power and the bellows the weight, with the pivot between 
 as the fulcrum. A crowbar as used for prying down top rock is a lever of 
 the second class. The hand is the power, the rock to be barred down 
 the weight, and the point in the roof against which the bar presses 
 ^ the fulcrum. The treadle of a grindstone is a lever of the third class. 
 The foot is the power, the hinge at the back of the foot is the fulcrum 
 and the moving of the machinery is the weight. ’ 
 
 A lever is in equilibrium when the arms balance each other. The dis- 
 tances through which the power and the weight move depend on the 
 comparative length of the arms. Let L represent power’s distance from 
 the fulcrum (C), l the weight’s distance, and a the distance between power 
 and weight; then, if L is twice l, the power will move twice as far as the 
 weight. Substituting these terms in the law of mechanics, we have 
 
92 
 
 elements oe mechanics. 
 
 th e short* arm of The firs! actfoule lonTa™ of the second, Cd^so^nto 
 
 the last. . . fniprnm be four times the distance from the 
 
 If the distance from A to the fulcrum d m to B then a power of 
 
 5 lb. at A will lift 20 lb. at B. 
 If the arms of the second lever 
 
 13 D r-a 
 
 V 
 
 A 
 
 5 ) 
 
 E 
 
 B 
 
 6 
 
 Fig. 1. 
 
 ;ie aiuJia ^ 
 
 are of the same comparative 
 length, the 20-lb. power obtained 
 at B will exert a pressure of 80 
 lb on E\ and if the third lever 
 has the same comparative 
 
 lengths this. 80 lh.at-E wiUlift 320 lb. at G. Thus, ^power ^ wei ght 
 A will balance a weight of 320 id. 
 
 ' EZX. ¥ 5 . ;«. .h. . r . Th. 
 
 windlass is a common form. Thepower is^apphedto the 
 
 the handle, and the short arm is the radius oi t ine ax 
 Thiis Tis the fulcrum, Fc the long arm, and Fb the 
 
 S^aiSSJSSSVSS' t. proportional » ti.i. «U1. « Have 
 
 P :W : 
 
 Fig. 2. 
 
 Fig. 3. 
 
 r : R. PR = Wr. 
 
 Wr - Wr 
 
 P = TT R ~ P ‘ 
 
 RP RP 
 W~—. r = ^ 
 
 A train. Fig. 3, consists of a series of 
 wheels and axles that act on one another 
 on the principle of a compound lever. 
 The driver is the wheel to which power 
 is applied The driven , or follower, is 
 the one that receives m'diorifrointhe 
 driver. The pinion is the small gear 
 wheel on the axle. , , , • 
 
 If the diameter of the wheel A is 16 
 
 in and of the pinion B 4 m., a pull oi 
 1 lb applied at P will exert a force of 
 4 lb. on the wheel C ; if the diameter 
 
 of C is 6 in., and of D 3 in., a force of 4 lb. on C e | f jn °£ will raise a 
 E. If E is 16 in. in diameter, and F 4 tin., a force o amQunt ace0 rding to 
 
 weight of 32 lb on F 2,m Jd The wright will only pass through A of the 
 
 law of mechanics, 
 
 jyyppr ^ P R R r R f ^_ 
 
 P = p R"' " — rr f r" 
 
 n : n" : : r'r" : RR'. 
 v : v' :: rr'r" : RR'R". 
 n n / n" = number of revolutions; 
 
 ’ y v' = velocity or speed of rotation; 
 r r' r", etc. = radii of the pinions; F 4 
 
 JB R' P", etc. = radii of the wheels. 
 
ELEMENTS OF MECHANICS. 
 
 93 
 
 The Inclined Plane. — In Fig-. 5 we see that the power must descend a dis- 
 tance equal to A C in order to elevate the weight to the height B C; hence 
 we have P X length of the inclined plane = WX the height of the inclined 
 plane, or P : W : : height of inclined plane : length of inclined plane; or, 
 
 p = w _ pi l 
 
 l h sin a 
 
 ™ !?• Find the. Weight Required to Balance Any Weight on Any Inclined Plane. 
 
 Multiply the given weight by the sine of the angle 
 of inclination. 
 
 Thus, to find the weight required to balance a 
 loaded car weighing 2,000 lb. on a plane pitching 
 18°, we multiply 2,000 by the sine of 18°, or 2,000 X 
 .3090170 = 618.034 lb. 
 
 Or, if the length of the plane and the vertical 
 height are given, multiply the load bv the quotient 
 of the vertical height divided by the length. 
 
 Thus, if a plane between two levels is 300 ft. 
 long and rises 92.7 ft., and the load is 2,000 lb., the balancing weight is 
 found as follows: 6 
 
 92 7 
 
 2,000X— = 6 18+ . 
 
 Case 1 .— To find the horsepower required to hoist a given load up an 
 inclined plane in a given time, use the formula 
 
 Fig. 5. 
 
 | Load (in lb.) + weight of hoisting rope (in lb.) J 
 
 ( vertical height 
 X < through which the 
 (load is raised (in ft.) 
 
 }. 
 
 33.000 X time of hoisting (in minutes) 
 
 Example. —Find the horsepower required to raise, in 3 minutes, a car 
 
 weighing 1 ton and containing 1 ton of material up an inclined plane 1,000 
 ft. long and pitching 30°, if the rope weighs 1,500 lb. 
 
 c The total load equals car + contents + rope = 2,000 + 2,000 + 1,500 = 
 5,500 lb. 
 
 The vertical height through which the load is hoisted equals 
 
 1.000 X sin 30° = 1,000 X .5 = 500 ft. 
 
 • H P = 5,500 X 500 _ 27 7 
 
 . . j-l. x . 33 ; 000 X 3 
 
 Case 2. — When the power acts parallel to the base, use the formula 
 W x height of inclined plane = P X length of base. 
 
 These rules are theoretically correct, but in practice an allowance of 
 about 30 i<> must be made for friction and contingencies. 
 
 The screw consists of an inclined plane wound around a cylinder. The 
 inclined plane forms the thread, and the cylinder, the body. It works in a 
 nut that is fitted with reverse threads to move on the thread of the screw. 
 The nut may run on the screw, or the screw in the nut. The power may be 
 applied to either, as desired, by means of a wrench or a lever. 
 
 When the power is applied at the end of a lever, it describes a circle of 
 which the lever is the radius r. The distance through which the power passes 
 is the circumference of the circle; and the height to which the weight is 
 lifted at each revolution of the screw is the distance between two of the 
 threads, called the pitch ( p ). Therefore we have P X circumference of 
 circle = W X pitch, or P : W : : p : 2 n r. 
 
 Wp 
 
 
 r P 
 
 The power of 
 
 lengthening the lever 
 distance between the 
 
 P : W : 
 
 2 7r r p 
 
 the screw may be increased by 
 or by diminishing the 
 threads. 
 
 Example. — How great a weight can be raised 
 by a force of 40 lb. applied at the end of a wrench 
 14 in. long, using a screw with 5 threads per inch? 
 W X * * = 40 X 28 X 3.1416. 
 
 W = 17,593 lb. 
 
 The wedge usually consists of two inclined 
 planes placed back to back. (Fig. 6.) In theory, 
 the same formula applies to the wedge as to the 
 inclined plane, Case 2. 
 thickness of wedge : length of wedge. 
 
ELEMENTS OF MECHANICS . 
 
 Friction, in the other mechanical P°^rs, i^lft^each^ wTh^ wedge 
 efficiency; in this it is essential, since, ou ^ in in the others the powder 
 
 isappl^^ steady* force; in this it is a sudden blow, and is equal to the 
 
 m °Tlie 1 pu He y is*shnply™nothCT v ^- pqf a sing^ 
 
 TvllrlW a™ an y d a is e used to change the 
 
 direction of toeforce^ ^ _ ty of p 
 
 P = w. V = v‘ m 
 
 Mnuahle Pullev — A form of the single pulley, 
 Movable Kuney. we ight, is shown m 
 
 %L e f ta“his one half of tlfe weight is sus- 
 
 Srned by the hook, and the other half by the 
 a,.-. .. laineu .““r thp ko We r is only one-half the 
 
 rw??cth^^ 
 
 ^Combinations of Pulleys SSTlM 
 
 ^we S r ^Yatacef lto ofweight. \‘i) In Fig. 10, a power of lib. will in the 
 
 8 421 _ 9 _ 
 
 Fig. 7. 
 
 Fig. 8. 
 
 Fig. 12. 
 
 f-n e ll^re^resents the OTdi^ry^a^le^bloc^ used bymechK 
 
 wtoch catTbe calculated by the continuous rope is used, a 
 
 Rule.— In any combination of V ui ffV^ ™ fh mnra hie block as many times 
 
 'ing the load, not counting the free ma. has a tension 
 
 K&5 t&SS.’S-ffwftS r-« r. a . - of 
 
 voneys, p = W w = T P. 
 
 Differential Pulley.— Fig. 13. 
 
 2 " 
 w = 
 
 2 PR 
 
 Fig. 13. 
 
FRICTION AND LUBRICATION. 
 
 95 
 
 In all combinations of pulleys, nearly one-half the effective force is lost 
 by friction. 
 
 Composition of Forces. — When two forces act on a body at different angles, 
 their result may be obtained by the following rule: 
 
 Rule .— Through a point draw two lines parallel to the directions of the lines of 
 action of the two forces. With any convenient scale , measure off, from the point 
 of intersection, distances corresponding to the magnitudes of the respective forces, 
 and complete the parallelogram. From the common point of application, draw the 
 diagonal of the parallelogram; this diagonal will he the resultant, and its magni- 
 tude can he measured with the same scale that was used to measure the two forces. 
 When more than two forces act on a body simultaneously, find the resultant of 
 any two of them as above; then, by the same method, combine this resultant with a 
 third force, and this resultant with the fourth force , and so on. 
 
 FRICTION AND LUBRICATION. 
 
 Friction. — Friction is the resistance to motion due to the contact of 
 surfaces. It is of two kinds, sliding and rolling. If the surface of a body 
 could be made perfectly smooth, there would be no friction; but, in spite of 
 the most exact polish, the microscope reveals minute projections and cavities! 
 We till these with oil or grease, and thus diminish friction. Since no surface 
 can be made perfectly smooth, some separation of the two bodies must, in all 
 cases, take place in order to clear such projections as exist on the surfaces. 
 Therefore, friction is always more or less affected by the amount of the 
 perpendicular pressure that tends to keep them together. 
 
 The ultimate friction is the greatest frictional resistance that one body 
 sliding over another is capable of opposing to any sliding force when at rest. 
 
 The coefficient of friction is the proportion that the ultimate friction in a 
 given case bears to the perpendicular pressure. The coefficient of friction is 
 usually expressed in decimals; but sometimes, as in the case of cars and 
 engines, it is expressed in pounds (of friction) per ton. 
 
 The coefficient of friction equals the ultimate friction divided by the 
 perpendicular pressure, and the ultimate friction equals the perpendicular 
 pressure multiplied by the coefficient of friction. Thus, if we have a block 
 weighing 100 lb. standing on another block, and it takes 35 lb. pressure to 
 slide it, the coefficient of friction = x 3 0 5 ^, or .35. 
 
 Table of Coefficients of Friction. 
 
 Materials. 
 
 Smooth, Clean, 
 and Dry Plane 
 Surfaces. 
 
 Smooth Plane Sur- 
 faces, Perfectly 
 Lubricated With 
 Tallow. 
 
 Oak on oak 
 
 .40 
 
 .079 
 
 Wrought iron on oak 
 
 .62 
 
 .085 
 
 Wrought iron on cast iron 
 
 .19 
 
 .103 
 
 Wrought iron on wrought iron 
 
 .14 
 
 .082 
 
 "Wrought iron on brass 
 
 .17 
 
 .103 
 
 Cast iron on cast iron 
 
 .15 
 
 .100 
 
 Cast iron on brass 
 
 .15 
 
 .103 
 
 Steel on cast iron 
 
 .20 
 
 .105 
 
 Steel on steel 
 
 .14 
 
 
 Steel on brass 
 
 .15 
 
 .056 
 
 Brass on cast iron - 
 
 .22 
 
 .086 
 
 Brass on wrought iron 
 
 .16 
 
 .081 
 
 Brass on brass 
 
 .20 
 
 
 Oak on cast iron 
 
 
 .080 
 
 Oak on wrought iron 
 
 
 .098 
 
 Cast iron on oak 
 
 
 .078 
 
 Steel on wrought iron 
 
 
 .093 
 
 The above coefficients are only approximate, for the coefficient will vary 
 with the intensity of the pressure and the velocity, and also with the condi- 
 tions of the atmosphere. But they are correct enough for practical purposes. 
 
96 
 
 ELEMENTS GF MECHANICS. 
 
 Thp friction of liauids moving in contact with solid bodies is independent 
 nf thp nressure because the forcing of the particles of the fluid over the pro- 
 
 the area of surface or contact. 
 
 Coefficients of Friction in Axles. _ 
 
 Axle. 
 
 Bell metal 
 
 Cast iron 
 Wrought iron 
 Wrought iron 
 
 Cast iron 
 
 Cast iron 
 
 Wrought iron 
 
 Bearing. 
 
 Ordinary 
 
 Lubrication. 
 
 Bell metal 
 
 .097 
 
 Bell metal 
 
 .07 
 
 Bell metal 
 
 .07 
 
 Cast iron 
 
 .07 
 
 Cast iron 
 
 .07 
 
 Lignum vitae 
 
 .10 
 
 Lignum vitae 
 
 .12 
 
 Lubricated 
 
 Continuously. 
 
 .049 
 
 .05 
 
 .05 
 
 .05 
 
 rnmmmm 
 
 Frictional Resistance of Shafting- 
 
 Let K = coefficient of friction; 
 
 J “ re^hf^^ftSgTnWleys + the resultant stress of belts; 
 
 H = horsepower absorbed; 
 
 D = diameter of journal in inches; 
 
 E = number of revolutions per minute. 
 
 Then ’ Ordinary Oiling. Continuous Oiling. 
 
 ... _ m R 9 v P V D- .0112 X PX D\ „ „ 
 
 K = :S»5MXPXi)XP; .000000339 XPXPXPl 
 
 As f rough approximation, 100 ft. of shafting, 3 in. diameter, making 
 
 Friction,” under Vend- 
 
 ^Friction of Mine Cers.-The Metier .of 'mine ^™s |0 “U^h mat^h 
 
 ^^tto^heT^e^otdition of g the track and the lubrication are 
 
 important factors in determimngthe^ requisites of good 
 
 In this connection, we _may, however state some ol me U _ n dry 
 
 ings should renewal 6 andso contracted that, while it may 
 
 ^3^"^ ™t open by jarring or by being acer- 
 
 dentally struck by as P ra £ ^tP^nS^f^ilinff wheels that are improvements 
 onM^^^lS^^r^^ undoubtedly some 
 
 annular oil chambers, and those ^th ended to a welfiubricated bearing is 
 
 SgSS SK£i££^- «» '«■«> 
 
 Wa \Vith d a view of adopting a sta ™ da * d ^number 6 of U years^fth different 
 of W T ilkesbarre, Pa., experimented for a number oi years 
 
FRICTION AND LUBRICATION. 
 
 97 
 
 styles of self-lubricating wheels, and as a result of the experiments it adopted 
 a wheel patented by its chief engineer, Mr. Jas. H. Bowden. 
 
 Mr. R. Van A. Norris, E. M., Assistant Engineer, made a series of 989 tests 
 with old-style wheels, some of which had patent removable bushings, and 
 others annular oil chambers, and the Bowden wheel. The old wheels were 
 found to be practically alike in regard to friction. All the wheels were of 
 the loose outside type, 16 in. in diameter, mounted on 2} in. steel axles, 
 with journals 5£ in. long. The axles passed loosely through solid cast 
 boxes, bolted to the bottom sills of the cars, and were not expected to 
 revolve. 
 
 The table of friction tests shows the results obtained with both old- and 
 new-style wheels, and is of interest to all colliery managers, inasmuch as 
 the figures given for the old-style wheels alone are the most complete in 
 existence, and, as stated before, they are good averages. 
 
 Tests were made on the starting and running friction of each style of 
 wheel, under the conditions of empty and loaded cars, level and grade 
 track, curves, and tangents. The instruments used were a Pennsylvania 
 Railroad spring dynamometer, graduated to 3,000 lb., with a sliding recorder, 
 a hydraulic gauge (not recording) reading to 10,000 lb., graduated to 25 lb., 
 and a spring balance, capacity 300 lb., graduated to 3 lb. All these were 
 tested and found correct previous to the experiments. 
 
 Most of the observations on single cars were made with the 300-lb. 
 balance. The two types of “ old-style ” wheels have been classed together in 
 the table. Each car was carefully oiled before testing, and several of each 
 type were used, the results being averages from the number of trials shown 
 in the table. 
 
 In the experiments on the slow start and motion, the cars were started 
 very slowly by a block and tackle, and the reading was taken at the moment 
 of starting. They were then kept just moving along the track for a 
 considerable distance, and the average tractive force was noted, the whole 
 constituting one experiment. 
 
 The track selected for these experiments was a perfectly straight and 
 level piece of 42 in. gauge, about 200 ft. long, in rather better condition than 
 the average mine track. The cars were 41£ in. gauge, 3* ft. wheel base, 10 ft. 
 long, capacity about 85 cu. ft., with 6-in. topping. 
 
 To ascertain the tractive force required at higher speeds, trips of one, four, 
 and twenty cars, both empty and loaded, were attached to a mine locomotive 
 and run about a mile for each test, the resistance at various points on the 
 track, where its curve and grade were known, being noted, care also being 
 taken to run at a constant speed. Unfortunately, only four of the “new- 
 style” cars were available on the tracks where these trials were made. 
 
 The remarkably low results for the twenty-car trips are attributed to 
 variations in the condition of the track, and the fact that the whole train 
 was seldom pulling directly on the locomotive, the cars moving by jerks, so 
 that correct observations were impracticable. The hydraulic gauge was used 
 for these twenty-car tests, and the needle showed vibrations from 1 to 4 tons 
 and back. The mean was taken as nearly as possible. The gauge was 
 rather too quickly sensitive for the work, and the Pennsylvania Railroad 
 dynamometer was not strong enough to stand the starting jerks and the 
 strain of accelerating speed. 
 
 The tests marked “rope haul” were made on an empty-car haulage 
 system, about 500 ft. long, with overhead endless rope running continuously 
 at a speed of 180 ft. per min., the cars being attached to the moving rope by 
 a chain, a ring at the end of which was slipped over a pin on the side of the 
 car. The increase of friction on the heavier grades was due to the rope 
 pulling at a greater angle across the car. Correction was not made for this 
 angularity at the time, and the rope has since been rearranged, so that the 
 correction cannot now be made. There were not enough curve experi- 
 ments to permit the deduction of any general formula for the resistance 
 of these cars on curves. 
 
 The experiments on grade agree fairly well with those on a level, the 
 rather higher values obtained being probably due more to the greater effort 
 required in moving them, and the consequent jerkiness of the motion, than 
 to any real increase in resistance. As the experiments on all styles of 
 wheels were made in an exactly similar manner, the comparative value 
 of the results is believed to be nearly correct, the probable error in each set of 
 experiments, as computed by the method of least squares, varying from about 
 4$ for slow start and motion to 12$ for the rapid motion and twenty-car trips. 
 
Summary of Friction Tests on Susquehanna 
 
 98 
 
 ELEMENTS OF MECHANICS. 
 
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Summary of Friction Tests on Susquehanna Coal Co.’s Mine Cars, April, 1889— ( Continued). 
 
 FRICTION AND LUBRICATION. 
 
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100 
 
 ELEMENTS OF MECHANICS. 
 
 Lubrication.— There is probably no factor that has a more direct bearing 
 on the cost of production per ton of coal and ores than the lubrication o_ 
 mine machinery and yet it is doubtful if there is another item connected 
 with t^e operation of a mine less understood by owners, their managers, and 
 
 e "fteira S pfants a Ire'equipped with boilers of the highest known efficiency; 
 ® team P- stpam will heat the feedwater for 
 
 heaters are used that, by utilizing 
 boilers to the highest point. Mor 
 
 • waste steam, will heat the feedwater for 
 iern engines that will develop a horsepower 
 ^^S^o^fsteama^e installed; bends, instead of elbows, are 
 placed in steam^ndexhaust pipes, so that the friction and back pressure 
 mav be deduced to a minimum In a word, everything is done in the equip- 
 ment of a plant to secure economy in its operation. After all this is done, 
 freauently a long step is taken in the opposite direction by the use of an oil 
 unsuited to the existing conditions, and those m charge of the plant are led 
 to believe that the lubrication is all that could be desired, simply because 
 the ermines and machinery run quietly and the temperature of the bearings 
 ^e^not^t^co^^^hirmingly high. The office of a lubricant is not merely to 
 setfure^h^re^lt^but, primarily, to reduce friction and wear to a nnnimum; 
 fnd an oil that will do this is the best oil to use, no matter what the price 
 
 Pe Xv?eahze y th e e great loss in power due to the friction of wearing parts. 
 r>np nf the greatest living authorities on lubrication writes. 
 
 ° “ It ma v^probably he fairly estimated that one-half the power expended 
 in the ^vJrSe case whether in mill, mine, or workshop, is wasted on lost 
 work being consumed in overcoming the friction of lubricated surfaces. 
 
 He adds g that Treduction of 504 in the work lost by friction has often been 
 
 See As e o d ne y of m\“ S in°sta“h^ing the loss that will occur by the use of 
 inforinr Oihripunts attention is called to two flour mills located m one ot the 
 Middle 1 S^ate^OneoTthe plants was equipped with a condensmpngme 
 
 and valves found to be properly adjusted and the engine working within 
 Itfribuf edTo R ?ie ra bofltrs S °buT ^ 
 
 “ c " ey ’ " 
 
 “ to. inferior lubrica- 
 
 Among tne expeiibeh ^ k power produced in excess of that 
 
 iiisll 
 
 (9) 6 W°^randTaV n §fXchXrTwhich is constantly doing more work per 
 ^"idiere^^l^^ai^ele^S^o^danger^that^ought to r ® c . e ' ve t se ^9^P j 2“g^r 
 
 WUMm 
 
 ^it^m^t^ffieulMrfsm'article of this character, to do much more than 
 
FRICTION AND LUBRICATION. 
 
 101 
 
 point out the danger due to the use of inferior lubricants, leaving it to the 
 purchaser himself to determine as to the intrinsic worth of the lubricants 
 offered to him. In making his selection he would do well to consult with 
 and heed the advice of some highly responsible manufacturer of lubricants 
 who has given to the question, in all its phases, the most careful study, and 
 who would most probably have the benefit of a wide experience in the applica- 
 tion as well as the manufacture of lubricants. Some buyers have, to then 
 ultimate regret, adopted, as a method of determining the merits of lubri- 
 cants, a schedule of laboratory tests. Such a method is not only useless, but 
 it is misleading to any one other than a manufacturer of lubricants, who 
 makes use of it merely as a means of insuring uniformity in his manu- 
 factured products, and not as a measure whereby to judge their practical 
 value. Indeed, many oils can be very properly described by practically the 
 same schedule of tests, and yet are widely apart when their utility for a 
 given service is considered. . . 
 
 As a general guide in purchasing cylinder oil for mine lubrication, it 
 might be said that a dark-colored oil is of greater value, as a rule, than one 
 that has been filtered to a red or light amber color, as the process of filtration 
 necessarily takes from the oil a considerable percentage of its lubricating 
 value, and at the same time the process is an expensive one. In short, if a 
 light-colored oil is insisted upon, a high price must be paid for an inferior 
 lubricant. As a word of caution, however, it would be well to add right 
 here that irresponsible manufacturers frequently take advantage of the fact 
 that the most efficient and best known cylinder oils are dark-colored, and 
 endeavor, with more or less success, to market as “cylinder oil” products 
 absolutely unsuited to the lubrication of steam cylinders, and that would 
 consequently be expensive could they be procured without cost. 
 
 For the lubrication of engine bearings, where modern appliances for 
 feeding are used, an engine oil of a free running nature is best, as it more 
 quickly reaches the parts requiring lubrication than an oil of a more sluggish 
 nature. It, of course, must not be an oil susceptible to temperature changes, 
 but must be capable of performing the service required of it under the most 
 severe conditions, where an oil of less “ backbone ” would fail. Such an oil 
 would also be suitable for the lubrication of dynamos, and should also give 
 satisfaction where used in lubricating the cylinders of air compressors. 
 Where the machinery is of an old type and loose-jointed, or when the bear- 
 ings are open and the oil is applied directly to them by means of an oiler, 
 an engine oil of a more sluggish, or viscid, nature is best. 
 
 Perhaps of equal importance to the lubrication of power machinery 
 must be considered the lubrication of the axles of mine cars. This is 
 important, first, because of the fact that perhaps three-fourths of the oil 
 used about a coal mine is used for this purpose, and, secondly, because there 
 is really a marked difference in the quality and, therefore, in the efficiency 
 of lubricants used for this purpose. Fully nine-tenths of the prominent 
 railroads of this country are today using car-axle oil, costing perhaps as 
 much per gallon as much of the so-called cylinder oil that is used in coal 
 mines, they having discovered, by exhaustive experiments, that the increased 
 efficiency gained bv using an oil of such quality many times offsets the 
 difference in the cost per gallon and enables them to secure a greater mile- 
 age without any increase in their power or other fixed charges. This, we are 
 certain, would apply just as forcibly to the lubrication of coal cars, no 
 matter whether the ‘power is derived from “long-eared mules” or electric 
 motors, and we believe this feature of lubrication of mine equipment should 
 receive more careful attention than it does receive, as a rule. 
 
 There is a considerable amount of waste in the lubrication of mine cars. 
 This waste is hard to avoid, and, naturally, makes the buyer hesitate before 
 adopting the use of a car oil that costs very much per gallon; but we believe 
 it can be demonstrated, even in the face of this waste, that the increased 
 efficiency secured by the use of a high-grade car oil would warrant its use. 
 Such waste is pretty hard to correct in mines where the old-fashioned style 
 of car axles is still in use, and where the oil is applied through an ordinary 
 spout oil can into the axle box, and allowed to drip off the axles and on to 
 the ground. When axles are equipped in the same manner as those of 
 freight cars, or where cars are equipped with one of the several different 
 stylefc of patent car wheels and axles that are coming into use quite 
 extensively, it is possible to regulate the feeding of the oil to the axles, so as 
 to reduce the waste to a minimum. One of these patent car wheels, which 
 is perhaps better known than any other, is constructed with a hollow hub 
 
102 
 
 STRENGTH OF MATERIALS. 
 
 that acts as a reservoir tor the oil, the oil passing from this reservoir through 
 small holes onto a telt washer. which it must saturate, and by which n is 
 applied to the axles. Such wheels require a limpid oil. as a heavy^ sluggish 
 oil would not so readilv saturate the felt washer referred to. A tight cap is 
 adjusted to the end of the axle, to prevent waste of oil. These wheels will 
 run quite a length of time without reoiling after the reservoir is once filled. 
 
 Of course, it costs something to equip mine cars with these patent axles, 
 but we are convinced that such an outlay would result in more economical 
 operation, particularly if at the same time the very best quality of car oil 
 obtainable is used. 
 
 BEST LUBRICANTS FOR DIFFERENT PU RPOSES (TH U RSTON). 
 
 Low temperatures, as in rock drills 
 driven by compressed air 
 
 Very great pressures, slow speed 
 
 Heavy pressures, with slow speed 
 
 Heavy pressures and high speed 
 
 Light pressures and high speed 
 
 Ordinarv machinery - 
 
 { oils. 
 
 Steam cylinders - - - j Heavy mineral oils. lard, tallow. 
 
 ( Clarified sperm, neat’s foot, por- 
 Watches and other delicate mechanism. < poise, olive, and light mineral 
 
 ( lubricating oils. 
 
 For mixture with mineral oils, sperm is best; lard is much used; olive 
 and cottonseed are good. 
 
 STRENGTH AND WEIGHT OF MATERIALS 
 
 Light mineral lubricating oils. 
 
 Graphite, soapstone, and other 
 solid lubricants. 
 
 The above, and lard, tallow, and 
 other gTeases. 
 
 Sperm oil. castor oil, and heavy 
 mineral oils. 
 
 Sperm, refined petroleum, olive, 
 rape, cottonseed. 
 
 Lard oil. tallow oil. heavy mineral 
 oils, and the heavier vegetable 
 
 WOODEN BEAMS. 
 
 To Find the Quiescent Breaking Load of a Horizontal Square or Rectangular Beam 
 Supported at Both Ends and Loaded at the Middle. -Multiply the br^dth in 
 
 inches bv the square of depth in inches, divide the product by distance in 
 feet between the supports, and multiply the quotient by the constant given 
 in the table on the next page. Take safe working load one-third of break- 
 
 ing To°Find the Quiescent Breaking Load of a Horizontal Cylindrical Beam.— Divide 
 the cube of the diameter in inches by the distance between the supports in 
 
 feet, and multiplv the quotient by the constant. . . . , 
 
 When the load is uniformly distributed on the beam, the results obtained 
 by the above rules should be doubled. n . . , , . 
 
 Example 1. — Find the quiescent breaking load and safe working load oi 
 a yellow-pine collar 8 in. square. 12 ft. between legs. 
 
 Breaking load = 8 X 500 — 21,333 lb. for seasoned, and 10,666 lb. for 
 
 ^Saf^working load = 7.111 lb. for seasoned, and 3,556 lb. for green timber. 
 
 Example 2.— Find the quiescent breaking load, and the safe working 
 load of a hemlock collar 10 in. diameter, 7 ft, between legs. 
 
 103 . , OO, /1-i 
 
 Breaking load = ^ X 236 = 33,714 lb. for seasoned timber, and 
 
 = 16,857 lb. for green timber. 
 
 03 71 4 , 33,714 11.238 
 
 Safe working load = — = 11,238 lb. for seasoned, and - 6 or —j- 
 
 = 5.619 lb. for green timber. _ _ , , . 
 
 To Find the Load a Rectangular Collar Will Support When Its Depth Is Increased. 
 
 When the length and width remain constant, the load vanes as the square ot 
 the depth. 
 
IRON AND STEEL BEAMS. 
 
 103 
 
 Example —A rectangular collar 10 in deep supports 15,000 lb. What will 
 it support if its depth Ans . 
 
 Havin*r the Length and Diameter of a Collar, to Find the Diameter of a Longer Collar 
 to Support the Same Weight.— For the same load, the strength of collars vanes 
 as the cubes of their diameters, and inversely as their lengths. . 
 
 Example —If a collar 6 ft. long and 8 in. diameter supports a certain 
 weight, what must be the diameter of a collar 12 ft. long to support the same 
 weight? ■ 0 . 
 
 if 6 : f 12 : : 8 in. : 10+ m. Ans. 
 
 Having the Loads of Two Beams of Equal Length and the Diameter of One, to 
 Find the^Diameter Of the Other. — When the lengths are equal, the diameters 
 vary as the cube roots of the loads, or the cubes of the diameters vary as the 
 
 ^Example 1.— A beam 11 in. in diameter supports a load of 32,160 lb. What 
 will be the diameter of another beam the same length, to support a load 
 
 of 19,440 lb.? 
 
 if 32,160 : if 19,440 : : 11 : 9. Ans. 
 
 Example 2.— A beam 8 in. in diameter will support a load of 10,240 lb, 
 What load will a beam the same length and 7 m m diameter support? 
 vv 8 3 : 7 3 : : 10,240 : 6,860. Ans. 
 
 Table of Constants. 
 
 Calculated for seasoned timber. For green timber, take one-half of these 
 constants. Safe working load is one-third of breaking load. 
 
 Woods. 
 
 Ash, white 
 
 Ash, swamp 
 
 Ash, black 
 
 Balsam, Canada 
 Beech, white 
 
 Beech, red 
 
 Birch, black 
 
 Birch, yellow ... 
 
 Cedar, white 
 
 Chestnut 
 
 Elm 
 
 Elm, rock 
 
 Hemlock 
 
 Hickory 
 
 Ironwood 
 
 Square or 
 Rectan- 
 gular. 
 
 Round. 
 
 Woods. 
 
 Square or 
 Rectan- 
 gular. 
 
 Round 
 
 650 
 
 383 
 
 Locust 
 
 600 
 
 353 
 
 400 
 
 236 
 
 Lignum vitae 
 
 650 
 
 383 
 
 300 
 
 177 
 
 Larch 
 
 400 
 
 236 
 
 350 
 
 206 
 
 Maple 
 
 550 
 
 324 
 
 450 
 
 265 
 
 Oak, red or black ... 
 
 550 
 
 324 
 
 550 
 
 324 
 
 Oak, white 
 
 600 
 
 353 
 
 450 
 
 265 
 
 Oak, live 
 
 600 
 
 353 
 
 450 
 
 266 
 
 Pine, white 
 
 450 
 
 265 
 
 250 
 
 147 
 
 Pine, yellow 
 
 500 
 
 295 
 
 450 
 
 265 
 
 Pine, pitch 
 
 550 
 
 324 
 
 350 
 
 206 
 
 Poplar 
 
 550 
 
 324 
 
 600 
 
 353 
 
 Spruce ■' 
 
 450 
 
 265 
 
 400 
 
 236 
 
 Sycamore 
 
 500 
 
 295 
 
 650 
 
 383 
 
 Willow 
 
 350 
 
 206 
 
 600 
 
 353 
 
 
 
 1 
 
 To Find the Diameter of a Collar When the Weight Increases in Proportion to the 
 Length —Find the required diameter to support the same weight as the short 
 collar. Then the length of the short collar is to the length of the long one 
 as the diameter found to support the original weight is to the required 
 
 Example. — If a collar 6 ft. long, 8 in. in diameter, supports a certain weight, 
 what must be the diameter of a collar 12 ft. long to support twice the weignt ? 
 
 1 : 2 : : — : 4^, or 1 : 2 : : 2 X 8 3 : ( ) 3 , 
 
 6 
 
 or 
 
 12 
 
 fl:if2 ::8f2 :( ) == 12.7. Ans. 
 
 IRON AND STEEL BEAMS. 
 
 Constants for use in calculating strength of iron and steel beams: 
 
 Cast iron 
 
 Wrought iron * 1 * 
 
 Steel 5 000 
 
Safe Loads Uniformly Distributed for Standard and Special I Beams. 
 
 ( Tons of 2,000 Pounds. ) 
 
 104 
 
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PILLARS OR PROPS. 
 
 105 
 
 Sard steel will break the same as cast iron. # Soft steel will bend like 
 wrought iron. The elastic limit of wrought iron is reached at about 2,200 lb. 
 a o it dop« not break, we use the limit of elasticity. . , 
 
 Tn FinH thp Quiescent Breaking Load of a Horizontal Square or Rectangular 
 
 lron or S ee Be?m Su C r,ed a t Both En<ls and Loaded at tie Middle.-Multiply 
 
 iff? snuareof its depth in inches by its breadth in inches; multiply this result 
 by 6 th^?onstant for the material used, and divide by the length in feet 
 between the supports. For the neat load, subtract one-half the weight of 
 
 thebe Quiescent Breaking Load of a Cylindrical Iron or Steel Beam. 
 
 Find the breaking load of a square beam the sides of which are equal to the 
 
 d ' a ^fe waking 6 loatUn each of th™preceding cases is one-third of the breaking 
 load. If the load is equally distributed over the beam, it will be twice as 
 great. 
 
 PILLARS OR PROPS. 
 
 To Find the Crushing Load of Either Square or Rectangular Wooden } 
 
 ProDS.— Call one side 8 of the square or the least side of the rectangle the 
 breadth. Divide the square of the length in inches by the square of the 
 breadth' in inches multiply the quotient by .004, add 1 to the product, and 
 divide the constant in the following table by the result Then multiply 
 this quotient by the number of square inches m the end of the prop. 
 
 Constant . , 
 
 Or, breaking load m lb. = -r» — : ; ~~ X 0 a > 
 
 ( f x -°“) +1 
 
 when l = length in inches, b = breadth in inches, and d = depth in inches. 
 
 Crushing Loads of Well-Seasoned American Woods. 
 
 Wood. 
 
 Ash 
 
 Beech 
 
 Birch 
 
 Cedar, red 
 
 Cedar, white 
 
 Chestnut 
 
 Hemlock 
 
 Hickory 
 
 Linden 
 
 Locust, black, yellow 
 Locust, honey 
 Maple, broad-leafed 
 Oregon 
 
 Crushing 
 Load. Lb. 
 per Sq. In. 
 
 6,800 
 
 7.000 
 
 8.000 
 6.000 
 4,400 
 
 5.300 
 
 5.300 
 8,000 
 
 5.000 
 9.800 
 
 7.000 
 
 5.300 
 
 Wood. 
 
 Crushing 
 Load. Lb. 
 per. Sq. In. 
 
 Maple, sugar, black 
 
 Maple, white, red 
 
 Oak, white, red, black 
 
 Oak, scrub, basket 
 
 Oak, chestnut, live 
 
 Oak, pin - 
 
 Pine, white 
 
 Pine, pitch 
 
 Pine, Georgia 
 
 Poplar 
 
 Spruce, black 
 
 Spruce, white 
 
 Willow 
 
 8,000 
 
 6,800 
 
 7.000 
 
 6.000 
 
 7.500 
 
 6.500 
 
 5.400 
 5,000 
 
 8.500 
 5,000 
 5,700 
 
 4.500 
 
 4.400 
 
 For green timber, take one-half of the constants or crushing strength. 
 Safe working load equals one-third of crushing load. ■ , -rvr,c+ 
 
 Example 8 — What is the breaking load of a well-seasoned hemlock post 
 
 10 in. by 8 in. and 12 ft. long? 
 
 5,300+ [l + (^T X - 004 )] = 2,308.4 lb. per sq. in. of area. 2,308.4 X 80 
 
 = Tcf Find 'the Breaking Load of a Cylindrical Wooden 
 
 load of a square prop whose ends are equal m area to those oi me cym 
 
 drical one and proceed according to foregoing rule. . . 
 
 Example.— W hat is the safe working load for a hemlock mine prop 10 m. 
 
 dia The e area of the^nd of the prop = 78.54 sq. in. A square of equal area 
 will have sides equal to 1/ 78.54 = 8.86+ in. 
 
 Then, 5,300 + [l + X .004)] = 3,058.3 lb. per each sq. In. of area. 
 
106 
 
 STRENGTH OF MATERIALS. 
 
 And 3,058.3 X 78.54 = 240,198 lb. This is the crushing strength of a similar 
 prop of seasoned timber, but, as mine timber is used in its green state, we 
 take one-half of 240,198 lb., or 120,099 lb., as the crushing load of the prop in 
 question. Then, the safe working load is one-third of this, or 40,033 lb. 
 
 The strength of similar props varies as the cubes of their diameters, and 
 inversely as their lengths. 
 
 Safe Loads, in Tons of 2,000 Pounds, for Hollow Cylindrical Cast-Iron 
 
 Columns. 
 
 (The Carnegie Steel Co ., Limited.) 
 
 Outside Diam. 
 Inches. 
 
 Thickness of 
 Metal. 
 
 Length of Columns in Ft. 
 
 Sectional Area. 
 Inches. 
 
 Weight in Lb. of 
 Columns per 
 Ft. of Length. 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 | 
 
 18 
 
 20 
 
 22 
 
 24 
 
 Tons. 
 
 Tons. 
 
 Tons. 
 
 Tons. 
 
 1 
 
 Tons. 
 
 Tons. 
 
 Tons. 
 
 Tons. 
 
 Tons. 
 
 6 
 
 £ 
 
 26.2 
 
 23.0 
 
 20.1 
 
 17.5 
 
 15.2 
 
 13.2 
 
 11.5 
 
 
 
 8.6 
 
 26.95 
 
 6 
 
 f 
 
 37.5 
 
 33.0 
 
 28.8 
 
 25.0 
 
 21.7 
 
 18.9 
 
 16.5 
 
 
 
 12.4 
 
 38.59 
 
 6 
 
 7 
 
 42.7 
 
 37.6 
 
 32.8 
 
 28.5 
 
 24.7 
 
 21.5 
 
 18.8 
 
 
 
 14.1 
 
 43.96 
 
 6 
 
 1 
 
 47.6 
 
 41.9 
 
 36.5 
 
 3i.8 
 
 27.6 
 
 24.0 
 
 21.0 
 
 
 
 15.7 
 
 49.01 
 
 6 
 
 1| 
 
 52.2 
 
 46.0 
 
 40.1 
 
 34.8 
 
 30.2 
 
 26.3 
 
 23.0 
 
 
 
 17.2 
 
 53.76 
 
 7 
 
 f 
 
 47.7 
 
 43.1 
 
 38.5 
 
 34.3 
 
 30.4 
 
 26.9 
 
 23.9 
 
 21.2 
 
 18.9 
 
 14.7 
 
 45.96 
 
 7 
 
 l 
 
 61.1 
 
 55.2 
 
 49.3 
 
 43.8 
 
 38.9 
 
 34.4 
 
 30.6 
 
 27.1 
 
 24.2 
 
 18.9 
 
 58.90 
 
 7 
 
 1} 
 
 67.2 
 
 60.8 
 
 54.3 
 
 48.3 
 
 42.8 
 
 37.9 
 
 33.7 
 
 29.9 
 
 26.7 
 
 20.8 
 
 64.77 
 
 8 
 
 2 
 
 57.9 
 
 53.3 
 
 48.6 
 
 44.1 
 
 39.7 
 
 35.8 
 
 32.2 
 
 28.9 
 
 26.1 
 
 17.1 
 
 53.29 
 
 8 
 
 l 
 
 74.6 
 
 68.7 
 
 62.5 
 
 56.7 
 
 51.1 
 
 46.0 
 
 41.4 
 
 37.3 
 
 33.6 
 
 22.0 
 
 68.64 
 
 8 
 
 1£ 
 
 89.9 
 
 82.8 
 
 75.5 
 
 68.4 
 
 61.7 
 
 55.5 
 
 49.9 
 
 44.9 
 
 40.5 
 
 26.5 
 
 82.71 
 
 9 
 
 f 
 
 68.1 
 
 63.6 
 
 58.9 
 
 54.2 
 
 49.6 
 
 45.2 
 
 41.2 
 
 37.5 
 
 34.1 
 
 19.4 
 
 60.65 
 
 9 
 
 l 
 
 88.0 
 
 82.3 
 
 76.2 
 
 70.0 
 
 64.1 
 
 58.4 
 
 53.2 
 
 48.4 
 
 44.1 
 
 25.1 
 
 78.40 
 
 9 
 
 H 
 
 106.6 
 
 99.6 
 
 92.2 
 
 84.8 
 
 77.6 
 
 70.8 
 
 64.4 
 
 58.7 
 
 53.4 
 
 30.4 
 
 94.94 
 
 9 
 
 U 
 
 123.8 
 
 115.7 
 
 107.1 
 
 98.5 
 
 90.1 
 
 82.2 
 
 74.8 
 
 68.1 
 
 62.0 
 
 35.3 
 
 110.26 
 
 9 
 
 H 
 
 139.6 
 
 130.5 
 
 120.8 
 
 111.1 
 
 101.6 
 
 92.7 
 
 84.4 
 
 76.8 
 
 69.9 
 
 39.9 
 
 124.36 
 
 10 
 
 1 
 
 101.4 
 
 95.9 
 
 89.8 
 
 83.6 
 
 . 77.4 
 
 71.5 
 
 65.8 
 
 60.5 
 
 55.5 
 
 28.3 
 
 88.23 
 
 10 
 
 H 
 
 123.3 
 
 116.5 
 
 109.1 
 
 101.6 
 
 94.1 
 
 86.8 
 
 79.9 
 
 73.4 
 
 67.5 
 
 34.4 
 
 107.23 
 
 10 
 
 1 £ 
 
 143.7 
 
 135.8 
 
 127.3 
 
 118.5 
 
 109.7 
 
 101.2 
 
 93.2 
 
 85.6 
 
 78.7 
 
 40.1 
 
 124.99 
 
 10 
 
 If 
 
 162.7 
 
 153.8 
 
 144.1 
 
 134.1 
 
 124.2 
 
 114.6 
 
 105.5 
 
 97.0 
 
 89.1 
 
 45.4 
 
 141.65 
 
 11 
 
 1 
 
 114.8 
 
 109.4 
 
 103.5 
 
 97.3 
 
 91.0 
 
 84.8 
 
 80.2 
 
 73.1 
 
 67.7 
 
 31.4 
 
 98.03 
 
 11 
 
 H 
 
 139.9 
 
 133.3 
 
 126.1 
 
 118.6 
 
 110.9 
 
 103.3 
 
 97.8 
 
 89.4, 
 
 82.5 
 
 38.3 
 
 119.46 
 
 11 
 
 U 
 
 163.5 
 
 155.9 
 
 147.5 
 
 138.6 
 
 128.7 
 
 120.8 
 
 114.3 
 
 104.1 
 
 96.4 
 
 44.8 
 
 139.68 
 
 11 
 
 1 $ 
 
 185.7 
 
 177.1 
 
 167.5 
 
 157.5 
 
 147.3 
 
 137.2 
 
 129.8 
 
 118.3 
 
 109.5 
 
 50.9 
 
 158.68 
 
 11 
 
 2 
 
 206.6 
 
 196.9 
 
 186.3 
 
 175.1 
 
 163.8 
 
 152.6 
 
 144.4 
 
 131.5 
 
 121.8 
 
 56.6 
 
 176.44 
 
 12 
 
 1 
 
 128.0 
 
 122.9 
 
 117.2 
 
 111.0 
 
 104.7 
 
 98.4 
 
 92.2 
 
 86.1 
 
 80.4 
 
 34.6 
 
 107.51 
 
 12 
 
 1£ 
 
 156.4 
 
 150.1 
 
 143.1 
 
 135.7 
 
 127.9 
 
 120.2 
 
 112.6 
 
 105.2 
 
 98.2 
 
 42.2 
 
 131.41 
 
 12 
 
 li 
 
 183.3 
 
 175.9 
 
 167.7 
 
 159.0 
 
 149.9 
 
 140.9 
 
 132.0 
 
 123.3 
 
 115.1 
 
 49.5 
 
 154.10 
 
 12 
 
 1$ 
 
 208.7 
 
 200.4 
 
 191.0 
 
 181.1 
 
 170.7 
 
 160.4 
 
 150.3 
 
 140.5 
 
 131.1 
 
 56.4 
 
 175.53 
 
 12 
 
 2 
 
 232.7 
 
 223.4 
 
 213.0 
 
 201.9 
 
 190.4 
 
 178.9 
 
 167.6 
 
 156.6 
 
 146.1 
 
 62.8 
 
 195.75 
 
 13 
 
 1 
 
 141.2 
 
 136.3 
 
 130.7 
 
 124.7 
 
 118.5 
 
 112.1 
 
 105.8 
 
 99.5 
 
 93.5 
 
 37.7 
 
 117.53 
 
 13 
 
 li 
 
 172.8 
 
 166.8 
 
 160.0 
 
 152.7 
 
 145.0 
 
 137.2 
 
 129.4 
 
 121.8 
 
 114.4 
 
 46.1 
 
 143.86 
 
 13 
 
 1£ 
 
 203.0 
 
 195.9 
 
 187.9 
 
 179.3 
 
 170.3 
 
 161.1 
 
 152.0 
 
 143.1 
 
 134.3 
 
 54.2 
 
 168.98 
 
 13 
 
 W 
 
 231.6 
 
 223.6 
 
 214.5 
 
 204.7 
 
 194.4 
 
 183.9 
 
 173.5 
 
 163.3 
 
 153.3 
 
 61.9 
 
 192.88 
 
 13 
 
 2 
 
 258.9 
 
 249.9 
 
 239.7 
 
 228.7 
 
 217.3 
 
 205.5 
 
 193.9 
 
 182.5 
 
 171.3 
 
 69.1 
 
 2x5.56 
 
 14 
 
 1 
 
 154.3 
 
 149.6 
 
 144.3 
 
 138.5 
 
 132.3 
 
 125.9 
 
 119.5 
 
 113.1 
 
 106.8 
 
 40.8 
 
 127.6C 
 
 14 
 
 1 £ 
 
 189.2 
 
 183.4 
 
 176.9 
 
 169.7 
 
 162.2 
 
 154.4 
 
 146.5 
 
 138.6 
 
 131.0 
 
 50.1 
 
 156.31 
 
 14 
 
 1 £ 
 
 222.6 
 
 215.8 
 
 208.1 
 
 199.7 
 
 190.8 
 
 181.7 
 
 172.3 
 
 163.1 
 
 154.1 
 
 58.9 
 
 183.67 
 
 14 
 
 If 
 
 254.4 
 
 246.7 
 
 237.9 
 
 228.3 
 
 218.1 
 
 207.6 
 
 197.0 
 
 186.5 
 
 176.2 
 
 67.4 
 
 21C.0G 
 
 14 
 
 2 
 
 284.8 
 
 276.2 
 
 266.4 
 
 255.6 
 
 244.2 
 
 232.4 
 
 220.6 
 
 208.8 
 
 197.2 
 
 75.4 
 
 235.12 
 
 15 
 
 1 
 
 167.4 
 
 162.9 
 
 157.8 
 
 152.1 
 
 146.0 
 
 139.7 
 
 133.3 
 
 126.8 
 
 120.4 
 
 44.0 
 
 137.28 
 
 15 
 
 1£ 
 
 205.5 
 
 200.0 
 
 193.7 
 
 186.7 
 
 179.3 
 
 171.5 
 
 163.6 
 
 155.7 
 
 147.9 
 
 54.0 
 
 168.48 
 
 15 
 
 1£ 
 
 242.1 
 
 235.7 
 
 228.2 
 
 220.0 
 
 211.2 
 
 202.1 
 
 192.8 
 
 183.5 
 
 174.2 
 
 63.6 
 
 198.74 
 
 15 
 
 If 
 
 277.2 
 
 269.8 
 
 261.3 
 
 251.9 
 
 241.9 
 
 231.4 
 
 220.7 
 
 210.1 
 
 199.5 
 
 72.9 
 
 227.45 
 
 15 
 
 2 
 
 310.8 
 
 302.5 
 
 293.0 
 
 282.5 
 
 271.2 
 
 259.5 
 
 247.5 
 
 235.5 
 
 223.6 
 
 1 81.7 
 
 254.90 
 
 j 
 
SPECIFIC GRA VITY. 
 
 107 
 
 Minimum Safe-Bearing Values of Masonry Materials. 
 
 Materials. 
 
 Tons per Sq. Ft. 
 
 Granite, capstone 
 
 Squared masonry 
 
 Sandstone, capstone 
 
 Squared masonry 
 
 Rubble, laid in lime mortar 
 
 Rubble, laid in cement mortar 
 
 Limestone, capstone 
 
 Squared masonry 
 
 Rubble, laid in lime mortar 
 
 Rubble, laid in cement mortar 
 
 Bricks, hard, laid in lime mortar 
 
 Hard, laid in Portland cement mortar 
 
 Hard, laid in Rosendale cement mortar ... 
 Concrete, 1 Portland, 2 sand, 5 broken stone 
 
 50 
 
 25 
 
 25 
 
 12 
 
 5 
 
 10 
 
 36 
 
 18 
 
 5 
 
 10 
 
 7 
 
 14 
 
 10 
 
 10 
 
 SPECIFIC GRAVITY, WEIGHT, AND PROPERTIES OF MATERIALS, ETC. 
 
 The specific gravity of a body is the ratio of its weight to the weight of 
 an equal balk of pure water, at a standard temperature (62° F. = 16.670° C.). 
 
 Some experimenters have used 60° F. as the standard temperature, others 
 32° and still others, 39.1°. To reduce a specific gravity, referred to water 
 at 39 1° F., to the standard of water at 62° F., multiply by 1.00112. 
 
 Given specific gravity referred to water at 62° F., multiply by 62.355 to 
 find the weight of a cubic foot of the substance. Given weight per cubic 
 foot, to find specific gravity, multiply by 0.016037. . . _ 
 
 Given specific gravity, to find the weight per cubic inch, multiply by 
 
 0. 036085. nd ^ Specific Gravity of a Solid Heavier Than Water.— Weigh the body 
 both in air and in water, and divide the weight in air by the difference of 
 the weights in air and water. 
 
 Example.— A piece of coal weighs, say, 480 grains. Loss of weight 
 when weighed in water, 398 grains. 
 
 1 , 000 . 
 
 1.206, specific gravity of the coal compared with water at 
 
 As a cubic foot of water weighs, approximately, 1,000 oz., the weight of a 
 cubic foot of any substance can be found by multiplying its specific gravity 
 
 by To°°Fi'nd the Specific Gravity of a Solid Lighter Than Water.— Attach to it 
 another body heavy enough to sink it; weigh severally the compound mass 
 and the heavier body in water, divide the weight of the body in air by the 
 weight of the body in air plus the weight of the sinker in water minus the 
 combined weight of the sinker and body in water. 
 
 To Find the Specific Gravity of a Fluid.— Weigh both in and out of the fluid 
 a solid (insoluble) of known specific gravity, and divide the product of the 
 weight lost in the fluid and the specific gravity of the solid by the weight of 
 
 th ¥he weight of a cubic foot of water at a temperature of 62° is about 
 1 000 oz. avoirdupois, and the specific gravity of a body, water being 1,000, 
 shows the weight of a cubic foot of that body in ounces avoirdupois. 
 Then, if the magnitude of the body be known, its weight can be com- 
 puted; or, if its weight be known, its magnitude can be calculated, provided 
 we know its specific gravity; or, of the magnitude, weight, and specific 
 gravity, any two being known, the third may be found. 
 
 To Find the Magnitude of a Body in Cubic Feet From its Weight.— Divide the 
 weight of the body in ounces by 1,000 times the specific gravity of the body. 
 
 To Find the Weight of a Body in Ounces From Its Magnitude.— Divide the 
 weight of the body in ounces by the specific gravity of the substance mul- 
 tiplied by 1,000. . , , ., . ,, . 
 
 Note— The specific gravitv of any substance is equal to its weight in 
 grams per cubic centimeter. (See table of metric weights and measures.) 
 
108 
 
 WEIGHT OF MATERIALS . 
 Specific Gravity of Substances- 
 
 Substance. 
 
 Air, atmospheric; at 60° F. under pressure of 1 at- 
 mosphere, or 14.7 lb. per sq. in. 
 
 Alcohol, pure 
 
 Alcohol, of commerce 
 
 Aluminum 
 
 Anthracite* coal - 
 
 Asphaltum 
 
 Brass, cast 
 
 Brass, rolled - 
 
 Bronze, gun metal " 
 
 Brick, best pressed 
 
 Brick, common hard 
 
 Carbonic-acid gas 
 
 Clay, dry, in lumps, loose 
 
 Clay, potters’, dry ... - 
 
 Coke, f loose, of good coal 
 
 Coal, bituminous % 
 
 Coal, bituminous, broken loose . 
 
 Coal, bituminous, moderately shaken 
 
 Copper, cast 
 
 Copper, rolled 
 
 Earth, common ioam, perfectly dry, loose 
 
 Earth, common loam, perfectly dry, shaken ■■■•" 
 
 Earth, common loam, perfectly dry , moderately packed 
 
 Earth, common loam, slightly moist, loose 
 
 Earth, common loam, more moist, loose 
 
 Earth, common loam, more moist, shaken 
 
 Earth, common loam, more moist, packed 
 
 Earth, common loam, as a soft flowing mud 
 
 Earth, common loam, as a soft mud packed 
 
 Gold, cast, pure or 24 carat 
 
 Gravel 
 
 Gutta percha 
 
 Gypsum (plaster of Pans) 
 
 Gypsum, in irregular lumps 
 
 Gypsum, ground, loose 
 
 Gypsum, ground, well shaken 
 
 Gvnsum, calcined, loose 
 
 Hydrogen gas, 14£ times lighter than air and lb times 
 
 lighter than oxygen 
 
 Ice - 
 
 Iron, cast 
 
 Iron, rolled bars 
 
 Iron, sheet 
 
 Iron, wrought 
 
 Lead - 
 
 Lime, q uick - - • - - - • ■ — • ■ - v ■ - • • • * vr • 
 
 Lime, quick, ground, loose, per struck bushel, 66 lb. 
 
 Mercury, at 32° F. 
 
 Mercury, at 60° F 
 
 Mercury, at 212° F ■ ■ ; - 
 
 Nitrogen gas, & part lighter than air 
 Oils, whale, olive 
 
 Average 
 Specific ^ 
 Gravity. ( 
 
 Average 
 Weight per 
 Du. Ft. Lb. 
 
 .00123 
 
 .0765 
 
 .793 
 
 49.43 
 
 .834 
 
 52.1 
 
 2.66 
 
 166.0 
 
 1.5 
 
 93.5 
 
 1.4 
 
 , 87.3 
 
 8.1 
 
 504.0 
 
 8.4 
 
 524.0 
 
 8.5 
 
 529.0 
 
 150.0 
 
 125.0 
 
 .00187 
 
 .1167 
 
 63.0 
 
 1.9 
 
 119.0 
 
 27.5 
 
 1.35 
 
 84.0 
 
 50.0 
 
 54.0 
 
 8.7 
 
 542.0 
 
 8.9 
 
 555.0 
 
 .25 
 
 15.6 
 
 76.0 
 
 87.0 
 
 95.0 
 
 78.0 
 
 80.0 
 
 90.0 
 
 96.0 
 108.0 
 115.0 
 
 19.26 
 
 1,204.0 
 
 98.0 
 
 .98 
 
 61.1 
 
 2.27 
 
 141.6 
 
 82.0 
 
 56.0 
 
 64.0 
 
 56.0 
 
 .92 
 
 7.21 
 
 7.65 
 
 7.77 
 
 11.38 
 1.5 
 
 13.62 
 
 13.58 
 
 13.38 
 
 .92 
 
 .00527 
 
 57.4 
 
 450.0 
 
 480.0 
 
 485.0 
 
 485.0 
 709.6 
 
 95.0 
 
 53.0 
 
 849.0 
 
 846.0 
 
 836.0 
 .0744 
 
 57.3 
 
 * Anthracite increases about 75 per cent, in bulk when broken to any market size. A ton 
 
 ’“n'A^e^SeTSoolw&sfrom 35 to 42 lb, A «ou occupies « M» M -« ; «• 
 
 1 1 heaped bushel, loose, weighs about 74 lb., and a ton occupies from 43 to 48 o. ft. Bttumt 
 nous coal, when broken, occupies 75 per cent, more space than in the solid. 
 
 L 
 
 - 
 
SPECIFIC GRAVITY. 
 
 Specific Gravity of Substances— (Contin ued). 
 
 109 
 
 Substance. 
 
 Oxygen gas, part heavier than air 
 
 Petroieum 
 
 Powder - 
 
 Rosin 
 
 Silver 
 
 Slate 
 
 Steel 
 
 Sulphur — »—*••• 
 
 Tallow 
 
 cast 
 
 Water, purej rain or distilled, at 32° F., Barom. 30 in. 
 Water, pure, rain or distilled, at 62° F., Barom. 30 in. 
 Water, pure, rain or distilled, at 212° F., Barom. 30 in. 
 
 Water, sea, average - 
 
 Zinc 
 
 Average 
 
 Specific 
 
 Gravity. 
 
 Average 
 Weight per 
 Cu. Ft. Lb. 
 
 .00136 
 
 .0846 
 
 .878 
 
 54.8 
 
 1.5-1.85 
 
 105.0 
 
 1.1 
 
 68.6 
 
 10.5 
 
 655.0 
 
 2.8 
 
 175.0 
 
 7.85 
 
 490.0 
 
 2.0 
 
 125.0 
 
 .94 
 
 58.6 
 
 7.35 
 
 459.0 
 
 
 62.417 
 
 1.00 
 
 62.355 
 
 
 59.7 
 
 1.028 
 
 64.08 
 
 7.00 
 
 437.5 
 
 The following table gives the specific gravities of various coals: 
 
 Name of Coal. 
 
 Sp. Gr. 
 
 Weight of a 
 Cu. Ft. Lb. 
 
 Weight of a 
 Cu.Yd. Tons. 
 
 Newcastle Hartley, England 
 
 1.29 
 
 80.6 
 
 .972 
 
 Wigan, 4 ft., England 
 
 1.20 
 
 75.0 
 
 .914 
 
 Portland, England 
 
 1.30 
 
 81.2 
 
 .978 
 
 Anthracite, Wales 
 
 1.39 
 
 86.9 
 
 1.047 
 
 Eglington, Scotland 
 
 1.25 
 
 78.1 
 
 .941 
 
 Anthracite, Irish 
 
 1.59 
 
 99.4 
 
 1.193 
 
 Anthracite, Pennsylvania 
 
 1.55 
 
 96.9 
 
 1.167 
 
 Bituminous, Pennsylvania 
 
 1.40 
 
 87.5 
 
 . 1.054 
 
 Block coal, Indiana 
 
 1.27 
 
 79.4 
 
 .956 
 
 SPECIFIC GRAVITY AND WEIGHT OF PREPARED ANTHRACITE COAL. 
 
 To Mr. Irving A. Stearns, General Superintendent of the Pennsylvania 
 Railroad Co.’s Coal Department, we are indebted for the following sum- 
 mary of tests made by the mining engineers of the company. 
 
 In a series of tests to ascertain the specific gravity of the coal from differ- 
 ent seams worked by the company, it was found that the average specific 
 gravity was 1.4784, and the average weight per cubic foot was 92.50 lb. This 
 was calculated for space filled at breaker without settling. Add 5 °jo for 
 packed spaces of large heaps. 
 
 Weight per Cubic Foot of Susquehanna Coal Co.’s White Ash Anthra- 
 cite Coal. 
 
 Size. 
 
 Size of Mesh. 
 
 Weight peri 
 Cu. Ft. 1 
 Pounds. 1 
 
 > 
 
 Cu.Ft. Froml 
 
 Over. 
 
 Through. 
 
 Cu. Ft. Solid. 
 
 Lump 
 
 Broken 
 
 4£" to 9 " 
 2f" to 2f ' 
 
 3i" to 4i" 
 
 57 
 
 53 
 
 1.614 
 , 1.755 
 
 Egg 
 
 11" to 2|" 
 
 2f" to 2£" 
 
 52 
 
 1.769 
 
 Large stove 
 
 H" to H" 
 
 1 *" to 2i" 
 
 51* 
 
 1.787 
 
 Small stove 
 
 1 " to H" 
 
 H" to H" 
 
 5H 
 
 1.795 
 
 Chestnut 
 
 I" to V' 
 
 1 " to li" 
 
 51 I 
 
 | 1.804 
 
 Pea 
 
 I" to f" 
 
 £"to 
 
 50* 
 
 1.813 
 
 No. 1 buckwheat 
 
 xV'to f" 
 
 f" to 
 
 •50* 
 
 1.813 
 
 .No. 2 buckwheat 
 
 T V'to §" 
 
 50* 
 
 1.813 
 
110 
 
 WEIGHT OF MATERIALS. 
 
 LINE SHAFTING. 
 
 Shafting 1 is usually made cylindrically true, either by a special rolling 
 process, when it is known as cold-rolled shafting oi it is turned u|p m a 
 machine called a lathe. In the latter case, it is called bright shafting. What 
 is known as black shafting is simply bar iron rolled by the ordinary process 
 and turned where it receives the couplings pulleys, bearings etc 
 
 Bright turned shafting varies m diameter by 4 m. up to about 3 2 in. m 
 diameter; above this diameter the shafting varies by 2 in- Tiie actual 
 diameter of a bright shaft is less than the commercial diameter it 
 
 being designated from the diameter of the ordinary round bar iron, from 
 whfch it is turned. Thus, a length of what is called 3" bright shafting is 
 
 ° n ^Cold-roiied shutting is designated by its commercial diameter; thus, a 
 leneth of what is called 3" shafting is 3 in. in diameter. 
 
 Cold-rolled iron is considerably stronger than ordinary turned wrought 
 iron* the increased strength being due to the process of rolling, which seems 
 to° compreffi tho metal and so make it denser-not merely skin deep, but 
 practically throughout the whole diameter. 
 
 STRENGTH OF SHAFTING. 
 
 D = diameter of shaft; 
 
 jt = revolutions per minute; 
 
 H = horsepower transmitted; 
 
 C = constant given in table. 
 
 Constants foe, Line Shafting. 
 
 in the accompanying table the bearings are WPOsed to » 
 
 excessive bending; also, 
 in the third vertical col- 
 umn, an average num- 
 ber and weight of pulleys 
 and power given off is 
 assumed. 
 
 In determining the 
 constants given in the 
 accompanying table, al- 
 
 
 No Pulleys 
 
 Pulleys 
 
 Material of Shaft. 
 
 Between 
 
 Between 
 
 
 Bearings. 
 
 Bearings. 
 
 Steel or cold-rolled iron 
 
 65 
 
 85 
 
 Wrought iron 
 
 Cast iron - 
 
 70 
 
 9C 
 
 95 
 
 120 
 
 ii 
 
 D 3 X R 
 
 1 ) = 
 
 s 1C X H 
 
 M R 
 
 R = 
 
 CXH 
 
 H * 
 
 Shafts are subject to forces that produce stresses of two kmds-transverse 
 and torsional. When the machines to bedriven are below the shaft, there 
 is a transverse stress on the shaft, due to the weight of the shaft itself, ■ 
 nullev and tension of the belt. Sometimes the power is taken off horizon 
 tally on one side, in which case the tension of the belt 
 transverse stress, while the weight of the pulley acts with the 
 shaft to produce a vertical transverse stress. When the machinery to oe 
 driven is placed on the floor above the shaft, the tension of the belt produces 
 a transverse stress in opposite direction to that due to the weight of the 
 
 S ^^The torsional strength of shafts, or their resistance to breaking by. twisting, 
 is nronortional to the cube of their diameter. Their stiffness or resistance to 
 bending is proportional to the fourth power of their diameters, ^d inversely 
 proportional to the cube of the lengths of their spans. No simple 
 formula can be given that will sately apply to engine and other shafting 
 that is subjected to the bending stresses produced by overhung cranks, the 
 weight of heavy flywheels, the poll of large helts or to ^evere^shocks p - 
 duced by the intermittent action of the power or load. The emulations 101 
 such shafts should always be based on the special conditions h 
 
 In the following table is given the maximum distance between the bear 
 ings of some continuous shafts that are used for the d^iwavs 
 
 Pulleys from which considerable power is to be taken should always 
 
 be Fen^hsof shafts composing a line of shaft- 
 
 ing may be proportional to the quantity of power delivered by 
 
 length. In this connection, the positions of the various pulleys depend 
 
WEIGHT OF CASTINGS. 
 
 Ill 
 
 on the distance between the pulley and the bearing, and on the amount of 
 power given off by the pulleys. Suppose, for example, that a piece of shaft- 
 ing delivers a certain amount of _____ 
 
 power; then, it is obvious that the 
 shaft will deflect or bend less if the 
 pulley transmitting that power be 
 placed close to a hanger or bear- 
 ing, than if it be placed midway 
 between the two hangers or bear- 
 ings. It is impossible to give any 
 rule for the proper distance of bear- 
 ings that could be used univer- 
 sally, as in some cases the require- 
 ments demand that the bearings 
 be nearer together than in others. 
 
 If the work done by a line of shaft- 
 ing is distributed quite equally 
 along its entire length, and the 
 power can be applied near the 
 middle, the strength of the shaft 
 need be only half as great as would 
 
 be required if the power were applied at one end of the shaft. 
 
 Diameter 
 of Shaft. 
 Inches. 
 
 Distance Between Bearings. 
 Feet. 
 
 Wrought-Iron 
 
 Shaft. 
 
 Steel Shaft. 
 
 2 
 
 11 
 
 11.50 
 
 3 
 
 13 
 
 13.75 
 
 4 
 
 15 
 
 15.75 
 
 5 
 
 17 
 
 18.25 
 
 6 
 
 19 
 
 20.00 
 
 7 
 
 21 
 
 22.25 
 
 8 
 
 23 
 
 24.00 
 
 9 
 
 25 
 
 26.00 
 
 WEIGHT OF CASTINGS. 
 
 To Find the Approximate Weight of a Casting.— For iron, multiply the weight 
 of the pattern by 20. Copper is £ heavier; lead, f heavier; brass, £ heavier; 
 and zinc is T % 7 <y as heavy. _ 
 
 Thickness 
 
 or 
 
 Diameter. 
 
 Inches. 
 
 Weight of 
 a Square 
 Foot. 
 Pounds. 
 
 Weight of a 
 Square Bar 
 1 Ft. Long. 
 Pounds. 
 
 Weight of a 
 Round Bar 
 1 Ft. Long. 
 Pounds. 
 
 Thickness 
 
 or 
 
 Diameter. 
 
 Inches. 
 
 Weight of 
 a Square 
 Foot. 
 Pounds. 
 
 Weight of a 
 Square Bar 
 1 Ft. Long. 
 Pounds. 
 
 Weight of a 
 Round Bar 
 1 Ft. Long. 
 Pounds. 
 
 i 
 
 9.375 
 
 .195 
 
 .154 
 
 4£ 
 
 168.7 
 
 63.33 
 
 49.71 
 
 | 
 
 14.06 
 
 .440 
 
 .346 
 
 4| 
 
 173.4 
 
 66.86 
 
 52.52 
 
 8 
 
 £ 
 
 18.75 
 
 . .781 
 
 .610 
 
 4f 
 
 178.1 
 
 70.52 
 
 55.39 
 
 £ 
 
 23.44 
 
 1.221 
 
 .959 
 
 4£ 
 
 182.8 
 
 74.27 
 
 58.34 
 
 £ 
 
 28.12 
 
 1.758 
 
 1.381 
 
 5 
 
 187.5 
 
 78.12 
 
 61.37 
 
 7 . 
 
 32.81 
 
 2.393 
 
 1.880 
 
 5£ 
 
 196.9 
 
 86.14 
 
 67.65 
 
 I 8 
 
 37.50 
 
 3.125 
 
 2.455 
 
 5£ 
 
 206.2 
 
 94.54 
 
 74.26 
 
 
 42.19 
 
 3.955 
 
 3.107 
 
 51 
 
 215.6 
 
 103.3 
 
 81.16 
 
 i| 
 
 46.87 
 
 4.883 
 
 3.835 
 
 6 
 
 225.0 
 
 112.5 
 
 88.36 
 
 if 
 
 51.57 
 
 5.909 
 
 4.640 
 
 6£ 
 
 234.4 
 
 122.1 
 
 95.89 
 
 n 
 
 56.26 
 
 7.033 
 
 5.523 
 
 6£ 
 
 243.8 
 
 132.0 
 
 103.7 
 
 if 
 
 60.94 
 
 8.253 
 
 6.484 
 
 
 253.1 
 
 142.4 
 
 111.9 
 
 if 
 
 65.63 
 
 9.572 
 
 7.518 
 
 7 
 
 262.5 
 
 153.2 
 
 120.2 
 
 i| 
 
 70.32 
 
 10.99 
 
 8.630 
 
 7* 
 
 271.9 
 
 164.2 
 
 129.0 
 
 2 
 
 75.01 
 
 12.50 
 
 9.821 
 
 7£ 
 
 281.3 
 
 175.8 
 
 138.1 
 
 2 £ 
 
 79.70 
 
 14.11 
 
 11.09 
 
 72 
 
 290.7 
 
 187.7 
 
 147.4 
 
 
 84.40 
 
 15.83 
 
 12.43 
 
 8 
 
 300.0 
 
 200.1 
 
 157.0 
 
 2f 
 
 89.07 
 
 17.63 
 
 13.85 
 
 8i 
 
 309.4 
 
 212.7 
 
 167.0 
 
 2| 
 
 93.75 
 
 19.54 
 
 15.34 
 
 8£ 
 
 318.8 
 
 225.8 
 
 177.3 
 
 2£ 
 
 98.44 
 
 21.54 
 
 16.56 
 
 82 
 
 328.2 
 
 239.3 
 
 187.9 
 
 2f 
 
 103.2 
 
 23.64 
 
 18.56 
 
 9 
 
 337.4 
 
 253.1 
 
 198.8 
 
 2 £ 
 
 107.8 
 
 25.84 
 
 20.29 
 
 92 
 
 346.8 
 
 267.4 
 
 210.0 
 
 3 
 
 112.6 
 
 28.13 
 
 22.10 
 
 k 
 
 356.2 
 
 282.1 
 
 221.5 
 
 3£ 
 
 117.3 
 
 30.52 
 
 23.97 
 
 92 
 
 365.6 
 
 297.0 
 
 233.3 
 
 31 
 
 121.8 
 
 33.01 
 
 25.93 
 
 10 
 
 375.0 
 
 312.5 
 
 245.5 
 
 3f 
 
 126.5 
 
 35.60 
 
 27.95 
 
 102 
 
 384.4 
 
 328.4 
 
 257.8 
 
 3i 
 
 131.2 
 
 38.28 
 
 30.07 
 
 102 
 
 393.7 
 
 344.5 
 
 270.6 
 
 3| 
 
 135.9 
 
 41.07 
 
 32.25 
 
 102 
 
 403.1 
 
 361.2 
 
 283.7 
 
 31 
 
 140.6 
 
 43.95 
 
 34.51 
 
 11 
 
 412.5 
 
 378.2 
 
 297.0 
 
 31 
 
 145.3 
 
 46.93 
 
 36.85 
 
 112 
 
 421.9 
 
 395.5 
 
 310.6 
 
 4 
 
 150.0 
 
 50.01 
 
 39.27 
 
 1H 
 
 431.2 
 
 413.3 
 
 324.6 
 
 41 
 
 154.7 
 
 53.18 
 
 41.77 
 
 112 
 
 440.6 
 
 431.4 
 
 338.8 
 
 41 
 
 . 41 
 
 159.3 
 
 164.0 
 
 56.46 
 
 59.82 
 
 44.33 
 
 46.99 
 
 12 
 
 450.0 
 
 450.0 
 
 353.4 
 
112 
 
 WEIGHT OF MATERIALS. 
 
 WEIGHTS OF SHEETS AND PLATES OF STEEL, WROUGHT IRON, 
 COPPER, AND BRASS. 
 
 ( Cambria Steel Co.) 
 
 American, or Brown & Sharpe, Gauge. 
 
 No. 
 
 of 
 
 Gauge. 
 
 Thickness. 
 
 Inch. 
 
 Weight Per Square Foot. 
 
 Steel. 
 
 Iron. 
 
 Copper. 
 
 Brass. 
 
 0000 
 
 .460000 
 
 18.7680 
 
 18.4000 
 
 20.8380 
 
 19.6880 
 
 000 
 
 .409642 
 
 16.7134 
 
 16.3857 
 
 18.5568 
 
 17.5327 
 
 00 
 
 .364796 
 
 14.8837 
 
 14.5918 
 
 16.5253 
 
 15.6133 
 
 0 
 
 .324861 
 
 13.2543 
 
 12.9944 
 
 14.7162 
 
 13.9041 
 
 1 
 
 .289297 
 
 11.8033 
 
 11.5719 
 
 13.1052 
 
 12.3819 
 
 2 
 
 .257627 
 
 10.5112 
 
 10.3051 
 
 11.6705 
 
 11.0264 
 
 3 
 
 .229423 
 
 9.3605 
 
 9.1769 
 
 10.3929 
 
 9.8193 
 
 4 
 
 .204307 
 
 8.3357 
 
 8.1723 
 
 9.2551 
 
 8.7443 
 
 5 
 
 .181940 
 
 7.4232 
 
 7.2776 
 
 8.2419 
 
 7.7870 
 
 6 
 
 .162023 
 
 6.6105 
 
 6.4809 
 
 7.3396 
 
 6.9346 
 
 7 
 
 .144285 
 
 5.8868 
 
 5.7714 
 
 6.5361 
 
 6.1754 
 
 8 
 
 .128490 
 
 5.2424 
 
 5.1396 
 
 5.8206 
 
 5.4994 
 
 9 
 
 .114423 
 
 4.6685 
 
 4.5769 
 
 5.1834 
 
 4.8973 
 
 10 
 
 .101897 
 
 4.1574 
 
 4.0759 
 
 4.6159 
 
 4.3612 
 
 11 
 
 .090742 
 
 3.7023 
 
 3.6297 
 
 4.1106 
 
 3.8838 
 
 12 
 
 .080808 
 
 3.2970 
 
 3.2323 
 
 3.6606 
 
 3.4586 
 
 13 
 
 .071962 
 
 2.9360 
 
 2.8785 
 
 3.2599 
 
 3.0800 
 
 14 
 
 .064084 
 
 2.6146 
 
 2.5634 
 
 2.9030 
 
 2.7428 
 
 15 
 
 .057068 
 
 2.3284 
 
 2.2827 
 
 2.5852 
 
 2.4425 
 
 16 
 
 .050821 
 
 2.0735 
 
 2.0328 
 
 2.3022 
 
 2.1751 
 
 17 
 
 .045257 
 
 1.8465 
 
 1.8103 
 
 2.0501 
 
 1.9370 
 
 18 
 
 .040303 
 
 1.6444 
 
 1.6121 
 
 1.8257 
 
 1.7250 
 
 19 
 
 .035890 
 
 1.4643 
 
 1.4356 
 
 1.6258 
 
 1.5361 
 
 20 
 
 .031961 
 
 1.3040 
 
 1.2784 
 
 1.4478 
 
 1.3679 
 
 21 
 
 .028462 
 
 1.1612 
 
 1.1385 
 
 1.2893 
 
 1.2182 
 
 22 
 
 .025346 
 
 1.0341 
 
 1.0138 
 
 1.1482 
 
 1.0848 
 
 23 
 
 .022572 
 
 92094 
 
 90288 
 
 1.0225 
 
 .96608 
 
 24 
 
 .020101 
 
 82012 
 
 80404 
 
 .91058 
 
 .86032 
 
 25 
 
 .017900 
 
 .73032 
 
 .71600 
 
 .81087 
 
 .76612 
 
 26 
 
 .015941 
 
 .65039 
 
 .63764 
 
 .72213 
 
 .68227 
 
 27 
 
 .014195 
 
 .57916 
 
 .56780 
 
 .64303 
 
 .60755 
 
 28 
 
 .012641 
 
 .51575 
 
 ,50564 
 
 .57264 
 
 .54103 
 
 29 
 
 .011257 
 
 .45929 
 
 .45028 
 
 .50994 
 
 .48180 
 
 30 
 
 .010025 
 
 .40902 
 
 .40100 
 
 .45413 
 
 .42907 
 
 31 
 
 .008928 
 
 .36426 
 
 .35712 
 
 .40444 
 
 .38212 
 
 32 
 
 .007950 
 
 .32436 
 
 .31800 
 
 .36014 
 
 .34026 
 
 33 
 
 .007080 
 
 .28886 
 
 .28320 
 
 .32072 
 
 .30302 
 
 34 
 
 .006305 
 
 .25724 
 
 .25220 
 
 .28562 
 
 .26985 
 
 35 
 
 .005615 
 
 .22909 
 
 .22460 
 
 .25436 
 
 .24032 
 
 36 
 
 .005000 
 
 .20400 
 
 .20000 
 
 .22650 
 
 .21400 
 
 37 
 
 .004453 
 
 .18168 
 
 .17812 
 
 .20172 
 
 .19059 
 
 38 
 
 .003965 
 
 ,16177 
 
 .15860 
 
 .17961 
 
 .16970 
 
 39 
 
 .003531 
 
 .14406 
 
 1 .14124 
 
 .15995 
 
 .15113 
 
 40 
 
 .003144 
 
 1 
 
 .12828 
 
 .12576 
 
 .14242 
 
 .13456 
 
CAST-IRON PIPE. 
 
 113 
 
 WEIGHT OF CAST-IRON PIPE PER FOOT IN POUNDS. 
 
 These weights are for plain pipe. For hautbov pipe, add 8 in. in length 
 for each joint. For copper, add for lead, for welded iron, T? . 
 
 Diam- 
 eter of 
 Bore. 
 Inches. 
 
 
 
 
 Thickness of Pipe. 
 
 Inches. 
 
 
 
 
 
 i 
 
 4 
 
 3. 
 
 8 
 
 a 
 
 8 
 
 f 
 
 •7 
 
 8 
 
 1 
 
 H 
 
 H 
 
 If 
 
 lk 
 
 If 
 
 2 
 
 1 
 
 3.07 
 
 5.07 
 
 7.38 
 
 
 
 
 
 
 
 
 
 
 
 li 
 
 3.69 
 
 6.00 
 
 8.61 
 
 
 
 
 
 
 
 
 
 
 
 li 
 
 4.30 
 
 6.92 
 
 9.84 
 
 
 
 
 
 
 
 
 
 
 
 If 
 
 4.92 
 
 7.84 
 
 11.10 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 5.53 
 
 8.76 
 
 12.30 
 
 16.2 
 
 
 
 
 
 
 
 
 
 
 2£ 
 
 6.15 
 
 9.69 
 
 13.50 
 
 17.7 
 
 
 
 
 
 
 
 
 
 
 2£ 
 
 6.76 
 
 10.60 
 
 14.80 
 
 19.2 
 
 24.0 
 
 
 
 
 
 
 
 
 
 2* 
 
 7.37 
 
 11.50 
 
 16.00 
 
 20.8 
 
 25.9 
 
 
 
 
 
 
 
 
 
 3 
 
 7.98 
 
 12.50 
 
 17.20 
 
 22.3 
 
 27.7 
 
 33.4 
 
 
 
 
 
 
 
 
 3i 
 
 9.21 
 
 14.30 
 
 19.70 
 
 25.4 
 
 31.4 
 
 37.7 
 
 
 
 
 
 
 
 
 4 
 
 10.30 
 
 16.10 
 
 22.20 
 
 28.5 
 
 35.1 
 
 42.0 
 
 
 
 
 
 
 
 
 4£ 
 
 11.70 
 
 18.00 
 
 24.60 
 
 31.5 
 
 38.8 
 
 46.3 
 
 
 
 
 
 
 
 
 5 
 
 12.90 
 
 19.80 
 
 27.10 
 
 34.6 
 
 42.5 
 
 50.6 
 
 
 
 
 
 
 
 
 5i 
 
 14.20 
 
 21.70 
 
 29.50 
 
 37.7 
 
 46.1 
 
 54.9 
 
 
 
 
 
 
 
 
 6 
 
 15.40 
 
 23.50 
 
 32.00 
 
 40.8 
 
 49.8 
 
 59.2 
 
 68.9 
 
 84.4 
 
 
 
 
 
 
 6£ 
 
 16.60 
 
 25.40 
 
 34.50 
 
 43.8 
 
 53.5 
 
 63.5 
 
 73.8 
 
 
 
 
 
 
 7 
 
 17.80 
 
 27.20 
 
 36.90 
 
 46.9 
 
 57.2 
 
 67.8 
 
 78.7 
 
 89.4 
 
 
 
 
 
 
 7£ 
 
 19.10 
 
 29.10 
 
 39.40 
 
 50.0 
 
 60.9 
 
 72.1 
 
 83.7 
 
 95.5 
 
 108 
 
 127 
 
 
 
 
 8 
 
 20.30 
 
 30.90 
 
 41.80 
 
 53.1 
 
 64.6 
 
 76.4 
 
 88.6 
 
 101.0 
 
 114 
 
 148 
 
 
 
 8£ 
 
 21.50 
 
 32.80 
 
 44.30 
 
 56.1 
 
 68.3 
 
 80.7 
 
 93.5 
 
 107.0 
 
 120 
 
 134 
 
 
 
 9 
 
 22.80 
 
 34.60 
 
 46.80 
 
 59.2 
 
 72.0 
 
 85.1 
 
 98.4 
 
 112.0 
 
 126 
 
 140 
 
 155 
 
 
 
 9£ 
 
 24.00 
 
 36.40 
 
 49.20 
 
 62.3 
 
 75.7 
 
 89.3 
 
 103.0 
 
 118.0 
 
 132 
 
 147 
 
 163 
 
 202 
 
 
 10 
 
 25.10 
 
 38.30 
 
 51.70 
 
 65.3 
 
 79.4 
 
 93.6 
 
 108.0 
 
 123.0 
 
 138 
 
 164 
 
 170 
 
 
 11 
 
 27.60 
 
 42.00 
 
 56.60 
 
 71.5 
 
 86.7 
 
 102.0 
 
 118.0 
 
 134.0 
 
 151 
 
 168 
 
 185 
 
 220 
 
 275 
 
 12 
 
 30.00 
 
 45.70 
 
 61.50 
 
 77.7 
 
 94.1 
 
 111.0 
 
 128.0 
 
 145.0 
 
 163 
 
 181 
 
 199 
 
 237 
 
 13 
 
 32.50 
 
 49.40 
 
 66.40 
 
 83.8 
 
 102.0 
 
 120.0 
 
 138.0 
 
 156.0 
 
 175 
 
 195 
 
 214 
 
 254 
 
 294 
 
 14 
 
 35.00 
 
 53.10 
 
 71.40 
 
 89.4 
 
 109.0 
 
 128.0 
 
 148.0 
 
 168.0 
 
 188 
 
 208 
 
 229 
 
 271 
 
 314 
 
 15 
 
 37.40 
 
 56.70 
 
 76.30 
 
 96.1 
 
 116.0 
 
 137.0 
 
 158.0 
 
 179.0 
 
 200 
 
 222 
 
 244 
 
 289 
 
 334 
 
 16 
 
 39.10 
 
 60.40 
 
 81.20 
 
 102.0 
 
 124.0 
 
 145.0 
 
 167.0 
 
 190.0 
 
 212 
 
 235 
 
 258 
 
 306 
 
 353 
 
 17 
 
 42.30 
 
 64.10 
 
 86.10 
 
 108.0 
 
 131.0 
 
 154.0 
 
 177.0 
 
 201.0 
 
 225 
 
 249 
 
 273 
 
 323 
 
 373 
 
 18 
 
 44.80 
 
 67.80 
 
 91.00 
 
 115.0 
 
 139.0 
 
 163.0 
 
 187.0 
 
 212.0 
 
 237 
 
 262 
 
 288 
 
 340 
 
 393 
 
 19 
 
 47.30 
 
 71.50 
 
 96.00 
 
 121.0 
 
 146.0 
 
 171.0 
 
 197.0 
 
 223.0 
 
 249 
 
 276 
 
 303 
 
 357 
 
 412 
 
 20 
 
 49.70 
 
 75.20 
 
 101.00 
 
 127.0 
 
 153.0 
 
 180.0 
 
 207.0 
 
 234.0 
 
 261 
 
 289 
 
 317 
 
 375 
 
 432 
 
 21 
 
 52.20 
 
 78.90 
 
 106.00 
 
 133.0 
 
 161.0 
 
 188.0 
 
 217.0 
 
 245.0 
 
 274 
 
 303 
 
 332 
 
 392 
 
 452 
 
 22 
 
 54.60 
 
 • 82.60 
 
 111.00 
 
 139.0 
 
 168.0 
 
 196.0 
 
 227.0 
 
 256.0 
 
 286 
 
 316 
 
 347 
 
 409 
 
 471 
 
 23 
 
 57.10 
 
 ' 86.30 
 
 116.00 
 
 145.0 
 
 175.0 
 
 206.0 
 
 236.0 
 
 267.0 
 
 298 
 
 330 
 
 362 
 
 426 
 
 491 
 
 24 
 
 59.60 
 
 i 89.90 
 
 121.00 
 
 1 152.0 
 
 183.0 
 
 214.0 
 
 246.0 
 
 278.0 
 
 311 
 
 343 
 
 375 
 
 444 
 
 511 
 
 25 
 
 62.00 
 
 i 93.60 
 
 126.00 
 
 1 158.0 
 
 190.0 
 
 223.0 
 
 256.0 
 
 ' 289.0 
 
 323 
 
 357 
 
 391 
 
 461 
 
 531 
 
 26 
 
 64.50 
 
 i 97.30 
 
 ' 131.00 
 
 1 164.0 
 
 198.0 
 
 231.0 
 
 266.0 
 
 i 300.0 
 
 ' 335 
 
 370 
 
 406 
 
 478 
 
 550 
 
 27 
 
 66.90 
 
 1 101.00 
 
 1 135.00 
 
 1 170.0 
 
 • 205.0 
 
 240.0 
 
 ' 276.0 
 
 i 311.0 
 
 i 348 
 
 384 
 
 421 
 
 495 
 
 570 
 
 28 
 
 69.40 
 
 1 105.00 
 
 1 140.00 
 
 1 176.0 
 
 i 212.0 
 
 249.0 
 
 1 286.0 
 
 i 323.0 
 
 i 360 
 
 397 
 
 436 
 
 512 
 
 590 
 
 29 
 
 71.80 
 
 1 109.00 
 
 1 145.00 
 
 1 182.0 
 
 i 220.0 
 
 i 257.0 
 
 i 295.0 
 
 > 334.0 
 
 i 372 
 
 411 
 
 450 
 
 530 
 
 609 
 
 30 
 
 74.2C 
 
 > 112.00 
 
 1 150.00 
 
 1 188.0 
 
 l 227.0 
 
 i 266.0 
 
 i| 305.0 
 
 1 345.0 
 
 l 384 
 
 424 
 
 465 
 
 | 547 
 
 | 629 
 
 DIAMETER- AND NUMBER OF WOOD SCREWS. 
 
 Formulas for Wood Screws. 
 
 No. 
 
 Diameter. 
 
 No. 
 
 Diameter. 
 
 No.* 
 
 Diameter. 
 
 N = 
 
 number 
 
 0 
 
 .056 
 
 11 
 
 .201 
 
 22 
 
 .347 
 
 I) =- 
 
 diameter 
 
 1 
 
 .069 
 
 12 
 
 .215 
 
 23 
 
 .361 
 
 D - 
 
 {NX -01325) + .056 
 
 2 
 
 .082 
 
 13 
 
 .228 
 
 24 
 
 ,374 
 
 
 D - .056 
 
 3 
 
 .096 
 
 14 
 
 .241 
 
 25 
 
 .387 
 
 N — 
 
 .01325 
 
 4 
 
 5 
 
 .109 
 
 .122 
 
 15 
 
 16 
 
 .255 
 
 .268 
 
 26 
 
 27 
 
 401 
 
 414 
 
 
 
 6 
 
 .135 
 
 17 
 
 281 
 
 28 
 
 427 
 
 
 
 7 
 
 .149 
 
 18 
 
 .293 
 
 29 
 
 440 
 
 
 
 8 
 
 .162 
 
 19 
 
 ,308 
 
 30 
 
 453 
 
 
 
 9 
 
 .175 
 
 20 
 
 .321 
 
 
 
 
 
 10 
 
 .188 
 
 21 
 
 .334 
 
 
 
114 
 
 WEIGHT OF MATERIALS. 
 
 WEIGHT OF WROUGHT IRON. 
 
 The following table is for wrought iron. Multiply by .95 for weight of cast 
 iron; by 1.02 for steel; by 1.16 for copper; by 1.09 for brass; by 1.48 for lead. 
 
 Thickness 
 
 or 
 
 Diameter. 
 
 Inches. 
 
 Weight 
 of a 
 
 Square Foot. 
 Pounds. 
 
 Weight of a 
 Square Bar 
 1 Ft. Long. 
 Pounds. 
 
 Weight of a 
 Round Bar 
 1 Ft. Long. 
 Pounds. 
 
 Thickness 
 
 or 
 
 Diameter. 
 
 Inches. 
 
 Weight . 
 of a 
 
 Square Foot. 
 Pounds. 
 
 Weight of a 
 Square Bar 
 1 Ft. Long. 
 Pounds. 
 
 Weight of a 
 Round Bar 
 1 Ft. Long. 
 Pounds. 
 
 
 5.052 
 
 .0526 
 
 .0414 
 
 4f 
 
 176.8 
 
 64.47 
 
 50.63 
 
 1 
 
 10.10 
 
 .2105 
 
 .1653 
 
 4k 
 
 181.9 
 
 68.20 
 
 53.57 
 
 | 
 
 15.16 
 
 .4736 
 
 .3720 
 
 4f 
 
 186.9 
 
 72.05 
 
 56.59 
 
 
 20.21 
 
 .8420 
 
 .6613 
 
 4f 
 
 192.0 
 
 75.99 
 
 59.69 
 
 | 
 
 25.26 
 
 1.316 
 
 1.033 
 
 4f 
 
 •197.0 
 
 80.05 
 
 62.87 
 
 f 
 
 30.31 
 
 1.895 
 
 1.488 
 
 5 
 
 202.1 
 
 84.20 
 
 66.13 
 
 7 _ 
 
 35.37 
 
 2.579 
 
 2.025 
 
 5f 
 
 212.2 
 
 92.83 
 
 72.91 
 
 l 6 
 
 40.42 
 
 3.368 
 
 2.645 
 
 5f 
 
 222.3 
 
 101.9 
 
 80.02 
 
 lj 
 
 45.47 
 
 4.263 
 
 3.348 
 
 5f 
 
 232.4 
 
 111.4 
 
 87.46 
 
 It 
 
 50.52 
 
 5.263 
 
 4.133 
 
 6 
 
 242.5 
 
 121.3 
 
 95.23 
 
 If 
 
 55.57 
 
 6.368 
 
 5.001 
 
 6f 
 
 252.6 
 
 131.6 
 
 103.3 
 
 1£ 
 
 60.63 
 
 7.578 
 
 5.952 
 
 6i 
 
 262.7 
 
 142.3 
 
 111.8 
 
 If 
 
 65.68 
 
 8.893 
 
 6.985 
 
 6f 
 
 272.8 
 
 153.5 
 
 120.5 
 
 If 
 
 70.73 
 
 10.31 
 
 8.101 
 
 7 
 
 282.9 
 
 165.0 
 
 129.6 
 
 If 
 
 75.78 
 
 11.84 
 
 9.300 
 
 7j 
 
 293.0 
 
 177.0 
 
 139.0 
 
 2 
 
 80.83 
 
 13.47 
 
 10.58 
 
 7i 
 
 303.1 
 
 189.5 
 
 148.8 
 
 2£ 
 
 85.89 
 
 15.21 
 
 11.95 
 
 7f 
 
 313.2 
 
 202.3 
 
 158.9 
 
 2f 
 
 90.94 
 
 17.05 
 
 13.39 
 
 8 
 
 323.3 
 
 215.6 
 
 169.3 
 
 2f 
 
 95.99 
 
 19.00 
 
 14.92 
 
 8i 
 
 333.4 
 
 229.3 
 
 180.1 
 
 2\ 
 
 101.0 
 
 21.05 
 
 16.53 
 
 8£ 
 
 343.5 
 
 243.4 
 
 191.1 
 
 2f 
 
 106.1 
 
 23.21 
 
 18.23 
 
 8f 
 
 353.6 
 
 247.9 
 
 202.5 
 
 2f 
 
 111.2 
 
 25.47 
 
 20.01 
 
 9 
 
 363.8 
 
 272.8 
 
 214.3 
 
 2f 
 
 116.2 
 
 27.84 
 
 21.87 
 
 9| 
 
 373.9 
 
 288.2 
 
 226.3 
 
 3 
 
 121.3 
 
 30.31 
 
 23.81 
 
 9f 
 
 384.0 
 
 304.0 
 
 238.7 
 
 3f 
 
 126.3 
 
 32.89 
 
 25.83 
 
 9f 
 
 394.1 
 
 320.2 
 
 251.5 
 
 3f 
 
 131.4 
 
 35.57 
 
 27.94 
 
 10 
 
 404.2 
 
 336.8 
 
 264.5 
 
 3f 
 
 136.4 
 
 38.37 
 
 30.13 
 
 m 
 
 414.3 
 
 353.9 
 
 277.9 
 
 3f 
 
 141.5 
 
 41.26 
 
 32.41 
 
 10k 
 
 424.4 
 
 371.3 
 
 291.6 
 
 3| 
 
 146.5 
 
 44.26 
 
 34.76 
 
 lOf 
 
 434.5 
 
 389.2 
 
 305.7 
 
 3f 
 
 151.6 
 
 47.37 
 
 37.20 
 
 ii 
 
 444.6 
 
 407.5 
 
 320.1 
 
 3f 
 
 156.6 
 
 50.57 
 
 39.72 
 
 ni 
 
 454.7 
 
 426.3 
 
 334.8 
 
 4 
 
 161.7 
 
 53.89 
 
 42.33 
 
 lii 
 
 464.8 
 
 445.4 
 
 349.8 
 
 41 
 
 166.7 
 
 57.31 
 
 45.01 
 
 iif 
 
 474.9 
 
 465.0 
 
 365.2 
 
 41 
 
 171.8 
 
 60.84 
 
 47.78 
 
 12 
 
 485.0 
 
 485.0 
 
 380.9 
 
 Spikes and Nails. 
 
 
 Starn 
 
 Length. 
 
 lard Steel-Wire 1 
 Common. 
 
 ST ails. 
 
 Finishing. 
 
 Steel-Wire Spikes. 
 
 Common Iron Nails. 
 
 Sizes. 
 
 In. 
 
 Diam. 
 
 No. per 
 
 Diam. 
 
 No. per 
 
 Length. 
 
 Diam. 
 
 No. per 
 
 Size* 
 
 Length. 
 
 No. per 
 
 
 
 In. 
 
 Lb. 
 
 In. 
 
 Lb. 
 
 In. 
 
 In. 
 
 Lb. 
 
 
 In. 
 
 Lb. 
 
 2d 
 
 1 
 
 .0524 
 
 1,060 
 
 .0453 
 
 1,558 
 
 3 
 
 .1620 
 
 41 
 
 2d 
 
 1 
 
 800 
 
 3d 
 
 li 
 
 .0588 
 
 640 
 
 .0508 
 
 913 
 
 3i 
 
 .1819 
 
 • 30 
 
 3d 
 
 li 
 
 400 
 
 4d 
 
 li 
 
 .0720 
 
 380 
 
 .0508 
 
 761 
 
 4 
 
 .2043 
 
 23 
 
 4d 
 
 li 
 
 300 
 
 5d 
 
 If 
 
 .0764 
 
 275 
 
 .0571 
 
 500 
 
 4i 
 
 .2294 
 
 17 
 
 5d 
 
 If 
 
 200 
 
 6d 
 
 2 
 
 .0808 
 
 210 
 
 .0641 
 
 350 
 
 5 
 
 .2576 
 
 13 
 
 6d 
 
 2 
 
 150 
 
 7d 
 
 2i 
 
 .0858 
 
 160 
 
 .0641 
 
 315 
 
 5i 
 
 .2893 
 
 11 
 
 7d 
 
 2i 
 
 120 
 
 8d 
 
 2i 
 
 .0935 
 
 115 
 
 .0720 
 
 214 
 
 6 
 
 .2893 
 
 10 
 
 8d 
 
 2i 
 
 85 
 
 9d 
 
 2f 
 
 .0963 
 
 93 
 
 .0720 
 
 195 
 
 6i 
 
 .2249 
 
 7i 
 
 9d 
 
 2f 
 
 75 
 
 lOd 
 
 3 
 
 .1082 
 
 77 
 
 .0808 
 
 137 
 
 7 
 
 .2249 
 
 7 
 
 lOd 
 
 3 
 
 60 
 
 12d 
 
 3f 
 
 .1144 
 
 60 
 
 .0808 
 
 127 
 
 8 
 
 .3648 
 
 5 
 
 12d 
 
 3i 
 
 50 
 
 16d 
 
 3i 
 
 .1285 
 
 48 
 
 .0907 
 
 90 
 
 9 
 
 .3648 
 
 4i 
 
 16d 
 
 3i 
 
 40 
 
 20d 
 
 4 
 
 .1620 
 
 31 
 
 .1019 
 
 62 
 
 
 
 
 20d 
 
 4 
 
 20 
 
 30d 
 
 4i 
 
 .1819 
 
 22 
 
 
 
 
 
 
 30d 
 
 4i 
 
 16 
 
 40d 
 
 5 
 
 .2043 
 
 17 
 
 
 
 
 
 
 40d 
 
 5 
 
 14: 
 
 50d 
 
 5i 
 
 .2294 
 
 13 
 
 
 
 
 
 
 50d 
 
 5i 
 
 11 
 
 60d 
 
 6 
 
 .2576 
 
 11 
 
 
 
 
 
 
 60d 
 
 6 
 
 8 
 
WROUGHT IRON. 
 
 115 
 
 Weight, in Pounds, of 1 Lineal Foot of Wrought Iron— Flat. 
 
 Multiply by .95 for weight of cast iron; by 1.02 for weight of steel; by 1.16 
 for copper; by 1.09 for brass; by 1.48 for lead. 
 
 Size. 
 
 Inches, 
 
 Weight. 
 
 Pounds. 
 
 Size. 
 
 Inches. 
 
 Weight. 
 
 Pounds. 
 
 Size. 
 
 Inches. 
 
 Weight. 
 
 Pounds. 
 
 1 X* 
 
 0.85 
 
 5£Xt 
 
 6.65 
 
 4 XI 
 
 8.45 
 
 ttx £ 
 
 1.06 
 
 5£ X 1 
 
 6.97 
 
 4£ X 1 
 
 8.98 
 
 U X J 
 
 1.27 
 
 5f X I 
 
 7.29 
 
 4£ X 1 
 
 9.51 
 
 if X £ 
 
 1.48 
 
 6X| 
 
 7.60 
 
 4f X I 
 
 10.03 
 
 2 X £ 
 
 1.69 
 
 
 
 5 XI 
 
 10.56 
 
 2£ X £ 
 
 1.90 
 
 i X£ 
 
 1.69 
 
 5£ X 1 
 
 11.09 
 
 2s X 4 
 
 2.11 
 
 *-4 X 1 
 
 2.11 
 
 5£ X I 
 
 11.62 
 
 2f X | 
 
 2.32 
 
 -*£j X ? 
 
 2.53 
 
 5f XI 
 
 12.15 
 
 3 X£ 
 
 2.53 
 
 If Xi 
 
 2.96 
 
 6 XI 
 
 12.67 
 
 3£ X £ 
 
 2.75 
 
 2 Xi ! 
 
 ! 3.38 
 
 
 
 3£ X £ 
 
 2.96 
 
 2£ X 1 | 
 
 -5.80 
 
 1 Xf 
 
 2.53 
 
 3f X £ 
 
 3.17 
 
 2£X£ 
 
 4-22 
 
 1£ X f 
 
 3.17 
 
 4 X£ 
 
 3.38 
 
 2f X £ 
 
 4.65 
 
 1£ X f 
 
 3.80 
 
 4£X£ 
 
 3.59 
 
 3 X£ 
 
 5.07 
 
 If Xf 
 
 4.44 
 
 4£ X £ 
 
 3.80 
 
 3£ X £ 
 
 5.49 
 
 2 Xf 
 
 5.07 
 
 4f X £ 
 
 4.01 
 
 3£ X £ 
 
 5.92 
 
 2£Xf 
 
 5.70 
 
 5 X £ 
 
 4.22 
 
 3f X £ 
 
 6.33 
 
 2£ X f 
 
 6.33 
 
 5£x£ 
 
 4.44 
 
 4 X £ 
 
 6.76 
 
 2f Xf 
 
 6.97 
 
 X £ 
 
 4.65 
 
 4£ X £ 
 
 7.18 
 
 3 X f 
 
 7.60 
 
 5f X £ 
 
 4.86 | 
 
 4£ X £ 
 
 7.60 
 
 3£ X f 
 
 8.24 
 
 6 X £ 
 
 5.07 
 
 4f X £ 
 
 8.03 
 
 3£ X f 
 
 8.87 
 
 
 
 5 X £ 
 
 8.45 
 
 3f X f 
 
 9.51 
 
 1 XI 
 
 1.27 
 
 5£ X £ 
 
 8.87 
 
 4 X f 
 
 10.14 
 
 1£ X | 
 
 1.58 
 
 5£ X £ 
 
 9.30 
 
 4£X f 
 
 10.77 
 
 1| X f 
 
 1.90 
 
 5f X £ 
 
 9.72 
 
 4£ X f 
 
 11.41 
 
 If XI 
 
 2.22 
 
 6 X £ 
 
 10.14 
 
 4f Xf 
 
 12.04 
 
 2 XI 
 
 2.53 
 
 
 
 5 X f 
 
 12.67 
 
 2£ X ! 
 
 2.85 
 
 1 XI 
 
 2.11 
 
 5£ X f 
 
 13.31 
 
 2i X 1 
 
 3.17 
 
 l£ X t 
 
 2.64 
 
 5£ X f 
 
 13.94 
 
 2f XI 
 
 3.49 
 
 1£ X 1 
 
 3.17 
 
 5f X f 
 
 14.57 
 
 3 XI 
 
 3.80 
 
 If XI 
 
 ‘ 3.70 
 
 6 Xf 
 
 15.21 
 
 3£ X | 
 
 4.12 
 
 2 XI 
 
 4.22 
 
 
 
 31X1 
 
 4.44 . 
 
 2£ X ! 
 
 4.75 
 
 11X1 
 
 5.07 
 
 3f X 1 
 
 4.75 
 
 2£ X 1 
 
 5.28 
 
 2 XI 
 
 6.76 
 
 4 XI 
 
 5.07 
 
 2f XI 
 
 5.81 
 
 3 XI 
 
 10.14 
 
 4£ X 1 
 
 5.39 
 
 3 XI 
 
 6.33 
 
 4X1 
 
 13.52 
 
 4£ X t 
 
 5.70 
 
 3£ X 1 
 
 6.87 
 
 5 XI 
 
 16.90 
 
 4f X f 
 
 6.02 
 
 3£ X I 
 
 7.39 
 
 6X1 
 
 20.28 
 
 5 X | 
 
 633 
 
 3f X I 
 
 7.92 
 
 7. X 1 
 
 23.66 
 
 Strength of Metals in Pounds per Square Inch. 
 
 Material. 
 
 Ultimate 
 
 Tensile. 
 
 Ultimate 
 
 Compres- 
 
 sion. 
 
 Ultimate 
 
 Shearing. 
 
 Modulus 
 
 of 
 
 Rupture. 
 
 Modulus 
 
 of 
 
 Elasticity. 
 
 Millions. 
 
 Wrought iron 
 
 50,000 
 
 44,000 
 
 44,000 
 
 48,000 
 
 27 
 
 Shape iron 
 
 48,000 
 
 
 
 
 26 
 
 Structural steel j 
 
 60,000 
 
 
 
 
 
 
 65,000 
 
 52,000 
 
 52,000 
 
 60,000 
 
 29 
 
 Cast iron 
 
 18,000 
 
 81,000 
 
 25,000 
 
 45,000 
 
 12 
 
 Steel, castings 
 
 70,000 
 
 70,000 
 
 60,000 
 
 70,000 
 
 30 
 
 Brass, cast 
 
 24,000 
 
 *30,000 
 
 36,000 
 
 20,000 
 
 9 
 
 Bronze, phosphor 
 
 50,000 
 
 
 
 
 14 
 
 Bronze, aluminum 
 
 75,000 
 
 120,000 
 
 
 
 
 Aluminum, commercial 
 
 15,000 
 
 12,000 
 
 12,000 
 
 1 
 
 11 
 
 * Unit stress producing 10 $ reduction in original length. 
 
116 
 
 WEIGHT OF MATERIALS. 
 
 Weight of Wrought-Iron Boltheads, Nuts, and Washers. 
 
 Diameter of 
 Bolt. 
 Inches. 
 
 Hexagon Heads 
 and Nuts. 
 Per Pair. 
 
 Square Heads 
 and Nuts. 
 Per Pair. 
 
 Round Washers. 
 Per Pair. 
 
 j. 
 
 20 to a lb. 
 
 16 to a lb. 
 
 20 to a lb. 
 
 4 
 
 A 
 
 10 to a lb. 
 
 8i to a lb. 
 
 10 to a lb. 
 
 i 
 
 5 to a lb. 
 
 to a lb. 
 
 5 to a lb. 
 
 a 
 
 A 
 
 2t to a lb. 
 
 2£ to a lb. 
 
 3 to a lb. 
 
 8 
 
 3 
 
 2 to a lb. 
 
 0.56 lb. 
 
 0.63 lb. 
 
 7 
 
 0.77 lb. 
 
 0.88 lb. 
 
 0.77 lb. 
 
 8 
 
 1 
 
 1.251b. 
 
 1.31 lb. 
 
 1.25 lb. 
 
 1! 
 
 1.75 lb. 
 
 2.10 lb. 
 
 1.751b. 
 
 h 
 
 2.13 lb*. 
 
 2.56 lb. 
 
 2.25 lb. 
 
 It 
 
 3.00 lb. 
 
 3.60 lb. 
 
 3.25 lb. 
 
 3.75 lb. 
 
 4.42 lb. 
 
 4.25 lb. 
 
 J-2 
 
 It 
 
 It 
 
 H 
 
 2 
 
 4.75 lb. 
 
 5.70 lb. 
 
 5.25 lb. 
 
 5.75 lb. 
 
 7.00 lb. 
 
 6.50 lb. 
 
 7.27 lb. 
 
 8.72 lb. 
 
 8.00 lb. 
 
 8.75 lb. 
 
 10.50 lb. 
 
 9.60 lb. 
 
 Weight of 100 Bolts With Square Heads and Nuts. 
 (The Carnegie Steel Co Limited.) 
 
 ! Diameter of Bolts. 
 
 _L,engtu 
 Under 
 Head to 
 Point. 
 
 1 
 
 E 
 
 pi 
 
 — ■ — 
 
 fW 
 o' - 
 
 K 
 
 & 
 
 E 
 
 f" 
 
 Lb. 
 
 Lb. 
 
 1" 
 
 Lb. 
 
 11 
 
 4.0 
 
 7.0 
 
 10.5 
 
 15.2 
 
 22.5 
 
 39.5 
 
 63.0 
 
 
 
 1A 
 
 4.4 
 
 7.5 
 
 11.3 
 
 16.3 
 
 23.8 
 
 41.6 
 
 66.0 
 
 
 
 -Lf 
 
 o 
 
 4.8 
 
 8.0 
 
 12.0 
 
 17.4 
 
 25.2 
 
 43.8 
 
 69.0 
 
 109.0 
 
 163 
 
 u 
 
 9A 
 
 5.2 
 
 8.5 
 
 12.8 
 
 18.5 
 
 26.5 
 
 45.8 
 
 72.0 
 
 113.3 
 
 169 
 
 91 
 
 o*5 
 
 9.0 
 
 13.5 
 
 19.6 
 
 27.8 
 
 48.0 
 
 75.0 
 
 117.5 
 
 174 
 
 93 
 
 5.8 
 
 9.5 
 
 14.3 
 
 20.7 
 
 29.1 
 
 50.1 
 
 78.0 
 
 121.8 
 
 180 
 
 *4 
 
 Q 
 
 6.3 
 
 10.0 
 
 15.0 
 
 21.8 
 
 30.5 
 
 52.3 
 
 81.0 
 
 126.0 
 
 185 
 
 O 
 
 Qi 
 
 7.0 
 
 n.o 
 
 16.5 
 
 24.0 
 
 33.1 
 
 56.5 
 
 87.0 
 
 134.3 
 
 196 
 
 O3 
 
 A 
 
 7.8 
 
 12.0 
 
 18.0 
 
 26.2 
 
 35.8 
 
 60.8 
 
 93.1 
 
 142.5 
 
 207 
 
 
 8.5 
 
 13.0 
 
 19.5 
 
 28.4 
 
 38.4 
 
 65.0 
 
 99.1 
 
 151.0 
 
 218 
 
 a 
 
 9.3 
 
 14.0 
 
 21.0 
 
 30.6 
 
 41.1 
 
 69.3 
 
 105.2 
 
 159.6 
 
 229 
 
 0 
 
 10.0 
 
 15.0 
 
 22.5 
 
 32.8 
 
 43.7 
 
 73.5 
 
 111.3 
 
 168.0 
 
 240 
 
 o a 
 
 A 
 
 10.8 
 
 16.0 
 
 24.0 
 
 35.0 
 
 46.4 
 
 77.8 
 
 117.3 
 
 176.6 
 
 251 
 
 u 
 
 Ai 
 
 
 
 25.5 
 
 37.2 
 
 49.0 
 
 82.0 
 
 123.4 
 
 185.0 
 
 262 
 
 Da 
 
 7 
 
 
 
 27.0 
 
 39.4 
 
 51.7 
 
 86.3 
 
 129.4 
 
 193.7 
 
 273 
 
 1 
 
 71 
 
 
 
 28.5 
 
 41.6 
 
 54.3 
 
 90.5 
 
 135.0 
 
 202.0 
 
 284 
 
 * a 
 Q 
 
 
 
 30.0 
 
 43.8 
 
 59.6 
 
 94.8 
 
 141.5 
 
 210.7 
 
 295 
 
 O 
 
 Q 
 
 
 
 
 46.0 
 
 64.9 
 
 103.3 
 
 153.6 
 
 227.8 
 
 317 
 
 V 
 
 in 
 
 
 
 
 48.2 
 
 70.2 
 
 111.8 
 
 165.7 
 
 224.8 
 
 339 
 
 1U 
 
 1 1 
 
 
 
 
 50.4 
 
 75.5 
 
 120.3 
 
 177.8 
 
 261.9 
 
 360 
 
 11 
 1 0 
 
 
 
 
 52.6 
 
 80.8 
 
 128.8 
 
 189.9 
 
 278.9 
 
 382 
 
 14 
 
 1 Q 
 
 
 
 
 
 86.1 
 
 137.3 
 
 202.0 
 
 296.0 
 
 404 
 
 lo 
 
 1 A 
 
 
 
 
 . 
 
 91.4 
 
 145.8 
 
 214.1 
 
 313.0 
 
 426 
 
 14 
 
 1 K 
 
 
 
 
 
 96.7 
 
 154.3 
 
 226.2 
 
 330.1 
 
 448 
 
 lo 
 
 1 A 
 
 
 
 
 
 102.0 
 
 162.8 
 
 238.3 
 
 347.1 
 
 470 
 
 16 
 1 *1 
 
 
 
 
 
 107.3 
 
 171.0 
 
 250.4 
 
 364.2 
 
 492 
 
 17 
 
 1 0 
 
 
 
 
 
 112.6 
 
 179.5 
 
 262.6 
 
 381.2 
 
 514 
 
 18 
 1 n 
 
 
 
 
 
 117.9 
 
 188.0 
 
 274.7 
 
 398.3 
 
 536 
 
 iy 
 
 20 
 
 
 
 
 
 123.2 
 
 206.5 
 
 286.8 
 
 415.3 
 
 558 
 
 Per Inch 
 
 1.4 
 
 2.1 
 
 3.1 
 
 4.2 
 
 5.5 
 
 8.5 
 
 12.3 
 
 16.7 
 
 21.8 
 
 Addit’ al. 
 
 
 
 
 
 
 
 — 
 
 
 rrr-J 
 
RAILROAD IRON. 
 
 117 
 
 IRON REQUIRED FOR ONE MILE OF TRACK. 
 
 Tons of Iron. 
 
 Rule. — To find the number of tons of rails to the mile , divide the weight per yard 
 by 7, and multiply by 11. Thus, for 56-pound rail, divide 56 by 7 equal 8 , multi- 
 plied by 11 equal 88 tons, for 1 mile of single track. 
 
 . Weight of 
 Rail per Yard. 
 Pounds. 
 
 Tons per Mile. 
 
 Weight of 
 Rail per Yard. 
 Pounds. 
 
 Tons per Mile. 
 
 Tons. 
 
 Pounds. 
 
 Tons. 
 
 Pounds. 
 
 12 
 
 18 
 
 1,920 
 
 45 
 
 70 
 
 1,600 
 
 .14 
 
 22 
 
 
 48 
 
 75 
 
 960 
 
 16 
 
 25 
 
 320 
 
 50 
 
 78 
 
 1,280 
 
 18 
 
 28 
 
 640 
 
 52 
 
 81 
 
 1,600 
 
 20 
 
 31 
 
 960 
 
 56 
 
 88 
 
 
 22 
 
 34 
 
 1,280 
 
 57 
 
 89 
 
 1,280 
 
 25 
 
 39 
 
 640 
 
 60 
 
 94 
 
 640 
 
 26 
 
 40 
 
 1,920 
 
 62 
 
 97 
 
 960 
 
 27 
 
 42 
 
 960 
 
 64 
 
 100 
 
 1,280 
 
 28 
 
 44 
 
 
 65 
 
 102 
 
 * 320 
 
 30 
 
 47 
 
 320 
 
 68 
 
 106 
 
 1,920 
 
 33 
 
 51 
 
 1,920 
 
 70 
 
 110 
 
 
 35 
 
 55 
 
 
 72 
 
 113 
 
 320 
 
 40 
 
 62 
 
 1,920 
 
 76 
 
 119 
 
 960 
 
 Number of Rails, Splices, and Bolts per Mile of Track. 
 
 Length of 
 Rail. Feet. 
 
 No. of Rails 
 per Mile. 
 
 No. of Splices. 
 
 No. of Bolts, 
 4 to Each 
 Joint. 
 
 No. of Bolts, 
 6 to Each 
 Joint. 
 
 18 
 
 586 
 
 1,168 
 
 2,336 
 
 3,504 
 
 20 
 
 528 
 
 1,056 
 
 2,112 
 
 3,168 
 
 21 
 
 503 
 
 1,006 
 
 2,012 
 
 3,018 
 
 22 
 
 480 
 
 960 
 
 1,920 
 
 2,880 
 
 24 
 
 440 
 
 880 
 
 1,760 
 
 2,640 
 
 25 
 
 422 
 
 844 
 
 1,688 
 
 2,532 
 
 26 
 
 406 
 
 812 
 
 1,624 
 
 2,436 
 
 27 
 
 391 
 
 782 
 
 1,564 
 
 2,346 
 
 28 
 
 377 
 
 754 
 
 1,508 
 
 2,262 
 
 30 
 
 352 
 
 704 
 
 1,408 
 
 2,112 
 
 Railroad Spikes per Mile of Track. 
 
 Size Measured 
 Under Head. 
 
 Average No. 
 per Keg 
 of 200 Lb. 
 
 Ties 2 Ft. Between Centers, 
 4 Spikes to a Tie. 
 
 Rails Used. 
 Pounds per 
 
 Inches. 
 
 Pounds. 
 
 Kegs. 
 
 Yard. 
 
 X t 9 b 
 
 375 
 
 5,870 
 
 291 
 
 45 to 70 
 
 5 X & 
 
 400 
 
 5,170 
 
 26 
 
 40 to 56 
 
 5 X* 
 
 450 
 
 4,660 
 
 231 
 
 35 to 40 
 
 44X4 
 
 530 
 
 3,960 
 
 20 
 
 28 to 35 
 
 4 X4 
 
 600 
 
 3,520 
 
 17f 
 
 24 to 35 
 
 HWS 
 
 XX 
 
 680 
 
 720 
 
 ' 3,110 
 2,910 
 
 154 
 
 14| 
 
 1 
 
 i 20 to 30 
 
 34 X ts 
 4X1 
 
 900 
 
 1,000 
 
 2,350 
 
 2,090 
 
 11 
 
 104 
 
 
 t 16 to 25 
 
 34 Xf 
 3 X§ 
 
 1,190 
 
 1,240 
 
 1,780 
 
 1,710 
 
 9 
 
 84 
 
 
 j- 16 to 20 
 
 24 X 1 
 
 1,342 
 
 1,575 
 
 74 
 
 12 to 16 
 
118 
 
 WIRE ROPES . 
 
 WIRE ROPES. 
 
 Wire rot>es for mine use are made of either iron or steel, and are g e uer- 
 allv round Flat wire ropes are sometimes used, but the round rope is the 
 favorite for many reL>n^ and is generally . used in American practice, 
 excentina in some of the deep metal mines having small compartment shafts. 
 
 6X Taner^ooes are sometimes used, the idea being to produce a rope of um- 
 fv-.T'n-i Erpriffth that is to have it less strong and of less diameter at the cage 
 fo T St £ e ?f is leTs^and greater in strength and diameter- at the 
 
 djum^end? vd^re^he 8 lc^d^^is^^eatSt. The theory is correct . and some 
 weight of rope is saved; but practically there is not much ^ 
 
 is doubtful whether taper ropes will ever be generally used. The loug- 
 established conviction that the best of all ropes for colliery use is a round 
 one made of steel or iron has never been overcome and probably ne\er 
 
 Wll Sfeei rones are in most respects superior to iron ropes, and are therefore 
 crninimr in fa?or lvervTear. The principal advantage of a steel rope is that 
 fr has ! greater stre iigt h th a n an iron rope of equal diameter; consequently 
 it ^can be made lighter and can pass around pulleys and drums with less 
 
 ^Mfenfng’ tof dmm therefs often a grievous error made. Men 
 who will not Stink of passing a rope around a pulley of too small diameter 
 will insert it in the drum rim in such a way 
 as to make a very sharp curve, and make a 
 weak point in the rope that would not other- 
 wise exist. In the accompanying cut v a) 
 shows the right and (6) the wrong way of 
 passing the rope through the drum rim. 
 
 The securing of the rope to the drum or --i 
 
 the drum shaftly several Soils around 
 
 around either the drum or the ; shaft ? a pull ^ 
 
 ^Sn^^^^e^s^of^ur^lu^l^cases^auSe^margiif'be^veeu^th'Jb^^ 
 
 loaded'fs unduly W 
 
 wrnmmm 
 
 i? 1 sns W 
 
 th Tro P e' s houl,l not be changed from a large dn,m to 
 
 sumebub 
 
 mmmmmsrn 
 
 «mn“cb^ e &idtoexmU wear for short distances at intervals 
 
WIRE ROPES. 
 
 119 
 
 along the rope. In the ordinary lay, Fig. (a), the wires are twisted in 
 the opposite direction to the strands. This method prevents the rope from 
 untwisting when in use, and the wires from unraveling when they are worn 
 through or broken at the surface. In the Lang lay, Fig. (6), the wires 
 are twisted in the same direction as the strands, thus giving each wire a 
 greater wearing surface, while the rope is smoother and will wear longer 
 A fter the wires begin to break, unraveling becomes troublesome, and it is 
 more difficult to splice a Lang lay rope than an ordinary lay one. Hoisting 
 ropes, especially those used to raise and lower men, should not be spliced 
 The locked wire rope, a cross-section of which is shown in Fig. (c) consists 
 of wires of special cross-section formed in concentric layers. The lav of 
 the inner wires is opposite to that of the outer ones, and somewhat longer. 
 This prevents untwisting, and brings the greater stress upon the outside 
 
 layer, which is supposed to give way first. The inside layer, although inac- 
 cessible, and therefore cannot be inspected or oiled, can be relied upon 
 until the external portion of the rope wears out. This form of rope has a 
 smooth cylindrical surface, but it is not so flexible as the other forms, and is 
 most suitable for haulage purposes or bucket transportation. The life of a 
 steel rope depends largely on the conditions to which it is subjected, and the 
 care it receives. At some mines the ropes must be changed every six 
 months, while at others the ropes last for one year and longer. Where the 
 rope enters the socket by which it is attached to the cage is perhaps the 
 place where signs of weakness will first appear. This point should be fre- 
 quently inspected, and a new connection made every two or three months 
 by cutting a few feet off the end and paying it out from the drum end. 
 
 WEIGHTS AND STRENGTHS OF WIRE ROPES. 
 
 Flat Ropes, 
 
 ( Trenton Iron Co ., Trenton , N.J.) 
 
 Size. Inches. 
 
 Approximate 
 Weight per Foot. 
 Pounds. 
 
 Breaking Stress. 
 (Approximate.) Pounds. 
 
 
 Iron. 
 
 Cast Steel. 
 
 2 X| 
 
 1.35 
 
 20,000 
 
 40,000 
 
 2|X| 
 
 1.70 
 
 25,000 
 
 50,000 
 
 3X1 
 
 2.05 
 
 30,000 
 
 60,000 
 
 3| X I 
 
 2.40 
 
 35,000 
 
 70,000 
 
 4 XI 
 
 2.75 
 
 40,000 
 
 80,000 
 
 5 X | 
 
 3.45 
 
 50,000 
 
 100,000 
 
 6 X | 
 
 4.15 
 
 60,000 
 
 120,000 
 
 3X1 
 
 2.40 
 
 37,500 
 
 75,000 
 
 3|X | 
 
 2.S5 
 
 43,750 
 
 87,500 
 
 4 XI 
 
 3.30 
 
 50,000 
 
 100,000 
 
 5 X I 
 
 4.20 
 
 62,500 
 
 125,000 
 
 6 Xi 
 
 5.10 
 
 75,000 
 
 150,000 
 
 7 XI 
 
 6.00 
 
 87,500 
 
 175,000 
 
 8 XI 
 
 6.90 
 
 100,000 
 
 200,000 
 
 L safe working load allow from one-fifth to one-seventh of the break 
 
 ing stress. 
 
120 
 
 WIRE ROPES. 
 
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WIRE ROPES. 
 
 121 
 
 
 
 
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 The wire-rope table given on pages 
 120 and 121 is a rearrangement of the 
 standard tables published in the cata- 
 logues of most of the American manu- 
 facturers of wire rope. 
 
 The proper working load given in 
 this table is one-fifth of the ap- 
 proximate breaking stress, that is, 
 a factor of safety of 5 is used, and 
 when the values given in this table 
 are used this factor is supposed to 
 allow for the bending stress. The 
 sizes of sheaves or drums given in 
 this fable are largely empirical, but 
 they are based on a long experience 
 in the use of wire ropes, and in most 
 cases represent the minimum diam- 
 eter recommended by the rope makers. 
 The factor of safety of 5 assumes ordi- 
 nary conditions of working; where 
 the conditions are extraordinary, and 
 particularly in cases where men are 
 to be hoisted, a larger factor than 5 is 
 used, varying from 5 to 10. Many ele- 
 vator specifications require a factor of 
 10 to be used. When any factor but 
 5 is used, the proper working load is 
 obtained by dividing the approxi- 
 mate breaking stress by the factor of 
 safety assumed, or, if we know the 
 load to be lifted and multiplv this 
 by the assumed factor, we obtain a 
 certain value, and by comparing 
 this with the values given in the 
 table for the approximate breaking 
 stress we can determine the size of 
 rope, and the minimum diameter of 
 sheave to use. 
 
 For example, suppose it is desired 
 to determine the size of rope needed 
 to hoist a load of 8 tons through a 
 height of 300 feet. Multiplying 8 by 5, 
 we have 40 tons, and from the table 
 we find that the approximate breaking 
 strength of a lf-inch, crucible-steel 
 hoisting rope is 42 tons. Such a rope 
 weighs 2 pounds per foot; hence, the 
 weight of the rope is 300 X 2 = 600 
 pounds, or .3 of a ton, and the total 
 load on the rope is therefore 8.3 tons. 
 8.3 X 5 = 41.5, which is less than the 
 approximate breaking stress for the 
 given rope. The proper size of sheave 
 or drum for this rope is given by the 
 table as 4| feet. 
 
 Galvanized wire is sometimes used 
 in the manufacture of rope to prevent 
 corrosion of the iron or steel of the 
 wire. Galvanizing accomplishes this 
 in the case of standing ropes but is 
 not effective for running ropes, in 
 which the friction of the rope against 
 sheaves or drums soon wears off a por- 
 tion of the zinc, and with both zinc 
 and iron exposed and in contact with 
 water, corrosion proceeds more rapidly 
 than it would if the zinc were not 
 present. 
 
122 
 
 WIRE ROPES. 
 
 flattened-strand rope. 
 
 Fig 
 
 r x Fig. 2. FlG - 3 * 
 
 able saving in the wear of pulleys and sheaves. 
 
 Patent Flattened-Strand Rope. 
 
 ( From, A. Leschen & Sons Rope Co., St. Louis , Mo.) 
 
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 4.32 
 
 3.92 
 
 3.53 
 
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 2.74 
 
 2.35 
 
 1.96 
 
 1.57 
 
 Swedish Iron, 
 
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 14 
 
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 Crucible 
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 as ordinary ^ire rope, thus making unnecessary any change in plant. 
 
WIRE ROPES. 
 
 122a 
 
 MINES AND MINERALS. 
 
mb 
 
 WIPE ROPES. 
 
 SUSPENSION CABLEWAYS. 
 
 \ suspension eablewav is a hoisting and conveying device using a sus- 
 pended cable for a traekwav. There are two types : the inclined, or semi- 
 gravitv. Fig 1. and the horizontal. Fig. 2. page 122a. . 0 
 
 The inclined cableway consists of a cable inclined 20° to 22° to the 
 horizontal, and passing over a cast-iron addle B on top of a tower or 
 frame A It is anchored bv logs D buried about 5 feet underground, or from 
 iron plu<*s secured in the' rock, when the rock is near the surface. The 
 trolley carriage G runs down the incline of the cableway by gravity until it 
 
 reaches a stop. A hoisting rope E operated 
 by a winding drum F leads over a sheave 
 puller e , thence to a pulley in the carriage G, 
 thence to a fall block if, upwards again to a 
 second pulley in the carriage, and downwards 
 again to the fall block. Winding in the rope 
 heists the fall block to the carriage, the car- 
 riage remaining at the lower stop. W hen the 
 fall block collides with the carriage, both the 
 carriage and the fall block are pulled up 
 the incline cable, and when the carriage 
 arrives at the head-tower, a gate, or hook. O is 
 lowered to hold it in place. The fall block is 
 then lowered and the load discharged. The 
 eneine F has usually a 10" X 12" double cylin- 
 der and a single friction drum 37 inches in 
 diameter. „ _ - 
 
 An endless rope H takes several turns 
 around the sheave J to prevent it from slip- 
 ping. and both ends are passed over sheaves h 
 at the top of the derrick, one end being 
 secured to the front of the carriage, while 
 the other end is taken through the carriage 
 and around the return sheave I and fastened 
 to the rear end of the carriage G. The end- 
 less-rop*e wheel J is provided with a band 
 brake, which, when applied, holds the car- 
 riage securelv at any point on the cable. 
 
 All ropes pass through the supporting 
 trolleys A', which are connected by a chain L. 
 These' trolleys follow the carriage by gravity, 
 and the chain may or may not be fast to the 
 carriage. , . . 
 
 Instead of chain-connected trolleys, patent 
 button-stop, fall-rope carriers, which are 
 lighter, mav be used. These are spaced 
 along an auxiliary rope on which buttons are 
 screwed The carriers are picked up by a 
 horn on the front of the carriage. These are 
 said to be cheaper for operation than the 
 chain trolleys. . _ .. 
 
 The length of the span for inclined cable- 
 wavs varies from 200 to 1.200 feet. The main 
 rope is from It to 2£ inches in diameter, the 
 hoisting rope from f to # inch, and the end- 
 less rope and £ inch. The rope mostly used 
 is dx strands, nineteen wires to the strand, 
 crucible cast steel. The hoisting rope lasts 
 from 1 vear to 2 vears. and the main cable from 5 years to 10 years. These 
 eablewav? are widely used about slate quarries in Eastern Pennsylvania, 
 where the operating* expense for each cable, where two or more are con- 
 nected with one boiler, is about §5 per day of 10 hours. This includes the 
 engineer <team. and maintenance of the cableway. - v . 
 
 The horizontal cableway requires a double friction drum and r^ersible 
 link-motion engine. It may be operated at any inclination of the carrying 
 cable and either from the high or low point of the support, though, if posa" 
 bfe. it should be from the higher end. The endless, ot tractaoiu 
 attached to one of the drums of the engine and the engineer thus^has com 
 pU^ control of the carriage ; hence, because of its greater applicability, this 
 
WIRE ROPES , 
 
 122c 
 
 system is supplanting the inclined, although the inclined costs one-fourth 
 less for installation. The method of operation is similar to the inclined. 
 The amount of rope required is the same in each system. A horizontal 
 cableway of the Hamilton Coal Co., near Tarentum, Pa., has a span of 2,200 
 feet. The stationary rope is 2,500 feet long and 2£ inches in diameter. The 
 hoisting rope is 4,500 feet long and £ inch in diameter. The head-tower is 80 
 feet high, the tail-tower is 100 feet, and the rope deflects 80 feet. The skip 
 used holds 3 tons of coal and makes 10 trips per hour. Five men operate the 
 plant and it takes 2 tons of coal for the engine. Based on a capacity of 
 100 tons per day, the cost of carrying the coal is 13 cents per ton. For a 
 cableway . of average length, 1,000-1,500 feet, the cost of operation should not 
 be one-half the above cost. A cableway 2,140 feet long was used in con- 
 structing the dam at the power plant at Glens Falls, N. Y. 
 
 One or both towers of a cableway may be mounted on wheels capable of 
 moving on a track at right angles to the cable and the cableway then made 
 to cover a wide territory. 
 
 WIRE-ROPE TRAMWAYS. 
 
 A tramway, in America, is a cableway of the horizontal type consisting 
 of a number of spans. In England, the term cableway includes tramways. 
 
 Single Tramways. — Single wire-rope tramways have a single moving rope, 
 which serves to support and advance the load at one and the same time, 
 Fig. 3. This rope passes over suitable sheaves at the intermediate supports, 
 and the load is carried in buckets suspended from it by gooseneck or 
 i straight hangers. The hangers are usually attached to the cable by means 
 of a clip, which is either inserted in the center of the cable or strapped to it. 
 The carriers are often loaded and unloaded while in motion, the loading 
 being accomplished by a traveling mechanical hopper and the unloading by 
 a drop bottom to the bucket. If the line is level, or the grade light, the 
 hangers are provided with box heads filled with wood or leather and rub- 
 ber, which rest on the rope; the rubber or wood providing sufficient friction 
 to prevent the hanger’s slipping. With this system, long spans are evi- 
 dently out of the question, because with a long span the angle of the 
 rope in the vertical plane, at the supports, becomes so great that the friction 
 will not hold the box head. For all practical purposes, grades exceeding 
 1 : 4 are to be avoided; and for steeper grades, to prevent slipping, a clamp or 
 a elip inserted in the rope is used to fasten the hanger to the rope. 
 
 The single moving-rope tramways carry loads not exceeding 200 pounds. 
 The speed of the rope for the variety in which the hangers are fastened to 
 the rope may be as high as 450 feet per minute, and for one in which the 
 hanger is loose, 200 feet per minute. The single moving-rope tramway has 
 a capacity up to 200 tons per day, and may be built, say, H to 2 miles long. 
 
 Double Tramways. — The more satisfactory and substantial kind of wire- 
 rope tramways has one or more fixed ropes which constitute the permanent 
 
122d 
 
 WIRE HOPES. 
 
 - 
 
 wav and an endless traction rope. The loaded carrier travels outwards on 
 one fixed cable and returns by a parallel one suspended from the opposite 
 side of the same supports. The terminals have suitable appliances for load- 
 
 ing and unloading the buckets, either by hand or automatically. 
 
 The intermediate supports are built of wood or steel framing, with sad- 
 dles of cast iron a. Fig. 4, in which the fixed cables b rest The traction 
 rope c is supported (in the absence of a bucket) by the rollers d, set con- 
 veniently on the supports. The load is carried in buckets e, or other con- 
 trivances suitable for the purpose, which are suspended from a trolley /, 
 which runs on the fixed cables, the wheels of which are large enough to 
 pass over the rope couplings, and also to clear the saddles. Grips g attach 
 the carriers to the traction rope. These grips may be operated by hand or 
 
 ^T hi s i n dU > f tramway is capable of carrying individual loads up to 1,400 
 or 1,500 pounds, not including the weights of the bucket and hanger itself. 
 The speed of traction rope may be from 150 to 350 feet per minute. The 
 capacity is from 200 to 1,000 tons per day of 10 hours. These figures represent 
 good, safe practice, but they are not, of course, inflexible. 
 
 The maximum length of line which may be built in one section varies 
 largely with conditions of load, spacing of supports, contour of ground, etc. 
 Wire-rope tramwavs work under great difficulties, and probablv 2* to 4 miles 
 is the economical iimit. This has been exceeded, but the trouble is that for 
 a much greater distance the friction becomes too great for economical work- 
 ing of the traction rope. This does not, however, limit the length of tram- 
 wav which may be built, as the power station may be located at a convenient 
 intermediate point, dividing the line into sections. Several intermediate 
 power stations may be used, and the length of the line greatly increased 
 above the limit given. A tramway at Grand Encampment, Wyoming, is 16 
 miles long and carries 40 tons of ore per hour. 
 
 TRANSMISSION OF POWER BY WIRE ROPES. 
 
 The term transmission, as here used, applies simply to the modification of 
 belt driving, using grooved wheels or sheaves at each end of the line, lhe 
 power is applied to one sheave and taken off from the other . 
 
 The friction between rope and sheaves depends directly on the weight 
 and tension of the rope and on the nature of the surfaces in contact. This 
 pressure is better obtained by using a large, heavy rope at a low tension than 
 by using a smaller rope at a high tension. . ^ - 
 
 The deflection or sag of the rope, between the sheaves, is the same for 
 both upper and lower parts of the rope when the transmission is not running 
 and should be, according to John A. Roebling’s Sons Co., equal to about one- 
 thirty-sixth of the span. The deflection may be calculated by the formula 
 from the Trenton Iron Co.: 
 
 . w s 2 
 h _ 8 1 * 
 
 in which h = the deflection in feet; 
 
 w = weight of the rope per foot, in pounds; 
 
 8 — span, in feet; 
 t = tension, in pounds. 
 
 When driving from the under side, this part of the rope will be tightened 
 and its deflection decreased, while the upper part of the rope becomes slack- 
 ened and its deflection increased. Under proper conditions the deflection ot 
 the lower rope should be about one-fiftieth, and that of the upper about one- 
 twenty- fifth of the span. The difference in the tensions of the two parts 
 of the rope is the effective pull of the driving sheaves, enabling power to be 
 transmitted. 
 
 Transmission ropes are subject to three stresses, viz.: .. * 
 
 (1) The direct tension, due to the power transmitted, plus the friction ana 
 weight of the rope; (2) the bending stress, due to the bending of the rope 
 around the sheaves; (3) the centrifugal tension, due to the centrifugal force 
 
 in the rapidly running rope. . . •, _ Mr wm 
 
 The following data on stresses m transmission ropes are given by Mr. V\ m. 
 Hewitt, of the Trenton Iron Co. . . . , 
 
 In transmitting power by wire rope, working tension should not exceed 
 
WIRE ROPES. 
 
 122e 
 
 1 
 
 the difference between the maximum safe stress and the bending stress. 
 It may be greater, therefore, as the bending stress is less, but to avoid slipping 
 a certain ratio must exist between the tensions in taut and slack portions of 
 the rope when running, which is determined by the formula 
 
 T= Sefm; 
 
 in which T = tension in the taut portion of the rope; 
 
 S = tension in the slack portion; 
 
 e = base of the Naperian system of logarithms — 2.7182818; 
 n = number of half laps of rope about sheaves or drums at either 
 end of line; 
 
 7r = 3.1416; 
 
 f = coefficient of friction depending on the kind of filling in the 
 grooves of the sheaves, or character of the material on 
 which the rope tracks. 
 
 The useful effort of transmitting force is the difference between the tension 
 of the taut and slack portions of the rope. T — S = & (ef n ^ — 1), and to 
 obtain this, the initial tension, or tension when the rope is at rest, must be 
 one-half the sum of the two tensions. 
 
 e fn 7T _L 1 
 
 TP S = + i) = __ r ^_ (T-S). 
 
 The following are some of the values of / : 
 
 Dry rope on a grooved iron drum 120 
 
 Wet 'rope on a grooved iron drum 085 
 
 Greasy rope on a grooved iron drum 070 
 
 Dry rope on wood-filled sheaves 235 
 
 Wet rope on wood-filled sheaves .170 
 
 Greasy rope on wood-filled sheaves 140 
 
 Dry rope on rubber and leather filling 495 
 
 Wet rope on rubber and leather filling 400 
 
 Greasy rope on rubber and leather filling 205 
 
 e f n TT -. — i 
 
 above values of/, for one up to six half laps of the rope, are as follows: 
 
 Values of efm *. 
 
 f = 
 
 n = Number of Half Laps About Sheaves or Drums at Either 
 End of Line. 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 .070 
 
 1.246 
 
 1.552 
 
 1.934 
 
 2.410 
 
 3.003 
 
 3.741 
 
 .085 
 
 1.306 
 
 1.706 
 
 2.228 
 
 2.910 
 
 3.801 
 
 4.964 
 
 .100 
 
 1.369 
 
 1.875 
 
 2.566 
 
 3.514 
 
 4.810 
 
 6.586 
 
 .120 
 
 1.458 
 
 2.125 
 
 3.099 
 
 4.518 
 
 6.586 
 
 9.602 
 
 .130 
 
 1.504 
 
 2.263 
 
 3.405 
 
 5.122 
 
 7.706 
 
 11.593 
 
 .140 
 
 1.552 
 
 2.410 
 
 3.741 
 
 5.808 
 
 9.017 
 
 13.998 
 
 .150 
 
 1.602 
 
 2.566 
 
 4.111 
 
 6.586 
 
 10.551 
 
 16.902 
 
 .170 
 
 1.706 
 
 2.910 
 
 4.964 
 
 8.467 
 
 14.445 
 
 24.641 
 
 .200 
 
 1.875 
 
 3.514 
 
 6.586 
 
 12.346 
 
 23.140 
 
 43.376 
 
 .205 
 
 1.904 
 
 3.626 
 
 6.904 
 
 13.146 
 
 25.031 
 
 47.663 
 
 .235 
 
 2.092 
 
 4.378 
 
 9.160 
 
 19.166 
 
 40.100 
 
 * 83.902 
 
 .250 
 
 2.193 
 
 4.810 
 
 10.551 
 
 23.140 
 
 50.637 
 
 111.318 
 
 .265 
 
 2.299 
 
 5.286 
 
 12.153 
 
 27.941 
 
 64.239 
 
 147 693 
 
 .300 
 
 2.566 
 
 6.586 
 
 16.902 
 
 43.376 
 
 111.318 
 
 285.680 
 
 .350 
 
 3.001 
 
 9.017 
 
 27.077 
 
 81.307 
 
 244.152 
 
 733.145 
 
 .400 
 
 3.514 
 
 12.346 
 
 43.376 * 
 
 152.405 
 
 535.488 
 
 1,849.140 
 
 .410 
 
 3.626 
 
 13.146 
 
 47.663 
 
 172.814 
 
 626.577 
 
 2,271.775 
 
 .450 
 
 4.111 
 
 16.902 
 
 69.487 
 
 285.680 
 
 1,174.480 
 
 4,828.510 
 
 .495 
 
 4.716 
 
 22.425 
 
 106.194 
 
 502.881 
 
 2,381.400 
 
 
 .500 
 
 4.810 
 
 23.140 
 
 111.318 
 
 535.488 
 
 2,575.940 
 
 
 A 
 
122f 
 
 WIRE ROPES. 
 
 Values of 
 
 ef n 7T + l 
 e t n 7T _ 1* 
 
 / = 
 
 n = Number of Half Laps About Sheaves or Drums at Either 
 End of Line. 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 .070 
 
 9.130 
 
 4.623 
 
 3.141 
 
 2.418 
 
 1.999 
 
 1.729 
 
 .085 
 
 7.536 
 
 3.833 
 
 2.629 
 
 2.047 
 
 1.714 
 
 1.505 
 
 .100 
 
 6.420 
 
 3.287 
 
 2.280 
 
 1.795 
 
 1.525 
 
 1.358 
 
 .120 
 
 5.345 
 
 2.777 
 
 1.953 
 
 1.570 
 
 1.358 
 
 1.232 
 
 .130 
 
 4.968 
 
 2.584 
 
 1.832 
 
 1.485 
 
 1.298 
 
 1.189 
 
 .140 
 
 4.623 
 
 2.418 
 
 1.729 
 
 1.416 
 
 1.249 
 
 1.154 
 
 .150 
 
 4.322 
 
 2.280 
 
 1.643 
 
 1.358 
 
 1.209 
 
 1.126 
 
 .170 
 
 3.833 
 
 2.047 
 
 1.505 
 
 1.268 
 
 1.149 
 
 1.085 
 
 .200 
 
 3.287 
 
 1.795 
 
 1.358 
 
 1.176 
 
 1.090 
 
 1.047 
 
 .205 
 
 3.212 
 
 1.762 
 
 1.338 
 
 1.165 
 
 1.083 
 
 1.043 
 
 .235 
 
 2.831 
 
 1.592 
 
 1.245 
 
 1.110 
 
 1.051 
 
 1.024 
 
 .250 
 
 2.676 
 
 1.525 
 
 1.209 
 
 1.090 
 
 1.010 
 
 1.018 
 
 .265 
 
 2.539 • 
 
 1.467 
 
 1.179 
 
 1.072 
 
 1.032 
 
 1.014 
 
 .300 
 
 2.280 
 
 1.358 
 
 1.126 
 
 1.047 
 
 1.018 
 
 1.007 
 
 .350 
 
 2.000 
 
 1.249 
 
 1.077 
 
 1.025 
 
 1.008 
 
 1.003 
 
 .400 
 
 1.795 
 
 1.176 
 
 1.047 
 
 1.013 
 
 1.004 
 
 1.001 
 
 .410 
 
 1.765 
 
 1.164 
 
 L043 
 
 1.012 
 
 1.003 
 
 1.001 
 
 .450 
 
 1.643 
 
 1.126 
 
 1.029 
 
 1.007 
 
 1.002 
 
 1.000 
 
 .495 
 
 1.538 
 
 1.093 
 
 3 .019 
 
 1.004 
 
 1.001 
 
 1.000 
 
 .500 
 
 1.525 
 
 1.090 
 
 1.018 
 
 1.004 
 
 1.001 
 
 1.000 
 
 For a given diameter of sheave, and a variable diameter of wire, a ratio 
 exists between these diameters corresponding to a maximum working 
 tension. This ratio results, approximately, in a working tension of one- 
 third and bending stress of two-thirds of the maximum safe tension, which 
 is from one-third to two-fifths of the ultimate stress, and practically deter- 
 mines the minimum diameter of sheave for any rope. The ratio for any 
 size of wire varies slightly, according to the number of wires composing the 
 rope, and in terms of rope diameter is, 
 
 For 7-wire rope 
 
 For 12-wire rope - 
 
 For 19-wire rope * 
 
 from which we derive the following table: 
 
 Diam- 
 eter of 
 Rope. 
 
 T5 
 
 i 
 
 Steel. 
 
 T5 
 
 5 
 
 ft 
 
 7- Wire. 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 55 
 
 60 
 
 70 
 
 80 
 
 12- Wire. 
 
 15 
 
 19 
 
 22 
 
 26 
 
 30 
 
 33 
 
 37 
 
 41 
 
 44 
 
 52 
 
 59 
 
 19- Wire. 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 32 
 
 35 
 
 41 
 
 47 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 140 
 
 160 
 
 
 
 Steel 
 
 Iron 
 
 79.6 
 
 160.5 
 
 59.3 
 
 120.0 
 
 47.2 
 
 95.8 
 
 IN INCHES. 
 
 Iron. 
 
 12-Wire. 
 
 19-Wire. 
 
 30 
 
 24 
 
 38 
 
 30 
 
 45 
 
 36 
 
 53 
 
 42 
 
 60 
 
 48 
 
 68 
 
 54 
 
 75 
 
 60 
 
 83 
 
 66 
 
 90 
 
 72 
 
 105 
 
 84 
 
 120 
 
 96 
 
WIRE ROPES. 
 
 122g 
 
 Sheaves. — To decrease the bending stresses the sheaves for wire-rope 
 transmissions are generally of as large diameter as is practicable to give the 
 required speed to the rope. Large sheaves are also advantageous because 
 with them the rope is run at a high velocity allowing of a lower tension, 
 and permitting a rope of smaller diameter to be used than would be pos- 
 sible with smaller sheaves, provided, of course, that the span is of suffi- 
 cient length to give the necessary weight. 
 
 Sheaves are generally made of cast iron when not exceeding 12 feet in 
 diameter, and when larger than this they are usually built up with wrought- 
 iron arms. Sheaves, upon which the rope is to make but a single half-turn, 
 are made with V-shaped grooves in their circumference. The bottom part 
 of the groove is widened to receive the tilling, which consists of some sub- 
 stance to give a bed for the rope to run on and protect it from wear, and to 
 increase the friction so that the rope will not slip. This filling is made of 
 blocks of wood, rubber, and leather, or other material. Rubber and leather 
 have been used separately, but blocks of rubber separated by pieces of 
 leather have been found to give the best results. 
 
 Power Transmitted. — The horsepower transmitted is equal to the resistance 
 overcome (the effective null), in pounds, multiplied by the speed of the 
 rope, in feet per minute, and divided by 33,000, that is (formula from John A. 
 Roebling’s Sons Co.), 
 
 T V 
 
 H= w,m' 
 
 in which H = horsepower transmitted; 
 
 T = difference in tension between the driving and driven sides of 
 the rope; 
 
 V = speed of the rope, in feet per minute. 
 
 In applying this formula, V is either given or assumed. T is equal to 
 the weight of the rope suspended between the sheaves multiplied by three 
 (for the proportion of deflection stated). 
 
 To transmit a given horsepower, the speed of the rope may be increased 
 and the tension (effective pull) correspondingly decreased, and a smaller 
 rope may be used provided other considerations will allow it. 
 
 For determining the horsepower that can be transmitted over a given 
 transmission, the following formula is given by the Trenton Iron Co.: 
 H=[cd 2 - .000006 ( W + g' + g")] s ; 
 in which H — horsepower that can be transmitted; 
 
 c = constant depending on material of rope, the filling in the 
 grooves of the sheaves, and the number of half laps about 
 the sheaves or drums at either end of the line; 
 d = diameter of rope, in inches; 
 
 W — weight of rope, in pounds; 
 
 ' g f = weight of terminal sheaves and shafts; 
 
 g" = weight of intermediate sheaves and shafts. 
 
 The following table gives the value of c for ropes on different materials: 
 
 VALUE OF C FOR ROPES ON DIFFERENT MATERIALS. 
 
 c — for 
 Steel Rope on 
 
 Number of Half Laps About Sheaves or Drums at 
 Either End of Line. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 Iron 
 
 5.6L 
 
 8.81 
 
 10.62 
 
 11.65 
 
 12.16 
 
 12.56 
 
 Wood 
 
 6.70 
 
 9.93 
 
 11.51 
 
 12.26 
 
 12.66 
 
 12.83 
 
 Rubber and 
 Leather 
 
 9.29 
 
 11.95 
 
 12.70 
 
 12.91 
 
 12.97 
 
 13.00 
 
 The values of c for iron rope are one-half of the above. 
 
 It is evident from the above figures that when more than three laps are 
 made it is immaterial what the surface is on which the rope tracks, as far as 
 frictional adhesion is concerned. 
 
122h 
 
 WIRE ROPES. 
 
 From the foregoing formula, assuming the sheaves to be of equal diameter, 
 and of a size not less than the minimum diameter given in the table, we 
 deduce the following: 
 
 HORSEPOWER CAPABLE OF BEING TRANSMITTED BY A STEEL 
 ROPE MAKING A SINGLE LAP ON WOOD-FILLED SHEAVES 
 
 Diam- 
 eter of 
 Rope 
 in 
 
 Inches. 
 
 Velocity of Rope in Feet per Second. 
 
 T6 
 
 i 
 
 T6 
 
 f 
 
 ii 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 4 
 
 8 
 
 13 
 
 17 
 
 21 
 
 25 
 
 28 
 
 32 
 
 37 
 
 40 
 
 7 
 
 13 
 
 20 
 
 26 
 
 33 
 
 40 
 
 44 
 
 51 
 
 57 
 
 62 
 
 10 
 
 19 
 
 28 
 
 38 
 
 47 
 
 56 
 
 64 
 
 73 
 
 80 
 
 89 
 
 13 
 
 26 
 
 38 
 
 51 
 
 63 
 
 75 
 
 88 
 
 99 
 
 109 
 
 121 
 
 17 
 
 34 
 
 51 
 
 67 
 
 83 
 
 99 
 
 115 
 
 130 
 
 144 
 
 159 
 
 22 
 
 43 
 
 65 
 
 86 
 
 106 
 
 128 
 
 147 
 
 167 
 
 184 
 
 203 
 
 27 
 
 53 
 
 79 
 
 104 
 
 130 
 
 155 
 
 179 
 
 203 
 
 225 
 
 247 
 
 32 
 
 63 
 
 95 
 
 126 
 
 157 
 
 186 
 
 217 
 
 245 
 
 
 
 38 
 
 76 
 
 103 
 
 150 
 
 186 
 
 223 
 
 
 
 
 
 52 
 
 104 
 
 156 
 
 206 
 
 
 
 
 
 
 
 68 
 
 135 
 
 | 
 
 202 
 
 
 
 
 
 
 
 
 The horsepower that may be transmitted by iron ropes is one-half of the 
 
 The foregoing table gives the maximum amount of power capable of 
 being transmitted under the conditions stated, so that in using wood-lined 
 sheaves, it is well to make some allowance for the stretching of the rope, 
 and to advocate somewhat heavier equipments than the above table would 
 give- that is, if it is desired to transmit 20 horsepower, for instance, to put m 
 a plant that would transmit 25 to 30 horsepower, thus avoiding the necessity 
 of having to take up a comparatively small amount of stretch. On rubber and 
 leather filling however, the amount of power capable of being transmitted 
 is considerablv greater than on wood, so that this filling is generally used; 
 and in this case no allowance need be made for stretch as such sheaves^ will 
 likely transmit the power given by the table, under all possible deflections 
 of the rope. 
 
 Thet 
 sheaves, 
 
 rope makes but a single lap, or a half lap at either end of the line. 
 
 GLOSSARY OF ROPE TERMS. 
 
 Anneaf°d Wire Rope.— A wire rope made from wires that have been softened 
 bvannealing and the tensile strength thereby lowered. 
 
 Bending Stress.— 1 The stress produced in the outer fibers of a rope by bending 
 over a sheave or drum. , +v . 
 
 Breaking Strain Breaking Strength , Breaking Stress.—' The least load that 
 will break a rope. These terms are used indiscriminately to mean the 
 load which will break a rope. The stress on a rope at tne moment of 
 breaking is the breaking stress , and the strain or deformation produced in 
 the material bv this stress is the breaking strain. 
 
 Bright Rope.— Rope of any construction, whose wires have not been gal- 
 vanized, tinned, or otherwise coated. , . o 
 
 Cable-Laid Rope— A term applied to wire cables made of several ropes 
 twisted together, each rope being composed of strands twisted together 
 without limitation as to the number of strands or direction of twist A 
 fiber cable-laid rope is a rope having three strands of hawser-laid rope, 
 twisted right-handed. 
 
WIRE ROPES. 
 
 122i 
 
 Cable —Same as cable-laid rope ; a fiber cable consists of three hawsers laid 
 up left-handed. . . _ , 
 
 Cast Steel— Steel that has been melted, cast into ingots, and rolled out 
 
 into bars. . , * . . 
 
 Clamp— A device for holding two pieces or parts of rope together by 
 pressure. , „ 
 
 Clip — a device similar to a clamp but smaller and for the same purpose. 
 
 Coir. — Coconut-husk fiber. 
 
 Compound.— A lubricant applied to the inside and outside of ropes pre- 
 venting corrosion and lessening abrasion of the rope when in contact 
 with hard surfaces. 
 
 Core— The central part of a rope forming a cushion for the strands. In wire 
 ropes it is sometimes made of wire, but usually it is of hemp, jute, or 
 some like material. 
 
 Coupling. — A device for joining two rope ends without splicing. 
 
 Crucible Steel.— A fine quality of steel made by the crucible process. 
 
 Drum. — The part of a hoisting engine on which the rope is wound. 
 
 Elastic Limit.— The elastic limit is-that point at which the deformations in 
 the material cease to be proportional to the stresses. 
 
 Elevator Rope.— A rope used to operate an elevator. 
 
 Endless Rope— A rope which moves in one direction, one part of which 
 carries loaded cars from a mine at the same time that another part 
 brings the empties into the mine. 
 
 Fiber— A single thread-like filament. 
 
 Flat Rope.— A rope in which the strands are woven or sewed together to 
 form a flat, braid-like rope. 
 
 Flattened-Strand Rope.— A wire rope whose strands are flattened or oval and 
 which therefore presents an increased wearing surface over that of the 
 ordinary round-strand rope. 
 
 Flattened-Strand Triangular Rope— A wire rope of the flattened-strand con- 
 struction in which the strands are triangular in shape. 
 
 Fl ee t— Movement of a rope sideways in winding on a drum. 
 
 Fleet Wheel.— A grooved wheel or sheave that serves as a drum and about 
 which one or more coils of a haulage rope pass. 
 
 Galvanized Rope.— Rope made of wires that have been galvanized or coated 
 with zinc to protect them from corrosion. 
 
 Grip Wheel.— A wheel, the periphery of which is fitted with a series of 
 toggle-jointed, cast-steel jaws that grip the rope automatically. 
 
 Guy.— A strand or rope used to support a pole, structure, derrick, or chim- 
 ney, etc. 
 
 Haulage Rope.— A rope used for haulage purposes. 
 
 Hawser.— A term applied to any wire rope used for towing on lake or sea. 
 A fiber hawser consists of three strands laid up right-handed. 
 
 Hawser- Laid Rope has three strands of yarn twisted left-handed, the yarns 
 being laid up right-handed. Synonymous with cable-laid rope as 
 applied to wire ropes. 
 
 Hawser Wire Rope.— Galvanized ropes of iron or steel usually composed of 
 six strands, 12 wires each, principally used in marine work for towing 
 purposes. 
 
 Hemp.— A tough, strong fiber obtained from the hemp plant. 
 
 Hoisting Rone— A rope composed of a sufficient number of wires and strands 
 to insure flexibility. Such ropes are used in shafts, elevators, quarries, 
 etc. 
 
 Idler.— A sheave or pulley running loose on a shaft to guide or support 
 
 Jute.— A fiber obtained from the inner bark of two Asiatic herbs: Corchorus 
 
 capsularis and C. olitorius. 
 
 Lang-Lay Rope.— A rope in which the wires in each strand are twisted in 
 the same direction as the strands in the rope. 
 
 Lay— A term indicating the direction, or length, of twist of the wires and 
 strands in a rope. 
 
 Live Load.— A load which is variable in distinction from a constant load. 
 
 Load Stress. — The stress produced by the load. 
 
 Locked- Wire Rope— A rope with a smooth cylindrical surface, the outer 
 wires of which are drawn to such shape that each one interlocks with 
 the other and the wires are disposed in concentric layers about a wire 
 core instead pf in strands. Particularly adapted for haulage and rope- 
 transmission purposes. 
 
122j 
 
 WIRE ROPES. 
 
 Manila.— The fiber of Musa textilis; Manila hemp. „ _ 
 
 Modulus of Elasticity. — The ratio between the amount of extension or com- 
 pression of a material and the load producing this same extension or 
 compression. , . 
 
 Plow Steel.— & select grade of steel of high tensile strength. First used in 
 rope for plowing fields. , . . , , ... , 
 
 Proper Working Load.— The maximum load that a rope should be permitted 
 to support under working conditions. (See working load.) 
 
 Regular-Lay Rope— A. rope in which the wires in each strand are twisted in 
 ODposite direction to the strands in the rope. 
 
 Round-Strand Rope.— A rope made of round twisted strands. 
 
 Running Rope.— A flexible rope that will pass througn blocks and used for 
 hoisting on shipboard. The term is also often used for any moving rope. 
 
 Sheave.— A wheel or pulley around or over which a rope passes. 
 
 Shroud Laid , or Four-Strand , Rope has four strands laid around a core. 
 
 Sisal— A hemp. The fiber of the Agave Sisalona. . 
 
 Socket.— A device fastened to the; end of a rope by means of which the rope 
 mav be attached to its load. The socket may be opened or closed. 
 
 Splice —The joining of two ends of rope by interweaving the strands. 
 
 Step Socket.— A special form of socket for use on locked-wire rope. 
 
 Stirrup— An adjustable bale of a socket _ . 
 
 Stone Wire.— A term applied to wire smaller than No. 14 put up m 12-pound 
 coils, which are about 8 inches inside' diameter. 
 
 Strand— A term applied to a varying number of wires or fibers twisted 
 together. The strands in turn are twisted together, forming a rope. 
 
 Stress. -A force or combination of forces tending to change the shape of a 
 
 Strain.— A chahge of shape produced in a body. (Stress and strain are 
 often used incorrectly as synonymous terms.) 
 
 Surging.— The flapping of a moving rope. 
 
 Swedish Iron.— A soft and comparatively pure iron. 
 
 Switch Rope.— A short length of rope fitted with a hook on one end and a 
 link on the other, used for the switching of freight cars. 
 
 Tail-Rope— (1) The rope that is used to draw the empties back into a 
 mine in a tail-rope haulage system. (2) A rope attached beneath the 
 cage when the cages are hoisting in balance. 
 
 Taper Rope. — A rope which has a gradually diminishing diameter frorn the 
 upper to the lower end. The diameter of the rope is decreased by 
 1 dropping one wire at a time at regular intervals. Both round and flat 
 ropes may be made tapered, and such ropes are intended for deep-shaft 
 hoisting with a view to proportioning the diameter of the rope to the 
 load to be sustained at different depths. 
 
 Tensile Strength— The stress required to break a rope by pulling it in two. 
 
 Thimble. — An oval iron ring around which a rope end is bent and fastened 
 to form an eye. ■ ,, „ 
 
 Tiller Rope.— A very flexible wire rope composed of six small ropes, usually 
 of seven-wire strands laid about a hemp core. . 
 
 Tinned Rope.— Rope made of wires that have been coated with tin to protect 
 them from corrosion . , 
 
 Torsion. — The process of twisting a wire thereby showing its ductility. 
 
 Traction Rope— A rope used for transmitting the power in a wire-rope tram- 
 way and to which the buckets are attached. 
 
 Transmission Rope. — A rope used for transmitting power. 
 
 Traveler.— A truck rolling along a suspended rope for supporting a load to 
 be transported. „ „ . , ^ 
 
 Turnbuckle— A form of coupling so threaded or swiveled that by turning it 
 the tension of a rope or rod may be regulated. 
 
 Ultimate Tensile Strength.— Same as tensile strength. 
 
 Universal Lay— Another term for lang lay. 
 
 Whipping.— The flopping of a moving rope. 
 
 Wire Gauge.— Standard sizes or diameters for wire. 
 
 Wire Rope— A rope whose strands are made of wires, twisted or woven 
 together 
 
 Working Load.- The maximum load that a rope can carry under the con- 
 ditions of working without danger of straining. (Same as proper work- 
 ingload.) . 
 
 Wrought Iron — A comparatively pure and malleable iron. , 
 
 Yarn.— 1 Twisted fiber of which rope strands are made. 
 
 Ife. 
 
00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 r IRE AND SHEET-METAL GAUGES. 
 
 OF WIRE AND SHEET-METAL GAUGES. 
 
 British 
 
 Wire 
 Gauge. 
 (Legal 
 Standard 
 n Great 
 Britain.) 
 Millim. 
 
 127 
 
 11.78 
 
 10.97 
 
 10.16 
 
 9.45 
 
 8.84 
 
 8.23 
 
 7.62 
 
 7.01 
 6.4 
 5.89 
 5.38 
 4.88 
 4.47 
 4.06 
 3.66 
 3.26 
 2.95 
 2.64 
 2.34 
 2.03 
 1.83 
 
 1.63 
 1.42 
 1.22 
 
 1.01 
 .91 
 .81 
 .71 
 .61 
 .56 
 .51 
 .45 
 .42 
 .38 
 .35 
 .31 
 .29 
 .27 
 .25 
 .23 
 .21 
 .19 
 .17 
 .15 
 .13 
 .12 
 .11 
 .10 
 .09 
 .08 
 .07 
 .06 
 .05 
 04 
 .03 
 .025 
 
 Birming- 
 
 ham 
 
 Gauge. 
 
 Inch. 
 
 American 
 or Brown . 
 and 
 Sharpe 
 Gauge, 
 inch. 
 
 Roebling's 
 
 Gauge. 
 
 Inch. 
 
 Trenton 
 Iron 
 Co.’s 
 ‘Wire ! 
 Gauge. 
 Inch. 
 
 English 
 
 Legal 
 
 Standard. 
 
 Inch. 
 
 
 
 .49 
 
 .46 
 
 
 .500 ( 
 
 .464 
 
 
 
 .43 
 
 .45 
 
 .432 
 
 .454 
 
 .46 
 
 .393 
 
 .40 
 
 .4 
 
 .425 
 
 .40964 
 
 .362 
 
 .36 
 
 .372 
 
 .38 
 
 .3648 
 
 .331 
 
 .33 
 
 .348 
 
 .34 
 
 .32486 
 
 .307 
 
 .305 
 
 .324 
 
 .3 
 
 .2893 
 
 .283 
 
 .285 
 
 .3 
 
 .284 
 
 .25763 
 
 .263 
 
 .265 
 
 .276 
 
 .259 
 
 .22942 
 
 .244 
 
 .245 
 
 .252 
 
 .238 
 
 .20431 
 
 .225 
 
 .225 
 
 .232 
 
 .22 
 
 .18194 
 
 .207 
 
 .205 
 
 .212 
 
 .203 
 
 .16202 
 
 .192 
 
 .19 
 
 .192 
 
 .18 
 
 .14428 
 
 .177 
 
 .175 
 
 .176 
 
 .165 
 
 .12849 
 
 .162 
 
 .16 
 
 .16 
 
 .148 
 
 .11443 
 
 .148 
 
 .145 
 
 .144 
 
 .134 
 
 .10189 
 
 .135 
 
 .13 
 
 .128 
 
 .12 
 
 .09074 
 
 .12 
 
 .1175 
 
 .116 
 
 .109 
 
 .08081 
 
 .105 
 
 .105 
 
 .104 
 
 .095 
 
 .07196 
 
 .092 
 
 .0925 
 
 .092 
 
 .083 
 
 .06408 
 
 .08 
 
 .08 
 
 .08 
 
 .072 
 
 .05707 
 
 .072 
 
 .07 
 
 .072 
 
 .065 
 
 .05082 
 
 .063 
 
 .061 
 
 .064 
 
 .058 
 
 .04526 
 
 .054 
 
 .0525 
 
 .056 
 
 .049 
 
 .0403 
 
 .047 
 
 .045 
 
 .048, 
 
 .042 
 
 .03589 
 
 .041 
 
 .04 
 
 .04 
 
 .035 
 
 .03196 
 
 .035 
 
 .035 
 
 .036 
 
 .032 
 
 .02846 
 
 .032 
 
 .031 
 
 .032 
 
 .028 
 
 .02535 
 
 .028 
 
 .028 
 
 .028 
 
 .025 
 
 .02257 
 
 .025 
 
 .025 
 
 .024 
 
 .022 
 
 .0201 
 
 .023 
 
 .0225 
 
 .022 
 
 .02 
 
 .0179 
 
 .02 
 
 .02 
 
 .02 
 
 .018 
 
 .01594 
 
 .018 
 
 .018 
 
 .018 
 
 .016 
 
 .01419 
 
 .017 
 
 .017 
 
 .0164 
 
 .014 
 
 .01264 
 
 .016 
 
 .016 
 
 .0148 
 
 .013 
 
 .01126 
 
 .015 
 
 .015 
 
 .0136 
 
 .012 
 
 .01002 
 
 .014 
 
 .014 
 
 .0124 
 
 .01 
 
 .00893 
 
 .0135 
 
 .013 
 
 .0116 
 
 .009 
 
 .00795 
 
 .013 
 
 .012 
 
 .0108 
 
 .008 
 
 .00708 
 
 .011 
 
 .011 
 
 .01 
 
 .007 
 
 .0063 
 
 .01 
 
 .01 
 
 .0092 
 
 .005 
 
 .00561 
 
 .0095 
 
 .0095 
 
 .0084 
 
 .004 
 
 .005 
 
 .009 
 
 .009 
 
 .0076 
 
 
 .00445 
 
 .0085 
 
 .0085 
 
 .0068 
 
 
 .00396 
 
 .008 
 
 .008 
 
 .006 
 
 
 .00353 
 
 .0075 
 
 .0075 
 
 .0052 
 
 
 .00314 
 
 .007 
 
 .007 
 
 f 
 
 .0048 
 .0044 
 .004 
 .0036 
 .0032 
 . .0028 
 .0024 
 .002 
 .0016 
 .0012 
 .001 
 
1221 
 
 WIRE ROPES. 
 
 STRESS IN HOISTING 
 
 ROPES ON INCLINED PLANES OF VARIOUS 
 DEGREES. 
 
 (From “Wire-Rope Transportation “ published by Trenton Iron Co.) 
 
 The following table is based upon an allowance of 40 lb. per ton for 
 rolling friction, but there will be an additional stress due to the weight of 
 the rope and inclination of the plane. 
 
 Rise per 
 100 Ft. 
 Horizontal. 
 Ft. 
 
 Angle 
 
 of 
 
 Inclination. 
 
 Stress in Lb. 
 
 per Ton 
 of 2.000 Lb. 
 
 5 
 
 9° 52' 
 
 140 
 
 10 
 
 5° 43' 
 
 240 
 
 15 
 
 8° 32' 1 
 
 336 
 
 20 
 
 11° lO' 
 
 432 
 
 25 
 
 14° 03' 
 
 527 
 
 30 
 
 16° 42' 
 
 613 
 
 35 
 
 19° 18' 
 
 700 
 
 40 
 
 21° 49' 
 
 782 
 
 45 
 
 24° 14' 
 
 860 
 
 50 
 
 26° 34' 
 
 933 
 
 55 
 
 28° 49' 
 
 1.003 
 
 60 
 
 30° 58' 
 
 1,067 
 
 65 
 
 33° 02' 
 
 1,128 
 
 • 70 
 
 35° 00' 
 
 1.185 
 
 75 
 
 36° 53' 
 
 1,238 
 
 80 
 
 38° 40' 
 
 1.287 
 
 85 
 
 40° 22' 
 
 1.332 
 
 90 
 
 42° 00' 
 
 1.375 
 
 95 
 
 43° 32' 
 
 1.415 
 
 100 
 
 45° 00' 
 
 1.450 
 
 Rise per 
 100 Ft. 
 Horizontal. 
 Ft. 
 
 Angle 
 
 of 
 
 Inclination. 
 
 i 
 
 Stress in Lb. 
 
 per Ton 
 of 2, 000 Lb. 
 
 105 
 
 46° 24' | 
 
 1,484 
 
 110 
 
 47° 44' 
 
 1.516 
 
 115 
 
 49° 00' 
 
 1,535 
 
 120 
 
 50° L* 
 
 1,573 
 
 125 
 
 51° 21' 
 
 1.597 
 
 130 
 
 52° 26' 
 
 1.620 
 
 135 
 
 53° 29' 
 
 1,642 
 
 140 
 
 54° 28' 
 
 1.663 
 
 145 
 
 55° 25' 
 
 1.682 
 
 150 
 
 56° 19 7 
 
 1.699 
 
 155 
 
 57° 11' 
 
 1.715 
 
 160 
 
 58° 00' 
 
 1 1.730 
 
 165 
 
 58° 47' 
 
 1.744 
 
 170 
 
 j 59° 33' 
 
 1,758 
 
 175 
 
 60° 16' 
 
 1,771 
 
 180 
 
 60° 57' 
 
 1.782 
 
 185 
 
 61° 37' 
 
 1.794 
 
 190 
 
 ! 62° 15' 
 
 1,804 
 
 195 
 
 62° 52' 
 
 1.813 
 
 200 
 
 63° 27' 
 
 ; 1.822 
 
 RELATIVE EFFECTS OF VARIOUS SIZED SHEAVES OR DRUMS ON 
 THE LIFE OF WIRE ROPES. 
 
 Aline officials and other users of wire ropes have often felt the want of a 
 table or set of tables that would enable them to determine at a glance what 
 effect the use of various sized sheaves would have on various sized ropes. 
 The following tables have been specially prepared for the Coal and Metal 
 Aimer's Poeketbook bv Air. Thomas E. Hughes, of Pittsburg, Pa. 
 
 cast-steel ropes for inclines. 
 
 Made of 6 Strands of 7 Wires Each, Laid Around a Hemp Core. 
 
 Diameters of Sheaves or Drums in Feet, Showing Percentages 
 Diameter 0 f Life for Various Diameters. 
 
 of _ = 
 
 Rope. 
 
 Inches. 
 
 100? j 
 
 90* * 
 
 80£ 
 
 75£ | 
 
 601 
 
 501 
 
 25* 
 
 11 
 
 16.00 
 
 14.00 
 
 12.00 
 
 11.00 
 
 9.00 
 
 7.00 
 
 4.75 
 
 a 8 
 
 li 
 
 14.00 
 
 12.00 
 
 10.00 
 
 8.50 
 
 7-00 
 
 6.00 
 
 1 £2 
 
 •*■8 
 
 11 
 
 12.00 
 
 10.00 
 
 8.00 
 
 7.25 
 
 6.00 
 
 1 5.50 
 
 \ 4.2 d 
 
 I4 
 
 11 
 
 10.00 
 
 8.50 
 
 7.75 
 
 7.00 
 
 6-00 ! 
 
 ! 5.00 
 
 ; 4.00 
 
 a 8 
 
 1 
 
 8.50 
 
 7.75 
 
 6.75 
 
 6.00 
 
 5.00 
 
 4.50 
 
 3.75 
 
 J. 
 
 * 
 
 7.75 
 
 7.00 
 
 6.25 
 
 5.75 
 
 4.50 
 
 3. /D 
 
 3.25 
 
 8 
 
 3 
 
 7.00 
 
 6.25 
 
 5.50 
 
 5.00 
 
 I 4.25 
 
 3.50 
 
 2.75 
 
 * 
 
 6.Q0 
 
 5.25 
 
 4.50 
 
 4.00 
 
 3-25 
 
 i 3.00 
 
 2.50 
 
 8 
 
 £ 
 
 5.00 
 
 4.50 
 
 4.00 
 
 3.50 
 
 2.75 
 
 j 2.25 
 
 ; 1.75 
 
 X OTE _\y e do not publish a table of iron ropes for inclines as the use of 
 iron ropes for this purpose has been generally abandoned, steel ropes being 
 far more satisfactory and economical. 
 
WIRE ROPES. 
 
 m 
 
 CAST-STEEL HOISTING ROPES. 
 
 Made of 6 Strands of 19 Wires Each, Laid Around a Hemp Core. 
 
 _ . , Diameters of Sheaves or Drums in Feet, Showing Percentages 
 
 Diameter 0 f Li f e for Various Diameters. 
 
 of 
 
 Hope. 
 
 Inches. 
 
 100/* 
 
 90/ 
 
 80/ 
 
 75/ 
 
 60/ 
 
 50/ 
 
 25/ 
 
 H 
 
 14.00 
 
 12.00 • 
 
 10.00 
 
 8.50 
 
 7.00 
 
 6.00 
 
 4.50 
 
 If 
 
 12.00 
 
 10.00 
 
 8.00 
 
 7.00 
 
 6.00 
 
 5.25 
 
 4.25 
 
 If 
 
 10.00 
 
 8.50 
 
 7.50 
 
 6.75 
 
 5.50 
 
 5.00 
 
 4.00 
 
 If 
 
 9.00 
 
 7.50 
 
 6.50 
 
 5.50 
 
 5.00 
 
 4.50 
 
 3.75 
 
 1 
 
 8.00 
 
 7.00 
 
 6.00 
 
 5.50 
 
 4.50 
 
 4.00 
 
 3.50 
 
 2 . 
 
 7.50 
 
 6.75 
 
 5.75 
 
 5.00 
 
 4.25 
 
 3.50 
 
 3.00 
 
 1 
 
 f 
 
 2 
 
 3.' 
 
 8 
 
 5.50 
 
 4.50 
 
 4.00 
 
 3.75 
 
 3.25 
 
 3.00 
 
 2.25 
 
 4.50 
 
 4.00 
 
 3.75 
 
 3.25 
 
 3.00 
 
 2.50 
 
 2.00 
 
 4.00 
 
 3.00 
 
 3.00 
 
 3.00 
 
 2.75 
 
 2.00 
 
 2.25 
 
 2.00 
 
 1.50 
 
 1.50 
 
 IRON HOISTING ROPES. 
 
 Made of 6 Strands of 19 Wires Each, Laid Around a Hemp Core. 
 
 . Diameters of Sheaves or Drums in Feet, Showing Percentages 
 Diameter of Life for y ar i 0 us Diameters. 
 
 of 
 
 Rope. 
 
 Inches. 
 
 100/ 
 
 90/ 
 
 80/ 
 
 75/ 
 
 60/ 
 
 50/ 
 
 25/ 
 
 If- 
 
 12.00 
 
 11.00 
 
 9.00 
 
 7.50 
 
 6.00 
 
 5.00 
 
 3.00 
 
 If 
 
 10.00 
 
 9.00 
 
 7.50 
 
 7.00 
 
 5.25 
 
 4.75 
 
 4.00 
 
 If 
 
 9.00 
 
 7.75 
 
 6.50 
 
 5.75 
 
 4.50 
 
 4.00 
 
 3.50 
 
 If 
 
 8.00 
 
 6.75 
 
 5.50 
 
 5.00 
 
 4.25 
 
 3.50 
 
 3.00 
 
 1 
 
 6.75 
 
 6.00 
 
 5.00 
 
 4.75 
 
 4.00 
 
 3.25 
 
 2.75 
 
 z 
 
 6.75 
 
 6.00 
 
 5.00 
 
 4.50 
 
 4.00 
 
 3.00 
 
 2.50 
 
 6 
 
 f 
 
 5.00 
 
 4.75 
 
 4.00 
 
 3.75 
 
 3.00 
 
 2.75 
 
 2.00 
 
 f 
 
 4.50 
 
 3.75 
 
 3.25 
 
 3.00 
 
 2.75 
 
 2.25 
 
 1.75 
 
 
 3.50 
 
 3.25 
 
 3.00 
 
 2.75 
 
 2.00 
 
 1.50 
 
 1.00 
 
 I 
 
 3.00 
 
 
 2.00 
 
 
 
 1.25 
 
 1.00 
 
 Wire rope is as pliable as new hemp rope of the same strength; the former 
 will therefore run on the same sized sheaves and pulleys as the latter. But 
 the greater the diameter of the sheaves, pulleys, and drums, the longer wire 
 rope will last. In the construction of machinery for wire rope, it will be 
 found good economy to make the drums and sheaves as large as possible. 
 
 The tables of wire-rope manufacturers give “ proper diameters of drum or 
 sheave” at from 50 to 65 times the rope diameter; but the expression would 
 more properly be the “minimum admissible diameter.” For ordinary ser- 
 vice, by using sheaves and drums from 75 to 100 times the diameter of the 
 rope, the average life of hoisting ropes would be materially lengthened. 
 For rapid hoisting, during which abnormal strains are most likely to occur, 
 or where a low factor of safety is employed, a sheave diameter of 150 times 
 that of the rope is to be recommended. 
 
 Experience has demonstrated that the wear increases with the speed. It 
 is therefore better to increase the load than the speed. Wire rope is manu- 
 factured either with a wire or a hemp center. The latter is more pliable 
 than the former, and will wear better where there is short bending. 
 
 Wire rope must not be coiled or uncoiled like hemp rope. When mounted on 
 a reel, the latter should be mounted on a spindle or flat turntable to pay off 
 the rope. When forwarded in a small coil, without reel, roll it over the 
 ground like a wheel, and run off the rope in that way. All untwisting or 
 kinking must be avoided. 
 
124 
 
 WIRE ROPES. 
 
 EXTRA STRAIN ON A HOISTING ROPE WITH A FEW INCHES 
 
 OF SLACK CHAIN 
 
 Dynamometer Tests. 
 
 First Test 
 
 Empty cage lifted gently : 
 
 Empty cage with 2\ inches slack chain 
 
 Empty cage with 6 inches slack chain : 
 
 Empty cage with 12 inches slack chain 
 
 Second Test 
 
 Cage and four empty cars weighed by machine 
 
 Cage and four empty cars lifted gently ..................... 
 
 Cage and four empty cars with 3 inches slack chain 
 Cage and four empty cars with 6 inches slack chain 
 Cage and four empty cars with 12 inches slack chain 
 
 Third Test 
 
 Cage and full cars weighed by machine 
 
 No. 1, lifted gently 
 
 No. 2, lifted gently 
 
 No. 1, with 3 inches slack chain 
 
 No. 2, with 3 inches slack chain 
 
 No. 1, with 6 inches slack chain 
 
 No. 2, with 6 inches slack chain 
 
 No. 1, with 9 inches slack chain - 
 
 No. 2, with 9 inches slack chain 
 
 Tons. 
 
 Cwts. 
 
 1 
 
 16 
 
 2 
 
 10 
 
 4 
 
 0 
 
 5 
 
 10 
 
 2 
 
 17 
 
 3 
 
 0 
 
 5 
 
 0 
 
 5 
 
 10 
 
 7 
 
 0 
 
 5 
 
 1 
 
 5 
 
 1 
 
 5 
 
 3 
 
 8 
 
 10 
 
 8 
 
 10 
 
 10 
 
 10 
 
 11 
 
 10 
 
 12 
 
 10 
 
 11 
 
 10 
 
 WIRE-ROPE CALCULATIONS. 
 
 The working load, also called the proper working load, is the maximum 
 ioad that a rope should be permitted to support under working conditions. 
 The stress on a rope to which a load is attached, and which bends over a 
 sheave, is made up of two parts: (1) That due to the load on the rope, known 
 as the load stress; (2) that due to the bending of the rope about a sheave or 
 drum, known as the bending stress. That is, if 
 S = total safe stress; 
 
 S b = bending stress; 
 
 Si = load stress; 
 
 S = S b + S, and Si = S — S b . 
 
 The total stress must not equal the elastic limit of the material composing 
 the rope and is usually taken as from one-third to one-fourth the approxi- 
 
 ma it is only quite recently that account has been taken of this second stress 
 in wire-rope problems, and it is riot, as yet, by any means universal practice 
 to consider it in calculating the size of rope needed for a given purpose. 
 
 If we have a given weight to hoist with a wire rope, the 
 to use may be taken directly from the tables on pages 120, 121, 122, but this 
 does not take account of the bending stress, except by allowing for it in the 
 factor of safety assumed. t 
 
 A second method calculates the bending stress. 
 
 The following formulas and the diagram based on them, given on 
 page 125, are given bv Mr. E. T. Sederholm, former chief engineer for Fraser 
 & Chalmers, and will be found in the hoisting-engine catalogue of the 
 Allis-Chalmers Co. 
 
 The general formula for the bending stress is 
 
 _EaA, 
 bb D > 
 
 in which S h = bending stress; 
 
 E = modulus of elasticity; 
 a = diameter of each wire; 
 
 D = diameter of drum or sheave; . . 
 
 A = total area of the wire cross-section, in inches. 
 
WIRE ROPES. 
 
 124a 
 
 For a rope of 19 wires to the strand the diameter of each wire is about one- 
 fifteenth (exactly ) of the diameter of the rope. That is, if d = diameter 
 
 of rope, a = and by substituting this in the above formula S b 
 
 E Ad 
 15.527/ 
 
 15.52’ 
 
 The modulus of elasticity for the different kinds of wire is given different 
 values by different authorities. Mr. Sederholm uses 29,400,000 in his formula 
 and diagram, and Mr. Hewitt 28,500,000, the same modulus being used for 
 the different materials of which ropes are made. 
 
 The cross-section of metal A in a wire rope is approximately .4 d 2 , or it 
 may be more accurately calculated by multiplying the cross-section of each 
 wire, as given by a wire table, by the number of wires in the rope. 
 
 Find the bending stress in a 19-wire, cast-steel hoisting rope 2 in. in diam- 
 eter, winding on an 8-ft. drum, if A — .4 d 2 , and E = 29,400,000, 
 
 „ 29,400,000 X 23 X .4 
 
 S b = — . . ^ . . ~~ - . = 31.51 tons. 
 
 is 124 tons, and if 
 
 10 X 15.52 X 96 X 2,000 
 
 The approximate breaking stress for such a rope 
 124 
 
 we assume a factor of 3, — = 41+ tons for the safe working stress, and 
 
 O 
 
 41 — 32 = 9 tons, for the safe lifting load under the given conditions. 
 
 The diagram given on page 125 is based on the above formula. 
 
 Mr. Wm. Hewitt, of the Trenton Iron Co., has given a similar but more 
 complicated formula for the bending stress, which is supposed to give some- 
 what more accurate results, as he has introduced terms which allow for the 
 actual radius of the bend at the outside fiber of the rope, while the Sederholm 
 formula assumes the radius of the bend to be the radius of the sheave. 
 
 Mr. Hewitt’s formula is as follows: 
 
 EA 
 
 & b — 
 
 2.06 ~ + C 
 
 in which 
 
 S b = 
 E = 
 
 bending stress, in pounds; 
 modulus of elasticity (28,500,000); 
 
 A — aggregate area of wires, in square inches ; 
 
 R te radius of drum or sheave, in inches ; 
 d = diameter of individual wires, in inches; 
 
 C = a constant depending on number of wires in strands. 
 
 The values of d and C are : 
 
 7-Wire Rope 19-Wire Rope 
 
 d = h diameter of rope d = T V diameter of rope 
 
 C = 9.27 C = 15.45 
 
 For 12-wire and 16-wire ropes the values are intermediate in proportion to 
 the number of wires In the case of ropes having strands composed of dif- 
 ferent sizes of wires, take the larger of the outer layer for the value of d. 
 
 Mr. Hewitt assumes one-third of the approximate breaking stress as the 
 maximum safe stress and uses 28,500,000 for the modulus of elasticity. 
 
 If the problem given under the Sederholm formula is worked out by the 
 Hewitt formula the safe working load will be 11£ tons, while the table on 
 pages 120 and 121 gives 24.8 tons. 
 
 * The Sederholm diagram on page 125 gives for a load of 24.8 tons a sheave 
 between 16 and 17 ft. diameter, the Sederholm formula gives a sheave of 
 15 ft. in diameter, while the table on pages 120 and 121 gives 8 ft. It is evi- 
 dent that there is a wide difference of opinion among the wire-rope authori- 
 ties and a good opportunity for experimental work along this line. 
 
 In using the Sederholm or Hewitt formulas there are two unknown 
 quantities, the diameter of the rope d and the diameter D or radius R of the 
 drum, d varies inversely as D, that is, for a given load the smaller d is 
 taken the larger D must be to give the same conditions of safety. 
 
 If we could assume a certain ratio between S h and S t in the* formula 
 S.—- S b + Si the problem could be easily solved, but an examination of this 
 ratio in a number of cases where good results have been obtained from 
 
 S S 2 
 
 hoisting ropes shows this ratio to vary from ^ = 1 to 
 
 b b ts b 
 
 S 1 
 
 mission of power by wire ropes, Mr. Hewitt assumes ~ = -, but this relation 
 
 will scarcely hold in a hoisting problem, and the above problem must be 
 solved by the cut-and-try method. 
 
 = Inthetrans- 
 
 5 
 
124b 
 
 WIRE ROPES. 
 
 Wear of Wire Ropes— The deterioration of wire ropes may he either 
 
 external or internal, and may be due (1) t0 abra ^°£’ t ^ Ue o i 0 t o h the U internal 
 the outside surface of the rope against other objects, or the internal 
 chafing of the wires composing the strands against one another, (2) to 
 iniurv g from overloading, to shock due to sudden starting of the load, or to 
 reDeated ^ndings about too sharp angles or over sheaves or rollers of too 
 small a diameter for the size of the rope; (3) to rust or corrosion of the wi e 
 from acid waters or to decay of the hemp cores. 
 
 As a result of abrasion, the wires in a rope are either flattened or torn 
 ar»art With properly designed drums and head-trame and properly placed 
 sheaves a hofithig rope is but slightly abraded, and the wear is due chiefly 
 S hendina or to overloading. A haulage rope is subjected to constant 
 ahrasionin passta° over rollers and sheaves and from dragging along the 
 hottom and S of the haulage wavs and from the grips. It is also often 
 subject to severe shocks and abrasion from the lashing or vibration when 
 
 “‘Vhewea? a£§“ua rope increases as its velocity is increased; hence, 
 conditiZs lermittfng?it is better to increase the output by increasing the 
 load within allowable limits rather than by increasing the velocity of 
 
 “Inspection of Ropes ,_The life of a hoisting rope. depends not only onits 
 duality but also on the conditions under which it is used and en.the eare- 
 
 fulness'of the engineer in handling the load hoisUng ropes are 
 
 nffpn and at regular intervals. At some mines the hoisting ropes aie 
 
 insnected everv morning before lowering the men. The cage is slow y 
 lowered and then raised, each rope being carefully examined b> an 
 insnector to detect any broken wires. Particular attention should be given 
 to the nart of the rope where it is attached to the socket at the cage, 
 r»art is more subject to corrosion and sharp bending than any other. When 
 tlm core fails at any noint the rope should be discarded at once, usthe wires 
 are likely to kink and break internally as the rope passes over the ^sheave. 
 At some mines hoisting ropes are discarded at regular intervals, whether 
 
 while the rone is run will indicate loose wire ends. . 
 
 Lubrication of Ropes -Mine water has a very corrosive action on wire 
 
 comfngta YoXc'I 1 whh n thl "/of the 
 
 W °For MstoVroP>es"one bushel of freshly slacked lime to one barrel of pine 
 nr eoAl tar makes a good lubricant; with pine tar, which contains no acid, 
 tallow mav be used instead of lime. Another mixture contains tar, summer 
 oil^xl™ greas^and a fittle pulverized mica, mixed to such a consistency 
 that it wifi penetrate thoroughly between the wires and will n 9 t dry ? h 
 strip off The lubricant should not be thick enough to render difficult the 
 thorough inspection of the rope, and all lubricants of this ^ature should be 
 nse^smrin^lv after the first application, as the rope should be kept clean 
 and SfS Graphite is also used for the purpose. Lubricants may 
 he applied bv running the engine slowly and allowing the rope to pass 
 through a bunch of waste saturated with lubricant, by ruling the lubricant 
 into the rone bv means of a brush, or by pouring the oil into the groove oi 
 tbe sheaveas tL ropeis run slowly ba6k and forth. A new hoisting rope 
 should be passed through a bath of hot lubricant and thus be thoroug y 
 
 1Ub HaSla^e roues are not usually lubricated as thoroughly as hoisting ropes 
 on Account of ?he grease causing slipping of grips and gathering of dirt and 
 dusl but they can be treated with raw linseed oil thickened with lamp 
 black boiled ^vith an equal portion of pine tar, and the mixture applied 
 while hot. Ordinary black oil, such as is used to oil mine cars and hoisting 
 rones can be used on haulage ropes where no friction grips are emplo>ed. 
 These mixtures^ if fluid can be poured on the rope as it is run over the 
 Theave^r appUed from a leather-lined box filled with oil. Patent lubricants 
 known’ as cable shields or rope fillers, which fill the interstices between the 
 strands, are often used on tail and mam ropes. 
 
 
WIRE ROPES. 
 
 125 
 
 PROPER WORKING LOAD 
 
126 
 
 WIRE ROPES. 
 
 Starting Strain on Hoisting Rope.— In selecting a hoisting rope, due allow- 
 ance must be made for the shock and extra strain imposed on the rope when 
 the load is started from rest. Experiments made by placing a dynamometer 
 between the rope and the cage have shown that starting stress may be from 
 two to three times the actual load. 
 
 Experiment 1. 
 
 Strain in Rope. 
 Pounds. 
 
 Empty cage, lifted gently 
 
 Empty cage, started with 2k in. of slack rope 
 Empty cage, started with 6 in. of slack rope . 
 Empty cage, started with 12 in. of slack rope 
 
 4,030 
 
 5,600 
 
 8,950 
 
 12,300 
 
 Experiment 2. 
 
 Strain in Rope. 
 Pounds. 
 
 Cage and loaded cars, as weighed 
 
 Cage and loaded cars, lifted slowly and gently 
 
 Cage and loaded cars, started with 3 in. of slack rope 
 Cage and loaded cars, started with 6 in. of slack rope 
 Cage and loaded cars, started with 9 in. of slack rope 
 
 11,300 
 
 11,525 
 
 19,025 
 
 24,625 
 
 26,850 
 
 Horsepower of Manila Ropes. 
 ( Link-Belt Engineering Co.) 
 
 Diam. of 
 Rope. 
 
 Weight per 
 Foot. 
 
 Breaking 
 
 Strain. 
 
 Working 
 
 Strain. 
 
 1,000 Ft. 
 per Min. 
 
 2,000 Ft. 
 per Min. 
 
 3,000 Ft. 
 per Min. 
 
 4,000 Ft. 
 per Min. 
 
 5,000 Ft. 
 per Min. 
 
 H.P. 
 
 Tens. 
 
 Wt. 
 
 H.P. 
 
 Tens. 
 
 Wt. 
 
 H.P. 
 
 Tens. 
 
 Wt. 
 
 H.P. 
 
 Tens. 
 
 Wt. 
 
 H.P. 
 
 Tens. 
 
 Wt. 
 
 I 
 
 0.15 
 
 4,000 
 
 121 
 
 2* 
 
 90 
 
 4* 
 
 90 
 
 6* 
 
 80 
 
 7* 
 
 80 
 
 8* 
 
 70 
 
 * 
 
 0.18 
 
 5,000 
 
 151 
 
 2| 
 
 110 
 
 5* 
 
 110 
 
 7* 
 
 100 
 
 9* 
 
 100 
 
 10* 
 
 90 
 
 7 
 
 8 
 
 0.27 
 
 7,500 
 
 227 
 
 4* 
 
 170 
 
 8* 
 
 170 
 
 11* 
 
 160 
 
 14* 
 
 150 
 
 16 
 
 130 
 
 1 
 
 0.33 
 
 9,000 
 
 272 
 
 5 
 
 200 
 
 10 
 
 200 
 
 14 
 
 180 
 
 17* 
 
 170 
 
 19 
 
 150 
 
 1* 
 
 0.45 
 
 12,250 
 
 371 
 
 7 
 
 280 
 
 13* 
 
 270 
 
 19 
 
 250 
 
 23* 
 
 230 
 
 26 
 
 210 
 
 1* 
 
 0.50 
 
 14,000 
 
 424 
 
 8 
 
 320 
 
 15* 
 
 310 
 
 22 
 
 290 
 
 27 
 
 270 
 
 29* 
 
 240 
 
 if 
 
 0.65 
 
 18,062 
 
 547 
 
 10* 
 
 410 
 
 20 
 
 400 
 
 28* 
 
 370 
 
 34* 
 
 350 
 
 38* 
 
 310 
 
 1* 
 
 0.73 
 
 20,250 
 
 613 
 
 11* 
 
 460 
 
 22 
 
 440 
 
 31* 
 
 420 
 
 39 
 
 390 
 
 43* 
 
 350 
 
 H 
 
 0.82 
 
 25,000 
 
 760 
 
 14* 
 
 570 
 
 27* 
 
 550 
 
 39* 
 
 520 
 
 49 
 
 490 
 
 554 
 
 448 
 
 1* 
 
 1.08 
 
 30,250 
 
 916 
 
 17 
 
 680 
 
 33* 
 
 660 
 
 47* 
 
 630 
 
 58* 
 
 580 
 
 64* 
 
 520 
 
 2 
 
 1.27 
 
 36,000 
 
 1,000 
 
 20* 
 
 810 
 
 40 
 
 790 
 
 56* 
 
 740 
 
 69* 
 
 670 
 
 77* 
 
 620 
 
 WIRE-ROPE FASTENINGS. 
 
 Thimble spliced, in ordinary style, is shown in Fig. 1 (a). In this method, 
 the wires, after being frayed out at the end and the rope bent around the 
 thimble, are laid snugly about the main portion of the rope and securely 
 fastened by wrapping with stout wire, the extreme ends that project below 
 this wrapping being folded back, as shown. 
 
 Another style of thimble splicing is shown in Fig. 1(5). In this case the 
 strands are interlocked as in splicing, and the joint is wrapped with wire as 
 in the former method. The socket fastening is sho wn*in Fig. 1 (c). The hole 
 in which the rope end is fastened is conical in shape. The rope is generally 
 secured by fraying out the wires at the end, the interstices being filled up 
 with spikes driven in tightly. The whole is finally cemented by pouring in 
 molten Babbitt metal. This makes a much neater fastening than either 
 of thos6 shown in (a) and (5), but it does not possess anything like as 
 much strength. The thimble possesses a serious disadvantage; it is 
 usually made of a piece of curved metal bent around into an oval shape, 
 
WIRE-ROPE SPLICING. 
 
 127 
 
 Fig. 1. 
 
 ns shown in (a) and (6), with the groove, in which the rope lies, outside, 
 the ends coming together in a sharp point. When weight is placed on the 
 rope the strain on the thimble is apt to cause one end to wedge itself 
 beyond or past the other, and with its sharp edge 
 it cuts the strands in the splice Mr. William 
 Hewitt, of Trenton, N. J., while testing the 
 strength of wire ropes, discovered this tendency, 
 and experimented with sockets with the idea 
 of devising some method of fastening the rope 
 securely in the socket. He found that by adopt- 
 ing the following plan he secured good, results: 
 
 The wires, after being frayed out at the end, 
 were bent upon themselves in hook fashion, the 
 prongs of some being longer than others, so that 
 the bunch would conform to the conical aperture 
 of the socket, and the melted Babbitt metal was 
 finally run in as usual. The rope was subjected 
 
 the a socket The sfmpHcUy of this method commends 
 
 itself to practical men. . 
 
 RAPID METHOD OF SPLICING A WIRE ROPE.* 
 
 The only tools needed are a cold cutter and hammer for cutting and 
 trimming- the strands, and two needles 12 in. long, made of good steel and 
 tapered'ovaUy toa point. Cut off the ends of the ropes to be ^phcedand 
 unlav three adjacent strands of each back 15 ft., cut out the nemp 
 center to this point and relay the strands for 7 ft. and cut thenroff. Pull 
 the ropes by each other until they have the position shown m Fig. 2 (a), 
 cut off a and d' 5 and c', Fig. 2 (6), making their lengths approximately 10 
 and 12£ ft respectively, measured from the point where the hemp centers 
 were^cut. Pllce theses together, Fig. 2 (O^unlay 
 
 \ mp *. lfc'?Fi| n 2 (hj^SimilOTlyl 
 
 /, it 
 
 until the rope appears as in 
 
 2 (c). Next run the 
 
 u Hemp Center strands into the middle of 
 
 (a> the rope. To do this, cut 
 
 (b) /'«■' b 
 
 Fig. 2. 
 
 off the end of the strand e', Fig. 2 (c), so that when it is put in place it will just 
 reach to the end x of the hemp core, and then push _ the needle A , , Fig. 2 1(c) 
 through the rope from the under side, leaving two strands at the front of the 
 needle, as shown. Push the needle B through from the upper side and as 
 close to the needle A as possible, leaving the strands e and e between them; 
 place the needle A on the knee and turn the needle B around with the coil of 
 the rope, and force the strand e' into the center of the rope. Repeat this 
 
 # W. H. Morris, “ Mines and Minerals,” September, 1898. 
 
WIRE ROPES. 
 
WIRE-ROPE SPLICING. 
 
 129 
 
 ODeration with the other ends and cut them off* so that the ends coming 
 together in the center of the rope will butt against each other as nearly as 
 
 possible. . 
 
 ORDINARY LONG SPLICE. 
 
 Tools Reauired— One pair wire nippers, for cutting off strands; one pair 
 nliers for pulling through and straightening ends of strands; two niarline- 
 snikes one round and one oval, for opening strands; one knife to cut hemp 
 center- two clamps, to untwist rope to insert ends of strands, or, m place of 
 them two short hemp-rope slings, with a stick for each as a lever; a wooden 
 mallet and some rope twine. Also, a bench and yise are handy. 
 
 The length of the splice depends on the size of the rope. The larger ropes 
 require the longer splices. The splice of ropes from f hi. to * m. m diameter 
 should not be less than 20 ft.; from ¥ m. to 1 ¥ in., 30 ft., and from l a in. 
 
 UP, To splice a rope, tie each end with a piece of cord at a distance equal to 
 one-half the length of the splice, or 10 ft. back from the end, for a $ rope, after 
 which unlay each end as far as the cord. Then cut out the hemp center 
 and bring the two ends together as close as possible, placing the strands of 
 the one end between those of the other, as shown m Fig. 3 (a). Now remove 
 the cord k from the end M of the rope, and unlay any strand, as a, and follow it 
 up with the strand of the other end M' of the rope that corresponds to it as 
 a /P Fig- 3 (a) About 6 in. of a are left out, and a is cut off about 6 m from the 
 rone thus leaving two short ends, as shown at P in Fig. 3 (h), which must be 
 tied for the present by cords as shown. The cord k should again be wound 
 around the end M of the rope, Fig. 3 (a), to prevent the unraveling of the 
 strands- after which remove the cord k' on the other or if end of the rope, 
 and unlay the strand 6; follow it up, as above, with the strand b , leaving 
 the ends out, and tying them down for the present, as before described m 
 the case of strands a and a ', see Q, Fig. 3 (6) ; also, replac ie the cord fc for the 
 same purpose as stated above. Now, again remove the cord k and unlay the 
 next strand as c, Fig. 3 (a), and follow it up with c', stopping, however, this 
 time within 4 ft. of the first set. Continue this operation with the remaining 
 6 strands stopping 4 ft. short of the preceding set each time. The strands 
 are now in their proper places, with the ends passing each other at intervals 
 of 4 ftas shownin Fig. 3(c). To dispose of the loose ends, clamp the rope m a 
 vise at the left of the strands a and a', Fig. 3 (c), and fasten a clamp to the rope 
 at the right of these strands; then remove the cords tied around the rope 
 that hold these two strands down; after which turn the clamp in the oppo- 
 site direction to which the rope is twisted, thereby untwisting the rope, as 
 shownin Fig 3 (d) . The rope should be untwisted enough to allow its hemp 
 cores' to be pulled out with a pair of nippers. Cut off 24 in. of the hemp core 
 12 in at each side from the point of intersection of the strands a and a,and 
 push the tSds ofthe strands P in their place as shown in Fig. 3 (d). Then 
 allow the rope to twist up to its natural shape, and remove the clamps. 
 After the rope has been allowed to twist up, the strands tucked m generally 
 bulge out somewhat. This bulging may be reduced by lightly tapping the 
 bulged part of the strands with a wooden mallet, which will force their ends 
 farther into the rope. Proceed in the same manner to tuck m the other ends 
 of the strands. 
 
 CHAINS. 
 
 The links of iron chains are usually made as short as is consistent with 
 easy play, so as to make them less liable to kink, and also to prevent bending 
 
 when wound around drums, sheaves, etc. . ■ . , . « 
 
 The weight of close-link chain is about three times the weight of bar from 
 
 which it is made, for equal lengths. , ' . , . , 
 
 Karl von Ott, comparing weight, cost, and strength of three materials, 
 hemp iron wire, and chain iron, concludes that the proportion between cost 
 of hemp rope, wire rope, and chain is as 2 : 1 : 3, and that, therefore, for 
 equal resistances, wire rope is only half the cost of hemp rope, and a third 
 
 Chains of warranted superior iron will stand 25 } more strain before 
 breaking The report of the U. S. Test Board, 1881, shows that the ultimate 
 strength of chains may be taken at 1.6 that of the iron from which the links 
 are made. 
 
130 
 
 hydrostatics. 
 
 The strength of chains varies, owing to the nature of the iron from which 
 they are made, and their mechanical construction. The following table is 
 approximately correct for ordinary iron chains: 
 
 Table of Weight an© Strength of Chains. 
 
 Diameter of 
 
 Weight of 
 
 
 
 Diameter of 
 
 Weight of 
 
 
 
 Rod of 
 Which the 
 Links Are 
 Made. 
 
 Chain 
 
 per 
 
 Running 
 
 Foot. 
 
 Working 
 
 Strength. 
 
 Tons. 
 
 Breaking 
 
 Strain. 
 
 Tons. 
 
 Rod of 
 Which the 
 Links Are 
 Made. 
 
 Chain 
 
 per 
 
 Running 
 
 Foot. 
 
 Working 
 
 Strength. 
 
 Tons. 
 
 Breaking 
 
 Strain. 
 
 Tons. 
 
 Inches. 
 
 Pounds. 
 
 
 
 Inches. 
 
 Pounds. 
 
 
 
 t 3 s 
 
 ■ A 
 •4 
 
 h 
 
 T5 
 
 1 
 
 ll 
 
 .325 
 
 .19 
 
 .773 
 
 § 
 
 7.10 
 
 4.40 
 
 16.80 
 
 .579 
 
 .36 
 
 1.37 
 
 it 
 
 8.14 
 
 5.00 
 
 19.32 
 
 .904 
 
 .45 
 
 2.14 
 
 1 
 
 9.26 
 
 5.71 
 
 22.00 
 
 1.30 
 
 .85 
 
 3.09 
 
 
 11.70 
 
 7.23 
 
 26.44 
 
 1.78 
 
 1.09 
 
 4.20 
 
 li 
 
 14.50 
 
 9.00 
 
 32.64 
 
 2.31 
 
 1.43 
 
 5.50 
 
 if 
 
 17.50 
 
 10.80 
 
 39.42 
 
 2.93 
 
 1.80 
 
 6.96 
 
 l£ 
 
 20.80 
 
 13.00 
 
 47.00 
 
 3.62 
 
 2.23 
 
 8.58 
 
 if 
 
 24.40 
 
 15.24 
 
 55.14 
 
 4.38 
 
 2.70 
 
 10.39 
 
 n 
 
 28.40 
 
 17.65 
 
 63.97 
 
 1 6 
 3 
 
 5.21 
 
 3.21 
 
 12.36 
 
 if 
 
 32.60 
 
 20.27 
 
 73.44 
 
 4 
 
 13 
 
 TS 
 
 6.11 
 
 3.80 
 
 14.42 
 
 2 
 
 37.00 
 
 23.10 
 
 83.55 
 
 HYDROSTATICS. 
 
 Hydrostatics treats of the equilibrium of liquids, and of their pressures on 
 the walls of vessels containing them; the science depends on the way in 
 which the molecules of a liquid form a mass under the action of gravity 
 and molecular attraction, the latter of which is so modified in liquids as to 
 give them their state of liquidity. While the particles of a liquid cohere, 
 they are free to slide upon one another without the least apparent friction; 
 and it is this perfect mobility that gives them the mechanical properties 
 considered in hydrostatics. . 
 
 The fundamental property may be thus stated: When a pressure is exerted 
 on any part of the surface of a liquid , that pressure is transmitted undiminished 
 to all parts of the mass , and in all directions. This is a physical axiom, and on 
 it are based nearly all the principles of hydrostatics. 
 
 Equilibrium of Liquids.— This is a property of liquids that can be easily 
 demonstrated, and examples are frequently seen. Thus, if two barrels are 
 connected at the bottom with a pipe, and water is poured in one until it 
 reaches within a foot of the top, the water in the other will be found to have 
 
 attained the same height. 
 
 Pressure of Liquids on Surfaces. — The general proposition on this point is as 
 follows* The pressure of a liquid on any surface immersed m it is equal to the 
 weiqht of a column of the liquid whose base is the surface pressed , and whose 
 heiqht is the perpendicular depth of the center of gravity of the surface below the 
 surface of the liquid. The pressure thus exerted is independent of the shape 
 or size of the vessel or cavity containing the liquid. 
 
 The pressure of a liquid against any point of any surface, either curved 
 
 or plane, is always perpendicular to the surface at that point. 
 
 At any given depth the pressure of a liquid is equal in every direction, 
 and is in direct proportion to the vertical depth below the surface. 
 
 The weight of a cubic foot of fresh water, at ordinary temperature of the 
 atmosphere, that is, from 32° F. to 80° F., is usually assumed at 62.5 lb. This 
 is a trifle more than the actual weight, but is sufficiently close for purposes 
 of calculation. . . f « 
 
 To Find the Pressure Exerted by Quiet Water Against the Side of a Gangway or 
 Heading —Multiply the area of the side in square feet by the perpendicular 
 distance from the surface of the water to a point equidistant between the 
 top and bottom of the submerged heading or gangway, and multiply the 
 product bv 62.5. The result will be the pressure in pounds, exclusive of 
 atmospheric pressure. This latter need not be considered in ordinary 
 mining work. 
 
HYDROSTATICS . 
 
 131 
 
 Example.— If an abandoned colliery, opened by a slope on a pitch of 25° 
 and 100 yd. long, is allowed to fill with water, what is the average pressure 
 exerted on each square foot of the lower rib of the gangway, assuming 
 that the gangways were driven dead level, and that the length of the slope 
 was measured to a point on the lower rib equidistant between top and 
 bottom of gangway. 
 
 We here have a perpendicular height of water = 300 X sine of 25° 
 = 126.78 ft. Then, the pressure on each square foot of the lower rib of 
 gangway = 126.78 X 62.5 lb., or the weight of 1 cu. ft., or a pressure on each 
 square foot of surface of 7,923.75 lb., or over 3i gross tons. The total pressure 
 exerted along the gangwav may readily be found by multiplying the 
 7,923.75 lb. by the number of square feet of the lower rib against which 
 it rests. 
 
 To find the total pressure of quiet water against and perpendicular to any 
 surface whatever, as a dam, embankment, or the bottom, side or top of any 
 containing vessel, water pipe, etc., no matter whether said surface be 
 vertical, horizontal, or inclined; or whether it be flat or curved; or whether 
 it reach to the surface of the water or be entirely below it: 
 
 Multiply the area, in square feet , of the surface pressed, by the vertical depth in 
 feet of its center of gravity below the surface of the water, and this product by 62.5. 
 The result will be the pressure in pounds. 
 
 Thus, assuming that in the annexed three figures the depth of water in 
 each dam is 12 ft., and the wall or embankment is 50 ft. long, then in 
 Fig. 1 the total pressure will equal 12 X 50 X 6 X 62.5 ==, 225,000 lb. 
 
 In Figs. 2 and 3 the walls or embankments, being inclined, expose a 
 greater surface to pressure, say 18 ft. from A to B. Then the total pressure 
 equals 18 X 50 X 6 X 62.5 = 337,500 lb. 
 
 Now, the results obtained are the total pressures without regard to direction. 
 
 In Fig. 1 the total 
 pressure calculated, 
 or 225,000 lb., is hori- 
 zontal, tending 
 either to overturn 
 the wall or make it 
 slide on its base. 
 The center of pres- 
 sure is at C, or one-third of the vertical depth from the bottom. 
 
 In Fig. 2 the pressure is exerted in two directions; one pressure, acting 
 horizontally, tends to overthrow or slide the wall, while the other, acting 
 vertically, tends to hold it in place. 
 
 In Fig. 3 the pressure is also exerted in two directions; one pressure, 
 acting horizontally, tends to overthrow or slide the wall, while the other 
 tends to lift. , , _ . _ 
 
 So long as the vertical depth of water remains the same, the horizontal 
 pressure remains the same, no matter what inclination is given the wall. 
 Thus, in Figs. 2 and 3, the horizontal pressure is the same as in Fig. 1, 
 or 225,000 lb. ^ ^ 
 
 The total pressure of the water is distributed over the entire depth of the 
 submerged part of the back of the wall, and is least at the top, gradually 
 increasing toward the bottom. But so far as regards the united action of 
 every portion of it, in tending to overthrow the wall, considered as a single 
 mass of masonry, incapable of being bent or broken, it may all be assumed 
 to be applied at C, which is one-third of the vertical depth from the bottom 
 in Fig. 1, or, what is the same thing, one-third of the 
 slope distance from the bottom in Figs. 2 and 3. 
 
 No matter how much water is in a dam or vessel, 
 the pressure remains the same, so long as the area 
 pressed and the vertical depth of its center of gravity 
 below the level surface of the water remains un- 
 changed. Thus, if the water in dam shown in Fig. 1 
 extended back 1 mile, it would exert no more pressure 
 against the wall than if it extended back only 1 ft. 
 
 In any two vessels having the same base, and con- 
 taining the same depth of water, no matter what 
 quantity, the pressures on the bases are equal. Thus, 
 if Figs. 4 and 5 have the same base and be filled with 
 water to the same depth, the pressure on the bases 
 Will be equal. This fact, that the pressure on a given surface, at a given 
 
 Fig. 4. 
 
 Fig. 5. 
 
132 
 
 HYDROSTATICS. 
 
 depth, is independent of the quantity of water, is called the hydrostatic 
 
 P<1 As the pressure of water against any point is at right angles to the surface 
 at that point, it follows that props or other strengthening material for the 
 strengthening of such structures as a sloping dam, should be so placed as to 
 offer the greatest resistance in a line at right angles to the sloping surface, 
 and these supports should be strongest and closest together at the bottom. 
 For the same reason, the hoops on a circular tank should be strongest and 
 closest at the bottom. . ... . . 
 
 Transmission of Pressure Through Water-Water, in common with other 
 Kauids possesses the important property of transmitting pressure equally in 
 4 all directions. Thus, if a vessel is constructed with two 
 
 cylinders, the area of one being 10 sq. in., and that of the 
 other 100 sq. in., and the vessel is filled with water (Fig. 6), 
 and pistons fitted to the cylinders, a pressure of 100 lb. 
 applied at the smaller will balance 1,000 lb. at the larger. 
 This is the principle of the hydrostatic press. Air and other 
 gaseous fluids transmit pressure equally in all directions, 
 like liquids, but not as rapidly. 
 
 To Find the Pressure on a Plane Surface at Any Given Depth 
 of Water.— For pounds per square inch, multiply depth in 
 feet by .434. For pounds per square foot, multiply depth 
 in feet by 62.5. For tons per square foot, multiply depth 
 in feet by .0279. The pressure per square foot at different 
 depths increases directly as the depths. The total pressure 
 against a plane 1 ft. wide at different depths increases as the square of the 
 depths. 
 
 Pressure in Pounds per Sq. Ft. at Different Vertical Depths, and 
 Also the Total Pressure Against a Plane 1 Ft. Wide Extending 
 Vertically From the Surface of the Water to the Same Depths. 
 
 Fig. 6. 
 
 Depth. 
 
 Pressure. 
 
 Pouuds 
 
 Total 
 
 Pressure. 
 
 Depth. 
 
 Pressure. 
 
 Pounds 
 
 Total 
 
 Pressure. 
 
 Depth. 
 
 Pressure. 
 
 Pounds 
 
 Total 
 
 Pressure. 
 
 Feet. 
 
 per Sq. Ft. 
 
 Pounds. 
 
 Feet. 
 
 per Sq. Ft. 
 
 Pounds. 
 
 r cci. 
 
 per Sq. Ft. 
 
 Pounds. 
 
 i 
 
 62.5 
 
 31 
 
 27 
 
 1,687 
 
 22,781 
 
 65 
 
 4,062 
 
 132,025 
 
 X 
 
 2 
 
 125 
 
 125 
 
 28 
 
 1,750 
 
 24,500 
 
 70 
 
 4,375 
 
 153,124 
 
 3 
 
 187 
 
 281 
 
 29 
 
 1,812 
 
 26,281 
 
 75 
 
 4,687 
 
 175,779 
 
 4 
 
 250 
 
 500 
 
 30 
 
 1,875 
 
 28,125 
 
 80 
 
 5,000 
 
 200,000 
 
 K 
 
 312 
 
 781 
 
 31 
 
 1,937 
 
 30,031 
 
 85 
 
 5,312 
 
 225,775 
 
 o 
 
 Q 
 
 375 
 
 1,125 
 
 32 
 
 2,000 
 
 32,000 
 
 90 
 
 5,625 
 
 253,124 
 
 7 
 
 437 
 
 1,531 
 
 33 
 
 2,062 
 
 34,031 
 
 95 
 
 5,937 
 
 282,025 
 
 i 
 
 g 
 
 500 
 
 2,000 
 
 34 
 
 2,125 
 
 36,125 
 
 100 
 
 6,250 
 
 312,500 
 
 9 
 
 562 
 
 2,531 
 
 35 
 
 2,187 
 
 38,281 
 
 110 
 
 6,875 
 
 378,124 
 
 10 
 
 625 
 
 3,125 
 
 36 
 
 2,250 
 
 40,500 
 
 120 
 
 7,500 
 
 450,000 
 
 11 
 
 687 
 
 3,781 
 
 37 
 
 2,312 
 
 42,781 
 
 130 
 
 8,125 
 
 528,100 
 
 12 
 
 750 
 
 4,500 
 
 38 
 
 2,375 
 
 45,125 
 
 140 
 
 8,750 
 
 612,496 
 
 13 
 
 812 
 
 5,281 
 
 39 
 
 2,437 
 
 47,531 
 
 150 
 
 9,375 
 
 703.116 
 
 14 
 
 875 
 
 6,125 
 
 40 
 
 2,500 
 
 50,000 
 
 160 
 
 10,000 
 
 800 j 000 
 
 15 
 
 937 
 
 7,031 
 
 41 
 
 2,562 
 
 52,531 
 
 170 
 
 10,625 
 
 903,100 
 
 16 
 
 1,000 
 
 8,000 
 
 42 
 
 2,625 
 
 55,125 
 
 180 
 
 11,250 
 
 1,012,496 
 
 17 
 
 1,062 
 
 9,031 
 
 43 
 
 2,687 
 
 57,781 
 
 190 
 
 11,875 
 
 1,128,100 
 
 18 
 
 1,125 
 
 10,125 
 
 44 
 
 2,750 
 
 60,500 
 
 200 
 
 12,500 
 
 1,250,000 
 
 19 
 
 1,187 
 
 11,281 
 
 45 
 
 2,812 
 
 63,281 
 
 225 
 
 14,062 
 
 1,582,025 
 
 20 
 
 1,250 
 
 12,500 
 
 46 
 
 2,875 
 
 66,125 
 
 250 
 
 15,625 
 
 1,953,100 
 
 21 
 
 1,312 
 
 13,781 
 
 47 
 
 2,937 
 
 69,031 
 
 275 
 
 17,187 
 
 2,363,275 
 
 22 
 
 1,375 
 
 15,125 
 
 48 
 
 3,000 
 
 72,000 
 
 300 
 
 18,750 
 
 2,812,500 
 
 23 
 
 1,437 
 
 16,531 
 
 49 
 
 3,062 
 
 75,031 
 
 350 
 
 21,875 
 
 3,828,100 
 
 24 
 
 1,500 
 
 18.000 
 
 50 
 
 3,125 
 
 78,125 
 
 400 
 
 25,000 
 
 5,000,000 
 
 25 
 
 1,562 
 
 19,531 
 
 55 
 
 3,437 
 
 94,531 
 
 450 
 
 28,120 
 
 6,328,100 
 
 26 
 
 1,625 
 
 21,125 
 
 60 
 
 3,750 
 
 112,500 
 
 500 
 
 31,250 
 
 7,812,500 
 
 Pressure of Water in Pipes.— As water exerts a pressure equally in al. 
 directions, it is important that in pipe lines the pipe should be sufficiently 
 thick to assure strength enough to resist a bursting pressure. In ordinary 
 
H YD ROST A TICS. 
 
 133 
 
 practice, the thickness of cast-iron water pipes of different bores is calculated 
 by Mr. J. T. Fanning’s formula, given in his Hydraulic Engineering, which 
 is as follows: 
 
 . _ (pressure in lb. per sq. in. + 1 00) X bore in in. 
 
 Thickness m inches — .4 X ultimate tensile strength 
 
 + .333 
 
 bore in in.\ 
 Too )' 
 
 This formula, worked out for different heads and different sizes of bore, 
 yields the following results: 
 
 Thickness of Pipe for Different Heads and Pressures. 
 
 Head in Ft 
 
 50 
 
 100 
 
 200 
 
 300 
 
 500 
 
 1,000 
 
 Pressure in Lb. 
 
 21.7 
 
 43.4 
 
 86.8 
 
 130 
 
 217 
 
 434 
 
 per Sq. In. 
 
 
 
 
 
 
 
 Bore. Inches. 
 
 Thickness of Pipe. Inches. 
 
 2 
 
 .36 
 
 .37 
 
 .38 
 
 .39 
 
 .42 
 
 .48 
 
 3 
 
 .37 
 
 .38 ■ 
 
 .40 
 
 .42 
 
 .45 
 
 .54 
 
 4 
 
 .39 
 
 .40 
 
 .42 
 
 .45 
 
 .50 
 
 .61 
 
 6 
 
 .41 
 
 .43 
 
 .47 
 
 .50 
 
 .57 
 
 .75 
 
 8 
 
 .45 
 
 .47 
 
 .52 
 
 .57 
 
 .66 
 
 .90 
 
 10 
 
 .47 
 
 .50 
 
 .56 
 
 .62 
 
 .74 
 
 1.04 
 
 12 
 
 .49 
 
 .53 
 
 .60 
 
 .67 
 
 .82 
 
 1.18 
 
 16 
 
 .55 
 
 .60 
 
 .70 
 
 .79 
 
 .98 
 
 1.46 
 
 18 
 
 .57 
 
 .63 
 
 .74 
 
 .85 
 
 1.06 
 
 1.60 
 
 20 
 
 .61 
 
 .67 
 
 .79 
 
 .91 
 
 1.15 
 
 1.75 
 
 24 
 
 .66 
 
 .73 
 
 .87 
 
 1.02 
 
 1.30 
 
 2.03 
 
 30 
 
 .74 
 
 .83 
 
 1.01 
 
 1.19 
 
 1.55 
 
 2.46 
 
 36 
 
 .82 
 
 .93 
 
 1.15 
 
 1.36 
 
 1.80 
 
 2.88 
 
 48 
 
 .98 
 
 1.13 
 
 1.42 
 
 1.70 
 
 2.28 
 
 3.73 
 
 In the above table, the ultimate tensile strength of cast iron is taken at 
 18 000 lb. per sq. in. The addition of 100 lb. to the pressure is made to allow 
 for water ram. The valves of water pipes should he closed slowly , and the 
 necessity of this increases as the pipes increase in diameter. If this rule is 
 not observed, the momentum of the running water is arrested suddenly, and 
 a great pressure is created against the pipes in all directions, and through- 
 out the entire length of the line above the valve, even if it be many miles, 
 and there is danger of their bursting at any point. For this reason, stop- 
 gates are shut by screws, because they prevent any very sudden closing; but 
 in pipes of large diameters, even the screws must be worked very slowly to 
 
 ^^Com prelss fifiUty of Liquids.— Liquids are not entirely incompressible, but 
 they are so nearly so, that for most purposes they may be considered as 
 incompressible. The bulk of water is diminished about by a pressure of 
 324 lb per sq. in., or 22 atmospheres; varying slightly with its temperature. 
 It is perfectly elastic, regaining its original bulk when the pressure is 
 r emo v ed 
 
 Construction of Dams in Mines.— Dams may be constructed in mines, either 
 to isolate a portion of the workings so that they can be flooded to extinguish 
 fires or, in cases where an extremely wet formation has been penetrated, it 
 is sometimes expedient to construct a dam so as to prevent the water from 
 flowing into the workings. Mine dams should be of sufficient strength to 
 resist any column of water that will be likely to come against them. The 
 dam should be arched toward the direction from which the pressure comes, 
 and should be given a good firm bearing in both walls and in the floor and 
 roof Fig 7 illustrates a brick dam that was constructed in Kehley’s Run 
 Colliery at Shenandoah, Pa., to isolate a portion of the seam so that it might 
 
134 
 
 HYDROSTATICS. 
 
 be flooded to extinguish a mine fire. This is one of the largest mine dams 
 that has ever been constructed. It is composed of three brick arches, each 
 having a thickness of 5 ft., that are placed one against the other so that they 
 act as one solid structure. The gangway at this point is about 20 ft. wide, 
 and the distance to the next upper level is about 119 ft. It was intended 
 that this should be the maximum head of water that the dams would ever \ 
 have to resist, though they were made sufficiently strong to resist a head of 
 water reaching to the surface. The separate walls were constructed one at 
 a time, and the cement allowed to set before the next wall was placed. The 
 back wall was carried to a greater depth and height than the others, so as to 
 make sure of the fact that all slips or partings had been closed. The total 
 pressure upon the dam when the water was in the mine was about 70,000 lb. 
 per sq. ft. 
 
 Dams constructed to permit the flooding of a mine usually require no 
 passages through them, but where dams are constructed to confine the 
 water to certain parts of the workings, and so reduce pumping charges, it 
 may be necessary to provide both manways and drain pipes through the 
 
 Fig. 7. Fig. 8. 
 
 dams. Fig. 8 illustrates a plan and cross-section of a dam in the Curry 
 Mine, at Norway, Mich. (“Mines and Minerals,” Vol. 18, page 177; Trans. 
 A I. M. E., XXVII, 402), constructed to keep the water that came from some 
 exploring drifts out of the mine workings. As originally constructed, it 
 consisted of a sandstone dam 10 ft. thick and arched on the back face 
 with a radius of 6 ft. A piece of 20" pipe provided a manway through the 
 masonry and was held in place by three sets of clamps and bolts passing 
 through the stonework. A 5" drain pipe was also carried through the dam 
 and secured by clamps. When the pressure came upon the dam it was 
 found to leak, so the water was drained off and a 22" brick wall built 
 2 ft. 4 in. back of the dam, the space between being filled with concrete, and 
 the manway and drain pipe extended through the brick wall. Before closing 
 the drain pipe, horse manure was fastened against the face of the brick wall 
 by means of a plank partition. After this the manway and drain pipe were 
 closed, and when the pressure came on, the dam was found to leak a small 
 
HYDRAULICS. 
 
 135 
 
 amount, but this soon practically ceased, showing that the manure had 
 closed the leaks. A gauge in the head of the manway on this dam showed a 
 pressure of 211 lb., which corresponded to a static head of 640 ft. of water. 
 The total pressure against the dam was something over 800 tons, which it 
 successfully resisted. 
 
 HYDRAULICS. 
 
 Hydraulics treats of liquids in motion, and in this, as in hydrostatics, 
 water is taken as the type. In theory its principles are the same as those of 
 falling bodies, but in practice they are so modified by various causes that 
 they cannot be relied on except as verified by experiment. The discrepancy 
 arises from changes of temperature that vary the fluidity of the liquid, from 
 friction, the shape of the orifice, etc. As we shall deal with water only, the 
 first cause may be thrown aside as of little account. 
 
 In theory the velocity of a jet is the same as that of a body falling from the 
 surface of the water. 
 
 To Find the Theoretical Velocity of a Jet of Water.— Let v = the velocity, 
 g = the acceleration of gravity (32.16 ft.), and d = the distance of the orifice 
 below the surface of the water. 
 
 Then, v = V 2 g d, or v = the square root of twice the product of g x d. 
 
 Example.— The depth of water above the orifice is 64 ft.; what is the 
 velocity ? 
 
 Substituting 64 for d , and 32.16 for g , we have, v = ]/ 2 X 32.16 X 64, or 
 64.16. 
 
 To Find the Theoretical Quantity of Water Discharged in a Given Time.— Multi- 
 ply the area of the orifice by the velocity of the water, and that product by 
 the number of seconds. 
 
 Example.— What quantity of water will be discharged in 5 seconds from 
 an orifice having an area of 2 sq. ft., at a depth of 16 ft.? 
 
 \/ 2 X 32.16 X 16 X 2 = 64.16 cu. ft,, or the amount discharged in 1 
 second, and in 5 seconds the amount will be 5 X 64.16 = 320.8 cu. ft. 
 
 The above rules are only theoretical, and are only useful as foundations 
 on which to build practical formulas. 
 
 Flow of Water Through Orifices.— The standard orifice, or an orifice so 
 arranged that the water in flowing from it touches only a line, as would be 
 the case in flowing through a hole in a very thin plate, is used for the 
 measurement of water. The contraction of the jet, which always occurs 
 when water issues from a standard orifice, is due to the circumstance that 
 the particles of water as they approach the orifice move in converging 
 directions, and that these directions continue to converge for a short distance 
 beyond the plane of the orifice. This contraction causes only the inner 
 corner of the orifice to be touched by the escaping water, and takes place 
 in orifices of any shape, its cross-section being similar to the orifice until 
 the place of greatest contraction is passed. Owing to this contraction, 
 the actual discharge from an orifice is always less than the theoretical 
 discharge. 
 
 The Coefficient of Contraction. — The coefficient of contraction is the number 
 by which the area of the orifice is to be multiplied in order to find the area 
 of the least cross-section of the jet. In this way by experiment this coeffi- 
 cient has been found to be about .62 (Merriman’s “Hydraulics”); or, in 
 other words, the minimum cross-section of the jet is 62$ of the cross-section 
 of the orifice. 
 
 The Coefficient of Velocity.— The coefficient of velocity is the number by 
 which the theoretical velocity of flow from the orifice is to be multiplied in 
 order to find the actual velocity at the least cross-section of the jet. This 
 may be taken for practical work as .98; or, in other words, the actual flow 
 at the contracted section is 98$ of the theoretical velocity. 
 
 The Coefficient of Discharge — The coefficient of discharge is the number by 
 which the theoretical discharge is to be multiplied in order to obtain the 
 actual discharge. This has been found by thousands of experiments to be 
 equal to the product of the coefficients of contraction and velocity, and for 
 practical work it may be taken as .61; or, the actual discharge from standard 
 orifices is 61$ of the theoretic discharge. 
 
136 
 
 HYDRAULICS. 
 
 Note —While the coefficients for standard orifices with sharp edges have 
 been determined fairly close, those for the more complicated cases of weirs, 
 and especially for the flow of water through long pipes, are simply the 
 nearest approximation to the truth that it has been possible to obtain. In 
 all cases the coefficient should be one that has been determined under con- 
 ditions similar to those in the problem in hand. For instance, it is not prac- 
 ticable to use the coefficient for small pipes m solving problems relating to 
 large ones, or for short pipes in solving problems relating to long ones. 
 
 Suppression of the Contraction. — When a vertical orifice has its lower edge 
 at the bottom of a reservoir, the particles of water flowing through its lower 
 portion move in lines nearly perpendicular to the plane of the orifice, and 
 the contraction of the jet does not form on the lower side. The same thing 
 occurs in a lesser degree when the lower edge of the orifice is within a dis- 
 tance of three times its least diameter from the bottom. The suppression of 
 contraction will occur on the side if it is placed within a distance of three 
 times its least diameter from the side of a reservoir the suppression of 
 contraction being the greater the nearer the orifice is to the side. By round- 
 ino- the edge of the orifice sufficiently, the contraction can be completely 
 suppressed and the discharge can be increased. As stated before the value 
 of the coefficient of contraction for a standard square-edged orifice is .62, but 
 with a rounded orifice it may have any value between .62 and 1.00, depend- 
 Sff on the degree of rounding. The coefficient of discharge for square- 
 edled orifices has a mean value of .61; this is increased with rounded edges 
 and may have any value between .61 and 1.00, although it is not probable 
 that values greater than .95 can be obtained except by the most careful 
 adtustment of the rounded edges to the exact curve of a completely con- 
 tracted jet. A rounded interior orifice is therefore always a source of error 
 when the object of the orifice is the measurement of the discharge. 
 
 GAUGING WATER. 
 
 Water is sold by two methods; i. e., the flowing unit and the capacity 
 unit The flowing unit is a cubic foot per second. In the western part of 
 North America the miners' inch has come into use .quite largely, while in 
 Australia and New Zealand the cubic foot per second is the common measure, 
 Australia ai being x head » and 10 heads of water would be 10 cu ft. 
 
 oer second regardless of the actual hydrostatic head under which the water 
 was delivered: Water is sometimes sold for irrigation by the capacity unit, 
 that is so much land covered to a certain depth, as, for instance, the acre- 
 foot ” which means that 1 acre has been covered to a depth of 1 foot, or that 
 an amount of 43,560 cu. ft. of water has been furnished. 
 
 Miners’ Inch -The miners’ inch may be roughly defined as the quantity of 
 water that wili flow in 1 minute through a vertical standard orifice having a 
 section of lsq ! in and a head of 6* in. above the center of the orifice. This 
 ouantitv equals 1.53 cu. ft., and the mean quantity may be taken at, approxi- 
 matelv 15 q cu ft. per minute. The laws or customs defining the miners 
 inch in different districts vary so that the amount of water actually delivered 
 varies from 1.2 to 1.76 cu. ft. per minute, the principal reasons for these varia- 
 tions being the method adopted for measuring, the water where large quan- 
 tities are used" as, for instance, at Smartsville, m California, an opening 4 in. 
 deep 250 in f long with a head’ of 7 in. above the .top edge is said to furnish 
 i non miners’ inches, while it would actually furnish considerably over 1,000. 
 Tu othe? nlaces the size of the opening for measuring the amounts is 
 restricted and may actually furnish less than the rated amount. In Montana 
 the common method of measurement was formerly through a vertical rect- 
 anele TiS hiSi . with a head on the center of the orifice of 4 m. The num- 
 bef of minem’ inches discharged was considered to be the same as the 
 number of linear inches in the length of the orifice; thus, under the given 
 head an orifice 1 in. deep and 60 in. long could discharge 60 miners ^hes. 
 
 The State Legislature of Montana has now passed a law defining the 
 miners’ inch as the number of gallons of water discharged m a given time, 
 regardless of the character of the openings or methods of measurement. 
 Tffie statement is as follows; “Where water rights, expressed m miners 
 inches hive been granted, 100 miners’ inches shall be. considered equivalent 
 to a flow of 2* cu ft. (18.7 gal.) per second, and this proportion shall be 
 observed in determining the equivalent flow represented by any number of 
 miners’ inches.” 
 
MINERS’ INCH. 
 
 137 
 
 If this amount is reduced to cubic feet per minute, it will be found to 
 be equal to a flow of 1.5 cu. ft. per minute, which is the value given above 
 for the miners’ inch. 
 
 Duty or Work Performed by a Miners’ Inch of Water.— Few tests have been 
 made in regard to the duty of a miners’ inch of water, but the North 
 Bloomfield mine and the La Grange mine, in California, have carried on 
 a series of experiments extending over several years. At the La Grange 
 mine the observations were carried on simultaneously upon several differ- 
 ent claims, hence parallel dates appear. The accompanying tables give 
 the results of these experiments. In general it is governed by the size, 
 capacity, character of pavement, and grade of sluices, together with the 
 supplv of water. A heavy grade will compensate for a limited supply. 
 With an abundant supply of water and material, the capacity of the sluices, 
 will depend on: First , the character of the material washed; second , the 
 size and minimum grade of the sluices; third , the character of the riffles used. 
 
 Duty of Miners’ Inch. 
 
 ( Risdon Iron Works , Evans's Elevator Catalogue.) 
 North Bloomfield Mine. 
 
 Years. 
 
 Cubic Yards of Gravel 
 Washed. 
 
 Miners’ 24-Hour 
 Inches. 
 
 Grades. 
 
 Cubic Yards Washed 
 per Miners’ Inch. 
 
 Cubic Feet of Water Used 
 per Cubic Feet 
 of Gravel Moved. 
 
 Height of Bank. 
 
 Remarks. 
 
 1870-74 
 
 1875 
 
 1876 
 
 1877 
 
 3.250.000 
 
 1.858.000 
 2,919,700 
 2,993,930 
 
 710,987 
 
 386,972 
 
 700.000 
 
 595.000 
 
 6^ in. to 12 ft. 
 6H in. to 12 ft. 
 6J^ in. to 12 ft. 
 6J^ in. to 12 ft. 
 
 4.60 
 
 4.80 
 
 4.17 
 
 3.86 
 
 18 
 
 17 
 
 20 
 
 21 
 
 100 ft. 
 100 ft. 
 200 ft. 
 265 ft. 
 
 
 Sluices 6 ft. wide, 
 32 in. deep. 
 
 Riffles principally 
 blocks (wood), but 
 > rock riffles in tail 
 sluices. 
 
 The larger portion 
 of the material 
 moved was top 
 gravel. 
 
 Totals 
 
 11,021,630 
 
 2,392,959 
 
 
 4.60 
 
 18 
 
 
 La Grange Mine. 
 
 1874- 76 
 
 1875- 76 
 
 1874- 76 
 
 1875- 78 
 1880-81 
 
 676,968 
 
 683,244 
 
 284,932 
 
 459,570 
 
 329,120 
 
 624.745 
 
 375,155 
 
 207,010 
 
 302,960 
 
 203,325 
 
 4 in. to 16 ft. 
 4 in. to 16 ft. 
 4 in. to 16 ft. 
 4 in. to 16 ft. 
 4 in. to 16 ft. 
 
 1.08 
 
 1.82 
 
 1.37 
 
 1.52 
 
 1.57 
 
 74.0 
 43.9 
 
 58.0 
 
 52.0 
 
 50.0 
 
 10 to 48 ft. 
 
 6 ft. 
 50 to 80 ft. 
 40 to 50 ft. 
 10 to 80 ft. 
 
 
 Sluices 4 ft. wide 
 and 30 in. deep, 
 > paved with 
 blocks. 
 
 Totals 
 
 2,433,834 
 
 1,713,195 
 
 
 1.42 
 
 56.0 
 
 
 
 The right-angled V notch is frequently used for gauging the flow of compara- 
 tively small streams. The notch is usually fitted into a box provided with 
 baffle boards, Fig. 9, or where this is not practicable the water should be 
 so impounded above the notch as to remove all possibility of surface cur- 
 rents producing a perceptible velocity of approach. The distance a of the 
 surface of the water below the top of the box is taken at a point some dis- 
 tance back from the notch (at least 18 to 20 in.), where the surface of the 
 water is unaffected by the flow through the notch. The distance a, sub- 
 tracted from the total depth of the notch H, gives the head h of the water 
 passing over the notch. The discharge in cubic feet per second may be 
 found by the formula _ __ 
 
 Q = .306 / hP = .306 ft*/ ft, 
 
 in which Q equals the quantity in cubic feet per minute and h equals the 
 
138 
 
 HYDRAULICS. 
 
 head in inches. The accompanying table gives the discharge in cubic feet 
 per minute through a right-angled V notch, as shown m Fig. 9, for heads 
 varying from 1.05 in. to 12 in. 
 
 TABLE I. 
 
 Discharge of Water Through a Right-Angled V Notch. 
 
 h 
 
 Head. 
 
 Inches. 
 
 Q 
 
 Quantity 
 per Min. 
 Cu. Ft. 
 
 h 
 
 Head. 
 
 Inches. 
 
 Q 
 
 Quantity 
 per Min. 
 Cu. Ft. 
 
 h 
 
 Head. 
 
 Inches. 
 
 Q 
 
 Quantity 
 per Min. 
 Cu. Ft. 
 
 h 
 
 Head. 
 
 Inches. 
 
 Q 
 
 Quantity 
 per Min. - 
 Cu. Ft. 
 
 h 
 
 Head. 
 
 inches. 
 
 Q 
 
 Quantity 
 per Min. 
 Cu. Ft. 
 
 1 05 
 
 .3457 
 
 3.25 
 
 5.827 
 
 5.45 
 
 21.22 
 
 7.65 
 
 49.53 
 
 9.85 
 
 93.18 
 
 1 10 
 
 .3884 
 
 3.30 
 
 6.054 
 
 5.50 
 
 21.71 
 
 7.70 
 
 50.34 
 
 9.90 
 
 94.37 
 
 1 15 
 
 .4340 
 
 3.35 
 
 6.285 
 
 5.55 
 
 22.20 
 
 7.75 
 
 51.16 
 
 9.95 
 
 95.56 
 
 1 20 
 
 .4827 
 
 3.40 
 
 6.523 
 
 5.60 
 
 22.70 
 
 7.80 
 
 51.99 
 
 10.00 
 
 96.77 
 
 1 25 
 
 .5345 
 
 3.45 
 
 6.765 
 
 5.65 
 
 23.22 
 
 7.85 
 
 52.83 
 
 10.05 
 
 97.98 
 
 1 30 
 
 5896 
 
 3.50 
 
 7.012 
 
 5.70 
 
 23.74 
 
 7.90 
 
 53.67 
 
 10.10 
 
 99.20 
 
 1 35 
 
 .6480 
 
 3.55 
 
 7.266 
 
 5.75 
 
 24.26 
 
 7.95 
 
 54.53 
 
 10.15 
 
 100.43 
 
 1 40 
 
 .7096 
 
 3.60 
 
 7.524 
 
 5.80 
 
 24.79 
 
 8.00 
 
 55.39 
 
 10.20 
 
 101.67 
 
 1 45 
 
 .7747 
 
 3.65 
 
 7.788 
 
 5.85 
 
 25.33 
 
 8.05 
 
 56.26 
 
 10.25 
 
 102.92 
 
 1 50 
 
 .8432 
 
 3.70 
 
 8.058 
 
 5.90 
 
 25.87 
 
 8.10 
 
 57.14 
 
 10.30 
 
 104.18 
 
 
 .9153 
 
 3.75 
 
 8.332 
 
 5.95 
 
 26.42 
 
 8.15 
 
 58.03 
 
 10.35 
 
 105.45 
 
 1 60 
 
 9909 
 
 3.80 
 
 8.613 
 
 6.00 
 
 26.98 
 
 8.20 
 
 58.92 
 
 10.40 
 
 106.73 
 
 * i 65 
 
 1.0700 
 
 3.85 
 
 8.899 
 
 6.05 
 
 27.55 
 
 8.25 
 
 59.82 
 
 10.45 
 
 108.02 
 
 1.70 
 
 1.1530 
 
 3.90 
 
 9.191 
 
 6.10 
 
 28.12 
 
 8.30 
 
 60.73 
 
 10.50 
 
 109.31 
 
 1.75 
 
 1.2400 
 
 3.95 
 
 9.489 
 
 6.15 
 
 28.70 
 
 8.35 
 
 61.65 
 
 10.55 
 
 110.62 
 
 L80 
 
 1.3300 
 
 4.00 
 
 9.792 
 
 6.20 
 
 29.28 
 
 8.40 
 
 62.58 
 
 10.60 
 
 111.94 
 
 L85 
 
 1.4240 
 
 4.05 
 
 10.100 
 
 6.25 
 
 29.88 
 
 8.45 
 
 63.51 
 
 10.65 
 
 113.26 
 
 1.90 
 
 1.5220 
 
 4.10 
 
 10.410 
 
 6.30 
 
 30.48 
 
 8.50 
 
 64.45 - 
 
 10.70 
 
 114.60 
 
 L95 
 
 1.6250 
 
 4.15 
 
 10.730 
 
 6.35 
 
 31.09 
 
 8.55 
 
 65.41 
 
 10.75 
 
 115.94 
 
 2^00 
 
 1.7310 
 
 4.20 
 
 11.060 
 
 6.40 
 
 31.71 
 
 8.60 
 
 66.37 
 
 10.80 
 
 117.29 
 
 2.05 
 
 1.8410 
 
 4.25 
 
 11.390 
 
 6.45 
 
 32.33 
 
 8.65 
 
 67.34 
 
 10.85 
 
 118.65 
 
 2.10 
 
 1.9550 
 
 4.30 
 
 11.730 
 
 6.50 
 
 32.96 
 
 8.70 
 
 68.32 
 
 10.90 
 
 120.02 
 
 2.15 
 
 2.0740 
 
 4.35 
 
 12.070 
 
 6.55 
 
 33.60 
 
 8.75 
 
 69.30 
 
 10.95 
 
 121.41 
 
 2/20 
 
 2.1960 
 
 4.40 
 
 12.420 
 
 6.60 
 
 34.24 
 
 8.80 
 
 70.30 
 
 11.00 
 
 122.81 
 
 2.25 
 
 2.3230 
 
 4.45 
 
 12.780 
 
 6.65 
 
 34.89 
 
 8.85 
 
 71.30 
 
 11.05 
 
 124.21 
 
 2.30 
 
 2.4550 
 
 4.50 
 
 13.140 
 
 6.70 
 
 35.56 
 
 8.90 
 
 72.31 
 
 11.10 
 
 125.61 
 
 2.35 
 
 2.5900 
 
 4.55 
 
 13.510 
 
 6.75 
 
 36.23 
 
 8.95 
 
 73.33 
 
 11.15 
 
 127.03 
 
 2.40 
 
 2.7300 
 
 4.60 
 
 13.890 
 
 6.80 
 
 36.89 
 
 9.00 
 
 74.36 
 
 11.20 
 
 128.45 
 
 2.45 
 
 2.8750 
 
 4.65 
 
 14.270 
 
 6.85 
 
 37.58 
 
 9.05 
 
 75.40 
 
 11.25 
 
 129.90 
 
 2.50 
 
 3.0240 
 
 4.70 
 
 14.650 
 
 6.90 
 
 - 38.27 
 
 9.10 
 
 76.44 
 
 11.30 
 
 131.35 
 
 2.55 
 
 3.1770 
 
 4.75 
 
 15.040 
 
 6.95 
 
 38.96 
 
 9.15 
 
 77.49 
 
 11.35 
 
 132.81 
 
 2.60 
 
 3.3350 
 
 4.80 
 
 15.440 
 
 7.00 
 
 39.67 
 
 9.20 
 
 78.55 
 
 11.40 
 
 134.27 
 
 A 65 
 
 3.4980 
 
 4.85 
 
 15.850 
 
 7.05 
 
 40.38 
 
 9.25 
 
 79.63 
 
 11.45 
 
 135. / 5 
 
 2.70 
 
 3.6660 
 
 4.90 
 
 16.260 
 
 7.10 
 
 41.10 
 
 9.30 
 
 80.71 
 
 11.50 
 
 137.23 
 
 2.75 
 
 3.8380 
 
 4.95 
 
 16.680 
 
 7.15 
 
 41.83 
 
 9.35 
 
 81.80 
 
 11.55 
 
 138.73 
 
 2.80 
 
 4.0140 
 
 5.00 
 
 17.110 
 
 7.20 
 
 42.56 
 
 9.40 
 
 82.90 
 
 11.60 
 
 140.23 
 
 2.85 
 
 4.1960 
 
 5.05 
 
 17.540 
 
 7.25 
 
 43.30 
 
 9.45 
 
 84.01 
 
 11.65 
 
 141.75 
 
 2.90 
 
 4.3820 
 
 5.10 
 
 17.970 
 
 7.30 
 
 44.06 
 
 9.50 
 
 85.12 
 
 11.70 
 
 143.28 
 
 2.95 
 
 4.5740 
 
 5.15 
 
 18.420 
 
 7.35 
 
 44.82 
 
 9.55 
 
 86.24 
 
 11.75 
 
 144.82 
 
 3.00 
 
 4.7700 
 
 5.20 
 
 18.870 
 
 7.40 
 
 45.58 
 
 9.60 
 
 87.37 
 
 11.80 
 
 146 36 
 
 3.05 
 
 4.9710 
 
 5.25 
 
 19.320 
 
 7.45 
 
 46.36 
 
 9.65 
 
 88.52 
 
 11.85 
 
 147.91 
 
 3.10 
 
 5.1780 
 
 5.30 
 
 19.790 
 
 7.50 
 
 47.14 
 
 9.70 
 
 89.67 
 
 11.90 
 
 149.48 
 
 3.15 
 
 5.3880 
 
 5.35 
 
 20.260 
 
 7.55 
 
 47.92 
 
 9.75 
 
 90.83 
 
 11.95 
 
 151.05 
 
 3.20 
 
 5.6050 
 
 5.40 
 
 20.730 
 
 7.60 
 
 48.72 
 
 9.80 
 
 92.00 
 
 12.00 
 
 152.64 
 
 1 cu. ft. contains 7.48 U. S. gallons; 1 U. S. gallon weighs 8.34 lb. 
 
 Gauging bv Weirs.— A weir is an obstruction placed across a stream for the 
 nurpose of diverging the water so as to make it flow through a desired chan- 
 nel which may be l notch or opening in the weir itself The term usually 
 applies to rectangular notches in which the water touches only ^he bottom 
 and ends, the opening being a notch without any upper edge. Weirs are of 
 two general classes: weirs with end contractions , Fig. 10 (a), and weirs without 
 
GAUGING BY WEIRS. 
 
 139 
 
 Fig. 9. 
 
 Let l 
 H 
 
 h = 
 
 c 
 Q 
 Q' 
 
 end contractions , Fig. 10 (6). The crest and edges of the weir with end con- 
 tractions should be sharp, as shown at a , Fig. 10 (c) and (d). The head 
 of water H producing the flow over the weir should be measured at a suffi- 
 cient distance from the crest to avoid the effects of the curve of the surface 
 as it flows over the crest. The 
 water above the weir should 
 be motionless, or if it has any 
 perceptible current toward 
 the weir, this should be deter- 
 mined and taken into account 
 in the formula. Fig. 11 illus- 
 trates a weir constructed across 
 a small stream for measuring 
 its flow. The head is measured 
 from the stake E some distance 
 
 back of the weir, the top of the stake being level with the crest of the weir 
 B. The discharge over the weir may be calculated from tlje following 
 formula: 
 
 length of weir in feet; 
 head in feet; 
 
 velocity with which the water approaches the weir in feet; 
 
 head equivalent to the velocity with which the water 
 approaches the weir; 
 coefficient of discharge; 
 theoretic discharge in cubic feet per second; 
 actual discharge in cubic feet per second. 
 
 For weirs with end contractions and a velocity of approach, the actual 
 
 discharge is 
 
 Q = 5.347 cl]/ (£T + 1.4 h) 3 . 
 
 Where the water has no 
 
 velocity of approach, _ 
 
 Q = 5.347 cl ]/ H 3 . 
 
 For weirs without end con- 
 tractions, but with a velocity 
 of approach, the actual dis- 
 charge is ______ 
 
 Q = 5.347 c l ]/\H + | h) s . 
 
 Where the water has no 
 velocity of approach, 
 
 Q = 5.347 c 1 1 /H*. 
 
 The velocity with which 
 the water approaches the weir 
 may be found by determining 
 the approximate discharge 
 from the stream without any allowance for velocity of approach, and then 
 dividing this discharge in cubic feet per second by the area of the stream in 
 square feet where it approaches the weir, which will give the velocity of 
 approach in feet per 
 second. Having ob- 
 
 tained the value oft;, ^ ^ 
 
 the equivalent head h « 
 may be found by the 
 formula 
 
 h = 0.01555 v-. 
 
 Since v is small in 
 a properly constructed 
 weir, it is usually neg- 
 lected unless great 
 accuracy is required. 
 
 The values of coeffi- 
 cients of discharge, as 
 determined from ex- 
 periments for weirs 
 with end contractions, 
 are given in Table II, 
 and for weirs without end contractions in Table III. The values of the 
 coefficients in Tables II and III are given in feet and tenths. Frequently 
 
140 
 
 HYDRA VLICS. 
 
 in measuring water where only a close approximation is required it is 
 desired to take all of the measurement in feet and inches. See Table IV. 
 
 TABLE II. 
 
 Values of the Coefficient of Discharge for Weirs With 
 End Contractions. 
 
 Length of W 7 eir. Feet. 
 
 Enecnve neau. 
 Feet. 
 
 1 
 
 .66 
 
 i 1 
 
 2 
 
 3 
 
 5 
 
 10 | 
 
 19 
 
 .10 
 
 .632 
 
 .639 
 
 .646 
 
 .652 
 
 .653 
 
 .655 
 
 .656 
 
 .15 
 
 .619 
 
 .625 
 
 .634 
 
 .638 
 
 .640 
 
 .641 
 
 .642 
 
 .20 
 
 .611 
 
 .618 
 
 .626 
 
 .630 
 
 .631 
 
 .633 
 
 .634 
 
 *25 
 
 .605 
 
 .612 
 
 .621 
 
 .624 
 
 .626 
 
 .628 | 
 
 .629 
 
 .30 
 
 .601 
 
 .608 
 
 .616 
 
 .619 
 
 .621 
 
 .624 ' 
 
 .625 
 
 .40 
 
 .595 
 
 .601 
 
 .609 
 
 .613 
 
 .615 
 
 .618 , 
 
 .620 
 
 .50 
 
 .590 
 
 .596 
 
 .605 
 
 .608 
 
 .611 
 
 .615 
 
 .617 
 
 .60 
 
 .587 
 
 .593 
 
 .601 
 
 .605 
 
 .608 
 
 .613 | 
 
 .615 
 
 ; 70 
 
 
 .590 
 
 .598 
 
 .603 
 
 .606 
 
 .612 
 
 .614 
 
 ftfl 
 
 
 
 .595 
 
 .600 
 
 .604 
 
 .611 
 
 .613 
 
 •OU 
 
 90 
 
 
 
 .592 
 
 .598 
 
 .603 
 
 .609 
 
 .612 
 
 1 00 
 
 
 
 .590 
 
 .595 
 
 .601 
 
 .608 
 
 .611 
 
 l.V/U 
 1 20 
 
 
 
 .585 
 
 .591 
 
 .597 
 
 .605 
 
 .610 
 
 1 . AAJ 
 
 1 40 
 
 
 
 .580 
 
 .587 
 
 .594 
 
 .602 
 
 .609 
 
 1.60 
 
 
 
 
 .582 
 
 I 
 
 .591 
 
 1 
 
 .600 
 
 .607 
 
 TABLE III. 
 
 Values of the Coefficient of Discharge for Weirs Without 
 End Contractions. 
 
 Effective Head. 
 Feet. 
 
 
 
 Length of W'eir. 
 
 . Feet. 
 
 
 
 19 f 
 
 10 1 
 
 7 
 
 5 ! 
 
 4 
 
 3 
 
 2 
 
 10 
 
 .657 
 
 .658 
 
 .658 
 
 .659 
 
 
 
 
 .lu 
 
 .15 
 
 .643 
 
 .644 
 
 .645 
 
 .645 
 
 .647 i 
 
 .649 
 
 .652 
 
 [20 
 
 .635 
 
 .637 
 
 .617 
 
 .638 
 
 .641 I 
 
 .642 
 
 .645 
 
 *25 
 
 .630 
 
 .632 
 
 .633 
 
 .634 
 
 .636 j 
 
 .638 | 
 
 .641 
 
 .30 
 
 .626 
 
 .628 
 
 .629 
 
 .631 
 
 .633 ! 
 
 .636 
 
 .639 
 
 .40 
 
 .621 
 
 .623 
 
 .625 
 
 .628 
 
 .630 
 
 .633 
 
 .636 
 
 "'SO 
 
 .619 
 
 .621 
 
 .624 
 
 .627 
 
 .630 
 
 .633 
 
 .637 
 
 .60 
 
 .618 
 
 .620 
 
 .623 
 
 .627 
 
 .630 
 
 .634 
 
 .638 
 
 .70 
 
 .618 
 
 .620 
 
 .624 
 
 .628 
 
 .631 
 
 .635 
 
 .640 
 
 "on 
 
 .618 
 
 .621 
 
 .625 
 
 .629 
 
 .633 
 
 .637 
 
 .643 
 
 .oU 
 
 00 
 
 .619 
 
 .622 
 
 .627 
 
 .631 
 
 .635 
 
 .639 
 
 .645 
 
 • «7 \j 
 1.00 
 
 .619 
 
 .624 
 
 .628 
 
 .633 
 
 .637 
 
 .641 
 
 .648 
 
 1.20 
 
 .620 
 
 .626 
 
 .632 
 
 .636 
 
 .641 
 
 .646 
 
 
 1.40 
 
 .622 
 
 .629 
 
 .634 
 
 .640 
 
 .644 
 
 
 
 L60 
 
 .623 
 
 .631 
 
 .637 
 
 .642 
 
 .647 
 
 
 
 L 
 
CONVERSION FACTORS. 
 
 141 
 
 TABLE IV. 
 
 Weir Table Giving Cubic Feet Discharged per Minute for Each Inch 
 in Length of Weir for Depths From £ In. to 25 In. 
 
 This table should not be used unless the length of the crest is at least 
 B or 4 times the depth of water passing over the weir, for if this is not the 
 case, there will be serious errors caused by end contractions. 
 
 Inches. 
 
 0 
 
 8 
 
 1 
 
 4 
 
 1 
 
 
 £ 
 
 £ 
 
 7 
 
 8 
 
 0 
 
 
 .01 
 
 .05 
 
 .09 
 
 .14 
 
 .20 
 
 .26 
 
 .33 
 
 1 
 
 .40 
 
 .47 
 
 .55 
 
 .65 
 
 .74 
 
 .83 
 
 .93 
 
 1.03 
 
 2 
 
 1.14 
 
 1.24 
 
 1.36 
 
 1.47 
 
 1.59 
 
 1.71 
 
 1.83 
 
 1.96 
 
 3 
 
 2.09 
 
 2.23 
 
 2.36 
 
 2.50 
 
 2.63 
 
 2.78 
 
 2.92 
 
 3.07 
 
 4 
 
 3.22 
 
 3.37 
 
 3.52 
 
 3.68 
 
 3.83 
 
 3.99 
 
 4.16 
 
 4.32 
 
 5 
 
 4.50 
 
 4.67 
 
 4.84 
 
 5.01 
 
 5.18 
 
 5.36 
 
 5.54 
 
 5.72 
 
 6 
 
 5.90 
 
 6.09 
 
 6.28 
 
 6.47 
 
 6.65 
 
 6.85 
 
 7.05 
 
 7.25 
 
 7 
 
 7.44 
 
 7.64 
 
 7.84 
 
 8.05 
 
 8.25 
 
 8.45 
 
 8.66 
 
 8.86 
 
 8 
 
 9.10 
 
 9.31 
 
 9.52 
 
 9.74 
 
 9.96 
 
 10.18 
 
 10.40 
 
 10.62 
 
 9 
 
 10.86 
 
 11.08 
 
 11.31 
 
 11.54 
 
 ' 11.77 
 
 12.00 
 
 12.23 
 
 12.47 
 
 10 
 
 12.71 
 
 13.95 
 
 13.19 
 
 13.43 
 
 13.67 
 
 13.93 
 
 14.16 
 
 14.42 
 
 11 
 
 14.67 
 
 14.92 
 
 15.18 
 
 15.43 
 
 15.67 
 
 15.96 
 
 16.20 
 
 16.46 
 
 12 
 
 16.73 
 
 16.99 
 
 17.26 
 
 17.52 
 
 17.78 
 
 18.05 
 
 18.32 
 
 18.58 
 
 13 
 
 18.87 
 
 19.14 
 
 19.42 
 
 19.69 
 
 19.97 
 
 20.24 
 
 20.52 
 
 20.80 
 
 14 
 
 21.09 
 
 21.37 
 
 21.65 
 
 21.94 
 
 22.22 
 
 22.51 
 
 22.79 
 
 23.08 
 
 15 
 
 23.38 
 
 23.67 
 
 23.97 
 
 24.26 
 
 24.56 
 
 24.86 
 
 25.16 
 
 25.46 
 
 16 
 
 25.76 
 
 26.06 
 
 26.36 
 
 26.66 
 
 26.97 
 
 27.27 
 
 27.58 
 
 27.89 
 
 17 
 
 28.20 
 
 28.51 
 
 28.82 
 
 29.14 
 
 29.45 
 
 29.76 
 
 30.08 
 
 30.39 
 
 18 
 
 30.70 
 
 31.02 
 
 31.34 
 
 31.66 
 
 31.98 
 
 32.31 
 
 32.63 
 
 32.96 
 
 19 
 
 33.29 
 
 33.61 
 
 33.94 
 
 34.27 
 
 34.60 
 
 34.94 
 
 35.27 
 
 35.60 
 
 20 
 
 35.94 
 
 36.27 
 
 36.60 
 
 36.94 
 
 37.28 
 
 37.62 
 
 37.96 
 
 38.31 
 
 21 
 
 38.65 
 
 39.00 
 
 39.34 
 
 39.69 
 
 40.04 
 
 40.39 
 
 40.73 
 
 41.09 
 
 22 
 
 41.43 
 
 41.78 
 
 42.13 
 
 42.49 
 
 42.84 
 
 43.20 
 
 43.56 
 
 43.92 
 
 23 
 
 44.28 
 
 44.64 
 
 45.00 
 
 45.38 
 
 45.71 
 
 46.08 
 
 46.43 
 
 46.81 
 
 24 
 
 47.18 
 
 47.55 
 
 47.91 
 
 48.28 
 
 48.65 
 
 49.02 
 
 49.39 
 
 49.76 
 
 CONVERSION FACTORS. 
 
 Cubic feet Into Gallons: 
 
 1 728 
 
 1 cu. ft. = 1,728 cu. in. = gal. = 7.4805194 gal. 
 
 Gallons Into Cubic Feet: 
 
 231 
 
 1 United States liquid gal. = 231 cu. in. = — cu. ft. = .133680555 cu. ft. 
 
 1| /2o 
 
 Feet per Second Into Miles per Hour: 
 
 3 600 15 
 
 1 ft. per sec. = 3,600 ft. per hr. = , or miles per hour. 
 
 D } Z£5\) ZZ 
 
 Miles per Hour Into Feet per Second: 
 
 5 280 22 
 
 1 mi. per hr. = 5,280 ft. per hr. = r, or — ft. per sec. 
 
 * UjOUi) lv 
 
 Second-Feet per Day Into Gallons: 
 
 lsecond-foot, or 7.4805194 gal. per sec. for 1 day, or 86,400 sec. = 646,316.87616 gal. 
 
 Millions of Gallons Into Second-Feet per Day:j 
 
 231 000 000 
 
 ] ,000,000 gal. per 24 hr. = r728 X 86 400 CU ’ ft ’ persec -’ or 1.5472286 second-feet. 
 
 Second-Feet per Day Into Acre-Feet: 
 
 1 second-foot flow for 1 day = 86,400 cu. ft. = or 1.983471 acre-feet. 
 
142 
 
 HYDRAULICS. 
 
 Acre-Feet Into Second-Feet Flow for 24 Hours: 
 
 One acre-foot each 24 hr. = 43,560 cu. ft. each 86,400 sec. 
 
 = 43 ’--, or second-foot flow for 24 hr. 
 
 86,400 ’ 240 
 
 Acre-Feet Into Gallons: 
 
 1 acre-foot = 43,560 cu. ft. = 
 
 43,560 X 1,728 75,271,680 
 
 231 ’ 231 
 
 or 325,851.428 gal. 
 
 Millions of Gallons Into Acre-Feet: 
 
 1,000,000 United States liquid gal., or 231,000,000 cu. in. = 
 
 or 133,6 ~ = 3.0688832 acre-feet. 
 43,560 
 
 133,680.555 cu. ft., 
 
 Second-Feet Into Minute Gallons: 
 
 Factors: 1 cu. ft. contains 1,728 cu. in.; 1 gal. has a ca Pac“y of 231 cu. iir; 
 1 second-foot equals [(1,728 - 4 - 231) X 60] gal. per min., or 448.831164 minute- 
 gallons. 
 
 Minute-Gallons Into Second-Feet: 
 
 Tf' a ptotis* • 1 ffal contains 231 cu. in.; 1 cu. ft. contains 1,728 cu. in.; 1 gal. 
 per F mh™equal! [(231 4 - 1,728) -*■ 60] second-feet, or .0022280092 second-foot. 
 
 FLOW OF WATER IN OPEN CHANNELS. 
 
 Ditches —In the case of hydraulic mining and irrigation, water is usually 
 conveyed through ditches. The ditch line should be carefully surveyed and 
 ail brush and trees removed, and the underbrush cut away and burned, 
 
 ^^T^e^ftSlowtag^tetters ^ifl be useiTin the formulas for determining the 
 various factors rela tin be j W g en en( j s 0 f canal or ditch, or between two 
 
 uoints under consideration; ,. 
 
 I — horizontal length of portion of canal or ditch under consideration; 
 
 s = slope = ratio - = sin of slope; 
 
 n = area of water cross-section in square feet; . _ A 
 
 “ = we t perimeter = portion of outline of cross-section of stream m 
 p contact with channel, in feet; ^ 
 
 r _ hydraulic radius, or hydraulic mean depth = ratio 
 
 C’ = coefficient, depending on nature of surface of the ditch; 
 c = coefficient depending on nature of surface of ditch, as determined by 
 Kutter’s formula; 
 
 v = mean velocity of flow m feet per second; 
 v' = surface velocity of a stream; 
 
 V x = bottom orune side of a section, the form of which is half a regular 
 hexagon, in feet; , . _ . 
 
 o = quantity of water flowing, m cubic feet per second; 
 
 n = coefficient of roughness in Kutter’s formula. 
 
 The form of ditch and its grade will depend largely on the > amount of 
 water to be conveyed and the character of the soil in the section ll ^der 
 consideration As a general rule, the average flow of water in a ditch 
 should not be less than 2 ft. per second, and under most circumstances 
 should not exceed 4 ft., though in rare cases where the formation is suitable, 
 mean velocities of 5 ft. per second are employed. Sand will deposit from a 
 Sirrent flowing at the rate of H ft. per second, and if the current does not 
 have a velocity of at least 2 ft. per second, vegetation is liable to block the 
 ditch line. „ 
 
 Safe Bottom Velocity.— The bottom velocity of a stream may be obtained 
 
 from the average velocity by the following formula: v h = v — 10.87 y rs. 
 
 The following table gives values of safe bottom and mean velocities, cor 
 responding with certain materials, as given by Ganguillet and Kutter: 
 
FLOW OF WATER IN CHANNELS. 
 
 143 
 
 Material of Channel. 
 
 Safe Bottom Velocity v b . 
 Feet per Second. 
 
 Mean Velocity v. 
 Feet per Second. 
 
 Soft brown earth 
 
 .249 
 
 .328 
 
 Soft loam 
 
 .499 
 
 .656 
 
 Sand 
 
 1.000 
 
 1.312 
 
 Gravel 
 
 1.998 
 
 2.625 
 
 Pebbles 
 
 2.999 
 
 3.938 
 
 Broken stone, flint 
 
 4.003 
 
 5.579 
 
 Conglomerate, soft slate 
 
 4,988 
 
 6.564 
 
 Stratified rock 
 
 6.006 
 
 8.204 
 
 Hard rock 
 
 10.009 
 
 13.127 
 
 Resistance of Soils to Erosion by Water. — W. A. Burr, “ Engineering News,” 
 February 8, 1894, gives a diagram showing the resistance of various soils to 
 erosion by water. The following values have been selected from Mr. Burr’s 
 work for different kinds of soil: 
 
 Pure sand resists erosion by flow of 1.10 ft. per sec. 
 
 Sandy soil, 15 0 clay 1.20 ft. per sec. 
 
 Sandy loam, 40 0 clay 1.80 ft. per sec. 
 
 Loamy soil, 650 clay 3.00 ft. per sec. 
 
 Clay loam, 850 clay 4.80 ft. per sec. 
 
 Agricultural clay, 950 clay 6.20 ft. per sec 
 
 Clay 7.35 ft. per sec. 
 
 Carrying Capacity of Ditches.— Ditches should never be run full, but should 
 be constructed large enough so that they will carry the desired amount of 
 water when from £ to i full. For any given cross-section, the greatest flow 
 will be attained when the hydraulic radius or hydraulic mean depth is equal 
 to one-half of the actual depth of the channel. The cross-section of a ditch 
 or conduit that has the greatest possible carrying capacity is a half circle, 
 and the nearest practical approach to this is a half hexagon. Knowing 
 the cross-section of a ditch, its dimensions may be found by the formula: 
 
 / 2a 
 X ~ \ 2.598 
 
 As the obtuse angle between the side and bottom of the ditch is 120°, the 
 form can be easily laid off. The carrying capacity of ditches generally 
 increases after they have been in use some time, as the ditch becomes lined 
 with a fine scum that closes the pores in the soil and prevents leakage. This 
 may increase the amount to as much as 100. 
 
 Grade.— The grade of the ditch must be sufficient to create the desired 
 velocity of flow, and depends largely on the character of the material com- 
 posing the surfaces of the ditch. If the surface is smooth, as, for instance, 
 where the ditch is cut through clay or is lined with masonry, the grade can 
 be considerably less than where the surface is rough, or when cut through 
 coarse gravel or when lined with rough stone. In mountainous countries, 
 where the ground is hard, deep narrow ditches with steep grades are gener- 
 ally preferred to larger channels with gentle slopes, as the cost of excavation 
 is considerably less; but steep grades and narrow ditches are suitable only 
 when the banks can resist the rapid flow. In California, grades of from 16 to 
 20 ft. per mile are used, and 10 ft. per mile is quite common. Water channels 
 of a uniform cross-section should have a uniform grade, otherwise, the flow 
 will be checked in places, which will result in deposits of sand or silt in 
 some portions of the ditch, which are liable to cause the banks to be o ver- 
 flowed and the ditch to be ultimately destroyed. In designing any given 
 ditch, the grade is generally assumed to correspond to the formation of the 
 country and the velocity figured from the grade. In case v is found to be so 
 great that it would cut the banks, it will be necessary either to reduce the 
 grade or to change the form of the ditch so as to reduce the velocity. 
 
 Ditch banks, when possible, should be composed of solid material, but 
 frequently it is necessary to use excavated material. Where this is the case, 
 care must be taken to see that the material is so placed as to avoid settling 
 
144 
 
 HYDRAULICS. 
 
 sSv iUs beltto build them of masonry, provided the expense is not too 
 
 Stonework ■ lafd up in. order and carefully bonded together. Sometimes the 
 
 able innue rnnoh greater average velocity can be attained in a 
 
 deeD stream * 5 than SaSwC wlthSut causing* an excessive velocity 
 oft^fwateP to “intact with the wet perimeter. For toWBW/ m cases 
 where banks will stand it, it is best to use narrow deep ditches rather than 
 wide flat ditches, though each location has to be tre , a * e ,^/? /f®” rdan th 
 
 its own s to' 0 i ; dc “ di ‘Vwater“i <: i Chafneto-TheTws^ to toe resistance to the 
 flow may b! expressed by the relation (see page 142 for significance of letters) : 
 
 ha = c'lpv* or, v = X | ^ X * X r. 
 
 If C = the formula becomes v = c V' n 
 
 The coefficient c is usually found by means of Kutter’s formula, one form 
 
 of which is as follows: * 
 
 1 .00155 
 
 23 + b — — 
 
 c = 
 
 / „ .00155 \ n 
 
 .5521 + (23 + — ) 
 
 The values for n, the coefficient of roughness, under various conditions, 
 are as follows: 
 
 Character of Channel. 
 
 Clean, well-planed timber ------ 
 
 fOpan smooth, glazed iron, and stoneware pipes 
 
 Masonry, smoothly plastered with cement, and for very clean, 
 
 Unvaried timber, ordinary cast-iron pipe, and selected pipe 
 
 sewers, well laid and thoroughly flushed -■■■■— - 7 ," 
 
 Rough iron pipes and ordinary sewer pipes, laid under the 
 
 usual conditions - v - 
 
 Tirpsscd masonry and well-laid brickwork vw'-'V \ 
 
 Good rubble masonry and ordinary rough or fouled brickwork 
 Coarse rubble masonry and firm, compact gravel 
 
 £v“ canals in“moc 1 era tel y perfbcUy *£ 
 
 RWeS antPcMials in rather bad condition and somewhat 
 
 Rivers and canals in bad condition, overgrown with vegeta- 
 tion and strewn with stones and other detritus, according to 
 condition - 
 
 Value of 7i. 
 
 .009 
 
 .010 
 
 .011 
 
 .012 
 
 .013 
 
 .015 
 
 .017 
 
 .020 
 
 .0225 
 
 .025 
 
 .030 
 
 .035 to .050 
 
FLUMES . 
 
 145 
 
 As it is quite difficult to obtain the value of c by Kutter’s formula, the fol- 
 lowing three approximate formulas for v are given: 
 
 For canals with earthen banks, 
 
 If the ditch is lined with dry stonework, 
 
 100,000 r 2 s 
 9 r -f 35 
 
 v 
 
 ; 100, 000 r 2 s 
 * = ■>/ 8r + 16" 
 ,100,000 r 2 s 
 
 If the ditch is lined with rubble masonry, t» = . y 
 
 To find the quantity Q of water flowing through any channel in a given 
 time, multiply the velocity by the area, or Q = a v. 
 
 Flow in Brooks and Rivers.— When a stream is so large that it becomes 
 impracticable to employ a weir for measuring its flow, fairly accurate 
 results may be arrived at by determining the velocity of the current at 
 various points in a carefully surveyed cross-section of the stream, thus 
 determining both v and a. The greatest velocity of current occurs at a point 
 some distance below the surface, in the deepest part of the channel. When 
 determining the current velocities in the different portions of a stream, it is 
 freauentlv advantageous to divide the stream into divisions. Ihis may be 
 accomplished by stretching a wire across and tying strings or rags about the 
 wire at various points. The mean velocity of the current between these 
 points can be determined by current meters, or by floats. The points for 
 observation should be chosen where the channel is comparatively straight 
 and the current uniform. Surface floats may be used, in which case the 
 mean velocity of the point where the float is used may be found as follows: 
 If equals the observed velocity, then the mean velocity will be v = .9 v'. 
 
 Bv taking observations of the velocity of the current m each section of 
 a stream the amount of water flowing may be determined for each separate 
 section ’ The total amount of water flowing in the stream will be the sum 
 of the amounts in each section. The average velocity of the entire stream 
 mav be found by dividing the total amount of water flowing by total area of 
 the cross-section of the stream. The correction necessary to reduce surface 
 velocity to mean velocity may be made as follows: Measure off x % of the 
 ordinary distance, and figure the time as though for the full distance. For 
 instance, if only 90 ft. were employed, the time would be taken and the 
 problem figured as though it were 100 ft., on account of the fact that the 
 mean velocity is only T % of the surface velocity. 
 
 FLUMES. 
 
 Flumes are used for conveying water when a ditch line would be 
 abnormally long, or when the material to be excavated is very hard. They 
 may be constructed of timber or of metal, but metal flumes are compara- 
 tively rare as piping can be used instead. The line of the proposed flume 
 should be carefully cleared of all standing timber, and the brush burned for 
 at least 20 ft. each side of the flume line to prevent danger from fire. The 
 life of an ordinary flume, which is supported on or constructed of timber, is 
 always short, varying, as a rule, from 10 to 20 years, depending on whether 
 the flume is allowed to run dry a portion of the year or is always full of 
 water, the care with which it was originally constructed, and the attention 
 
 paid to repairs. „ , , , ,, 
 
 Grade of Flumes.— Flumes are usually set on a much steeper grade than is 
 possible in ditches, the grade frequently being as much as 25 to 30 ft. per 
 mile and in special cases even more. The result of this is that the carrying 
 capacity of flumes is much greater than that of ditches of the same size. 
 
 The form of flume depends largely on the material of which it is constructed. 
 Metal flumes may have a semicircular form, while wooden flumes are either 
 rectangular or V-shaped. The former is used almost exclusively for con- 
 veying water, and the latter quite extensively for fluming timber or 
 cord wood from the mountains to the shipping point in the valley. 
 
 Timber flumes should be so constructed that the water will meet with but 
 small resistance, and the bottom and side should be enclosed in a frame of 
 timbers so braced or secured that there is no possible chance of the sides 
 spreading or lifting from the bottom, and thus cause leakage. As a rule, all 
 mortised and tenoned joints should be avoided in flume construction, 
 
m mm * « 
 
 W ■* 
 
 Thai Inmw -nar liare -hen- hil -ar— tnc ranarit y^ hey ha ve ha <* 
 
 n -n mlM if i Parser m a wr tiitn u mfh *he -niranr* amt the* f t. In 
 
 mrt --r»n ntm*» i 31*. ' le iese T» rniunrut "he Inm* 3 aiTna<ir than it deeo. 
 » n *nwt ▼-rarher -He jtk 31 'tie aarm<r lume 'mat *mnrf wmw 
 
 lw* airtw * "tin* imiemna -tie water brim :Sirther irtum <*f the *nimi» 
 uut taraoBty ',rr,Mru?:ne *he 4 i*w hmmm the tiime fhr *ver*i ee*e 
 Ttuln rit>* Aailfiv Imm-i r:Il nnt in *he flrfrr • 'jmrki.jr .it will 
 
 tp-eze n tnm t*> uvt ates inrii "he-r an? nraearmily a *>t*f*a»‘W 
 
 When a 'tnnie * awi in hie rrinmt an-ne a T a n k it eu uni «aaf aw 
 vi hu? «*na t» payable. •> a» si imteri it ftnm «wa <ir Lamt-i ^ and 
 m rttmr n tie r nier tie* wu»w wtH trtft .n imier amf balum* th«» 
 vrprrenf.ne he nmiacnn if he ur innnc the 1 nn». Thia will pcact the 
 inme* mrt tuv imiomr he low hr *une line liter mid weather ■* im. 
 
TUNNELS. 
 
 147 
 
 TUNNELS. 
 
 Tunnels are sometimes used for conveying water, in connection with 
 flume or ditch lines. Where a tunnel is unlined, it is best to give the roof 
 the shape of the Gothic arch, owing to the fact that this stands better and 
 resists scaling to a greater extent than the round arch, which usually scales 
 off until it has the form of the Gothic arch. If tunnels are to be used as 
 water conduits, without lining, care should be taken to make the inside of 
 the tunnel as smooth as possible. In some cases, in order to increase the 
 carrying capacity of the tunnel, they have been lined with wooden-stave 
 pipe, backed with concrete, the pipe requiring no metal bands, but depend- 
 ing on the concrete to keep it in place. When such linings are employed, it 
 is not practicable to have them exposed to the alternate action of the water 
 and the atmosphere, hence the tunnel should be kept continually full of 
 water. To accomplish this, the tunnel may be dropped below the grade of 
 the ditch or flume line, so that it is always under a slight hydrostatic 
 pressure, and even if the water were turned off from the line, the tunnel 
 would remain full of water, the same as an inverted siphon. Sometimes 
 tunnels are lined with cement, being given either a circular or oval form, 
 or they may have a flat bottom, with flat sides and an arched roof. The 
 cement may be placed directly on the country rock composing the walls of 
 the tunnel, or the tunnel may be lined with brick or stone, and then 
 cemented on the inside. 
 
 Flow Through Tunnels.— The flow of water through tunnels, when they are 
 only partially filled, is calculated by the formulas for flow in open channels, 
 while in the case of lined tunnels that are run full, the flow is calculated by 
 formulas for calculating the flow through pipes. 
 
 FLOW THROUGH PIPES. 
 
 Hydraulic Gradient.— If a pipe of uniform cross-section be connected with 
 a reservoir, and water allowed to discharge through its open end, it has been 
 found that the pressure on the pipe at any point is equal to the vertical dis- 
 tance from the center of the pipe at that point to an imaginary line, called 
 the hydraulic gradient or hydraulic grade line. This is a line drawn from a 
 point slightly below the 
 surface of the water in the 
 reservoir to the outlet of 
 the pipe, as ab , Fig. 14. 
 
 The distance from the sur- 
 face of the water to the 
 point a is equal to the head 
 lost in overcoming the fric- 
 tion at the entrance to the 
 pipe, and is rarely over 1 ft. 
 
 If the pipe were laid along 
 the line a b, it would carry 
 exactly the same amount 
 of water as when laid hori- 
 zontally, as shown, but 
 there would be practically 
 no pressure tending to 
 burst the pipe at any point 
 along this line; while if it 
 were laid along the line from the point a' (the reservoir being made 
 deeper), it would still deliver exactly the same amount of water, but the 
 pressure tending to burst the pipe would be greatly increased. In order 
 that a pipe may have a maximum discharge, no point in the line must rise 
 above the hydraulic gradient, and it makes no difference in the discharge 
 how far below the gradient it may fall. 
 
 In Fig. 15, the pipe rises above the hydraulic gradient a c, and in this case 
 a new hydraulic gradient ab would have to be established, and the flow 
 calculated for this head, the pipe b c simply acting to carry off the water 
 delivered to it at b. If the upper side of the pipe were ©pen at the point b , 
 the water would have no tendency to escape, but, on the contrary, air would 
 probably enter, and the pipe flow only partially full from b to c. 
 
 Flow in Pipes.— Darcy, a French engineer, made a series of experiments on 
 different diameters of cast-iron pipe, with different degrees of internal 
 
148 
 
 HYDRA ULICS. 
 
 roughness, from which he calculated a series of formulas. The following 
 are some of these formulas, as arranged by Mr. E. Sherman Gould. C. E., E. M., 
 one of the most experienced hydraulic engineers in America. Darcy found 
 
 that the character of the inside 
 surface of the pipe played a 
 very important part in its dis- 
 charge, and he deduced a 
 formula and determined a 
 series of coefficients for it, but 
 Mr. Gould calls attention to the 
 fact that the coefficients for 
 pipes from 8" to 48" in diam- 
 eter practically cancel the 
 numerical factor employed in 
 Darcy’s formula, and that a 
 slightly different factor applies to pipes from 3 to 8 in., so that we may have 
 the following simple formulas, in which the factors given apply: 
 
 Q = amount of water in cubic feet per second; 
 q = U. S. gallons per minute; 
 
 D = diameter of pipe in feet; 
 d = diameter of pipe in inches; 
 
 H = total head in feet; 
 h = head per 1,000 ft.; 
 
 V = velocity in feet per second. 
 
 Pipes above 8 in. in diameter, rough inside surface, 
 
 Q = = D^VDh\ y = 1*271 /Dh. 
 
 Fig. 15. 
 
 For diameter in inches, 
 
 Pipes between 3 and 8 in. in diameter, rough inside surface, 
 Q = 0.89 \/Wh = 0.89 DVM; V = 1.13 /ZT/i. 
 
 Large pipes, smooth inside surface, 
 
 Q = 1.4/x^/i = 1.4 dV^J y = 1*781 /'Dh. 
 
 Small pipes, smooth inside surface, 
 
 Q = 0.89/2 H>h = 1.25 D 2 /d7; V = l.G]/ Dh. 
 
 As a rule, it is best to calculate any pipe line by the formula for pipes 
 having a rough internal surface, for if this is not done the results are liable 
 to be disappointing, since all pipes become more or less rough with use. 
 
 Eytetwein’s Formula for the Delivery of Water in Pipes: 
 
 D = diameter of pipe in inches; 
 
 H = head of water in feet; 
 
 L = length of pipe in feet; 
 
 IF = cubic feet of water discharged per minute 
 
 IF =4.71^^-. D 
 
 .538 
 
 5 L X IF 2 
 
 \ II ■ 
 
 Hawksley’s Formula: 
 
 G 
 L 
 H 
 D 
 
 number of gallons delivered per hour: 
 length of pipe in yards; 
 head of water in feet; 
 diameter of pipe in inches. 
 
 D 
 
 = 4 “ 
 
 (15 Df II 
 
 Neville’s General Formula: 
 
 v = velocity in feet per second; 
 r = hydraulic mean depth in feet; 
 
 s = sine of inclination, or total fall divided by total length. 
 v = 140 / r s — 11 / r s. 
 
 In cylindrical pipes, v multiplied by 47.124 d 2 gives the discharge per 
 minute in cubic feet, or v multiplied by 293.7286 d* gives the discharge per 
 minute in gallons, d being the diameter of the pipe in feet. 
 
COMPARISON OF FORMULAS . 
 
 149 
 
 COMPARISON OF FORMULAS. 
 
 R = mean hydraulic depth in feet = area -i- wet perimeter = - for 
 circular section of pipe; 4 
 
 S = sine of slope = j-; 
 
 v = velocity in feet per second; 
 d = diameter of pipe in feet; 
 
 L = length of pipe in feet; 
 
 H = head of water in feet. 
 
 Prony, v 
 
 Eytelwein, 
 
 Eytelwein, 
 
 Hawksley, 
 
 Neville, 
 
 Darcy, 
 
 = 97.05]/ RS — .08; or, v = 99.88 \/RS- .154. 
 
 f dH 
 L + 50 d' 
 
 v = 108]/.RS-.13. 
 V = + 
 
 v = U0\/RS — llfr RS. 
 
 v = C]/ R S; for value of C, see following Table. 
 
 Diameter of pipe (inches) 
 
 1 
 
 2 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 Value of C. 
 
 65 
 
 80 
 
 93 
 
 99 
 
 102 
 
 103 
 
 105 
 
 106 
 
 107 
 
 Diameter of pipe (inches) 
 
 9 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 Value of C 
 
 108 
 
 109 
 
 109.5 
 
 110 
 
 110.5 
 
 110.7 
 
 111 
 
 111.5 
 
 111.5j 
 
 Maximum value of C for very large pipes, 113.3. 
 v = C \Zr~s, 
 
 Kutter, 
 
 where 
 
 Weisbach, 
 
 181 + 
 
 .00281 
 
 C = 
 
 .026/ . , .00281 \ 
 l + -7=( 416 + -S ~) 
 
 y' d'- 
 
 V 2 
 
 ft = ^.0036 + ^V 
 
 »' V Vv 
 
 where h = head necessary to overcome the friction in a pipe; r = the mean 
 radius of the pipe in feet; and g = gravity = 32.2. 
 
 Siphons.— When any part of the pipe line rises above the source of supply, 
 such a line is called a siphon. If this rise is greater than the height of the 
 water barometer (34 ft. at sea level), water will not flow through the siphon. 
 The flow through the siphon will be the same as that through any pipe line 
 so long as there is no accumulation of air at the highest point of the line; 
 but such an accumulation will decrease or entirely stop the flow. All 
 siphons should be provided at their highest points with valves for discharging 
 the air and introducing water to fill the siphon, and it is usually best to trap 
 the lower end of the pipe so that air cannot enter it, and to enlarge the 
 upper end so as to reduce the loss of the stream in entering. For a siphon 
 to work well, the fall between the intake and the discharge end should be 
 considerable, if the rise amounts to much. 
 
Table Showing the Actual Amount, or 80$ of the Theoretical Flow in Pipes From $ In. to 30 In. Diameter. 
 
 ( Supplement to “Industry” No. 45, April, 1892.) 
 
 150 
 
 HYDRAULICS. 
 
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 Slope. 
 
 1 in 1,000 
 
 2 in 1,000 
 
 3 in 1,000 
 
 4 in 1,000 
 
 5 in 1,000 
 
 6 in 1,000 
 
 7 in 1,000 
 Sin 1,000 
 
 9 in 1,000 
 
 1 in 100 
 
 2 in 100 
 
 3 in 100 
 
 4 in 100 
 
 5 in 100 
 
 0 in 100 
 
 7 in 100 
 
 8 in 100 
 
 9 in 100 
 
 1 in 10 
 
 2 in 10 
 
 3 in 10 
 
 4 in 10 
 
 5 in 10 
 
 6 in 10 
 
 7 in 10 
 
 8 in 10 
 
 9 in 10 
 
 10 in 10 
 
 The quantities above are American gallons per minute. For cubic feet, divide by 7.5. 
 
FRICTION IN PIPES. 
 
 151 
 
 LOSS OF HEAD IN PIPE BY FRICTION 
 
 In each 100 ft. in length of different diameters, when discharging the follow- 
 ing quantities of water per minute, as given by Pel ton Water Wheel Co. 
 
 Inside Diameter of Pipe. Inches. 
 
 Velocity. 
 Ft. per Sec. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu.' Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 ! 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. | 
 
 1 
 
 Cu. Ft. 
 per Min. 
 
 2.0 
 
 2.37 
 
 .65 
 
 1.185 
 
 2.62 
 
 .791 
 
 5.89 
 
 .593 
 
 10.4 
 
 .474 
 
 16.3 
 
 .395 
 
 23.5 
 
 2.2 
 
 2.80 
 
 .73 
 
 1.404 
 
 2.88 
 
 .936 
 
 6.48 
 
 .702 
 
 11.5 
 
 .561 
 
 18.0 
 
 .468 
 
 25.9 
 
 2.4 
 
 3.27 
 
 .79 
 
 1.639 
 
 3.14 
 
 1.093 
 
 7.07 
 
 .819 
 
 12.5 
 
 .650 
 
 19.6 
 
 .547 
 
 28.2 
 
 2.6 
 
 3.78 
 
 .86 
 
 1.891 
 
 3.40 
 
 1.260 
 
 7.65 
 
 .945 
 
 13.6 
 
 .757 
 
 21.3 
 
 .631 
 
 30.6 
 
 2.8 
 
 4.32 
 
 .92 
 
 2.160 
 
 3.66 
 
 1.440 
 
 8.24 
 
 1.080 
 
 14.6 
 
 .864 
 
 22.9 
 
 .720 
 
 32.9 
 
 3.0 
 
 4.89 
 
 .99 
 
 2.440 
 
 3.92 
 
 1.620 
 
 8.83 
 
 1.220 
 
 15.7 
 
 .978 
 
 24.5 
 
 .815 
 
 35.3 
 
 3.2 
 
 5.47 
 
 1.06 
 
 2.730 
 
 4.18 
 
 1.820 
 
 9.42 
 
 1.370 
 
 16.7 
 
 1.098 
 
 26.2 
 
 .915 
 
 37.7 
 
 3.4 
 
 6.09 
 
 1.12 
 
 3.050 
 
 4.45 
 
 2.040 
 
 10.00 
 
 1.520 
 
 17.8 
 
 1.220 
 
 27.8 
 
 1.021 
 
 40.0 
 
 3.6 
 
 6.76 
 
 1.19 
 
 3.380 
 
 4.71 
 
 2.260 
 
 10.60 
 
 1.690 
 
 18.8 
 
 1.350 
 
 29.4 
 
 1.131 
 
 42.4 
 
 3.8 
 
 7.48 
 
 1.26 
 
 3.740 
 
 4.97 
 
 2.490 
 
 11.20 
 
 1.870 
 
 19.9 
 
 1.490 
 
 31.0 
 
 1.250 
 
 44.7 
 
 4.0 
 
 8.20 
 
 1.32 
 
 4.100 
 
 5.23 
 
 2.730 
 
 11.80 
 
 2.050 
 
 20.9 
 
 1.640 
 
 32.7 
 
 1.370 
 
 47.1 
 
 4.2 
 
 8.97 
 
 1.39 
 
 4.490 
 
 5.49 
 
 2.980 
 
 12.30 
 
 2.240 
 
 22.0 
 
 1.790 
 
 34.3 
 
 1.490 
 
 49.5 
 
 4.4 
 
 9.77 
 
 1.45 
 
 4.890 
 
 5.76 
 
 3.250 
 
 12.90 
 
 2.430 
 
 23.0 
 
 1.950 
 
 36.0 
 
 1.620 
 
 51.8 
 
 4.6 
 
 10.60 
 
 1.52 
 
 5.300 
 
 6.02 
 
 3.530 
 
 13.50 
 
 2.640 
 
 24.0 
 
 2.110 
 
 37.6 
 
 1.760 
 
 54.1 
 
 4.8 
 
 11.45 
 
 1.58 
 
 5.720 
 
 6.28 
 
 3.810 
 
 14.10 
 
 2.850 
 
 25.1 
 
 2.270 
 
 39.2 
 
 1.900 
 
 56.5 
 
 5.0 
 
 12.33 
 
 1.65 
 
 6.170 
 
 6.54 
 
 4.110 
 
 14.70 
 
 3.080 
 
 26.2 
 
 2.460 
 
 40.9 
 
 2.050 
 
 58.9 
 
 5.2 
 
 13.24 
 
 1.72 
 
 6.620 
 
 6.80 
 
 4.410 
 
 15.30 
 
 3.310 
 
 27.2 
 
 2.650 
 
 42.5 
 
 2.210 
 
 61.2 
 
 5.4 
 
 14.20 
 
 1.78 
 
 7.100 
 
 7.06 
 
 4.730 
 
 15.90 
 
 3.550 
 
 28.2 
 
 2.840 
 
 44.2 
 
 2.370 
 
 63.6 
 
 5.6 
 
 15.16 
 
 1.85 
 
 7.580 
 
 7.32 
 
 5.060 
 
 16.50 
 
 3.790 
 
 29.3 
 
 3.030 
 
 45.8 
 
 2.530 
 
 65.9 
 
 5.8 
 
 16.17 
 
 1.91 
 
 8.090 
 
 7.58 
 
 5.400 
 
 17.10 
 
 4.040 
 
 30.3 
 
 3.240 
 
 47.4 
 
 2.700 
 
 68.3 
 
 6.0 
 
 17.23 
 
 1.98 
 
 8.610 
 
 7.85 
 
 5.740 
 
 17.70 
 
 4.310 
 
 31.4 
 
 3.450 
 
 49.1 
 
 2.870 
 
 70.7 
 
 7.0 
 
 22.89 
 
 2.31 
 
 11.450 
 
 9.16 
 
 7.620 
 
 20.60 
 
 5.720 
 
 36.6 
 
 4.570 
 
 57.2 
 
 3.810 
 
 82.4 
 
 Inside Diameter of Pipe. Inches. 
 
 . o 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 Velocity 
 Ft. per S< 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 -w S3 
 
 *3 
 
 * 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 2.0 
 
 .338 
 
 32.0 
 
 .296 
 
 41.9 
 
 .264 
 
 53.0 
 
 .237 
 
 65.4 
 
 .216 
 
 79.2 
 
 .198 
 
 g 94.2 
 
 2.2 
 
 .401 
 
 35.3 
 
 .351 
 
 46.1 
 
 .312 
 
 58.3 
 
 .281 
 
 72.0 
 
 .255 
 
 87.1 
 
 .234 
 
 103.0 
 
 2.4 
 
 .468 
 
 38.5 
 
 .410 
 
 50.2 
 
 .365 
 
 63.6 
 
 .327 
 
 78.5 
 
 .297 
 
 95.0 
 
 .273 
 
 113.0 
 
 2.6 
 
 .540 
 
 41.7 
 
 .473 
 
 54.4 
 
 .420 
 
 68.9 
 
 .378 
 
 85.1 
 
 .344 
 
 103.0 
 
 .315 
 
 122.0 
 
 2.8 
 
 .617 
 
 44.9 
 
 .540 
 
 58.6 
 
 .480 
 
 74.2 
 
 .432 
 
 91.6 
 
 .392 
 
 111.0 
 
 .360 
 
 132.0 
 
 3.0 
 
 .698 
 
 48.1 
 
 .611 
 
 62.8 
 
 .544 
 
 79.5 
 
 .488 
 
 98.2 
 
 444 
 
 119.0 
 
 .407 
 
 141.0 
 
 3.2 
 
 .785 
 
 f 1.3 
 
 .686 
 
 67.0 
 
 .609 
 
 84.8 
 
 .549 
 
 105.0 
 
 .499 
 
 127.0 
 
 .457 
 
 151.0 
 
 3.4 
 
 .875 
 
 54.5 
 
 .765 
 
 71.2 
 
 .680 
 
 90.1 
 
 .612 
 
 111.0 
 
 .557 
 
 134.0 
 
 .510 
 
 160.0 
 
 3.6 
 
 .969 
 
 57.7 
 
 .848 
 
 75.4 
 
 .755 
 
 95.4 
 
 .679 
 
 118.0 
 
 .617 
 
 142.0 
 
 .566 
 
 169.0 
 
 3.8 
 
 1.070 
 
 60.9 
 
 .936 
 
 79.6 
 
 .831 
 
 101.0 
 
 .749 
 
 124.0 
 
 .680 
 
 150.0 
 
 .624 
 
 179.0 
 
 4.0 
 
 1.175 
 
 64.1 
 
 1.027 
 
 83.7 
 
 .913 
 
 106.0 
 
 .822 
 
 131.0 
 
 .747 
 
 158.0 
 
 .685 
 
 188.0 
 
 4.2 
 
 1.280 
 
 67.3 
 
 1.122 
 
 87.9 
 
 .998 
 
 111.0 
 
 .897 
 
 137.0 
 
 .816 
 
 166.0 
 
 .749 
 
 198.0 
 
 4.4 
 
 1.390 
 
 70.5 
 
 1.220 
 
 92.1 
 
 1.086 
 
 116.0 
 
 .977 
 
 144.0 
 
 .888 
 
 174.0 
 
 .815 
 
 207.0 
 
 4.6 
 
 1.510 
 
 73.7 
 
 1.320 
 
 96.3 
 
 1.177 
 
 122.0 
 
 1.059 
 
 150.0 
 
 .963 
 
 182.0 
 
 .883 
 
 217.0 
 
 4.8 
 
 1.630 
 
 76.9 
 
 1.430 
 
 100.0 
 
 1.270 
 
 127.0 
 
 1.145 
 
 157.0 
 
 1.040 
 
 190.0 
 
 .954 
 
 226.0 
 
 5.0 
 
 1.760 
 
 80.2 
 
 1.540 
 
 105.0 
 
 1.370 
 
 132.0 
 
 1.230 
 
 163.0 
 
 1.122 
 
 198.0 
 
 1.028 
 
 235.0 
 
 5.2 
 
 1.890 
 
 83.3 
 
 1.650 
 
 109.0 
 
 1.470 
 
 138.0 
 
 1.320 
 
 170.0 
 
 1.200 
 
 206.0 
 
 1.104 
 
 245.0 
 
 5.4 
 
 2.030 
 
 86.6 
 
 1.770 
 
 113.0 
 
 1.570 
 
 143.0 
 
 1.410 
 
 177.0 
 
 1.280 
 
 214.0 
 
 1.183 
 
 254.0 
 
 5.6 
 
 2.170 
 
 89.8 
 
 1.890 
 
 117.0 
 
 1.680 
 
 148.0 
 
 1.510 
 
 183.0 
 
 1.370 
 
 222.0 
 
 1.260 
 
 264.0 
 
 5.8 
 
 2.310 
 
 93.0 
 
 2.010 
 
 121.0 
 
 1.800 
 
 154.0 
 
 1.610 
 
 190.0 
 
 1.460 
 
 229.0 
 
 1.340 
 
 273.0 
 
 6.0 
 
 2.460 
 
 96.2 
 
 2.150 
 
 125.0 
 
 1.920 
 
 159.0 
 
 1.710 
 
 196.0 
 
 1.560 
 
 237.0 
 
 1.430 
 
 283.0 
 
 7.0 
 
 3.260 
 
 112.0 
 
 2.850 
 
 146.0 
 
 2.520 
 
 185.0 
 
 2.280 
 
 229.0 
 
 2.070 
 
 277.0 
 
 1.910 
 
 330.0 
 
 Example. — Have 200 ft. head and 600 ft. of 11" pipe, carrying 119 cu. ft. of 
 water per minute. To find effective head: In right-hand column, under 11" 
 pipe, find 119 cu. ft. Opposite this will be found the coefficient of friction for 
 this amount of water, which is .444. Multiply this by the number of hun- 
 dred feet of pipe, which is 6, and you will have 2.66 ft., which is the loss of 
 head. Therefore, the effective head is 200 — 2,66 = 197,34. 
 
152 
 
 HYDRA ULICS. 
 
 LOSS OF HEAD IN PIPE BY FRICTION 
 
 Tn each 100 ft in length of different diameters, when discharging the follow- 
 ing quaXies of water per minute, as given by Pelton Water Wheel Co. 
 
 Inside Diameter of Pipe. Inches. 
 
 
 % a 
 
 13 
 
 
 14 
 
 15 
 
 16 
 
 18 
 
 20 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 
 | per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 £.5 
 
 3 t- 
 O <u 
 
 0 . 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 .183 
 .216 
 .252 
 .290 
 .332 
 .375 
 .422 
 ; .471 
 
 .522 
 .576 
 .632 
 .691 
 .751 
 .815 
 .881 
 .949 
 1.020 
 j 1.092 
 j 1.167 
 1 1.245 
 i 1.325 
 ! 1.750 
 
 110 
 
 121 
 
 133 
 
 144 
 
 156 
 
 .169 
 
 128 
 
 . 158 : 
 
 147 
 
 .147 
 
 167 
 
 .132 j 
 
 212 
 
 .119 
 
 262 
 
 .200 1 
 
 141 
 
 .187 | 
 
 162 
 
 .175 
 
 184 
 
 .156 I 
 
 233 
 
 .140 j 
 
 288 
 
 234 
 
 154 
 
 .218 j 
 
 176 
 
 .205 
 
 201 
 
 .182 
 
 254 
 
 .164 | 
 
 314 
 
 .270 
 
 167 
 
 .252 
 
 191 
 
 .236 ! 
 
 218 
 
 .210 
 
 275 
 
 
 340 
 
 1308 
 
 179 
 
 .288 
 
 206 
 
 .270 1 
 
 234 
 
 .240 
 
 297 
 
 .216 1 
 
 366 
 
 166 
 
 177 
 
 .349 
 
 192 
 
 .325 
 
 221 
 
 .306 | 
 
 251 
 
 .271 
 
 318 
 
 .245 
 
 393 
 
 .392 
 
 205 
 
 .366 1 
 
 235 
 
 .343 
 
 268 
 
 .305 
 
 339 
 
 .275 
 
 419 
 
 188 
 
 .438 
 
 218 
 
 .408 
 
 250 
 
 .383 
 
 284 
 
 .339 
 
 360 
 
 .306 
 
 445 
 
 199 
 
 .485 
 
 231 
 
 .452 
 
 265 
 
 .425 
 
 301 
 
 .377 
 
 382 
 
 .339 
 
 471 
 
 210 
 
 .535 
 
 243 
 
 .499 
 
 280 
 
 .468 
 
 318 
 
 .416 
 
 403 
 
 .374 
 
 497 
 
 221 
 
 .587 
 
 256 
 
 .548 
 
 294 
 
 .513 
 
 335 
 
 .456 
 
 424 
 
 .410 
 
 523 
 
 232 
 
 .641 
 
 269 
 
 .598 
 
 309 
 
 .561 
 
 352 
 
 .499 
 
 445 
 
 .449 
 
 550 
 
 243 
 
 .698 
 
 282 
 
 .651 
 
 324 
 
 .611 
 
 1 368 
 
 .542 
 
 466 
 
 .488 
 
 576 
 
 254 
 
 .757 
 
 295 
 
 .707 
 
 339 
 
 .662 
 
 385 
 
 .588 
 
 488 
 
 .529 
 
 602 
 
 265 
 
 .818 
 
 308 
 
 .763 
 
 353 
 
 .715 
 
 ! 402 
 
 .636 
 
 509 
 
 .572 
 
 628 
 
 276 
 
 .881 
 
 321 
 
 .822 
 
 368 
 
 .770 
 
 419 
 
 .685 
 
 530 
 
 .617 
 
 654 
 
 287 
 
 .947 
 
 333 
 
 .883 
 
 383 
 
 .828 
 
 435 
 
 .736 
 
 551 
 
 .662 
 
 680 
 
 298 
 
 1.014 
 
 346 
 
 i .947 
 
 397 
 
 .888 
 
 452 
 
 .788 
 
 572 
 
 .710 
 
 707 
 
 309 
 
 1.083 
 
 359 
 
 1.011 
 
 412 
 
 | .949 
 
 469 
 
 .843 
 
 594 
 
 ! .758 
 
 733 
 
 321 
 
 1.155 
 
 372 
 
 1.078 
 
 427 
 
 1.011 
 
 486 
 
 .899 
 
 I 615 
 
 .809 
 
 759 
 
 332 
 
 1.229 
 
 385 
 
 1.148 
 
 442 
 
 1.076 
 
 502 
 
 .957 
 
 ; 636 
 
 .861 
 
 785 
 
 387 
 
 1.630 
 
 449 
 
 1.520 
 
 515 
 
 1 
 
 1.430 
 
 586 
 
 1.270 
 
 j 742 
 
 j 
 
 1.143 
 
 916 
 
 » i i-j 
 
 2.0 
 
 2.2 
 
 2.4 
 2.6 
 2.8 
 
 3.0 
 
 3.2 
 
 3.4 
 
 3.6 
 
 3.8 
 
 4.0 
 
 4.2 
 
 4.4 
 
 4.6 
 
 4.8 
 
 5.0 
 
 5.2 
 
 5.4 
 
 5.6 
 
 5.8 
 
 6.0 
 
 7.0 
 
 Inside Diameter of Pipe. Inches. 
 
 Velocity. 
 Ft. per Sec. 
 
 22 
 
 24 
 
 2 < 
 
 Loss of 
 Head. Ft. 
 
 5f| 
 
 0 g. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 2.0 
 
 .108 
 
 316 
 
 .098 
 
 377 
 
 .091 
 
 2.2 
 
 .127 
 
 348 
 
 .116 
 
 414 
 
 .108 
 
 2.4 
 
 .149 
 
 380 
 
 .136 
 
 452 
 
 .126 1 
 
 2.6 
 
 .171 
 
 412 
 
 .157 
 
 490 
 
 .145 I 
 
 2.8 
 
 .195 
 
 443 
 
 .180 
 
 528 
 
 .165 ; 
 
 3.0 
 
 .222 
 
 475 
 
 .204 
 
 565 
 
 .188 
 
 3.2 
 
 .249 
 
 507 
 
 .229 
 
 603 
 
 .211 
 
 3.4 
 
 .278 
 
 538 
 
 .255 
 
 641 
 
 .235 
 
 3.6 
 
 .308 
 
 570 
 
 .283 
 
 678 
 
 .261 
 
 3.8 
 
 .340 
 
 601 
 
 .312 
 
 716 
 
 .288 
 
 4.0 
 
 .373 
 
 633 
 
 .342 
 
 754 
 
 .315 
 
 4.2 
 
 .408 
 
 665 
 
 .374 
 
 791 
 
 .345 
 
 4.4 
 
 .444 
 
 697 
 
 .407 
 
 829 
 
 .375 
 
 4.6 
 
 .482 
 
 728 
 
 .441 
 
 867 
 
 .407 
 
 4.8 
 
 .521 
 
 760 
 
 .476 
 
 905 
 
 .440 
 
 5.0 
 
 .561 
 
 792 
 
 .513 
 
 942 
 
 .474 
 
 5.2 
 
 .602 
 
 823 
 
 .552 
 
 980 
 
 .510 
 
 5.4 
 
 .645 
 
 855 
 
 .591 
 
 1,018 
 
 .546 
 
 5.6 
 
 .690 
 
 887 
 
 .632 
 
 1 ,055 
 
 .583 
 
 5.8 
 
 .735 
 
 918 
 
 .674 
 
 1.093 
 
 .622 1 
 
 6.0 
 
 .782 
 
 950 
 
 .717 
 
 1,131 
 
 .662 
 
 7.0 
 
 1.040 
 
 1 
 
 1,109 
 
 .953 
 
 ! 1 ,319 
 
 .879 , 
 
 
 O 
 
 CO 
 
 00 
 
 cs 
 
 36 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 Loss of 
 Head. Ft. 
 
 Cu. Ft. 
 per Min. 
 
 442 
 
 .084 
 
 513 .079 ; 
 
 589 j 
 
 .066 
 
 848 
 
 486 
 
 .099 
 
 564 .093 
 
 648 , 
 
 .078 
 
 933 
 
 531 
 
 .116 
 
 616 .109 
 
 707 { 
 
 .091 
 
 1,018 
 
 575 
 
 .134 
 
 667 ; .126 
 
 766 ! 
 
 .104 
 
 1,100 
 
 619 
 
 .153 
 
 718 .144 
 
 824 1 
 
 .119 
 
 1,188 
 
 663 
 
 .174 
 
 770 .163 
 
 883 I 
 
 .135 
 
 1 ,273 
 
 708 
 
 .195 
 
 821 ' .182 
 
 942 l 
 
 .152 
 
 1.357 
 
 752 
 
 .218 
 
 872 .204 
 
 1,001 ! 
 
 .169 
 
 1.442 
 
 796 
 
 .242 
 
 923 .226 
 
 1,060 
 
 .188 
 
 1,527 
 
 840 
 
 .267 
 
 974 j .249 
 
 1,119 
 
 .207 
 
 1,612 
 
 885 
 
 .293 
 
 1,026 .273 
 
 1,178 
 
 .228 
 
 1 ,697 • 
 
 929 
 
 .320 
 
 1,077 : .299 
 
 1,237 
 
 .249 
 
 1,782 
 
 973 
 
 .348 
 
 1,129 .325 
 
 1,296 
 
 .271 
 
 1,866 
 
 1,017 
 
 .378 
 
 1,180 I .353 
 
 1,355 
 
 .294 
 
 1,951 
 
 1,062 
 
 .409 
 
 1,231 .381 
 
 1,414 
 
 .318 
 
 2,036 
 
 1,106 
 
 .440 
 
 1,283 , .411 
 
 1,472 
 
 .342 
 
 2,121 
 
 1.150 
 
 .473 
 
 1.334 .441 
 
 1,531 
 
 .368 
 
 2,206 
 
 1,194 
 
 .507 
 
 1,385 ! .473 
 
 1,590 
 
 .394 
 
 2,291 
 
 1,239 
 
 .542 
 
 1.437 .506 
 
 1,649 
 
 .421 
 
 2,376 
 
 1,283 
 
 .578 
 
 1,488 .540 
 
 1,708 
 
 .450 
 
 2,460 
 
 1,327 
 
 .615 
 
 1.539 .574 
 
 1,767 
 
 .479 
 
 2,545 
 
 1,548 
 
 .817 
 
 1,796 .762 
 
 2,061 
 
 .636 
 
 2,968 
 
FRICTION IN PIPES. 
 
 153 
 
 The following formula, deduced by William Cox, gives practically 
 the same results as the foregoing table and will be found useful in many 
 
 instances: F = fo^2> (4r2 + 5F_2 ), where/ 1 = friction head; L = length 
 
 of pipe in feet; D = diameter of pipe in inches; V = velocity in feet per 
 second. 
 
 Friction of Knees and Bends. -This subject has not been investirated suffi- 
 ciently to enable the- engineer to make exact allowance for this factor, but 
 
 the following formulas may be taken as giving close approximate results. 
 It is well to bear in mind that right angles should be avoided whenever 
 possible, and that bends should be made with as large a radius as circum- 
 
 A = angle of bend or knee with forward line 
 of direction; 
 
 V = velocity of water in feet per second; 
 
 R == radius of center line of bend; 
 r = radius of bore of pipe (or i diameter); 
 
 K = coefficient for angles of knees; 
 
 L = coefficient for curvature of bends; 
 
 H = head of water in feet necessary to over- 
 come the friction of the bends, or knees. 
 H = .0155 V 2 K. 
 
 The value of iTis as follows for different angles: 
 
 20° 
 
 40° 
 
 60° 
 
 I 80° 
 
 90° 
 
 100° 
 
 120° 
 
 .046 
 
 .139 
 
 .364 
 
 1 ' 74 
 
 1 .98 
 
 1.26 
 
 1.86 
 
 stances will allow. 
 
 For bends, 
 
 H = .0155 V 2 (Ai). 
 
154 
 
 HYDRA ULICS. 
 
 Values of L with various ratios of the radius of bend to radius of bore: 
 
 When = 
 
 j .1 ' 
 
 .2 ! 
 
 .3 
 
 •4 | 
 
 1 .5 
 
 .6 j 
 
 • 4 
 
 .8 | 
 
 .9 i 
 
 1 
 
 1.0 
 
 In circular section L 
 
 .131 ! 
 
 .138 
 
 .158 
 
 . '206 ! 
 
 .294 
 
 .44 
 
 .66 
 
 .98 
 
 1.4 
 
 2.0 
 
 In rectangular L 
 
 .124 ; 
 
 .135 
 
 .18 
 
 .25 
 
 .4 
 
 .64 
 
 1.01 
 
 1.55 , 
 
 2.3 | 
 
 3.2 
 
 RESERVOIRS. 
 
 Reservoir Site.— In selecting a site for a reservoir, the following points 
 should be observed: 
 
 1 . A proper elevation above the point at which the water is required. 
 
 2. The total supply available, including observations as to the rainfall 
 and snowfall. 
 
 3. The formation and character of the ground, with reference to the 
 amount of absorption and evaporation. 
 
 The most desirable formation of ground for a reservoir site is one of com- 
 pact rock, like granite, gneiss, or slate; porous rocks, like sandstones and 
 limestones, are not so desirable. Steep bare slopes are best for the country 
 surrounding a reservoir, as the water escapes from them quickly. The 
 presence of vegetation above the reservoir causes a considerable amount of 
 absorption; but, at the same time, the rainfall is usually greater in a region 
 covered with vegetation than in a barren region, hence the streams have a 
 more uniform flow. A reservoir must be made large enough to hold a 
 supplv capable of meeting the maximum demand. The area of a reservoir 
 should be determined, and a table made showing its contents for every foot 
 in depth, so that the amount of water available can always be known. 
 
 DAMS. 
 
 Dams are used for retaining water in reservoirs, for diverting streams in 
 placer mining, and for storing debris coming from placer mines in canons 
 or ravines. 
 
 Foundations for dams must be solid to prevent settling, and water-tight to 
 prevent leakage under the base of the dam. Whenever possible, the founda- 
 tion should be solid rock. Gravel is better than earth, but when gravel is 
 emploved it will be necessary to drive sheet piling under the upper toe of 
 the dam, to prevent water from seeping through the formation under the 
 dam. Vegetable soil should be avoided, and all porous material, such as 
 «and, gravel, etc. should be stripped off until hard pan or solid rock is 
 reached. In case springs occur in the area covered by the foundation of the 
 dam, it will be necessary to trace them up, and if they originate on the 
 upper side to confine their flow to that side of the dam. so that they will 
 have no tendency to ultimately become passageways for water from the 
 upper face to the lower face of the dam, thus providing holes which may 
 ultimatelv destrov the entire foundation of the structure. 
 
 Wooden Dams.— Wooden dams are constructed of round, sawed, or hewn 
 logs. The timbers are usually at least 1 ft. square, or, if round, from 18 to 
 ^4 in in diameter. A series of cribs from 8 to 10 ft. square are constructed by 
 building up the logs log-house fashion and securing them together with 
 treenails. The individual cribs are secured to one another with treenails 
 or by means of bolts. The cribs are usually filled with loose rock to keep 
 them in place, and in many cases are secured to the foundation by means 
 of bolts A laver of planking on the upper face of the dam makes it water- 
 tight. and if the spillway is over the crest of the dam it will be necessary to 
 plank the top of the cribs, and. in most cases, to provide an apron for the 
 water to fall on. The apron may be set on small cribs, or on timbers pro- 
 ‘ecting from the cribs of the dam itself. 
 
 J Abutments and Discharge Gates.— Abutments are structures at the ends of a 
 dam They mav be constructed from timber, masonry, or dry stonework. 
 If possible' abutments should have a curved outline, and should be so 
 placed that there is no possibility of the water overflowing them, or getting 
 behind them during floods. If the regular discharge from a dam takes place 
 from the main face, the gates may be arranged in connection with one of 
 the abutments, or by means of a tunnel and culvert through the dam. In 
 
DAMS. 
 
 155 
 
 either case, some structure should he constructed above the outlet so as to 
 prevent driftwood, brush, and other material from stopping the discharge 
 gates. When the discharge gates are placed at one side o£ the dam, they are 
 usually arranged outside of the regular abutment, between it and another 
 special abutment, the discharge being through a series of gates into a flume, 
 
 dlt Spillways P or Waste Ways— These are openings provided in a dam for the 
 discharge of water during floods or freshets, or tor the dischargeofaportioii 
 not being used at any time. The spillway may be over the crest of the dam, 
 or where the topography favors such a construction, the mam dam may be 
 of’ sufficient height to prevent water from ever passing its crest, the spillway 
 being arranged at another outlet over the lower dam. Waste ways, proper, 
 are openings through the dam, and are intended for the discharge of the 
 large quantities of water that come down during freshets or floods. In the 
 case of timber dams, the waste ways are usually surrounded by heavy cribs, 
 and have an area of from 40 to 50 sq. ft. each. There are two general 
 forms of construction employed for waste ways. One consists of a compara- 
 tively narrow opening in the dam, extending to a considerable depth (8 or 
 
 Fig. 18. 
 
 10 ft ) Water is allowed to discharge through this during flood time, but 
 when it is desired to stop, the flow planks are placed across the up-stream 
 face of the opening in such a manner as to close it. The opening, which is 
 usually not over 8 or 4 ft. wide, is provided with guides on the upper face of 
 the dam, and between which the planks are slid down, the individual pieces 
 of planking being at least 1 ft. longer than the opening that they are to 
 cover. The other device frequently used consists m providing the waste 
 way at one side of the regular spillway, with a crest 2 or 3 ft. lower than the 
 rejmlar spillway. The 
 and 4 or 
 
 spillway. 
 5 ft. abc 
 
 . e VV<X^ io 
 
 >ove there is arranged a parallel timber, the space between 
 ’ ■ ” flash boards. These are made from 
 
 the two being closed by what are called uasn ^ — ~rr 
 
 nieces of 2" or 3" plank, 6 or 8 in. wide. The planks are placed against both 
 timbers so as to close the space. The individual p anks are made long 
 enough so that they extend from 1 to 2 ft. above the upper timber, and 
 through the upper end of each plank is bored a hole through which a piece 
 of rope is passed and a knot tied in the end of the rope. These ropes are 
 secured by staples to the upper timber. When it becomes necessary to open 
 the waste way, men go under with peevies, cant hooks, or pmch bars, and 
 pry up the planks in such a way as to draw the longer end out of contact 
 with the lower timber, when the force of the water will immediately carry 
 the plank down the stream as far as the rope will allow it to go. After the 
 first plank has been loosened, the succeeding ones can be pulled up with 
 comparative ease, and two men can open a 25' or 30 section of waste way in 
 a very few minutes. The ropes keep the plank from being lost, and the 
 opening can be closed again by passing the plank down into the water to 
 one side of the opening and moving them into the current. Some skill is 
 required, both in opening and closing the waste ways. 
 
 Stone Dams— Where cement or lime is expensive and suitable rubble 
 stone can be obtained, dams are frequently constructed without the use of 
 mortar. The upper and lower faces of the dam should be of hammer- 
 dressed stone, carefully bonded, and the stones m the lower face of the dam 
 are sometimes anchored by means of bolts. The dam can be made^ter- 
 tight by means of a skin of planking on the upper face. In case water 
 should ever pass over the crest of such a dam, much of it would settle 
 through the openings in the stone into the interior of the dam, and this 
 would subject the stones in the lower portion of the face to a hydrostatic 
 
156 
 
 HYDRAULICS. 
 
 pressure, provided an opening was not made for the escape of such water. 
 For this reason, culverts or openings should be made through the lower 
 portion of the dam, to discharge any such water. When such dams as this 
 are constructed, the regular spillway is not placed over the face of the dam, 
 but at some other point, and usually over a timber dam. 
 
 Earth Dams.— Earth dams are used for reservoirs of moderate height. 
 They should be at least 10 ft. wide on top, and a height of more than 60 ft. is 
 unusual When the material of which the dam is composed is not water- 
 tight as for instance, gravel, sand, etc., it is sometimes necessary to con- 
 struct a puddle wall of clav in the center of the regular dam. This consists 
 of a narrow dam of clay mixed with a certain proportion of sand. The 
 nuddle wall should not be less than from 6 to 8 ft. thick at the top of the 
 dam and should be given a slight batter on each side. It is constructed 
 during the building of the dam, and should be protected from contact with 
 the water bv a considerable thickness of earth on the upper face. The 
 unner face of an earthen dam is frequently protected by means of plank or 
 a pavement of stone. The lower face is frequently protected by means of 
 cod or sod and willow trees. Sometimes earth dams are provided with a 
 masonrv core in place of the puddle wall, to render them water-tight. This 
 consists of a masonrv wall carried to an impervious stratum, and up through 
 the center of the dam. The masonry core should never be less than 2 or 3 ft. 
 thick at the top, and should be given a batter of at least 104 on each side. 
 At the regular water level, earthen dams are liable to have a small bench 
 or shelf formed, and on this account, during the construction, such a bench 
 or shelf is sometimes built into the earth dam. Fig. 18 shows a dam with a 
 masonry core, with the upper face covered with rubble and the lower face 
 covered with grass. . , , , 
 
 Debris Dams —These are dams or obstructions placed across the bed of 
 streams to hol’d back tailings from mines, and to prevent damage to the 
 vallevs below. They are made of stone, timber, or brush. No attempt is 
 made to render the debris dam water-tight, the only object being that it 
 should retard the flow of the stream and give it a greater breadth of dis- 
 charge co that the water naturally drops and deposits the sediment that- 
 it is carrying The sediment soon silts or fills up against the face of the 
 dam the ‘area above the dam becoming a flat expanse or plain over which 
 the water finds its way to the dam. When these darns are constructed of 
 stone the individual stones on the lower face and crest of the dam should 
 be so laro-e that the current will be unable to displace them, while the upper 
 face and core of the dam mav be composed of finer material. In case a 
 breach should occur in the debris dam, it will not necessarily endanger the 
 region farther down the stream, as is the case when a break occurs m a 
 water dam The reason for this is that the debris dam is not made water- 
 tight and hence there is never much pressure against it, or a large volume 
 of water held back that can rush suddenly down the stream should a break 
 occur The only result of the break would be that more or less of the gravel 
 behind the dam* would be washed through the breach. 
 
 wing Dams —Wing dams are used for turning streams from their courses, 
 so as to expose all or a portion of the bed for placer mining or other pur- 
 poses They are usually of a temporary nature, and are constructed of 
 brush and stones, light cribs filled with stones, and of large stones or timber 
 Sometimes the course of a stream is turned by an obstruction inade of sand 
 bao-s and a wing dam constructed behind this of frames of timoer, the inter- 
 vening space being filled with gravel or earth, and. in some cases, the timber 
 being covered with stone, and the surface nprapped so that if the flow ever 
 comes over the top of the structure it will not destroy it. 
 
 Masonrv Dams —When high masonry dams are to be employed they should 
 be designed bv a competent hydraulic engineer. Masonry dams are not, as 
 a rule used for hydraulic mining, owing to the fact that the length of time 
 during which the dam is required rarely warrants the expense ot the con- 
 struction of a masonry dam. 
 
 WATER-POWER. 
 
 The Theoretical Efficiency of the Water-Power.-The gross power of a fall of 
 
 water is the product of the weight of water discharged in a unit of time, and 
 the total head or difference in elevation of the surface of the water, above 
 and below the fall. The term head, used in connection with waterwheels, 
 
WATER-POWER. 
 
 157 
 
 is the difference in height between the surface of water in the penstock and 
 that in the tailrace, when the wheel is running. 
 
 If q = cubic feet of water discharged per minute, 
 
 W = weight of a cubic foot of water = 62.5 lb., 
 and H = total head in feet, 
 
 then WQH = gross power in foot-pounds per minute, 
 and = the horsepower. 
 
 Substituting the value for W, we have 
 
 = .00189 Q H. as the horsepower of a fall. 
 
 83 000 
 
 The total power can never be utilized by any form of motor, owing to the 
 fact that there is a loss of head, both at the entrance to, and exit from, the 
 wheel, and there are also losses of energy, due to friction of the -water m 
 passing through the wheel. The ratio of the power developed by the wheel 
 to the gross power of the fall, is the efficiency of the wheel. A head of water 
 can be made use of in any one of the following ways: 
 
 1 By its weight, as in the water balance, or overshot wheel. 
 i. By its pressure, as in the hydraulic engine, hydraulic presses, cranes, 
 etc., or in a turbine water wheel. , , , . . , , , 
 
 3. By its impulse, as in the undershot and impulse wheels, such as 
 
 Peltons, etc. 
 
 4. By a combination of the above. 
 
 The Horsepower of a Running Stream.— The gross horsepower, as seen above, is 
 
 H. P. = — .00189 Q if, 
 
 33,000 . ... 
 
 in which Q is the quantity in cubic feet per minute actually impinging on 
 the float or bucket, and if the theoretical head added to the velocity of the 
 
 stream, or 
 
 H 2 g 64.4’ 
 
 in which v is the velocity in feet per second. , 
 
 For example, if the floats of an undershot waterwheel were 2 ft. X 10 ft. .and 
 the stream had a velocity of 3 ft. per second, i. e., v = 3, we would have 
 
 H = = .139, and Q = 2 x 10 X 3 x 60 = 3,600 cu. ft. per min. 
 
 From this, H. P. = 3,600 X .139 X .00189 = .945 H. P., or a gross horse- 
 power for practically .05 sq. ft. of wheel surface; but, under ordinary circum- 
 stances, it would be impossible to attain more than 40^ of this, or practically 
 .02 horsepower per sq. ft. of surface, which would require 50 sq. ft. of float 
 
 surface to each horsepower furnished. 
 
 Current Motors.— A current motor fully utilizes the energy of a stream 
 only when it is so arranged that it can take all of the velocity out of 
 the water; that is, when the water leaves the floats or vanes with no 
 velocity. It is evident that in practice we can never even obtain a close 
 approximation to these results, and hence only a small fraction of the 
 energy of a running stream can be utilized by the current motor. Current 
 motors are frequently used to obtain small amounts of power from a large 
 stream, as, for instance, for pumping a limited amount of water for irrigation. 
 For this work, an ordinary undershot wheel having radial paddles is usually 
 employed. At one end of the wheel a series of small buckets are placed, and 
 so arranged that each bucket will dip up water at the bottom of the wheel 
 and discharge it into the launder, near the top of the wheel. The shape of 
 the buckets should be such that only the amount of water which the bucket 
 is capable of carrying to the launder will be dipped up, for if the bucket 
 is constantly slopping or pouring water as it ascends, a large amount 
 of useless work is performed in raising this extra water and then pouring it 
 out again, as only the portion that reaches the launder can be of any service. 
 Current motors are not practicable for furnishing large amounts of power. 
 
 UTILIZING THE POWER OF A WATERFALL. 
 
 The power of a waterfall may be utilized by a number of different styles 
 
 of motors, but each has certain advantages. 
 
 Breast and Undershot Wheels.— When the head is low (not over 5 or 6 ft ), 
 breast or undershot wheels are frequently employed. If these are properly 
 
158 
 
 HYDRAULICS . 
 
 proportioned, it is possible to realize from 25# to 50# of the theoretical power 
 of the fall, but the wheels are large and cumbersome compared with the duty 
 they perform, and not often installed at present, especially near manufac- | 
 turing centers. „ , A , . 
 
 Overshot Wheels.— For falls up to 40 or 50 ft., overshot wheels are very 
 commonly employed, and they have been used for even greater heads than 
 this. The overshot wheel derives its power both from the impulse of the 
 water entering the buckets, and from the weight of the water as it descends 
 on one side of the wheel in the buckets. The latter factor is by far the, more 
 important of the two. When properly proportioned, overshot wheels may 
 realize from 70# to 90# of the power of the waterfall, but they are large and 
 cumbersome compared with the power that they give, and are not often 
 installed except in isolated regions, where they are made from timber by 
 local mechanics. „ . 
 
 Impulse Wheels.— For heads varying from 60 ft. up, impulse wheels are 
 very largely used. These are also sometimes called hurdy gurdies, and are 
 usually of the Felton type, consisting of a wheel provided with buckets, so 
 arranged about its periphery that they receive an impinging jet of water and 
 turn it back upon itself, discharging it with practically no velocity, and con- 
 verting practically all the energy into useful work. The efficiency of these 
 wheels varies from 85# to 90# under favorable circumstances. This style of 
 wheel is especially adapted for very high heads and comparatively ^small 
 amounts of water. There are a number of instances where wheels are 
 operating under a head of as much as 2,000 ft. This style of impulse wheel i 
 is an American development; in Europe, a style of impulse turbine has been li 
 used to some extent, but has not found very much favor in the United States. 
 
 Turbines.— Turbines or reaction wheels are very largely employed, espe- 
 cially for moderate heads. When properly designed to fit the working 
 conditions, they can be used for heads varying from 4 or 5 ft. up to consider- 
 ably over 100 ft., and when properly placed are capable of utilizing the 
 entire head, a factor that gives them a decided advantage over any other 
 style of waterwheel. Turbines are capable of returning 85# to 90# of the 
 theoretical energy as useful power, and are largely used, especially where a 
 considerable volume of water at a low head, or a smaller volume at a 
 moderate head, can be obtained. 
 
 PUMP MACHINERY. 
 
 Pumps are employed for unwatering mines, handling water at placer 
 mines, irrigation, water-supply systems, boiler feeds, etc. 
 
 For un watering mines, two general systems of pumping are employed. 
 (1) The pump is placed in the mine and is operated by a motor on the sur- 
 face, the power being transmitted through a line of, moving rods. (2) Both 
 the motor and pump are placed in the mine, the motor being an engine 
 driven by steam, compressed air, hydraulic motor, or an electric motor. 
 
 Cornish Pumps.— Any method of operating pumps by rods is commonly 
 called a Cornish system. Formerly, the motor in the Cornish system con- 
 sisted of a steam engine placed over the shaft head, which operated the 
 pump by a direct line of rods. With this arrangement, there is great danger 
 of accident to the engine from the settling of the ground around the shaft, 
 or from fire in the shaft; also, the position of the motor renders access to the 
 shaft difficult. To overcome these objections, the engine is frequently 
 placed at one side of the shaft, and the rods operated by a bob; this has 
 become the common practice, and is generally called the Cornish rig. The 
 engine employed in the most modern plants is generally of the Corliss 
 type, and is provided with a governor to guard against the possibility of the 
 engine running away, in case the rods should break. 
 
 This svstem requires no steam line down the shaft, and is independent of 
 the depth of water .in the mine, so that the pump is not stopped by the 
 drowning of a mine, but the moving rods are a great inconvenience in the 
 shaft, and they absorb a great amount of power by friction. 
 
 Simple and Duplex Pumps.— In the simple pump, a steam cylinder is con- 
 nected directly to a water cylinder, and the steam valves are operated by 
 tappets. Such a pump is more or less dependent on inertia at certain points 
 of the stroke to insure the motion of the valves, hence will not start from 
 
P VMP M A C MINER Y. 
 
 159 
 
 anyplace, but is liable to become stalled at times. In the duplex pump, 
 two steam cylinders and two water cylinders are arranged side by side, 
 and the valves so placed that when one piston is at mid-stroke it throws 
 the steam valve for the other cylinder, etc. With this arrangement, the 
 pump will start from anv point, and can never be stalled tor lack of steam, 
 due to the position of the valves. Ordinarily, duplex pumps are to be pre- 
 ferred for mine work. , ... , . , 
 
 The packing for the water piston of a pump maybe either inside or outside. 
 Any form of packing that is inside the cylinder, either upon a moving piston 
 or surrounding the ram, 
 and so situated that any 
 wear will allow communi- 
 cation between the oppo- 
 site ends of the cylinder, 
 is called inside packing. 
 
 It may consist simply of 
 piston rings about the pis- 
 ton, as in the case of an 
 ordinary steam-engine pis- 
 ton G, Fig. 19, or stationary 
 rings may be employed 
 about the outside of a mov- 
 
 Fig. 19. 
 
 In^either case, the cylinder heads have to be removed before the condition 
 of the packing can be inspected, and any leak does not make itself visible. 
 
 When outside packing is employed, separate rams are used in opposite 
 ends of the cylinder, there being no internal communication between the 
 chambers in which the rams work. The rams are packed by ordinary 
 outside stuffingboxes and glands. The arrangement consists practically ot 
 two single-acting pumps arranged to work alternately, so that one is forcing 
 water while the other is drawing water. Fig. 20 shows a horizontal section 
 of a cylinder so arranged, together with the yoke rods that operate the ram 
 at the farther end of the cylinder. . , , . . , 
 
 As a rule, inside-packed pumps should be avoided m mines, on account of 
 the fact that acid or gritty waters are liable to cut the packing, and make the 
 pumps leak in a very short time. For dipping work m single stopes or 
 entries, small single or duplex outside-packed pumps may be employed. 
 It is generally best to operate such pumps by compressed air, for the exhaust 
 will then be beneficial to the mine air. If steam is employed, it is frequently 
 necessary to introduce a trap and remove entrailed water from the steam 
 before it enters the pump, and to dispose of the exhaust by piping it out or 
 condensing it. Such isolated steam pumps are about the most waste! ul form 
 of steam-driven motor in existence. 
 
 For sinking, center-packed single or duplex pumps are usually employed, 
 the duplex style being the better. For station work, where much water is to 
 be handled, large compound, or triple-expansion, condensing, duplex pump- 
 ing engines are employed. They may, or may not, be provided with cranks 
 and a flywheel. Engineers differ greatly upon this point, and, as a rule, for 
 very high lifts and great pressures, the flywheel is employed. 
 
 The main points in consideration are the first cost of the pump, and the 
 amount that will be saved by using the more expensive engine. The large 
 flywheel pumping engines are several times as expensive as the direct- 
 acting steam pumps, and the question is as to whether their greater efficiency 
 6 * * > ^ will more than coun- 
 
 terbalance the in- 
 creased outlay. Most 
 engineers favor fly- 
 wheel pumps for 
 handling large vol- 
 umes of water where 
 the work is approxi- 
 mately constant, and direct-acting pumps, without fly wheels or cranks, 
 for handling small amounts of water, or for very irregular service, owing 
 to the fact that if the flywheel pump is driven below its normal speed it does 
 not govern properly, nor work economically. Until recently, water was 
 removed from mines in lifts of about 300 to 350 ft., pumps being placed at 
 stations along the shaft. 
 
 While a series of station pumps are still employed, m some cases they are 
 
 Fig. 20. 
 
160 
 
 PUMP MACHINERY. 
 
 generallv intended to take care of water coming into the shaft, or workings 
 at or near their level, and are not employed for handling water in successive 
 stages or lifts. For handling the bulk of the water from the bottom of the 
 shaft large pumping engines are employed that frequently force the water 
 to the surface from depths of over 1,000 ft. These high-dutv pumping plants, 
 when near the shaft and operated by steam with a condenser, frequently 
 show a verv high efficienev. When air is employed to operate such a plant 
 a much higher efficienev can be obtained if the compressed air is heated 
 before using in the high-pressure cylinder, and during its passage from the 
 high-pressure to the low-pressure cylinder. This has been very successfully 
 accomplished bv means of a steam reheater, the small amount of steam 
 necessarv being ‘conveyed to the station in the small pipe, and entirely con- 
 densed in the reheater* from which it is trapped as water. . 
 
 The dutv of steam pumps is approximately as follows: For small-sized 
 steam pumps, the steam consumption is from 180 to 200 lb. per horsepower 
 uer hour when operating in the workings of a mine at some distance from 
 the boiler. For larger sizes of simple steam pumps, the consumption runs 
 from SO to 130 lb. of steam per horsepower per hour. Compound-condensing 
 numps, such as are commonly used as station pumps, consume from 40 to 
 70 lb. of steam per horsepower per hour. Tnple-expansion. condensing, 
 hi°h-class pumping engines consume from 24 to 26 lb. per hoi^epowerper 
 hemr The Cornish pump consumes varied amounts of steam in proportion 
 to the water delivered, depending largely on the friction of the geanng 
 bobs, rods, etc., but its efficiency is usually considerably below the best class 
 
 0f CTo n f wTtfr 1 Through Valves, Pipes end Pump 
 
 through the valves and passages of a pump should not exceed ^A) ft. per 
 minute and care should be taken to see that the passages are not too 
 abruntlV deflected. The flow of water through the discharge pipe should 
 not exceed 500 ft per minute, but for single-cylindered pumps it is usually 
 figured Id between 250 and 400 ft. per minute. In the case of very large 
 SSS ^reafer velocities maybe allowed. The suction pipe for the pump 
 Should be larger than the discharge pipe. Ordinarily, the suction pipe for a 
 D^ns^uld not exceed 250 ft. in length, and should not contain more than 
 two elbows. The following formula gives the diameter of the suction and 
 
 discharge pipes of a pump: 
 
 G = U. S. gallons per minute; _ 
 
 d' = diameter of suction pipe in inches: 
 
 d"= diameter of discharge pipe in inches; 
 
 I G 
 
 d' = -4.95^^; 
 
 <*"= 495 \'-^- . . . 
 
 = velocity of water in feet per minute in the suction pipe - from 
 
 50 v" to 75 
 
 v" = velocity of water in feet per minute in the discharge pipe. 
 
 RATIO OF STEAM AND WATER CYLINDERS IN A 
 DIRECT-ACTING PUMP. 
 
 A = area of steam cylinder; 
 
 D = diameter of steam cylinder: 
 
 P = steam pressure in pounds per 
 
 p = pressure per’ square inch, corresponding to the head H 
 ** ^ wnrk done in nump cylinder 
 
 H = head of water = 2.309 p; 
 a = area of pump cylinder; 
 d = diameter of pump cylinder; 
 
 .433 H\ 
 
 E = 
 
 A = 
 
 a = 
 
 pressure per squar. 
 
 efficiency of pump — work done in steam cylinder 
 ap 
 
 D = d 
 
 EP ’ 
 EAP . 
 
 P 
 
 I P . 
 
 D 
 
 P = 
 
 
 P = 
 
 iE P 
 
 M~V' 
 
 op. 
 EA’ 
 EAP . 
 
 a 
 
 JL_ 
 
 EP 
 
 .433 H 
 
 EP ’ 
 
 H= 2.309 E P X — 
 
CAPACITY AND HORSEPOWER OF PUMPS. 
 
 161 
 
 If E = 75 $, then H = 1.732 PX^. 
 
 E is commonly taken at from .7 to .8 for ordinary direct-acting pumps. 
 For the highest class of pumping engines it may amount to .9. The steam 
 pressure P is the mean effective pressure, according to the indicator dia- 
 gram; the pressure p is the mean total pressure acting on the pump plunger 
 or piston, including the suction, as would be shown by the indicator dia- 
 gram of the water cylinder. The pressure on the pump cylinder is frequently 
 much greater than that due to the height of the lift, on account of the 
 friction in the valves and passages, which increases rapidly with the velocity 
 of the flow. . . 
 
 Piston Speed of Pumps.— For small pumps, it is customary to assume a 
 speed of 100 ft. per minute, but, in the case of very small short-stroke pumps, 
 this is too high, owing to the fact that the rapid reverses make the flow 
 through the valves and change in the direction of the current too frequent. 
 When the stroke of the pump is somewhat longer (18 in. or more), higher 
 speeds can be employed, and in the case of large pumping engines having 
 long strokes, speeds of as much as 200 to 250 ft. per minute are successfully 
 used without jar or hammer. 
 
 Boiler Feed-Pumps.— In practice, it has been shown that a piston speed 
 greater than 100 ft. per minute results in excessive wear and tear on a 
 boiler feed-pump, especially when the water is warm. This is due to the 
 fact that vapor forms in the' cvlinders t and results in a water hammer. In 
 determining the proper size of a pump for feeding a steam boiler, not only 
 the steam employed in running the engine, but that necessary for the 
 pumps, heating system, etc. must be taken into consideration. 
 
 THEORETICAL CAPACITY OF PUMPS AND THE HORSEPOWER 
 REQUIRED TO RAISE WATER. 
 
 Q 
 
 G 
 
 G> 
 
 d 
 
 l 
 
 N 
 
 v 
 
 W 
 
 P 
 
 £ 
 
 Let Q = cubic feet of water per minute; 
 
 = U. S. gallons per minute; 
 
 = U. S. gallons per hour; 
 
 = diameter of cylinder in inches; 
 
 = stroke of piston in inches; 
 
 = number single strokes per minute; 
 
 = speed of piston in feet per minute; 
 
 == weight moved in pounds per minute; 
 
 = pressure in pounds per square feet = 62.5 X H; 
 
 = pressure in pounds per square inch = .433 X H • 
 
 = height of lift in feet; 
 
 H. P. = horsepower. 
 
 Then, Q = \ ■ ^ ~ = .0004545 Nd- 1. 
 
 O = "■ = .0034 Nd- 1. G> = .204 Nd"- 1. 
 
 The diameter of piston required for a given capacity per minute will be 
 
 d = 46 - 9 Vj n = 17 ' 15 \fe ord - 13 - 54 V* = 4 - 95 Vf- 
 
 The actual capacity of a pump will vary from 60$ to 95$ of the theoretical 
 capacity, depending on the tightness of the piston, valves, suction pipe, etc. 
 
 QP QHX 144 X .433 = QH = Gp 
 33,000 33,000 529.2 1,714.5* 
 
 The actual horsepower required will be considerably greater than the 
 theoretical, on account of the friction in the pump; hence, at least 20$ should 
 be added to the power for friction and usually about 50$ more is added to 
 cover leaks, etc., so that the actual horsepower required by the pump is 
 about 70$ more than the theoretical. 
 
 Example 1.— If it is desired to find the size of a pump that will throw 
 30 gal. of water per minute up 125 ft., from the bottom of a pit or prospect 
 shaft to the station pump at the main shaft, it may be accomplished as 
 follows: 
 
 An allowance of probably 25$ should be made with a small pump of this 
 character, to overcome slippage or leaking through the valves, past the piston, 
 
162 
 
 PUMP MACHINERY. 
 
 etc and hence we will call the total amount of water to he handled 40 gal 
 ner mfnute? The formula for the diameter of piston is 
 
 F G 
 
 d = 4.95 yj-. 
 
 Assuming that v = 100 ft. per minute, we have 
 
 d = 4.95]/ .4 = 4.95 X .63 = 3.13. 
 
 Tn -nractice a 31" pump would probably be employed. 
 
 Fx?mple 2 -If it is desired to find the approximate horsepower necessary 
 to THtwSl /per minute in the above example, without determimng the size 
 of the pump, it can be done as follows: 
 
 w p __ Q X P = 30 X -433 X 125 __ ^ or prac ti C ally 1 H. P. 
 
 H,P, ~ 1,714.5 1,714-5 
 
 Tn order to cover leakage through valves, friction, etc., an addition of at 
 leas? ^^should°be r nmde to a very" small pump like this, and so we would 
 
 co “ Auction —Theoretically, a perfect pump will raise water to a 
 , Pepth f f ^bucnon ^ sea level; but, owing to the fact that a perfect 
 
 ti^Son it is never 'possible to reach this theoretical limit, and, in practice, 
 wlfe^tKter il°coW W Wa“ ter wlter“ 
 
 pn£F£ir« e o d r ai 
 
 with a clack valve in the plunger, the amount lifted may actually exceed 
 
 more water through the valve than would pass through it on account of the 
 ^ce passed through This increases as the speed or number of strokes 
 
 inc |“ v»lves.-As a rule, a large number of small valveshaving a compar- 
 ativelvsmall opening are preferable to a small number of large valves with 
 a Jreatironening and most modern pumps are built upon these lines. A 
 small valve P represents a proportionately larger surface of discharge with the 
 same lift than the large valve, hence whatever the total area o[ the \ah e 
 onening its full contents can be discharged with less lilt througn 
 numerous s&all valves than through one large valve. Cornish pumps 
 
 gel power' V Pumps —Where comparatively small amounts of water are to be 
 handfed, and P power is available, belt-driven power pumps are very much 
 
 "‘'Eleohic^y DHven^e^P^s^bere water is to be delivered from 
 isolated wordings to the sumps for the large station pumps, electrically 
 driven power pumps are far more efficient than steam pumps. In some 
 cases It Fs°mobSy best to equip the entire mine with electnc pumps both 
 ?n the isolated workings and at the stations, on account of the fact that they 
 can^e^rivenl^ 0 ^ high-class compound-condensing ^n^^e °n the ^irface, 
 directly connected to a generator, and furnishing electricity through con- 
 
 ^ U The S f^al 1 efflciency?f I a^series of small, electric pumps that aggregate a 
 sufficient amount of power to enable this arrangement to be used, is very 
 sufficient amount y e fficiencv of a number of small isolated steam or 
 
 ra^uresfe^-ai^ wmps^Sroduced C lnto a the workings. With compound- 
 condensing engines upon the surface, operating electric pumps underground 
 ?he st^amconsumption per pump horsepower per hour for the smaller 
 2“ e would onlvbe about 40 lb. per horsepower per hour; for medium-sized 
 electric° pump?, about 30 lb. of steam per hour, and larger sizes from 20 to 
 30 lb. per horsepower per hour. It will be seen from these figures that for 
 
PUMP AND WATER MEMORANDA, 
 
 163 
 
 pumping from isolated portions of the mine the electric pump is much more 
 efficient than the steam pump, and owing to the fact that the current can 
 frequently be obtained from the lines operating the underground haulage 
 system, furnishing light, etc., it is evident that this system of pumping has 
 a great future before it in connection with mining. 
 
 The following table gives the gallons per minute delivered from various 
 sized pumps operating at different piston speeds: 
 
 Pump and Water Memoranda. 
 
 Gallons per Minute Delivered at Indicated Piston Speeds. 
 
 500 
 
 ^occooooooooqoooqqqqqqqqooqooqooqooq 
 
 dHcodd^ddxw^ddddcccocooocooQOOOOoqto^dqqqoq 
 
 (NOOXOHCO<OO^O^i0^^iQ!0 01COOO^H05C5 03QNiCCR'tHOOcOiO!CiO 
 iH<reBl5-a6©rHCOH<CC>GCOCMH<COa5rH^lHC5CM 1 005 CM lO 00 CM COcqcO IH rH IH 
 r-T rH tH i-T tH t-T CM cm" CM CM~ Cm" CO CO CO CO tjT ■ ■ rf lO UO lO CO CD CO I> t> 00 rH 
 
 400 
 
 coM,ooooooqqqqqqqqqqqqqqqqqqqqqqqqqqq 
 
 cotdi>rHodo6ooo6^<^cM'eocM‘aiioodooo6TjHoirHCMOco^co^oo^cM3oqo 
 
 H(O^COOOOOOH^t-(NI>COONiOiOiOiOt-OM^CMt-^HOCOqOC5q(MO 
 rHCM-'tfiOCO00O5©rHCOH<COlHa5tHC0iOlHO5tHHiCOa5rHH<lH OS_{N »0 00 CM O 
 
 THr^THrHiHrHr^CMCMCMCMCMCOCOCOCOTh^^Tt lO iOlO CO CO oT 
 
 350 
 
 coHOOoooooooqqqqqqqqqqqqqqqqqqqqqqoq 
 ^l>OS0dl>^00OC0^CMJt^050Q^0Q<^cdrH(»C0a5CMC0OTj?oil>rHI>cdcd05CMiQ 
 HiO^NibHOOOHCOlOOOClNCloOiOCOHOOOHCOlOOOCMNlNXlOCMH^ 
 tHCMCOiOCOlHGC05©tHCMH< lO l> O0C5 CM ^CO^L^O C^^^OOrH CO CO OOrH t^|> CM 
 v r-TrH rH rH HH tH CM CM CM CM" - CM~CO COCO CO COrJHrtfH 3 '' TfTiOuOlO 00 
 
 300 
 
 cMoooocqoooocqoooooooooooooooooooooooo 
 CM05©cdc0rHt>Oaic0TfiCMl0^C}rH05C0C0airHcic0Tj5©CMC01>l>C0C0cic0C0O 
 H'^HtKO'^HOOOOOOOOOlNTHOOHCOi-iiOCOONlO'^COCOCO-^COOOHiOOiO 
 tH tH CO ^ lO CO CO IH 00 05 tH CM CO cqiH_05_© CM^CO^uO r^<prH_cq^ipi> 05 rH CO GO © 
 
 rH r-T rH r-T rH rH rH CM CM~ C<f CM CM~ of CO CO CO~ CO CO rjT rf rf rf IH 
 
 275 
 
 cm os q pppppppppppppppppppppppppppppqpo 
 
 rHT^rHOrHH^T^OrHOOrHoicOCMr^OO^cdcOcdlCO^oilOlCrHlOCMTjicdo'jrHlOodcO 
 H'^ONOOONiOCOHHOHCMCOiOOOHiOOHCiiOOIOit'iOHCOCOMiOCOQOCO 
 1 ^riHCl^'Ct , i0CDI>0005OHCM CO CO 00 © tH CO lO© CO © CM r* COGO © 0M ^ ^ 
 
 rHrHrHrHrHrHrHrHCMCMCMCMCMCMCOCOCOCOCOrfTlI'TjrcO 
 
 o 
 
 m 
 
 CVI 
 
 m 
 
 CM 
 
 CSI 
 
 cMoooqpppppppoppppppppppppppppppoppoop 
 
 ddHcciONHd^'coNdHd^Hdci’t^o6c>io>o'ddooo6ciiodcoo6dio 
 
 HHOiOiOCOCOONLOCOCMCMIMCMCOHCOOCMiOOHCSiOHN^iMoOQOr'OON 
 tHCMCOH<lO»OCOlHa005 © tH CM CO tMOIH OOO^rH CM T^COIH O^rH CO ^COGO© 00 
 r-T r-T r-T r-T r-t r-T r-T rH t-T cl CMCM~ CM CM CM CO CO CO CO CO lO 
 
 ~Cm!h CO O O O C> O O © © OO pppppppppppppppppppppp 
 oicdcM’tHOrHOOOCOOOCOH^ododcMrH^oiH^rHCO'asOCOiOOOCOOiCrHTfOCMQO 
 COOOCOCOCOOOiOHCO'O^NHHHHMCOiOt'OCOCDOHOiOrtNTfHOr^OO 
 HNCOCO^iOiCOt^QOOOH CM CO CO O CM CO p^CO 00O rH CO CM 
 
 tH* rH rH HHHHHH C^CM" CM CM' CM" CM" CM'cO CO CO CO lC 
 
 O 
 
 o 
 
 CM 
 
 CMcoHOoqqoooqqqqooqqqqqqqqqqqqqqqqqqq 
 odcMCOrHH5cOlOOaiCMOrHcdc001>05lOl005I>-'05cdcdOG(6cM‘ododH5cM’cdcMrJHO 
 COt'(NOC5 , ctlOiO(NaiCOWHOOOt'l''Nt'OOC5HCOCOCOCMiOO^C5HOCOO 
 iH CM CM CO Tft-iO rOCOl'-OOOOOH CM CO ^lO l^OOO O CM CO r^CO O rH CM IH 
 
 rH rH rH rH rH rH rH r-T rH CM*~ CM" CM" cm" CM" CM" Cm" CO CO" -H 
 
 175 
 
 Hcocooopppppppppooop o o p p p p o o o o o o q o o o o 
 
 l> 00 H 3 H 3 O I> CM © CM IH CO GO H 3 H? I'.' Tt? H 3 O0 CO l> rH 05 rH id Tf 3 CO lO H 3 CO* GO H 3 CC > CM* 
 C!OOMOOiCOiOHN'rfHCOCDHMHOOCiOOH(N^COOOH^>HiOH 
 iHrHCMCOCO^'^iOiOCOt^t^OOOS O rH CM CO CO lO CO GO O O tH CO t^GOrH 
 
 rHrHrHrHrHrHrHrHrHrHCMCMCMCMCMCMCMTtr 
 
 150 
 
 HiOHOOoooqppppppppppppppppppppqpqppp 
 COHiONCodcid^lNCicDddiOHdHCDrtiuidNN'ocdNcicOCOHONOOiQ 
 Cn-OOOiOCliOO^O^OiCHl-'^OCOLOCOHOOC^NCOCOCOOQOaiHfNHlN 
 rHCMCMCOCOCOTfHfUOCOCOXHGOGOCi O rHC^ C^ CO r)H_ipcpi>_<pp_0 CM CO TP lO 
 
 rH rH rH rH tHtH rH rH rH rH rH CM CM CM CM CO 
 
 125 
 
 rH'HCiCOOOOOOOppppppppppppppppppppOOOpp 
 
 iodiOHo6^cdoNcodcoddcii>i6HNddd(No6inicoi^’HcoidHc:dco 
 
 CMt}fl>ClCOHiOQO''lCOHCDHOHNCOCiCOCOOI>H^qQONCOiOH^CO'^CO 
 rH rH CM CM CM CO CO tP tP lO lO CO CO t> I> GO 05 © © H CM CO CO ' H^uOCO I> 00 p © 05 
 
 HHrHHHHHHHHH rH CM CM 
 
 001 
 
 Hcot>cooooooopopppppppppppppppppoooppp 
 HrcdcdiOCM^CMOOrHiOrHCOGOOH^OGOCOOTjrOGOGOOH^rHOSoiCMCOCOrHCMO 
 HMCOO^HOCOCOOCOCOOiOO^OOCOO^OiOHOO^HN^INONiOCOiO 
 rHtHrHCMCMCMCMCOCOTP^TpLOiCCOCOI>COa00505 © tH^CM COCO rj^ipco CO 
 
 HHHHHHHHHM 
 
 Contents for 
 1 Ft. Length 
 
 Gallons. 
 
 OOCMCMQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 
 O CO IH OI © 05 tP 05 lO H GO lO CM © GO IH CO lO iO »0 CO Ih GCt O H O OOOOOOOOO 
 HcOCDiOCMcOCMC50HHOOOOOC5COO>NH05f005NGOOHH020(NCOeOH(NO 
 OHCocoo^t^05Ncoo5cocoo^qMcoMoq^qqHoo^HH^qqHiocoiO 
 
 * * ’ ' rH rH rH rH CM CM CM* CO CO H 3 H 3 H 3 lO lO CO CO IH IH 00 05 05 © rH rH CM* CO CO H 3 lO CO CO 
 
 THrHrHrHrHrHrHrHrHCM 
 
 Cubic 
 
 Feet. 
 
 8Ss!eS®SeSs!3SaS383SSS88gg8S88SRg8S8a 
 
 OCM'HQ0C005C0C0OHtH05'H05'HOC0CMG0i0CM05C0TtrCMrH05G0lHtHCCC0C0lHG0'H 
 
 OOOOHHWCMCOCOM'T^iO®qi>l>OOqqpH(NCOMrjJiqqi>OOqpHH 
 
 rH rH rH rH rH rH rH rH rH tH rH CM CM CO 
 
 Area. 
 Sq. In. 
 
 .7854 
 
 3.1416 
 
 7.0686 
 
 12.5660 
 
 19.6350 
 
 28.2740 
 
 33.1830 
 
 33.4850 
 
 44.1790 
 
 50.2650 
 
 56.7450 
 
 63.6170 
 
 70.8820 
 
 78.5400 
 
 86.5900 
 
 95.0330 
 
 103.8700 
 
 113.1000 
 
 122.7200 
 
 132.7300 
 
 143.1400 
 
 153.9400 
 
 165.1300 
 
 176.7100 
 
 188.6900 
 
 201.0600 
 
 213.8200 
 
 226.9800 
 
 240.5300 
 254.4700 
 268.8000 
 
 283.5300 
 298.6500 
 314.1600 
 452.3900 
 
 | japmi^o 
 j jaioureia 
 
 ^ ^ i v Hst^; Her^: -Isrt; Her^ Hcr^; He>t; 2|cr^ Hsr^ v 
 
 ^ i; ^ ^ i ^ — iloiir -le.5- -taOOHH(M(NCOCOH'ctMOlCCOCDNHOOX05C50H 
 
 HCMM^iC)CDCOIh1HOOG005C5HHHHHHHHHHHHHHHHHHHHCMN 
 
 1 gal. = 231 cu. in. = .13368 cu. ft. 1 gal. of water at 39.2° = 8.33888 lb. 
 1 cu. ft. of water = 7.48052 gal., and weighs 62.423 lb. 
 
164 
 
 PUMP MA C II IS EE Y. 
 
 MISCELLANEOUS FORMS OF WATER ELEVATORS. 
 
 Jet Pump. — In this form, the energy of the jet of water is utilized for 
 raising a larger volume through a small distance, or a mixture of w ater a 
 solid material through a short distance. 
 
 Vacuum Pump.—' The pulsometer, which is the most important representa- 
 tive of this class, consists of two chambers in a large casting, with smtable 
 automatic valves arranged at the top and bottom of the chambers, ^eain is 
 fnttcS into o^of the chambers, then the valve at the top closed. This 
 steam will condense, forming a vacuum that draws tv ater from the suction 
 into^th” chamber, when tie chamber is filled with water, steam is again 
 introduced and forces the water out through the discharge pipe. The 
 oneratioTis then repeated, more water being drawn in by the condensation 
 
 thp steam The two chambers work alternately, one being engaged m 
 driving wSer in while the other forces it out. The total steam efeciency 
 of this form of pump is s ma ll, though it may actually be abo\e that of -ma 
 steam nummseumloyed in isolated portions of a mine. The advantages are 
 that th^ ^pSLp Assesses no intricate mechanism,no reciprocating , parts, 
 reauires no lubrication, and is not injured by gntty or acid materials. On 
 this account it may be emploved for pumping water in concentration lvork s, 
 coal-washing plants, and similar places where the water is liable to contain 
 
 gnt A i r-Lift^P u m p s. — By introducing compressed air at 
 submerged in anv liquid, the air in the pipe rises as bubbles, 
 the^iecific gravity of the fluid in the pipe. This causes the fluid m the pipe 
 to rise above the' level of that surrounding the pipe. The difference m 
 soecific gravitv can never be great, and hence the fluid can never be 
 llevated to anv considerable height withput having the lower endunmersed 
 to a correspondingly^ great depth. On this account it is frequently necessary 
 to drill a well considerably below the water-bearing strata, so as to obtain 
 the proper ratio between the submerged portion of the pipe 
 to which the water is to be lifted. Some advantages of this form of pump 
 are that there are no moving parts, no lubrication is required, and gntty 
 material does not interfere with the operation. If the pump is constructed 
 of suitable material, it may be employed for handling acids or solutions in 
 electrolvtic or chemical works. This style of pump is also quite extensiy ely 
 employed fo? pumping water from Artesian wells. It has not been successful 
 as a mine pump, owing to the ratio between the part immersed and the lift. 
 
 Centrifugal Pumps.-The height of lift depends on the tangential velocity 
 of the revolving disk of pump and the quantity of water discharged, and is 
 proportional to the area of the discharge orifices at the circumference of the 
 disk The most efficient total lift for the centrifugal pump is. approximately , 
 17 ft' and for small lifts the centrifugal pump is much more efficient than 
 anv stvle of piston pump. For a given lift, the total efficiency of a centrif- 
 ugal pump increases with the size of the pump. Centrifugal pumps are 
 always designated bv the size of their outlet, as. for instance, a 2" or 4 pump 
 meaning with a 2"' or 4" discharge pipe. Centrifugal pumps are not at 
 all effective for dealing with great heads, and hence have never come 
 into competition with piston pumps for this class of work. For lifting 
 large volumes of water against a low head, as in irrigation or drainage 
 
 problems, they are remarkably efficient. Under th ^ m( ^ ' ■^hait is" 
 
 stances the efficiency of the centrifugal pump may be practically /(K. that is, 
 the^ pump ^ma/v do an amount of work upon the water that is theoretically 
 equal to 7(K of the power furnished to the pump, Pumping engines work- 
 ing against high heads, and operated by the most improved class of engines, 
 may attain an efficiency of practically 85ft. 
 
 Centrifueal Pump as a Dredge.— When dredging is done by means of centrif- 
 ugal pumps, a greater amount of power is necessary, and the pump 
 be run at a greater speed than when pumping water, owing to the i fact ; that 
 the fluid being handled has a greater density than water When dealing 
 with fine sand, as much as 50 ft of the bulk of the material handled may be 
 sand, though, as a rule, the amount of solid material m the water dredged 
 only runs from 30# to 35# of the total. n , . 
 
 Water Buckets.— Where only a limited amount of water collects m the 
 mine workings it is frequentlv removed by means of a special water bucket 
 ^r ^ater ca? during the houre that the hoisting engine would otherwise be 
 i^e Where very large amounts of water are to be removed, it has also 
 
SINKING PUMPS. 
 
 165 
 
 been found economical to remove them by means of special water buckets. 
 This is especially true in the case of deep shafts. 
 
 One of the best illustrations of this class of work is the Gilberton water 
 shaft, which has been equipped at the Gilberton Colliery of the Phila- 
 delphia and Reading Coal & Iron Co. The collieries draining to this shaft 
 require the removal of 6,000,000 gal. of water per 24 hours during the wet 
 season, and this has to be lifted from a depth of 1,100 ft. In order to accom- 
 plish the work by means of steam pumps, it required a number of pump 
 stations in different parts of the mine, each of which had to be attended by 
 a pumpman, and a large number of steam lines were required in the mine. 
 In order to remove the danger of fire caused by these steam lines, and to 
 dispense with the large amount of labor otherwise necessary, it was decided to 
 hoist the water, and a shaft 22 ft. x 26 ft. 8 in. outside of timbers, was sunk. 
 This shaft contains two compartments 7 ft. X 7 ft., in which the water 
 buckets are operated, and two compartments 7 ft. X 11 ft. 8 in. that are 
 utilized for cages to lower men, timber, and other supplies. The water 
 tanks employed in the special water compartments are 5 ft. 6 in. in diameter, 
 and 14 ft. long. They are provided with a special device sliding on regular 
 cage guides, and empty themselves automatically at the surface by means 
 of a trip or sliding valve. Two pairs of direct-acting hoisting engines, with 
 45" X 60" cylinders, operating drums 14 ft. 8 in. in diameter by 15. ft. face, 
 are employed. These operate the water buckets in cages by means of 2" 
 crucible steel ropes, at 50 revolutions per minute, which is equivalent to a 
 piston speed of 500 ft. per minute. The drums will hoist two tanks of 2,400 
 gal. per minute. This gives an output of 7,000,000 gal. per 24 hours. By 
 slightly increasing the speed of the engine this amount can be increased 10#, 
 which is 25# in excess of the calculated maximum demand on the shaft. 
 The cages in the cage compartments are so arranged that they can be discon- 
 nected, and water buckets substituted for them. This would be a total 
 output of over 14,000,000 gal. per 24 hours at the normal speed of the 
 engine. One great advantage of this style of pumping plant is that there is 
 absolutely no fear of drowning the pumps. 
 
 Some years ago the Hamilton iron mine, in Michigan, was drowned by 
 a sudden inrush of water that drove the pumpmen from the pumps. In 
 order to remove this large volume of water, special bailing buckets were 
 substituted for the ordinary mine skips. These bailing buckets ran on the 
 inclined skip road, and un watered the mine in a remarkably short time. 
 
 Sinking Pumps.— Sinking pumps may be either single or duplex in their 
 action, and may be inside or outside packed. Outside-packed single-acting 
 pumps are in many ways preferable, owing to the fact that they are less 
 liable to get out of order. One requisite of any sinking pump is that it should 
 have as few exposed parts as possible, and that these parts should be so placed 
 that they will be protected from injury by blasting to as great an extent as 
 possible. Sinking pumps are usually provided with a telescopic section in the 
 suction pipe, and sometimes also in the discharge pipe, so that they can be 
 moved down several feet without having to break the joints of the piping. 
 
 Pumps for Acid Waters.— Where mine waters are acid in their nature, brass 
 or brass-lined pumps are usually employed, 
 and in some cases even wooden pumps 
 have been used, as, for instance, in the 
 Swedish copper mines, though this prac- 
 tice is disappearing in favor of the use of 
 brass or copper linings. The pipes for such 
 pumps should be of brass or copper tubing, 
 or should be lined with some substance 
 that will not be affected by the acid of the 
 water. Sometimes wooden linings are em- 
 ployed, placed as shown in Figs. 21 and 22, 
 
 Fig. 21 being a section of the pipe with the 
 lining complete, and Fig. 22 a cross-section 
 of one of the individual boards used in the 
 lining. These are usually made of pine 
 about | in. thick, and are grooved on each 
 end as shown. They are sprung in so as to complete a circle on the inside 
 of the pipe, and then long, thin, wooden keys driven into the grooves. 
 When the water is allowed to go into the pipes, the linings swell and make 
 all joints perfectly tight. Elbows and other crooked sections are lined with 
 sheet lead beaten in with a mallet, 
 
 Fig. 21. 
 
166 
 
 FUELS. 
 
 FUELS. 
 
 The value of any fuel is measured by the number of heat units that its 
 combustion will generate, a unit of heat being the amount required to heat 
 1 lb of water 1° F. The fuels used in generating steam are composed mainly 
 of carbon and hydrogen, ash, and moisture, with sometimes small quantities 
 of other substances not materially affecting their value. 
 
 Combustible is that portion which will burn; the ash or residue vanes 
 from 2$ to 36$ in different fuels. , , 
 
 The following table gives, for the more common combustibles, the. air 
 reauired for complete combustion, the temperature with different proportions 
 of air the theoretical value, and the highest attainable value under a steam 
 boiler assuming that the gases pass off at 320°, the temperature of steam at 
 75 lb pressure, and the incoming draft to be at 60°; also that with chimney 
 draft twice and with blast only the theoretical amount of air is required for 
 combustion. 
 
 Table of Combustibles. 
 
 Kind of 
 Combustible. 
 
 ft 3 
 < & 
 
 oft 
 ft ft 
 
 t- 1 3 
 
 |I 
 
 o o 
 ft 
 
 Temperature of 
 Combustion. 
 
 i Highest 
 
 Theoretical Attainable 
 
 ;ite.... J 
 
 Hydrogen .... 
 
 Petroleum . .. 
 
 ( Charcoal 
 Carbon -< Coke 
 
 (Anthracite 
 Coal, Cumberland, 
 
 Coal, Coking bituminous 
 
 Coal, Cannel 
 
 Coal, Lignite 
 
 Peat, Kiln dried 
 
 Peat, Air dried, 25$ water 
 
 Wood, Kiln dried 
 
 Wood, Air dried, 20$ water 
 
 36.00 5,750 
 15.43 5,050 
 
 | 
 
 12.13,4,580 
 
 . 
 
 O Sh 
 
 S3 
 
 So 
 
 ft >> 
 ^ ft 
 
 ft ft 
 ft 3 
 
 © . 
 
 0”-\ 
 
 ©<1 
 
 in . 
 
 © >> 
 
 3ft 
 
 ft^ 
 73 o 
 
 3,860 
 
 3,515 
 
 3,215 
 
 aj f-i 
 
 ft 
 
 ©v* 
 
 - <X> r— ; 
 
 O ft 
 
 © t-' 
 
 ft- 
 © O 
 
 3ft 
 
 ft ft 
 ^ ft 
 © 3 
 
 ft o 
 ^O 
 
 2.8601,940 
 
 2,710;1,850 
 
 Value. 
 
 3®© 
 
 f-l 'ft Q 
 
 Sb 
 
 c3 CO 
 
 ■SSS 
 
 CO Q. O 
 
 •So « 
 
 f 
 
 62,032 
 
 21,000 
 
 2,440 1,650 14,500 
 
 12.06,4,900 3,360 
 11.73 5,140 3,520 
 11.80 4,850 3,330 
 9.30 4,600 3,210 
 7.6814,47013,140 
 5.76 4,000 2,820 
 6.00 4,08012,910 
 4.80 3, 700 ; 2, 607 1 2, 100 j 1 ,490 
 
 2,550|l,730 
 2,680,1,810 
 2,5401,720 
 2,490:1,670 
 2,4201,660 
 2,2401,550 
 2,260; 1,530 
 
 c5 0 © 
 
 ft^ft 
 
 ■ga 
 
 o 
 
 o 
 
 ^ C l— t 
 
 £|£ rH 
 
 °ft ft 
 
 ,dftft 
 
 - © > 
 
 u C 
 © o3 
 
 64.20 
 
 21.74 
 
 15.00 
 
 Value Un- 
 der Boiler. 
 
 o3 
 
 Ut 
 
 ft 
 
 © 
 
 a 
 
 3 
 
 ft 
 
 o 
 
 ft 
 
 15,370 15.90 
 15,837 16.00 
 15,080 15.60 
 11,745! 12.15 
 
 9,660 
 7,000 
 7,245 
 5,600 | 
 
 10.00 
 
 7.25 
 
 7.50 
 
 5.80 
 
 18.55 
 
 13.30 
 
 14.28 
 
 14.45 
 
 14.01 
 
 10.78 
 
 8.92 
 
 6.41 
 
 6.64 
 
 4.08 
 
 19.90 
 
 14.14 
 
 15.06 
 
 15.19 
 
 14.76 
 
 11.46 
 
 9.42 
 
 6.78 
 
 7.02 
 
 4.39 
 
 The effective value of all kinds of wood per pound when dry, is substan- 
 tially the same. This is usually estimated at .4, the value of the same weight 
 of coal. The following are the weights and comparative values of different 
 woods by the cord: 
 
 Wood. 
 
 Hickory (shell bark) 
 Hickory (red heart) 
 
 White oak 
 
 Red oak.. 
 
 Spruce 
 
 New Jersey pine 
 
 Weight. 
 
 4,469 
 
 3,705 
 
 3,821 
 
 3,254 
 
 2,325 
 
 2.137 
 
 Wood. 
 
 Beech 
 
 Hard maple.... 
 Southern pine. 
 Virginia pine . 
 Yellow i>ine.. ... 
 White pine 
 
 Weight. 
 
 3,126 
 
 2,878 
 
 3,375 
 
 2,680 
 
 1,904 
 
 1,868 
 
SLACK. 
 
 167 
 
 Much is said nowadays about the wonderful saving that is to be expected 
 from the use of petroleum for fuel. This is all a myth, and a moment’s 
 attention to facts is sufficient to convince any one that no such possibility 
 exists. Petroleum has a heating capacity, when fully burned, equal to from 
 21,000 to 22,000 B. T. U. per lb., or, say, 50$ more than coal. But, owing to 
 the ability to burn it with less losses, it has been found, through extended 
 experiments by the pipe lines, that under the same boilers, and doing the 
 same work, 1 lb. of petroleum is equal to 1.8 lb. of coal. The experiments on 
 locomotives in Russia have shown practically the same value, or 1.77. Now, 
 a gallon of petroleum weighs 6.7 lb. (though the standard buying and selling 
 weight is 6.5 lb.), and therefore an actual gallon of petroleum is equivalent 
 under a boiler to 12 lb. of coal, and 190 standard gallons are equal to a gross 
 ton of coal. It is very easy with these data to determine the relative cost. 
 At the wells, if the oil is worth, say, 2 cents a gallon, the cost is equivalent 
 to $3.80 per ton for coal at the same place, while at, say, 3 cents per gallon, 
 the lowest price at which it can be delivered in the vicinity of New York, it 
 costs the same as coal at $5.70 per ton. The Standard Oil Company estimates 
 that 173 gal. are equal to a gross ton of coal, allowing for incidental savings, 
 as in grate bars, carting ashes, attendance, etc. 
 
 Sawdust can be utilized for fuel to good advantage by a special furnace 
 and automatic feeding devices. Spent tan bark is also used, mixed with 
 some coal, or it may be burned without the coal in a proper furnace. Its 
 value is about one-fourth that of the same weight of wood as it comes from 
 the press, but, when dried, its value is about 85$ of the same weight of wood 
 in same state of dryness. 
 
 It has been estimated that, on an average, 1 lb. of coal is equal, for steam- 
 making purposes, to 2 lb. dry peat, 2} to 2\ lb. dry wood, 2\ to 3 lb. dried tan 
 bark, 2f to 3 lb. cotton fetalks, 3i to 3£ lb. wheat or barley straw, and 6 to 8 lb. 
 wet tan bark. 
 
 Natural gas varies in quality, but it is usually worth 2 to 2£ times the same 
 weight of coal, or about 30,000 cu. ft. are equal to a ton of coal. 
 
 Slack, or the screenings from coal, when properly mixed— anthracite and 
 bituminous— and burned by means of a blower on a grate adapted to it, is 
 nearly equal in combustible value to coal, but its percentage of refuse is 
 greater. 
 
 The accompanying table of proximate analyses and heating values of 
 American coals was compiled by Mr. William Kent, for the 1898 edition of 
 the Babcock & Wilcox Co.’s book, “ Steam.” The analyses are selected 
 from various sources, and, in general, are averages of many samples. The 
 heating values per pound of combustible are either obtained from direct 
 calorimetric determinations or calculated from ultimate analyses, except 
 those marked (?), which are estimated from the heating values of coals 
 of similar composition. The figures in the last column are obtained # by 
 dividing the figures in the preceding column by 965.7, the number of heat 
 units required to evaporate 1 lb. of water - at 212° into steam of the same 
 temperature. 
 
 The heating values per pound of combustible given in the table, except 
 those marked (?), are probably within 3$ of the average actual heating values 
 of the combustible portion of the coals of the several districts. When the 
 percentage of moisture and ash in any given lot of coal is known, the 
 heating value per pound of coal may be found, approximately, by multi- 
 plying the heating value per pound of combustible of the average coal of the 
 district by the difference between 100$ and the sum of the percentages of 
 moisture and ash. 
 
 The heating effect is calculated on the basis of the coal burned to carbon 
 dioxide and liquid water at 100° C., and is stated either in calories per kilo- 
 gram or English heat units per pound. The theoretical evaporative effect is 
 calculated by dividing the number of calories per kilogram by 536, or the 
 number of English heat units per pound by 965. In either case, it expresses 
 the theoretical number of kilograms or pounds of water converted into steam 
 from and at 100° C., by 1 kilogram or 1 lb. of coal. 
 
 A committee of the Western Society of Engineers, of Pittsburg, report 
 that 1 lb. of good coal = 7£ cu. ft. of natural gas. When burned with just 
 enough air, its temperature of combustion is 4,200° F. The Westinghouse 
 Air Brake Co. found from experiment that 1 lb. Youghiogheny coal 
 = 12£ cu. ft. natural gas, or 1,000 cu. ft. natural gas = 81.6 lb. coal. Indiana 
 natural gas gives 1,000,000 B. T. U. for 1,000 cu. ft. and weighs .045 lb. per 
 cu. ft. 
 
168 
 
 FUELS. 
 
 Proximate Analyses and Heating Values of American Coals. 
 
 Coal. 
 
 Anthracite. 
 Northern Coal Field . 
 East Middle Coal Field 
 West Middle Coal Field 
 Southern Coal Field . 
 
 Semianthracite. 
 Loyalsock Field . . 
 
 Bernice Basin . . . 
 
 Semibituminous. 
 Broad Top, Pa. . . . 
 
 Clearfield Co., Pa. . . 
 
 Cambria Co., Pa. . . 
 Somerset Co., Pa. . . 
 Cumberland, Md. . . 
 
 Pocahontas, Ya. . . . 
 
 New River, W. Ya. . 
 
 Bituminous. 
 Connellsville, Pa. . . 
 
 Youghiogheny, Pa. . 
 Pittsburg, Pa. . . . 
 
 Jefferson Co., Pa. . . 
 
 and Ohio .... 
 Thacker, W. Va. . . 
 
 Jackson Co., Ohio . . 
 
 Brier Hill, Ohio . . 
 Hocking Valley, Ohio . 
 Yanderpool, Ky. . . 
 Muhlenberg Co., Ky. . 
 Scott Co., Tenn. . . 
 
 Jefferson Co., Ala. . . 
 
 Big Muddy, 111. . . • 
 
 Mt. Olive, 111 
 
 Str^ator, 111 
 
 Missouri . . . . • • 
 
 Lignite and Lignitic Cot 
 
 Iowa 
 
 Wyoming 
 
 Utah 
 
 Oregon lignite . . . 
 
 Moisture. 
 
 Volatile Matter. 
 
 Fixed Carbon. 
 
 Ash. 
 
 Sulphur. 
 
 Heating Value per Lb. 
 Coal, B. T. U. 
 
 1 
 
 Volatile Matter Per Cent, 
 of Combustible. 
 
 Fixed Carbon. 
 
 Per Cent, of Combustible. 
 
 Heating Value per Lb. 
 Combustible. 
 
 1 Theoretical Evaporation 
 
 3.42 
 
 4.38 
 
 83.27 
 
 8.20 
 
 .73 
 
 13,160 
 
 5.00 
 
 95.00 
 
 14,900 1 
 
 3.71 
 
 3.08 
 
 86.40 
 
 6.22 
 
 .58 
 
 13,420 
 
 3.44 
 
 96.56 
 
 14,900 1 
 
 3.16 
 
 3.72 
 
 81.59 
 
 10.65 
 
 .50 
 
 12,840 
 
 4.36 
 
 95.64 
 
 14,900 1 
 
 3.09 
 
 4.28 
 
 83.81 
 
 8.18 
 
 .64 
 
 13,220 
 
 4.85 
 
 95.15 
 
 14,900 1 
 
 1.30 
 
 8.10 
 
 83.34 
 
 6.23 
 
 1.63 
 
 13,920 
 
 8.86 
 
 91.14 
 
 15,500 1 
 
 .65 
 
 9.40 
 
 83.69 
 
 5.34 
 
 .91 
 
 13,700 
 
 10.98 
 
 89.02 
 
 15,500 1 
 
 .79 
 
 15.61 
 
 77.30 
 
 5.40 
 
 .90 
 
 14,820 
 
 17.60 
 
 82.40 
 
 15,800 1 
 
 .76 
 
 22.52 
 
 71.82 
 
 3.99 ; 
 
 .91 
 
 14,950 
 
 24.60 
 
 75.40 
 
 15,700 ] 
 
 .94 
 
 19.20 
 
 71.12 
 
 7.04 
 
 1.70 
 
 14,450 
 
 22.71 
 
 77.29 
 
 15,700 
 
 1.58 
 
 16.42 
 
 71.51 
 
 8.62 
 
 1.87 
 
 14,200 
 
 20.37 
 
 79.63 
 
 15,800 ’ 
 
 1.09 
 
 17.30 
 
 73.12 
 
 7.75 
 
 .74 
 
 14,400 
 
 19.79 
 
 80.21 
 
 15,800 
 
 1.00 
 
 21.00 
 
 74.39 
 
 3.03 
 
 .58 
 
 15,070 
 
 22.50 
 
 77.50 
 
 15,700 
 
 .85 
 
 17.88 
 
 77.64 
 
 3.36 
 
 .27 
 
 15,220 
 
 18.95 
 
 81.05 
 
 15,800 
 
 1.26 
 
 30.12 
 
 59.61 
 
 8.23 
 
 .78 
 
 14,050 
 
 34.03 
 
 65.97 
 
 15,300 
 
 1.03 
 
 36.50 
 
 59.05 
 
 2.61 
 
 .81 
 
 14,450 
 
 38.73 
 
 61.27 
 
 15,000 
 
 1.37 
 
 35.90 
 
 52.21 
 
 8.02 
 
 1.80 
 
 13,410 
 
 41.61 
 
 58.39 
 
 14,800 
 
 1.21 
 
 32.53 
 
 60.99 
 
 4.27 
 
 1.00 
 
 14,370 
 
 35.47 
 
 64.53 
 
 15,200 
 
 L. 1.81 
 
 35.33 
 
 53.70 
 
 7.18 
 
 1.98 
 
 13,200 
 
 40.27 
 
 59.73 
 
 14,500 
 
 ' 1.93 
 
 35.90 
 
 50.19 
 
 9.10 
 
 2.89 
 
 13,170 
 
 43.59 
 
 56.41 
 
 14,800 1 
 
 1.38 
 
 35.04 
 
 56.03 
 
 6.27 
 
 1.28 
 
 14,040 
 
 39.33 
 
 60.67 
 
 15,200 
 
 , 3.83 
 
 32.07 
 
 57.60 
 
 6.50 
 
 
 13,090 
 
 35.76 
 
 64.24 
 
 14,600 
 
 4.80 
 
 34.60 
 
 56.30 
 
 4.30 
 
 
 13,010 
 
 38.20 
 
 61.80 
 
 14,300 
 
 6.59 
 
 34.97 
 
 48.85 
 
 8.00 
 
 1.59 
 
 12,130 
 
 42.81 
 
 57.19 
 
 14,200 
 
 4.00 
 
 34.10 
 
 54.60 
 
 7.30 
 
 
 12,770 
 
 38.50 
 
 61.50 
 
 14.400 
 
 4.33 
 
 33.65 
 
 55.50 
 
 4.95 
 
 1.57 
 
 13,060 
 
 38.86 
 
 61.14 
 
 ' 14,400(?) | 
 
 1.26 
 
 35.76 
 
 53.14 
 
 8.02 
 
 1.80 
 
 13,700 
 
 34.17 
 
 65.83 
 
 1 15,100(?) | 
 
 1.55 
 
 34.44 
 
 59.77 
 
 2.62 
 
 1.42 
 
 13,770 
 
 37.63 
 
 62.37 
 
 14,400(?) | 
 
 7.50 
 
 30.70 
 
 53.80 
 
 8.00 
 
 
 12,420 
 
 36.30 
 
 63.70 
 
 ! 14,700 
 
 . 11.00 
 
 35.65 
 
 37.10 
 
 13.00 
 
 
 10,490 
 
 i 47.00 
 
 53.00 
 
 13,800 
 
 12.00 
 
 33.30 
 
 40.70 
 
 14.00 
 
 
 10,580 
 
 i 45.00 
 
 55.00 
 
 14,300 
 
 . 6.44 
 
 37.57 
 
 47.94 
 
 8.05 
 
 
 12,230 
 
 i 43.94 
 
 56.06 
 
 14,300(?) 
 
 s. 
 
 8.45 
 
 37.09 
 
 35.60 
 
 18.86 
 
 
 8,720 
 
 ) 51.03 
 
 48.97 
 
 ! 12,000(?) 
 
 8.19 
 
 38.72 
 
 41.83 
 
 11.26 
 
 
 10,390 
 
 > 48.07 
 
 51.93 
 
 12,900(?) 
 
 9.29 
 
 41.97 
 
 44.37 
 
 3.20 
 
 1 .18 
 
 i 11,030 
 
 ) 48.60 
 
 51.40 
 
 1 ]2,600(?) 
 
 . 15.25 
 
 42.98 
 
 33.32 
 
 7.11 
 
 1.6( 
 
 5 8,540 
 
 ) 54.95 
 
 45.05 
 
 11,000(?) 
 
 % I 
 
 15.84 
 
 15.53 
 
 15.32 
 
 15.74 
 
 15.01 
 
 15.32 
 
 15.74 
 
 15.11 
 
 14.91 
 
 14.91 
 
 15.63 
 
 14.91 
 
 15.22 
 
 14.29 
 
 14.80 
 
 14.80 
 
 12.42 
 
 13.35 
 
 13.04 
 
 11.39 
 
 A British therma un t (B. T. U.) is me quantity ui neat — v. 
 
 temperature ofl lb. of water 1° F. at or near the temperature of maximum 
 density, 89.1° F. _ , . , 
 
 A calorie is the quantity of heat required to raise the temperature of 
 1 kilogram of water 1° C. at or about 4° C. 
 
 A pound calorie is the quantity of heat necessary to raise the temperature 
 
 of 1 lb. of water 1° C. . . . . . 
 
 1 French calorie = 3.968 British thermal units. 
 
 1 B T U. = -252 calorie. 
 
 lib. calorie = | B. T. U. = .4536 calorie. 
 
 The heating value of any coal may be calculated from its ultimate 
 analysis, with I probable error not exceeding by Dulong s formula: 
 
 Heating value per lb. = 146 C + 620 ( H — — ^ , 
 in which C ; H, and 0 are, respectively, the percentages of carbon, hydrogen, 
 and oxygen. 
 
CLASSIFICATION OF COALS. 
 
 169 
 
 ?>• 
 
 in which C, 0, H, and S represent the weights of carbon, oxygen, hydro- 
 gen, and sulphur in 1 lb. of the substance. 
 
 Composition of Fuels. 
 
 ( Mechanical Draft, B. F. Sturtevant Co.) 
 
 Heat in pound calorie = 8,080 C + 34,462 (^H — 
 or = 8,080 C + 34,462 ) + 2,250 S. 
 
 T. U. — 14,650 C -f 62,100 (ll — y ) 
 
 Heat in B. 
 
 Description. 
 
 Carbon. 
 
 Hydro- 
 
 gen. 
 
 Oxy- 
 
 gen. 
 
 Nitro- 
 
 gen. 
 
 Sul- 
 
 phur. 
 
 Ash. 
 
 Anthracite. 
 
 
 
 
 
 
 
 France 
 
 90.9 
 
 1.47 
 
 1.53 
 
 1.00 
 
 .80 
 
 4.3 
 
 Wales 
 
 91.7 
 
 3.78 
 
 1.30 
 
 1.00 
 
 .72 
 
 1.5 
 
 Rhode Island 
 
 85.0 
 
 3.71 
 
 2.39 
 
 1.00 
 
 .90 
 
 7.0 
 
 Pennsylvania 
 
 78.6 
 
 2.50 
 
 1.70 
 
 .80 
 
 .40 
 
 14.8 
 
 Semibituminous. 
 
 
 
 
 
 
 
 Maryland 
 
 80.0 
 
 5.00 
 
 2.70 
 
 1.10 
 
 1.20 
 
 8.3 
 
 Wales 
 
 88.3 
 
 4.70 
 
 .60 
 
 1.40 
 
 1.80 
 
 3.2 
 
 Bituminous. 
 
 
 
 
 
 
 
 Pennsylvania 
 
 75.5 
 
 4.93 
 
 12.35 
 
 1.12 
 
 1.10 
 
 5.0 
 
 Indiana 
 
 69.7 
 
 5.10 
 
 19.17 
 
 1.23 
 
 1.30 
 
 3.5 
 
 Illinois 
 
 61.4 
 
 4.87 
 
 35.42 
 
 1.41 
 
 1.20 
 
 5.7 
 
 Virginia 
 
 57.0 
 
 4.96 
 
 26.44 
 
 1.70 
 
 1.50 
 
 8.4 
 
 Alabama 
 
 53.2 
 
 4.81 
 
 32.37 
 
 1.62 
 
 1.30 
 
 6.7 
 
 Kentucky 
 
 49.1 
 
 4.95 
 
 41.13 
 
 1.70 
 
 1.40 
 
 7.2 
 
 Cape Breton 
 
 67.2 
 
 4.26 
 
 20.16 
 
 1.07 
 
 1.21 
 
 6.1 
 
 Vancouver Island 
 
 66.9 
 
 5.32 
 
 8.76 
 
 1.02 
 
 2.20 
 
 15.8 
 
 Lancashire gas coal 
 
 80.1 
 
 5.50 
 
 8.10 
 
 2.10 
 
 1.50 
 
 2.7 
 
 Boghead cannel 
 
 63.1 
 
 8.90 
 
 7.00 
 
 .20 
 
 1.00 
 
 19.8 
 
 Lignite. 
 
 
 
 
 
 
 
 California brown 
 
 49.7 
 
 3.78 
 
 30.19 
 
 1.00 
 
 1.53 
 
 13.8 
 
 Australian brown 
 
 73.2 
 
 4.71 
 
 12.35 
 
 1.11 
 
 .63 
 
 8.0 
 
 Petroleum. 
 
 
 
 
 
 
 
 Pennsvl vania (crude) 
 
 84.9 
 
 13.70 
 
 1.40 
 
 
 
 
 Caucasian (light) 
 
 86.3 
 
 13.60 
 
 .10 
 
 
 
 
 Caucasian (heavy) 
 
 86.6 
 
 12.30 
 
 1.10 
 
 
 
 
 Refuse 
 
 87.1 
 
 11.70 
 
 1.20 
 
 
 
 
 CLASSIFICATION, COM POSITIO: J, AN D PROPERTIES OF COALS. 
 
 Coals may be broadly divided into two classes: Anthracite, or hard, coal; 
 and bituminous, or soft, coal. 
 
 Anthracite, or Hard, Coal.— Specific gravity, 1.30 to 1.70. This is the densest, 
 hardest, and most lustrous of all varieties. It burns with little flame and 
 no smoke, but gives a great heat. Contains very little volatile combustible 
 matter. Color, deep black, shining; sometimes iridescent. Fracture, 
 conchoidal. 
 
 Semianthracite coal is not so dense nor so hard as the true anthracite. 
 Its percentage of volatile combustible matter is somewhat greater, and it 
 ignites more readily. 
 
 Bituminous, or Soft, Coal.— Specific gravity, 1.25 to 1.40. It is generally 
 brittle; has a bright pitchy or greasy luster, and is rather fragile as compared 
 with anthracite. It burns with a yellow smoky flame, and gives, on distil- 
 lation, hydrocarbon oils or tar. 
 
 Under the term “bituminous” are included a number of varieties of coal 
 that differ materially under the action of heat, giving rise to the general 
 classification: Coking or caking coals, and free-burning coals. 
 
 Semibituminous coal has the same general characteristics as the bituminous, 
 although it is usually not so hard, and its fracture is more cuboidal. The 
 
170 
 
 FUELS. 
 
 nercentage of volatile combustible matter is less. It kindles readily, and 
 burns quickly with a steady fire, and is much valued as a steam coal. 
 
 Coking coals are those that become pasty or semiviscid in the fire, and, 
 when heated in a close vessel, become partially fused and agglomerate into a 
 mas? of ^coherent coke. This property of coking may, however, become 
 Seatly impaired, if, indeed, not entirely destroyed, by weathering. 
 
 Free-burning coals have the same general characteristics as the cokm^ 
 coals but they burn freely without softening, and do not fuse or cake 
 
 t0S slkl o?arZaduUblacTcolor, and is much harder and less frangible 
 than "he coMng coal. It is readily fissile, like slate, but breaks with 
 difficulty on cross-fracture. It ignites less readily, but makes a hot fire, 
 constituting a good house coal. 
 
 Weights and Measurements of Coal. 
 
 ( Coxe Bros. & Co., Chicago , III.) 
 
 Coal. 
 
 Lehigh lump 
 
 Lehigh cupola 
 
 Lehigh broken 
 
 Lehigh egg 
 
 Lehigh stove 
 
 Lehigh nut 
 
 Lehigh pea 
 
 Lehigh buckwheat... 
 Lehigh dust 
 
 Weight per 
 Cubic Foot. 
 Pounds. 
 
 Cubic Feet 
 per Ton, 
 2,000 Lb. 
 
 55.26 
 
 36.19 
 
 55.52 
 
 36.02 
 
 56.85 
 
 35.18 
 
 57.74 
 
 34.63 
 
 58.15 
 
 34.39 
 
 58.26 
 
 34.32 
 
 53.18 
 
 37.60 
 
 54.04 
 
 37.01 
 
 57.25 
 
 34.93 
 
 Coal. 
 
 Free-burning egg... 
 Free-burning stove 
 Free-burning nut ... 
 
 Pittsburg 
 
 Illinois ' - 
 
 Connellsville coke 
 
 Hocking 
 
 Indiana block 
 
 Erie 
 
 Ohio cannel 
 
 * 1.8 
 
 US 
 
 ’S-Si 
 
 A 2 
 
 56.07 
 56.33 
 56.88 
 46.48 
 47.22 
 
 26.30 
 
 49.30 
 43.85 
 
 48.07 
 49.18 
 
 - . 
 
 0^0 
 f-i o 
 ^ ®o 
 
 35.67 
 
 35.50 
 
 35.16 
 
 43.03 
 42.35 
 
 76.04 
 40.56 
 
 45.61 
 
 41.61 
 40.66 
 
 Cannel coal differs from the ordinary bituminous coal in its texture. It is 
 compact with little or no luster and without any appearance of a banded 
 structure. It breaks with a smooth conchoidal fracture, kindles readily, and 
 burns with a dense smoky flame. It is rich in volatile matter, and makes an 
 excellent gas coal. Color, dull black and grayish black. 
 
 Uenffe or brown coal, often has a lamellar or woody structure; is some- 
 times pitch black, but more often rather dull and brownish black. It kindles 
 readily and burns rather freely with a yellow flame and comparatively little 
 smoke but it gives only a moderate heat. It is generally non-coking. The 
 nercentage of moisture present is invariably high from 10^ to 30/). 
 
 The subdivisions given above are entirely arbitrary, as the different 
 varieties of coal are found to shade insensibly into one another. The follow- 
 ing are two classifications according to percentages of volatile combustible 
 matter: „ 
 
 Classification of Coal According to Volatile Combustible. 
 
 Coal. 
 
 Per Cent. 
 
 Kent. 
 Per Cent. 
 
 
 2.5 to 6 
 
 0 to 7 
 
 An till dci ic * 
 
 7 to 10 
 
 7.5 to 12 
 
 
 
 12 to 20 
 
 12.5 to 25 
 
 oGiniulluiiLiiiuuo 
 
 over 20 
 
 25 to 50 
 
 I) 1 LlIILLlllU 
 
 
 over 50 
 
 
 
 - 
 
 _ . . 
 
 The ComDosition of Coals.-A proximate analysis determines tne proport 
 of those products of a coal having the most important bearing on its uses. 
 Th^subsCces as usually presented are: Moisture, or water, volatile com- 
 
PROPERTIES OF COALS . 
 
 171 
 
 bustible matter, fixed carbon, sulphur, and ash. In addition to these, the 
 following physical properties are generally given: Color of ash, specific 
 gravity, and strength or hardness. The determination of these eight factors 
 gives a fair general idea of the adaptabilities of a coal. 
 
 Moisture, or water, in coal, has no fuel value, is an inert constituent, dug, 
 handled, and hauled, and finally expelled at a cost of fuel. Each per cent, 
 of moisture means 20 lb. less fuel for each ton of coal. 
 
 Volatile combustible matter is an important constituent of coal, the amount 
 and quality deciding whether a coal is suitable for the manufacture of 
 illuminating gas. The coking of coal also is largely dependent on this 
 constituent. When a large percentage of volatile combustible matter is 
 present, coals ignite easily and burn with a long yellow flame, and, in 
 ordinary combustion, give out dense smoke, and form soot. This quality 
 makes a fuel objectionable for railway and sometimes for naval use. 
 
 The fixed carbon is the principal combustible constituent in coal, and, in 
 bituminous and semibituminous coals, the steaming value is in proportion to 
 the percentage of fixed carbon. Though the fixed carbon of a coal evapo- 
 rates much less water than an equivalent weight of the volatile combustible 
 matter when properly burnt, in practice, so much of the latter is lost through 
 careless firing, or improper furnace construction, that the relative steaming 
 value of a coal may be fairly approximated by assuming the carbon to be 
 the only useful constituent. 
 
 Sulphur will burn and develop heat, and is not inert like moisture and 
 as}i. But it corrodes grates and boilers; in the blast furnace it injures iron, 
 and produces a hot short pig, and is objectionable in coal for forge use. In 
 gas making, the sulphur must be removed. It usually occurs in coal in the 
 form of iron pyrites, which, oxidizing, causes disintegration, and sometimes 
 spontaneous combustion. It is then an element of danger and loss. 
 
 Ash is an inert constituent, which means 20 lb. of weight to be handled 
 and 20 lb. loss per ton of coal for each per cent, present. Water in coal is 
 removed at the cost of fuel, while ashes are removed at extra cost of labor. 
 It is estimated that if the cost of stoking coal is 6 f0 of the cost of coal (coal 
 at $3.00 per ton, and labor at $1.00 per day), and with cost of handling ashes 
 double that of stoking coal, 5$ of ash will lessen the fuel value of coal over 
 6$; 10 0 ash, over 120; and so on. 
 
 The color of the ash furnishes a rough estimate of the amount of iron con- 
 tained in a fuel. Iron in an ash makes it more fusible, and increases its 
 tendency to clinker. In domestic consumption, where the temperature is 
 low, the quantity of ash is of more importance than its fusibility, but for 
 steam purposes, where an excessive heat is required, ashes of a clinkering 
 coal will fuse into a vitreous mass and accumulate upon the grate bars and 
 exclude the passage of necessary air. The practicability of employing a coal 
 will often be determined by the quality of the clinkering of the ashes. 
 Under such conditions, such coals are best whose ashes are nearly pure 
 white and which contain little or no alkali nor any lime, and do not contain 
 silica and alumina. 
 
 The specific gravity is an important factor when there is restriction of 
 I space, as on railway cars and in ship bunkers. A given bulk of anthracite 
 ! coal will weigh from 100 to 150 more than the same bulk of bituminous coal, 
 so that from 100 to 150 more pounds of fuel can be carried in the same place. 
 The average specific gravity of anthracite coal is 1.5, and a cubic yard weighs 
 about 2,531 lb. 
 
 The average specific gravity of American bituminous coals, and of grades 
 intermediate between them and anthracite, is about 1.325, and 1 cu. yd. 
 weighs about 2,236 lb. 
 
 Strength or hardness is valuable in preventing waste. In soft coal, much is 
 ground to dust in mining and at the tipple. In railway transportation, soft 
 coal is crushed, which further increases the loss, and the coal reaches market 
 in bad condition. A very soft coal is shipped m lump, and is not in so wide 
 demand. For marine use, a soft coal is objectionable, because of disintegra- 
 tion by the motion of the ship. Strength is a requisite for the use of raw 
 coal in the blast furnace, and also to prevent excessive loss of coal through 
 the grates in ordinary furnaces. 
 
 Steaming Coals— For steam making, the superiority of coals high in com- 
 bustible constituents is admitted, and those with the higher percentage of 
 fixed carbon are the most desirable. But the consideration of the steaming 
 qualities of a coal involves, also, a consideration of the form of furnace and 
 of all the conditions of combustion. The evaporative power of a coal in 
 
172 
 
 FUELS. 
 
 Dractice cannot be stated without reference to the conditions of combustion, 
 and every practical test of a coal, to be thorough, should lead to ^determi- 
 nation of the best form of furnace for that coal, anu slmuld furnish knowl- 
 edge as to what class of furnaces in actual use such coal is specially adapted. 
 
 Tt fs not sufficient that in comparative tests of coals the same conditions 
 should ex?s? w!?h each, but there should also be determined the best 
 
 co: Ofraals 1 high a in fixed carbon, the semianthracites and the semibitumi- 
 nous rank as high as the anthracites in meeting the various requirements of 
 
 a ^Fo? railway < use, t these^co^ls'^ave been found to excel anthracites in 
 evaporating power. The comparative absence, m semibitummous coals, of 
 smoke which means loss of combustible matter as well as discomfort to the 
 traveler is sufficient to suggest the superiority of these coals over bituminous 
 coals for such use In fact the high rate of combustion and the strong drait 
 necessary ^n locomotiv e s is particularly unfavorable to the economic com- 
 hnstTon of bituminous coal. Such semibitummous coals are also specially 
 well suited ^for^mall tubular boilers, firebox steam boilers, or other forms 
 with small uifiined combustion chambers, in which the gases from bitumi- 
 nous coals become cooled, are not burnt, and deposit soot m nT1( q 
 
 Steaming coal should kindle readily and burn quickly but steadily , and 
 should contain only enough volatile matter to insure rapid combustio n . I 
 should be low in ash and sulphur, should not clinker, and when it is t 
 
 tra Coafsfo? Ircm 'Makfng^.— J th?manufarture^rf iron and forme— cal 
 •nnrrioses coal is chiefly used after being converted into coke, though it is 
 also P used to a limited extent in the raw state. Coal directly used must be 
 «tronp- and not swell nor disintegrate so as to choke the furnace. It .hould 
 be cafable of producing a high heat and should not contain a large amount 
 
 0f S Co\ P e h -Cok^s tCfaed carbon of a coal, a.fused and porous product pro- 
 duced by the distillation of the gaseous constituent. For metallurgical use, 
 t ^hoi fl be firm tough, and bright, with a sonorous ring, and should 
 InuSn not over ’k of sulphur. For blast-furnace use, a dense coke is 
 nhWtLable Ind the best i^the one with the largest cell structure and the 
 hardest cell wall. A high percentage of volatile hydrocarbon is, as a rule, 
 
 ne TMXfit?of C the in cMboi; the amount of disposable hydrogen, the 
 tenacftv whh wL?h the gaseous’ constituents are held, all affect the recite 
 
 iected'^ the^nflue’nee a of^^e 1S \^Sher”loses its^apacityfor coking. The 
 
 antH r^the 'prcsCTit^rajde^of manufacture the 
 ccontial mi a ii ties of a good coking coal are: that it shall contain not less than 
 
 a manner f to form {“T^ofTolaSfmatter^ it will not fuse properly 
 wmmi^it has more Ithan 3CH the porous structure will be unduly developed 
 
 Wsf S ail.^and 0 therefOTe^ do* riot 
 coke whife others fuse properly but give off their gas so as to form large 
 
 “rSSIS ffiSwaafK 
 
 the "lttsDurg o+ntidard coking coal, but coals whose analysis differ very 
 instance, the Pocahontas coke, Virginia. 
 
ANALYSIS OF. COAL. 
 
 173 
 
 Domestic Coals. — In domestic use, coal is burned in open grates, in closed 
 stoves with ordinary fire bowls and flat grates, or with basket grates in small 
 furnaces for hot-air heating, and in cooking stoves. In all these, the coal 
 that sustains a mild, steady combustion, and remains ignited at a low tem- 
 perature with a comparatively feeble draft, is the best. A coal burning with 
 a smoky flame is objectionable as producing much soot and dirt, especially 
 for open grates or cooking purposes. For self-feeding stoves, or for base 
 burners, a dry non-coking coal is necessary. A very free and fiercely burn- 
 ing coal is not desirable, particularly in stoves, as the temperature cannot be 
 easily regulated. A sulphurous coal is also bad, as it produces stifling gases 
 with a defective draft, and corrodes the grates and fire bowls. The difficulty 
 from clinkering is not so great in domestic uses, as the temperature is not 
 generally high enough to fuse the ash. A stony, hard ash that will not pass 
 between the grate bars is bad, and light pulverulent ash is best. 
 
 Gas Coals.— Mr. H. C. Adams, of The American Gas Light Association, 
 says: “ The essentials of a good gas coal are a low percentage of ash, say 5 $, 
 and of sulphur, say £ of 1$, a generous share, say 37$ to 40$ of volatile matter, 
 charged with rich illuminating hydrocarbons. And it should yield, under 
 present retort practice, 85 candle-feet to the pound carbonized. It should be 
 sufficiently dense to bear transportation well, so that, when carried long 
 distances, it may not arrive at its destination largely reduced to slack or fine 
 coal of the consistency of sand. And it should possess coking qualities that 
 will bring from the retorts, after carbonization, about 60$ of clean, strong-, 
 bright coke.” 
 
 Blacksmith Coals.— A good coal for blacksmith purposes should have a 
 high heating power, should contain a very small amount of sulphur, if 
 any, should coke sufiflciently to form an arch on the forge, and should 
 also be low in ash. 
 
 From the above, it is readily seen that the analysis of a coal does not 
 necessarily determine its value or the uses to which it can be put. How- 
 ever, by examining the analyses given in the table on page 168, certain 
 standards may be adopted as showing in a general way about what the 
 analysis of coal should be for certain purposes. For steam purposes, the 
 semibituminous coals have established reputations. For gas coals, that 
 from Youghiogheny, Pa., is well known. For blacksmiths, Broad Top and 
 Tioga County, Pennsylvania, coals are standards; while for coking, Connells- 
 ville is recognized as a standard. 
 
 The sizes of anthracite coal vary. The sizes of screen mesh and bar open- 
 ings used for separating, range as follows: 
 
 Lump, over bars placed 7 to 9 in. apart. 
 
 Steamboat, over bars placed 3£ to 5 in. apart and through bars 7 in. apart. 
 
 Grate, over 2£ in. and through 4£ in. square mesh. 
 
 Egg, over 2 in. and through 2£ in. square mesh. 
 
 Stove, over If in. and through 2 in. square mesh. 
 
 Chestnut, over £ in. and through If in. square mesh. 
 
 Pea, over £ in. and through £ in. square mesh. 
 
 Buckwheat, over £ in. and through £ in. square mesh. 
 
 No. 2 Buckwheat, or Bird’s-eye, over £ in. and through x 8 ff in. square mesh. 
 
 The sizes of bituminous coal are Lump, Nut, and Slack. 
 
 All coal that passes over bars 1£ in. apart is called Luw,p. 
 
 All coal that passes through bars 1£ in. apart and over bars £ in. apart is 
 called Nut. 
 
 All coal that passes through bars £ in. apart is called Slack. 
 
 ANALYSIS OF COAL. 
 
 The following is the outline of the method recommended for the 
 analysis of coal by a committee of the American Chemical Societv. Messrs 
 . VV. F. Hillebrand, C. B. Dudley, and W. A. Noyes: 
 
 Sampling.— At least 5 lb. of coal should be taken for the original sample, 
 with care to secure pieces that represent the average. These should be 
 broken up and quartered down to obtain the smaller sample, which is to 
 be reduced to a fine powder for analysis. The quartering and grinding 
 should be carried out as rapidly as possible, and immediately after the 
 original sample is taken, to prevent gain or loss of moisture. The pow- 
 dered coal should be kept in a tightly stoppered tube, or bottle, until 
 analyzed. Unless the coal contains less than 2$ of moisture, the shipment 
 of large samples in wooden boxes should be avoided. 
 
174 
 
 FUELS. 
 
 In boiler tests shovelfuls of coal should be taken at regular intervals and 
 out in a tight covered barrel, or some air-tight cbvered receptacle, and the 
 Fatter sLufd be placed whereit is protected from the heat of the furnace 
 In sampling from a mine , the map of the mine should be carefully 
 examined and points for sampling located m such a manner as to fairly 
 rporesent the body of the coal. These points should be placed close to the 
 ^kinS facl hefore sampling, make a fresh cut of the face from top to 
 bottom^o a depth that will insure the absence of possible changes or of 
 sulphur and smoke from the blasting powders. Clean the floor and spread 
 Tmece of canvas to catch the cuttings. Then, with a chisel, make a cutting 
 from floor to roof, say 3 in. wide and about 1 in. deep. Do not chisel out 
 the shale or other impurities that it is the practice at that flame to reject. 
 Measure the length of the cutting made, but do not include the impurities 
 ^tMs measurement? With a piece of flat iron and a hammer break all 
 nieces to quarter-inch cubes or less, without removing from the cloth. 
 Quarter down and transfer to a sealed bottle or jar. For the r\ 
 
 samnle samples taken at several points in this manner should be mixed 
 and^iuartered down. If the vein varies in thickness at different points, the 
 ^mnles taken at each point should correspond in amount to the thickness 
 vein For instance, a small measure may be filled as many times 
 with the coal of the sample as the vein is feet m thickness. Should the e 
 appear differences in the nature of the coal, it will be more satisfactory to 
 tekrfn adSon to the general sample, samples of such portions of the 
 
 V6i ■oK-Srlg’ ofthe d coal“ open porcelain or platinum crucible 
 at 104° to 107° C. for 1 hour, best in a double-walled bath containing pure 
 
 t0l Volatile Combustible ^ Matter.— Placelgfof fresh, undried coal in a platinum 
 crucible weiehing 20 to 30 g„ and haying a tightly fitting coyer Heat over 
 thpTll flame of a Bunsen burner for 7 minutes. The crucible should be 
 the lull name o triangle with the bottom 6 to 8 cm. above the top 
 
 n7t? 1P burner %he flame use! should be fully 20 cm. high when burning 
 free Ind the determinSfon made in a place free from drafte. The upper 
 Jfrfkce of the cover should burn clear, but the under surface should 
 remain covered with carbon. To find “ volatile combustible matter, 
 «nhtrnet the percentage of moisture from the loss found here. 
 
 SU ^sh — Burr^Se^ortfon of coal used for the determination of moisture at 
 first over a very low flame, with the crucible open and inclined, untilfree 
 from carbon If properly treated, this sample can be burned much more 
 quSkly than the dJnse carbon left from the determination of volatile 
 
 ma Fixed Carbon.-This is found by subtracting the percentage of ash from 
 the percentage of coke. _ Mix thoroughly 1 g. of the finely powdered 
 coa S lw P hh lg of magnesium oxide and.£ g. of dry sodium carbonateinathm 
 7 ?toToOc c platinum dish or crucible. The magnesium oxide should be 
 
 triangle ove^analcoho? lT£p P heid KTS affirsf)^ mu^not be 
 
 ?nals The flame is kept in motion and barely touching the dish, at first, 
 
 fhen’lL carbon 
 
 60 c. c. oi warer. a™ '' , . through a filter, boil a second and third 
 
 ?iSe n wUh 3 o"oTwfter%n e rlsh unfil the filtrate gives only a slight 
 opalescence with Over hydrochloric acid 
 
 Snl SCfn^is^ 
 
 especially at^first, ^ with oonstant^irring, 10 
 
 iS ln r the case of coals containing much pyrites or calcium sulphate, the 
 
STEAM. 
 
 175 
 
 > 
 
 residue of magnesium oxide should be dissolved in hydrochloric acid and 
 the solution tested for sulphuric acid. 
 
 When the sulphur in the coal is in the form of pyrites, that compound is 
 converted almost entirely into ferric oxide in the determination of ash, and, 
 since 3 atoms of oxygen replace 4 atoms of sulphur, the weight of the ash 
 is less than the weight of the mineral matter in the coal by f the weight of 
 the sulphur. While the error from this source is sometimes considerable, 
 a correction for “proximate” analyses is not recommended. When 
 analyses are to be used as a basis for calculating the heating effect of the 
 coal, a correction should be made. 
 
 The analysis of a coal may be reported in three different forms, as per- 
 centages of the moist coal, of the dry coal, or of the combustible. Thus, 
 suppose 1 g. of coal is analyzed, and the first heating shows a loss of 
 weight of .1 g., the second of .3 g., the third .5 g., the remainder, or ash, 
 weighing .1 g., the complete report would be as follows: 
 
 
 Per Cent. 
 
 of the 
 Moist Coal. 
 
 Per Cent. 
 
 of the 
 Dry Coal. 
 
 Per Cent, 
 of the 
 
 Combustible. 
 
 Moisture 
 
 10 
 
 
 
 Volatile matter 
 
 30 
 
 33.33 
 
 37.50 
 
 Fixed carbon 
 
 50 
 
 55.56 
 
 62.50 
 
 Ash 
 
 10 
 
 11.11 
 
 
 Total 
 
 100 
 
 100.00 
 
 100.00 
 
 STEAM. 
 
 A calculation of the power that coal possesses, compared with the useful 
 work which steam engines exert, shows that probably in the very best 
 engines not one-tenth of the power is converted into useful work, and in 
 some very bad engines, probably not one one-hundredth. There are many 
 causes for this; some we can never remedy, because to do so it would be 
 necessary to exhaust the steam at a lower temperature than is practical. 
 There are other causes that can and ought to be removed. We want good 
 engines, good boilers, high-pressure steam, expansive working, and con- 
 densing appliances. 
 
 High-Pressure Steam.— Why should we use high-pressure steam? There 
 are several good reasons. Whatever pressure we have available at the 
 steam boiler, a certain amount is absorbed in overcoming the resistances of 
 the engine and without doing any useful work. Suppose our available 
 steam pressure is 20 lb., and 10 lb. are so absorbed; that leaves us only 
 one-half; but, if we have 100 lb. available, it would leave us nine-tenths. 
 High-pressure steam means fewer boilers and smaller engines, with founda- 
 tions and houses of less dimensions. Then, again, the amount of work that 
 it is possible to get out of a given quantity of steam depends on the differ- 
 ence between the temperature at the commencement of the stroke and the 
 temperature at the end of the stroke. 
 
 Now, there is a limit as to how low the temperature can be at the end, 
 and as we raise the commencing temperature, we enlarge the available 
 difference. We may put the advantages of high-pressure steam in this way. 
 By taking a fixed temperature in the condenser of, say, 100° F., and initial 
 temperatures when the steam enters the cylinder, of varying amounts, the 
 theoretic efficiency of that steam can be determined. Commencing with 
 atmospheric pressure, we have an efficiency of 16.6$. 
 
 Lb. Per Cent. 
 
 10 20.0 
 
 20 22.1 
 
 30 23.7 
 
 40 25.0 
 
 50 26.1 
 
 60 27 .0 
 
 §0 28.6 
 
 Lb. Per Cent 
 
 100 29.8 
 
 125 31.1 
 
 150 32.2 
 
 200 33.9 
 
 250 35.3 
 
 300 36.5 
 
176 
 
 BOILERS. 
 
 Wp oan only get in practice with steam a certain proportion of the 
 theoretic power, and that proportion varies with the pressure of the steam. 
 
 In early days we used steam at atmospheric pressure, the efficiency 
 hpinp - 16 6 afterwards, we had, in compound engines of two cylinders, 
 steam of^Olb., the efficiency being 27/*. we have^ ^ 
 
 pnp-irips usine* stG&in £it 150 lb., ttiG GflticiGiicy being 32.2,0. It 
 observed that although the efficiency increases as the steam pressure 
 increases the amount of that increase is a diminishing quantity, and it 
 becomes so small at and beyond 150 lb. pressure that probably any gain m 
 efficiency is not a satisfactory set-off to the additional expense of strength- 
 eSng the parts of the engine. But then, how very few of our engines work 
 
 n6 ¥he advantages^ high-pressure steam are not yet sufficiently appreciated. 
 Tt S not meVelf t he difference between 60 lb. and 120 lb. Suppose we use 
 steam at 60 lb.f probably we shall get 50 lb. at engine, and ^sirtances of 
 enHne will absorb 10 lb., leaving 40 lb. Now, suppose we use 120 lb., we can 
 gefat engine 110 lb., and if resistances of engine absorb 10 lb., we shall have 
 
 100 EKoan S s ro? l ofsu 0 am.-By “ expansion of steam ” we mean that at a certain 
 noint of the stroke we shut off steam supply from the boiler to the 
 Irnrl the steam already within the cylinder performs the remainder of the 
 s^ok^un^ded aiI ^h)W, Suppose we do not expand at all. Suppose we allow 
 fee ^^dmitsion' of steam into the cylinder all through the stroke; we shall 
 have at the end of the stroke pressure exactly similar to the Pressure with 
 which we commenced. Now, we cannot work a seam of coal and smi nave 
 the coal left* we cannot get work out of steam and still have the work left 
 in it and so’, if our steam pressure is the same at the end of the stroke as at 
 the beginning, we simply discharge twice m each revolution a whole 
 cylinder full of steam that has done no work at all, and waste it just the 
 same as if we had discharged it from the boiler without passing through the 
 SStafat aTl? But some one mil say, work has been done upon the engine 
 while that steam was in the cylinder. True — and the explanation is, that 
 while the steam is performing work its heat and pressure must dimmish, 
 and so long as the communication with the boiler is open, fresh heat comes 
 from the boiler into the cylinder to take its place, and at the end of the 
 strcSe we have expended heat represented by the capacity of two cylinders, 
 and have performed work as represented by the capacity of one cylinder. 
 Nffw suppose we close the communication, and beyond a certain point of 
 the stroke allow no more steam to enter, we get an amount of work from 
 the steam already in the cylinder, represented by the diminishing pressure 
 
 ° f Condensers —The Effective power of an engine does not depend on, and 
 is not measured by, the pressure pushing the piston. There] 
 is termed a back pressure holding the piston back, and the real effective 
 nressure is evidently the difference between the two. Suppose we have a 
 locomotive engine, or a winding engine, throwing exhaust into the open 
 air The back pressure cannot be less than the pressure of the open air, 
 and indeed to overcome it, it must be something more. But if we can 
 d?scha?ge our exhaust into some vessel from which atmospheric pressure 
 and all^ other pressure has been removed, we ku°w 
 
 “ e amounts to about 15 lb., and the removal of that from the front of the 
 piston is as good as adding 15 lb. behind. 
 
 BOILERS. 
 
 The steam boiler that will be the most suitable for a certain mine will 
 denend on the nature of the feedwater, the cost of fuel, and the amount of 
 steam required. When the acid water from the mine is used for feedwater, 
 and^uel is cheap the type of boiler generally used is either the plain 
 rvllndrical or flue boiled because it is simple in construction and can 
 therefore be easily cleaned and cheaply replaced when eaten by the mine 
 water The tubular or locomotive type is used where good water can be 
 offiffined 1 except in the best equipped plants, where the water-tube boiler is 
 used Feedwater taken from the mine, or containing acid, should be 
 neutralized by lime or soda before being used. In case it contains minerals 
 fn solution, a feedwater separator should be employed to Precipitate the 
 mineral substance before the water is allowed to enter the boiler. 
 
HORSEPOWER OF BOILERS. 
 
 177 
 
 We always calculate the strength of a boiler in the direction of its 
 diameter, because, theoretically, a boiler is twice as strong in the direction 
 of length as direction of diameter. Many causes may bring about boiler 
 explosions. First, bad materials; second, bad workmanship; third, bad 
 water, which eats away the plates by internal corrosion; fourth, water lying 
 upon plates, bringing about external corrosion; fifth, overpressure: sixth, 
 safety valves sticking; seventh, water getting too low; eighth, excessive 
 tiring; ninth, hot gases acting on plates above water level; tenth, choking of 
 feedpipes; eleventh, insufficient provision for expansion and contraction: 
 twelfth, insufficient steam room and too sudden a withdrawal of a large 
 quantity of steam; thirteenth, getting up steam, or knocking off a boiler 
 too suddenly; fourteenth, allowing wet ashes to lie in contact with plates. 
 The probable causes suggest their several remedies. 
 
 Wherever possible and except under certain circumstances, stear^ 
 engines should not be placed in the mine, and certainly steam boilers 
 should be in all cases placed upon the surface. Steam injures the ventiia* 
 tion, increasing the temperature where already too high, doing injury and 
 causing inconvenience by condensation, and many tires in mines have been 
 caused by underground boilers. 
 
 The Lancashire Boiler.— The colliery boiler that finds much fa^or m Eng- 
 land is that class of Lancashire boiler which is 28 or 30 ft. long ana 7 or 8 ft. 
 in diameter, and has two large flues running through. There is no doubt 
 that the marine type will generate more steam with a given amount of 
 coal, and, consequently, is gaining ground, and will gain grouna where 
 coal is dear. But the Lancashire boiler is a good steam generator, and will 
 not only work longer without repairs, but is less troublesome and expensive 
 to repair. The favorite construction some few years ago was wrought iron 
 with double-riveted horizontal joints and Galloway tubes (Galloway 
 tubes are simply taper tubes running across the flues in the boiler), and 
 expansion weldless hoops strengthening the flues and allowing for expan- 
 sion and contraction. The dimensions were 7 ft. diameter, and from 
 28 to 30 ft. long, with internal flues each 2 ft. 9 in. diameter, the circular 
 plates being about i in. and the end plates about f in. The safe working 
 pressure was about 60 lb. per sq. in. Now the conditions are somewhat 
 altered. Steel has taken the place of iron, giving increased strength, and 
 allowing increased diameter and increased pressure. Ring plates have also 
 abolished a great source of weakness in a boiler, namely, horizontally 
 riveted joints. A good Lancashire boiler now will measure 8 ft. in 
 diameter and 30 ft. long, with ring plates f in. thick, end plates probably 
 £ in., and will work very well at 120 lb. pressure per sq. in. 
 
 Horsepower of Boilers.— The horsepower of a boiler is a measure of its 
 capacity for generating steam. Boilermakers usually rate the horsepower 
 of their boilers as a certain fraction of the heating surface; but this is a 
 very indefinite method, for with the same heating surface, different boilers 
 of the same type may, under different circumstances, generate different 
 quantities of steam. 
 
 In order to have an accurate standard of boiler power, the American 
 Society of Mechanical Engineers has adopted as a standard horsepower 
 an evaporation of 30 lb. of water per hour from a feedwater temperature of 100° F. 
 into steam at 70 lb. gauge pressure , which is considered equivalent to 
 34.5 units of evaporation; that is, to 34.5 lb. of water evaporated from a 
 feedwater temperature of 212° F. into steam at the same temperature. 
 
 Example.— A boiler evaporates per hour 1,980 lb. of water from a feed 
 temperature of 100° into steam at 70 lb. gauge pressure. What is the 
 horsepower of the boiler? 
 
 Since, under the given conditions, an evaporation of 30 lb. is equivalent 
 to 1 horsepower, the number of horsepower is 1,980 -r- 30 = 66. 
 
 In the various types of boilers there is a nearly constant ratio between 
 the water-heating surface and the horsepower, and also between the heating 
 surface and the grate area. These ratios are given in the table on page 178. 
 
 If the heating surface of a boiler is known, the horsepower can be found 
 roughly; thus, if a return-tubular boiler has a heating surface of 900 sq. ft., 
 its horsepower lies between - 9 T ° g °- = 50 H. P. and 9 X ° 5 °- = 64.3 H. P., say about 
 57 H. P. 
 
 The heating surface of a boiler is the portion of the surface exposed to the 
 action of flames and hot gases. This includes, in the case of the multi- 
 tubular boiler, the portions of the shell below the line of brickwork, the 
 exposed heads of the shell, and the interior surface of the tubes. In the 
 
178 
 
 BOILERS. 
 
 case of a water-tube boiler, the heating surface comprises the portion ol 
 the shell below the brickwork, the outer surface of the headers, and outer 
 surface of tubes. In any given case, the heating surface may be calculated 
 
 Ratio of Heating Surface to Horsepower and of Heating Surface 
 
 to Grate Area. 
 
 Type of Boiler. 
 
 _ A ._ Heating Surface 
 
 Ratio — Horsepower 
 
 Heating Surface 
 Ratio - Grate Area 
 
 Plain cylindrical 
 Elue 
 
 6 to 10 
 8 to 12 
 
 12 to 15 
 20 to 25 
 
 Return - tubular 
 V ertical 
 
 14 to 18 
 
 15 to 20 
 
 25 to 35 
 25 to 30 
 
 Water-tube 
 
 Locomotive 
 
 10 to 12 
 1 to 2 
 
 35 to 40 
 50 to 100 
 
 bv the rules of mensuration, rne ionowmg ewimuc 
 
 nf calculating the heating surface of a return-tubular boiler. 
 
 Example — A horizontal return-tubular boiler has the followmg dimen- 
 sion Diameter 60 in.; length of tubes, 12 ft.; internal diameter of tubes^ 
 3 iS number of tubes ’ 82. Assume that f of the shell is in contact with 
 hnt or flame and f of the two heads are heating surface. _ . 
 
 S fH ' ofshe11 - - m5in " say ' 
 
 . 
 
 Heating surface of tubes = v* TOM - 4 2827 44 sq^n. ^ 
 
 Area of one head — ® X ™ h = V7fiq'92 sa in 
 
 the d helds mu^t be sSbtracted\wice the area cut out by the tubes; 
 this is 82 X 32 X .7854 X 2. = 1,159.26. 
 
 Total heating surface m square feet — 
 
 18,096 + 111,290.4 + 3,769.92-1,159.26 
 
 144 
 
 = 916.64 sq.ft. Ans. 
 
 Pporarle Maximum Work of a Plain Cylindrical Boiler of 120 Sq.Ft. 
 ^ ^HeaSw” Bubfacb and 12 Sq. Ft. Gbate Surface, at 
 
 Different Rates of Driving. 
 
 Rate of driving; lb. 
 
 
 
 
 
 
 
 
 
 
 water evaporated | 
 
 
 
 
 
 
 
 
 
 
 per sq. ft. of heating 
 surface per hour .... 
 
 2 
 
 3 
 
 3.5 
 
 4 
 
 4.5 
 
 5 
 
 6 
 
 7 
 
 8 
 
 Total water evapora- 
 
 
 
 
 
 
 
 
 
 
 ted by 120 sq. ft. 
 
 
 
 
 
 
 
 
 
 
 heating surface peri 
 hour lb 
 
 240.00 
 
 360.00 
 
 420.00; 
 
 o 
 
 © 
 
 oB 
 
 540.00 
 
 600.00 
 
 720.00 
 
 840.00 
 
 960.00 
 
 Horsepower; 34.5 lb. 
 per hour = 1 H. P. 
 
 6.96 
 
 10.43 
 
 12.17 
 
 13.91 
 
 1 
 
 15.65 
 
 1 
 
 17.39 
 
 20.87 
 
 24.35 
 
 27.83 
 
 Pounds water evapo- 
 
 
 
 
 
 
 
 
 
 
 rated per lb. com- 
 
 V^nctihlp 
 
 10.88 
 
 11.30 
 
 11.36 
 
 11.29 
 
 11.20 
 
 11.05 
 
 10.48 
 
 9.48 
 
 i 
 
 8.22 
 
 Uuoiiuiv ? 
 
 Pounds combustible 
 burned per hour ... 
 
 22.10 
 
 31.90 
 
 37.00 
 
 42.50 
 
 48.20 
 
 54.30 
 
 68.70 
 
 88.60 
 
 116.80 
 
 Pounds combustible 
 
 
 
 
 
 
 
 
 
 
 per hour per sq.ft. 
 
 nf OTQt.P 
 
 1.85 
 
 2.65 
 
 3.08 
 
 3.55 
 
 4.02 
 
 4.52 
 
 5.72 
 
 7.38 
 
 9.73 
 
 Pounds combustible 
 
 
 
 
 
 
 
 
 
 
 per hour per horse- 
 
 
 3.17 
 
 3.05 
 
 i 3.04 
 
 3.06 
 
 3.08 
 
 ; 3.12 
 
 3.30 
 
 i 3.64 
 
 4.16 
 
 pu w 
 
 
 
 
 
 1 
 
 
 
 
 
 urAm fhh fi mires in the last line, we see that the amount of fuel required 
 for^given horsepower is nearly 37^ greater when the rate of evaporation is 
 8 lb. Sian when it is 3-5 lb. 
 
DANGER OF EXPLOSION. 
 
 179 
 
 The figures in the preceding table that represent the economy of fuel, viz., 
 “Pounds water evaporated per lb. combustible” and “Pounds combustible 
 per hour per horsepower,” are what may be called “maximum” results, 
 and they are the highest that are likely to be obtained with anthracite coal, 
 with the most skilful tiring and with every other condition most favorable. 
 Unfavorable conditions, such as poor tiring, scale on the inside of the 
 heating surface, dust or soot on the outside, imperfect protection of the top 
 of the boiler from radiation, leaks of air through the brickwork, or leaks of 
 water through the blow-off pipe, may greatly reduce these figures. 
 
 Choice of a Boiler.— Questions that arise under this head in regard to any 
 boiler are: 
 
 1. Is the grate surface sufficient for burning the maximum quantity of 
 coal expected to be used at any time, taking into consideration the availa- 
 ble draft, the quality of the coal, its percentage of ash, whether or not the 
 ash tends to run into clinker, and the facilities, such as shaking grates, for 
 getting rid of the ash or clinker? 
 
 2. Is the furnace of a kind adapted to burn the particular kind of coal 
 used? 
 
 3. Is the heating surface of extent sufficient to absorb so much of the 
 heat generated that the gases escaping into the chimney shall be reasonably 
 low in temperature, say not over 450° F. with anthracite, and 550° F. with 
 bituminous coal? 
 
 4. Are the gas passages so designed and arranged as to compel the gas 
 to traverse at a uniform rate the whole of the heating surface, being not so 
 large at any point as to allow of the gas finding a path of least resistance, or 
 short-circuiting, or, on the other hand, so contracted at any point as to 
 cause an obstruction to the draft? 
 
 These questions being settled in favor of any given boiler — and they may 
 be answered favorably for boilers of many of the common types — the 
 relative merits of the different types may now be considered with reference 
 to their danger of explosion; their probable durability; the character and 
 extent of repairs that may be needed from time to time, and the difficulty, 
 delay, and expense that these may entail; the accessibility of every part of 
 the boiler to inspection, internal and external; the facility for removal 
 of mud and scale from every portion of the inner surface, and of dust and 
 soot from the exterior; the water and steam capacity; the steadiness of 
 water level, and the arrangements for securing dry steam. 
 
 Each one of the points referred to above should be considered carefully 
 by the intending purchaser of any type of boiler with which he is not 
 familiar by experience. The several points may be considered more in 
 detail. 
 
 Danger of Explosion.— All boilers may be exploded by overpressure, such as 
 might be caused by the combination of an inattentive fireman and an 
 inoperative safety valve, or by corrosion weakening the boiler to such 
 an extent as to make it unable to resist the regular working pressure; but 
 some boilers are much more liable to explosion than others. In consider- 
 ing the probability of explosion of any boiler of recent design, it is well to 
 study it to discover whether or not it has any of the features that are known 
 to be dangerous in the plain cylinder, the horizontal tubular, the vertical 
 tubular, and the locomotive boilers. The plain cylinder boiler is liable to 
 explosion from strains induced by its method of suspension, and by changes 
 of temperature. Alternate expansion and contraction may produce a line 
 of weakness in one of the rings, which may finally cause an explosion. A 
 boiler should be so suspended that all its parts are free to change their posi- 
 tion under changes of temperature without straining any part. The 
 circulation of water in the boiler should be sufficient to keep all parts 
 at nearly the same temperature. Cold ffiedwater should not be allowed to 
 come in contact with the shell, as this will cause contraction and strain. 
 The horizontal tubular boiler, and all externally-fired shell boilers, are 
 liable to explosion from overheating of the shell, due to accumulation of 
 mud, scale, or grease, on the portion of the shell lying directly over the fire, 
 to a double thickness of iron with rivets, together with some scale, over the 
 fire, or to low water uncovering and exposing an unriveted part of the shell 
 directly to the hot gases. Vertical tubular boilers are liable to explosion 
 from deposit of mud, scale, or grease, upon the lower tube-sheet, and from 
 low Water allowing the upper part of the tubes to get hot and cease to act 
 as stays to the upper tube-sheet. Locomotive boilers may explode from 
 deposits On the crown sheet, from low water exposing the dry crown sheet 
 
180 
 
 BOILERS. 
 
 to the hot gases, and from corrosion of the staybolts. Double-cylinder 
 boilers, such as the French elephant boiler, and the boilers used at some 
 American blast furnaces, have exploded on account of the formation oi a 
 “steam pocket” on the upper portion of the lower drum, the steam being 
 prevented from escaping from out of the rings of the drum by the lap joint 
 of the adjoining ring, thus making a layer of steam about 4 inch micx 
 against the shell, which was directly exposed to the hot gases. 
 
 Questions to Be Asked Concerning New Boilers. — The causes mentioned 
 above are only a few of the causes of explosions, but they are the principal 
 ones that are due to features of design. These features should be looked for 
 in any new style of boiler, and if they are found they should be considered 
 elements of danger. Such questions as the following may be asked: is tne 
 method of suspension of the boiler such as to allow its parts to be tree to 
 move under changes of temperature? Is the circulation such as to keep all 
 parts at practically the same temperature? Is there a shell with Riveted 
 seams exposed to the fire? Is there a shell exposed to the hre that may at 
 any time be uncovered by water? Is there a crown sheet on which scale 
 mav lodge? Are there vertical or inclined tubes acting as stays to an upper 
 sheet, the upper part of which tubes may become overheated m case 01 
 low water? Are there any stayed sheets, the stays of which are liable to 
 become corroded? Is there any chance for a steam pocket to be formed on a 
 sheet that is exposed to the fire? _ _ , . , . 
 
 In addition to the above-mentioned features of design, which are 
 elements of danger, all boilers, as already stated, are liable 
 to corrosion. Internal corrosion is usually due to acid feedwater, and all 
 boilers are equally liable to it. External corrosion, however, is more liable 
 to take place in some designs of boilers than others, and m 
 rather than others. If any portion of a boiler is in a cold and damp place, it 
 is liable to rust out. For this reason the mud-drums of many modern forms 
 of boilers are made of cast iron, and resist rusting better than either wrought 
 iron or steel. If any part of a boiler, other than a part made of cast iron is 
 liable to be exposed to a cold and damp atmosphere, or covered with damp 
 or ashes or exposed to drip from ram or from leaky pipes, and 
 especially if such part is hidden by brickwork or otherwise so that it can- 
 
 “,:as sasssa ssss: », w . x 
 
 3—ed by interi or external corrosion, but they may also be burned 
 nut It mav “i e ded ag imposs ible to burn a plate or tube of iron or 
 rin matter how high the temperature of the flame, provided one side of 
 ft* Staffs ^cove?S with water. If a steam pocket is formed, so that the 
 wntS does not* touch the metal, or if there is a layer of grease or hard scale 
 thpn the nlate or tube may be burned. In a water tube that is horizontal, o 
 ^trlv so and in which the circulation of water is defective, it is Possible to 
 nearly so, anu , . m drive the water away from the metal, and 
 
 fallow the tabe to ^ burn out In considering the probable durability of 
 a Soiler we may ask the same questions as those that have been asked 
 dimeer of explosion. There are, however, many chances of 
 Wntn^ out a mfnor part of a boiler without serious danger, to one chance 
 nf ^difastrous explomon. Thus the tubes of a water-tube boiler, if allowed 
 £ thicklvcovered with scale, might be burned out again and again 
 
 without causing ^any further destruction at any one time than the rupture 
 Tnew type of boiler should be questioned in regard to the 
 
 u 4 ohhnod of frequent small repairs being necessary, as well as in regard to 
 likelihood ot irequein ^ P We ^ gk; Is the circulation through 
 
 all rnr’ S \he ta er s4h Sarthe water cannot be driven out of any tube 
 anv nortion of a plate, so as to form a steam pocket exposed to high 
 temperature? 0 Are thereproper facilities for removing the scale from every 
 
 P° oin?i« th Thiquts a t?ons U of e dL and of repairs are, in some respects, 
 
 ,*•£1 "to 7a ch other The more infrequent and the less extensive the 
 relat >^ tb p 6 ffreater the durability. The tubes of a boiler, where corroded or 
 ™P a JZv bt replaced and made as good as new. The shell, when it 
 burnt out, may P tched an( j i s then likely to be far from as good as 
 springs a leak, y P d badlv it must be replaced, and to replace the 
 new . When ^the ^shel oau Herei ^ is the advantage of the 
 
 seclional wX-Ue toilerl The sections, or parts of a section, may be 
 
WATER AND STEAM CAPACITY. 
 
 181 
 
 renewed easily, and made as good as new while the shell being far reared 
 from the fire and easily kept dry externally, is: H J 
 out or external corrosion. In considering .the merits o a new; style ol 
 boiler with reference to repairs, we may ask what parts ot tne Doner are 
 most likelv to give out and need to be repaired or replaced? Are these 
 repairs easily effected, how long will they require, and, after they are made, 
 
 iS t F* e eHU» e fof Removal ofYcale and for Inspeotion.-These questions have 
 already lleen discussed to some extent under the head of durability. Some 
 w«t2r-tube boilers now dead and gone, were some years ago put on the 
 market which had no facilities for the removal of scale. It was claimed by 
 SeirproCtersthat they did not need any, because th^utobonwMso 
 raDid Every few years boilers of these types are reinvented, and the same 
 claim is madl for them, that their rapid circulation prevents the formation 
 of scale The fact is that if there is scale-forming material m the water it 
 will be deposited when the water is evaporated, and no amount or kind of 
 circulation will keep it from accumulating on every part of the boiler, jmd 
 in every kind of tubes, vertical, horizontal, and inclined. I have seen 
 the nearly vertical circulating tubes of a water-tube boiler, 
 circulation is nine times as fast as the average circulation m the inclined 
 tubes! nearly full of scale; that is, a 4" tube had an opening m it of less than 
 1 in in diameter. This was due to carelessness m blowing off the boiler, or 
 exceptionally bad feedwater, or both. If circulation would prevent scaling 
 
 at Water^nd^tf^CaJadty^-B is claimed for some forms of boilers that 
 they are better than others because they have a larger water or steam 
 capacity. Great water capacity is useful where the demands for steam are 
 extremely fluctuating, as in a rolling mill or a sugar refinery, where it is 
 desirable^to store up heat in the water in the boilers during the periods of 
 the least demand, to be given out during periods of greatest demand. Large 
 water capacity is obiectionable in boilers for factories, usually, especially if 
 ?hev do not run at night, and the boilers are cooled down, because there is a 
 
 lar Je Quantity of w | te r to be heated before starting each morning. If 
 -mnid steaming” or the ability to get up steam quickly from cold water, or 
 to rSse the pressure quickly, is desired, large water capacity is a. detriment. 
 Thl dSWe steam capacity is usually overrated. It is useful to 
 enable the steam to be drained from water before it escapes into the steam 
 r>i ne but the same result can be effected by means of a dry Pipe, as m 
 fo?omotive and SIrine practice, in which the steam space m the boiler is 
 very small hi proportioS to the horsepower. Large steam space in the 
 io no irrmortance for storing energy or equalizing the pressure 
 ^irinVfhfstrokeXan^ enjine The water in the boiler is the place to store 
 heat and if the steam pipe leading to an engine is of such small capacity 
 that it reduces the pressure, the remedy is a steam reservoir close to the 
 
 eng St n eldTn«s a oTwate?Te«eL-This requires either a large area of water sur- 
 Steadiness oTwaxer Leve. h slow i y by fluctuations in the demand 
 
 fofstlam or in the of thffl-pumpT or else constant, and preferably 
 a^itomaUc° regulation 1 of the feedwater supply to suit the steam demand 
 A ranidlv lowering water level is apt to expose dry sheets or tubes to the 
 ^ctfon of thl hot gases, and thus be a source of danger A rapidly rising 
 level may before it is seen by the fireman, cause water to be carried over 
 
 ° f f^ilfp'boifor 'of a 
 
 boiler is important for two reasons: (1) To keep all parts of the toiler of a 
 uniform temperature, and (2) to prevent the adhesion of steam bubbles to 
 ihe f SStee whfch may cause overheating of the metal It is claimed by 
 some manufacturers that the rapid circulation of water m their boilers 
 tends to make them more economical than others. I have as yet, however, 
 to findanv proof that increased rapidity of circulation of water beyond 
 t h at usual 1 v found in any boiler will give increased economy. We know 
 that increased rate of flow of air over radiating surfaces increases the 
 amount of heat transmitted through the surface, but 
 
 increased circulation, cold air is continually brought into contact with 
 the surface making an increased difference of temperature on the two sides, 
 which causes increased transmission. But by increasing the rapidity oi 
 circulation^ a steam boiler we cannot vary the difference of temperature to 
 any appreciable extent, for the water and the steam in the boiler are at 
 
182 
 
 BOILERS . 
 
 about the same temperature throughout. The ordinary or “Scotch” form 
 of marine boiler shows an exception to the general rule of uniformity of 
 temperature of water throughout the boiler, but the temperature above the 
 level of the lower fire tubes is practically uniform. 
 
 INCRUSTATION AND SCALE. 
 
 Nearly all waters contain foreign substances in a greater or less degree, 
 and though this may be a small amount in each gallon, it becomes of 
 importance where large quantities are evaporated. For instance, a 100 H. P. 
 boiler evaporates 30,000 lb. of water in 10 hours, or 390 tons per month; in 
 comparatively pure water there would be 88 lb. of solid matter in that 
 quantity, and in many kinds of spring water as much as 2,000 lb. 
 
 The nature and hardness of the scale formed of this matter will depend 
 on the kind of substances held in solution and suspension. Analyses of a 
 great variety of incrustations show that carbonate and sulphate of lime 
 form the larger part of all ordinary scale, that from carbonate being soft 
 and granular, and that from sulphate, hard and crystalline. Organic 
 substances in connection with carbonate of lime will also make a hard and 
 troublesome scale. 
 
 The presence of scale or sediment in a boiler results in loss of fuel, 
 burning and cracking of the boiler, predisposes to explosion, and leads to 
 extensive repairs, it is estimated that the presence of in. of scale causes 
 a loss of 13 $ of fuel; i in., 38^; and ± in., 60$. The Railway Master 
 Mechanics’ Association of the United States estimates that the loss of fuel, 
 extra repairs, etc., due to incrustation, amount to an average of 8750 per 
 annum for every locomotive in the Middle and Western States, and it must 
 be nearly the same for the same power in stationary boilers. 
 
 Causes of Incrustation.— 
 
 1. Deposition of suspended matter. 
 
 2. Deposition of salts from concentration. 
 
 3. Deposition of carbonates of lime and magnesia, by boiling off carbonic 
 acid, which holds them in solution. 
 
 4. Deposition of sulphates of lime, because sulphate of lime is soluble in 
 cold water, less soluble in hot water, insoluble above 270° F. 
 
 5. Deposit of magnesia, because magnesium salts decompose at high 
 temperatures. 
 
 6. Deposition of lime soap, iron soap, etc., formed by saponification of 
 grease. 
 
 Method of Preventing Incrustation.— 
 
 1. Filtration. 
 
 2. Blowing off. 
 
 3. Use of internal collecting apparatus, or devices, for directing the 
 circulation. 
 
 4. Heating feedwater. 
 
 5. Chemical or other treatment of water in boiler. 
 
 6. Introduction of zinc in boiler. 
 
 7. Chemical treatment of water outside of boiler. 
 
 Troublesome Substance. 
 
 Trouble. 
 
 Remedy or Palliation. 
 
 Sediment, mud, clay, etc. 
 
 Readily soluble salts. 
 
 Bicarbonates of lime, magnesia, and iron. 
 Sulphate of lime. 
 
 Chloride and sulphate of magnesium. 
 Carbonate of soda in large amounts. 
 Acid (in mine water). 
 
 Dissolved carbonic acid and oxygen. 
 
 Incrustation. 
 
 Incrustation. 
 
 Incrustation. 
 
 Incrustation. 
 
 Corrosion. 
 
 Priming. 
 
 Corrosion. 
 
 Corrosion. 
 
 Grease (from condensed water). 
 
 Corrosion. 
 
 Organic matter (sewage). 
 
 Priming. 
 
 Organic matter. 
 
 Corrosion. 
 
 Filtration; blowing off. 
 
 Blowing off. 
 
 Heating feed; addition of caustic soda, 
 lime, or magnesia, etc. 
 
 Addition of carbonate of soda, barium 
 chloride, etc. 
 
 Addition of carbonate soda, etc. 
 
 Addition of barium chloride, etc. 
 
 Alkali. , 
 
 Heating feed; addition of caustic soda, 
 slaked lime, etc. 
 
 Slaked lime and filtering. Substitute 
 mineral oil. 
 
 Precipitate with alum or ferric chloride, 
 and filter. 
 
 Precipitate with alum or ferric chloride, 
 and filter. 
 
PREVENTION OF SCALE. 
 
 183 
 
 Means of Prevention. — It is absolutely essential to the successful use of any 
 boiler except in pure water, that it be accessible for the removal of scale, 
 for though a rapid circulation of water will delay the deposit, and certain 
 chemicals will change its character, yet the most certain cure is Periodical 
 inspection and mechanical cleaning. This may , however, be rendered less 
 frequently necessary, and the use of very bad water more practical bv the 
 employment of some preventives. The following are fair samples ol those 
 
 in use. with their results: . . , ,, , . . 
 
 M Bidard’s observations show that “ anti-mcrustators containing 
 organic matter help rather than hinder incrustations, and are therefore 
 
 t0 Oak hemlock, and other barks and woods, sumac, catechu, logwood, etc. 
 are effective in waters containing carbonates of lime or magnesia, by reason 
 of their tannic acid, but are injurious to the iron and not to be recom- 
 
 me Mo!asses, cane juice, vinegar, fruits, distillery slops etc. have been used 
 With success so far as scale is concerned, by reason of the acetic acid that 
 they contain, but this is even more injurious to the iron than tannic acid, 
 while the organic matter forms a scale with sulphate of lime when it is 
 
 Pre MUk of lime and metallic zinc have been used with success in waters 
 charged with bicarbonate of lime, reducing the bicarbonate to the insoluble 
 
 Ca Barium chloride and milk of lime are said to be used with good effect at 
 Krupp’s works, in Prussia, for waters impregnated with gypsum. 
 
 Soda ash and other alkalies are very useful in waters containing sulphate 
 of lime bv converting it into a carbonate, and so forming a soft scale 
 easily cleaned. But when used in excess they cause foaming, particularly 
 where there is oil coming from the engine, with which they form soap. All 
 soapv substances are objectionable for the same reason. 
 
 Petroleum has been much used of late years. It acts best m waters m 
 which sulphate of lime predominates. Sulphate of lime is the injurious 
 Stance in nearly all mine waters, and petroleum when properly 
 prepared, is a good preventive of scale and pitting. Crude petroleum 
 Should not be used, as it sometimes helps in forming a very injurious scale. 
 Refined petroleum, on the other hand, is useless, as it vaporizes at a 
 temperature below that of boiling water. Therefore, only such prepara- 
 tions should be used as will not vaporize below 500° F. 
 
 Tannate of soda* is a good preparation for general use, but m waters con- 
 taining much sulphate, it should be supplemented by a portion of carbonate 
 
 of ^rom the leaves of the eucalyptus is found to work well in 
 
 ^^o^muddy 1 water^^rticularly if it contain salts ? f lime, no preventive of 
 incrustation will prevail except filtration, and in almost every instance the 
 use of a filter, either alone or in connection with some means of precipita- 
 ting the solid matter from solution, will be found very desirable. . ,, 
 
 In all cases where impure or hard waters are used, frequent blowing 
 from the mud-drum is necessary to carry off the accumulated matter, 
 which if allowed to remain would form scale. ....... 
 
 When boilers are coated with a hard scale, difficult to remove, it will be 
 found that the addition of i lb. caustic soda per horsepower, and steaming 
 for some hours, according to the thickness of the scale, just before cleaning, 
 will ffreatlv facilitate that operation, rendering the scale soft and loose. 
 This should be done, if possible, when the boilers are not otherwise m use. 
 
 COVERING FOR BOILERS, STEAM PIPES, ETC. 
 
 The losses by radiation from unclothed pipes and vessels containing 
 steam are considerable, and in the case of pipes leading to steam engines, are 
 magnified by the action of the condensed water m the cylinder. It there- 
 fore is important that such pipes should be well protected The following 
 table ffives the loss of heat from steam pipes naked, and clothed with wool or 
 hair felt, of different thickness, the steam pressure being assumed at 75 lb., 
 
 There is a wide difference in the value of different substances for protec- 
 tion from radiation, their values varying nearly in the reverse ratio to their 
 conducting power for heat, up to their ability to transmit as much heat as 
 
184 
 
 BOILERS. 
 
 the surface of the pipe will radiate, after which they become detrimental, 
 rather than useful, as covering. This point is reached nearly at baked clay 
 or brick. 
 
 Table of Loss of Heat From Steam Pipes. 
 
 Outside Diameter of Pipe, Without Felt 
 
 
 Thickness of Covering. In< 
 
 2 In. Diameter. 
 
 4 In. 
 
 Diameter. 
 
 6 In. Diameter. 
 
 8 In. Diameter. 
 
 12 In. Diameter. 
 
 Loss in Units 
 per Foot Run per 
 Hour. 
 
 Ratio of Loss. | 
 
 • 1 
 
 Feet in Length per 
 H. P. Lost. 
 
 Loss in Units 
 per Foot Run per 
 Hour. 
 
 Ratio of Loss. 
 
 Feet in Length per 
 H. P. Lost. 
 
 Loss in Units 
 per Foot Run per 
 Hour. 
 
 Ratio of Loss. 
 
 Feet in Length per 
 H. P. Lost. 
 
 Loss in Units 
 per Foot Run per 
 Hour. 
 
 Ratio of Loss. 
 
 Feet in Length per 
 H. P. Lost. 
 
 Loss in Units 
 per Foot Run per 
 Hour. 
 
 | Ratio of Loss. 
 
 Feet in Length per 
 H. P. Lost. 
 
 0 
 
 219.0 
 
 1.00 
 
 132 
 
 390.8 
 
 1.00 
 
 75 
 
 624.1 
 
 1.000 
 
 46 
 
 729.8 
 
 1.000 
 
 40 
 
 1,077.4 
 
 1.000 
 
 26 
 
 i 
 
 100.7 
 
 .46 
 
 288 
 
 180.9 
 
 .46 
 
 160 
 
 
 
 
 
 
 
 
 
 
 i 
 
 65.7 
 
 .30 
 
 441 
 
 117.2 
 
 .30 
 
 247 
 
 187.2 
 
 .300 
 
 154 
 
 219.6 
 
 .301 
 
 132 
 
 301.7 
 
 -.280 
 
 92 
 
 ] 
 
 43.8 
 
 .20 
 
 662 
 
 73.9 
 
 .18 
 
 392 
 
 111.0 
 
 .178 
 
 261 
 
 128.3 
 
 .176 
 
 225 
 
 185.3 
 
 .172 
 
 157 
 
 2 
 
 28.4 
 
 .13 
 
 1,020 
 
 44.7 
 
 .11 
 
 648 
 
 66.2 
 
 .106 
 
 438 
 
 75.2 
 
 .103 
 
 385 
 
 98.0 
 
 .091 
 
 294 
 
 4 
 
 19.8 
 
 .09 
 
 1,464 
 
 28.1 
 
 .07 
 
 1,031 
 
 41.2 
 
 .066 
 
 703 
 
 46.0 
 
 .063 
 
 630 
 
 60.3 
 
 .056 
 
 486 
 
 6 
 
 
 
 
 23.4 
 
 .06 
 
 1,238 
 
 33.7 
 
 .054 
 
 860 
 
 34.3 
 
 .047 
 
 845 
 
 45.2 
 
 .042 
 
 642 
 
 A smooth or polished surface is of itself a good protection, polished tin 
 or Russia iron having a ratio, for radiation, of 53 to 100 for cast iron. Mere 
 color makes but little difference. 
 
 Table of Conducting Power of Various Substances. 
 
 ( From Peclet.) 
 
 Substance. 
 
 Conducting 
 
 Power. 
 
 Substance. 
 
 * 
 
 Conducting 
 
 Power. 
 
 Blotting paper 
 
 .274 
 
 Wood, across fiber 
 
 .83 
 
 Eiderdown 
 
 .314 
 
 Cork 
 
 1.15 
 
 Cotton or Wool, | 
 
 QOQ 
 
 Coke, pulverized 
 
 1.29 
 
 any density j 
 
 .oZo 
 
 India rubber 
 
 1.37 
 
 Hemp, canvas 
 
 .418 
 
 Wood, with fiber 
 
 1.40 
 
 Mahogany dust 
 
 .523 
 
 Plaster of Paris 
 
 3.86 
 
 Wood ashes 
 
 .531 
 
 Baked clay 
 
 4.83 
 
 Straw 
 
 .563 
 
 Glass 
 
 6.60 
 
 Charcoal powder 
 
 .636 
 
 Stone 
 
 13.68 
 
 Hair or wool felt has the disadvantage of becoming soon charred from 
 the heat of steam at high pressure, and sometimes of taking fire therefrom. 
 This has led to a variety of “cements” for covering pipes— composed gen- 
 erally of clay mixed with different substances, as asbestos, paper fiber, 
 charcoal, etc. A series of careful experiments, made at the Massachusetts 
 Institute of Technology in 1871, showed the condensation of steam in a pipe 
 covered by one of them, as compared with a naked pipe, and one clothed 
 with hair felt, was 100 for the naked pipe, 67 for the “cement” covering, 
 and 27 for the hair felt. . . . 
 
 The presence of sulphur in the best coverings and its recognized injurious 
 effects make it imperative that moisture be kept from the coverings, for, 
 if present, it will surely combine with the sulphur, thus making it active. 
 Stated in other words, keep the pipes and coverings in good repair. Much 
 of the inefficiency of coverings is due to the lack of attention given them; 
 they are often seen hanging loosely from the pipe which they are supposed 
 to protect. 
 
CARE OF BOILERS. 
 
 185 
 
 Table of Relative Value of Non-Conductors. 
 ( From Chas. E. Emery, Ph. D.) 
 
 Non-Conductor. 
 
 Value. 
 
 Non-Conductor. 
 
 Value. 
 
 Wnnl felt 
 
 1.000 
 
 Loam, dry and open 
 
 .550 
 
 Mineral wool No 2 
 
 .832 
 
 Slaked lime 
 
 .480 
 
 lYllllvlcU W wl i’v. ^ 
 
 Mineral with tar 
 
 .715 
 
 Gas-house carbon 
 
 .470 
 
 1VIII1CI cXA W ini 
 
 Qo wflnat. 
 
 .680 
 
 Asbestos 
 
 .363 
 
 .345 
 
 oaWUUSl 
 
 Mineral wool No. 1 
 
 PViotpaqI 
 
 .676 
 
 .632 
 
 f'lnal qkLpk 
 
 Coke in lumps 
 
 .277 
 
 T)i y-i a q PTOSS fiber 
 
 .553 
 
 Air space undivided 
 
 .136 
 
 JL 111“ W UUU dblvoo AX MUX 
 
 
 Carbonate of magnesia, as compared with wool felt at 1.000, has a rela- 
 tive value of 472. This is determined from tests by Prof. Ordway, of Boston, 
 
 an '^^ ^g, - a 
 
 ^°^rk°clSps, n cemented C together with water glass, make one of the best 
 eoverin^s known but a very efficient one, may be 
 
 annlied as ^follows First, wrap the pipe in asbestos paper, though this may 
 be dispenLd wUh; then lay slips of wood lengthways from 6 to 12, accord- 
 ino- to size of pipe, binding them in position with wire or cord, and 
 around the framework thus constructed wrap roofing paper, fastening it 
 bv naste or twine. For flanged pipe, space may be left for access to the 
 bolts which space should be filled with felt. If exposed to weather, use 
 tkrrld naper T paint the exterior. A French plan is to cover the surface 
 with a rough flour paste, mixed with sawdust until it forms a nmderately 
 stiff dough Apply with a trowel in layers of about 4 in. thick, give 4 or 5 
 layers iifall. If P iron surfaces are well cleaned from grease, the adhesion is 
 perfect. For copper, first apply a hot solution of clay m water. A coating 
 of tar renders the composition impervious to the weather. 
 
 DATA FOR PROPORTIONING AN ECONOMIZER. 
 
 ( The Green Fuel Economizer Co., Matteawan, N. Y.) 
 
 The following estimate is given for the amount of heating surface to be 
 provided in an Iconomizer to be used in connection with a given amount of 
 
 b01 Bv allowing 4 sq. ft. of heating surface per boiler horsepower (Centennial 
 rating^W lb of water evaporated from and at 212° = I E P.), we are able 
 to raise the feedwater 60° for every 100° reduction m the temperature enter- 
 ing the economizer with gases from 450° to 600°. These results are cor- 
 
 r ° b WH^?lm^t^p?rature S o?the gases entering the economizer at 600° to 700°, 
 we have allowed 4£ to 5 sq. ft. of heating surface per boiler horsepower, 
 and for every 100° reduction of gases we have obtained about 65 rise m 
 temperature of the water; the temperature of the feedwater entering aver- 
 
 agl w1thToOO° sq°ft. 2 of ’ boiler heating surface (plain cylinder boilers) develop- 
 ing 1 000 H. P., we should recommend using 5 sq. ft of economizer heating 
 surface per B. H. P., or an economizer of about 500 tubes, and it should heat 
 the feedwater about 300°. 
 
 CARE OF BOILERS. 
 
 1 Safetv Valves —Great care should be exercised to see that these valves 
 are ample in size and in working order. Overloading or neglect frequently 
 leads to the most disastrous results. Safety valves should be tried at least 
 once every day, to see that they act freely. 
 
 2 Pressure Gauge.-The steam gauge should stand at zero when the 
 pressure is off, and it should show same pressure as the safety valve when 
 that is blowing off. If not, then one is wrong, and the gauge should be 
 tested by one known to be correct. 
 
186 
 
 BOILERS. 
 
 3. Water Level.— The first duty of an engineer before starting, or at the 
 beginning of his watch, is to see that the water is at the proper height. Do 
 not rely on glass gauges, floats, or water alarms, but try the gauge-cocks. 
 If they do not agree with water gauge, learn the cause and correct it. 
 
 4. Gauge-cocks and water gauges must be kept clean. Water gauges should 
 be blown out frequently, and the glasses and passages to them kept clean. 
 The Manchester, England, Boiler Association attributes more accidents to 
 inattention to water gauges than to all other causes put together. 
 
 5. Feed-Pump or Injector. — These should be kept in perfect order, and be 
 of ample size. No make of pump can be expected to be continuously 
 reliable without regular and careful attention. It is always safe to have two 
 means of feeding a boiler. Check-valves and self-acting feed-valves should 
 be frequently examined and cleaned. Satisfy yourself frequently that the 
 valve is acting when the feed-pump is at work. 
 
 6. Low Water.— In case of low water, immediately cover the fire with 
 ashes (wet if possible) or any earth that may be at hand. If nothing else is 
 handy, use fresh coal. Draw fire as soon as it can be done without increas- 
 ing the heat. Neither turn on the feed, start nor stop engine, nor lift safety 
 valve until fires are out and the boiler cooled down. 
 
 7. Blisters and Cracks.— These are liable to occur in the best plate iron. 
 When the first indication appears, there must be no delay in having it 
 carefully examined and properly cared for. 
 
 8. Fusible plugs, when used, must be examined when the boiler is cleaned, 
 and carefully scraped clean on both the water and fire sides, or they are 
 liable not to act. 
 
 9. Firing.— Fire evenly and regularly, a little at a time. Moderately 
 thick fires are most economical, but thin firing must be used where the 
 draft is poor. Take care to keep grates evenly covered, and allow no air 
 holes in the fire. Do not “ clean” fires oftener than necessary. With 
 bituminous coal, a “coking fire,” i. e., firing in front and shoving back 
 when coked, gives best results if properly managed. 
 
 10. Cleaning.— All heating surfaces must be kept clean outside and in, or 
 there will be a serious waste of fuel. The frequency of cleaning will depend 
 on the nature of fuel and water. When a new feedwater supply is 
 introduced, its effect upon the boiler should be closely observed, as this new 
 supply may be either an advantage or a detriment as compared with the 
 working of the boiler previous to its introduction. As a rule, never allow 
 over T y r scale or soot to collect on surfaces between cleanings. Handholes 
 should be frequently removed and surfaces examined, particularly in 
 case of a new boiler, until proper intervals have been established by 
 experience. 
 
 The exterior of tubes can be kept clean by the use of blowing pipe and 
 hose through openings provided for that purpose. In using smoky fuel, it is 
 best to occasionally brush the surfaces when steam is off. 
 
 11. Hot Feedwater.— Cold water should never be fed into any boiler when 
 it can be avoided, but when necessary it should be caused to mix with the 
 heated water before coming in contact with any portion of the boiler. 
 
 12 Foaming.— When foaming occurs in a boiler, checking the outflow of 
 steam will usually stop it. If caused by dirty water, blowing down and 
 pumping up will generally cure it. In cases of violent foaming, check the 
 draft and fires. 
 
 13 Air Leaks.— Be sure that all openings for admission of air to boiler or 
 flues, except through the fire, are carefully stopped. This is frequently an 
 unsuspected cause of serious waste. 
 
 14 Blowing Off.— If feedwater is muddy or salt, blow off a portion fre- 
 quentlv, according to condition of water. Empty the boiler every week or 
 two and fill up afresh. When surface blow cocks are used, they should be 
 often opened for a few minutes at a time. Make sure no water is escaping 
 from the blow-off cock when it is supposed to be closed. Blow-off cocks and 
 check-valves should b'e examined every time the boiler is cleaned. Never 
 empty the boiler while the brickwork is hot. 
 
 15. Leaks.— When leaks are discovered, they should be repaired as soon 
 as possible. 
 
 16 Filling Up.— Never pump cold water into a hot boiler. Many times 
 leaks, and, in shell boilers, serious weaknesses, and sometimes explosions 
 are the result of such an action. 
 
THICKNESS OF BOILER IRON. 
 
 187 
 
 \ 
 
 17 Dampness.— Take care that no water comes in contact with the 
 exterior of P the boiler from any cause, as it tends to corrode and weaken 
 the boiler. Beware of all dampness in seatmgs and coverings. 
 
 18 Galvanic Action.— Examine frequently parts in contact with copper or 
 
 brass where water is present, for signs of corrosion. If water is salt or acid, 
 some’ metallic zinc placed in the boiler will usually prevent corrosion, but it 
 will need attention and renewal from time to time. , 
 
 19 Rapid Firing.— In boilers with thick plates or seams exposed to the 
 fire steam should 8 be raised slowly, and rapid or intense firing avoided. 
 With thin water tubes, however, and adequate water circulation, no dam- 
 
 ag ^ 0 an stand?ng r Unu sel— If a boiler is not required for some time, empty and 
 dry it thoroughly* If this is impracticable, fill it quite full of water and put 
 in a quantity of common washing soda. External parts exposed to damp- 
 
 ne ?l S X d a Aould be looked after at least once 
 
 a vear given necessary repairs, refitted to the pipe, and the spaces due to 
 ffiLfe taken up Little can be expected from the best non-conductors if 
 they are allowed to become saturated with water, or if air-currents are 
 nermitted to circulate between them and the pipe. . i_, vrv f 
 
 P 22. General Cleanliness.— All things about the boiler room should be kept 
 clean and in good order. Negligence tends to waste and decay. 
 
 THICKNESS OF BOILER IRON REQUIRED AND PRESSURE 
 ALLOWED BY THE LAWS OF THE UNITED STATES. 
 
 Pressure Equivalent to the Standard for a Boiler 42 In. in Diam- 
 eter and £ In. Thick. 
 
 Thickness. 
 16ths. • 
 
 5 
 
 4* 
 
 4 
 
 3» 
 
 3£ 
 
 3 
 
 Diameter. 
 
 34 In. 
 Lb. 
 
 36 In. 
 Lb. 
 
 38 In. 
 Lb. 
 
 40 In. 
 Lb. 
 
 42 In. 
 Lb. 
 
 44 In. 
 Lb. 
 
 46 In. 
 Lb. 
 
 169.9 
 
 158.5 
 
 135.9 
 
 124.5 
 113.2 
 
 101.9 
 
 160.4 
 
 149.7 
 
 128.3 
 
 117.6 
 
 106.9 
 
 96.2 
 
 152.0 
 
 141.8 
 
 121.6 
 
 111.4 
 
 101.3 
 
 91.2 
 
 144.4 
 134.7 
 
 115.5 
 105.9 
 
 96.2 
 
 82.6 
 
 137.5 
 
 128.3 
 
 110.0 
 
 100.8 
 
 91.7 
 
 82.5 
 
 131.2 
 
 122.5 
 
 105.0 
 
 96.2 
 
 87.5 
 
 78.7 
 
 125.5 
 
 117.2 
 
 100.0 
 
 92.0 
 
 83.0 
 
 75.1 
 
 The rule for finding the proper sectional area for the narrowest part of 
 the nozzle is given by Rankine, S. E., page 477, as follows: 
 
 cubic feet per hour gross feedwater 
 Area in square inches — 800 V pressure in atmospheres 
 
 Diameter of 
 Throat. Decimals 
 of an Inch. 
 
 .10 
 
 .15 
 
 .20 
 
 .25 
 
 .30 
 
 Delivery in Gallons per Hour with a Pressure 
 per Square Inch of 
 
 30 Lb. 
 
 45 Lb. 
 
 60 Lb. 
 
 75 Lb. 
 
 56 
 
 69 
 
 80 
 
 89 
 
 127 
 
 156 
 
 180 
 
 201 
 
 226 
 
 278 
 
 321 
 
 360 
 
 354 
 
 434 
 
 502 
 
 561 
 
 505 
 
 624 
 
 722 
 
 807 
 
 90 Lb. 
 
 98 
 
 221 
 
 393 
 
 615 
 
 884 
 
188 
 
 BOILERS. 
 
 Pressure of Steam at Different Temperatures. 
 
 ( Results of Experiments Made by the Franklin Institute.) 
 
 Pressure. 
 Inches of 
 Mercury. 
 
 Tempera- 
 
 ture. 
 
 Degrees F. 
 
 Pressure. 
 Inches of 
 Mercury. 
 
 Tempera- 
 
 ture. 
 
 Degrees F. 
 
 Pressure. 
 Inches of 
 Mercury. 
 
 Tempera- 
 
 ture. 
 
 Degrees F. 
 
 30 
 
 212.0 
 
 135 
 
 298.5 
 
 225 
 
 . 331.0 
 
 45 
 
 235.0 
 
 150 
 
 304.5 
 
 240 
 
 336.0 
 
 60 
 
 250.0 
 
 165 
 
 310.0 
 
 255 
 
 340.5 
 
 75 
 
 264.0 
 
 180 
 
 315.5 
 
 270 
 
 345.0 
 
 90 
 
 275.0 
 
 195 
 
 321.0 
 
 285 
 
 349.0 
 
 105 
 
 284.0 
 
 | 210 
 
 326.0 
 
 300 
 
 352.5 
 
 120 
 
 291.5 
 
 
 
 
 
 Maximum Economy of Plain Cylinder Boilers. 
 
 Pounds of Water Evaporated From and at 212°. 
 
 Per sq. ft. heating 
 surface per hour 
 
 1.70 
 
 2.00 
 
 2.60 
 
 
 3.50 
 
 4.00 
 
 4.50 
 
 5.00 
 
 6.00 
 
 7.00 
 
 8.00 
 
 Per lb. combustible, 
 maximum of other 
 boilers, Centennial 
 tests 
 
 11.90 
 
 12.00 
 
 12.10 
 
 12.05 
 
 12.00 
 
 11.85 
 
 11.70 
 
 11.50 
 
 10.85 
 
 9.80 
 
 8.50 
 
 Subtract extra radia- 
 tion loss for cylin- 
 der boilers 
 
 1.32 
 
 1.12 
 
 .87 
 
 .75 
 
 .64 
 
 .56 
 
 .50 
 
 .45 
 
 .37 
 
 .32 
 
 .28 
 
 Probable maximum 
 per lb. combus- 
 tible, cylinder 
 boilprs 
 
 10.58 
 
 10.88 
 
 11.23 
 
 11.30 
 
 11.36 
 
 i 
 
 11.29 
 
 11.20 
 
 11.05 
 
 10.48 
 
 9.48 
 
 8.22 
 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 ■ 
 
 Scheme for Boiler Test. 
 
 1 
 
 Mn mhpr nf t 
 
 
 1 
 
 o 
 
 Mn/lp hv 
 
 
 Z 
 
 Q 
 
 Tvnp nf* Vinllpr - 
 
 
 O 
 
 A 
 
 Tjotp of t,pst, - 
 
 
 1 
 
 Rnratinn of t,PSt 
 
 Hr. 
 
 O 
 
 6 
 
 n 
 
 Dimensions and Proportions. 
 
 Number of boilers tested 
 
 In. 
 
 / 
 
 Q 
 
 E/laLllclCi, uuiici -------- 
 
 Ft. In. 
 
 O 
 
 Q 
 
 
 Ft. In. 
 
 V 
 
 in 
 
 T.PnO’th orrc itP 
 
 Ft. In. 
 
 1U 
 
 JUCllgUl) g 1 
 
 No. 
 
 1 9 
 
 T^iorYidtpr nf tllhps 
 
 In. 
 
 1Z 
 
 IQ 
 
 
 Ft. In. 
 
 lo 
 
 1 A 
 
 T'rfctcil wqIpt 1 Y \ pn t i n p* siirfn op 
 
 Sq. Ft. 
 
 
 1 Utdl Wdtt/l llucttiiig 
 
 Tnfol ctoom liPQtiTiO' cn T*"fVl PP 
 
 Sq. Ft. 
 Sq. Ft. 
 
 * 1> 
 
 ID 
 1 A 
 
 1 Utdl otCdlll DUlldvv 
 
 turfopp t\ot hnilpp 
 
 lo 
 
 1 7 
 
 Ijrldtv 0 U .1 idCv uui 1 vi 
 
 Par norit air C'AQPP 1TI OTQtp 
 
 1 / 
 18 
 1 Q 
 
 it/l belli, dll opdbe 111 gicite 
 
 Ratio water heating to grate surface - 
 
 Sq. Ft. 
 
 on 
 
 
 Ft. 
 
 ZU 
 
 21 
 
 09 
 
 ilGlglll SlaUi. dUUYc UedU piaieo 
 
 Ratio stack area to grate surface - 
 
 Average Pressures. 
 
 A irw aotvVi orn Taxt Kq vayyi ptpp 
 
 In. 
 
 zz 
 
 9Q 
 
 A Lino s pile i c uy udi uiiiei^i 
 
 GtooTYi nroccnrh hv o*Q 11 o*P - 
 
 Lb. 
 
 Zo 
 
 OXalll pieooUlG uy gdUge 
 
 
24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 CHIMNEYS. 
 
 Scheme for Boiler Test— ( Continued). 
 
 Force chimney draft, inches water 
 
 Force blast in ash pit, inches water •--- 
 
 Average Temperatures. 
 
 Of external air 
 
 Oftireroom 
 
 Of steam 
 
 Of feedwater before heater 
 
 Of feedwater after heater 
 
 Of stack gases •• •••: 
 
 Fuel. 
 
 Kind of coal 
 
 Total coal consumed 
 
 Moisture in coal 
 
 Total dry coal consumed - 
 
 Total ash and refuse - 
 
 Per cent, ash and refuse in dry coal 
 
 Total combustible consumed - 
 
 Calorimetric Tests. 
 
 Per cent, moisture in steam 
 
 Degrees superheat in steam 
 
 "Water . 
 
 Total water pumped into boiler ----- 
 
 Water evaporated corrected for quality of steam 
 
 Equiv. water evap. to dry steam from and at 212 - 
 
 Equiv. water evap. to dry steam from and at 212° per hour 
 Economic Evaporation. 
 
 Water evap. per lb. dry coal actual pressures and temp 
 
 Equiv. water evap. lb. dry coal from and at 212 
 
 Equiv. water evap. lb. combustible from and at 212 
 
 Rate of Combustion. 
 
 Dry coal burned per hr. per sq. ft. grate surface 
 
 Combustible burned per hr. per sq. ft. grate surface 
 
 Dry coal per hour per H. P. developed 
 
 Rate of Evaporation. 
 
 Water evap. from and f Per sq. ft. grate surface 
 
 at 212° per hour j Per sq. ft. heating surface 
 
 Commercial Horsepower. 
 
 Basis 20 lb. water from 100° feed to 70 lb. steam per hour.... 
 
 Horsepower builders rating - - 
 
 Heating surface to one horsepower developed 
 
 Per cent, total horsepower d ^e to feed heater 
 
 189 
 
 \ 
 
 In. 
 
 In. 
 
 °F. 
 
 °f" 
 
 °F. 
 
 °f’ 
 
 °F. 
 
 °f! 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 i 
 
 Lb. 
 
 °F. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 Lb. 
 
 H. P. 
 H. P. 
 Sq. It. 
 
 ]o 
 
 CHIMNEYS. 
 
 Chimneys have two important duties to perform, the first being to carry 
 off the waste furnace gases, which requires size, and the second to produce 
 a draft sufficient to insure the complete combustion of the ^uei, which 
 requires height. The area of a chimney is usually made from - f to is as large 
 as the area of the furnace grates, or of about the same cross-section as the 
 cross-sectional area of the flues or tubes; we have, therefore ^ a c omparaUvely 
 simple method of determining one of the required dimensioi^ of a ch^ 
 and, when this is known, it becomes an easy matter to determine the height 
 of the chimney when the horsepower of the boiler has been ascertained. 
 
 The horsepower of a boiler being given and the necessary chimney area 
 having been determined, the following rule gives the required height that 
 
 the chimney must be to produce the necessary draft: , 
 
 Rule —From 3.33 times the area of the chimney in square feet, subtract twice the 
 rxfvho nron nffhfi nbim/npn in. Ran, are feet, and divide the given horsepower 
 
 Ru | e —From 3.33 times the area oj me chimney in squuicjv », 
 square root of the area of the chimney in square feet, and divide the given hoi sepower 
 by the remainder. The square of the quotient will be the height of the chimney in feet. 
 A — area of chimney; 
 horsepower of boil 
 height of chimney 
 
 / x— Jy 
 
 \3.33 A — 2]/ A/ 
 
 Then, 
 
190 
 
 STEAM ENGINES. 
 
 Example. — What must be the height of a chimney that is to have a cross- 
 sectional area of 7 sq. ft., and to supply the draft for a 141-horsepower boiler? 
 
 h = \3.33 X 7 — 2]/ 7 ) = - (.3.33 X 7 - (2 X 2.65 ) ) = 613 ft AnS - 
 
 Forced Draft.— The use of forced draft as a substitute for, or as an aid to, 
 natural chimney draft is becoming quite common in large boiler plants. 
 Its advantages are that it enables a boiler to be driven to its maximum 
 capacity to meet emergencies without reference to the state of the weather 
 or to the character of the coal; that the draft is independent of the tempera- 
 ture of the chimney gases, and that therefore lower flue temperatures may 
 be used than with natural draft; and in many cases that it enables a poorer 
 quality of coal to be used than is required with natural draft. Forced draft 
 may be obtained: First, by a steam jet in the chimney, as in locomotives and 
 steam fire-engines; second, by a steam-jet blower under the grate bars; 
 third, by a fan blower delivering air under the grate bars, the ash-pit doors 
 being closed; fourth, by a fan blower delivering air into a closed fireroom, 
 as in the “ closed stoke-hold ” system used in some ocean-going vessels; and 
 fifth, by a fan placed in the flue or chimney drawing the gases of combustion 
 from the boilers, commonly called the induced-draft system. Which one of 
 these several systems should be adopted in any special case will usually 
 depend on local conditions. The steam jet has the advantage of lightness 
 and compactness of apparatus, and is therefore most suitable for locomotives 
 and steam fire-engines, but it also is the most wasteful of steam, and there- 
 fore should not be used when one of the fan-blower systems is available, 
 except for occasional or temporary use, or when very cheap fuel, such as 
 anthracite culm at the coal mines, is used. 
 
 STEAM ENGINES. 
 
 What Is a Good Steam Engine?— It should be as direct acting as possible; 
 that is, the connecting parts between the piston and the crank-shaft should 
 be few in number, as each part wastes some power. Formerly, beam 
 engines were all the rage. They were well enough in their time for pump- 
 ing, when the pump was at one end of the beam and the piston at the other. 
 Few of our modern colliery engines have such an appendage, except in 
 some instances for pumping, and even for that kind of work the better 
 engines have no beams. The moving parts of an engine should be strong, to 
 resist strains, and light, so as to offer no undue resistance to motion; parts 
 moving upon each other should be well and truly and smoothly finished, to 
 reduce resistances to a minimum; the steam should get into the cylinder 
 easily at the proper time, and the exhaust should leave the cylinder as 
 exactly and as easily. The steam pipes supplying steam should have 
 an area one-tenth the combined areas of the cylinders they supply, and 
 exhaust pipes should be somewhat larger. The cylinder and the steam pipes 
 and the boiler should be well protected. The engine should be capable of 
 being started and stopped and reversed easily and quickly. 
 
 Rule. — To find the indicated horsepower developed by an engine , multiply 
 together the M. E. P. per square inch , the area of the piston, the length of stroke, 
 and the number of strokes per minute. This gives the work per minute in foot- 
 pounds. Divide the product by 33,000; the result wiU be the indicated horse- 
 power of the engine. 
 
 Let I. H. P. = indicated horsepower of engine; 
 
 P = M. E. P. in pounds per square inch; 
 
 A = area of piston in square inches; 
 
 L = length of stroke in feet; 
 
 N = number of strokes per minute. 
 
 Then, the above rule may be expressed thus: 
 
 I. H. P. = 
 
 PLAN 
 33,000 * 
 
 The number of strokes per minute is twice the number of revolutions per 
 minute. For example, if an engine runs at a speed of 210 revolutions per 
 minute, it makes 420 strokes per minute. A few types of engines, however, 
 are single acting; that is, the steam acts on only one side of the piston. 
 In this case, only 1 stroke per revolution does work, and, consequently, the 
 
RULES FOR ENGINE DRIVERS. 
 
 191 
 
 number of strokes per minute to be used in the above rule is the same as the 
 
 number of revolutions per minute. . 
 
 Example— The diameter of the piston of an engine is 10 in. and the 
 length of stroke 15 in. It makes 250 revolutions per minute, with a M. E. P. 
 of 40 lb ner sa in. What is the horsepower? 
 f As it £ not stated whether the engine is single or double acting, assume 
 that it is double acting. Then, the number of strokes is 250 X 2 — 500 per 
 minute. Hence, 
 
 I. H. P. = 
 
 PLAN 40 X if X (10^ X .7854) X 500 
 
 = 59.5 H. P. 
 
 33,000 33,000 
 
 Approximate Determination of M. E. P.-To approximately determine the 
 M. E. P. of an engine, when the point of apparent cut-off is known and the 
 boiler pressure, or the pressure per square inch m the boiler from which 
 the supplv of steam is obtained, is given: . , .. . 
 
 Rule.— Add U.7 to the gauge pressure, and multiply the result by the number 
 ovvosite the fraction indicating the point of cut-off in the following table. Subtract 
 17 from the product , and multiply by .9. The result is the M. E. P. for good , 
 simple non-condensing engines. 
 
 Or, letting 
 
 p gauge pressure; 
 k -= a constant (see following table); 
 
 M. E. P. = mean effective pressure. 
 
 Then, 
 
 M. E. P. = .9[k(p 14.7) -17]. 
 
 Table. 
 
 Cut-Off. 
 
 Constant. 
 
 Cut-Off. 
 
 Constant. 
 
 Cut-Off. 
 
 Constant. 
 
 b 
 
 I 
 
 £ 
 
 .566 
 
 f 
 
 .771 
 
 2 
 
 3 
 
 .917 
 
 .603 
 
 .4 
 
 .789 
 
 .7 
 
 .926 
 
 .659 
 
 £ 
 
 .847 
 
 I 
 
 .937 
 
 .3 
 
 .708 
 
 .6 
 
 .895 
 
 .8 
 
 .944 
 
 h 
 
 .743 
 
 5 
 
 5 
 
 .904 
 
 7 
 
 5 
 
 .951 
 
 If the engine is a simple condensing one, subtract the pressure in the 
 condenser instead of 17. * The fraction indicating the point of cut-off is 
 obtained by dividing the distance that the piston has. traveled when the 
 steam is cut off by the whole length of the stroke. For a f cut-off, and 92 lb. 
 gauge pressure in the boiler, the M. E. P. is, by the formula just given, 
 .9 [.917(92 4- 14.7) — 17] = 72.6 lb. per sq. in. a . . 
 
 Example.— Find the approximate 1. H. P. of a 9" X 12" non-condensmg 
 engine, cutting off at £ stroke, and making 240 revolutions per minute. The 
 boiler pressure is 80 lb. gauge. 7 _ ... 
 
 80 4- 14.7 = 94.7. The constant for £ cut-off is .847, and .847 X boiler 
 pressure = .847 X 94.7 = 80.21. M. E. P. = (80.21 - 17) X .9 = 56.89 lb. per 
 sq. in. Then, 
 
 P L A N _ 56.89 X if X (.7854 X 9 2 ) X 240 X 2 
 33,000 _ 33,000 
 
 52.64 H. P. Ans. 
 
 RULES FOR ENGINE DRIVERS. 
 
 If a gauge glass breaks, turn off the water first and then the steam, to 
 avoid scalding yourself. 
 
 Don’t buy oil or waste simply because it is very cheap; it will cost more 
 than a good article in the end. 
 
192 
 
 STEAM ENGINES. 
 
 In cutting rubber for gaskets, etc., have a dish of water handy, and keep 
 wetting the knife blade; it makes the work much easier. 
 
 Don’t forget that there is no economy in employing a poor fireman. He 
 can, and probably will, waste more coal than would pay the wages of a first- 
 class man. 
 
 An ordinary steam engine having two cylinders connected at right angles 
 on the same shaft consumes one-third more steam than a single-cylinder 
 engine, while developing only the same amount of power. 
 
 A fusible plug ought to be renewed every three months, bv removing the 
 old metal and refilling the case; and it should be scraped clean and bright 
 on both ends every time that the boiler is washed out, to keep it in good 
 working order. 
 
 When you try a gauge-cock, don’t jerk it open suddenly, for if the water 
 happens to be a trifle below the cock, the sudden relief from pressure at that 
 point may cause it to lift and flow out, deceiving you in regard to its height. 
 Whereas, if you open it quietly, no lift will occur, and you ascertain surely 
 whether there is water or steam at that level. 
 
 Always open steam stop-valves between boilers very gently, that they 
 may heat and expand gradually. By suddenly turning on steam a stop- 
 valve chest was burst, due to the expansive power of heat unequally 
 applied. The same care is also recommended when shutting off stop- valves. 
 A fearful explosion once occurred by shutting a communicating stop-valve 
 too suddenly— due to the recoil. 
 
 In order to obtain the driest possible steam from a boiler, there should be 
 an internal perforated pipe (dry pipe, so called) fixed near the top of the 
 boiler, and suitably connected to the steam pipe. The perforations in this 
 pipe should be from one-quarter to one-half greater in area than that of the 
 steam pipe. Domes are of no use as steam driers; they only add a very little 
 to the steam space of a boiler, and are often a source of loss by radiation. 
 
 if a glass gauge tube is too long, take a triangular file ana wet, 11 with 
 turpentine; hold the tube in the left hand, with the thumb and forefinger at 
 the place where you wish to cut it, saw it quickly and lightly two or three 
 times with the edge of the file, and it will mark the glass. Now take the 
 tube in both hands, both thumbs being on the side opposite the mark, and 
 an inch or so apart, and then try to bend the glass, using your thumbs as 
 ful crums, and it will break at the mark, w r hich has weakened the tube. 
 
 A stiff charge of coal all over a furnace will lower the temperature 200° 
 or 300° in a very short time. After the coal is well ignited the temperature 
 will rise about 500°, and as it continues burning will gradually drop about 
 200°, until the fireman puts in another charge, when the sudden fall before 
 mentioned takes place again. This sudden contraction and expansion 
 frequently causes the bursting of a boiler, and it is for this reason that light 
 and frequent charges of coal, or else firing only one-half of the furnace at a 
 time, should be always insisted on. 
 
 Be careful when using a wrench on hexagonal nuts that it fits snugly, or 
 the edges of the nut will soon become rounded. 
 
 Be careful how you use a monkeywrench, for if it is not placed on the nut 
 properly the strain will often bend or fracture the wrench. 
 
 The area of grate for a boiler should never be less than | sq. ft. per I. H. P. 
 of the engine, and it is seldom advisable to increase this allowance beyond 
 I sq . ft. per I. H. P. 
 
 The area of tube surface for a boiler should not be less than 2£ sq. ft. per 
 I. H. P. of the engine. 
 
 The ratio of heating surface to grate area in a boiler should be 30 to 1 as a 
 minimum, and may often be increased to 40 to 1, or even more, with 
 advantage. 
 
 Lap-welded pipe of the same rated size has always the same outside 
 diameter, whether common, extra, or double extra, but the internal diame- 
 ter is of course decreased with the increased thickness. 
 
 A good cement for steam and water joints is made by taking 10 parts, by 
 weight, of white lead, 3 parts of black oxide of manganese, 1 part of litharge, 
 and mixing them to the proper consistency with boiled linseed oil. 
 
 To harden a cutting tool, heat it in a coke fire to a blood-red heat and 
 plunge it into a solution of salt and water (1 lb. of salt to 1 gal. of water), 
 then polish the tool, heat it over gas, or otherwise, until a dark straw and 
 purple mixed color shows on the polish, and cool it in the salt water. 
 
 Small articles can be plated with brass by dipping them in a solution of 
 9£ gr. each of sulphate of copper and chloride of tin, in If pt. of water. 
 
BELTING AND VELOCITY OF PULLEYS. 
 
 193 
 
 Don’t be eternally tinkering about your engine, but let well enough 
 
 al °Don’t forget that with a copper hammer you can drive a key just as well 
 as with a steel one, and that it doesn’t leave any marks. 
 
 Keep on hand slips of thin sheet copper, brass, and tin, to use as liners, 
 and if you shape some of them properly, much time will be saved when you 
 
 net A few wooden skewer pins, such as butchers use, are very useful for 
 many purposes in an engine room. Try them. . . , 
 
 In running a line of steam pipe where there are certain rigid points, 
 make arrangements for expansion on the line between those points, or you 
 
 will come to grief. ' „ , , . , 
 
 Arrange the usual work of the engine and firerooms systematically, and 
 
 adhere to it. It pays well. 
 
 Don’t forget that cleanliness is next to godliness. , ._ 
 
 Rubber cloth kept on hand for joints should be rolled up and laid away 
 by itself, as any oil or grease coming in contact with it will cause it to soften 
 aiid give out when put to use. , 
 
 When using a jet condenser, let the engine make three or four revolutions 
 before opening the injection valve, and then open it gradually, letting the 
 engine make several more revolutions before it is opened to the full amount 
 
 reuuired. * 
 
 Open the main stop-valve before you start the fires under the boilers. 
 When starting fires, don’t forget to close the gauge-cocks and safety valve 
 as soon as steam begins to form. . 
 
 An old "Turkish towel, cut in two lengthwise, is better than cotton waste 
 for cleaning brass work. , 
 
 Always connect your steam valves m such a manner that the valve 
 closes against the constant steam pressure. , , . - . 
 
 Turpentine well mixed with black varnish makes a good coating for iron 
 smoke T)it)6S 
 
 Ordinary lubricating oils are not suitable for use in preventing rust. 
 
 You can make a hole through glass by covering it with a thm coating of 
 wax, warming the glass and spreading the wax on it. Scrape off the wax 
 where you want the hole, and drop a little fluoric acid on the spot with a 
 wire. The acid will cut a hole through the glass, and you can shape the 
 hole with a copper wire covered with oil and rottenstone. 
 
 A mixture of 1 oz. of sulphate of copper, j oz. of alum, ^ teaspoonful of 
 powdered salt, 1 gill of vinegar and 20 drops of nitric acid will max® a hole 
 in steel that is too hard to cut or file easily. Also, if applied to steel and 
 washed off quickly, it will give the metal a beautiful frosted appearance. 
 
 BELTING AND VELOCITY OF PULLEYS. 
 
 Belts should not be made tighter than necessary. Over half the trouble 
 from broken pulleys, hot boxes, etc. can be traced to the fault of tight belts, 
 while the machinery wears much more rapidly than when loose belts are 
 
 em Thespeed of belts should not be more than 3,000 or 3,750 ft. per minute 
 The motion of driving should run with and not against the laps of the 
 
 ^Leather belts should be run with the stronger or flesh side on the outside 
 and the grain (hair) side on the inside, nearest the pulley, so that the 
 stronger plirt of the belt may be subject to the least wear. It will also drive 
 30$ more than if run with the flesh side nearest the pulley. The grain side 
 adheres better because it is smooth. Do not expose leather belts to the 
 
 We Wh!n the length of a belt cannot be conveniently . ascertained by 
 measuring around the pulleys with a tape line, the following rule will be 
 
 semceabTe: diameterg of the 2 pulley s together and divide by 2; multiply 
 this quotient by 3i, and to the product add twice the distance between the 
 centers of the shafts; the sum will be the length required. 
 
194 
 
 COMPRESSED AIR. 
 
 COMPRESSED AIR.* 
 
 By Prof. Robert Peele. 
 
 An air compressor consists essentially of a cylinder in which atmospheric 
 air is compressed by a piston, the driving power being steam or water. 
 
 Classification of Compressors.— Steam-driven compressors in ordinary use 
 may be classed as follows: . , , . A , . .. , 
 
 (a) Straight-line type, in which a single horizontal air cylinder is set 
 
 tandem with its steam cylinder, and provided with two flywheels. This 
 pattern is generally adapted for compressors of small size. . 
 
 ( b ) Duplex type, in which there are two steam cylinders, each driving an 
 air cylinder, and coupled at 90° to a crank-shaft carrying a fly wheel. 
 
 (c) Horizontal , cross-compound engines, each steam cylinder set tandem 
 
 with an air cylinder, as in (b). . , 
 
 ( d ) Vertical , simple , or compound engines, with the air cylinders set 
 
 above the steam cylinders. . . .. , ,, , 
 
 (e) Compound or stage compressors , m which the air cylinders themselves 
 
 are compounded. The compression is carried to a certain point in one 
 cylinder and successively raised and finally completed to the desired pres- 
 sure in the others. They may be either of the straight-line or duplex form, 
 with simple or compound steam cylinders. , _ _ 
 
 Classes (a), ( b ), (c), and (e) are those commonly employed for mine 
 service The principle of compound, or two-srage, air compression is 
 recognized as applicable for even the moderate pressures required in 
 mining, and the compressors of class (e) are frequently employed. 
 
 Construction of Compressors.— Compressors are usually built with a short 
 stroke as this is conducive to economy in compression as well as the attain- 
 ment of a proper rotative speed. In ordinary single-stage compressors, the 
 usual ratio of length of stroke to diameter of steam cylinders is U to 1 or 1£ 
 to 1. In some makes, such as the Rand, the ratio is considerably greater, 
 varying from H to If to 1, as in several large plants built for the Calumet 
 & Hecla Mining Co. Many compressors have length and diameter of steam 
 cylinders equal. The relative diameters of the air and steam cylinders 
 depend on the steam pressure carried, and the air pressure to be produced. 
 In mining operations, there is usually but little variation in these con- 
 ditions. For rock-drill work, the air pressure is generally from 60 to 80 lb. 
 
 In using water-power, a compressor is driven most conveniently by a 
 bucket impact wheel, such as the Pelton or Knight. The waterwheel is 
 generally mounted directly on the crank-shaft, without the use of gearing. 
 Since the power developed is uniform throughout the revolution of the 
 wheel, the compressor should be of duplex form, in order to equalize the 
 resistance so far as possible. The rim of the wheel is made extra heavy, to 
 supply the place of a flywheel. When direct-connected, the wheel is of 
 relatively large diameter, as its speed of rotation must of necessity be slow. 
 With small high-speed wheels, the compressor cylinders may be operated 
 through belting or gearing. In most cases, however, the waterwheel may 
 be large enough to render gearing unnecessary. Impact wheels may be 
 employed with quite small heads of water, by introducing multiple nozzles. 
 To prevent the water from splashing over the compressor, the wheel is 
 enclosed in a tight iron or wooden casing. The force of the water is regu- 
 lated usually by an ordinary gate valve. If the head be great, it may be 
 necessary to introduce means for deflecting the nozzle, so that, wnen the 
 compressor is to be stopped suddenly, danger of rupturing the water 
 main will be avoided. . _ 
 
 Theory of Air Compression.— The useful effect or efficiency of a compressor is 
 the ratio of the force stored in the compressed air to the work that has been 
 expended in compressing it. This probably never reaches 80$, and often 
 falls below 60$. 
 
 # gee “Mines and Minerals," Vols. XIX and XX, for complete discussion of this subject by the 
 same author. 
 
RATING OF COMPRESSORS. 
 
 195 
 
 Free air is air at ordinary atmospheric pressure as taken into the com- 
 pressor cylinder. As commonly used, this means air at sea-level pressure 
 (14.7 lb. per sq. in.) at 60° F. 
 
 The absolute pressure of air is measured from zero, and is equal to the 
 assumed atmospheric pressure plus gauge pressure. Air-compression calcu- 
 lations depend on the two well-known laws: 
 
 1. Boyle's Law— The temperature being constant, the volume varies 
 inversely as the pressure; or P V = P' V' — a constant; in which V equals 
 volume of given weight of air at the freezing point, and the pressure P; V' 
 equals the volume of the same weight of air at the same temperature and 
 under the pressure P'. 
 
 2. Gay-Lussac's Law. —The volume of a gas under constant pressure, when 
 heated, expands, for each degree of rise in temperature, by a constant pro- 
 portional part of the volume that it occupied at the freezing point; or, 
 V’ = V (1 + a £°), in which a equals for centigrade degrees, or 5 ^ T for 
 Fahrenheit degrees. 
 
 Theoretically, air may be compressed in two ways, as follows: 
 
 1. Isothermally , when the temperature is kept constant during compres- 
 sion, and in this case, the formula P V = P r V' is true. 
 
 2. Adiabatically, when the temperature is allowed to rise withdut check 
 during the compression. 
 
 Since the pressure rises faster than the volume diminishes, the equation 
 
 pr / y\n 
 
 P V — P' V' no longer holds, and we have -p- = ( -yj ) > in which n equals 
 
 1.406. The specific heat of air at constant pressure is .2375, and at constant 
 2375 
 
 volume .1689, and n = = 1.406. 
 
 In practice, compression is neither isothermal nor adiabatic, but inter- 
 mediate between the two. The values of n for different conditions in 
 practice are as follows, as determined from a 2,000-horsepower stage com- 
 pressor at Q,uai de la Gare, Paris. 
 
 For purely adiabatic compression, with no cooling arrangements, 
 n = 1.406; in ordinary single-cylinder dry compressors, provided with a 
 water-jacket, n is roughly 1.3; while in the best wet compressors (with spray 
 injection), n becomes 1.2 to 1.25. In the poorest forms of compressor, the 
 value n = 1.4 is closely approached. For large, well-designed compressors 
 with compound air cylinders, the exponent n may be as small as 1.15. 
 
 Rating of Compressors. — Compressors are rated as follows: (1) In terms of 
 the horsenower developed by the steam end of the compressor, as shown bv 
 indicator cards taken when running at full speed, and when the usual 
 volume of air is being consumed. (2) Compressors for mines are often 
 rated roughly as furnishing sufficient air to operate a certain number of rock 
 drills; a 3" drill requires a volume of air at 60 lb. pressure, equal to 100 or 
 110 cu. ft. of free atmospheric air per minute. (3) In terms of cubic feet of 
 free air compressed per minute to a given pressure. 
 
 As the actual capacity of a compressor depends on the density of the 
 intake air, it will obviously be reduced in working at an altitude above sea 
 level, because of the diminished density of the atmosphere. The following 
 table gives the percentages of output at different elevations: , 
 
 Example.— Calculate the volume of air 
 furnished by an 18" X 24" compressor work- 
 ing at an elevation of 5,000 ft. above sea 
 level, revolving 95 times per minute, and 
 having a piston speed of 380 ft. per minute. 
 9 2 x 3.14 = 254.3 sq. in. = piston area. 
 254 3 
 
 -tttt X 380 = 668.8 cu. ft. = volume dis- 
 144 
 
 placed per minute by the piston; deducting 
 10$ for loss gives 602 cu. ft. At sea level at 
 
 15 
 
 80 lb. gauge pressure, this equals ^ ^ X 
 
 602 = 95 cu. ft. At an elevation of 5,000 
 ft., the output of a compressor would be 
 95 X 85$ — 80.7 cu. ft. per minute. 
 
 Cooling.— Compressor cylinders may be 
 cooled by either of the following methods: 
 
 Altitude. 
 
 Feet. 
 
 Atmospheric 
 
 Pressure. 
 
 Pounds. 
 
 Percentages 
 of Output 
 at Sea Level. 
 
 0 
 
 14.7 
 
 100.0 
 
 1,000 
 
 14.2 
 
 97.2 
 
 2,000 
 
 13.6 
 
 93.5 
 
 3,000 
 
 13.1 
 
 90.8 
 
 4,000 
 
 12.7 
 
 88.4 
 
 5,000 
 
 12.2 
 
 85.0 
 
 6,000 
 
 11.7 
 
 82.0 
 
 7,000 
 
 11.3 
 
 79.3 
 
 8,000 
 
 10.9 
 
 77.0 
 
 9,000 
 
 10.5 
 
 75.0 
 
 10,000 
 
 10.1 
 
 72.0 
 
196 
 
 COMPRESSED AIR. 
 
 (1) by injecting water into the cylinder, known as wet compressors; or (2) by 
 jacketing the cylinder in water, known as dry compressors. 
 
 Dry Versus Wet Compressors.— Up to about the year 1885 there seemed to 
 be little doubt among mechanical engineers that wet compressors were, on 
 the whole, superior to dry, because, by bringing the air into direct contact 
 with water, the heat is most effectually absorbed. This view is correct, so 
 far as heat loss alone is concerned, presided the water in the cylinder is 
 properly applied. But the question of heat loss is not the only consideration. 
 Low first cost and simplicity of construction are often more advantageous 
 than a close approximation to isothermal compression. Latterly, the wet 
 system has lost ground, owing to the fact that moisture is objectionable in 
 the air, as it forms frost in the exhaust ports of the drills, and stops them up, 
 and probably no wet compressors are now being built in the United States. 
 In Europe, also, dry compressors have grown in favor, at least for mining 
 plants and others of moderate size. 
 
 c = 
 
 TRANSMISSION OF AIR IN PIPES. 
 
 The actual discharge capacity of piping is not proportional to the cross- 
 sectional area alone, that is, to the square of the diameter. Although the 
 periphery is directly proportional to the diameter, the interior surface 
 resistance is much greater in a small pipe than in a large one, because, as 
 the pipe becomes smaller, the ratio of perimeter to area increases. 
 
 To pass a given volume of compressed air, a 1" pipe of given length re- 
 quires over three times as much head as a 2" pipe of the same length. 
 The character of the pipe, also, and the condition of its inner surface, have 
 much to do with the friction developed by the flow of air. Besides imper- 
 fections in the surface of the metal, the irregularities incident on coupling 
 together the lengths of pipe must increase friction. There are so few relia- 
 ble data that the influences by which the values of some of the factors may 
 be modified are not fully understood; and, owing to these uncertain condi- 
 tions the results obtained from formulas are only approximately correct. 
 
 Among the formulas in common use, perhaps the most satisfactory is that 
 of D’Arcy. As adopted for compress ed-air transm ission, it takes the form: 
 
 j d 5 (pi — p 2 j 
 D ” \ Wi l 
 
 in which D = volume of compressed air in cubic feet per minute discharged 
 at final pressure; „ . A . 
 
 coefficient varying with diameter of pipe, as determined by 
 
 d — dTameter^f pipe in inches (the actual diameters of If" 
 and pipe are 1.38" and 1.61", respectively; the nomi- 
 nal diameters of all other sizes may be taken for calcu- 
 lations); 
 
 l = length of pipe in feet; . 
 
 pi = initial gauge pressure in pounds per square inch; 
 p, = final gauge pressure in pounds per square inch; 
 w[ = density of air, or its weight in pounds per cubic foot, at 
 initiar pressure pi. . . , 
 
 The values of the coefficients c for sizes of piping up to 12 are: 
 
 1" 4 45.3 5" 59.0 9" 
 
 2" 52.6 6" 59.8 10" 
 
 3"" 56.5 7" 60.3 11" 
 
 4 // 58.0 8" 60.7 12" v 
 
 Some apparent discrepancies exist for sizes larger than 9", but they cause 
 no very 'material differences in the results. . 
 
 Another formula, published by Mr. Frank Richards, is as follows: 
 
 H = V * L 
 
 10,000 IP a 
 
 in which H = head or difference of pressure required to overcome friction 
 
 and maintain the flow of air; 
 
 y _ ydume of compressed air delivered in cubic feet per 
 minute; 
 
 L = length of pipe in feet; 
 
 D = diameter of pipe in inches; . 
 
 a = coefficient, depending on the size of pipe. 
 
 .61.0 
 
 .61.2 
 
 .61.8 
 
 .62.0 
 
TRANSMISSION OF AIR IN PIPES. 
 
 197 
 
 Values of a for nominal diameters of wrought-iron pipe: 
 
 1" 
 
 .350 
 
 3" 
 
 730 
 
 8" 
 
 1.125 
 
 1±// 
 
 500 
 
 34 /' 
 
 787 
 
 10" 
 
 1.200 
 
 -1-4 
 
 H" 
 
 662 
 
 4" 
 
 .840 
 
 12" 
 
 1.260 
 
 2" 
 
 565 
 
 5" 
 
 .934 
 
 
 
 2£" 
 
 650 
 
 6" 
 
 1.000 
 
 
 
 The values of a fur l\" and 1\" pipe are not consistent with those for 
 other sizes, for the reason stated above. In using this formula with its 
 constants, the calculated losses of pressure are found to be smaller, and, 
 conversely, the volumes of air discharged are larger, under the same condi- 
 tions, than those obtained from D’Arcy’s formula. 
 
 It must be remembered that, within certain limits, the loss of head or 
 pressure increases with the square of the velocity. To obtain the best results, 
 it is found in practice that the velocity of flow in the main air pipes should 
 not exceed 20 or 25 ft. per second. When the initial velocity much exceeds 
 50 ft. per second, the percentage loss becomes very large; and, conversely, 
 by using piping large enough to keep down the velocity, the friction loss 
 may be almost eliminated. For example, at the Hoosac tunnel, in transmit- 
 ting 875 cu. ft. of free air per minute, at an initial pressure of 60 lb., through 
 an 8 " pipe, 7,150 ft. long, the average loss including leakage was only 2 lb. 
 A volume of 500 cu. ft. of free air per minute, at 75 lb., can be transmitted 
 through 1,000 ft. of 3" pipe with a loss of 4.1 lb., while if a 5" pipe were used 
 the loss would be reduced to .24 lb. The velocity of flow in the latter case 
 is only 10 ft. per second. 
 
 In driving the Jeddo mining tunnel, at Ebervale, Pa., two 34 " drills were 
 used in each heading, with a 6 " main, the maximum transmission distance 
 being 10,800 ft. This pipe was so large in proportion to the volume of air 
 required fur the drills (230 cu. ft. free air per minute) that the loss was 
 reduced to an extremely small quantity. A calculation shows a loss of 
 .002 lb., and the gauges at each end of the main were found to record 
 practically the same pressure. 
 
 A due regard for economy in installation, however, must limit the use ot 
 very large piping, the cost of which should be considered in relation to the 
 cost of air compression in any given case. Diameters of from 4 to 6 in. for 
 the mains are large enough for any ordinary mining practice. Up to a 
 length of 3,000 ft., a 4" pipe will carry, per minute, 480 cu. ft. of free air 
 compressed to 82 lb., with a loss of 2 lb. pressure. This volume of air will 
 run four 3" drills. Under the same conditions, a 6 " pipe, 5,000 ft. long, will 
 carry 1,100 cu. ft. of free air per minute, or enough for 10 drills. 
 
 A mistake is often made in putting in branch pipes of too small a diam- 
 eter. For a distance of, say, 100 ft., a 1£" pipe is small enough for a single 
 drill, though a 1" pipe is frequently used. While it is, of course, admissible 
 to increase the velocity of flow in short branches considerably beyond 20 ft. 
 per second, extremes should be avoided. To run a 3" drill from a 1" pipe 
 100 ft. long, would require a velocity of flow of about 55 ft. per second, 
 causing a loss of 10 lb. pressure. . 
 
 The piping for conveying compressed air may be of cast or wrought iron. 
 If of wrought iron, as is customary, the lengths are connected either by 
 sleeve couplings or by cast-iron flanges into which the ends of the pipe 
 are screwed or expanded. Sleeve couplings are used for all except the 
 large sizes. The smaller sizes, up to 1£ in., are butt-welded, while all from 
 li in. up are lap-welded, to insure the necessary strength. Wrought-iron 
 spiral-seam riveted or spiral-weld steel tubing is sometimes used. It is 
 made in lengths of 20 ft., or less. For convenience of transport in remote 
 regions, rolled sheets in short lengths may be had. They are punched 
 around the edges, ready for riveting, and are packed closely — 4, 6 , or more 
 sheets in a bundle. . . _ _ 
 
 All joints in air mains and branches should be carefully made. Air leaks 
 are more expensive than steam leaks because of the losses already suffered 
 in compressing the air. The pipe may be tested from time to time by allow- 
 ing the air at full pressure to remain in the pipe long enough to observe the 
 gauge. In case a leak is indicated, it should be traced and stopped imme- 
 diately. In putting together screw joints, care should be taken that none of 
 the white lead or other cementing material is forced into the pipe. This 
 would cause obstruction and increase the friction loss. Also, each length as 
 put in place should be cleaned thoroughly of all foreign substances that may 
 have lodged inside. To render the piping readily accessible for inspection 
 
198 
 
 COMPRESSED AIR. 
 
 and stoppage of leaks, it should, if buried, be carried in boxes sunk just 
 below the surface of the ground; or, if underground, it should be supported 
 upon brackets along the sides of the mine workings. Low points in pipe 
 lines, which would form “pockets” for the accumulation of entrained 
 water, should be avoided, as they obstruct the passage of the air. In long 
 pipe lines, where a uniform grade is impracticable, provision may be made 
 near the end for blowing out the water at intervals, when the air is to be 
 used for pumps, hoists, or other stationary engines. 
 
 For long lengths of piping, expansion joints are required, particularly 
 when on the surface. They are not often necessary underground, as the 
 temperature is usually nearly constant, except in shafts, or where there may 
 be considerable variations of temperature between summer and winter. 
 
 LOSSES IN THE TRANSMISSION OF COMPRESSED AIR. 
 
 By E. Hill, Norwalk Iron Works Co. 
 
 The increasing use that is being made of compressed-air engines for mine 
 and underground work stimulates the inquiry regarding their efficiency. 
 
 The situation is apparently very simple. An engine drives an air com- 
 pressor, which forces air into a reservoir. The air under pressure is led 
 through pipes to the air engine, and is there used after the manner of steam. 
 
 The resulting power is frequently a small percentage of the power 
 expended. In a large number of cases the losses are due to poor designing, 
 and are not chargeable as faults of the system or even to poor workmanship. 
 
 The losses are chargeable, first, to friction of the compressor. This will 
 amount ordinarily to 15# or 20 #, and can be helped by good workmanship, 
 but cannot probably be reduced below 10#. Second, we have the loss 
 occasioned by pumping the air of the engine room, rather than air drawn 
 from a cooler place. This loss varies with the season, and amounts to from 
 3# to 10#. This can all be saved. The third loss or series of losses arises in 
 the compressing cylinder. Insufficient supply, difficult discharge, defective 
 cooling arrangements, poor lubrication, and a host of other causes, perplex 
 the designer and rob the owner of power. The fourth loss is found in the 
 pipe. This has heretofore received by no means the consideration that the 
 subject demands. The loss varies with every different situation, and is sub- 
 ject to somewhat complex influences. The fifth loss is chargeable to fall of 
 temperature in the cylinder of the air engine. Losses arising from leaks are 
 often serious, but the remedy is too evident to require demonstration. No 
 leak can be too small to require immediate attention. An attendant who 
 is careless about packings and hose couplings will permit losses for which 
 no amount of engineering skill can compensate. 
 
 We can only realize 100# efficiency in the air engine, leaving friction out 
 of our consideration, when the expansion of the air and the changes of its 
 temperature in the expanding or air-engine cylinder are precisely the 
 reverse of the changes that have taken place during the compression of the 
 air in the compressing cylinder. But these conditions can never be realized. 
 The air during compression becomes heated, and during expansion it 
 becomes cold. If the air immediately after compression, before the loss of 
 any heat, was used in an air engine and there perfectly expanded back to 
 atmospheric pressure, it would, on being exhausted, have the same tem- 
 perature it had before compression, and its efficiency would be 100#. 
 
 But the loss of heat after compression and before use cannot be pre- 
 vented, as the air is exposed to such very large radiating surfaces in the 
 reservoir and pipes, on its passage to the air engine. The heat, which 
 escapes in this way, did, while in the compressing cylinder, add much to 
 the resistance of the air to compression, and since it is sure to escape, at 
 some time, either in reservoir or pipes, it is evidently the best plan to 
 remove it as fast as possible from the cylinder, and thus remove one element 
 of resistance. Hence, we find compressors are almost universally provided 
 with cooling attachments more or less perfect in their action, the aim being 
 to secure isothermal compression, or compression having equal temperature 
 throughout. Where the temperature rises, without check, during com- 
 pression, the term adiabatic compression is employed. 
 
 If air compressed isothermaliy is used with perfect expansion and the 
 fall of temperature during expansion be prevented, then we will have 100# 
 efficiency. But air will grow cold on being expanded in an engine, and 
 hence we conclude that warming attachments have the same economic 
 place on an air engine that cooling attachments have on an air compressor. 
 In fact, we find attachments of this kind more particularly in large and 
 
TRANSMISSION OF COMPRESSED AIR. 
 
 199 
 
 permanently located engines, but, for practical reasons, their use on most of 
 the engines for mine work is dispensed with, and the engines expand the 
 air adiabaticallv, or without receiving heat. _ . ,, . 
 
 The practical engineer, therefore, has. to deal with nearly isothermal 
 compression, and nearly adiabatic expansion, and must also consider that 
 thfair in reservoirs and pipes becomes of the same temperature as surround- 
 ing objects. Consideration must also be had for the friction of the com- 
 prlssor and the air engine. For the pressure of 60 lb., which is that most 
 commonly used, the decrease in resistance to compression secured by the 
 cooling attachments, is almost exactly equaled by the friction of the com- 
 pressor. Hence it is safe, in calculating the efficiency of the air engine, to 
 consider the compressor as being without cooling* attachments, and also as 
 working without friction. The results of such calculations will be too high 
 efficiencies for light pressures, which are little used; about correct for medium 
 pressures, which ; are commonly employed; and too low for higher pressures 
 and will thus have the advantage of not being overestimated. This result 
 is occasioned by the fact that, owing to the slight heat m compressing low 
 pressures of air, the saving of power by the cooling attachments is not 
 equal to the friction of the machine, but at high pressures on account of the 
 great heat, the cooling attachments are of great value and save very much 
 
 more power than friction consumes. . , „ r QO 
 
 In the expanding engines, the expansion never falls as low as the 
 adiabatic law would indicate, owing to a number of reasons, but w e will 
 consider the expansion as being adiabatic, as an error m calculations 
 caused thereby will be on the “ safe side ’’ and the actual power will exceed 
 the calculated power. We therefore consider the compressor and engine as 
 following the adiabatic law of compression and expansion, and as working 
 
 Wi wnh ttoview of the case, the efficiency of an air engine, working with 
 perfect expansion, stated in percentages of the power required to operate 
 the compressor, can be placed as below for the various pressures above the 
 
 atmosphere. , < , ftn1 . 
 
 Pressure above the atmosphere, 2.9 lb. 
 
 Pressure above the atmosphere, 14.7 lb. 
 
 Pressure above the atmosphere, 29.4 lb. 
 
 Pressure above the atmosphere, 44.1 lb. 
 
 Pressure above the atmosphere, 58.8 
 Pressure above the atmosphere, 73.5 lb. 
 
 Pressure above the atmosphere, 88.2 lb. 
 
 Wc observe that the efficiencies for the lower pressures are very much 
 ..reYfer iffian for the high pressures, and the conclusion is almost irresistible 
 fhsf to secure economical results we must design our air engines to run 
 “St lhrht measures And, in fact, the consideration of tables similar to the 
 above ? g heretXe published by writers on this subject, has led many 
 
 en ^he e ffinehas?een entirely neglected. We notice that a pressure of 2.9 lb. 
 
 • Sh i efficiency of 94.85$. It is clear that if the air were 
 
 is credited with an length of the pipe and the velocity of flow 
 
 conveyed through a P P , | . friction, then its efficiency, instead 
 
 zero It is,’ therefore the power that 
 leSttom the air, after it has passed the pipe and lost a part of its 
 pressure* by friction, which we must consider when we state the efficiency 
 
 of Ctor table of eSencies with a loss of 2,9 lb. in the pipe, now gives us dif- 
 ferent values for the efficiencies at the various pressures. 
 
 94.85$ efficiency. 
 81.79$ efficiency. 
 72.72$ efficiency. 
 66.90 $ efficiency. 
 62.70$ efficiency. 
 59.48$ efficiency. 
 56.88$ efficiency. 
 
 Pressure above the atmosphere, 2.9 lb. 
 
 Pressure above the atmosphere, 14.7 lb. 
 
 Pressure above the atmosphere, 
 
 Pressure above the atmosphere, 
 
 Pressure above the atmosphere, 
 
 Pressure above the atmosphere, 
 
 Pressure above the atmosphere, 
 
 Tt will hP noticed that the light pressures have lost most by the pipe 
 friction 2 9 lb having lost W: 14.7 lb. 11* and 88.2 lb. 9 nly a trifle over - of 
 w We’ see that now 14.7 lb. is apparently the economical pressure to use 
 Put a further careful analysis of the subject shows, that when the loss m the 
 pi^e is 2.9 lb., then 20.5 lb. is the most economical pressure to use, and that 
 
 29.4 lb. 
 44.1 lb. 
 58.8 lb. 
 
 73.5 lb. 
 5.2 lb. 
 
 00.00$ efficiency. 
 70.44$ efficiency. 
 68.81$ efficiency. 
 64.87$ efficiency. 
 61.48$ efficiency. 
 58.62$ efficiency. 
 56.23$ efficiency. 
 
200 
 
 COMPRESSED AIR. 
 
 the efficiency is 71#. But 2.9 lb. is a very small loss between compressor and 
 air engine, and cases are extremely exceptional where the friction of valves, 
 pipes, elbows, ports, etc. does not far exceed this. Yet, with these con- 
 ditions, which are very difficult to fill, we see that 20.5 lb. is the lightest 
 pressure that should probably ever be used for conveying power, and that 
 71# is an efficiency scarcely to be obtained. . . 
 
 Continuing our investigation and taking examples where the pipe friction 
 amounts to 5.8 lb., we find the following efficiencies to correspond to the 
 
 stated pressure: 
 
 Pressure above the atmosphere, 14.7 lb. 
 Pressure above the atmosphere, 29.4 lb. 
 Pressure above the atmosphere, 44.1 lb. 
 Pressure above the atmosphere, 58.8 lb. 
 Pressure above the atmosphere, 73.5 lb. 
 Pressure above the atmosphere, 88.2 lb. 
 
 57.14# efficiency. 
 64.49# efficiency. 
 62.71# efficiency. 
 60.12# efficiency. 
 57.73# efficiency. 
 55.59# efficiency. 
 
 We again notice that as friction increases, or in other word s, when we 
 beffin to use more air and make greater demands on the carrying capacity 
 of the pipe, then we must increase pressure very considerably to attain the 
 most economical results. If the demands are such as to increase the friction 
 and loss in pipe to 14.7 lb., the air of 14.7 lb. pressure at the compressor is 
 entirelv useless at the air engine. 
 
 The* table will stand thus: 
 
 Pressure above the atmosphere, 14.7 lb. 00.00^ efficiency. 
 
 Pressure above the atmosphere, 29.4 lb. 48.53# efficiency. 
 
 Pressure above the atmosphere, 44.11b. 55.13# efficiency. 
 
 Pressure above the atmosphere, 58.8 lb. 55.64# efficiency. 
 
 Pressure above the atmosphere, 73.5 lb. 54.74# efficiency. 
 
 Pressure above the atmosphere, 88.2 lb. 53.44# efficiency. 
 
 It is to be noticed that 88.2 lb. pressure has lost only about 3f# of its 
 efficiency by reason of as high a friction as 14.7 lb., while the efficiency of 
 the lower pressures has been greatly affected. 
 
 As the friction increases we see that the most efficient, and, consequently , 
 most economical, pressure increases. In fact, for any given friction in a 
 pipe, the pressure at the compressor must not be carried below a certain 
 limit. The following table gives the lowest pressures that should be used at 
 the compressor, with varying amounts of friction in the pipe: 
 
 2.9 lb. friction. 20.5 lb. at compressor. 70.92# efficiency. 
 
 5.8 lb. friction. 29.4 lb. at compressor. 64.49# efficiency. 
 
 8.8 lb. friction. 38.2 lb. at compressor. 60.64# efficiency. 
 
 11.7 lb. friction. 47.0 lb. at compressor. 57.87# efficiency. 
 
 14.7 lb. friction. 52.8 lb. at compressor. 55.73# efficiency. 
 
 17.6 lb. friction. 61.7 lb. at compressor. 53.98# efficiency. 
 
 20.5 lb. friction. 70.5 lb. at compressor. 52.52# efficiency. 
 
 23.5 lb. friction. 76.4 lb. at compressor. 51.26# efficiency. 
 
 26.4 lb. friction. 82.3 lb. at compressor. 50.17# efficiency. 
 
 29.4 lb. friction. 88.2 lb. at compressor. 49.19# efficiency. 
 
 So long as the friction of the pipe equals the amounts given above, an 
 
 efficiency greater than the corresponding sums stated in the table cannot be 
 expected. If we should have a case that corresponded to any of these cited 
 in the table, we could only increase efficiency by reducing the friction. 
 
 An increase in the size of pipe will reduce friction by reason of the lower 
 velocity of flow required for the same amount of air. But many situations 
 will not admit of large pipes being employed, owing to considerations of 
 economy outside of the question of fuel or prime motor capacity. 
 
 An increase of pressure will decrease the bulk of air passing the pipe, and 
 in that proportion will decrease its velocity. This will decrease the loss by 
 friction, and, as far as that goes, we have a gain. But we subject ourselves 
 to a new loss, and that is the diminishing efficiencies of increasing pressures. 
 Yet as each cubic foot of air is at a higher pressure, and. therefore, carries 
 more power, we will not need as many cubic feet as before for the same 
 work. It is obvious that with so many sources of gain or loss the question 
 of selecting the proper pressure is not to be decided hastily . 
 
 As an illustration of the combined effect of these different elements, we 
 will suppose a very common case. 
 
 Compressor 102 revolutions, pressure 52.8 lb., loss in pipe 14.7 lb., machine 
 in mine running at 38.2 lb., efficiency 55.73#. 
 
FRICTION OF AIR IN PIPES, 
 
 201 
 
 So Ions 1 as the friction of the pipe amounts to 14.7 lb., we have seen that 
 52 8 lb. is the best pressure and 55.73$ the greatest efficiency. We will 
 reduce the friction by reducing the bulk of air passing through the pipe. We 
 reduce the cylinder of the air engine so that it requires 47 lb. pressure to do 
 the same work as before. We find now that the inction of pipe drops to 
 11 7 lb. The pressure on the compressor rises to 58.8 lb., its number of revolu- 
 tions falls to 100, and the resulting efficiency is 57.22$. 
 
 Another change of pressure on compresscir to 64.7 lb. would decrease its 
 revolutions to 93, friction to 8.8 lb., and efficiency would rise to 57.94$. Still 
 again increasing the pressure to 73.5 lb., we have only 84 revolutions of com- 
 pressor, 5.8 lb. loss in pipe, and efficiency of 57./ 3$. In this last case the 
 efficiency begins to fall off a little, and higher pressures would now 
 show less efficiency; but, in comparison with the first example, we find we 
 are doing the same work in the mine with a trifle less power and with a 
 decrease of nearly 20$ in the speed oi the compressor. 
 
 Other common examples can be shown where an increase of pressure 
 would result in wonderful increase in efficiency and economy. There are 
 many cases where light pressures and high velocity in the pipe will convey 
 a given power with greater economy than higher air pressures and lower 
 speed of flow through the pipe. But these cases arise mostly when the 
 higher air pressures become very much greater than are at present m com- 
 
 m °Therefore, in estimating the efficiency of the complete outfit, we find that 
 the pipe and the pressure are very important elements, and must be deter- 
 mined with care and skill to secure the most satisfactwy results. As the 
 volume and power of air vary with its pressure, the size and consequent 
 cost of compressor for a certain work would also be affected by the pressure. 
 To plan an outfit for a mine, due regard must be had to cost of fuel or prime 
 motor power, and also to cost of compressor, pipes, and machinery, as the 
 saving in one is often secured by a sacrifice in the other. . 
 
 Next to determining the size of pipe, the skilful engineer has need of 
 further care in the proper position of reservoirs, branches, drains, and other 
 attachments, as only by the exercise of good judgment m this can satis- 
 factory working be secured. • 
 
 The fact that, on account of the diminished density of the atmosphere at 
 high altitudes, air compressors do not give the same results as at sea level, 
 should also be taken into consideration when a compressor is to be installed 
 in a mountainous region. . . . . 
 
 Friction of Air in Pipes.— Air in its passage through pipes is subject to 
 friction in the same manner as water or any other fluid. Tim pressure at 
 the compressor must be greater than at the point of consumption m order 
 to overcome this resistance. The power that is needed to produce the extra 
 pressure representing the friction of the pipe is lost, as there can be no use- 
 ful return for it. The friction is affected by very many circumstances, but 
 chiefly to be noted is the fact that it increases in direct proportion to the 
 length of the pipe and also as the square of the velocity of the flow of air. 
 The pressure of the air does not affect it. . 
 
 The losses bv friction may be quite serious if the piping system -is poorly 
 designed, and, on the other hand, extravagant expenditure in pipe may 
 result from a timid overrating of the evils of friction. A thorough knowl- 
 edge of the laws governing the whole matter, as well as a ripe experience, is 
 necessary to secure true economy and mechanical success. 
 
 The loss of power in pipe friction is not always the most serious result. 
 When a number of machines are in use in a mine, and the pipes are so small 
 as to cause a considerable loss of pressure by friction, then there will be 
 sudden and violent fluctuations in pressure whenever a machine is started 
 or stopped. Breakages will be of common occurrence, as the changes are too 
 quick to be entirely guarded against by the attendant. Perfectly even pres- 
 sure at the compressor is no safeguard against this class of accidents. The 
 trouble arises in the pipe, and the remedy must be applied there. A system 
 of reservoirs and governing valves will regulate these matters and allow 
 successful work to be done with pipes, which would otherwise be entirely 
 
 i n a Th e S ordin ary formulas for calculating the volume of air transmitted 
 through a pipe do not take into account the increase of volume due to 
 reduction of pressure, i. e., loss of head. To transmit a given volume of air 
 at a uniform velocity and loss of pressure, it would be necessary to construct 
 the pipe with a gradually increasing area. This, of course, is impracticable, 
 
202 
 
 COMPRESSED AIR. 
 
 and in pipe of uniform section both volume and velocity must increase as 
 the pressure is reduced by friction. The loss of head in properly propor- 
 tioned pipes is so small, however, that in practice the increase in volume is 
 usually neglected. 
 
 Loss of Pressure in Pounds per Square Inch, by Flow of Air in Pipes. 
 
 Calculated for pipes 1,000 ft. long; for other lengths, the loss varies directly 
 as the length. 
 
 Velocity of Air 
 at Entrance 
 to Pipe. 
 
 1" Pipe. 
 
 2" 
 
 ' Pipe. 
 
 
 2h" Pipe 
 
 L 
 
 Meters per Second. 
 
 Feet per Second. 
 
 Loss of Pressure. 
 Pounds. 
 
 Cubic Feet of Free Air 
 Passed per Minute When 
 Compressed to 60 Lb. 
 Above the Atmosphere. 
 
 Cubic Feet of Free Air 
 Passed per Minute When 
 Compressed to 80 Lb. 
 Above the Atmosphere. 
 
 Loss of Pressure. 
 Pounds. 
 
 Cubic Feet of Free Air 
 Compressed to 60 Lb. 
 
 Cubic Feet of Free Air 
 Compressed to 80 Lb. 
 
 Loss of Pressure. 
 Pounds. 
 
 Cubic Feet of Free Air 
 Compressed to 60 Lb. 
 
 Cubic Feet of Free Air 
 Compressed to 80 Lb. 
 
 1 
 
 3.28 
 
 .1435 
 
 6 
 
 7 
 
 .0794 
 
 23 
 
 29 
 
 .0574 
 
 32 
 
 41 
 
 2 
 
 6.56 
 
 .6405 
 
 12 
 
 15 
 
 .3050 
 
 46 
 
 59 
 
 .2562 
 
 65 
 
 82. 
 
 3 
 
 9.84 
 
 1.4545 
 
 18 
 
 22 
 
 .7216 
 
 69 
 
 88 
 
 .5818 
 
 97 
 
 124 
 
 4 
 
 13.12 
 
 2.5620 
 
 24 
 
 29 
 
 1.2566 
 
 93 
 
 117 
 
 1.0248 
 
 130 
 
 165 
 
 5 
 
 16.40 
 
 3.9345 
 
 29 
 
 37 
 
 1.9642 
 
 116- 
 
 146 
 
 1.5738 
 
 163 
 
 207 
 
 6 
 
 19.68 
 
 5.4225 
 
 35 
 
 44 
 
 2.7120 
 
 139 
 
 175 
 
 2.1690 
 
 195 
 
 247 
 
 8 
 
 26.24 
 
 10.2480 
 
 47 
 
 59 
 
 5.0264 
 
 185 
 
 234 
 
 4.0992 
 
 260 
 
 330 
 
 10 
 
 32.80 
 
 15.7380 
 
 59 
 
 74 
 
 7.8568 
 
 232 
 
 294 
 
 6.2952 
 
 326 
 
 413 
 
 
 
 3" Pipe. 
 
 4" Pipe. 
 
 5" Pipe. 
 
 1 
 
 3.28 
 
 .0463 
 
 48 
 
 60 
 
 .0347 
 
 86 
 
 109 
 
 .0287 
 
 134 
 
 169 
 
 2 
 
 6.56 
 
 .2092 
 
 96 
 
 121 
 
 .1525 
 
 172 
 
 217 
 
 .1281 
 
 268 
 
 239 
 
 3 
 
 9.84 
 
 .4880 
 
 144 
 
 182 
 
 .3608 
 
 258 
 
 326 
 
 .2909 
 
 402 
 
 509 
 
 4 
 
 13.12 
 
 .8381 
 
 193 
 
 243 
 
 .6283 
 
 343 
 
 436 
 
 .5124 
 
 537 
 
 678 
 
 5 
 
 16.40 
 
 1.3176 
 
 241 
 
 304 
 
 .9821 
 
 429 
 
 544 
 
 .7869 
 
 671 
 
 844 
 
 6 
 
 19.68 
 
 1.8080 
 
 289 
 
 364 
 
 1.3560 
 
 515 
 
 653 
 
 1.0845 
 
 805 
 
 1,017 
 
 8 
 
 26.24 
 
 3.3525 
 
 386 
 
 486 
 
 2.5132 
 
 687 
 
 871 
 
 2.0496 
 
 1,073 
 
 1,357 
 
 10 
 
 32.80 
 
 5.2704 
 
 480 
 
 607 
 
 3.9284 
 
 859 
 
 1,088 
 
 3.1476 
 
 1,342 
 
 1,696 
 
 
 
 
 6" Pipe. 
 
 
 8" Pipe. 
 
 10" Pipe. 
 
 
 3.28 
 
 .0232 
 
 193 
 
 244 
 
 .0173 
 
 343 
 
 434 
 
 .0143 
 
 537 
 
 680 
 
 2 
 
 6^56 
 
 .1046 
 
 386 
 
 488 
 
 .0762 
 
 687 
 
 864 
 
 .0640 
 
 1,073 
 
 1,359 
 
 3 
 
 9.84 
 
 .2440 
 
 579 
 
 633 
 
 .1805 
 
 1,030 
 
 1,303 
 
 .1455 
 
 1,610 
 
 2,039 
 
 4 
 
 13.12 
 
 .4190 
 
 772 
 
 977 
 
 .3141 
 
 1,373 
 
 1,736 
 
 .2562 
 
 2,146 
 
 2,719 
 
 5 
 
 16.40 
 
 .6588 
 
 965 
 
 1,221 
 
 .4910 
 
 1,717 
 
 2,171 
 
 .3934 
 
 2,683 
 
 3,399 
 
 5 
 
 19.68 
 
 .9040 
 
 1,158 
 
 1,466 
 
 .6780 
 
 2,060 
 
 2,605 
 
 .5423 
 
 3,220 
 
 4,079 
 
 8 
 
 26.24 
 
 1.6762 
 
 1,544 
 
 1,954 
 
 1.2556 
 
 2,747 
 
 3,473 
 
 1.0248 
 
 4,293 
 
 5,438 
 
 10 
 
 32.80 
 
 .2.6352 
 
 1,931 
 
 2,443 
 
 1.9642 
 
 3,434 
 
 4,342 
 
 1.5738 
 
 5,367 
 
 6,798 
 
 The resistance is not varied by the pressure, only so fer as changes in 
 pressure vary the velocity. It increases about as the square of the velocity , 
 
 an E^ows^ho^ leaks in pipes all tend to reduce the pressure in 
 
 addition to the losses given in the table. 
 
ELECTRICITY. 
 
 203 
 
 Table of Loss by Friction in Elbows. 
 
 An elbow with a radius 
 as can be made. 
 
 Radius of elbow, 5 diams. 
 Radius of elbow, 3 diams. 
 Radius of elbow, 2 diams. 
 Radius of elbow, H diams. 
 Radius of elbow, H diams. 
 Radius of elbow, 1 diam. 
 Radius of elbow, f diam. 
 Radius of elbow, £ diam. 
 
 of one-half the diameter of the pipe is as short 
 
 Equivalent length of straight pipe, 7.85 diams. 
 Equivalent length of straight pipe, 8.24 diams. 
 Equivalent length of straight pipe, 9.03 diams. 
 Equivalent length of straight pipe, 10.36 diams. 
 Equivalent length of straight pipe, 12.72 diams. 
 Equivalent length of straight pipe, 17.51 diams. 
 Equivalent length of straight pipe, 35.09 diams. 
 Equivalent length of straight pipe, 121.20 diams. 
 
 ELECTRICITY. 
 
 . PRACTICAL UNITS. 
 
 In electrical work it is necessary to have units in terms of which to 
 express the different quantities entering into calculations. The four most 
 important of these are used to express strength of current; electrical pressure , 
 or electromotive force; resistance; power. . 
 
 The strength of current bowing in a wire may be measured in several 
 wavs. If a compass needle be held under or over a wire, it will be deflected 
 and will tend to stand at right angles to the wire. The stronger the current, 
 the greater the deflection of the needle. If the wire carrying the current be 
 cut and the ends dipped into a solution of silver nitrate, silver will be 
 deposited on the end of the wire toward which the current is flowing, and 
 the amount of silver deposited in a given time will be directly proportional 
 to the average strength of current flowing during that time. When the 
 current flowing in a wire is spoken of, the strength of the current is meant. 
 
 Unit Strength of Current.— The unit used to express the strength of a cur- 
 rent is called the ampere. If a current of 1 ampere be sent through a bath 
 of silver nitrate, .001118 gram of silver will be deposited per second. The 
 expression of the flow of current through a wire as so many amperes is 
 analogous to the expression of the flow of water through a pipe as so many 
 gallons per second. 
 
 Electromotive Force.— In order that a current may flow through a wire, 
 there must be an electrical pressure of some kind to cause the flow. In 
 hydraulics, there must always be a head or pressure before water can be 
 made to flow through a pipe. It is also evident that there may be a pressure 
 or head without there being any flow of water, because the opening in the 
 pipe might be closed; the pressure would, however, exist, and, as soon as 
 the valve closing the pipe was opened, the current would flow. In the 
 same way, an electrical pressure or electromotive force (usually written 
 E. M. F.) may exist in a circuit, but no current can flow until the circuit is 
 closed or until the wire is connected so that there will be a path for the 
 current. 
 
 Unit Electromotive Force (E. M. F.).— The practical unit of electromotive 
 force is the volt. It is the unit of electrical pressure, and fulfils somewhat 
 the same purpose as “pounds per square inch” in hydraulic and steam 
 engineering. The E. M. F. furnished by an ordinary cell of a battery usually 
 varies from .7 to 2 volts. A Daniell cell gives an E. M. F. of 1.072 volts. A 
 pressure of 500 volts is generally used for street-railway work, and, for incan- 
 descent lighting, 1 10 volts is common. 
 
 Resistance.— All conductors offer more or less resistance to the flow of a 
 current of electricity, just as water encounters friction in passing through a 
 pipe. The amount of this resistance depends on the length of the wire, the 
 diameter of the wire, and the material of which the wire is composed. The 
 resistance Of all metals also increases with the temperature. 
 
 Unit of Resistance — The practical unit of resistance is the ohm. A con- 
 ductor has a resistance of 1 ohm when the pressure required to set up 
 1 ampere through it is 1 volt. In other words, the drop , or fall, in pressure 
 through a resistance of 1 ohm, when a current of 1 ampere is flowing, 
 is 1 volt. 1,000 ft. of copper wire .1 in. in diameter has a resistance ot 
 nearly 1 ohm at ordinary temperatures- 
 
204 
 
 ELECTRICITY. 
 
 Ohm’s Law.— The law governing the flow of current in an electric circuit 
 was first stated by Dr. G. S. Ohm, and is known as Ohm's law. This law has 
 since stood the test of exhaustive experiment, and has been found correct. 
 Ohm’s law may be briefly stated as follows: The strength of the current in 
 
 any circuit is directly proportional to the electromotive force in the circuit , and 
 inversely proportional to the resistance of the circuit. 
 
 This means that if the resistance of a circuit were fixed, and the E. M. F, 
 varied, the current would be doubled if the E. M.F. were doubled. Also, if 
 the E. M. F. were fixed, and the resistance doubled, the current would be 
 halved. 
 
 Let E = electromotive force in volts; 
 
 R = resistance in ohms; 
 
 C = current in amperes. 
 
 Then, C = -=, or R = — , or E = C R. 
 
 R C 
 
 The last two forms are useful in many cases where the usual form 
 E 
 
 C — -== is not directly applicable. 
 
 R 
 
 Example. — A dynamo D which generates 110 volts, is connected to a coil 
 of wire C, Fig. 1, which has a resistance of 20 ohms; what current will 
 flow, supposing the resistance of the rest of the circuit to be negligible ? 
 
 We have E = 110 volts; R — 20 ohms; hence, C = ^ = 5.5 amperes. 
 
 were 8 amperes and the E. M. F. of the dynamo 110 volts, the resistance of 
 the circuit must be 
 
 Tf lift 
 
 R = ^; R = ~ = 13.75 ohms. 
 C o 
 
 Electrical Power.— The electrical power expended in any circuit is found 
 by multiplying the current flowing in the circuit by the pressure required 
 to force the current through the circuit. In other words, W = E C; where 
 W is the power expended, E is the E. M. F., and C is the current. When E 
 is expressed in volts and C in amperes, then W is expressed in watts. The 
 watt is the unit of electrical power, and is equal to the power developed 
 w T hen 1 ampere flows under a pressure of 1 volt. The watt is equal to 
 horsepower. We have, then, the following general relations: 
 
 Let E = electromotive force in volts; 
 
 C = current in amperes; 
 
 R = resistance in ohms; 
 
 W — power in watts; 
 
 H. P. = horsepower. 
 
 Then, W = E C, but E = C R; hence, W = C 2 R. That is, the power in 
 watts expended in any conductor of which the resistance is R, and through 
 which a current C is flowing, is equal to the product of the squares of the 
 current and the resistance. The energy used in forcing a current through 
 the wire reappears in the form of heat; hence, we may say that the heating 
 effect of a current flowing in a conductor is proportional to the square of the 
 current. From the preceding, we also have 
 
 np_!i = I 
 I 746 746' 
 
 This relation is very useful for calculating power in terms of electrical 
 units The watt is too small a unit for convenient use in many cases, so 
 that the kilowatt, or 1,000 watts, is frequently used. This is sometimes 
 abbreviated to K.W, 
 
CIRCUlfS. 
 
 205 
 
 The Unit of Work Is the Watt-Hour.— This is the total work done when 1 watt 
 is expended for 1 hour. For example, if a current of 1 ampere were made flow 
 for 1 hour through a resistance of 1 ohm, the total amount of work done would 
 be 1 watt-hour. A kilowatt-hour is the total work done when 1 kilowatt is 
 expended for 1 hour. It is about equivalent to 1} horsepower for 1 hour. 
 
 ELECTRICAL EXPRESSIONS AND THEIR EQUIVALENTS. 
 
 {Arranged for Convenient Reference by C. W. Hunt.) 
 
 One 
 
 Watt 
 
 f A rate of doing work. 
 
 1. ampere per sec. at one volt. 
 .7373 foot-pound per second. 
 44.238 foot-pounds per minute. 
 
 ^ 2,654.28 foot-pounds per hour. 
 .5027 mile-pound per hour. 
 .00134 horsepower. 
 
 755 horsepower. 
 
 One 
 
 Kilowatt 
 
 A rate of doing work. 
 
 737.3 foot-pounds per second. 
 44,238. foot-pounds per minute. 
 502.7 mile-pounds per hour. 
 1.34 horsepower. 
 
 One 
 
 Horse- 
 
 power 
 
 r A rate of doing work. 
 
 I 550. foot-pounds per second. 
 
 ! 33,000. foot-pounds per minute. 
 
 \ 375. mile-pounds per hour, 
 
 i 746. watts. 
 
 L .746 kilowatt. 
 
 One 
 
 Watt- 
 
 Hour 
 
 One 
 
 Horse- 
 
 POWKR- 
 
 Hour 
 
 1 A quantity of work. 
 
 2,654.28 foot-pounds. 
 
 503 mile-pound. 
 
 1. ampere hour X one volt. 
 
 .00134 horsepower-hour, 
 horsepower-hour. 
 
 r A quantity of work. 
 
 I 1,980,000. foot-pounds. 
 
 375. mile-pounds. 
 
 ! 746. watt-hour. 
 
 I .746 kilowatt-hour. 
 
 One 
 
 Ampere- 
 
 Hour 
 
 A quantity of current. 
 
 One ampere flowing for one hour, 
 irrespective of the voltage. 
 Watt-hour -5- volts. 
 
 r Force moving in a circle. 
 
 Torque ^ A force of one pound at a radius of 
 t one foot. 
 
 CIRCUITS. 
 
 The path through which a current flows is generally spoken of as an 
 electric circuit. This path may be made up of a number of different parts. 
 For example, the line wires may constitute part of the circuit, and the 
 remainder may be composed of lamps, motors, resistances, etc. In practice, 
 
 the two kinds of circuits most commonly met with are (1) those in which the 
 different parts of the circuit are connected in series; (2) those in which 
 the different parts of the circuit are connected in multiple or parallel. 
 
 1. Series Circuits.— In this kind of circuit, all the component parts are con- 
 nected in tandem, so that the current flowing through one part also flows 
 
206 
 
 ELECTRICITY. 
 
 through the other parts. Fig. 2(a) represents such a circuit made up of a 
 different number of parts. The current leaves the dynamo D at the + side 
 and flows through the arc lamps aaaa , thence through the incandescent 
 lamps 1 1 1, thence through the motor m and resistance r, back to the dynamo, 
 thus making a complete circuit. All these different parts are here connected 
 in series, so that the current flowing through each of the parts #nust be the 
 same unless leakage takes place across from one side of the circuit to the 
 other, and this would be impossible if the lines were properly insulated. 
 The pressure furnished by the dynamo must evidently be the sum of the 
 pressures required to force the current through the different parts, /the 
 most common use of this system is in connection with arc lamps. These 
 lamps are usually connected in series, as shown in Fig. 2 (&). The objections 
 to this system of distribution for general work are that the breaking of the 
 circuit at any point cuts off the current from all parts of the circuit; also 
 the pressure generated by the dynamo has to be very high if many pieces of 
 apparatus are connected in series. In such a system, the dynamo is pro- 
 vided with an automatic regulator that increases or decreases the voltage : of 
 the machine, so that the current in the circuit is kept constant, no matter 
 how many lamps or other devices are in operation. For this reason, such 
 circuits are often spoken of as constant-current circuits. . 
 
 2 Parallel Circuits.— In this type of circuit, the different pieces of appa- 
 ratus are connected side by side, or in parallel, across the mam wires from 
 the dynamo, as shown in Fig. 2 (c). In this case, the dynamo D supplies 
 current through the mains to the arc lamps a, incandescent lamps J and 
 motor m. This system is more widely used, and it will be seen at once from 
 the figure that the breaking of the circuit through any one piece of apparatus 
 will not prevent the current from flowing through the other parts. Incan- 
 descent lamps are connected in this way almost exclusively. The lamps are 
 connected directly across the mains, as shown in Fig. 2 (d). Street cars and 
 mining locomotives are operated in the same way, the trolley wire consti- 
 tuting one main and the track the other, as shown in Fig. 2(e). By adopting 
 this system, any car can move independently of the others, and the current 
 mav be turned off and on at will. In all these systems of parallel distribu- 
 tion the pressure generated by the dynamo is maintained constant, no matter 
 what current the dynamo may be delivering. For example, m the lamp 
 svstem Fig- 2 (d). the dynamo would maintain a constant E. M. J?. ot 110 
 volts. ’Each lamp has a fixed resistance, and will take a certain current 
 
 amperes^ when connected across the mains. As the lamps are turned 
 
 on the current delivered by the dynamo increases, the pressure remaining 
 constant. In street-railway work, the pressure between trolley and track is 
 kept in the neighborhood of 500 volts, the current varying with the number 
 of cars in operation. In mine-haulage plants, the pressure is usually 250 or 
 500 volts, the former being generally preferred as being less dangerous. 
 Lamps may also be connected in series multiple, as shown in Fig 2 (c). 
 Here the two 125-volt lamps 1 1 are connected m series across the 250-volt 
 circuit. Such an arrangement is frequently used in mines when lamps are 
 
 circuits 1 a s those just described are called constant-potential or con- 
 stant-pressure circuits, to distinguish them from the constant-current circuit 
 mentioned previously. 
 
 RESISTANCES IN SERIES AND MULTIPLE. 
 
 Resistances in Lines.— If two or more resistances are connected in series, 
 Fiff 2 ( f ) their total combined resistance is equal to the sum of their sepa- 
 rate resistances. If R equal total combined resistance, and R U R>, R s are the 
 separate resistances connected in series, then, R = -Ri + ^ . nhTT1 
 
 Example.— I f the separate resistances were Ri = 10 ohms, i? 2 — 1 ohm, 
 and = 30 ohms, then these three combined would be equivalent to a 
 
 ^"Rlthtancerin “pVralleK-lf a = numbeTof resistances are . connected in 
 parallel the reciprocal of their total combined resistance is equal to the 
 sum of the reciprocals of the separate resistances. In Fig. 2 (g), three resist- 
 ances are shown connected in parallel. It is evident that the total resistance 
 of such a Combination must be lower than that of the lowest resistance 
 entering into the combination. If the resistances m this case were all equal, 
 
ELECTRIC WIRING. 
 
 207 
 
 the resistance of the three combined would be one-third the resistance of 
 one of them, because a current passing through the three combined could 
 split up between three equal paths, instead of having only one path to pass 
 through. If R represents the combined resistance, and R u R 2 , and R 3 the 
 separate resistances, the following relation is true: 
 
 + -L + JL 
 
 R Ri R 2 R& 
 
 from which 
 
 R = 
 
 R\ Ro R$ 
 
 R 2 Rz + Lti R,i + Ri R-z 13 R 
 
 If the three resistances were all equal, we would have or R = 
 
 Example.— Three resistances of 3, 10, and 5 ohms are connected in par- 
 allel. What is their combined resistance? We have 
 
 l = i+s + i- orie = 
 
 150 
 
 50 + 15 + 30 
 
 150 
 
 95 
 
 1.58 ohms. 
 
 Shunt.— When one circuit B, Fig. 2(h), is connected across another A, so 
 as to form, as it were, a by-pass, or side track, for the current, such a circuit 
 is called a shunt, or it is said to be in shunt with the other circuit. 
 
 ELECTRIC WIRING (CONDUCTORS). 
 
 Materials.— Practically all conductors used in electric lighting or power 
 work are of copper, this metal being used on account of its low resistance. 
 Iron wire is used to some extent for conductors in telegraph lines, and steel 
 is largely used as the return conductor in electric-railway or haulage plants 
 wherti the current is led back to the power station through the rails. The 
 resistance of iron or steel varies from six to seven times that of copper, 
 depending on the quality of the metal. Aluminum is coming into use as a 
 material for conductors, and in future may play an important part in electric 
 transmission. It is so much lighter than copper that it is able to compete 
 with it as a conductor, even though its cost per pound is higher and its 
 
 conductivity only about 6(K that of copper. 
 
 Forms of Conductors.— Most of the conductors used are m the form of 
 copper wire of circular cross-section. Conductors of large cross-section are 
 made up of a number of strands of smaller wire twisted together. For 
 electrolytic plants, copper-refining plants, etc., copper bars of rectangular 
 cross-section are frequently used. . A . . , . 
 
 Wire Gauge.— The gauge most generally used in America to designate 
 the different sizes of copper wire is the American, or Brown & Sharpe 
 (B & S.). The sizes as given by this gauge range from No. 0000, the 
 largest 460 in. diameter, to No. 40, the finest, .003 in. diameter. Wire 
 drawn ’to the sizes given by this gauge is always more readily obtained 
 than sizes according to other gauges; hence, in selecting line wire for 
 anv purpose it is always desirable, if possible, to give the size required as 
 a wire of the B. & S. gauge. A wire can usually be selected from this gauge, 
 which will be verv nearly that required for any specified case. . 
 
 Estimation of Cross-Section of Wires.— The diameter of round wires is 
 usually given in the tables in decimals of an inch, and the area of cross- 
 section is given in terms of a unit called a circular mil. This is done simply 
 for convenience in calculation, as it makes calculations of the cross-section 
 much simpler than if the square inch were used as the unit area. A mil 
 is i of an inch, or .001 in. A circular mil is the area (in decimals of a 
 square inch) of a circle, the diameter of which is in., or 1 mil. The 
 
 circular mil is therefore equal to (.001) 2 = .0000007854 sq. in. 
 
 If the diameter of the conductor were 1 in., its area would be .7854 sq. in.,. 
 
 and the number of circular mils in its area would -be 0000007 ^54 = 1,000,000; 
 
 but 1 in. = 1,000 mils, and (1,000) 2 = 1,000,000; hence the following is true: 
 CM — d-\ or the area of cross-section of a wire in circular mils is equal to the 
 
 square of its diameter expressed in mils. . . 
 
 Example.— A wire has a diameter of .101 in. What is its area in circular 
 ^llOlin. = 101 mils. Hence, CM = (101) 2 = 10,201. 
 
vide 
 
 imbt 
 
 >lum 
 
 R0P1 
 
 cj 
 
 be 
 
 u 3 
 
 <V <3 
 
 |qd 
 
 PQ 
 
 0000 
 
 000 
 
 00 
 
 0 
 
 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 
 
 ELECTRICITY. 
 
 table gives the dimensions, weight, and resistance of pure 
 6 weights given are, of course, for bare wire. The first 
 B. & S. gauge number, the second the diameter in mils, 
 inches would be the number as given in this column, 
 The third column gives the area in circular mils, the 
 olumn being equal to the squares of those in the second 
 e carrying capacity is also given. 
 
 Iopper Wire. American, or Brown & Sharpe, Gauge. 
 
 Area in Circular 
 Mils. C. M. - d 2 . 
 
 Weight. 
 
 Pounds. 
 
 Resistance per 
 1,000 Ft. 
 International 
 Ohms. 68° F. 
 
 Current Capacity 
 (Amperes). 
 National Board 
 Fire Underwriters. 
 
 Per 
 
 1,000 Ft. 
 
 Per 
 
 Mile. 
 
 Weath- 
 
 er-Proof. 
 
 Rubber- 
 
 Covered. 
 
 211,600.0 
 
 640.50 
 
 3,381.4 
 
 .0489 
 
 312 
 
 210 
 
 167,805.0 
 
 508.00 
 
 2,682.2 
 
 .0617 
 
 262 
 
 177 
 
 133^079.4 
 
 402.80 
 
 2,126.8 
 
 .0778 
 
 220 
 
 150 
 
 105,534.5 
 
 319.50 
 
 1,686.9 
 
 .0981 
 
 185 
 
 127 
 
 83,694.2 
 
 253.30 
 
 1,337.2 
 
 .1237 
 
 156 
 
 107 
 
 66,373.0 
 
 200.90 
 
 1,060.6 
 
 .1560 
 
 131 
 
 90 
 
 52; 634.0 
 
 159.30 
 
 841.1 
 
 .1967 
 
 110 
 
 76 
 
 41,742.0 
 
 126.40 
 
 667.4 
 
 .2480 
 
 92 
 
 65 
 
 33,102.0 
 
 100.20 
 
 529.1 
 
 .3128 
 
 77 
 
 54 
 
 26,250.5 
 
 79.46 I 
 
 419.5 
 
 .3944 j 
 
 65 
 
 46 
 
 20,816.0 
 
 63.02 1 
 
 332.7 
 
 .4973 1 
 
 46 
 
 33 
 
 16,509.0 
 
 49.98 
 
 263.9 
 
 .6271 1 
 
 13.094.0 
 
 39.63 
 
 209.2 
 
 • .7908 
 
 32 
 
 24 
 
 10,381.0 
 
 31.43 
 
 165.9 
 
 .9972 
 
 8,234.0 
 
 24.93 
 
 131.6 
 
 1.257 
 
 23 
 
 
 6,529.9 
 
 19.77 
 
 104.4 
 
 1.586 
 
 17 
 
 5,178.4 
 
 15.68 
 
 82.8 
 
 1.999 
 
 
 12 
 
 4,106.8 
 
 12.43 
 
 76.2 
 
 2.521 
 
 16 
 
 3,256.7 
 
 9.86 
 
 52.0 
 
 3.179 
 
 8 
 
 
 2,582.9 
 
 7.82 
 
 41.3 
 
 4.009 
 
 6 
 
 2,048.2 
 
 6.20 
 
 32.7 
 
 5.055 
 
 
 
 1,624.3 
 
 4.92 
 
 25.9 
 
 6.374 
 
 5 
 
 3 
 
 1,288.1 
 
 3.90 
 
 20.6 
 
 8.038 
 
 
 
 1,021.5 
 
 3.09 
 
 16.3 
 
 10.14 
 
 
 
 810.1 
 
 2.45 
 
 12.9 
 
 12.78 
 
 
 
 642.4 
 
 1.94 
 
 10.3 
 
 16.12 
 
 
 
 509.4 
 
 1.54 
 
 8.1 
 
 20.32 
 
 
 
 404.0 
 
 1.22 
 
 6.4 
 
 25.63 
 
 
 
 320.4 
 
 .96 
 
 5.1 
 
 32.31 
 
 
 
 254.1 
 
 .76 
 
 4.0 
 
 40.75 
 
 
 
 201.5 
 
 .61 
 
 3.2 
 
 51.38 
 
 
 
 159.7 
 
 .48 
 
 2.5 
 
 64.79 
 
 
 
 126.7 
 
 .38 
 
 2.0 
 
 81.7 
 
 
 
 100.5 
 
 .30 
 
 1.6 
 
 103.0 
 
 
 
 79.70 
 
 .24 
 
 1.27 
 
 129.9 
 
 
 
 63.21 
 
 .19 
 
 1.01 
 
 163.8 
 
 
 
 50.13 
 
 .15 
 
 .801 
 
 206.6 
 
 
 
 39.75 
 
 .12 
 
 .635 
 
 1 260.5 
 
 
 
 31.52 
 
 .095 
 
 .504 
 
 328.4 
 
 
 
 25.00 
 
 .075 
 
 .400 
 
 1 414.2 
 
 
 
 19.83 
 
 .060 
 
 .317 
 
 ' 522.2 
 
 
 
 15.72 
 
 .047 
 
 .251 
 
 658.5 
 
 
 
 12.47 
 
 .038 
 
 .190 
 
 ► 830.4 
 
 
 
 9.89 
 
 .030 
 
 .15£ 
 
 1 
 
 ; 1,047.0 
 
 
 
ELECTRIC WIRING . 
 
 209 
 
 The following table gives a comparison of the properties of aluminum 
 and copper: 
 
 Comparison of Properties of Aluminum and Copper. 
 
 
 Aluminum. 
 
 Copper. 
 
 Conductivity (for equal sizes) 
 
 Weight (for equal sizes) - 
 
 Weight (for equal length and resistance).... 
 Price (per pound) aluminum, 29 cents; cop- 
 
 per, 16 cents (bare wire) 
 
 Price (equal length and resistance, bare 
 
 .54 to .63 
 .33 
 .48 
 
 1.81 
 
 .868 
 
 .002138 
 
 18.73 
 
 2.5 to 2.68 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 .002155 
 
 10.5 
 
 8.89 to 8.93 
 1 
 
 Temperature coefficient per degree F 
 
 Resistance of mil-foot (20° C.) 
 
 Cnnnififi crfiivitv 
 
 opt3CXX10 gXdVXl^ 
 
 Breaking strength (equal sizes) 
 
 In case a conductor larger man uuu given m ^ A 
 
 stranded cables are used. These are made m various sizes. The table 
 below gives some of the more common sizes, with their allowable current 
 capacity. 
 
 Carrying Capacity of Cables. 
 
 I 
 
 Area. 
 
 Current. 
 
 Amperes. 
 
 Area. 
 
 Circular 
 
 Mils. 
 
 Current. 
 
 Amperes. 
 
 Circular 
 
 Mils. 
 
 Exposed. 
 
 Concealed. 
 
 Exposed. 
 
 Concealed. 
 
 200,000 
 
 299 
 
 200 
 
 1,200,000 
 
 1,147 
 
 715 
 
 300,000 
 
 405 
 
 272 
 
 1,300,000 
 
 1,217 
 
 756 
 
 400,000 
 
 503 
 
 336 
 
 1,400,000 
 
 1,287 
 
 796 
 
 500,000 
 
 595 
 
 393 
 
 1,500,000 
 
 1,356 
 
 835 
 
 600,000 
 
 682 
 
 445 
 
 1,600.000 
 
 1,423 
 
 873 
 
 700,000 
 
 765 
 
 494 
 
 1,700,000 
 
 1,489 
 
 910 
 
 800,000 
 
 846 
 
 541 
 
 1,800,000 
 
 1,554 
 
 946 
 
 900,000 
 
 924 
 
 586 
 
 1,900,000 
 
 1,618 
 
 981 
 
 1,000,000 
 
 1,000 
 
 630 
 
 2,000,000 
 
 1,681 
 
 1,015 
 
 1,100,000 
 
 1,075 
 
 673 
 
 
 
 
 Estimation of Resistance.— The resistance ot any conductor is directly pro- 
 portional to its length, and inversely proportional to its area' of cross-sec- 
 tion, or R = K j, where K is a constant. If L is expressed in feet and 
 
 A is expressed in circular mils, then the constant K must be the resistance 
 of a foot of the wire in question of 1 circular mil cross-section. The resist- 
 ance of 1 mil-foot of copper wire at 75° F. is about 10.8 ohms. Hence, 
 
 for copper wire, we have R = 1^4 but A = <P when d is the diameter 
 
 ^ but A 
 
 , . r) 10.8 L 
 
 in mils; hence, we also have R = — • 
 
 This formula is easily remembered, and is very convenient for estimating 
 the resistance of any length of wire of given diameter when a wire table is 
 not at hand, or when the diameter of the given wire does not correspond to 
 flnvthinsr given in the table. . ^ . . _. 
 
 EXAMPLE.-Find the resistance of 1 mile of copper wire .20 m. m diameter. 
 1 mile = 5 280 ft. .20 in. = 200 mils. Area of cross-section = (200) 2 =* 
 l mue _ 10 .8 X L 10.8 X 5,280 _ i „ 0 
 
 40,000 circular mils. Hence, R — — 1,42 oiims * 
 
 <P 
 
 40,000 
 
210 
 
 electricity. 
 
 CALCULATION OF WIRES FOR ELECTRIC TRANSMISSION. 
 
 Direct-Current Circuits.— No matter how large a wire may be, some energy 
 must always be expended in forcing a current through it, because no con- 
 ductor can be entirely devoid of resistance. It is true that the loss may be 
 made as small as we please by using a very large conductor but m practice, 
 this would not pay, because the interest on the cost of the copper would 
 more than counterbalance the gain in the efficiency of transmission. In 
 starting out, then, to estimate the size of wire to transmit a given 
 of power over a given distance, one of the first things to be decided is the 
 amount of power that may be allowed for loss m the line, because it is evi- 
 dSit that the greater the power lost, the higher may be the line resistance 
 and hence the smaller the wire. The pressure required to force a current 
 C through a wire of resistance R is C X R- This pressure is generally spoken 
 of as the drop , for the reason that the pressure necessary to set up thecurrent 
 through the line is lost, and, consequently, the pressure falls off or drops 
 from tke dynamo to the receiving end of the line. In all cases, the pressure 
 at the end of the line, or point where the power is delivered, is equal to the 
 nressure at the dynamo less the drop in the line, and conversely, the pres- 
 sure that must be maintained by the dynamo m order to obtain a given 
 nressure at the end will be equal to the pressure at the receiving end plus 
 the drop in the line. To illustrate the above, take the case shown m Fig. 3, 
 where a dynamo D supplies current to a motor if situated 1 mile distant. 
 Tn order that the motor may operate properly, the pressure at its terminals 
 must be kept 'constant at, say', 50& volts It “entt^ 
 between a and b (the dvnamo terminals) must be more than 500 volts, bj the 
 drop o? pressure necessary to force the current through the line If the 
 motor is taking very little current, i. e., if it is running on a very light load, 
 the current will be small, and hence the drop in the line will be small. In 
 order then that the pressure at the motor may remain constant, or nearly 
 
 !^jtr S ^e e :ly\n 
 
 present, we are concerned only with the caleuhttionof The toe 
 
 with regard to the maximum 
 current it has to carry. We 
 will suppose, for the sake of 
 illustration, that the motor 
 takes 50 amperes at full load 
 and that the line wire is of 
 such size that it has a resist- 
 ance of .2 ohm per mile. The 
 current has to pass through 2 
 miles of wire (because it has 
 to flow out through 1 mile and 
 back through 1 mile), and 
 hence encounters a resistance 
 of .4 ohm. The drop in the 
 line will then be .4 X 50 = 20 volts, and in order to obtain a pressure of 
 500*volts at the motor, the pressure at the dynamo would have to be 520 
 volts The loss of power in the line would be current X drop — 50 X 20 — 
 1 000 watts or about U horsepower. The drop in an electrical transmission 
 line iJ analogous to the loss in pressure due to the friction encountered 
 
 size of wire were used such that its 
 resistance would be .1 ohm per mile, it is evident that the loss in the line 
 would be halved, but the weight of copper required doubled, because the 
 wire would have to be double the cross-section. The question as to whether 
 it would pay better to invest more money m the line or to put up with the 
 larger loss is something that must be determined m each case by the relative 
 
 C0S In 0 many e ca a se^ the^oss allowed in the line is about 10* of the power to be 
 delivered, though sometimes the loss may be allowed to run as high as 15$ 
 or 25^ This applies only to transmission lines. For local electric-hght or 
 power-distributing systems, the amount of drop allowed is usually about 2$ 
 for the former and 5$ for the latter. 
 
 f The problem of calculating line wires usually presents itself in the 
 following form: Given, a certain amount of power to transmit over a known 
 
CALCULATION OF WIRE. 
 
 211 
 
 distance with a certain allowable loss, to determine the cross-section of the 
 wire required. 
 
 Let P = power to be delivered, expressed in watts; P will be equal 
 to horsepower delivered at end of line multiplied by 746; 
 
 </o — allowable percentage of loss in line, i. e., percentage of 
 power delivered that may be lost in transmission; 
 
 E = voltage at end of line where power is delivered; 
 
 C = current at full load; 
 
 L — length of wire through which current flows. 
 
 The cross-section of the copper conductor will then be given by the 
 following formula: 
 
 . 10.8 X LX <7X100 
 
 A = e x :< ' (1) 
 
 A will be expressed in circular mils, and the corresponding size of wire 
 may be found by consulting the wire table. It should be noticed, particu- 
 larly, that in this formula, L is the average length of conductor through 
 which the current C flows. The application of distance of transmission in 
 the formula will be understood Irom what follows. 
 
 Example.— A mine pump, driven by an electric motor, is situated 2 miles 
 from the power station. The electrical input of the motor at full load is 
 50 H. P., and the voltage at its terminals is to be 500. • Estimate the size of 
 line wire necessary to supply the motor, the allowable loss in the line being 
 15^6 of the power delivered. 
 
 The actual length of line through which the current will flow will be 
 4 miles, because the current has to flow out to the motor and back again. 
 W e have 
 
 watts 50 X 746 
 ~ E ~ 500 
 
 Applying formula (1), we have 
 
 74.6 amperes. 
 
 10.8 X 2 X 2 X 5,280 X 74.6 X 100 
 500 X 15 
 
 226,880 circular mils, nearly. 
 
 By consulting the wire table it is found that this calls for a wire a little 
 larger than No. 0000, which has a cross-section of 211,600 circular mils; 
 No. 0000 wire would probably be used in this case, as it is near enough to the 
 calculated size for all practical purposes. In case the calculated size comes 
 out larger than any size given in the table, a number of wires may be used 
 in multiple to make up the required cross-section, or, what is better, a 
 stranded cable may be used. These heavy stranded cables may now be 
 obtained in different sizes, up to 2,000,000 circular mils cross-section. 
 
 It is evident that, in the above examples, if it were allowable to waste 
 twice as much power in the line, or what is equivalent to having a line drop 
 of 150 volts instead of 75 volts, the cross-section of wire required would have 
 been one-half of that found above. Such a large amount of loss would, 
 however, be objectionable unless power was very cheap. A large drop in 
 the line is in any case objectionable, because the voltage at the receiving 
 end of the circuit will fall off greatly unless the voltage at the generating 
 station is raised, as the load comes on, in order to compensate for the line 
 drop. Most of the uses to which electricity is put, in mines or other places, 
 requires that the pressure at the point where the power is utilized shall be 
 kept approximately constant. For example, in the case of incandescent 
 lights, the lamps will fall off greatly in brightness if the pressure decreases 
 even by a comparatively slight amount. Also, if motors are being operated, 
 the speed will vary considerably if the pressure is not kept constant, and it 
 may be stated, in general, that a large line loss tends to poor regulation at 
 the end of the circuit where power is delivered. 
 
 From the above considerations, it will be seen that, in the majority of 
 cases, the size' of wire to be used under given conditions is determined by 
 the allowable amount of drop. In some cases, however, especially if the 
 current is to be used near at hand, the size of wire so determined might not 
 be large enough to carry the current without overheating. Of course, in 
 such cases, the safe carrying capacity of the wire determines the size to be 
 used, and the drop will be correspondingly less. > 
 
 The amount of current that a given wire can carry without overheating 
 depends very largely on the location of the wire. For example, a wire 
 strung in the open air will carry a greater current, with a given temperature 
 rise, than the same wire would if boxed up in a molding or conduit. The 
 
212 
 
 ELECTRICITY. 
 
 table on page 209 giyes the approximate sate carrying capacity of wires 
 
 when strung in the air. , , 
 
 In order to keep down the size of wire required to transmit a given 
 amount of power over a given distance, with a certain allowable loss, the 
 current must be kept as small as possible. Now, for a given amount of 
 power the current can only be made small by increasing the pressure, 
 because the number of watts, or power delivered, is equal to the product of 
 the current and the pressure. As a matter of fact, if the pressure m any 
 given case be doubled, the amount of copper required will be only one- 
 fourth as great; in other words, for a given amount of power transmitted, 
 the weight of copper required decreases as the square of the voltage. It is 
 at once seen then, that if anv considerable amount of power is to be trans- 
 mitted over long distances, a high line pressure must be used or else the cost 
 of copper becomes prohibitory: The use of high pressures m power trans- 
 mission will be taken up in connection with alternating currents. 
 
 Insulated Wires.— For most overhead line work using modern voltages, 
 weather-proof insulated wire is used. This wire is covered with two or three 
 braids of cotton, and treated with insulating compound. For inside work 
 and in places where a better quality of insulation is required, rubber-covered 
 wires are used. The following table gives the approximate weight of 
 weather-proof line wire. The cost of the wire per pound vanes considerably 
 owing to the variations in the price of copper; about 18 cents per pound 
 may be taken as an approximate figure in making calculations. 
 
 Weather-Proof Line Wire (Roebling’s). 
 
 Number. 
 
 B.&S. 
 
 Gauge. 
 
 Double Braid. 
 
 Triple Braid. 
 
 Outside 
 Diameter. 
 32ds Inch. 
 
 Weight. 
 
 Pounds. 
 
 Outside ; 
 Diameter. 
 32ds Inch. 
 
 Weight. 
 
 Pounds. 
 
 Per 
 
 1,000 Ft. ; 
 
 Per 
 
 Mile. 
 
 Per i 
 1.000 Ft. 
 
 Per 
 
 Mile. 
 
 0000 
 
 20 
 
 716 
 
 3,781 
 
 24 
 
 775 
 
 4,092 
 
 000 
 
 18 
 
 575 
 
 3,036 
 
 22 
 
 630 
 
 3,326 
 
 00 
 
 17 
 
 465 
 
 2,455 
 
 18 
 
 490 
 
 2.58/ 
 
 0 
 
 16 
 
 375 
 
 1.9S0 
 
 17 
 
 400 
 
 2,112 
 
 1 
 
 15 
 
 285 
 
 1,505 
 
 16 
 
 306 
 
 1,616 
 
 2 
 
 14 
 
 245 
 
 1.294 
 
 15 
 
 268 
 
 1,415 
 
 3 
 
 13 
 
 190 
 
 1.003 
 
 14 
 
 210 
 
 1,109 
 
 4 
 
 11 
 
 152 
 
 803 
 
 12 
 
 164 
 
 866 
 
 5 
 
 10 
 
 120 
 
 634 
 
 11 
 
 145 
 
 766 
 
 6 
 
 9 
 
 98 
 
 518 
 
 10 
 
 112 
 
 591 
 
 g 
 
 8 
 
 66 
 
 349 
 
 9 
 
 78 
 
 412 
 
 10 
 
 7 
 
 45 
 
 238 
 
 8 
 
 55 
 
 290 
 
 12 
 
 6 
 
 30 
 
 158 
 
 7 
 
 35 
 
 1 185 
 
 14 
 
 5 
 
 20 
 
 106 
 
 6 
 
 26 
 
 137 
 
 16 
 
 4 
 
 14 
 
 74 
 
 5 
 
 20 
 
 106 
 
 18 
 
 3 
 
 10 
 
 53 
 
 4 
 
 16 
 
 85 
 
 For high-tension lines, it is customary to use bare wires ana msuiaie 
 them thoroughlv on special porcelain insulators. The ordinary weather- 
 proof wire insulation is of little or no use as a protection when these high 
 pressures are used, and it only makes the line more dangerous because of 
 the appearance of false security that it gives. In many cases, ^ is also 
 better to use bare feeders for mine-haulage plants, because the ordinary 
 insulation soon becomes defective in a mine, and a wire in thisconditionis 
 really more dangerous than a bare wire, because the latter is known to be 
 dangerous and will be left alone. 
 
 CURRENT ESTIMATES. 
 
 Before calculating the size of wire required for any given case, it is 
 necessary to know the current, and the method of getting at this will depend 
 on what the current is to be used for. 
 
INCANDESCENT LAMPS. 
 
 213 
 
 Incandescent Lamps.— These are usually operated ou 110-volt circuits, Fig. A, 
 or on the three-wire system, as shown in Fig. 5. In the three- Wire system, 
 two 110-volt dynamos are connected in series so that the voltage across the 
 outside wires is 220. The neutral 
 wire a a connects to the point b 
 where the machines are connected 
 together. The wire a a merely 
 serves to carry the difference in 
 the currents on the two sides of 
 the system, in case more lamps 
 should he burning on one side 
 than on the other. The outside 
 wires for such a system are calcu- 
 lated as if the lights were operated . . .. . ,. . 
 
 two in series across 220 volts. The middle wire is usually made equal m size 
 to the outer wires. An ordinary 16 c. p. incandescent lamp requires about 
 
 55 watts for its operation; a 32 c. p. lamp 
 
 requires about 110 watts. 
 Hence, in the case of 
 ordinary parallel dis- 
 tribution, as shown 
 in Fig. 4, the dy- 
 namo will deliver 
 about i ampere for 
 each 16 c. p. lamp 
 operated, and 1 am- 
 pere for each 32 c. p. 
 lamp. In the case of 
 the three-wire sys- 
 tem, each pair of 16 
 c. p. lamps will take 
 \ ampere, and the 
 
 total number of amperes in the outside wires will be one-fourth the num- 
 ber of lamps operated. . , ... . , „ _ 
 
 Example.— A certain part of a mine is to be illuminated by fifty 16 c. p. 
 lamps and ten 32 c. p. lamps. This portion of the mine is 1,000 ft. from the 
 dynamo room, and the allowable drop in pressure is 5 °J. The lamps are to 
 be run on a 110-volt system. Find the size of wire required. 
 
 Fifty 16 c. p. lamps require - 25 amperes 
 
 Ten 32 c. p. lamps require 10 amperes 
 
 Total current 35 amperes 
 
 We have, then, 
 
 circular mils = 10 ' 8X * 35 * — = 137 - 454 circular mils ’ 
 
 or about a No. 00 B. & S. wire. , . , , ., 
 
 Example. — Take the same case as in the last example, but suppose the 
 lights to be operated on the three-wire system. There will then be twenty - 
 five 16 c. p. lamps and five 32 c. p. lamps on each side of the circuit, and 
 the total current in the outside wires will be 17.5 amperes. The voltage 
 between the outside wires will be 220, and we will have 
 
 10.8 X 1,0 00 X 2 X 17.5 X 100 _ 34 353 circular mils, 
 
 220 X 5 
 
 or about a No. 5 B. & S. wire. 
 
 If We make the central wire also of this size, it is seen that this system 
 would require three-eighths the amount of copper called for by the Pl al d 
 110-volt system. There is the disadvantage that two dynamos are needed. 
 
 Note.— The length to be used in the wiring formula is the average dis- 
 tance traversed by the current in the conductor. For example, it, a $ in 
 Fig. 6 (a), the lamps were all grouped or bunched at the end of the line, tne 
 length used in the formula would be twice that from G to A, because 
 the whole current has to flow out to A through one main and back through the 
 other. In other words, the whole current here passes through the whole 
 length of the line. In case the load is uniformly distributed all along the 
 line, as shown in Fig. 6 (5), it is evident that the current decreases step 'by 
 step from the dynamo to the end. In such a case, the length or distance to 
 be used in the formula is one-half that used in the former case, or simply 
 the distance from the dynamo to the end, instead of twice this distance. 
 
 circular mils 
 
214 
 
 ELECTRICITY. 
 
 Arc LamDS — Arc lamps are frequently run on constant-potential circuits, 
 and usually consume from 400 to 500 watts. There are so many types of 
 these lamps that it is difficult to give any current estimates that will be gen- 
 erally applicable. Enclosed arc lamps usually take from 3 to 5 amperes 
 when run on 110-volt circuits. 
 
 Mot ors —Practically all the motors used in mining work are run on the 
 constant-potential system, either at 250 or 500 volts. The efficiency of ordi- 
 narv motors will vary from 70 $ to 9 096 or higher, depending on the size. The 
 efficiency is greater with the larger machines, and, for the ordinary run o( 
 motors, it will probably lie between 1 80$ and 90/,. By efficiency is here meant 
 the ratio of the useful output at the pulley or pinion of the motor to the 
 total input. The accompanying table gives the efficiency of motors of 
 
 oidinary size. approximate Motor Efficiency. 
 
 f to 11 H. P., inclusive = 75 $ efficiency 
 3 to 5 H. P., inclusive = 80$ efficiency 
 71 to 10 H. P., inclusive = 85$ efficiency 
 15 H. P. and upwards = 90$ efficiency 
 If the required output in horsepower is known, the input will be 
 w _ H. P. X 746 
 
 efficiency 
 
 W 
 
 and the current required at full load will be C = where E is the voltage 
 between the mains at the motor. 
 
 Conductors for Electric-Haulage Plants.-In electric-haulage plants the rails 
 take the place of one of the conductors, so that, m calculating the * size of 
 feeders required, only the overhead conductors are taken into account. It 
 is a difficult matter to assign any definite value to the resisto 
 circuit, as it depends very largely on the quality of the rail bonding at the 
 
 ioints. If this bonding is well done, the resistance of the return circuit 
 should be very low, because the cross-section of the rails is comparatively 
 large. For calculating the supply feeders, we may use the approximate 
 
 fOTmUla ’ • . . .. 14XXXCX100 
 
 circular mils = £x ^ drop • 
 
 In this case, L is the average length of feeder over which the power is to 
 be transmitted. It will be noticed that the constant 10.8 appearing m the 
 previous formulas has here been increased to 14. This has been done to allow, 
 approximately, for the track resistance, but this cons tant might vary con- 
 siderablv, depending on the quality of the rail bonding. If 1 ; ^^ad is ali 
 bunched at the end of the feeder, X is the actual length of the feeder in fee 
 If the load is uniformly distributed all along the line, as it would be it a 
 number of locomotives were continually moving along the line, the dis- 
 tance L in the above formula would be taken as one-half that used in l the 
 case where the load was bunched at the end. In other words the whole 
 current C would only flow through an average of one-half the length of tne 
 line. 
 
DYNAMOS AND MOTORS. 
 
 215 
 
 Example.- In Fig. 7, a b represents a section of track 4 ,°° 0 ft. long. 
 thp dvnamo c to the beginning of the section, the distance is 1,200 ft. The 
 Jmllev wheis No 00 B & S., and is fed from the feeder at regular mterva Is. 
 Two mining locomotives are operated, each of which takes an average cur 
 W 75 ZZr The total allowable drop to the end of the line is to be 
 
 xrf 0 f +^0 terminal voltage, which is 500 volts. Calculate the size of feeder 
 required, assuming that the constant 14, in the formula, takes account of the 
 
 reS Since the locomotives^are moving from place to place ’iJ ie 4 c ™ te f j 0f 
 tribution for the load may be taken at tne center of the 4,000 ft. 1 e 
 
 distance L will then be 1,200 + 2,000 = 3,200 ft 
 150 amperes; hence, we have ^ ^ x 150 x 100 
 
 circular mils 
 
 The total current will be 
 
 = 268,800. 
 
 500X5 . 
 
 This would require either a stranded cable or the use of two No. 00 wires 
 in uarahSfrom^to o From a to b we have the No .00 trolley wire m 
 narallel with the 'feeder; hence, the section of feeder a b may be a J^g e 
 No 00 wire In many cases, the drop is allowed to run as high 10/n, 
 because the loads are usually heavier, and the distances longer, than m the 
 example given above. . 
 
 DYNAMOS AND MOTORS. 
 
 A dynamo is a machine for converting mechanical energy into electrical 
 energv bv moving conductors relatively to a magnetic field. riT . rr . r 
 
 An y electric motor is a machine for converting electrical energy into 
 mechanical energy by the relative motion between conductors carrying a 
 
 current and a ma^netic^eld. b of conductors are made to move 
 
 across a magnetic held by means of a steam engine or other prime mover, 
 and the result is that an E. M. F. is set up in the conductors, and this E. M. I . 
 will set up a current if the circuit is closed. • A , ,, , 
 
 In the case of a motor, a number of conductors are arranged so that they 
 are free to move across a magnetic field, and a current is sent through these 
 “nd"s from some source of electric current The current flowing 
 through these conductors reacts on the magnetic field and causes tne 
 conductors to move, thus converting the electrical energy delivered to the 
 
 m °As r fa^as^e^^ica^coni^uction goes, dynamos and motors are almost 
 identical, and the operation of the motor is exactly the reverse to that of the 
 
 dy DvSamos and motors may be divided into two general classes: 
 
 mos and motors for direct current ; (5) dynamos and motors for alternating 
 
 current. . 
 
 DIRECT-CURRENT DYNAMOS. 
 
 Prinrinle of Action. -Direct-current dynamos are those that furnish a 
 current that always flows in the same direction. This kind of dynamo is 
 largely used for incandescent lighting, and also for the operation of street 
 
 raU A dynamo generates an E. M. F. by the motion of conductors across a 
 magnetic field; hence, at the outset, it is seen that there must be at least two 
 essential parts to a dynamo; namely, a magnet of some kind to set up a 
 magnetic field, and a ‘series of conductors arranged so that they may be 
 moved or revolved in the magnetic field. The first part is known as the 
 
216 
 
 DYNAMOS AND MOTORS. 
 
 field magnet, or very often, simply as the field. The second part is known as 
 the armature. The field is supplied by means of a powerful electromagnet 
 which is magnetized by the current in the field coils. Fig. 8 shows a typical 
 six-pole magnet of this kind; B , B are the magnetizing coils, which, when a 
 current is sent through them, form powerful magnetic poles at N, S. The 
 framework A of such a field magnet is usually made of cast iron or cast steel. 
 
 These field magnets may 
 have any number of poles, 
 but machines of ordinary 
 size are usually provided 
 with from two to eight poles. 
 
 The armature usually con- 
 sists of a number of turns of 
 insulated copper wire, 
 arranged around the periph- 
 ery of a ring or drum built 
 up of soft iron sheets. Fig. 9 
 shows the construction of a 
 typical armature of the ring 
 type. The winding is 
 divided into a number of 
 sections, and the terminals 
 connected to the commutator. 
 
 This commutator consists 
 of a number of copper bars, 
 insulated from each other by 
 means of mica, the bundle of 
 bars being clamped firmly 
 into place and turned up to 
 form a true cylindrical sur- 
 face. The sections in the 
 commutator correspond with 
 those in the armature, and 
 the use and operation of the 
 commutator will be described later. The winding on the ring is endless, 
 i. e., it consists of a number of coils or sections c, the end of one section 
 
 Fig. 8. 
 
 COMPLETE ARMATURE. conductor 
 
 Fig. 9. 
 
 being joined to the beginning of the next, thus forming an endless coil, as 
 shown in Fig. 10. The construction of such a ring armature would be as 
 shown in Fig. 9. 
 
DIRECT-CURRENT DYNAMOS. 
 
 217 
 
 Qnrmnsp the rin<* shown in Fig. 10 with its endless winding to be rotated 
 
 enter^he FK^ 
 
 hand face of the ring are moving upwards, they will have an E. M. F. ge 
 
 erated to them in one direction, while the E. M. F. in the conductors on the 
 left lade will have an E. M. F. in the opposite direction, because all the 
 conductors on this side are 
 moving downwards, or in 
 the opposite direction, to 
 those on the other side. 
 
 These two opposing E. M. 
 
 F.’s will meet at a, as 
 shown by the arrowheads, 
 and will neutralize each 
 other so that no current 
 will flow through the 
 windings of the armature. 
 
 Suppose, however, that 
 taps are connected a£ the 
 points a and a', as shown 
 by the dotted lines, and 
 these taps connected to 
 two rings r, r', mounted so 
 as to revolve with the 
 armature. By allowing 
 brushes b, b f to press on 
 these rings, we can make 
 
 Fig. 10. 
 
 connection with an outside circuit d, which may consist of a number of 
 lamns or any other device through which we wish to send a current. By 
 nuUiL in the taps at a and a', we have allowed the two opposing E. M. F.’s 
 to set up a current through the common connections to the S 
 
 thence trough the outside circuit. Current now flows in each halt of the 
 armature winding, unites at a, flows out by means of ring r and brush b , 
 thence through the outside circuit d to brush b and ring r, from whence it 
 nasses to a ' and thus completes the circuit. When the ring 1 ?^ es a 
 revolution from the position shown in the figure, it is seen that the current 
 in the ouside circuit will flow in the opposite direction. In fact, an arrange- 
 ment of this kind would deliver a current t^wouldte 
 
 it would be what is known as 
 an alternating current. 
 
 Instead of simply bringing 
 out two terminals to rings, 
 suppose the winding to be 
 tapped at a fairly large number 
 of points, and connections 
 brought down to a number of 
 insulated strips, as shown in 
 Fig. 11. If the armature be 
 now revolved, it is seen that 
 the brushes will come in contact 
 with successive bars and keep 
 the outside circuit in such re- 
 lation to the armature winding 
 that the current will always 
 flow through it in the same 
 direction . Moreover, if the num- 
 ber of divisions in the armature 
 be large, the current will fluc- 
 tuate veiv little, being nearly as steady as that obtained from a battery. 
 The arrangement made up of insulated bars is called the commutator, 
 because it commutes or changes the relation of the outside circuit to the 
 armature winding so that the current in the outside circuit always flows 
 in^the same direction. All practical machines used for the generation 
 of direct current must be provided with such a commutator. When alter- 
 nating currents are used it is only necessary to use plain collector rings, as 
 
 y 
 
 -vAAAAAf* 
 
 Re. 
 
 Fig. 11. 
 
218 
 
 DYNAMOS AND MOTORS. 
 
 shown in Fig. 10. The foregoing brief description will give a general idea 
 as to the construction of an ordinary direct-current dynamo or motor. 
 Drum- wound armatures are more frequently used than the ring type shown, 
 but the action is the same in either case. 
 
 Factors Determining E. M. F. Generated— A dynamo should be looked upon 
 as a machine for maintaining an electrical pressure rather than as a machine 
 for generating a current. A pump does not manufacture water — it merely 
 maintains a head or pressure that causes water to flow wherever an outlet is 
 provided for it to flow through. In the same way, a dynamo maintains a 
 pressure, and this pressure will set up a current whenever the circuit is 
 closed, so that the current can flow. The important thing to consider, 
 therefore, is the E. M. F. that the dynamo is capable of generating. 
 
 The E. M. F. generated by an armature depends on the total number of 
 magnetic lines cut through per second by the armature conductors. This 
 means that, in the first place, the faster the armature runs, the higher will be 
 the E. M. F.; in the second place, the greater number of conductors or turns 
 there are on the armature, the higher will be the E. M. F.; and in the third 
 place, the stronger the magnetic field, the higher will be the E. M. F. The 
 E. M. F. in terms of these quantities may be written 
 
 = nCN 
 
 100,000,000’ 
 
 where n = speed in revolutions per second; 
 
 C = number of conductors oh face of the armature; 
 
 N = number of magnetic lines flowing from one pole. 
 
 The constant 100,000,000 is necessary to reduce the result to volts. This 
 equation enables us to make calculations relating to any two-pole dynamo, 
 and with slight modification it is applicable to machines with field magnets 
 
 having a number of poles. It will not be necessary to consider this formula 
 further here, as the main thing to fix in mind is that the E. M. F. is propor- 
 tional to the three quantities: speed, number of conductors, and strength of 
 field. 
 
 Field Excitation of Dynamos.— In the earliest form of dynamo, the magnetic 
 field in which the armature rotated was set up by means of permanent 
 magnets. Permanent magnets are, however, very weak compared with 
 electromagnets, which are excited by means of current flowing around 
 coils of wire wound on a soft-iron core, as shown in Fig. 13. As soon as the 
 current ceases flowing around the coils of an electromagnet, the magnetism 
 almost wholly disappears, but a small amount, known as the residual magnet- 
 ism remains.* It is to this residual magnetism that the dynamo owes its 
 ability to start up of its own accord and excite its own field magnets. 
 When the armature is first started to revolve, a very feeble E. M. F. is gener- 
 ated in it, but the armature is connected to the field coils in such a way 
 •that this small E. M. F. is able to force a small current through the field 
 coils, and thus set up a larger amount of magnetism in the field. This in 
 turn increases the E. M. F. in the armature, and the building-up process goes 
 on rapidly until the dynamo generates its full pressure. There are three 
 different methods in use for supplying the field coils with current, and con- 
 tinuous-current. dynamos are divided into three classes, according to the 
 method used for exciting their fields. These three classes are: (a) Series- 
 wound dynamos; (6) shunt-wound dynamos; (c) compound- wound dynamos. 
 
SHUNT- WOUND DYNAMOS. 
 
 219 
 
 fa'* Series-Wound Dynamos.— In this class of machine, the field coils are 
 connected in Series with the armature, and all the current that passes 
 through the armature also passes through the field and the outside circuit. 
 This arrangement is shown in Fig. 12, where JS 
 and 6’ represent the poles of the magnet, B 
 and — B the brushes, and Re the outside cir- 
 cuit, which may consist of lamps, motors, or 
 any other device in which it is desired to 
 utilize the current. It will be noticed that with 
 an arrangement of this kind, the E. M. F. will 
 increase as the current increases, because the 
 field will become stronger and the speed is sup- 
 posed to remain constant. This will be true up 
 to the point where the field carries all the mag- 
 netism it is capable of, or, in other words, until 
 it becomes saturated. After this point is 
 reached, the E. M. F. will increase very little 
 with increase of current. In most ot the 
 work connected with lighting or power trans- 
 mission, it is desirable to have the voltage „ pri - P Q method of 
 
 remain nearly constant. For this reason therefore, the senes method ot 
 
 “^erator^whic^it^s been 6 applied at all ? e “®™Uy i£ Regulator of 
 
 motors. , 
 
 (b) Shunt-Wound Dynamos.— This style of machine has n0 \ 
 largely of late years, although it was formerly very co^on Its use t 
 present confined more particularly to machines of smal hv nass to the 
 method of excitation, the field is connected as a sh unt ; or .by pass to me 
 ormatnrp* i p the field winding is connected in parallel with tne armature. 
 
 ?e“^^ current flows th rou|h £ 
 
 Halt shows \he connections for this kind of field excitation An adpt 
 
 SS& 
 
 the dynamo is well designed, the pressure at more or^kS 
 
 mately constant. The pressure will, however, always fall off more or less, 
 
 on account of the drop in the armature, due to fh£ field 
 
 the tendency that the current m the armature has of weakening tne e . 
 The shunt winding is used quite largely for motors. 
 
 />\ Pom nound -Wound Dvnamos.— 1 The compound-wound dynamo is the one 
 most largely used for direct-current power and light distribution, and it is so 
 SSsette winding used forfeiting ^ » a««ntagtonofft| 
 
 «pri p<? and shunt windings previously described. me senes yuuing 
 
 serves the purpose of keeping up the field ^^even^makes itfris^with 
 increased and thus keeps the pressure constant, or even makes it rise wiin 
 
 increased load, if so deSred, . When the series winding is so admsted that 
 the pressure rises as the load is increased the machine is said to be overcow 
 wounded Fig. 14 shows the connections for such a machine. It will oe see 
 ?hat the shunt winding is connected as before, a field resistance or rheosta , 
 Xhown in the figurl, being inserted for the ™f 
 age. One brush connects directly to one terminal 0 .tthe machine + wnue 
 the other brush connects to one end of the series wmding on tfie field r l he 
 other end of the series winding forms the other terimna . - ^t? 
 nnt«idp pircnit R e is connected. It is thus seen that the shunt con supplies 
 a certain amount of initial magnetization that is ^ ^that 
 
 ism supplied by the series coils. Of course, care, must be taken to^ see that 
 the current in the series coils circulates around the field in the same mrec 
 tion as that in the shunt coils, otherwise the effect would be to make tne 
 E. M. F. fall off with increasing toad instead of keeping it up. This lstim.tyie 
 of dynamo used almost exclusively for electric haulage plants, as well as 
 plants for direct-current illuminating purposes. 
 
220 
 
 DYNAMOS AND MOTORS. 
 
 DIRECT-CURRENT MOTORS. 
 
 Direct-current motors are in general almost identical, so far as construc- 
 tion goes, with direct-current dynamos. Motors are often required to oper- 
 ate under very trying conditions, as for example, in mine haulage or pump- 
 ing plants or on the ordinary street car. For this reason, their mechanical 
 construction often differs somewhat from that of the dynamo, the design 
 being modified in such a way as to enclose the working parts as completely 
 as possible, and thus protect them from dirt and injury. The two kinds of 
 motors most commonly used are the series and shunt varieties. Compound- 
 wound motors are only used for a few special kinds of work. Practically all 
 of the motors in use* are operated from constant-pressure mains; i. e., the 
 pressure at the terminals of the- motor is practically constant, no matter 
 what load it may be carrying. We will here consider constant-potential 
 motors only. 
 
 Principles of Operation.— If the fields of an ordinary constant-potential 
 dynamo are excited and a current supplied to the armature from some out- 
 side source, such as another dynamo D , Fig. 15, so that the current enters 
 
 at +B, and passing through 
 the winding in the direc- 
 tion indicated by the 
 arrowheads, leaves at brush 
 — B, it will be found that 
 all of the conductors under 
 the S pole face, b, c, d, e , /, 
 and g , will tend to move 
 downwards , and all those 
 under the N pole face, j, k , 
 Z, m, ? 2 , and o, will tend to 
 move upwards, as indicated 
 by the small arrows. 
 
 These forces combine to 
 produce a tendency of the 
 armature to rotate about 
 its axis as indicated by the 
 large arrows, which ten- 
 dency is called the torque 
 of the motor. 
 
 The amount of this 
 torque— which is usually 
 expressed in pound-feet; 
 that is, a certain number of pounds acting at a radius of a certain number 
 (usually 1) of feet— depends on (1) the strength of the field, (2) the number 
 of conductors, (3) their mean distance from the axis of the armature, and 
 (4) the amperes in each conductor. In any given machine, the second and 
 third conditions are constant, so that tho torque depends on the strength of 
 the field and the current. . , A _ A _ 
 
 If the armature is stationary, the E. M. F. required to send the current 
 through the winding is only that necessary to overcome the drop, Avhich is 
 due to the resistance of the winding. If the torque exerted by this current 
 is greater than the opposition to motion, so that it causes the armature to 
 revolve, the motion of the conductors through the field generates in them 
 an E. M. F. that is opposed to the E. M. F. that is sending the current through the 
 
 armature. . 
 
 This opposing E. M. F., or counter E. M. F. as it is called, then diminishes 
 the effect of the applied E. M. F., so that the current is reduced, reducing 
 the torque. Should the torque still be greater than the opposition to 
 motion, the speed of the armature will continue to increase, increasing the 
 counter E. M. F., and thereby further reducing the current and the cor- 
 responding torque, until the torque just balances the opposition to the motion , 
 when the speed will remain constant. ' _ , 
 
 At all times, the drop of potential through the armature is equal to the 
 difference between the counter and the applied E. M. F.'s , and as the product of 
 this drop and the current represents energy wasted, it is desirable to make it 
 as low as possible. In good motors of about 10 H. P. output, the drop m 
 the armature is seldom more than about 5$ of the applied E. M. F., and is 
 less in larger machines. , . . .. , 
 
 This being the case, it is evident that if the armature is at rest, so that it 
 
DIRECT-CURRENT MOTORS. 
 
 221 
 
 has no counter E. M. F., and is connected directly to the mains, a very large 
 current will flow through it, which would be liable to damage the armature. 
 On this account an external resistance, called a starting resistance , is con- 
 nected in series with the armature when it is to be started. This resistance 
 isma^Sreat enough to prevent more than about the normal current from 
 flowing through the armature when it is at rest; as the armature speeds up 
 and develops some counter E. M. F., this resistance is gradually cut out 
 until the armature is connected directly to the mams, and is running at 
 
 n ° The energy represented by the product of drop in the armature and the 
 current is wlsted; that represented by the product of the current and the 
 rest of the E. M. F., that is, the counter E. M. F., is the energy required to 
 
 ke< Aside fTom Secom^aratively small amount of current required to furnish 
 the toraue necessary for overcoming the frictional losses m the motor itself, 
 which are SS^SSfy constont, the Amount of current taken from the mams 
 is directly proportional to, and varies automatically with, the amount of the 
 external load- for, if this external load is increased the current which has 
 been flowing’ in the armature cannot furnish sufficient torque for this 
 increased loaxl, so that the machine slows down. This decreases the counter 
 E. M. F., which immediately allows more current to flow though plnaf^ad 
 ture increasing the torque to the proper amount. If the external load 
 
 is decreased, tie current flowing furnishes an excess oftorque. wjK'fl 
 causes the speed to increase, increasing the counter E. M. *., and ae 
 creasing the current until it again furnishes only the required amount 
 
 ° f Sincere counter E. M. F. is very nearly equal to the a PP^, it is only 
 necessarv for it to vary a small amount to vary the current within wide 
 limits For example, if the resistance of a certain armature is 1 ohm, and it 
 is supplied with current at a constant potentialof 250 volts then, when a 
 current of 10 amperes is flowing through it, the drop is 10 XI — 10 \oits 
 and the counter E M F. is 250 — 10 = 240 volts. Now, if the current is 
 reduced to ? ampere, the drop is 1 X 1 = 1 volt, and the counter E. M. F. is 
 
 9 
 
 250 _ ! = 249 volts; that is, the counter E. M. F. only varies ^ 5 , or 3.75$, 
 
 while the current varies or 90$. 
 
 As stated before, the field magnets of constant-potential motors are 
 nsiiallv either shunt-wound or series-wound. . , . 
 
 If shunt-wound, and supplied from a constant-potential circuit, the mag- 
 netizing force of the field coils is constant, giving a pracacally constant 
 field. This being the case, the counter E. M. F. is directly proportional to 
 the speed, so that variations of the load make only slight variation m 
 the speed. A shunt-wound motor is then (practically) a constant-speed 
 
 m °With series-wound motors, the strength of the field varies with the cur- 
 rent; if the load on such a motor is reduced, the excess of torque makes 1 the 
 armature speed up, but as the resulting decrease of the cu ^t fn 
 
 the field strength, the armature must speed up to a much greater extent, m 
 order to increase the counter E. M. F. to the right degree, than would be 
 necessary if the field were constant. If the load is increased, the increase 
 in the current so increases the field strength that the speed must decrease 
 considerably, in order to decrease the counter E. M. F. by the right amount. 
 The speed of a series-wound motor, then, varies largely with variations in 
 
 th An a advantage of the series motor is that if a torque greater than the 
 normal is required, it can he obtained with less current than with a shunt 
 motor, since the increased current increases the field strength, and tne 
 
 torque is proportional to both these factors. . „ , . „_x_, 
 
 It would not be practicable to make the field strength of a shunt motor 
 as great as is possible to get with a series motor, since it would .require a 
 very large magnetizing force, and with the shunt winding, this extra 
 magnetizing force would have to be expended all the time, whether the 
 strong field was required or not, which would be very wakeful, ^ the 
 series motor, however, this extra magnetizing force is only expended while 
 it is needed. 
 
222 
 
 DYNAMOS AND MOTORS. 
 
 A disadvantage of the series winding is that if all the load is taken off, 
 the current required to drive the motor is very small, making a weak held, 
 which requires such a high speed to generate the proper counter E. M. F. 
 that the armature is liable to be damaged. In other words, the motor will 
 race or run away, if the load is all removed. This cannot occur with the 
 shunt motor as long as the field circuit remains unbroken. 
 
 On account of the above features, shunt motors are used to drive 
 machinery that requires a nearly constant speed with varying loads, or 
 which would be damaged if the speed should become excessive, such as 
 ordinary machinery in shops and factories, pumps, etc. Series motors are 
 used on street cars, to operate hoists, etc., where, on account of the gearing 
 used, the load cannot be entirely thrown off, and the torque required at 
 starting and getting quickly up to speed is much greater than the normal 
 
 ^Speed Regulation of Motors.— The torque of a motor depends on the current; 
 that is, for a given current, the torque will be the same whatever may be 
 the speed, provided the field strength remains the same. . The speed at 
 which the armature runs is a matter of E. M. F. only; that is, with a given 
 current the speed will be proportional to the applied E M. F., or, more 
 strictlv, the counter E. M. F., other conditions remaining the same. 
 
 It has been shown that the torque will automatically regulate itself for 
 changes in the load. The speed, however, may be varied by varying the 
 applied E. M. F. or the strength of the field. A change m speed may or 
 may not result in a change in the torque required, depending on the 
 character of the work done by the motor. 
 
 The simplest way to vary the applied E. M. F. is to insert a resistance, in 
 series with the armature, similar to the starting resistance. By varying 
 this resistance, the applied E. M. F. at the terminals of the motor is also 
 varied, although the E. M. F. of the mams remains constant. It is evident 
 
 Trolley 
 
 Rail 
 
 f — UUUU " 
 
 ^ 250 Volts > 
 
 1- 250 Volts H 
 
 
 ow y oils 
 
 Fig. 16. 
 
 that the energy represented by the product of the current and the drop 
 through the resistance is converted into heat, and is thereby wasted, 
 therefore, for great variations in speed, this method is not economical, 
 
 though often very convenient. . , . at t? mo 
 
 The applied E. II. F. may also be varied by varying the E M. F. of the 
 generator supplving the current, but this can only be done iv here a single 
 generator is supplying a single motor, or several motors, 
 all oe varied at the same time; so that this method is only used in special 
 
 CRS6S 
 
 If ’the strength of the field is changed, the speed necessary to give a cer- 
 tain counter E. M. F. will also be changed, which gives a convenient 
 method of varying the speed. If the strength of the field is lessened, the 
 speed will increase* and if the field is strengthened, the speed will decrease 
 With shunt motors, the field may be weakened by inserting a suitable 
 resistance in the field circuit, as in shunt dynamos; with senes motor* 3 the 
 same result mav be obtained by cutting out some of the turns* of the field 
 coils or by placing a suitable resistance m parallel with the field coils. 
 
 This method of regulation Is also of limited range, since it is not econom- 
 ical to maintain the strength of the field much above or betow a certain 
 density. The resistance method described above being rather more 
 it is generally used. For special cases, such as street : railroad work, various 
 special combinations of the above methods of regulation are used. 
 
 One of the most common of these is known as the series-parallel method, 
 and is the method of regulation generally used at present for operating 
 street cars. This method is equivalent to the method of cutting down the 
 speed by reducing the E. M. F. applied to the motor, and is only applicab e 
 where at least two motors are used. It is also used, to some extent, m 
 haulage plants. When a low speed is desired, or when- the cm to be 
 started up the motors are thrown m series, as shown in Fig. 16, thus making 
 ?he voltaV across each motor equal to one-half 
 
 lines, and cutting down the speed accordingly. When a high speed is 
 
CONNECTIONS FOR MOTORS. 
 
 223 
 
 Trolley 
 
 
 -SOOVoltSr 
 
 Fig. 17. 
 
 Ground 
 
 desired the motors are thrown in multiple, as shown in Fig. 17, and each 
 motor runs 6 a? 1 full 1 speed because it gets the full ine pressure. ^ practice, 
 ctartino- resistances are used in connection with the above to maKe me 
 startini smooth but the two running positions are as shown, the motors 
 beiim connected in series in the one case, and in parallel in the other. 
 
 C®nnXn S tr Crntin U ou S .Curr ? nt Motors.-Fig. 18 shows the manner n 
 which a shunt motor is connected to the terminals + and — ot the circuit. 
 It will be seen that the current 
 
 through the shunt field does not /TWTT'' — — 
 
 pass through the resistance R KAAAJ 
 
 which is connected in the arma- 
 ture circuit. This is necessary, 
 since to keep the field strength 
 constant, the full difference of 
 potential must be maintained be- 
 tween the terminals of the field 
 
 weak field that an excessive current would be required to turnisn ine 
 
 ,ie Whtn’ connected afshown! hZcver, the field is brought up to its full 
 strength before any curreiU passes through the armature; so this difficulty 
 
 d °fin?e fnfseries motor the same current 
 
 field coils, the starting resistance may be placed l in any part trf “e^ciM 
 ^fhe'tffe terminlds 1 ? the motor nC6 ^ 
 
 " f To C reverse^ the 
 
 either Urn direction of the field or the direct™ of the 
 armature. It is usual to reverse the direction of the current ; in tne a 
 ture a switch being used to make the necessary changes in the conne Ctl rS!l~ 
 l on chows the connections of one form of reversing switch Two 
 
 “nTLpplied with "a 1 h^nd^e^^^d t the e two^ar^ ^Sn^togethe? by®a 
 
 “ft c are^Sanged’on't^e'base* of SK&? eS& 
 
 bars B and B l may rest either on a and b, as shown by the full 
 
 c as shown by the dotted lines. The line is connected to the terminals 
 T and T n , and the y motor armature between a and b, or vice versa, a an 
 
 be ^“elwttehfsm^the position shown by the full lin^ r is connected 
 
224 
 
 DYNAMOS AND MOTORS. 
 
 between c and a. The direction of the current through the motor 
 armature, or whatever circuit is connected between a and b , is thus reversed. 
 
 In order to reverse only the current in the armature, the reversing switch 
 must be placed in the armature circuit only. Fig. 21 represents the connec- 
 tion for a reversing-shunt motor (a) and a reversing- 
 series motor (6); + and — are the line terminals; R, 
 the starting resistance; B and B\, the brushes of the 
 motor, and F \ the field coil of the motor. Some man- 
 ufacturers combine the starting resistance and revers- 
 ing switch in one piece of apparatus. 
 
 In connecting up motors, some form of main switch 
 is used to entirely disconnect the motor from the line 
 when it is not in use. 
 
 To prevent an excessive current from flowing 
 through the motor circuit from any cause, short strips 
 of an easily melted metal, known as fuses, mounted on 
 suitable terminals, known as fuse boxes, are placed in 
 the circuit. These fuses are made of such a sectional 
 area that a current greater than the normal heats 
 them to such an extent that they melt, thereby breaking the circuit and 
 preventing damage to the motor from an excessive current. The length 
 of fuse should be proportioned to the voltage of the circuit, a high voltage 
 requiring longer fuses than a low voltage, in order to prevent an arc being 
 maintained across the terminals when the fuse melts. 
 
 If desired, measuring instruments (ammeter and voltmeter) may be 
 connected in the motor circuit, so that the condition of the load on the 
 motor may be observed while it is in operation. All these appliances, 
 regulating' resistance, reversing switch, fuses, instruments, etc., are placed 
 
 inside the main switch; that is, the current must pass through the main 
 switch before coming to any of these appliances, so that opening the main 
 switch entirely disconnects them from the circuit, when they may be 
 handled without fear of shocks. 
 
 ALTERNATING-CURRENT DYNAMOS. 
 
 An alternating-current dynamo is one that generates a current 
 that periodically reverses its direction of flow. It was shown in 
 connection with Fig. 10 that an armature provided simply with collector 
 rings produced an alternating current in the outside circuit. This 
 current may be represented by a curve such as that shown in Fig. 22. 
 The complete set of values that the 
 current or E. M. F. passes through 
 repeatedly is known as a cycle. For 
 example, the values passed through 
 during the interval of time represented 
 by the distance a c would constitute a 
 cycle. The set of values passed 
 through during the interval a b is 
 known as an alternation. An alternation is, therefore, half a cycle. The 
 number of cycles passed through per second is known as the frequency of the 
 current, or E. M. F. 
 
 Alternating-current dynamos are now largely used both for lighting and 
 power transmission, especially when the transmission is over long distances. 
 The reason that the alternating current is specially suitable for long-distance 
 
 Fig. 22. 
 
ALTERNATORS. 
 
 225 
 
 work is that it may be readily transformed from one pressure to another. 
 We have already seen that in order to keep down the amount of copper in 
 the line, a high line pressure must be used. Pressures much over 500 or 600 
 volts cannot be readily generated with direct-current machines, owing to the 
 troubles that are likely to arise due to sparking at the commutator. On the 
 other hand, an alternator requires no commutator or even collecting rings, 
 if the armature is made stationary and the field revolving, as is frequently 
 done. Alternators are now built that generate as high as 8,000 or 10,000 volts 
 directly. If a still higher pressure is required on the line, it can be easily 
 obtained by the use of transformers, to be explained later. It is thus seen 
 that where power is to be carried over long distances, the alternating current 
 is indispensable. _ . . , , 
 
 Alternating-current dynamos, like direct-current machines, consist ot 
 two main parts, i. e., the field and armature. Either of these parts may 
 be the revolving member, 
 and in many modern ma- 
 chines the armature, or the 
 part in which the current 
 is induced, is the revolving 
 member. Fig. 23 shows a 
 typical alternator of the 
 belt-driven type, having a 
 revolving armature. It is 
 not unlike a direct-current 
 machine as regards its gen- 
 eral appearance. The 
 number of poles is usually 
 large, in order to secure 
 the required frequency 
 without running the ma- 
 chine at a high rate of 
 speed. The frequencies 
 met with in practice vary 
 all the way from 25 to 150. 
 
 The higher frequencies 
 are, however, passing out 
 of use, and at present a 
 frequency of 60 is very , 
 
 common. This frequency is well adapted both for power and lighting pur- 
 poses. When machines are used almost entirely for lighting work, frequen- 
 cies of 125 or higher may be used. The frequency of any machine may be 
 readily determined when the number of poles and the speed is known, as 
 follows: ' 
 
 number of poles v , rev. per min. 
 
 Frequency = t> X ^ • 
 
 For example, if an eight-pole alternator were run at a speed of 900 R. P. M., 
 the frequency would be 
 
 8 _ 900 
 2 
 
 / = — x ^77 = 60 cycles per second. 
 
 60 
 
 Alternators may be divided into the two following classes: (a) Single* • 
 phase alternators; ( b ) Multiphase alternators. , , 
 
 (a) Single-Phase Alternators.— These machines are so called because they 
 
 generate a single alternating current (as represented by the curve shown in 
 Fig. 22). The armature is provided , with a single winding and the two 
 terminals are brought out to collector rings, as previously described. Single- 
 phase machines have been largely used in the past for lighting work, but 
 they are gradually being replaced by multiphase machines, because the 
 single-phase machines are not well suited for the operation of alternating- 
 current motors. . „ , , .. 
 
 ( b ) Multiphase Alternators.— These machines are so called because they 
 deliver two or more alternating currents that differ in phase; i. e., when one 
 current is, say, at its maximum value, the other currents are at some other 
 value. This is accomplished by providing the armature with two or more 
 distinct windings which are displaced relatively to each other on the 
 armature. One set of windings, therefore, comes under the poles at a later 
 instant than the winding ahead of it, and the current in its winding comes 
 
226 
 
 DYNAMOS AND MOTORS. 
 
 to its maximum value at a later instant than the current in the first wind- 
 ins. In practice, the two types of multiphase alternator most commonly 
 used are (1) two-phase alternators, (2) three-phase alternators. 
 
 Two-phase alternators are machines that deliver two alternating currents 
 that differ in phase by one-quarter of a complete cycle; i. e., when the 
 current in one circuit is at its maximum value, the current m the other 
 circuit is passing through its zero value. By tapping four equidistant points 
 of a regular ring armature, as shown in Fig. 24, and connecting these points 
 to four collector rings, a simple two-pole two-phase alternator is obtained. 
 One circuit connects to rings 1 and T , the other circuit connects to rings 
 2 and ft It is easily seen from the figure that when the part of the winding 
 connected to one pair of rings is in its position of maximum action, the 
 E M F in the other coils is zero, thus giving two currents m the two 
 different circuits that differ in phase by one-quarter of a cycle or one-half 
 
 Three-phase alternators are machines that deliver three currents that differ 
 in phase by one-third of a complete cycle; i. e., when one current is flowing 
 in one direction in one circuit, the currents in the other two circuits are one- 
 half as great, and are flowing in the opposite direction. By tapping three 
 equidistant points of a ring winding, as shown in Fig. 25, a simple three- 
 phase two-pole alternator is obtained. Three mams lead from the collecting 
 rinsrs 
 
 In order to have three distinct circuits, it would ordinarily be necessary 
 to have six collecting rings and six circuits; but this is not necessary in a 
 three-phase machine if the load is balanced in the three different circuits, 
 because one wire can be made to act alternately for the return of the other 
 
 tU< Uses of Multiphase Alternators— Multiphase alternators are coming largely 
 into use, because, by using them, alternating-current motors can be readily 
 operated. By using multiphase machines, motors can be operated that will 
 
 Fig. 24. 
 
 Fig. 25. 
 
 start from rest under load, whereas with single-phase machines the motor 
 has to be brought up to speed from some outside source of power before it can 
 be made to run. For this reason, such machines are used for the operation 
 of modern power-transmission plants. As far as the general appearance of 
 three-phase machines goes, they are similar to ordinary single-phase alter- 
 nators^ the only difference being in the armature winding and the larger 
 number of collector rings. The multiphase alternator is also-adapted tor 
 the operation of lights, so that by using these machines, both lights and 
 motors may be operated from the same plant. They are well adapted for 
 power-transmission purposes in mines, especially for the operation of pump- 
 ing and hoisting machinery, because the motors operated by them are very 
 simple in construction and therefore not liable to get out of order. 
 
 ALTERNATING-CURRENT MOTORS. 
 
 Alternating-current motors may be divided into two general classes: 
 
 (a) Synchronous motors; (5) Induction motors. .. 
 
 '(a) Synchronous motors are almost identical, so far as construction goes, 
 with the corresponding alternator. For example, a two-phase synchronous 
 motor would be constructed in the same way as a two-phase alternator. 
 
alternating-current motors. 
 
 227 
 
 They are called synchronous motors because they always run in synchro- 
 nism, or in step, with the alternator driving them. This means that the 
 motor runs at the same frequency as the alternator, and if the motor had 
 the same number of poles as the 
 alternator, it would run at the same 
 speed, no matter what load it might 
 be carrying. This type of motor has 
 many good points, and is especially 
 well suited to cases where the 
 amounts of power to be transmitted 
 are comparatively large and where 
 the motor does not have to be 
 started and stopped frequently. 
 
 Multiphase synchronous motors will 
 start up from rest and will run up to 
 synchronous speed without aid from 
 any outside source. They will not, 
 however, start with a strong starting 
 torque or effort, and will not, there- 
 fore, start up under load, and can- 
 not be used in places where a strong 
 starting effort is required. For this 
 reason synchronous motors are not 
 suitable for intermittent work. 
 
 (6) Induction motors are so called 
 because the current is induced in the 
 
 Fig. 26. 
 
 because the current is induced in me , „ hrsW55 
 
 armature instead of being led into it from some outside eS mle hin^i 
 
 a typical induction motor. There are two essential parts ^n .these ! machines, 
 viz the field into which multiphase currents are led from the line, and me 
 armature in which currents are induced by the magnetism set up by the 
 field .SherWe^SS may be the stationaryorrevolvmg member but 
 in most cases the field, or part that is connected to the hue, is stationary. 
 Fig. 27 shows the construction of. the stationary member or field. This con 
 sists of a number of iron laminations, built up to n ?titu«nff 
 
 with slots around the inner periphery. Theform- woundcoils constitu ting 
 the field winding are placed in these slots and connected the mains. This 
 
 winding is arranged in the same way as the ar ^ a ture nse absent 
 
 phase alternator. When the alternating currents _differing in .phase are 
 through the winding, magnetic poles are 
 
 the constant changing of the 
 currents causes these poles to 
 shift around the ring, thus set- 
 ting up what is known as a 
 revolving magnetic field. This 
 armature, Fig. 28, consists of a 
 laminated iron core provided 
 with a number of slots, in each 
 of which is placed a heavy 
 copper bar b. The ends of 
 these bars are all connected 
 together by two heavy short- 
 circuiting rings r, r running 
 around each end of the arma- 
 ture. The bars and end rings 
 thus form a number of closed 
 circuits. When such an arma- 
 ture is placed in the revolv- 
 ing field, the magnetism will 
 cut across the armature con- 
 ductors, inducing E. M. F.’s 
 in them, and since the conduc- 
 tors are joined up into closed 
 circuits, currents will flow in 
 mmr them. These currents will 
 react on the field, and the armature will be forced to revolve. Such an 
 armature will not run exactly in synchronism, because if it did, it would 
 revolve just as fast as the magnetic field, and there would be no cutting ot 
 
 i 
 
228 
 
 DYNAMOS AND MOTORS . 
 
 tines of force. The speed drops slightly from no load to full load, but if the 
 motor is well designed, this falling off in speed is slight. 
 
 Induction motors possess many advantages for mine work. One of the 
 chief of these is the absence of the commutator or any kind of sliding con- 
 1 tacts whatever. Such motors can 
 
 vide the armature with a winding simils 
 the terminals to collecting rings, so that 
 armature circuit. 
 
 therefore operate with absolutely 
 no sparking— a desirable feature 
 for mine work. The motors are 
 also very simple in construction, 
 and are therefore not liable to 
 get out of order. They have an 
 additional advantage over the syn- 
 chronous motor in that they start 
 up with a strong starting effort, 
 and, in fact, behave in most re- 
 spects like any good shunt-wound 
 direct-current motor. They are 
 used quite successfully for all 
 kinds of stationary work, such as 
 pumping, hoisting, etc., but so far 
 have not been used to any great 
 extent for haulage purposes. 
 When these motors are used for 
 purposes where a variable speed is 
 required, it is customary to pro- 
 r to that of the field and bring out 
 resistance may be inserted in the 
 
 TRANSFORMERS. 
 
 Reference has already been made to the use of transformers for changing 
 an alternating current from a higher to a lower pressure, or vice versa, with 
 a corresponding change in current. Transformers used for raising the volt- 
 age are known as step-up transformers; those used for lowering the pressure 
 are known as step-down transformers. 
 
 The transformer consists of a laminated iron core upon which two coils 
 of wire are wound. These coils are entirely distinct, having no connection 
 with each other. One of these coils, called the primary , is connected to the 
 mains; the other coil, called the secondary , is connected to the circuit to 
 which current is delivered. Fig. 29 shows the arrangement of coils and core 
 for a common type of transformer. The secondary 
 coil is wound in two parts S, S', and the primary coii, 
 also in two parts P, P', 
 is placed over the sec- 
 ondary. C is the core, 
 built up of thin iron 
 plates. Fig. 30 shows a 
 weather-proof cast-iron 
 case for this transformer. 
 
 When a current is sent 
 through the primary it 
 sets up a magnetism in 
 the core which rapidly 
 alternates with the 
 changes in the current. 
 
 This changing magnet- 
 ism sets up an alterna- 
 ting E. M. F. in the sec- 
 ondary, and this second- 
 ary E. M. F. depends 
 upon the number of turns in the secondary coil. If the secondary turns are 
 greater than the primary, the secondary E. M. F. will be higher than that 
 of the primary. The relation between the primary E. M. F. and secondary 
 E. M. F. is given by the following: 
 
 Secondary E. M. F. = primary E. M. F. X ^ ecQn( ^ ar y ^ urns . 
 
 primary turns 
 
 Fig. 29. 
 
BATTERIES. 
 
 22 , 
 
 , ^ ,, primary E. M. F. 
 
 or, secondary E. M. F. = - — . — - — - . 
 
 ’ . primary turns 
 
 secondary turns 
 
 . . primary turns . . , , 
 
 The ratio — — z is known as the 
 
 secondary turns 
 
 ratio of transformation of the transformer. 
 For example, if a transformer had 1,200 pri- 
 mary turns and 60 
 
 
 Phase 1. 
 
 
 
 
 
 
 
 
 Phase 
 
 2. 
 
 
 lOOO V. 
 P 
 
 JLSSJJLSL 
 
 
 * 1000 P* 
 P' 
 
 
 
 
 
 3— 
 
 3 
 
 f 000 0006 
 F-J OOP- 
 
 n 
 
 secondary turns, its 
 ratio of transforma- 
 tion would be 20 to 1, Fig. 32. 
 
 and the secondary 
 
 voltage would be one-twentieth that of the primary. 
 Transformers are made for a number of different 
 ratios of transformation, the more common ones 
 being 10 to 1 or 20 to 1. Of course, a transformer 
 never gives out quite as much power from the sec- 
 ondary as it takes in from the primary mains, 
 because there is always some loss in the iron core 
 and in the wire making up the coils. The efficiency 
 
 
 
 
 
 
 
 
 1000 V- 
 p 
 
 I 
 
 -1000 V 
 p' 
 
 mwiim 
 
 Fig. 33. 
 
 |— *100 V — 
 
 loo r- 
 
 Fig. 31. 
 
 C 5 
 
 lOO V — * 
 
 of transformers is, however, high, reaching as high as 97$ or 98$ in the 
 larger sizes. Transformers are always connected in parallel across the 
 mains, and if they are well designed, will furnish a very nearly constant 
 secondary pressure at all loads, when furnished with a constant primary 
 pressure. Fig. 31 shows transformers connected on a single-phase circuit, 
 Fig. 32 shows the connection for a two-phase circuit, and Fig. 33 shows 
 one method of connection for a three-phase circuit. 
 
 ELECTRIC SIGNALING. 
 
 BATTERIES. 
 
 Batteries are used for various purposes in connection with mining work, 
 principally for the operation of bells and signals. 
 
 The LeclancM cell is one that is widely used for 
 bell and telephone work. It is made in two or 
 three different forms, 
 one of the most com- 
 mon of these being as 
 shown in Fig. 34 (a). 
 
 The zinc element of 
 this battery is in the 
 form of a rod Z, and 
 weighs about 3 oz. The 
 other electrode is a car- 
 bon plate placed in a 
 porous cup and sur- 
 rounded with black 
 oxide of manganese, 
 mixed with crushed 
 coke or carbon. The 
 electrolyte used in 
 the battery is a satura- 
 ted solution of sal Fig. 34, 
 
 (a) 
 
0 
 
 ELECTRIC SIGNALING. 
 
 ammoniac. The E. M. F. of this cell is about 1.48 volts when the cell is in 
 good condition. In another form of the cell, known as the Gonda type, the 
 black oxide of manganese is pressed into the form of bricks and clamped 
 against each side of the carbon plate by means of rubber bands. This cell 
 will do good work if it is only used intermittently, i. e., on circuits where 
 the insulation is good and where there is no leakage causing the cell to give 
 
 out ^nrrpnt, mntinuouslv. If current is taken from it for any length of 
 
 The simple bell circuit is shown in Fig. 35, where p is the push button, b 
 the bell, and c, c the cells of the battery connected up in series When two 
 or more bells are to be rung from one push button, they may be joined up 
 
 in parallel across the battery wires, as in Fig. 37 at a and 6, or they may be 
 arranged in series, as in Fig. 36. The battery B is indicated in each diagram 
 by short parallel lines, this being the conventional method. In the parallel 
 arrangement of the bells, they are independent of each other, and the 
 failure of one to ring would not affect the others; but in the series grouping, 
 all but one bell must be changed to a single-stroke action, so that each 
 impulse of current will produce only one movement of the hammer. The 
 current is then interrupted by the vibrator in the remaining bell, the result 
 
 Fig. 36. 
 
 BELL WIRING. 
 
 e 
 
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 Fig. 37. 
 
 Fig. 38. 
 
batteries. 
 
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232 
 
 ELECTRIC SIGNALING. 
 
 
 si 
 
 Fig. 40. 
 
 being that each bell will ring with full power. The only change necessary 
 to produce this effect is to cut out the circuit-breaker on all but one bell by 
 
 connecting 
 the ends of 
 the magnet 
 wires directly 
 to the bell 
 terminals. 
 
 When it is 
 desired to 
 ring a bell 
 from one of 
 two places some distance apart, the 
 wires may be run as shown in Fig. 
 38. The pushes p, p' are located at 
 the required points, and the battery 
 and bell are put in series with each 
 other across the wires joining the 
 pushes. 
 
 A single wire may be used to ring 
 signal bells at each end of a line, the 
 connections being given in Fig. 39. 
 Two batteries are required, B and 
 B', and a key and bell at each sta- 
 tion. The keys k, k' are of the 
 double-contact type, making connections 
 normally between bell b or b' and line 
 wire L. When one key, as A:, is depressed, 
 a current from B flows along the wire 
 through the upper contact of k' to bell b' 
 and back through ground plates G', G. 
 
 When a bell is intended for use as an 
 alarm apparatus, a constant-ringing 
 attachment may be introduced, which 
 closes the bell circuit through an extra 
 wire as soon as the trip at door or window 
 is disturbed. In the diagram, Fig. 40, the 
 main circuit, when the push p is depressed, 
 is through the automatic drop d by way of 
 the terminals a, b to the bell and battery. 
 This current releases a pivoted arm which, 
 on falling, completes the circuit between 
 b and c, establishing a new path for the 
 current by way of e , independent of the 
 push p. 
 
 For operating electric bells, any good 
 type of open-circuit battery may be used. 
 The Leclanche cell is largely used for this 
 purpose, also several types of dry cells. 
 
 Annunciator System.— The wiring dia- 
 gram for a simple annunciator system is 
 shown in Fig. 42. The pushes 1, 2, 3, etc. 
 are located in various places, one side being con- 
 nected to the battery wire b, and the other to the 
 
 1 1 - • 7 ' IITlfll + Vl 77 071111111. 
 
 Fig. 41. 
 
 leadin 
 
 ciator 
 
 Fig. 42. 
 
 , wire l in communication with the annun- 
 irop corresponding to that place. A bat- 
 tery of two or three Leclanche cells is placed 
 at B in any convenient location. The size of 
 wire used throughout may be No. 18 annuncia- 
 tor wire. „ . 4 „ . 
 
 A return-call system is illustrated in Fig. 41, in 
 which there is one battery wire b, one return 
 wire r, and one leading wire l , lo, etc. for each 
 place The upper portion of the annunciator 
 board is provided with the usual drops, and below 
 these are the return-call pushes. These are 
 double-contact buttons, held normally against 
 the upper contact by a spring. When in this 
 
SELL WISING. 
 
 233 
 
 position, the closing of the circuit by the push button in any room, such as 
 No. 4, rings the office bell and releases No. 4 drop, the path of the current in 
 
 Fig. 43. 
 
 this case being from push 4 to ct-c-d-e-f-g-B-h-b 
 back to the push button. On the return sig- 
 nal being made by pressing the button at the 
 lower part of the annunciator board, the 
 office-bell circuit is broken at d, and a new 
 circuit formed through k as follows: From 
 the battery B to g-m-r-n-o-a-c-k-p to battery, 
 the room bell being in this circuit. A gen- 
 eral fire-alarm may be added to this system, 
 consisting of an automatic clockwork appa- 
 ratus for closing all the room-bell circuits at 
 once, or as many at a time as a battery can 
 ring. When this system is installed, the bat- 
 tery wire should be either No. 14 or No. 16. 
 Four or five Leclanch6 cells are usually re- 
 quired in this case. 
 
 It will be seen that the connections are 
 so arranged that the room bell will ring when 
 the push in that room is pressed. If this be 
 not desired, a double-contact push may be 
 substituted, so that the room-bell circuit is 
 broken at the same time that the circuit is 
 made through the annunciator. This double 
 push should be so connected that the circuit 
 is normally complete through the bell, the 
 leading wire being connected to the tongue, 
 and the battery wire being connected to the 
 second contact point, which is normally out 
 of circuit. 
 
 Telephones are also used for signaling and 
 communicating purposes. It has been found 
 that a first-class long-distance bridging tele- 
 phone is the best type to use. Bridging tele- 
 phones are so called because they are bridged 
 or connected in parallel across the line, and 
 are not connected in series. If one telephone 
 should get out of order, the others are not 
 likely to be disabled. Fig. 43 shows a com- 
 plete bell annunciator and telephone outfit, 
 
 
 
 
 MTft LEVEL 
 
 SBD. LEVEL 
 
234 
 
 ELECTRIC SIGNALING . 
 
 Hoi it mg Engine House. 
 
 p 
 
 Battery. 
 
 Surface. 
 
 as installed in one of the anthracite 
 coal mines of the D., L. & W. R. R. Co. 
 It will be noticed that bridging instru- 
 ments are used and that each bell in 
 the shaft is provided with a return- 
 call button. This bell wiring should 
 be put up in a substantial manner, 
 and it is best, if possible, to run all the 
 wires down the shaft in the shape of a 
 lead-covered cable. 
 
 Another shaft-signaling apparatus 
 is shown in Fig. 44, as used at the 
 West Vulcan mines, Mich. Fig. 45 
 
 
 
 
 
 ilFS 
 
 
 
 H 
 
 
 9 f -t‘ level 
 
 to*-* level 
 
 //'- Level 
 
 /£tE level 
 
 Fig. 44. 
 
 shows a form of waterproof push but- 
 ton used at the same mine. Fig. 46 
 shows the arrangement of flash signals 
 as used in Montana. This consists of 
 a switch cut into this main circuit at 
 
PROSPECTING. 
 
 235 
 
 L _ 
 
 firS 
 
 
 Fig. 47. 
 
 each level of the mines. By pulling out the handle bar 
 of the switch, the lights on this circuit can all be flashed 
 at once, and by a properly arranged code 9f flash sig- 
 nals, the system can be used for communicating between 
 the surface to any part 
 of the mine, and between 
 different portions of the 
 mine. 
 
 A system of signaling 
 by which signals can be 
 sent to the engine room 
 from any point along the 
 haulage road is shown in , 
 
 Fig. 47. The conductors a and 6, leading from the battery 
 run parallel to each other along the roadside, and about 
 6 in. apart. A short iron rod, placed across the wires a, o, 
 signals to the engineer, or by simply bringing the two wires 
 together a signal may be sent. , . . 
 
 When the engineer hauls from different roads, the signal- 
 ing system should be supplemented with indicators, so that 
 when the bell rings the indicator would show from which 
 point the signal came, and in case several signals were 
 given at the same time, the engineer should not heed any 
 until the indicator shows that a complete signal came 
 from one place. , . . . 
 
 A system of signaling for showing whether or not a 
 section of track is occupied by another motor is shown 
 
 / 27S 
 
 Lamp Circuit 
 
 Fig. 48. 
 
 Ma/ns 
 
 
 /oo’ 
 
 200 
 
 f 300 * 
 
 400 
 
 SOO' 
 
 in Fig. 48. White lights indicate a clear track and dark- 
 ness an occupied section. A single-center hinge, double- 
 handle switch at each signal station is used and a touch of Fig. 46. 
 
 the handle throws the switch in the desired direction. The 
 
 switches are placed in the roof, 4£ ft. above rails within easy reach of motor- 
 man. Fig. 48 shows the connections. Each switch is provided with a spring 
 (not shown in the figure) which, drawing across the center hinge, when the 
 handles are in their central position, insures a perfect contact when the 
 switch is inclined toward either the trolley or rail-terminal plug. 
 
 PROSPECTING. 
 
 The prospector should have a general knowledge of the mineral-bearing 
 strata, and should know from the nature of the ledges exposed whether to 
 expect to find mineral or not. He should also possess such a kno wledgeof the 
 use of tools as will enable him to construct simple structures, and a sufficient 
 experience in blacksmithing to enable him to sharpen picks and drills, or 
 to set a horseshoe, if necessary. . , . . , ... 
 
 Outfit Necessary.— The character of the prospecting being carried on will 
 have considerable effect on the outfit necessary, which should always be as 
 simple as possible. In general, when operating in a settled country, the 
 outfit is as follows: A compass and clinometer for determining the dip and 
 strike of the various measures encountered; a pick and shovel for excavating, 
 and, where rock is liable to be encountered, a set of drills, hammer, spoon 
 for cleaning the holes, tamping stick, powder and fuse, or dynamite fuse 
 and cap; a blowpipe outfit; a small magnifying glass; an aneroid barometer 
 for determining elevations, and a small hand pick; the latter should weigh 
 about If lb., and should have a pick on one end and a square-faced ham- 
 mer on the other, the handle being from 12 to 14 in. long. 
 
 If the region under consideration has been settled for some time, there 
 will probably be geological, county, railroad, or other maps available. 
 
236 
 
 PBOSPECTISG. 
 
 These mav not be accurate as to detail, but will be of great assistance in the 
 work on account of the fact that they give the course of the railroads, 
 streams, etc. 
 
 When operating in a mountainous region, away from a settled country, 
 and especially when searching for precious metals, the following materials, 
 in addition to that already mentioned, may be required: A donkey or pony 
 packed with a couple of heavy blankets, an A tent, cooking utensils, etc.;, a . 
 supply of flour, sugar, bacon, salt, baking powder, and coffee, sufficient for 
 at least a month. It is also well to take some fruit, but all fruit containing 
 stones or pits should be avoided, as they are only dead weight, and every 
 pound counts. For the same reason, canned goods should be avoided, on 
 account of the laTge amount of water they contain. A healthy man will 
 require about 3 lb. of solid food per day. Many prefer to vary the diet by 
 taking rice, corn meal, beans, etc., in place of a portion of the flour. 
 
 The additional tools necessary are an ax, a pan for washing gold ore, 
 making concentrating tests, etc., and, in some cases, an assay furnace 
 and outfit packed upon another animal. Where game is abundant, a shot- 
 gun or rifle will be found useful for supplying fresh meat. In regions 
 abounding in swamps it becomes necessary to operate from canoes, or to - 
 take men for porters or packers, who carry the outfit on their backs or 
 heads. These men will carry from 60 to 125 lb. 
 
 Plan of Operations.— When the presence of mineral is suspected in a tract 
 of land, a thorough exa min ation of the surface and a study of the exposed 
 rocks, in place, may result in its immediate discovery, or in positive proof 
 of its absence; or it mav result in still further increasing the doubt of, or the 
 belief that, it does exist. The first procedure in prospecting a tract of land 
 is to thoroughly traverse it. and note carefully any stains or traces of smut, 
 and all outcrops of every description; and. whenever possible, take the dip 
 and the course of the outcrop with a pocket compass. Any fossils should 
 also be carefully noted, to assist in determining the geological age of the 
 region. These outcrops are frequently more readily found along roads or 
 streams than anv other place on the tract. In traveling along the streams, 
 the prospector should pav particular attention to its bed and banks, to see 
 whether there are any small particles of mineral in the bed of the stream, 
 or any stains or smut exposed along the washed banks. If small pieces of 
 mineral are found in the stream, a search up it and its tributaries will show 
 where the outcrop from which the find came is located. When the ravines 
 and vallevs are so filled with wash that no exposures are visible, and nothing 
 is gained* bv a careful examination of them, the prospector must rely on 
 topographical features to guide him. . _ 
 
 Any gold present in the vein material usually remains in the float as free 
 or metallic gold, but other valuable metals are often leached out. The fact 
 that the float itself may be barren does not indicate that it may not have 
 come from a very rich deposit, and hence it will often pay to follow barren 
 float since the outcrop of the vein itself is often either entirely barren, low 
 grade, or of a different nature from the deeper deposits. In cases where 
 there are no outcrops or any other surface indications, it would become 
 necessary to sink shafts or test pits, or to proceed by drilling. 
 
 The absence of any indication of mineral in the soil may not prove that 
 there is not an outcrop near at hand, for the soil is frequently brought from 
 a distance, and bears no relation to the material underlying it. In like 
 manner, glacial soil often contains debris transported from deposits many 
 miles away; but such occurrences can usually be distinguished by the gen- 
 eral character of the associated wash material. 
 
 Frequently, the weathered outcrop of a deposit has been overturned or 
 dragged back upon itself, so as to indicate the presence of a very thick 
 deposit For this reason, anv openings made to determine the character of 
 the material should be continued until the. coal or other mineral is of a firm 
 character, and both floor and roof are well exposed. Sometimes, in the case 
 of steeplv pitching coal beds, the surface may be overturned for a consider- 
 able depth, so that it is difficult to tell which is the roof and which is the 
 floor Usually, if Stigmariee are found in the rocks of one wall, it is supposed 
 that this wall ’is the floor of the seam, while if Sigillariae. fern leaves, etc. are 
 found in the wall rock, it is probably the roof of the deposit. These indi- 
 cations are not positive proof, for both of these fossils may occur in either 
 the top or bottom wall of a coal deposit, though they are usually found in 
 the positions noted. Coal. clav. gypsum, salt. etc. usually occur m unaltered 
 deposits, i. e., in rocks that have not undergone metamorphism. 
 
PROSPECTING. 
 
 237 
 
 The accompanying table gives the names of the various geological periods, 
 both as they occur in America and their foreign equivalents, together with 
 the name of the principal form of life during each period. The various 
 terms employed in geology are defined in the glossary. 
 
238 
 
 PROSPECTING. 
 
 Mptals and metallic ores usually occur in rocks that have undergone 
 
 MSSSBIE 
 
 mmmm 
 
 siT=^g^@gi 
 
 SHIS 
 
 &S5MS «FtKn d ih^K^ 
 
 toward the top of the h f^l^¥ eTT{ice or indication of the outcrop of a 
 to sink a shallow shaft above the terrace. importance have as yet 
 
 1> eSrB^^’& , a5KSai<^^i»« gs^s&'asr'pSEa: 
 
 a [>y “ e P d ^ r ik o d fThe coal hat up to been found 
 
 although the bulk ot the nest coai ^xma, up metamorphic regions con- 
 
 s j| t£S > » jp-s-i&'K 
 
 gassa A’g. vvp yp. ^^sa .ntq 
 SMSS3SS&' "^cssrsisiSti* w. -»-» ■» 
 
 frequently associated with beds or iron 01 ^ lly foU nd in mountainous 
 
ORE DEPOSITS . 
 
 239 
 
 economic products as coal, petroleum, building stone, clays, etc., but not often 
 of the precious metals. The reason for this appears to be that the latter are 
 commonly found to be associated with evidences of more or less heat. In 
 the Rocky Mountains they are rarely found except where volcanic eruptions 
 have at some time been active, or where the strata have been changed or 
 metamorphosed and crystallized by heat. 
 
 As metallic ore bodies occupy fissures and other openings in the earth’s 
 crust, we must go to regions where the greatest disturbances and uplifts 
 have occurred, accompanied by the greatest rending and contortions of the 
 rocks, and eruptions of volcanic matter. 
 
 As a broad assertion, we may say that the greater part of any mountain 
 region is a prospecting field, with the exception of those areas we have 
 restricted as unpromising. But over this wide area of more or less metamor- 
 phosed and crystalline rocks, there are regions and localities where the 
 precious metals have already been found, and others where on geological 
 grounds they are most likely yet to be found, and those are generally where 
 eruptive forces have been especially active, where once molten eruptive 
 rocks are most abundant, and the disturbance and crystallizing of the strata 
 most pronounced. _ 
 
 Position of Veins and Ore Deposits.— Ores, as a rule, are to be looked for at 
 the junction of any two dissimilar rocks, rather than in the mass of those 
 rocks. However, there are many exceptions to this, where the mass of a 
 decomposed dike or sheet of porphyry has been impregnated by free gold or 
 gold-bearing pyrites, and the whole rock is practically a gold vein. In this 
 mass, the richest gold is often found in a network of little quartz veins run- 
 ning through the porphyry mass. Some of our richest gold mines are found 
 in “ rotten,” decomposed, oxidized dikes and sheets of porphyry; but this is 
 
 rarely the case with lead-silver ores, 
 which frequent rather the lines of con- 
 tact in limestones or in fissure veins in 
 granite. The Cambrian quartzites a few 
 years ago were rather avoided by the 
 prospectors, their extreme hardness pre- 
 senting great difficulties in mining, and 
 from the fact that they were generally 
 supposed to be barren. The late disco v- 
 b eries of very rich gold deposits in them, 
 and of similar deposits in quartzites of 
 a later age, have drawn more attention 
 to them. The gold has been found in 
 a free state associated with oxide of iron 
 in cavernous deposits, and in close prox- 
 imity to eruptive rocks. In the granitic 
 rocks, both gold and silver occur in fissure veins associated with pyrites, 
 galena, etc. These fissures, occupied by mineralized quartz veins, may occur 
 in the granite or gneiss alone, or be at the contact of these rocks with a 
 porphyry dike. 
 
 Veins in overflows of volcanic lava generally fill a fissure having a more 
 or less steep inclination, penetrating the lava sheets, caused probably by 
 shrinkage of the molten lava on cooling. These fissures, in some cases, are 
 likely to be limited in depth to the thickness of the lava sheet. Where, in a 
 few rare cases, the fissure has been traced down to the underlying granite or 
 some other rock, it has come abruptly to an end. 
 
 Underground Prospecting. — Frequently a seam or deposit becomes faulted 
 or pinched out underground, and it is necessary to continue the search by 
 means of underground prospecting. Underground prospecting is, to a large 
 extent, similar to surface prospecting, the underground exposures being 
 simply additional faces for the guidance of the engineer. In the case of 
 coal beds or similar seams, if a fault or dislocation is encountered, the man- 
 ner of carrying on the search will depend on the character of the fault. 
 Where sand faults or washouts are encountered, the drift or entry should 
 be driven forwards at the angle of the seam until the continuation of the 
 formation is encountered, when a little examination of the rocks will indi- 
 cate whether they are the underlying or overlying measures. In the case 
 of dislocations or throws, the continuation of the vein may be looked for by 
 Schmidt’s law of faults, which is as follows: Always follow the direction of the 
 greatest angle. It has been discovered by observation that, in the majority 
 of cases, the hanging-wall portion of the fault has moved down, and on this 
 
240 
 
 PROSPECTING . 
 
 account such faults are commonly called normal Jaults. For instance, ii the 
 bed a b, Fig. 1, were being worked from a toward the fault, upon encoun- 
 tering the fault, work would be continued down on the farther side of the 
 fault toward d, until the continuation of the bed toward b was encountered. 
 In like manner, had the work been proceeding from 6, the exploration 
 would have been carried up in the direction of the greatest angle, and the 
 continuation toward a thus discovered. A reverse fault is one in which 
 the movement has been in the opposite direction to a normal fault. Espe- 
 cially in the case of precious metal mines, where the material occurs as 
 perpendicular or steeply pitching veins, faults are liable to displace the 
 deposit, both horizontally and vertically, in which case it may be difficult 
 to determine the direction of the continuation of the ore body; but fre- 
 quently pieces of ore are dragged into the fault, and these serve as a guide 
 to the miner, and indicate the proper direction for exploration. Where a 
 bed or seam is faulted, its continuation can frequently be found by breaking 
 through into the measures beyond, when an examination of the formation 
 will indicate whether the rocks are those that usually occur above or below 
 
 the desired seam. „ , , , .. . 
 
 Prospecting for Placer Deposits.— Placers are fragmental deposits from water 
 in which the heavier minerals have been concentrated in certain portions, 
 usually next the underlying, or bed, rock. The materials that are recovered 
 from placer deposits are metallic gold, tinstone, monazite, sand, or precious 
 stones. Placer deposits are modern or ancient. Modern placers are 
 deposits of washed material, or debris, in the beds or along the banks of 
 streams that are either now in existence or existed in comparatively recent 
 times. Placer deposits may also occur in deposits along the seashore. 
 Ancient placers are fragmental accumulations, similar to the modern 
 placers, which have been buried under accumulations of strata or flows of 
 lava, and they may or may not have become consolidated into rock. 
 
 At times, placers are very compact, owing to the presence of large quanti- 
 ties of oxide of iron or calcium carbonate, or similar cementing material. 
 Often, in the case of modern placers, the streams, or other sources of water 
 that deposited the material, have changed their course so that the placer 
 deposit is now high up in the benches bordering the streams, or, possibly 
 even on the top of the present hills. Such deposits are commonly called 
 bench deposits , while those along the sides of the streams below the high- 
 water mark are called bar deposits , diggings , or placers. ..... 
 
 Frequently, a large portion of the gold or other valuable material is 
 found in pockets or irregularities in the bed rock, but the pot holes under 
 waterfalls are frequentlv barren of gold, on account of the fact that the 
 current there was sufficiently swift to wash everything out, either heavy or 
 light. When the soil is saturated with water, the mass may partake of the 
 nature of a semifluid through which the heavy particles of gold settle until 
 they accumulate on the bed rock. . ^ . . 
 
 When prospecting for placers, the miner examines the country for any 
 indications of present or ancient watercourses in which the deposits ot 
 placer material have been formed. He pans the dirt from any deposits dis- 
 covered, to see if it contains colors (small particles of metallic gold). It 
 colors are found, more extensive operations are in order, and hence he 
 sinks to bed rock and examines the material thoroughly, to see if it contains 
 a paying quantity of the valuable mineral. . e ^ .. . . 
 
 The form of placer deposit in dry or and regions differs from that in 
 regions where the rivers have a continuous flow, on account of the tact that 
 the deposits are largely the result of sudden rushes of water partaking of the 
 nature of cloudbursts*, hence the rich portions in the placer material are 
 very irregular, and are rarelv situated on bed rock, but are usually found on 
 any strata that formed the bottom of the ravine during the sudden rush of 
 water. During the rainy season in arid regions, the surface soil is some- 
 times softened for a few inches, so that it becomes practically a mud, and 
 particles of gold that it may contain tend to settle to the bottom of the sott 
 portion, thus rendering the surface barren. This barren surface may be 
 subsequently washed away by the rain,' or blown away as dust during the 
 dry season. The repeating of this process year after year results in. the 
 removal of considerable of the original surface and the formation of a rich 
 stratum iust below the grass roots. Prospectors in and regions, who have 
 been used to operating in an ordinarily well-watered country, are frequently 
 deceived by finding this rich ground so high up in the deposit, not knowing 
 that it is no indication as to the value of the material at a greater depth. 
 
 
 
OEMS AND PRECIOUS STONES. 
 
 241 
 
 In many cases, in the arid regions the portion of the deposit upon bed rock 
 is entirely barren. In like manner, frozen ground may play an important 
 part in the formation and distribution of the values in placer deposits. 
 
 Gems and precious stones are prospected for in a manner similar to that 
 employed in searching for placer material, and are usually found in alluvial 
 deposits, from which they are obtained by washing. In a few cases gems 
 are found in the rocks themselves; as, for instance, diamonds in the hard 
 matrix that occurs as pipes or chimneys in metamorphic rocks, and which, 
 upon exposure to the atmosphere, becomes decomposed, so that the stones 
 are easily removed. Some of the corundum minerals are found in lime- 
 stone and metamorphic or crystalline rocks. Turquoise usually occurs in 
 veins, the outcrop of which is stained with carbonate of copper. In most 
 cases, it does not pay to extract gems from rock formations when the rock is 
 extremely hard, owing to the fact that the gems are liable to become broken 
 in separating them from the rock matrix. 
 
 For gem prospecting, the following outfit has been recommended: A 
 shovel and pick; two sieves, one of 2 or 3 meshes to the linear inch, and the 
 other of 20 or more meshes to the inch (the coarse sieve should be arranged 
 to fasten on top of the finer one for use together); a tub in which the sieves 
 can be submerged in water; an oilcloth on which to sort the gravel; several 
 stones and crude gems as a scale of hardness; a small pocket magnifying 
 glass, and a dichroscope. In some cases, a portion of the outfit is dispensed 
 with. The use of the outfit may be explained as follows: The tub is partially 
 filled with water, the two sieves fastened together, and a shovelful of 
 material placed in the upper one, when they are submerged in water, the 
 large stones cleaned and examined, and all of the fine material worked 
 through the upper sieve, which is then removed, the material on it examined 
 and disposed of. The material in the fine sieve is then washed until 
 free from clay, when a little jigging motion in the water will carry the 
 lighter material to the top. The sieve is then quickly inverted and the 
 material dumped out on the oilcloth, thus bringing the heavier stones to 
 the top. The various pieces should now be examined with the magnifying 
 glass, scale of hardness, etc., and the identity of any doubtful colored gems 
 settled, by means of the dichroscope. Few precious stones are of sufficient 
 specific gravity to be concentrated in distinct beds, like gold or tinstone, but 
 they are usually fairly well concentrated and freed from much of the lighter 
 worthless material. 
 
 Value of Free Gold per Ton of Ore.— The accompanying table was prepared 
 by Mellville Atwood, F. G. S., and its use may be explained as follows' 
 If a 4-lb. sample of quartz be crushed, the gold separated by panning and 
 
 Value of Free Gold per Ton of Ore. 
 ( Risdon Iron Works.) 
 
 Weight, 
 Washed Gold. 
 4-Lb. Sample. 
 Grains. 
 
 Fineness, 
 
 780. 
 
 Value per Oz., 
 $16.12. 
 
 Fineness, 
 
 830. 
 
 Value per Oz., 
 $17.15. 
 
 Fineness, 
 
 875. 
 
 Value per Oz., 
 $18.08. 
 
 Fineness, 
 
 920. 
 
 Value perOz., 
 $19.01. 
 
 5.0 
 
 $83.97 
 
 $89.36 
 
 $94.20 
 
 $99.05 
 
 4.0 
 
 67.18 
 
 71.49 
 
 75.36 
 
 79.24 
 
 3.0 
 
 50.38 
 
 53.61 
 
 56.52 
 
 59.43 
 
 2.0 
 
 33.59 
 
 35.74 
 
 37.68 
 
 39.62 
 
 1.0 
 
 16.79 
 
 17.87 
 
 18.84 
 
 19.81 
 
 .9 
 
 15.11 
 
 16.08 
 
 16.95 
 
 17.82 
 
 .8 
 
 13.43 
 
 14.29 
 
 15.07 
 
 15.84 
 
 .7 
 
 11.75 
 
 12.51 
 
 13.19 
 
 13.86 
 
 .6 
 
 10.07 
 
 10.73 
 
 11.30 
 
 11.88 
 
 .5 
 
 8.40 
 
 8.93 
 
 9.42 
 
 9.90 
 
 .4 
 
 6.71 
 
 7.14 
 
 7.53 
 
 7.92 
 
 .3 
 
 5.03 
 
 5.36 
 
 5.65 
 
 5.94 
 
 .2 
 
 3.36 
 
 3.57 
 
 3.76 
 
 3.96 
 
 .1 
 
 1.68 
 
 1.78 
 
 1.88 
 
 1.98 
 
 amalgamation, the quicksilver volatilized by blowpiping or otherwise, and 
 the resulting button weighed, the value of the ore per ton of 2,000 lb. will 
 
242 
 
 PROSPECTING. 
 
 be found opposite the weight of the button. The values are given for fine- 
 ness of gold varying from 780 to 920. . 
 
 To determine the value of gravel, a 6-lb. sample will give the same 
 results as that obtained from a 4-lb. sample of quartz, on account of the 
 fact that 18 cu. ft. of gravel measured in a bank weigh 1 ton, or 2,000 lb.; 
 hence, a cubic vard of gravel measured in a bank weighs 3,000 lb., and 
 for this reason a sample one and one-half times as large as that required 
 for quartz must, be taken. In case the gravel is of low grade, a sample ten 
 times as large, or 60 lb., mav be taken, in which case the value opposite 
 the weight of the button will have to be divided by 10. 
 
 As an example, in the use of the table we may suppose that a button 
 from 4 lb. of ore or 6 lb. of gravel weighs 3.8 gr., and that the fineness of 
 the gold is 830. Opposite 3 in the table we will find $53.61 as the value of 
 the button in dollars containing 3 gr. of gold, and opposite .8 we will find 
 $14.29. The sum of these is $67.90, the value of the ore per ton, or the gravel 
 per cubic yard. 
 
 EXPLORATION BY DRILLING OR BORE HOLES. 
 
 Earth Augers.— When testing soil or searching for placer gold, sand, soft 
 iron, or manganese ores, and similar materials that usually occur compara- 
 tively near the surface, hand augers may be employed to great advantage. 
 A good form of hand auger consists of a piece of fiat steel or iron, with a 
 steel tip, twisted into a spiral about 1 ft. long, and having four turns. The 
 point is split and the tips sharpened and turned in opposite directions and 
 dressed to a standard width, usually 2 in. The auger is attached to a short 
 piece of 1" pipe, and is operated by joints of 1" pipe, which are coupled 
 together witlr common pipe couplings. The auger is turned by means of a 
 double-ended handle having an eye in the center through which the rod 
 
 D&SS6S 
 
 The handle is secured by means of a setscrew. In addition to the auger, 
 it is well to have a straight-edged chopping bit for use in comparatively hard 
 seams. This may be made from a piece of If" octagon steel, with a 2 cut- 
 ting edge. The upper end of the steel is welded on to a piece of pipe similar 
 to that carrying the auger. When the chopping bit is employed, it is 
 necessary to have a heavy sinking bar, which may be made from a piece ot 
 solid U" iron bar, fitted with ordinary 1" pipe threads ()n the ends. Pros- 
 pecting can be carried on to a depth of from 50 to 60 ft. with this outfit. The 
 number of men necessary to operate the rods varies from 2 to 4, depending 
 on the depth of the hole being drilled. When more than 30 ft. of rods are in 
 use, it is usually necessary to have a scaffold on which some of the men can 
 stand to assist in withdrawing the rods. When withdrawing the rods, 
 to remove the dirt, they are not uncoupled unless over 40 ft. of rods are in 
 use at one time, and sometimes as many as 50 or 60 ft. are drawn without 
 
 Un Percussion or churn drills are frequently employed in drilling for oil, 
 water, or gas, and were formerly much used in searching for coal and ores, 
 but, owing to the fact that they all reduce the material passed through to 
 small pieces or mud, and so do not produce a fair sample, and to the fact 
 that they can only drill perpendicular holes, they are at present little used 
 in prospecting for ‘either ore or coal. . , , 
 
 The cost and rate of drilling by means of a percussive or churn drill 
 
 varies greatly, being affected much more by 
 the character of the strata penetrated than is 
 the case with the diamond drill. In the case 
 of highly inclined beds of varying hardness, 
 the holes frequently run out of line and be- 
 come so crooked that the tools wedge, and 
 drilling has to be suspended. For drilling 
 through moderately hard formations, usually 
 encountered in searching for gas or water, 
 such as sandstones, limestones, slates, etc., the 
 accompanying costs, from the American Well 
 Works, Aurora, 111., may be taken per foot 
 for wells from 500 to 3,000 ft. deep for the 
 central or eastern portion of the United 
 This cost includes the placing of the casing, but 
 
 Cost of Well-Drilling. 
 
 Size of Well. 
 Inches. 
 
 Cost per Foot. 
 
 6 
 
 $1.50 
 
 8 
 
 2.25 
 
 10 
 
 3.00 
 
 12 
 
 5.00 
 
 15 
 
 8.00 
 
 States at present (1900). 
 not the casing itself. 
 
drilling or bore HOLES. 
 
 243 
 
 When drilling wells for oil or gas to a depth of approximately 1^0® ft^, 
 usinff the ordinary American rig with a cable, the cost is sometimes reduced 
 per foot i5r 6" or Sewells. Tto “ 
 
 usually njpch 
 
 ifjA mav be drilled in a single day, and at other times, when very hard 
 
 to make more than from 1 to 2 ft. per 
 
 ^ The diamond drill is the only form that has been universally successful in 
 drilling in any direction through hard, soft, or variable material. Even 
 rhp of the diamond drill, many difficulties present themselves, and 
 
 KISS Mate is smt JSBftfswa a 
 
 practice, Dy u. m ^ « > . *? ’ t econ0 mv to employ a machine of large 
 
 When a large machine is operating small rods on light work, the ariner 
 
 in which the feed is independent of the material being cut, as m the case 
 
 th TheflrIt e K’^advantageous when drilling through variable. measures 
 in sea?ch of fofrly firm material, which does not occur m very thm beds or 
 seams On account of the fact that this class of feed insure© the maximum 
 
 amount of advance of which the bit is capable in 1 rounl upland 
 danger is that the core from any thin soft seam may be grouna up ana 
 washed away without any indication of its presence having been given. 
 
 The second class or positive gear-feed, if properly operated, requires 
 somewhat greater skill but if used in connection with a thrust register, it 
 ffi^s^ehaffi^in^rmation as to the material being cAit, andisespeciaHy 
 
 useful when prospecting for soft deposits of ve y^ 1 ^J® T ^ t ^| ,1 rf ze of t he 
 Size of Tools —The size of tools and rods, and consequently the size oi ine 
 core extracted depends on the depth of the hole and the character of the 
 materialTeing prospected. When operating in firm measures such as 
 “nthracite coal, hardrock, etc., it is best to employ a rattw; small bU even 
 when drilling ud to 700 ft., or more, m depth. For such work, a core oi 
 U in. to 1* in. is usually extracted. The rate of driUing with a small outfit 
 is verv much greater than with a large one, owing to i the fact _ that there 
 is a small cutting surface exposed, and the rate of rotation of the rods can be 
 much treater When prospecting for soft materials, such as bitummo 
 coal valuable soft ores, or for disseminated ores, such as lead^copper, gold, 
 
 silver etc it is best to employ a larger outfit and extract a core 2 or 3 in. in 
 diameter, and sometimes even larger, even though a comparatively small 
 
 ma D rift of 1S d ia m ond° d°r H?h d es^o/the divergence from 
 
 becomes a serious matter. This trouble may b ® wito 
 
 tools about the bit as nearly up to gauge as possible. . Core oarnfia wim 
 spiral water grooves about them, answer this purpose very well if they are 
 renewed before excessive wear has taken place. of tw0 
 
 Surveying of diamond drill-holes may be carried on f b ,y o ei ^ t rict Where 
 methods, depending on the magnetic conditions of tte E F 
 
 there is no magnetic disturbance, the system developed by Mr. -b. . 
 
 MacGeorge of Australia, may be employed. This consists m introducing 
 into the hole, at various points, small tubes containing ^lted gelat , 
 which are suspended magnetic needles and small P^mmets. After t 
 gelatine has hardened the tubes are removed, and J the angle s ^^between me 
 
 center line of the tube, the plummet, and the ne e d ^ noted L’/ This methol 
 the data from which the course of the hole can be plotted, lhismetnoa 
 gives both the vertical and the horizontal dn.t. 
 
244 
 
 PROSPECTING. 
 
 Where there is magnetic disturbances the needle cannot be used, but a 
 system brought out by Mr. G. Nolten, of Germany, has been quite exten- 
 sively employed. In this case, tubes partly filled with hydrofluoric acid are 
 introduced into the hole, at various points, and the acid allowed to etch a 
 ring on the inside of the tube. After the acid has spent itself the tubes are 
 withdrawn, and by bringing the liquid into such a position that it corre- 
 sponds with the ring etched on the inside of the tube, the angle of the hole 
 at the point examined can be determined. This method gives a record of 
 the vertical drift of the hole only. 
 
 The value of the record furnished by the diamond drill depends largely on 
 the character of the material sought. The core extracted is always of very 
 small volume when compared with the large mass of the formation pros- 
 pected, and hence will give a fair average sample only in the case of very 
 uniform deposits. The value of the diamond drill for prospecting may be 
 stated as follows: More dependence can be placed on the record furnished by 
 the diamond drill when prospecting for materials that occur in large bodies of 
 uniform composition than when prospecting for materials that occur in small 
 bunches or irregular seams. To the first class belong coal, iron ore, low-grade 
 finely disseminated gold and silver ores, many deposits of copper, lead, zinc, 
 etc., as well as salt, gypsum, building stone, etc. To the latter class belong 
 small but rich bunches of gold, silver mineral, or rich streaks of gold telluride. 
 
 The arrangement of holes has considerable effect upon the results fur- 
 nished. If the material sought lies in beds or seams (as coal), the dip of 
 which is fairly well know T n, it is best to drill a series of holes at right 
 angles to the formation. If the material sought occurs in irregular bunches, 
 pockets, or lenses, it will be necessary to drill holes at two or more angles, so 
 as to divide the ground into a series of rectangles, thus rendering it prac- 
 tically impossible for any vein or seam of commercial importance to exist 
 without being discovered. Where the surface of the ground is covered with 
 drift and wash material, it may be best to sink a shaft or drill pit to bed 
 rock, and locate the machine on bed rock. After this, several series of fan 
 holes may be drilled at various angles from the bottom of the pit. Owing 
 to the upward drift of diamond-drill holes, the results furnished from a set 
 of fan holes drilled from a single position would make a flat bed appear as 
 an inverted bowl, or the top of a hill. On this account, it is best to drill 
 sets of fan holes from two or more locations, so that they will correct one 
 another. If fan holes from different positions intersect the same bed, a care- 
 ful examination of them will usually furnish a check on the vertical drift 
 of the holes. 
 
 The cost and speed of drilling depend greatly on the formation being 
 penetrated. As a rule, it is more expensive to sink the stand pipe than to do 
 the subsequent drilling. Stand pipes may cost $5 or more per foot to sink, 
 while the cost of drilling in firm rock varies from $0.50 to $2 per foot; in the 
 case of difficult drilling, the cost may run over $4 per foot. Where a large 
 amount of drilling has to be done, a fair average estimate for shallow holes 
 up to 700 ft. deep would be $2 per foot, under such conditions as exist in 
 most mineral districts of the United States. The cost of labor, fuel, etc., 
 enter into the problem, and frequently affect it to a considerable extent. 
 
 The rate of drilling varies considerably, but in firm rock an average of 
 1 ft. per hour, including all delays for changing rods, etc., would be a fair 
 average up to 700 ft. Greater speed than this could be made in soft shales or 
 sandstones, and somew T hat less in hard rock. The hardness of the rock 
 affects the rate of drilling much less than does its character. A conglomer- 
 ate rock containing loose pebbles that come out during the drilling, or a 
 crystalline rock containing angular pieces that come out during drilling, 
 will cause far greater trouble than the hardest material ever encountered in 
 diamond drilling. The following tables will give some idea as to the cost 
 of diamond drilling under various conditions. 
 
 The cost of drilling 2,084 ft. of hole in prospecting the ground through 
 which the Croton aqueduct tunnel was to pass is given as follows: 
 
 814 ft. of soft rock (decomposed gneiss), in w hich an average of 23.1 ft. per 
 dav w 7 as drilled, at a cost of $1.15 per ft. 
 
 347 ft. of hard rock (gneiss), in which an average of 11.1 ft. per day was 
 drilled, at a cost of $3.97 per ft. 
 
 923 ft. of clay, gravel, and boulders, in which from 6£ to 9 ft. per day were 
 drilled, at a cost of $4.07 per ft. 
 
 The average progress per day in drilling the entire 2,084 ft. was 10.2 ft 
 per day. 
 
245 
 
 1 
 
 DRILLING OR BORE HOLES. ^*5 
 
 In the Minnesota Iron Co.’s mines, at Soudan, Minn., the diamond drill is 
 used for drilling holes from 10 to 40 it. in depth in the back of the stopes, 
 practically all the work being done in iron ore. The average cost per foot 
 of drilling 13,512 ft. of hole was $0.7703, which was divided as follows. 
 
 Carbons 
 
 Supplies, oil, etc JJJc 
 
 Fuel 
 
 ifc::::::::::::::::::::; o:^ 
 
 Total $0.7703 
 
 The following tables give the cost of boring at two Ishpeming, Mich., 
 mines: 
 
 TABLE I. 
 
 f 400| days setter at $3.00 $1,200.75 1 
 
 T , J 372 days runner at 2.25 837.00 
 
 Labor -j 2 3 oa days runner at 2.00 460.50 
 
 [ 4i days laborer at 1.75 7.85 j 
 
 Carbon 68f carats at $15.144 .• 
 
 Bits, lifters, shells, barrels, and repairs 
 
 Oil, candles, waste, and supplies 
 
 Estimated cost compressed air 
 
 Total 
 
 Cost. 
 
 L $2,506.10 
 
 . 1,035.47 
 .. 433.81 
 
 128.09 
 374.60 
 
 Cost 
 per Ft. 
 
 $0,669 
 
 0.276 
 
 0.115 
 
 0.035 
 
 0.100 
 
 .. $4,478.07 
 
 $1,195 
 
 
 
 28 
 
 
 
 193 ft. 
 
 Ullllt/U 111 lltJlIlall 
 
 
 646 ft. 
 
 
 
 986 ft. 
 
 urnieu in ihiacu uic 
 
 Drilled in dioritic schist 
 
 Total drilling 
 
 
 1,921 ft. 
 3,746 ft. 
 
 Number of 10-hour shifts drill was running, including 
 
 moving and setting up - V?o «• 
 
 Amount drilling per 10-hour shift II * 
 
 TABLE II. 
 
 Underground drilling W5 £* 
 
 Surface drilling 5* 
 
 Stand pipe sunk — 4/U 
 
 Total distance run 7,959 ft. 
 
 Actual drilling time underground 672 shifts 
 
 Actual drilling time on surface "*2 shins 
 
 Time of foreman, setter, moving, and stand-piping 1,314 sums 
 
 Total time worked shifts 
 
 Average progress per man per shift 
 
 Average progress per drill per shift actually run- 
 ning - 
 
 Weight of carbon consumed - 
 
 Distance drilled per carat of carbon consumed 
 
 3.70 ft. 
 
 8.95 ft. 
 111.00 carats 
 67.38 ft. 
 
 Cost of carbon 
 
 Cost of supplies and oils.. 
 
 Cost of fuel 
 
 Cost of shop material, etc, 
 
 Pay roll 
 
 Total cost 
 
 Amount. 
 
 $1,887.00 
 
 134.13 
 
 360.73 
 
 663.36 
 
 4,000.03 
 
 $7,045.25 
 
 Per Ft. 
 $0,237 
 0.017 
 0.045 
 0.083 
 0.502 
 
 $0,884 
 
246 
 
 PROSPECTING. 
 
 Records of Cost per Foot in Diamond Drilling. 
 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 F 
 
 G 
 
 11 
 
 / 
 
 J 
 
 K 
 
 L 
 
 M 
 
 N 
 
 O 
 
 Labor 
 
 .707 
 
 1.040 
 
 2.483 
 
 1.150 
 
 .581 
 
 1.615 
 
 1.030 
 
 1.720 
 
 1.189 
 
 1.284 
 
 .721 
 
 1.200 
 
 .939 
 
 .812 
 
 .984 
 
 Fuel 
 
 .094 
 
 .270 
 
 .256 
 
 .019 
 
 .000 
 
 .216 
 
 .090 
 
 .214 
 
 .157 
 
 .339 
 
 .419 
 
 .329 
 
 .126 
 
 .182 
 
 .251 
 
 Camp account 
 Repairs .... 
 
 .373 
 
 .559 
 
 .789 
 
 .538 
 
 .295 
 
 .621 
 
 .384 
 
 .549 
 
 .516 
 
 .495 
 
 .519 
 
 .595 
 
 .644 
 
 .722 
 
 .636 
 
 .139 
 
 .110 
 
 .294 
 
 .171 
 
 .135 
 
 .144 
 
 .103 
 
 . .185 
 
 .154 
 
 .165 
 
 .040 
 
 .087 
 
 .138 
 
 .126 
 
 .116 
 
 Supplies . . . 
 
 .034 
 
 .065 
 
 .039 
 
 .074 
 
 .023 
 
 .032 
 
 .011 
 
 ; .039 
 
 .048 
 
 .097 
 
 .020 
 
 .092 
 
 .076 
 
 .097 
 
 .088 
 
 Carbon .... 
 
 .263 
 
 .658 
 
 .859 
 
 .860 
 
 .843 
 
 1.587 
 
 .934 
 
 ; .684 
 
 .684 
 
 .733 
 
 .227 
 
 .209 
 
 .553 
 
 .239 
 
 .330 
 
 Supt 
 
 .239 
 
 .322 
 
 .628 
 
 .040 
 
 .063 
 
 .192 
 
 .140 
 
 .305 
 
 .259 
 
 .172 
 
 .347 
 
 .220 
 
 .106 
 
 .196 
 
 .199 
 
 Total . . . 
 
 1.849 
 
 3.024 
 
 5.348 
 
 2.852 
 
 1.940 
 
 4.407 
 
 2.692 
 
 1 
 
 1 
 
 3.696 
 
 3.007 
 
 3.285 
 
 1 
 
 2.293 
 
 2.732 
 
 2.582 
 
 2.374 
 
 2.604 
 
 A 5 holes, 1,066 ft. I 
 
 Sandstone and marble. 
 
 B 1 hole, 1,293 ft. 
 
 Black slate and jasper. J 
 
 C 3 holes, 478 ft. 
 
 Jasper, very hard. K 
 
 D 5 holes, 780 ft. 
 
 Jasper, hard. L 
 
 E 1 hole, 216 ft. 
 
 Iron slates. M 
 
 F 1 hole, 174 ft. 
 
 Jasper and slate. N 
 
 G 2 holes, 267 ft. 
 
 Jasper and slate. 0 
 
 H 3 holes, 410 ft. 
 
 Jasper. 
 
 Average cost of total work of 
 drilling 21 hoies. Total of 
 4,684 ft. 
 
 2 holes, 634 ft. 
 
 Iron slates. 
 
 2 holes, 360 ft. 
 
 Schist and jasper. 
 
 6 holes, 1,350 ft. 
 
 Iron slates. 
 
 2 holes, 611 ft. 
 
 Schist, jasper, and quartzite. 
 
 6 holes, 2,091 ft. 
 
 Quartzite. 
 
 Average cost of drilling 18 holes, 
 5,046 ft. 
 
 The following figures, taken from a letter written by T. F. Richardson, 
 Departmental Engineer of Dam and Aqueduct Department, Metropolitan 
 Water Board of Boston, and published by the U. S. Geological Survey, are of 
 interest, as they show the rate and cost of diamond drilling under certain 
 conditions. The costs do not take into account depreciation of machinery 
 nor losses of time in moving machines, etc. The machines employed in 
 this work were a Badger drill, manufactured by the M. C. Bullock Manu- 
 facturing Co., of Chicago, 111., and an S-510 drill, manufactured by the 
 Sullivan Machinery Co., Claremont, N. H. 
 
 The total amount drilled was 2,814 ft., the deepest hole being 286 ft. deep, 
 and the average depths of holes about 60 ft. The amount accomplished per 
 day was from 0 to 32 ft., the average amount being probably about 10 or 12 
 ft. per day. The cost of drilling varied very largely, both with the hard- 
 ness of the rock and the condition of the rock as to being seamy. 
 
 The following was the cost of drilling 321.2 ft. of rather hard, tough 
 diorite rock: 
 
 Labor $341.25 
 
 Diamonds 74.30 
 
 Coal 17.50 
 
 Total $433.05 
 
 Cost per foot 1.34 
 
 (86.6 ft. of this was drilled with a If" bit, and 237.6 ft. was drilled with a 
 U" bit.) 
 
 Drilling 150.7 ft. of very hard syenite rock: 
 
 Labor $158.00 
 
 Diamonds 298.69 
 
 Coal 10.50 
 
 Total - $467.19 
 
 Cost per foot ' 3.10 
 
 (Size of drill, 1$ in.) 
 
1 
 
 drilling or bore holes. 
 
 247 
 
 The following was the cost of drilling 286.1 ft. of soft schist rockt^ 
 
 Labor.. 87/75 
 
 Diamonds 11.50 
 
 Coa !n';zv *m25 
 
 Cost per foot 
 
 (Size of drill, If in.) 
 
 The following figures will be of considerable interest, owing ; to the fact 
 that the work is practically all of the nature of sinking stand pip.es the 
 
 Cost of operation per month of bed-rock exploration: 
 
 TThtpiti Tl HpXOU.UU 
 
 6 laborers, at $1 ! 50 per day, 28 days 234.00 
 
 lcook 1— $429.00 
 
 240 rations, at 60 cents. - - 144,00 
 
 Total repairs, pipe and lumber for one 
 
 party for 10 months ■•••••• 5UU * UU 
 
 Total commissary charges for team, teed, 
 
 dOU.UO 
 
 Total simdry incident^ 
 
 Total supervision hm.w 
 
 Totals 10 months $2,070.00 2 ^ 000 
 
 Sundry expenses per month 
 
 Total cost per month 803.00 
 
 10 months, at $803 jymuu 
 
 Total number of feet sunk ^254.20 
 
 Total cost 2 4 g 
 
 Cost per hole, 7,227 -s- 52 154 42 
 
 The drills were purchased second-hand from the Nicaragua Canal Co^, 
 and the other apparatus was new. If the ongmai cost of all this machinery 
 were distributed over the work, the results would be as follows. 
 
 Machinery 
 
 Total cost $9,63 2 86 
 
 Or average cost per foot 
 
 Both machines are still in good repair, after having been used in Nicara- 
 gua and in various localities in Arizona and California. . .. 
 
 g xhe total depths penetrated in all materials at the various dam sites are 
 as follows : 
 
 The Buttes ... 
 Queen Creek. 
 
 Riverside 
 
 Dikes 
 
 San Carlos ... 
 
 Total 
 
 Covering. 
 
 1,621.2 
 
 357.8 
 
 729.8 
 80.0 
 
 143.2 
 
 2,932.0 
 
 Rock. 
 
 Total. 
 
 196.0 
 
 1,817.2 
 
 55.6 
 
 413.4 
 
 40.2 
 
 770.0 
 
 0.0 
 
 80.0 
 
 30.4 
 
 173.6 
 
 322.2 
 
 3,254.2 
 
248 
 
 PROSPECTING . 
 
 Fig. 2. 
 
 Magnetic Prospecting.— Bodies of magnetic iron ore are frequently discov- 
 ered or located on account of their magnetic properties. Two forms of 
 compasses are employed in this work: the dipping needle, or miners’ 
 compass, and the ordinary compass. The ordinary compass is used to find 
 the center of magnetic attraction in the horizontal plane, and after this has 
 been found the ground may be run over with the 
 dipping needle, to locate the center of attraction 
 by this means. The ordinary compass does not 
 give good results when operating over a mag- 
 netic deposit, but is only useful in determining 
 its outside edge, and thus locating its general 
 position. The dipping needle differs from the 
 ordinary compass in that the needle is hung in a 
 vertical plane in place of horizontally, so that 
 the needle is free to assume any position varying 
 from the horizontal, depending on the downward component of magnetic 
 attraction at that point. The vertical magnetic component at the point 
 should be compensated for by balancing the dipping needle so that it will 
 ordinarily stand horizontally when not affected by local disturbances. 
 
 The actual work of prospecting may be carried on as follows: If there 
 were an outcrop of a vein of magnetic material, as shown in Fig. 2, covered 
 with a capping of wash material, the preliminary prospecting would be 
 carried on as shown in Fig. 3, the dipping needle being carried backwards 
 and forwards zigzag across the deposit, noting the point of maximum dip in 
 each case and establishing a stake there as indi- 
 cated by the crosses. After these stakes had all 
 been established, an average straight line would 
 be struck through them that would follow the 
 course of the deposit as nearly as possible. Stakes 
 would be placed at the ends of this line, as at X and 
 Y, and the line X Y divided off into 100' dis- 
 tances by means of stakes marked A , B, etc. Lines 
 at right angles to the original line would then be 
 turned off at these 100' points, and stakes placed 
 every 10 ft. upon the branch lines. These points 
 on the branch lines would be lettered with small 
 letters, corresponding to the large letter on the line 
 X Y, as shown in Fig. 4, which represents the obser- 
 vations taken at the first station. The dip would 
 be noted at each one of the 10' stations, and 
 recorded in the note book. A convenient method 
 of keeping the notes is to have a vertical line down 
 the center of the page for the line X Y, and other 
 vertical lines to the right and left of it for the indi- 
 vidual stations 10 ft. apart, each side of the main 
 
 line, the horizontal lines across the page being lettered A, B , etc., the sta- 
 tions to the right and left being marked with primes and subscripts of the 
 small letters corresponding to the line. After the observations have been 
 taken, lines may be drawn through points of equal dip and equal deflection 
 
 (isogonic lines). By this 
 means the general form of 
 the bed is determined. 
 
 The maximum dip, in the 
 case of an inclined deposit 
 like that shown in Fig. 2, 
 would occur at c, over the 
 hanging wall of the outcrop, 
 the dip at b being consider- 
 ably less, and the dip at a 
 being less than that at b. 
 After the center of magnetic 
 attraction has been discov- 
 ered, prospecting may be continued by means of the diamond drill, or by 
 sinking shafts or test pits. Sometimes, where deposits of magnetic iron ore 
 have been eroded, the sands near the surface may contain such a considerable 
 amount of magnetic disturbance as to indicate the presence of a body of iron 
 ore. while in reality there may be such a small quantity disseminated 
 through the sand that it could not be made to pay for its removal. 
 
 Fig. 3. 
 
GEOLOGICAL MAPS AND CROSS-SECTIONS. 
 
 249 
 
 Any body of magnetic iron ore is affected by polarity, and one end of it 
 will attract one end of the dipping needle, while the other end will attract 
 the opposite end. Where the body is badly broken up, this dip oi the 
 needle may be reversed several times in a comparatively short distance. 
 
 Prospecting for Petroleum, Natural Ga3, and Bitumen.— Among the surface 
 indications of petroleum and bitumen may be mentioned white leached 
 shales or sandstones, shales burned to redness, fumaroles, mineral springs, 
 and deposits from mineral springs. Also natural gas, springs of petroleum 
 oil and naphtha, porous rocks saturated with bitumen, cracks in shale, and 
 other rock partly filled with bitumen. Petroleum is never found in any 
 Quantity in metamorphic rocks, but always in sedimentary deposits. 
 Bitumen can be told from coal, vegetable matter, iron, manganese, and 
 other minerals, which it sometimes closely resembles, by its odor and taste, 
 also by the fact that it melts in the flame of a match or candle, giving a 
 bituminous odor. (Iron and manganese do not fuse, and coal and vegetable 
 matter burn without fusion.) Bitumen is also soluble m bisulphide of 
 carbon, chloroform, and turpentine, usually giving a dark, black or brown 
 solution. Frequently, springs or ponds have an iridescent coating ot oil 
 upon the surface. Sometimes iron compounds give practically the same 
 appearance, but the iron coating can always be distinguished from the oil py 
 agitating the surface of the water, when the iron coating will break up like 
 a crust of solid material, while the oil will behave as a fluid, and tend to 
 remain over the entire surface even when it is agitated. , 
 
 Frequently, bubbles of gas are seen ascending from the bottoms ot pools 
 or creeks. These may be composed of carbureted hydrogen or natural gas, 
 which is a good indication of the presence of petroleum or bitumen; they 
 may be composed of sulphurated hydrogen or carbom c-acid gas. Garb - 
 reted hydrogen can be distinguished by the fact that it burns with a yellow 
 luminous flame, whereas sulphureted hydrogen burns with a bluish flame, 
 and carbon dioxide will not support combustion, but, on the contrary, is a 
 product of combustion. ,. ~ 
 
 When carbureted hydrogen gas is discovered ascending from water, the 
 bottom of which is not covered with decaying vegetation, it is almost a 
 certain sign that there is petroleum or bitumen somewhere in the underlying 
 or adiacent formations. „ _ , . ... , 
 
 If natural gas or bitumen is found upon the surface of shale, it is probable 
 that the material ascended vertically through cracks m these rocks from 
 porous strata below; while if it is found in connection with sandstones it is 
 probable that the material was derived from the porous sandstone itsell. 
 This is especially liable to be true if the sandstone has a steep pitch. 
 
 As a rule, deposits of bitumen or petroleum occur m porous formations 
 overlaid by impervious strata, such as shales, slates, etc. Anticlines are 
 more liable to contain such deposits, though they are not absolutely neces- 
 sary to retain them, as at times portions of the underlying porous strata 
 have been rendered impervious by deposits of calcium salts, silica, etc., ana 
 hence the petroleum or bitumen will be confined to the porous portmns. 
 Natural gas also occurs under similar conditions, but usually in anticlines 
 
 Construction of Geological Maps and Cross-Sections.— After the surface exam- 
 ination of a property is complete, the data should be entered on the best 
 map procurable, or a map constructed. The scale depends on the size of 
 the property, the complexity of the geological formation, the value of the 
 property, and the material to be mined from it. The amount of work that 
 it will pay to put on the survey will depend largely on the value ot the 
 property, more detail being justified in the case of high-grade properties. 
 If a property 1,200 ft. X 3,000 ft. (the size of four U. S. mining claims) were to be 
 surveyed and mapped with a scale of 1 in. equal to 100 ft., the map would 
 be 12 in. X 30 in. A vein of strata 10 ft. wide on this map would appear as T « ot 
 an inch wide, which is about the smallest division that could be shown 
 with its characteristic symbol; for greater detail, a larger scale, or larger 
 scaled sheets of the most important portions of the deposit, will be 
 necessary. If the geologist constructs the topographical contour map, he 
 can take notes on the geology at the same time. When the l^undanes of 
 the property are being surveyed, certain points should be established, botn 
 vertically and horizontally, as stations in future topographical work. If the 
 map is on government surveyed land, the government lines may be used 
 for horizontal locations, but it will be necessary to determine the elevation 
 of the different points. If the property is much broken, it is well to run a 
 
250 
 
 PROSPECTING . 
 
 few lines of levels across it, to establish points from which to continue the 
 work. This work is usually done with a Y level and chain, the other details 
 being subsequently filled in with a transit and stadia, the levels of the other 
 
 points being taken 
 either by using the 
 transit as a level, by 
 vertical angles, by bar- 
 ometric observations, 
 or by means of a hand 
 level. Where lines of 
 levels are run across 
 the property in various 
 directions, it is best to 
 run them in such a 
 direction that they will 
 cross the strike of the 
 strata as nearly at right 
 angles as possible, so 
 that the profile thus de- 
 termined may be used 
 in constructing a cross- 
 section. Sometimes, 
 for preliminary work, 
 simply a sketch map is 
 all that may be neces- 
 sary. All of the outcrops and exposures, together with their proper dip, 
 should be entered on the map. 
 
 To Obtain Dip and Strike From Bore-Hole Records.— Before the results 
 obtained from bore holes are available for use in map construction, the dip 
 and strike of the various strata must be ascertained. The process, in the 
 case of stratified rock, is as follows: If three holes were drilled, as at A, B, 
 and < 7 , Fig. 6, each intersecting a given bed, the strike and angle of dip of the 
 bed may be obtained by reducing the results from the three holes to a plane 
 passing through the highest point of intersection, which is at A. The hole B 
 intersected the bed at the distance Be , and C at the distance Cd below the 
 point A. By continuing the line CB indefinitely, and erecting two lines Be 
 and Cd perpendicular to it, each representing the distance from the hori- 
 zontal plane through A to the intersection of the strata, two points in the 
 line de are obtained, which line intersects CB produced at/; / is one point 
 in the line of strike through A. In order to find the angle of dip, the 
 perpendicular Cg is dropped from the deepest hole C unon the line of 
 
 Fig. 7. 
 
 strike Af. The distance Ch, equal to Cd , is laid off at right angles to Cg , 
 when the angle Cg li gives the maximum dip. The results obtained from 
 bore holes may thus be reduced to such form that the dips can be projected 
 on the surface to obtain the line of outcrop for each stratum. Bore holes also 
 furnish data for constructing underground curves in cross-sections of 
 stratified rocks, and in locating the probable outline of ore bodies in other 
 formations. 
 
SAMPLING AND ESTIMATING AVAILABLE MINERAL. 
 
 251 
 
 -r-r • r»« thp man all exoosures, whether surface or those 
 
 lines ran, op, gr are '^ontours. M the g| s s ^ L b eds or deposits would be 
 seam the crost ^cfion must he taken along the u “« P€^dioulw to the 
 
 surface >of the cross-section .u i noi a i th p s . section> on this cross-section, 
 accordingT^their SpM'S co'nsiderZe danger of exaggerating their 
 tl * i fn e m a r!s''” t hVIu wwsed course of the beds should he sketched in, 
 
 the property ^ ei ^ 0 “ e l should be tied to monuments or natural 
 
 iSSSsSfe r Si ss 
 gg^gSTfiSfiSSSSHaSSS 
 
 material as it will be extracted the coal, shSuld be 
 
 a«ta Sr3e° When 
 
 may^lira te'^ay^^nd^n^wag^obtm^^later^or th^different^^ples 
 
 BffigMBBBi 
 
 which may be described as follows. , throwing each shovel- 
 
 ful^ to'r'fpe^oY^co'nr Vl^h^he Sne'mly le reduced by 
 
252 
 
 PROSPECTING. 
 
 scraping it down with a shovel, passing slowly around it. If the amount 
 of material is small, a flat plate may be introduced into the cone, and the 
 pile flattened by revolving the plate. The pile is then divided into quarters 
 by drawing lines across it. After this, two alternate quarters are scraped 
 out and shoveled away, and the other two quarters are left as the sample. 
 The process may be repeated until the block has been sufficiently reduced. 
 In shoveling away the discarded portions, care should be taken to see that 
 the fine dust under them is brushed away also, as they often contain fine 
 and valuable mineral that would unduly increase the value of the resulting 
 sample. When the sample consists of only a few pounds, it may be reduced 
 by means of a riffle. Large samples consisting of several tons are sometimes 
 sent to sampling works to be reduced by automatic sampling machines. If 
 the property being examined is a mine in active operation, samples may 
 be taken from the working faces, and also from cars, loading chutes, etc. 
 Usually the samples from the face are kept separate from those from the cars 
 and loading chutes, the latter being intended as a check on the former. 
 In the case of ores of the precious metals, large samples are sometimes taken 
 and used for mill runs. 
 
 Stock piles, or dumps, may be roughly sampled by taking pieces from 
 intervals over the surface, being careful to obtain a fair avenge of coarse 
 and fine material, and of rock and ore. These samples are quartered djwn 
 and assayed, but if a close valuation is d esired, it will be necessary to drive 
 cuts or tunnels through the mass, and to take a certain amount, as every 
 fifth or tenth shovelful, for the sample. When sampling dumps of fine 
 material (as, for instance, tailings) it is possible to take samples from the 
 pile by means of a drill, an auger 1 in. or 2 in. in diameter usually being 
 employed for this purpose. 
 
 The human factor always plays a large part in the value of a sample as 
 finally selected, and hence it should be taken by a man who has had con- 
 siderable experience in this class of work. For this reason, it is best to 
 employ a mining engineer. One not accustomed to sampling very rarely 
 undervalues a property, owing to the fact that it seems to be human nature 
 to pick up a rich piece of ore or coal, rather than the barren gangue material 
 or slate. 
 
 When only surface exposures or shallow prospect openings are available, 
 it is impossible to determine the amount of ore in sight, or to form more 
 than a guess as to the size of the deposit. It is not safe to count any ore in 
 sight unless it is exposed on at least three faces. Ore that is exposed on one or 
 two faces can be counted as probable ore, while slight exposures can be 
 counted only as chances indicated. 
 
 The amount of material available in coal deposits can be estimated much 
 closer than in the case of ores. If a seam is penetrated by a number of bore 
 holes, or by workings extended over a considerable area, it is fair to esti- 
 mate that the material will run practically as exposed for a considerable 
 area; but especially in the case of bituminous coal, it is a comparatively 
 easy matter to form some estimate as to the amount of material available. 
 
 When dealing with ores, it is impossible to form reliable estimates, owing 
 to the fact that horses or other masses of rock may be exposed at any point, 
 and the ore bodies themselves are usually very irregular, hence it will be 
 necessary to do careful blocking out before making anv estimates. 
 
 When estimating the amount of mineral available, only that portion 
 which can actually be removed in stoping should be counted, and if the 
 seam is so narrow that it is necessary to break material from the walls, or if 
 there are masses of country rock that have to be removed with the ore, the 
 expense of removing them should be estimated and deducted from the value 
 of the ore. 
 
 DIAGRAM FOR REPORTING ON MINERAL LANDS. 
 
 The following diagram will be useful as a guide in making out a report 
 on a mining property: 
 
 1. Situation 
 and Sur- 
 roundings. 
 
 { 
 
 1 . 
 
 2 . 
 
 Name, i 
 Distance 
 
 1. Location, if on surveyed land. 
 
 2. Nearest town or village. 
 
 3. Mineral district. 
 
 . 4. County, state, or territory. 
 
 and direction from one or more points. 
 
REPORTS ON MINERAL LANDS. 
 
 253 
 
 DIAGRAM FOR REPORTING ON MINERAL LAN DS {Continued). 
 
 f 1. Hills or mountains. . , 
 
 2. Topogra- I 2. Character of surface, vegetation, and timber. 
 phy. 1 3. Streams and water supply. 
 
 L 4. Elevations. 
 
 3. Geology. 
 
 1. Struc- 
 ture. 
 
 1. Rocks. 
 
 2. Axes. 
 
 3. Faults. 
 
 4. Dikes. 
 
 { 
 
 5. Horses. 
 
 2. Geological period. 
 
 3. (a) Coal f 1. Number, 
 beds. \ 2. Thickness, 
 
 1. Stratified. 
 
 2. Crystalline. 
 
 3. Igneous. 
 
 Anticlines or synclines. 
 
 1. Number. 
 
 2. Strike. 
 
 3. Dip. 
 
 4. Throw. 
 
 1. Number. 
 
 2. Strike. 
 
 3. Dip. 
 
 4. Filling. 
 
 5. Throw. 
 
 1. Number and size. 
 
 2. Location. 
 
 3. Material. 
 
 I 
 
 or 
 
 (5) Ore 
 bodies. 
 
 1. Veins. 
 
 2. Beds or 
 lenses. 
 
 Reported. r 1 Reported. 
 
 Averaee 3 ' J 2 ' Accordin « 
 
 Aveiage. -j tQ meas , 
 
 Umformity. y urement. 
 Number. ^ 
 
 Character. 
 
 Strike. 
 
 Dip. 
 
 w . ,,, f 1- Maximum. 
 Width. 1 2 Av erage. 
 
 Vein filling. 
 
 Ore chutes. 
 
 Walls. 
 
 Throw of walls. 
 
 Number. 
 
 Walls. 
 
 Strike. 
 
 Dip. 
 
 Length. 
 
 Height. 
 
 Maximum width. 
 
 Average width. 
 
 4. (a) Quality of 
 coal, specimens, 
 appearance 
 in mine, in cars, 
 benches. 
 
 or 
 
 , (6) Ore. 
 
 ' l. Color, external, powder. 
 
 2. Luster. _ , . 
 
 3. Clearness from clay or sand, shale, 
 
 4. Sulphur. 
 
 5. Resin. . , . 
 
 6. Firmness, size of lumps, air slaking. 
 
 7. Cleavage or fiber. 
 
 8. Coking. 
 
 9. Color of ashes. . 
 
 10. Use: Gas, steam, domestic, forge, 
 metallurgy. 
 
 11. Analyses or assays. 
 
 : 1. Shipping. 
 
 2. Concentrating. 
 
 3. Metals or minerals. 
 
 4. Gangue. 
 
 5. Impurities. 
 
 6. Assays or analyses. 
 
254 
 
 PROSPECTING. 
 
 DIAGRAM FOR REPORTING ON MINERAL LAN DS — ( Continued). 
 
 4. Mining. 
 
 1. History. 
 
 2. Mine. 
 
 ,’Y>- 
 
 1. Dates of opening, abandoning, reopening, 
 number of mines and names. 
 
 2. Ownership. 
 
 3. Superintendence. 
 
 1. Shaft, slope, or tunnel. 
 
 f 1. Total depth. 
 
 2. Extent of I 2. Depth below water level, 
 workings, j 3. Number of levels. 
 
 t 4. Extent of levels. 
 
 3. Water pumps, size, and kind, water cars, 
 number and size, natural drainage. 
 
 4. Ventilation, natural, furnace, fan (for- 
 cing or drawing out), sufficient or insuf- 
 ficient. 
 
 5. Lighting, system used. 
 
 6. Powder, kind and grade used. 
 
 7. Explosive or noxious gases. 
 
 8. Coal-cutting machines and power drills. 
 
 9. (a) Mode of working, holding under, 
 
 shearing, blasting, or wedging. 
 
 1. Underhand stoping. 
 
 2. Overhead stoping. 
 
 ' 5. Rooming with or with- 
 out timber. 
 
 6. Square sets. 
 
 10. Rooms, or stopes, pillars, dimensions, and 
 general plan. 
 
 11. Timbering, timber trees. 
 
 12. Roof, or hanging wall, strong or weak, 
 air slakes or not. 
 
 13. Floor, or foot- wall, hard or soft, creeps 
 or not. 
 
 14. Roads, rails, and cars. 
 
 1. Men. 
 
 15. System of under- 
 ground tram- 
 ming. 
 
 16. System of hoisting 
 
 2. Mules. 
 
 3. Electricity. 
 
 4. Compressed air. 
 
 5. Wire rope. 
 
 6. Chain. 
 
 7. Locomotive. 
 Cage. 
 
 Skip. 
 
 Cars. 
 
 ■{I 
 
 5. Maps and 
 
 Drawings. 
 
 1. Of the whole region. 
 
 2. Of the underground workings. 
 
 1. Cross. 
 
 2. Longitudinal. 
 
 (1. General. 
 
 3. Columnar. < 2. Coal bed or other de- 
 ( posit. 
 
 4. Buildings, works, or machinery. 
 
 f 1. Scale. 
 
 2. North line, magnetic variations. 
 
 3. Date. 
 
 4. Maker. 
 
 5. Can buy, take, borrow, or have 
 copied. 
 
 3. Sections. 
 
 5. Explanation. 
 
 6. Concentra- 
 tion, 
 
 1. Hand picking. 
 
 2. Cobbing and picking. 
 
 3. Magnetic. 
 
 4. Mechanical, Siy. 
 
REPORTS OX MINERAL LANDS. 
 
 255 
 
 7. Coke 
 
 Ovens. 
 
 DIAGRAM FOR REPORTING ON MINERAL LANDS ( Continued ). 
 
 1. History, ownership, etc. 
 
 2. Number. 
 
 3. Character of ovens. 
 
 4. Dimensions. 
 
 5. Construction, materials, etc. 
 f l. Charge, quantity, etc. 
 
 6. Operations. \ 2. Working. 
 
 L 3. Discharging, quenching. 
 
 7. Repairs. .. , 
 
 8. Quality of product. (Assays, if any.) 
 
 9. Disposition of by products. 
 
 1. In heaps. 
 
 2. In stalls. 
 
 1. Roasted. 3. In kilns. 
 
 4. By mechanical calcmers. 
 
 5. Number &nd dimensions of roasters. 
 
 ' 1. Base bullion. 
 
 f 1. In shaft fur 
 nace to 
 
 8. Metallur- 
 gical Works 
 and Treat- 
 ment of Ore. 
 
 2. Smelted. 
 
 9. Disposition 
 of Product. ' 
 
 1. Shipped. 
 
 2. Shipment. 
 
 10. Statistics. 
 
 2. In reverber- 
 atory furnace 
 to 
 
 3. In retorts to 
 
 1. As mined to 
 
 2. As concen- 
 trates or coke. 
 
 3. Metal or 
 
 matte to 
 
 1. Distance. 
 
 2. Roads. 
 
 3. Railroads. 
 
 4. Navigation, 
 f 1. Capacity. 
 
 2. Matte. 
 
 3. Metal. 
 
 4. Number and dimen- 
 sions of furnaces. 
 
 1. Metal. 
 
 2. Matte. 
 
 3. Number and dimen- 
 sions of furnaces. 
 
 1. Metallic zinc. 
 
 2. Mercury. 
 
 3. Number and size of 
 retorts. 
 
 1. Smelter. 
 
 2. Concentrator. 
 
 3. Sampling works. 
 
 4. Jobber at 
 
 1. Smelter or furnace. 
 
 2. Sampling works. 
 
 3. Jobber at 
 
 1. Refinery. 
 
 , 2. Smelter. 
 
 ^ 3. Jobber at 
 
 1. Production. 
 
 2. Labor. 
 
 : f 1. Daily, weekly, or month- 
 
 2. Actual. , ^ in - to - nS - 
 
 3. Prices. 
 
 i 2. Yearly in tons.* 
 
 [3. Average. 
 
 1. Whole number of workers. 
 
 2. Number of workers in each class. 
 
 3. Number of horses or mules. 
 
 1. Timber. 
 
 2. Tools. 
 
 3. Fuel. 
 
 4 
 
 5 
 
 Oil. 
 
 Powder. 
 
 Day, different classes. 
 Contract or piece, yard or 
 ton. 
 
 Porrip (rp 
 
 8. Local saies of product. 
 
 r l. Machinery. 
 
 6. Labor, 
 
 r. 
 
 {* 
 
 9. Value of plant. 
 
 2. Buildings. 
 
 3. Roads, tracks, etc. 
 
 4. Rolling stock. 
 
 5. Supplies. 
 
256 
 
 PROSPECTING. 
 
 DIAGRAM FOR REPORTING ON MINERAL LAN DS — ( Continued), 
 
 11. Surface 
 
 Plant. 
 
 1. Power plant. 
 
 2. Shops. 
 
 3. Powder 
 
 houses. 
 
 4. Offices. 
 
 f 1. Boilers. 
 
 2. Waterwheels. 
 
 3. Air compressors. 
 
 4. Steam and gas engines. 
 
 5. Electric plants. 
 
 1. Power for. 
 
 2. No. of forges. 
 
 3. Steam hammers. 
 
 4. Other tools. 
 
 ” 1. Power for. 
 
 2. Saws. 
 
 3. Lathes. 
 
 4. Other machines. 
 t 5. Benches and vises. 
 
 1. Power. 
 
 2. Lathes. 
 
 3. Planers. 
 
 4. Shapers. 
 
 5. Drill presses. 
 
 6. Other tools. 
 
 7. Benches and vises. 
 
 1. Smith’s 
 shop. 
 
 2. Carpenter 
 shop. 
 
 3. Machine 
 shop. 
 
 5. Dry or change houses. 
 
 6. Storehouses. 
 
 7. Boarding and dwelling houses. 
 
 8. Stables. 
 
 9. Shaft houses. 
 
 10. Tipples. 
 
 11. Pockets or ore bins. 
 
 12. Company store. 
 
 13. Timber yard and plant for preparing timber. 
 
 1 Cifv nr com- L Q ualit y of water 
 i. lAiy or com- n 
 
 mercial * 
 
 14. Water. 
 
 service. 
 
 2. Company 
 service. 
 
 Sufficient or in- 
 sufficient. 
 
 3. Pressure. 
 
 '1. Quality of water. 
 
 2. Sufficient or in- 
 sufficient. 
 
 3. Gravity system. 
 
 f 1. Direct. 
 
 4. Pump- 
 ing sys- 
 tem. 
 
 Reser- 
 voir or 
 stand 
 pipe. 
 
 15. Lighting. 
 
 1. Character. 
 
 2. Origin. 
 
 16. Hoisting or wind 
 ing plant. 
 
 17. Surface trans- 
 portation. 
 
 1. Gas. 
 
 2. Electric. 
 
 1. Commercial plant. 
 
 2. Company plant. 
 Sufficient or insufficient. 
 
 1. Steam engines. 
 
 2. Compressed-air engines. 
 
 3. Oil or gasoline engines. 
 
 4. Electric motors. 
 
 5. Water motors. 
 
 1. Gauge. 
 
 2. Total length. 
 
 3. Size of cars. 
 
 4. No. of cars. 
 
 5. Power used. 
 
 6. No. of motors. 
 r 1. Character and 
 
 surface of. 
 
 2. Length. 
 
 3. No. of wagons 
 and teams. 
 
 1. Size and capac- 
 ity of. 
 
 2. No. of wagons 
 for. 
 
 1. Railroad. 
 
 2. Wagon 
 roads. 
 
 3. Traction 
 engines. 
 
OPENING A MINE. 
 
 257 
 
 Diagram for reporting on mineral LANDS— {Continued). 
 12. Miscellaneous. 
 
 1. Yearly income, last year, or for any year. Netf S 
 
 2. Average cost per lb., or ton of material. 
 
 [ 1. Quality of ore Or product. 
 
 1. Deposits ~ * * ^ 
 
 of value. 
 
 13. Conclu- 
 sions. 
 
 3. Merits of 
 property. 
 
 4. Advice. 
 
 2. Amount of ore or f 1. Gross, 
 material in sight. ( 2. Net. 
 v 3. Value of material in sight. 
 2. Value of plant and works. 
 
 1. Mining. 
 
 2. Disposition 
 of product. 
 
 ^ 5. Local considerations. 
 
 1. Continue present system. 
 
 2. Change system to . 
 
 1. Ship as mined. 
 
 2. Concentrate, or coke and 
 ship. 
 
 3. Smelt and ship. 
 
 1. Troubles. 
 
 2. Labor. 
 
 3. Supplies. 
 
 4. Climate. 
 
 5. Shipment facilities. 
 
 6. Markets. 
 
 OPENING A MINE. 
 
 The location of the surface plant and the mine opening depend on the 
 formation of the deposit primarily, and secondarily on the facilities for 
 transporting the product to market. It is impossible for <me not on the 
 ground and unfamiliar with natural or railroad transportation facilities in 
 the neighborhood, to give an idea as regards the second consideration. In 
 regard to the first consideration, the following observations will be found 
 
 ° f When the seam or vein outcrops within the limits of the property and is 
 flat, a water-level drift is the best method of opening it. If it has any con- 
 siderable inclination, it should be opened by a slope, or by a tunnel driven 
 across the intervening measures. Where the deposit has an inclination of 
 but from A* of 1° to H°, the water-level drift is generally used, and the main- 
 haulage entry is opened at the lowest accessible point on the outcrop, which 
 insures free drainage and a favorable grade for haulage. When the outcrop 
 dips into the hill, the drift is usually commenced a few feet below the out- 
 crop terrace, and driven on a slight up grade until the normal dip is reached. 
 
 When the inward dip is too strong, the better plan is to sink a shaft in the 
 center of the basin, provided the depth is not too great and the amount of 
 water to be pumped is comparatively small. If the inward dip to the center 
 of the basin does not exceed a total of 25 ft. difference in level, a drift may 
 be used and drainage be effected by a siphon. . 
 
 Water-level drifts are only profitable where the inclined seam is exposed 
 in ravines or gorges eroded across the strike of the measure, or where the 
 vein can be reached by a short tunnel from the surface, to the seam across 
 the measures. This is often the case when the seam dips with the hill, but 
 when the dip is against the hill, the tunnel is generally a long one. While 
 the expense of operating a mine opened by a long tunnel is less than one 
 opened by a slope or shaft, owing to cheaper drainage and haulage, when 
 the coal above water level is exhausted the tunnel is almost worthless. 
 When the seam is inclined and is accessible at no point along its outcrop 
 low enough to furnish sufficient lift or breast length, it should be opened 
 by a slope or shaft. Or, if the seam is flat and does not crop on the tract, 
 a shaft is the only method of working it, unless it lies so near the surface 
 that it can be stripped. 
 
 Where a seam has a dip of 20° or more, and is brought close to the surface 
 by an anticlinal axis or “saddle,” a “rock slope,” or, in other words, a 
 tunnel dipping the same as the seam may be started from the surface, and, 
 when the seam is reached, may be continued to the desired depth in the 
 
258 
 
 OPENING A MINE. 
 
 seam. In sinking slopes for coal mines, it is customary to sink an airway 
 alongside of and parallel with the slope, with a pillar of about 10 yd. between. 
 The slope for coal mines is usually sunk so that there is a “lift” of from 
 100 to 110 yd., and then gangways are turned off on each side. The term - 
 “lift” in this connection means the length on pitch that breasts or rooms, 
 driven at right angles to the gangway, can be driven in good coal. Subse- 
 quent lifts are usually from 80 to 100 yd. long. 
 
 Opening Up a Gold Mine.— The following description of the method of open- 
 ing up a gold mine, by Mr. S. A. Josephi (“ Mines and Minerals,” February, 
 1900), will also apply in general for the opening of any inclined narrow ore 
 deposit: The equipment for the top of shaft for the preliminary work con- 
 sists of three pieces of timber, either sawed or rough hewn, 12 to 16 ft. long, 
 
 8 to 10 in. in diameter, arranged as a tripod; they should be strongly bolted 
 together, a pulley hung from the center, through this a rope passed with a 
 bucket fastened to the end entering the shaft, and a horse hitched to the 
 other end This equipment is called a whip , and is sufficient for the first 
 100 ft. in depth. In locating the shaft, care should be exercised in placing 
 it where there is ample dump, or ground for waste or valueless vein matter. 
 Should the character of the surface not admit of sufficient ground for this 
 purpose, have collar (or top) of shaft elevated 10 or 12 ft., aDd throw waste 
 around the outside of same until filled up solid. 
 
 The timbers should be square sets made of rough or square timbers, 
 preferably the latter, 8 in. X 8 in.; divide the shaft into two compartments 
 4 ft X 4 ft. each, one for the bucketwav, the other for the ladderway. Where 
 air is bad, board up the ladderway to aid the circulation. The sets should 
 be not less than 4 ft. and not over 6 ft. apart, in the clear. 
 
 Sink on the dip of the vein, and keep a careful record of the location, 
 width and value of ore bodv, until the depth of 100 ft. is attained; here 
 place station sets 8 ft. high, start levels each side of shaft, and, if there is 
 water in the shaft, a sump 16 ft. to 80 ft. deep should be sunk. The sump 
 should be built in the same manner as the shaft, so that it will serve as a 
 continuation of same when greater depth is wanted. 
 
 Levels should be run on both sides of the shaft sufficiently long to deter- 
 mine length of the ore chute, and also to determine the existence of other 
 ore chutes in the vein. This development work should be an indicator . of 
 the strength, value, and permanence of the property; it is now ready for 
 another examination by competent authorities, to determine the above 
 conditions. Should their verdict be favorable, continue the shaft to the 
 depth of an additional 100 ft.; if w ater is found in but small quantities, this 
 can best be done by replacing the whip with a whim; this runs by the same 
 horsepower, costing in the neighborhood of $100. It is well adapted to put 
 the shaft down 250 ft. A , ., 
 
 When the shaft is down 200 ft., start levels as at the 100-ft. depth, provide 
 sump, and drift both wavs upon the vein, proving up ore bodies as before. 
 
 Thus far the cost has been slight. The shaft, including timbers, supplies, 
 and contingent expenses, should not cost over $20 a foot, a total of $4,000; the 
 drifts $6 a foot, say 200 ft, each way from shaft on both levels, a total of 
 800 ft, or $4,800; total, $8,800, less amount received from mineral extracted in 
 sinking and drifting, w T hich is usually small. 
 
 Now it is to be decided what further amount the owners are willing to 
 expend, and how extensively they desire the mine opened up betore the 
 actual extraction of ore is to commence. M e will put the depth of shaft at 
 500 ft., the drifts the full length of the claim usually 1 £ 00 ft. 
 
 For this purpose, a shaft house should be built, say 40 ft. X 60 ft., w ith ore 
 house, say, 40 ft. X 40 ft, The equipment should be a 40 H. P. engine and a 60 
 H. P. boiler (if a large flow of water is encountered, an additional boiler and 
 pump must be provided), the shaft continued to the above stated depth, and 
 levels extended at each 100 ft. It will also be advisable to make upraises at 
 the farthest point practical from the shaft, on each side, connecting each 
 level with the other, and extending to the surface. These upraises should 
 be made in ore, and are valuable both for ventilation and escape for men, 
 in case of accident to or near the shaft. They should be furnished with 
 ladders. The machinery shaft house, and skip, with which all incline shatts 
 should be equipped, will cost about $4,500. The additional 300 ft. of shaft, 
 including contingent expenses of engineers, fuel, etc. will cost $40 a toot, or 
 $12,000; the drifts, $6 a foot. The upraises, being on ore, should pay for the 
 
 labor . 
 
 It is prudent to estimate the cost of thoroughly opening up a gold mine to 
 
SHAFTS. 
 
 259 
 
 be between $40,000 and $50,000, which fact probably originated the remark 
 that “it takes a gold mine to make a goldmine.” This is practically true, 
 and no one should attempt to engage in the mining business, as a business, 
 without both money and a willingness to use it. More failures can be 
 attributed to insufficient capital for development than to any cause save 
 mismanagement. 
 
 SHAFTS. 
 
 Shafts and tunnels may be, first, temporary, or those that are simply 
 driven for exploration purposes, and are not to be used for any great length 
 of time; second , permanent, or those that are driven for a specific purpose 
 and usually have a definite predetermined capacity. 
 
 Form of Shaft.— In the United States, shafts are usually square or rectangu- 
 lar in form. This is largely due to the fact that timber is used in lining 
 such shafts. In Europe, round or oval shafts are frequently employed with 
 a lining of brick, iron, or masonry. _ . 
 
 Compartments.— The number of compartments in a shaft and their arrange- 
 ment depends largely on the use to which the shaft is to be put; also on the 
 number of shafts at the property, and the depth of the shaft. VS here the 
 material to be removed is comparatively near the surface, it is usuallv 
 cheaper to sink a number of 2- or 3-compartment shafts than it is to tram all 
 the ore to one large shaft; while, in the case of very deep mines, large 4- or 
 6-compartment shafts are sunk, and the underground haulage extends over 
 a greater area. When the shafts are lined with timber, a stronger con- 
 struction can be obtained by placing the compartments side by side, as 
 shown in Fig. 1, than by placing them in the solid block, as shown in Fig. 2. 
 When a body of material compara- 
 tively near the surface is being re- 
 moved through a number of shafts, 
 
 2-compartment shafts are frequently 
 employed, both compartments being 
 used for hoisting, and separate 
 shafts being provided for the pump 
 column and ladderways. This re- 
 duces both the size of the shaft and 
 the timbering necessary, and also 
 
 does away with the special danger from fire that always exists when there 
 is a ladderway in the shaft, for it is always difficult to fight fire in these 
 special compartments. 
 
 Shaft Sinking.— As a general thing, the loose material or wash above bed 
 rock is not thick enough to cause any serious trouble, and ordinary cribbing 
 of heavy timber or a masonry curbing is sufficient. But when the surface is 
 very thick or loose, and runs like quicksand, considerable difficulty is 
 experienced. The general method of overcoming this difficulty in the past 
 was to at once divide the shaft into the required number of compartments 
 by heavy timbers alternating or placed “skin to skin,” which had the effect 
 of bracing the cribbing against the lateral pressure of the loose material. 
 This method is effectual where the wash will remain solid or stand long 
 enough to allow the timbering and cribbing to be put in. But when the 
 surface is thick, loose, or watery, or of quicksand, some one of the following 
 special methods of sinking must be adopted: 
 
 Forepoling.” 
 
 Fig. 1. 
 
 Fig. 2. 
 
 Quicksand 
 
 Rock (hard or soft, but very wet) 
 
 Rock (hard or soft, but not very wet) — 
 
 Metal linings. ’ ’ (Forced 
 
 down without the use of 
 compressed air.) 
 
 l4 Pneumatic”method. (Limited 
 to about 100 ft. in depth.) 
 
 “Poetsch” process. (Freezing 
 method.) 
 
 “Kind-Chaudron” method. 
 
 “ Continuous, ”or“Long-Hole, M 
 method. 
 
 Size of Shafts.— Shafts vary greatly in size, depending on the number 
 of compartments desired and the size of the compartments. For coal 
 mines, they are generally from 10 to 12 ft. wide inside of timbers, and each 
 
OPENING A MINE. 
 
 2 0 
 
 compartment is from 6 to 7 ft. wide inside the guides. This would make the 
 outside dimensions of a double-compartment shaft about 13 to 15 ft. wide, 17 
 to 18 ft. long, and a triple-compartment shaft from 24 to 25 ft. long. Shafts at 
 metal mines are generally smaller than those at coal mines, but the prac- 
 tice in different localities varies so that it is impossible to give general 
 dimensions that would be of value. 
 
 The table on opposite page gives the dimensions of a few well-known 
 shafts in different localities. 
 
 Forepoling. — When the ground is so bad that it will not stand for several 
 days between excavation and the completion of the lining, it becomes 
 necessary to carry the timber to the bottom of the work. This may be 
 accomplished by using square-set shaft timbering and driving laths, or 
 forepoling behind the timber so as to keep the soft material from running 
 into the opening. The advantages of forepoling are that, if the shaft is 
 being lined with square sets, it can be commenced at any point, and, if the 
 ground is not too bad, the work can be continued by this means until solid 
 material is encountered. When the ground is particularly bad, it may 
 become necessary to use breast boards, which are simply boards braced 
 against the bottom of the shaft so as to keep the material from rising into 
 the opening, only one board at a time being removed while the material 
 behind it is excavated. 
 
 In particularly bad ground, where breast boards have to be used, tne 
 progress made is very slow. After the shaft has been put down by fore- 
 poling; it is sometimes very difficult to repair or replace the lining. When 
 the forepoling method is employed in quicksand, there is considerable risk 
 of losing the shaft altogether, owing to sudden rushes or “boils” of the 
 material that throw the timbering out of line and fill up the shaft. 
 
 Metal Linings Forced Down. — Metal linings forced down without the use of 
 compressed air are rarely resorted to, though in some cases they have been 
 quite successful. If the formation contains but few boulders, it is some- 
 times possible to force the lining down 
 by flushing out the material from the in- 
 side with jets of water. At other times, 
 men enter the shaft and excavate the ma- 
 terial as the work progresses. 
 
 The pneumatic method of shaft sinking was 
 developed from the system in use for put- 
 ting down foundations for bridge piers. 
 At the bottom of the shaft there is a small 
 chamber called a caisson, in which a suffi- 
 cient air pressure is maintained to exclude 
 the water at all times. The shaft lining 
 is built on above this chamber, and grad- 
 ually forced down into the soil. Men 
 enter the chamber and excavate the mate- 
 rial from under the caisson as it descends. 
 
 By this method the sinking commences 
 at once and is continued without interrup- 
 tion until the lining is completed to bed 
 rock, to which the lining is joined, as 
 shown in Fig. 3. An air compressor, which 
 is subsequently used, is the only auxiliary 
 machine necessary, while in the freezing 
 process an ice machine is required. 
 
 It is best to use electric lights in the 
 caisson; hence, it may be necessary to 
 Fig. 3. install a small dynamo if the company 
 
 does not have an electric-light system 
 in operation. In the pneumatic system, the bottom of the shaft is always 
 exposed to view, and the workmen* know when they reach a solid founda- 
 tion; while, in the freezing process, it is sometimes difficult to tell to what 
 depths the pipes should be sunk so as to reach below any fissures or seams in 
 the bed rock. In the pneumatic process, the fine material is aspirated out 
 of the caisson by the air pressure. The pneumatic process is limited to a 
 depth of about 100 ft., as it is impossible for men to work under a greater air 
 pressure than that which corresponds to about 100 ft. of hydrostatic pressure. 
 
 By the freezing process, pipes are sunk in the ground about the area to be 
 frozen, as a rule, not more than 3 or 4 ft. apart. The lower ends of the pipes 
 
Table of Well-Known Shafts. 
 
 SHAFT SINKING. 
 
 261 
 
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 02g 
 
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 d c 3 
 
 3 © 
 
 be bo 
 
 ^ v - G 
 
 r« § 
 
 <® X -fi 
 y^ o 
 Too /3 
 oo^ be 2 
 
 ^ g-S6 
 
 ftsi 
 
 02 ^ 
 
 M O 
 
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 g o' 
 
 g bpbcS'SSo 
 
 rj -H -H £ CM * © 
 
 -gftft ft ' 
 
 £ c 3 g d 
 
 ft ft rift'd 
 
 .a .a a <x> ^ 2 
 
 ^ 83 O p aS 
 
 o o 
 
 ,3 
 
 ftftft° 
 
 ©ftft™ 
 
 ficot» 
 
 b 
 
 X 
 
 b 
 
 ►» 
 
 o 3 
 
 £ 
 
 ft 
 
 a 
 
 g 
 
 ft 
 
 tfl 
 
 pH 
 
 c 3 
 
 02 
 
 3 
 
 a 
 
 ft 
 
 02 Tft 
 
 2o2 
 
 5 &c 
 
 ag. 
 
 $s' 
 
 CO 
 
 ft © 
 ft a; 
 
 02 Nr. 
 
 O* 
 
 05 CO CO CM ^5 o 
 
 CO <M l'- QO O rH 
 
 O CO iO <M o SO 
 
 ooo 
 
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 iO 05 CO^ 
 r-T rf TjT 
 
 o o © 
 
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 Tf 1 05 05 
 
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 CM CM 
 
 rH 
 
 CM rH 00 
 
 b 
 
 b 
 
 b 
 
 bb 
 
 b 
 
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 A A 
 
 b 
 
 b 
 
 X 
 
 X 
 
 X 
 
 XX 
 
 X 
 
 xxxxx 
 
 X 
 
 XX 
 
 X 
 
 XX 
 
 
 V 
 
 X 
 
 O 
 
 -H> 
 
 
 •< 
 
 
 ^ V 
 
 *< 
 
 v V V 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 ib 
 
 O 
 
 b 
 
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 bbb 
 
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 b 
 
 ob 
 
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 rH 
 
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 Cl 
 
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 b 
 
 bbb 
 
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 cb 
 
 bb 
 
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 7—1 
 
 2 Ci 
 
 
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 CO 
 
 
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 CM CM 
 
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 rH 
 
 CM 
 
 CM rH 
 
 
 
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 b 
 
 ffl TfTtl 
 
 moio) bb 
 
 b 
 
 bbbbb 
 
 b 
 
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 b 
 
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 lobbibb 
 
 b 
 
 Mb 
 
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 X 
 
 XXXXX 
 
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 ■V 
 
 b 
 
 to CO Cl ^00 
 
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 CO CO 
 
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 bX 
 
 l-b 
 
 XB 
 
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 SHiawnjBd 
 
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 •g'd 
 
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 ss»< 
 
 o o ' h 2 ^ n * u u 
 
 33 ftft 5 r !02 02 Q202 
 
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 gg jnft ftftftft 
 go .J^ocoo 
 
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 gpq c 3 
 
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 Wft 
 
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 3ft ft 
 Oft 02 
 
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 ,§ - g g 
 
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 +3 ft 02 kH IiH 
 
 ft 02 03 ^)^ 
 
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 %B 8 ZZ 
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 ftQHftft 
 
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 -M 02 
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 > ft 
 
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 ft g 
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 o8 ft 
 ft t-s 
 
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 c8 <J 
 
 S ■g 
 
 g ft 
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 : ft 
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 02 g 
 ft ft 
 
 02 >> 'd 
 
 gft .a 
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 ft) 02 h ^ 02 
 
 ft ft ft 02 ft 
 n 02 ft g g 
 ^ 02 02 02 
 ft ft ftO V 
 
 02ft o 3 r O 
 
 ft 02 'd g 
 O 02 cO g c 3 
 
 3 8 02 
 
 -p'd S tiij 
 
 g a; £ g g 
 
 ft £ 6 
 
 d ft^be 
 OTigS 
 
 ’ a 2 §.s 
 
 O 53 CO 
 
 dSs a 
 
 g 1—1 ft ft c3ft 
 
 ft-n o g ft . 
 
 : 02 
 
 !tP 
 
 ; c3 
 
 : ft 
 
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 :«a.a 
 
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 be 02 
 
 «s g 
 
 9 & 5*5-51,8 2.S 
 
 >5 g ft ft r ^rO O M 
 
 as o t * H fec.g g > & 
 
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 02ft V 
 
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 ft a, 
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 ft © 
 
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 * : Depth completed, 1,150 ft. tin the clear. 
 
262 
 
 OPENING A MINE . 
 
 are sealed or closed, and an inner tube introduced so that a freezing mixture 
 may be caused to circulate down through the inner tube, and up through 
 theouter tube. This freezing mixture may be either liquid ammonia gas, 
 which is allowed to expand in the outer tube, or it may be a solution of 
 calcium chloride that has previously been reduced to a very I9W tempera- 
 ture by means of an ordinary refrigerating machine. The circulation is 
 maintained in the pipes until the ground between them is frozen solid, after 
 which the work may be continued as though the formation were solid rock, 
 the material being blasted and hoisted in buckets. The freezing process 
 mavTe applied to any wet formation, whether hard or soft while the 
 pneumltfc process is applicable only to soft formations The. freezing 
 process may be carried to practically any depth. As a rule, the freezing pipes 
 are never sunk inside of the shaft area. , . 
 
 The Kind-Chaudron method is applicable only to round shafts, and is 
 suitable for shafts passing through very wet and at the same time com- 
 paratively soft formations. The excavation is carried on by means of a 
 large set of boring tools armed with steel teeth, and operated in a manner 
 sinnlar to that employed in drilling wells by the percussive system first, 
 a pit or shaft 4 or 5 ft. in diameter is drilled; this is followed by a reaming bit 
 that enlarges the hole to the desired diameter, or the work may be accom- 
 plished in three stages by using two reaming bits. The material removed by 
 the first bit is hoisted out by means of a sand bucket, or sludger while that 
 removed by the succeeding tools is hoisted out by buckets that are placed 
 in the bottom of the first pit and kept there while the tools are m operation. 
 No water is pumped from the shaft while it is being excavated or lined, 
 and hence practically all the tendency that tbe s ]. des w . oul d have to cave 
 is removed. After the shaft has been excavated down to and into a solid 
 formation, it is lined by lowering cast-iron tubbing into the hole and making 
 a tight ioint against the bottom by means of an expansive packing that is 
 forced out by the weight of the tubbing, or lining. After the lining is m 
 place the space between it and the sides of the excavation is filled with 
 cement. When the cement is thoroughly hardened, the water is pumped 
 from the inside of the lining, and men descend and examine the joint at 
 bed rock. In this method, no workmen enter the shaft until it is lined 
 through the troublesome formation. . 
 
 Lone-Hole Process.— The long-hole process consists m the drilling of a 
 series of diamond-drill holes over the area of the proposed shaft, then filling 
 the holes with sand, after which the work progresses by removing the first 
 5 or 6 ft of sand from the holes in the interior of the shaft, charging these 
 holes with explosives, and firing them by electricity. Next, the holes 
 around the boundary of the shaft are charged and fired m the same manner, 
 and the process is continued until the bottoms of the diamond-drill holes are 
 reached. This method is especially applicable to work m hard rock where 
 great speed in sinking is desired, for all the drilling is accomplished at one 
 dperation, after which the sinking progresses by simply cleaning out the 
 drill hole and blasting the material. 
 
 General Comparison of Methods of Shaft Sinking. Where a shaft is sunk by 
 forepoling, it is usually made rectangular in form. The pneumatic method 
 mav be used for either round or rectangular shafts, and the lining may be 
 either of metal or wood. The freezing process may be used for either round 
 or rectangular shafts, and the lining may be either timber, metal, or masonry, 
 as the entire opening can be left open until the solid rock is reached, when 
 the lining can be built upon it. The Kind-Chaudron method is applicable 
 only to round shafts, on account of the fact that the hole is bored. The 
 long-hole process is applicable to either round or rectangular shafts, but .was 
 originally introduced for sinking rectangular shafts. . 
 
 Sinking Head-Frames.— Head-frames of very simple form are used for sink- 
 ins The skeleton of the frame is formed of heavy squared timber (10" X 10" 
 or 12" X 12") mortised and pinned together, and braced by diagonal braces. 
 A good height from the surface to the center of the sheave is from 20 to 25 ft. 
 The sheave should be from 6 to 8 ft. in diameter. The sinking bucket should 
 be of boiler iron, or of heavy hard wood strengthened by iron bands, about 
 3 ft. in diameter at the top by from 2i to 3 ft. deep. It should be suspended 
 bv a handle pivoted a trifle below the center, and it should have a pm on 
 the rim of the bucket that will hold it in an upright position when a loose 
 ring on the handle is slipped over it. A chain fastened to the top of the 
 head-frame, with a hook on its loose end, is suspended so that when hang- 
 ing plumb, it is over a chute leading to the dump car. As the bucket is 
 
SHAFT SINKING. 
 
 263 
 
 hoisted out of the shaft, this chain is attached, and the engine reversed. 
 The bucket swings over the chute, the ring holding it upright is knocked off 
 the pin, and the rock is dropped into the chute. Rocks too large for the 
 bucket are suspended in chains and are hoisted in that way, and removed 
 on a truck that runs on a track inside of the head-frame, the gauge of 
 which is sufficiently wide to give plenty of clearance for the bucket. 
 
 Sinking Engines.— Most shafts and slopes are sunk with old engines, or 
 else by engines especially designed for such work, and so constructed that 
 they can easily be moved from place to place. In some cases where an old 
 engine can be readily had, it is set up on temporary timber foundations and 
 used until the shaft or slope is finished, when it is replaced by the perma- 
 nent engines, and the old one is dismantled and disposed of to the best 
 
 a<1 Tod^— The old method of hand drilling is still adhered to in many 
 instances, but it is gradually giving way to machine drilling, especially in 
 deep shafts. When properly managed, the work is done much more rapidly 
 and economically by the several excellent types of rock drills now on the 
 market. They are constructed in a variety of shapes by the makers, and 
 there are so many convenient accessories in the shape of fittings, etc. that 
 all contractors prominent in the various coal fields possess one or more of 
 their favorite type of drills. These drills are run either by compressed air, 
 steam, or electric power, and in large shafts two are usually employed, so 
 that work may not be delayed by a breakdown of one drill. The center or 
 one side of the shaft is usually kept in advance of the rest, so as to furnish 
 a sump for the collection of the water. The holes are drilled from 3 to 6 ft. 
 apart, and the depth varies with the character of the rock. When a suf- 
 ficient number of holes are drilled, the drill is removed, and a cartridge 
 made of dynamite, dualine, or some other form of high explosive is tamped 
 in each hole. These are all fired simultaneously by an electric battery, 
 detonating caps being placed in each charge. 
 
 To keep the shaft the required shape, if rectangular, a plumb-bob is sus- 
 pended in each corner, either from the flooring on top, or from a beam laid 
 across the cribbing, and these guide the miner in squaring the corners and 
 sides. If the shaft is a circular one, a plumb-line is let down in the center, 
 from time to time, and a rod cut the exact radius is revolved around it. If 
 it strikes the rib, the miner knows that at that point the shaft is not true. 
 
 Drainage and Ventilation.— When only a small amount of water is encoun- 
 tered while sinking, the best plan is to allow it to collect in a depression and 
 bail it from there into the bucket, hoisting it the same as the rock. Where 
 the water is excessive in quantity, a steam pump is necessary. All the 
 leading pump works make pumps especially designed for sinking purposes, 
 and it is not in the province of this work to mention the advantages pos- 
 sessed by one over the other. 
 
 When the shaft is of moderate depth, a fire burning m one corner will 
 supply ample ventilation. To rapidly clear away smoke, a good plan is to 
 burn a bundle of straw or shavings in one end of the shaft, and throw a 
 couple of buckets of water down the other end. When the shaft is very 
 deep, or when the sectional area is small, ventilation is produced either by a 
 steam jet, or by a small fan turned either by steam or by hand. In some 
 cases, a fire is used that draws into a board pipe. 
 
 Speed and Cost of Sinking.— Any attempt at a general estimate regarding 
 the speed and cost of sinking is impossible, for many reasons appreciated by 
 the practical miner. Shafts vary so much in size, and in the character of the 
 material through which they pass, that even if there were no other items to 
 be considered, a general estimate could not be made. But if the ground is 
 pretty well known, and the sectional area and the depth given, the experi- 
 enced contractor knows how much he can drive in a given time, and 
 he can consequently form a good estimate for each separate shaft. The 
 range of cost is so great that it may be anywhere from SI to S10 per cubic 
 yard of material excavated. , . , . 
 
 Slope Sinking.— A slope is an inclined plane driven down on the bed of the 
 seam, and is generally through coal or ore, though sometimes they are 
 driven through rock across measures to cut the seam that cannot be conve- 
 niently worked bv a slope. In the latter case, it is merely an “inclined 
 tunnel.” In the former it might be termed an “inclined gangway.” 
 
 A slope and an inclined plane, when mentioned hereafter, will mean an 
 inclined opening in coal or ore , used as a passageway for mine cars. 
 
 When tne location of the slope has been decided on, erect a temporary 
 
264 
 
 OPENING A MINE. 
 
 sinking plant; an old engine is generally used. For a short distanee, varying 
 with the nature of the ground, but usually ranging from 10 t<3 20 > ft. on the 
 pitch, an open cut is made, and the earth, rock, or crop coal is town ^out 
 by hand. As soon as sufficient cover is reached, the work of undermining 
 and timbering is commenced, and at the same time a double or single track 
 is laid, so that the material can be taken out in a car or self-dumping skip. 
 When the latter is used, the track is continued up a trestle some distance 
 above the surface, and a head-sheave so placed as to draw the skip up the 
 required distance and dump the material in a chute beneath tke tres^mg. 
 
 The width of the slope depends on the size of the cars and the number of 
 compartments. The most common arrangement is to divide the slope into 
 three compartments; two large ones for hoistways, and a smaller one for 
 pump rod, column pipe, steam pipe, and traveling way. This last is also 
 used as an airway while sinking is going on. 
 
 In some instances, slopes have but one hoistway, laid with three ratio 
 and a turnout at the middle of the hoist, and some have single track with a 
 central turnout. This may be economy in first cost, but is uot ni the long 
 run. Collisions are apt to occur, and the breaking of a rope or the falling ol 
 coal from an ascending car is apt to cause more damage than when two 
 
 C ° 1 When n Sveral lifts are simultaneously worked, a single-track slope is 
 used; but unless the pitch is light and several cars can be hoisted at once, 
 this method produces a comparatively small output. . , , , 
 
 When the dip of a slope is under 40°, the height of the should be 
 
 about 7 ft. in the clear. When the slope dips more than 40°, unless self- 
 dumping skips or gunboats are used, a cage is necessary, and then the 
 height must be made greater. . . , 4 , , n , „ QTw1 
 
 The sinking of a slope is similar to gangway driving, and the tracks and 
 timbering are kept well up to the face. . . . „„ noTO j 
 
 The timbering is very similar to gangway timbering, except that squared 
 timber is more frequently used (but it is not necessary) and .^e jmito are 
 cut with more care. On steep pitches, a heavy mud all is let into the nb 
 on each side, to prevent the road from slipping down the pitch. 
 
 The Sump.— When the shaft or slope is completed, among the 
 necessary is a sump in which to collect the drainage of the mine. This is an 
 openingTower in the vein, when it is a pitching one, or in the rock when it 
 is a flat seam reached bv a shaft. It should be large enough to hold an\ 
 excess of water that the pumps cannot handle; and the pumping machinery 
 should be powerful enough to handle the ordinary drainage by running not 
 over 10 hours per day. When this is the case, m an emergency, the pumps 
 ran be run continuouslv, and thus handle the surplus w ater. 
 
 Drivine the Gangway.— in bituminous coal seams, the height of the gangway 
 is governed bv tlie thickness of the seam, and this is also true, m a certain 
 sense in the anthracite regions. But in the anthracite regions thej are very 
 Hfdom less than 6 ft. in height. In the larger seams they are ^m 6 ft 6 in 
 to 7 ft. 6 in. high in the clear, and from 10 to Id ft. wide The gauge oftrack 
 varies from 24 to 48 in. The grade should rise at least 4 in. in 100 ft., and 
 a cutter 3 ft wide bv 18 in. deep should be cut in the coal on the low side. 
 T f is gutter should be a gutter, and not a receptacle for refuse. There is no 
 ecoSln a sMlow^tter/or in neglecting it ^n^t costs a few cents 
 a dav to keep it open. Some authorities advise a rise of from 6 in. to 1 ft . in 
 o V erV 100 ft., but thev evidently do not take into consideration that ^g^ 
 a riSe means a loss of from 26 to 53 ft in lift at ££ 
 
 lone or in other words, in the loss of from 68,000 to 13/, 000 sq. ft. of tne area 
 LHoaT^o be reached by the gangway. This app lies to Puffing seams^ 
 Where the seam is fiat, or nearly so, the gangway must, of course, be cmven 
 on a grade that best suits the formation Turnouts oneach side 
 
 of the shaft or slope, of a suitable length, are a necessity, if the slope or snait 
 to be ke pt constantl v supplied with coal. These turnouts vary m length 
 de^ndingonTe length of the cars, and the number necessary to keep the 
 machinery in motion between trips. They should ^^? e enough^abffiw 
 at least 3 ft. in the clear between the bodies of the cars, 5 ft. is even oette . 
 When possible to avoid it, there should be no center props between the 
 
 ^Levels in Metal Mines.— The cross-section of the level depends largely on 
 the character of the ore mined, and the desired output from the deposit. In 
 .i. _ mnfni minoc tytwI n pi n tr hi eh-grade mineral from narrow 
 
 lixt; cuaiauici ui me uic , — ~ , 
 
 the case of precious metal mines, producing high- 
 veins, the levels are driven as small as possible. 
 
 •grade mineral from narrow 
 Immediately adjoining the 
 
MINE TIMBER AND TIMBERING. 
 
 265 
 
 shaft there is a plat or station the full width of the shaft. This is heavily 
 timbered and provided with a double track, but, as a rule, the levels have 
 but a single track, and in some cases there is but a single track at the shaft, 
 there being a turnout or switch in the level a short distance from the shaft. 
 In this class of mines, 5 ft. X 6£ ft. in the clear would probably be the average 
 size of a level, it being driven as small as possible. In the case of mines 
 producing lower grade material and handling heavy tonnages from large 
 deposits, as, for instance, in some of the iron and copper mines, the levels are 
 driven larger, and in some instances are double-tracked, being from 7 ft. to 
 8 ft. high in the clear, and from 7 ft. to 12 ft. wide inside timbers; but, even m 
 this class of mines, in most cases single-track levels 7 ft. X 7 ft. to 8 ft. X 8 ft. 
 in the clear are employed with turnouts or passing points at intervals, and 
 a double or triple track at the shaft. The levels are usually driven with a 
 slight grade away from the shaft, so that they will drain to the shaft, and 
 the grade will be in favor of the loaded car. In some mines where electric 
 tramming is employed, the levels are so driven that the motor makes a 
 circuit through the mine, following the foot-wall in one direction, and 
 returning along the hanging wall, or one of the drifts may be in the 
 country rock. Such systems as this are employed only in large properties 
 handling a very great tonnage. 
 
 TUNNELS. 
 
 Mining tunnels are usually of small cross-section compared with those that 
 occur in railroad work, it being rare that their size is such that they cannot 
 be driven in full section, and if the ground is firm the operation of placing 
 the lining may follow behind the work of driving. They are generally 
 lined with timber, and in case the ground is of a soft or treacherous nature, 
 bridged square sets and forepoling are employed, with or without breast 
 boards, as the necessity of the case demands. When the material is firm 
 rock, the tunnel is sometimes not lined, the roof being given an arched 
 form. The various forms of timbering employed as tunnel linings are 
 shown in the sections on Timbering. 
 
 MINE TIMBER AND TIMBERING. 
 
 Choice of Timber.— Timber used for underground supports in mines should 
 be long-grained and elastic, and, at the same time, should not be too heavy. 
 Oak, beech, and similar woods are very strong, but are heavy to handle, and 
 when set in place are treacherous, owing to the fact that they are short- 
 grained and not elastic, so that, though strong, when they do break, they 
 break without warning. Mine timber is placed, not with the intention of 
 ultimately resisting the great pressure of the earth, but so that it may keep 
 any loose pieces in place and also to give warning to the workmen, thus 
 enabling them to escape before a fall occurs. For this reason, pine and fir 
 are, as a rule, better for mine timbering, as they combine a fair amount 
 of strength with considerable elasticity, and hence give warning long before 
 they break. Very elastic timbers, such as cypress, willow, etc., are, as a rule, 
 to be avoided, on account of the fact that they will simply bend like a bow, 
 without offering the necessary resistance to hold the material in place for a 
 short time. 
 
 Preservation of Timbers.— The character of the ventilation in a mine has 
 considerable effect on the life of any timber supports. Damp stagnant air 
 will cause mold and fungus growth, which will be followed by the destruc- 
 tion of the timber through decay or dry rot. All timbered openings should 
 be well ventilated, and provision made for the speedy removal of damp hot 
 air, such as commonly occurs around pump rooms and along steam lines.. 
 
 Water is a good preservative, as it washes off the spores of the fungi as 
 fast as they are formed, and sometimes shaft timbers are kept wet on 
 account of the preservative action of the water. 
 
 Timber may be also preserved (1) by a solution of common salt and 
 water; (2) by impregnating the wood with such metallic substances as sul- 
 phates of copper, iron, etc.; (3) by impregnation with the chloride of mag- 
 nesium or zinc; (4) by creosoting; (5) by coal tar; (6) by carbolineum. 
 
 A solution of 1 lb. of salt in 4 or 5 gal. of water gives a cheap and easily 
 armiied preservative with which the timber should be thoroughly soaked. 
 
266 
 
 MINE TIMBER AND TIMBERING. 
 
 Sulphate of iron is economical and effective. In the zinc process, ablution 
 of 1 gal. of liquid chloride of zinc (Sp Gr. 1.5) mixed with So gal. of 
 is forced into the wood by pressure. Impregnation with crude CTe^ote 
 oil is effective, but it has the disadvantage of making the timber very 
 inflammable. Creosote acts in a threefold manner: (1) WAhsthe pores 
 
 and prevents saturation by water; (2) it destroys organic life, ^ car- 
 bolic acid that it contains coagulates the albuminoids and P r ^Y^^^cay. 
 Painting with liquid tar is effective, but makes the wood, very mflMomable. 
 Painting with ordinary whitewash is also said to give good results. Car- 
 bolineum is said to be effective but is quite expensive. It is applied with 
 a brush, or bv steeping in a tank; 1 gal. will cover 300 to 400 ft. of timber. 
 
 Professor Louis, of England, has shown that preservatives decrease the 
 strength of timber from 84 to 204, depending on the P^ce^used. 
 
 The following table gives the results of tests made by different methods 
 of treating wood at Saint Elroy, France, and recorded in 1890: 
 
 Tests of Preservatives for Mine Timber. 
 
 Relative Preservative Effect. 
 
 Name of Preservative. 
 
 Oak. 
 
 Fir. 
 
 1 
 
 Pine. 
 
 Beech. 
 
 Birch. 
 
 Poplar. 
 
 Tar 
 
 Chloride of 2 dnc 
 
 Sulphate of copper 
 
 Sulphate of iron 
 
 Creosote 
 
 27.8 
 
 10.5 
 
 42.1 
 
 18.0 
 
 1.7 
 
 263.5 
 
 50.0 
 
 12.0 
 12.5 
 
 2.5 
 
 87.5 
 
 26.3 
 
 8.0 
 
 4.2 
 
 4.4 
 
 105.4 
 
 18.6 
 
 1.8 
 
 4.7 
 
 0.6 
 
 26.2 
 
 52.5 
 
 2.5 
 
 3.7 
 
 3.3 
 
 150.5 
 
 34.7 
 
 15.5 
 
 2.9 
 
 1.3 
 
 The simple removing of the bark, under some circumstances seems to be 
 advantageous, but, in some woods, if the bark is removed, the sap ’wood 
 should also be removed. In many cases, the sap wood of coniferous trees 
 is as strong or stronger than the heart wood, and for this reason it should 
 not be removed. Also, in the case of many coniferous trees, the bark seems 
 to act as a protection to the timber in the underground workings. If it 
 becomes necessarv to reduce the size of the individual sticks, it is usually 
 better to split them than to saw them, especially in the case of wood from 
 coniferous trees, as this does not destroy the sap wood or unduly injure the 
 grain or fibers of the stick. Generally speaking, mine timbers last longer 
 when kept wet, and, on this account, some of the mines m Europe have 
 introduced a system of pipes for spraying the timbers m dry portions of the 
 mine When timbers are alternately wet and dry, they are destroyed with 
 amazing rapiditv. Timber should be probed from time to time to ascertain 
 its condition, as’ timbers may appear sound on the outside when the heart is 
 completely destroyed bv dry rot. In selecting props, the principal points to 
 be observed are: Straightness, slowness of growth as indicated by narrow 
 annular rings, freedom from knots, indents, resin, gum, and sap. They 
 should also be well seasoned before use. With these ^cautions and proper 
 mine ventilation, fungus growth may generally be obviated and durability 
 
 ^Pladne of Timber.— The individual sticks should never be weakened by 
 cutting mortise and tenon joints. The pressure should be evenly distributed 
 over a number of sticks, and not concentrated or centered at one point. 
 Centers of revolution should be avoided. The individual sticks should be 
 placed in the direction of the strain that they are to resist, so thatthey will 
 be subject to compression along their length rather than to a transverse 
 strain J The individual sticks should be so placed, and the joints so formed, 
 that the pressure tends to strengthen rather than weaken the stm . c ture up to 
 the crushing strength of the timber. In the case of large tlie timber- 
 
 ing should be done according to some regular system, while at the face of 
 coal mines, single props or posts are usually found better, owing to the fact 
 that “duty is only to support the loose portion of the roof for a lifted 
 time. Probably the most important point is to timber in time, before the 
 rock becomes broken or begins to settle. 
 
MINE TIMBER AND TIMBERING . 
 
 267 
 
 It seems generally agreed that the main weight in mines comes nearly at 
 right angles to the bedding, and that the props should be mainly set in that 
 direction. If the deposit is horizontal, the weight generally comes vertically; 
 but if the deposit is inclined, the weight comes at a right angle to the inclina- 
 tion. Some authorities hold it as a principle that all props should be set at a 
 rectangle against the main pressure. Others, in order to guard against 
 possible side thrusts and a tendency of the ordinary weight to ride to the dip 
 m inclined deposits, purposely cause a sufficient number of props to be set 
 slightly deviating from the common axis. 
 
 Sawyer fixes a maximum and minimum slope for the props, varying with 
 the rate of dip. He makes this maximum slope of the props one-sixth that 
 of the dip, ana the minimum slope one-third of the one-sixth. 
 
 Props are usually set with the butt end downwards, but not always. Hav- 
 ing the butt end upwards adds a trifle to the weight on the lower end, but 
 the larger size at the top should lessen the liability of its being split by a 
 coupling resting on it, and also gives more surface for abrasion in hammer- 
 ing up against a rough roof. Both ways may therefore have advantages 
 according to the circumstances. The butt end downwards, with air circu- 
 lating, is the way Molesworth recommends for stocking. 
 
 Size of Timber.— The general tendency at all metal mines at present is 
 toward the use of systematic frames composed of small sizes of timber, rather 
 than toward the use of large individual sticks. The advantages are: (1) the 
 small timber is cheaper and easier to procure; (2) it is easier to handle, and 
 hence costs less to place in position. By making the frames according to 
 some regular system, the individual sticks can be framed on the surface by 
 machinery so that better joints are secured. The setting of timber can be 
 done by less experienced help when it is all alike. 
 
 Joints in Mine Timbering. — In all mine timbering, the object is to so form 
 the joints that no fastenings will be necessary and that the shape of the 
 pieces will be such that the pressure from the surrounding material will 
 keep the joints tight. The reason for this is that any metal joints usually 
 corrode rapidly in mines, and that, when it becomes necessary to replace 
 timbering, this can be done with greater ease if the sticks are so framed 
 that, by relieving them temporarily of the pressure from the sides and top, 
 they can be simply lifted out of place and new ones substituted. The use 
 of a framing machine renders it possible to frame the joints more exactly 
 than with hand framing. With hand-framed timbers, the joints are always 
 cut a little free to allow for any unevenness in the surface, but, if machine- 
 framed, they are sure to be of the same size. As timber does not shrink in 
 the direction of its grain, it is evident that where the posts meet, if the 
 caps shrink slightly, they will become loose in the space between the shoul- 
 ders; hence, if timbers are cut green and framed to the exact size, subse- 
 quent shrinking may open some of the joints. This may be obviated by 
 keeping the timber moist. . 
 
 The method of taking timbers into a mine depends on the size and 
 number of timbers used and on the character of the opening into the mine. 
 In drift or tunnel mines, timbers are sent in on flat cars built especially for 
 the purpose, or in the regular mine cars. In vertical shafts, they are usually 
 stood on end on the floor of the cage, and lashed together and also to the 
 supports of the cage. Where the opening is an incline, it is the common 
 practice to load the timbers into a skip and thus lower them into the mine. 
 Timber should, wherever possible, be framed on the surface. 
 
 Undersetting of Props.— Props at the working face should not be set at 
 right angles to the inclined floor of the seam, but should be underset, and 
 the greater the inclination, the greater the underset. The amount of under- 
 set should vary with the inclination of the seam, and should not be so 
 great that the props will fail out before the roof has tightened them. 
 
 Forms of Mine Timbering and Underground Supports. — The timbering of a mine 
 maybe divided into two heads: (1) timbering the working faces; (2) tim- 
 bering the roads. 
 
 The roof may be supported (a) by packing the waste places entirely 
 where sufficient material is obtainable for the purpose, and timbering the 
 faces and roads; (ft) by partially packing the waste, by buildings or stone 
 
 S illars with intervening spaces, and by timbering the face and roads; (c) 
 y timbering the face and roads and supporting the roof in the waste places 
 by wooden or stone pillars, but without any packing; ( d ) by timbering alone 
 without any packs or walls whatever; ( e ) by supporting the main roads 
 With brick arching, or by steel or iron supports. 
 
268 
 
 MINE TIMBER AND TIMBERING. 
 
 The accompanying plates include all the common forms of mine timber- 
 ing and underground supports. _ . . . . 
 
 Fig 1 shows a post a and breast cap b. The breast cap b is also sometimes 
 called’ cap, head-block , headboard , lid, or bonnet Sometimes the posts are 
 placed upon blocks of wood similar to the head-blocks or headboards, the 
 block being called a sole; at other times, two or more posts may be set upon 
 one long block of timber called a sill. When posts are used m inclines, 
 they should not be set perpendicular to the foot and hanging walls, but 
 should be underset slightly, so that any tendency of the hanging wall to 
 settle will bring the posts nearer at right angles to the walls, and so tighten 
 them- the amount of underset should never be more than one-sixth the 
 Ditch’ of the deposit. Where posts are set at an angle, they are usually 
 Dlaced on wedges, and, as the pressure comes on, the wedges are tightened 
 Fig 2 represents a stull a, which is used either to keep the walls, of 
 perpendicular or steeply inclined beds or veins apart, to support planking 
 or lagging as a working platform, or as a platform upon which to pile 
 or© or rock 
 
 Fig. 3 represents cockermegs, which are simply timber frames employed 
 in coal mines for holding the face of the coal in place while it is being 
 undercut They are composed of a pole c extending along the face and 
 supported by short stulls or braces a, the whole being tightened into place 
 
 by the long stulls b. ...... „ .. , , 
 
 Fig. 4 shows a crib, cog, chock, pillar, or shanty built up of timbers and 
 filled with waste rock. It is intended to serve as a pillar and to withstand 
 great vertical pressure, doing away sometimes with the necessity of leaving 
 
 pil 7^is ^cribbing framed from round timbers laid skin to skin, and used 
 
 in raises or ore chutes. , , . r , 
 
 Gangway or Level Timbers.— Fig. 5 is a set employed m the case of an extra- 
 wide gangway, there being a center post under the middle of the cap. This 
 form of set may be provided with a sill when the floor of the drift or 
 
 gangway s^so^ ^ form Qf drift set surrounded by bridging and used where 
 such bad ground is encountered as to necessitate forepoling. A are the 
 posts, B the caps, and C the sill of the regular set; D are upright bridge 
 pieces; E a horizontal bridge piece separated from the set proper by blocks 
 F so as to provide spaces H around the regular set through which the spiles 
 or forepoles can be driven. , . , 
 
 Fig 8 shows a form of drift set sometimes employed m very heavy or 
 swelling ground. This method of framing the timbers shortens each piece 
 and reduces the transverse strain on all the timbers. 
 
 Fig. 9 shows an ordinary drift set provided with a sollar for ventilation 
 purposes. An additional brace b is placed parallel to the cap c, and this is 
 covered with plank lagging a, so as to provide a passage above the regular 
 drift, which may be used as a return air-course. 
 
 Fig. 10 is a simple form of drift set employed when the roof and walls are 
 of soft material, but the floor material firm. It is composed of posts l, upon 
 which is placed the cap c. The joggle cut into the cap to receive the heads 
 of the post should never be less than 1 in. nor more than one-third the 
 thickness of the cap. The cap is usually made of such a length that the 
 posts l have an inclination or batter as shown in the illustration, thus giving 
 greater strength to resist side pressure without decreasing the floor area of 
 the drift, which may be necessary for drains, ditches, water pipes, etc. at the 
 sides of the track. When the floor is not composed of solid material, the 
 posts l may be set upon a sill that is framed to fit the legs in a manner 
 similar to that shown for the cap. The joggle cut in the sill should never be 
 less than 1 in. nor more than one-third the thickness of the sill. The sill is 
 usually composed of lighter material than the cap, is flattened .on one 
 or both sides, and is sometimes used as one of the ties to receive the 
 t; rac k 
 
 Fig. 11 shows a post l and the cap or collar c, used where one wall is of 
 firm material. On one end the cap is placed in a hitch. When the collar 
 is supported in a hitch, it is sometimes said to be needled, the operation being 
 called “ needling.” The bottom of the post a is also secured m a hitch, in 
 case there is any side pressure. To keep the surrounding material in place, 
 lagging is necessary, as shown behind the timbers in Figs. 5, 10, and 11. In 
 the case of running ground, the lagging is usually made from sawed material 
 apd driven close together. 
 
MINE TIMBER AND TIMBERING. 
 
 269 
 
270 
 
 mine timber and timbering. 
 
 Fie. 13 illustrates a method of spiling or forepoling. a are the I*>sts of the 
 regular set, b the caps, and e the top bridging. The front ends of the spiles 
 from any given set rest on the bridging of the next advanced set, and the 
 spiles for advancing the work are driven between the bridging and the set, 
 as shown in the illustration. To force the spiles out into the ground, so as 
 to provide room for the placing of the next set, tail-pieces i are employed, 
 these are placed behind the back end of the spiles as they are being driven. 
 After the spiles have been driven forward the desired amount^another set 
 is placed, the tail-pieces knocked out, and the front end of the spiles allowed 
 to settle against the bridging of a new set. Where the face is composed of 
 extremely bad material, it may be necessary to hold it in place with breast 
 boards , asshown at k, the breast boards being held in place hy props ^ which 
 rest against the forward set. When breast boards are used, it is usually 
 necessary to employ foot and collar braces between the sets, so as to transfer 
 
 the pressure of the breast back through several sets. ^ 
 
 Fig. 14 shows a method of placing drift sets m the case of very heavy or 
 swelling ground, a are the posts, c the sills, b the caps, d are the collar 
 braces that bear against both the caps and the posts, while e are foot oj hM 
 braces that bear against both the sills and the posts; / are diagonal braces that 
 are halved together and placed as shown. . . . , , 
 
 Shaft Timbering.— Fig. 12 shows square-set timbering, sometimes employed 
 for shaft lining. A are the wall plates, B the end plates, C the buntons, 
 and D the posts. The method of framing the different parts is plainly 
 
 Sh °Fig* 15 represents cribbing sometimes employed for shafts. It is composed 
 of heavy sawed material halved together at the ends ^ 
 pieces a are called wall plates , and the short pieces b end plates Between the 
 compartments a partition is built up of pieces c called buntons The end of 
 the buntons are let into the wall plates an inch or so, as shown m tne 
 illustration and should be so placed that they will break joints with the 
 individual pieces of' the wall plates, thus preventing the timbers of any 
 
 6in f g .1f, sh“ws l a^tfJr Method of ' framing. ^times em^oyed for the 
 
 formed at I). This construction necessitates the cutting of a j^non onthe 
 the Dost F as shown. S is a 2"X 2" strip nailed along the center of 
 the back of the wall and end plates as a support f ^^c lag^ng tha is 
 placed outside of the sets. The lagging is usually composed of 2 or 3 
 
 ^ 18 shows the use of hangers between the individual square sets. The 
 
 it 
 
 timber, to the sinking of a small prospecting mld^from 
 
 wS,t^"ted down by 
 
 about 45°. 
 
272 
 
 MINE TIMBER AND TIMBERING . 
 
 Landings Plats or Stations.— Fig. 22 is one method of timbering a pjat or 
 station. The regular square-set timbering of the shaft is continued past 
 the station and the heavy stull or reacher a put across at the bottom of the 
 station. The posts b are bolted against the posts of the sets and the cap c 
 placed on top of them. After this, the wall plates are cut out between the 
 posts b and the station opened and timbered as shown in the illustration. 
 The height of the station is gradually reduced to that of the drift or level 
 
 COI Fig Ct 23 ^epresents a method of timbering a level in a slope where the 
 ground is so firm that only stulls are employed m the slope and at the 
 station the timbers all being secured in hitches or by stulls. a represents 
 the stulls and c the timbers that are spiked to the stulls and carry the 
 stringers for the car track, b represents the car track from the level that is 
 
 brought across above the skip track. , . , 
 
 Special Forms of Supports.— Fig. 24 shows a stone arch which as a stull 
 RiiTYDorts the waste material in the level. . 
 
 Fig. 25 shows a stone arch when one wall of the formation requires 
 
 SU ^ig.*26 illustrates a passage lined by a combination of stone or brick walls 
 
 with wooden caps and lagging for the roof. ' _ _ - . 
 
 Fig 27 illustrates the lining of a drift or level supported by means of iron 
 or steel shapes bent into the form of an arch and employed for the support of 
 
 ^Fig^28 illustrates a cast-iron post or stull that has been successfully used 
 as a support in mines. It is composed of two pieces a and 5, held together 
 by a collar vC. By driving c down on the post, f |he two pieces can be taken 
 
 a ^^Fig a 29 illus?rates a masonry shaft lining supported by means of cast-iron 
 plates C set in bell-shaped cavities cut in the walls of the shaft. As the 
 masonry of a section from below is built up toward that above, the over- 
 hanging portion D is cut out a little at a time, and the masonry from below 
 built up under the plate so that the lining becomes continuous. 
 
 Fig 30 illustrates masonry shaft linings, supported by artificial stone or 
 cement foundations built in bell-shaped cavities cut m the walls of the shaft. 
 The blocks of artificial stone are provided with inclined bearings C, whicn 
 serve to transmit a portion of the downward thrust of the lining in the 
 
 direction^of the^a SuppQrts _ The use G f i r0 n or steel, either for vertical or 
 
 horizontal supports in mines, has not become at all general. In America, 
 timber is as yet comparatively cheap m most mining localities, but this situ- 
 ation is fast changing and the timber reserves are be ^g rapidly cut off, 
 so that many mines now using wood must, m the comparatively near 
 future, resort to some other form of support. Some of the disadvantages ot 
 metal supports are their greater initial cost, and on this account it is essen- 
 tial that all such supports should be recovered. As very little timber is 
 recovered in American mining, this objection is one toat ^1 proMbly co^ 
 tinue. The mine water is often of such a character that it v ill dissolve iron 
 or steel; particularly is this the case in copper mines, and ^ 
 where there is much pyrites. Metal mines keep their shaft sets open but a 
 short time compared with the pit bottoms of large coal mines, and hence 
 the extra cost of metal construction is frequently not warranted. The dis- 
 tricts in which metal mines are located are more likely to be disturbed than 
 is the ground over a coal mine, and if timbering is crushed, it is much easier 
 to repair than iron or steel. 'Another objection to metal supports is the fact 
 
 that they cannot be as easily framed and worked as timber. 
 
 On the other hand, the life of metal is, under ordinary circumstances, 
 much greater than that of timber, and while the inital cost may be greater, 
 whenever the metal can be recovered it can be used over and over again, 
 and it always has a certain value as scrap iron or steel. After a metal beam 
 has bent, it can still be used by simply turning ^ upside down Another 
 advantage for steel is that it occupies less space than timber or masonry, 
 and thus gives a larger effective area of roadway for the same cost of driving, 
 
 or else the amount of excavation may be reduced. . 
 
 Although metal has not been greatly used for props or upright supports, 
 it has been quite extensively used, both in America and abrosm, for sup- 
 porting shaft bottoms and landings, and m England it has been quite 
 successfully used for cross-bars in timbering roads, the bar tong set upon 
 wooden legs. In some of the European mines, a complete metal casing has 
 
MINE TIMBER AND TIMBERING. 
 
 273 
 
 been used. In locations where a constantly increasing pressure comes upon 
 the roof, an elastic bending material must be used, and in such case, soft 
 steel is greatly to be preferred to cast iron. 
 
274 
 
 MINE TIMBER AND TIMBERING. 
 
 Trestles.— Figs. 31 and 33 illustrate the various timbers and methods of 
 cutting the joints for ordinary railroad trestles. In Fig. 33 the portion (a) at 
 the left illustrates the manner of framing a pile trestle, while the portion ( b ) 
 at the right represents the manner of placing timbers and cutting the joint 
 for the framed trestle. Fig. 31 represents bents of a frame and pile trestle 
 
HEAD FRAMES. 
 
 275 
 
 and the side elevation of a low pile trestle. The various pieces in the figures 
 are numbered, and the accompanying table gives the names of the parts. 
 
 Fig 32 illustrates a bent of a frame trestle that is fastened together entirely 
 by means of drift bolts, no joints whatever being cut. 
 
 Figs 34 and 35 illustrate one manner of cutting the tenons and mortises 
 on the ends of the batter braces and posts and frame bents, and also the drain 
 holes that are bored in the mortise to prevent the timber from rotting. Usually 
 the sills are notched or boxed to receive the ends of the timbers, in addition 
 to having mortises formed in them. Figs. 36 and 37 show such joints for 
 receiving the batter brace and post. . 
 
 Fig 38 illustrates the manner in which a tenon is sometimes formed on 
 the top of the pile to secure the cap. When the cap is secured by a tenon, 
 the two are united by a wooden pin shown in the lower part of the figure, 
 and known as a treenail. 
 
 Fig. 39 illustrates a manner m which the cap may be placed upon a pile 
 trestle by splitting the cap into two pieces, a and b with the tenon c the full 
 width of the pile between them. . . . 
 
 Fig. 40 illustrates the manner in which the cap is sometimes secured to a 
 
 pile by means of a drift bolt, and Fig. 41 shows the manner in which the 
 same thing may be accomplished with the use of a dowel. 
 
 Figs. 42 and 44 show two methods of longitudinal bracing between the 
 bents of the trestles for inclined planes, such as are used at breakers or 
 concentrating mills. # _ , 
 
 Fig. 43 is an elevation of a high trestle, showing the cross-bracing and 
 framing of the structure. „ „ , 
 
 • Timber Head-Frames or Head-Gears— Fig. 45 is the simplest form of head- 
 gear, which consists of a vertical post to carry the weight of the sheave, etc., 
 and a diagonal post that approximately bisects the angle between the rope 
 from the drum and the rope hanging down the shaft, thus taking the 
 resultant pull upon the axle of the sheave. There is usually some extra 
 timbering, as shown, to support the cage guides and form a platform about 
 the sheave for convenience in oiling. 
 
 Fig. 46 shows a modified form of the same type of frame, in which the 
 main upright leg is vertical and in which there is also another vertical 
 
 Bent , Framed, 1. 
 
 Bent, Pile, 2. 
 
 Cap, 8. 
 
 Cross-Tie, 4. 
 
 Dapping , 5. 
 
 Gaining, see Dapping, 5. 
 Guard-Rail, 6. 
 
 Jack-Stnnger, 7. 
 
 Longitudinal Brace, 8. 
 
 Mortise, 9. 
 
 Mud Sill, 10. 
 
 Notching, Gaining, Dapping, 5. 
 
 Packing Block, 11. 
 Packing Bolts, 12. 
 
 Sill, 17. 
 
 Stringer, 18. 
 
 Sway-Brace, 19. 
 
 Tenon, 20. 
 
 Waling Strip , see Longitudinal 
 
 Brace, 8. 
 
276 
 
 MIKE TIMBER ASD TIMBERING. 
 
 member on the opposite side of the shaft. The inclined leg is also braced 
 and connected to the main vertical member. 
 
 Fig. 47 is a head-frame for an inclined shaft where the ore pocket is in the 
 structure carrying the sheaves. Such head-frames are sometimes enclosed 
 
 49 
 
 in their upper portions in a building so as to protect the men during winter. 
 
 Fig. 4S is a form ot framing quite common in the anthracite coal fields of 
 Pennsylvania, in which the timbers are further braced bv tie-rods, as shown. 
 
 Steel Shaft Bottoms.— Fig. 49 is a shaft bottom fitted with steel supports, 
 the posts being Z-bar columns and the caps being replaced by I beams, 
 which, in the station proper, are supported on stone or brick walls. This 
 metal construction is employed throughout all the portion of the bottom 
 landing and passages where the cars are handled after they are brought 
 from the workings or before they are returned to the workings. 
 
 Undersetting of" Props.— The following table, from Sawyers “Accidents in 
 Mines.” gives the maximum and minimum angles at which props should be 
 set for varying inclinations. This table can be taken as a general guide, 
 but it does n<7t take account of the length of prop nor the varying amounts 
 of movement of the top rock under different conditions. 
 
METHODS OF WORKING. 
 
 277 
 
 Undersetting of Props. 
 
 Rate of Inclination of 
 Seam. 
 
 Degrees. 
 
 Angle or Underset of Props. 
 
 Minimum 
 
 Degrees. 
 
 Maximum 
 
 Degrees. 
 
 6 
 
 0 
 
 1 
 
 12 
 
 0 
 
 2 
 
 18 
 
 J- 
 
 3 
 
 24 
 
 1 
 
 4 
 
 30 
 
 2 
 
 6 
 
 36 
 
 2 
 
 6 
 
 42 
 
 2 
 
 7 
 
 48 
 
 3 
 
 8 
 
 64 and upwards 
 
 3 
 
 9 
 
 METHODS OF WORKING. 
 
 No definite rules can be given for the selection of a method of mining 
 that will cover all the conditions that may exist at any given mine. Each 
 mine is a distinct and separate proposition, and each superintendent must 
 judge how he will adapt the general principles here given to the local 
 conditions at his own mine. Every system of mining aims to extract the 
 maximum amount of the deposit m the best marketable shape and at a 
 minimum cost and danger. 
 
 OPEN WORK. 
 
 Open work applies to the working of all deposits that have no overburden, 
 or to those in which the overburden or overlying material is stripped from 
 the portion of the deposit to be removed by hand, steam shovels, scrapers, 
 etc. It includes particularly all quarries and placer workings, and can be 
 applied to many mineral and coal deposits. 
 
 The advantages of this system are that no timber is required; unprofitable 
 underground workings do not have to be kept open and in repair; when 
 required, a simple hoisting plant is used; there is less danger to the work- 
 men from falls of roof and from blasting; there is practically no danger 
 from fire; artificial lights are not required; mining can Be done more 
 economically, as larger faces are open, larger blasts can be used, and the 
 amount of work accomplished per miner is greater, and better superintend- 
 ence can be had; the health of the men is usually much better when 
 working in the open; the deposits can be more easily extracted and the ore 
 more easily and more perfectly selected, and, under proper conditions, the 
 output can be increased almost indefinitely. 
 
 The disadvantages of open work are: A large amount of overburden often 
 has to be removed and a place for sorting this waste material provided; the 
 workmen are exposed to the weather; the expense of open work increases 
 rapidly with depth of covering. 
 
 Open work may be divided into two general classes: First, where the 
 whole or a greater part of the deposit is of value and has to be removed, as 
 in quarries and in ordinary mines; second , where the valuable portion is but 
 a small part of the whole, as in placers or fragmental deposits carrying gold, 
 platinum, etc. 
 
 Deposits of the first class may be worked as follows: (a) The deposit is 
 stripped, if necessary, and the material is removed by hoisting with derricks 
 or a cableway, or by drawing out in cars with the use of underground 
 passages. This class includes practically all quarries for building or orna- 
 mental stone, slate quarries, and most of the open-pit and steam-shovel 
 iron, phosphate, and similar mines. ( b ) The deposit is stripped, and drifts 
 or tunnels are extended through the material below the surface, either from 
 adjacent valleys or from shafts sunk outside of the deposit. The material, 
 
278 
 
 METHODS OF WORKING. 
 
 after being mined in the open pit, is thrown through openings to these drifts 
 or tunnels, through which it is trammed to the surface or to the foot of the 
 hoisting shaft. . „ . 
 
 Steam -shovel mines are those in which the material is, when necessary, 
 first shaken loose by big blasts of low-grade powder, and then loaded into 
 railroad cars with steam shovels, which lift the ore from its natural bed and 
 deposit it in cars to be taken directly to market or to a concentrating or 
 washing plant. Mining is thus done very cheaply, but the steam shovel, 
 from a mechanical standpoint, is not an economical machine and the costs 
 of repairs are high. T he expense for hauling material from the steam shovel 
 increases rapidly with adverse grades. Economy in steam-shovel mining 
 depends on the shovel being kept constantly at work. An output of 2,000 
 tons per day for a steam shovel and one locomotive has been reached and 
 even surpassed, but this cannot he taken as an average for a season s work. 
 Under favorable conditions, there is probably no cheaper method of mining 
 The cost of removing 97,854 yd. of material over a seam of anthracite coal 
 was 81 per ton of material stripped, and 80.516 per ton of coal obtained. The 
 average depth of the stripping was 75 ft. and about two-thirds of the 
 material removed was rock. The cost of stripping a bank 15 to 18 ft. high in 
 
 Western Pennsylvania was 80.30 per cu. yd. of stripping. 
 
 By milling system, the deposit is stripped, shafts are sunk outside of the 
 boundaries, and drifts are extended through the ore some distance from 
 the surface. From these drifts, raises are put up to serve as chutes, after 
 w r hich the material is simply blasted loose and worked into these raises, 
 through which it passes to the underground passages, and is trammed to the 
 shafts and hoisted to the surface. The advantages of this system over the 
 steam-shovel methods are: It is not necessary to make any long cut through 
 
 the overburden to bring the cars on the surface of the ore body. The 
 mining force can be employed underground in extending drifts and driving 
 new raises in bad or stormy weather. Very little handling of the material 
 by manual labor is required, the men simply working the loosened ore into 
 the chutes by means of bars or shovels, without having to lift any of it. 
 Some of the soft-ore iron mines have used this system very advantageously. 
 
 Cableways in Mining— Cableways are extensively used for stripping 
 deposits, for transporting material after it has been quarried, and also for 
 mining soft or loose deposits, such as clays, phosphates, and gravels, ine 
 cost of removing the overburden varies greatly with the nature of the 
 ground, and depends largely on the distance to which it is necessary to 
 carry the waste material before dumping. Frequently, a cableway can be 
 installed spanning both the place of mining and the dumping ground. In 
 other cases, one end of the cableway is fixed and attached to a washing or 
 gold-saving plant, while the other end revolves about this fixed point m a 
 circle until all of the material within this circumference has been exca- 
 vated; the entire plant is then moved to another location. The advantages 
 of cableways over steam shovels or dredges are that the load may be 
 delivered at a considerable distance from the point of excavation, while the 
 entire apparatus rests on banks entirely clear of the excavation. 
 
 Cablewavs have been constructed with single spans up to 1,650 ft., han- 
 dling 25-ton loads, and delivering an average daily capacity (10 hours) of 617 
 vd. of rock Mr. Spencer Miller places the following limitations on the 
 practical applications of cableways: Span (sdnglej^.OOO ft.; load, 2o tons, 
 
 speed of travel, 1,800 ft. per minute; speed of hoist, 900 ft. per minute. The 
 average practice, however, is about as follow’s: Span, 600 to 1.200 ft., loads, 
 3 to 7 tons, speed of travel, 500 to 1,000 ft. per minute; speed of hoist, loO to 
 
 300 ft. per minute. ... _ , f 
 
 Placer or fragmental deposits may be worked by means of a stream of water 
 from a pipe or nozzle directed against the bank (hydraulic mining), or the 
 material may be excavated by hand or by mechanical means, such as 
 dredges, steam shovels, etc. , , , . _ 
 
 Hydraulic Placer Mines.— The material is broken down by water Sowing 
 over the bank, as flume waterfalls, or along the ground so as to ground-sluice 
 the material. The material is frequently just loosened by means of picks or 
 shovels, the current being depended on to carry it away. This is commonly 
 called ground sluicing. Another method of excavating the matenaL is to 
 direct a stream of water against the bank from a pipe or nozzle. _ This is 
 true hydraulicking. After the material has been loosened by the water, it is 
 allowed to flow through sluices and over undercurrents or gold-saving 
 tables so as to recover the valuable portions. 
 
CLOSED WORK. 
 
 279 
 
 Placer Mines Worked by Mechanical Means— Where water is scarce, the 
 material may be excavated by steam shovels or other excavators, such as 
 grab, or scooping, buckets, operated by cableways. The material is then 
 washed by a limited supply of water that is frequently used over and over, 
 or it may be passed over a dry washer or concentrator. Both the steam 
 shovel and the cable excavator have proved very efficient means for 
 working certain classes of deposits. 
 
 Dredge Mining.— Where the gold-bearing material lies below the water 
 level, or where water can be introduced so as to float a boat, a dredge may 
 be employed. > 
 
 For gold dredging, a dredge should fulfil the following conditions: 
 (1) Speed and readiness in moving and taking up different positions; (2) an 
 adaptability for cleaning up rock, and for digging to a maximum depth; 
 (3) feasibility of working and of the banking or disposing of the tail- 
 ings; (4) cheapness in working, as most of the dredging propositions are of 
 low grade. , , ,, 
 
 Three types of dredges have been used: the hydraulic suction dredge, the 
 shovel dredge, and the continuous-bucket ladder dredge. The first of these 
 is well adapted for digging very small gravel and sand, but is not suited for 
 boulders or even large stones without a great loss of efficiency. The con- 
 tinuous-bucket dredge has proved the most successful under ordinary cir- 
 cumstances, as it is controlled by lines and not by spiles or spuds, and hence 
 can be shifted more rapidly and made to conform to irregularities m the bed 
 rock more readily than the dipper type. Also, the continuous-bucket 
 dredge furnishes a constant supply of material to the apparatus used for 
 recovering the gold. „ ^ 
 
 A continuous-bucket dredge can operate to a depth of 60 ft. According 
 to Mr. R. H. Postlethwaite’, of San Francisco, Cal., a decided advocate of 
 continuous-bucket dredges, the cost of working a shovel dredge runs from 
 7 cents per cubic yard upwards; but he claims that the bucket dredge can be 
 worked at a cost of from 3 to 5 cents per cubic yard, including a charge of 
 $100 per week for depreciation. The cost of running a small gold dredger 
 should not average over $200per week— that is allowing $125 for wages, $50 
 for fuel, and $25 for repairs, etc. If the dredger handles 10,000 cu. yd. per 
 week, that would be at a cost of 2 cents per cubic yard. If the material 
 averaged 6 cents per cubic yard, there should be an approximate profit of 
 $400 per week on an investment of from $25,000 to $40,000. 18 cu. ft. of gravel 
 in place will weigh 2,000 lb.; a cubic yard will weigh li short tons. 
 
 4 CLOSED WORK. 
 
 Under this general heading it is customary to divide the methods of 
 mining into coal-mining methods and metal-mining methods. This classifi- 
 cation is not entirely logical, for identical methods are applied to nat bedded 
 deposits of coal, iron ore, clay, salt, etc., and identical or very similar 
 methods to highly inclined coal seams and mineral veins. A more logical 
 classification is one based on the position, character, and thickness of the 
 deposit, but the older classification has become so firmly established that it 
 is not advisable to give it up entirely in a pocketbook. 
 
 Bedded Deposits.— The typical and most extensive bedded mineral .deposits 
 are of coal and iron ore, and of these the former is by far the more extensively 
 mined. A description of the several methods of mining coal beds will 
 therefore comprise not only all of the essential points in the mining of other 
 bedded deposits, but will include a number of points not usually considered 
 in mining such deposits. The chief of these is the presence of explosiwe gas 
 in such quantities as to influence the choice of a method of mining, r rom 
 the descriptions of the methods of coal mining here given it will therefore 
 be a comparatively simple matter for the miner of clay, iron ore, etc. to 
 adapt a method. 
 
 COAL MINING. 
 
 General Considerations.— The elementary causes affecting the extraction of 
 coal are (1) weight of overlying strata or depth of the deposit; (2) strength 
 and character of roof; (3) character of floor; (4) texture of bedded material; 
 (5J inclination and thickness of bed; (6) presence of gas in the seam or in 
 adjoining strata. 
 
280 
 
 METHODS OF WORKING. 
 
 Roof Pressure.— Of these causes, the roof pressure is the most impOTtant, 
 and a number of the other causes are directly affected by it. The weight of 
 the overlying cover will give a maximum roof pressure, but this may be so 
 variously modified that the determination of the actual pressure is practically 
 impossible, and estimates of this pressure must be based largely on practical 
 experience; hence, rules for its calculation are of comparatively little value. 
 One very essential point, however, must be borne in mind, 1 . e., that the 
 direction of pressure is perpendicular to the bedding plane. 
 
 Strength and Character of Roof.— The strength of roof refers to the power of 
 being self-supporting over smaller or larger areas. A strong roof permits 
 larger openings, but increases the load on the pillars, thereby necessitating 
 larger pillars. A weak roof requires smaller openings, and permits smaller 
 
 The character of floor influences largely the size of pillars. A soft bottom 
 requires large pillars and narrow openings, especially when the roof is strong. 
 
 Texture of Coal and Inclination and Thickness of Seam.— Soft, friable coal 
 requires large pillars, while a hard, compact coal requires only small pillars. 
 The inclination and thickness of the deposit increase the size of pillars 
 required, and also influence the haulage, drainage, timbering, method of 
 working, arrangement of breasts, etc. 
 
 The presence of gas in the seam or in the enclosing strata affects the system 
 of working, as ample air passages must be provided, and provision must fre- 
 quently be made for ventilating separately the different sections of the mine. 
 Where the gas pressure is strong, and outbursts are of frequent occurrence, 
 narrow openings are necessitated that render the workings safe until the 
 gas has escaped. ___ 
 
 SYSTEMS OF WORKING COAL. 
 
 There are two general systems of working coal seams: (1) room-and-pillar, 
 
 and (2) longwall. There 
 are, however, a great num- 
 ber of modifications of 
 each, and it is often diffi- 
 cult to exactly classify a 
 given method under either 
 of these two systems. 
 
 The room-and-pillar sys- 
 te/n, also known as the pil- 
 lar-and-chamber or bord- 
 and-pillar. and which may 
 include the pillar-and-stall , 
 svstem, is the oldest of the 
 systems, and the one very 
 generally used in the mines 
 of the United States. By 
 this system, coal is first 
 mined from a number of 
 comparatively small places 
 called rooms, chambers, 
 stalls, bords . etc., which are 
 driven either square from 
 or at an angle to the haul- 
 ageway. These openings 
 may be wide or narrow, 
 and may be either a road- 
 way, incline, or chute, 
 according to existing con- 
 ditions. The pillars that 
 are left between the open- 
 ings in the original work- 
 ings support the roof, and 
 are usually subsequently 
 removed. All forms of 
 
 room-and-pillar workings become impracticable when the ^l ck ^y e f ^ 
 pillars necessary to support the roof pressure much exceeds aouDie me 
 width of the breast openings. 
 
LONG WALL METHOD. 
 
 281 
 
 The pillar-and-stall system is similar to the room-and-pillar system, but in 
 the former the stalls are opened off from the entry their full width, while in 
 the latter the rooms or chambers are turned narrow, and widened inside to 
 their regular width. Fig. 1 shows a typical room-and-pillar method for 
 working an approximately horizontal seam of coal of moderate thickness 
 (4 to 10 ft.), and with a fairly good roof and bottom. Main headings A are 
 usually driven perpendicular to the strike, unless this direction is changed 
 by the cleat in the coal, as explained later. Cross-headings, or entries B, B, 
 are turned off at regular intervals, and at an angle of 90 to these main 
 headings, the distance between any two pairs of cross-entries being deter- 
 mined, in flat seams, by twice the length to which a room can be driven, 
 which in turn is determined by the character of the roof, floor, and seam. 
 The rooms are turned to the right and left of each pair of butt headings, and 
 driven until they meet, or one-half the distance between two pairs of entries. 
 After the rooms are driven up, the pillars between the rooms are drawn, and 
 later the room stumps along these entries, and the entry pillars themsel ves, 
 are drawn, unless it should be necessary to keep some of these cross-entries 
 open for purposes of ventilation. A large chain pillar is left to protect the 
 
 main headings. , , 
 
 When cross-entries have been extended a considerable distance, roads are 
 often driven between them parallel to the main heading A. The object of 
 these subroads is to reduce to a minimum the air-courses and roadways to be 
 maintained; or such a subroad may be necessary on account of a squeeze 
 crossing any pair of cross-entries. The extent of the territory worked out to 
 the right and left of each main entry is a matter for local determination. 
 
 The room openings are made suitable to prevailing conditions, and Fig. 1 
 shows several of the common methods. The width of the room and the form 
 of the opening depend on the character of the roof and the extent to which 
 it is necessary to leave a pillar to support the cross-heading, it being advan- 
 tageous, of course, to open out the room to its full width at the earliest 
 
 PO TongwanMethod of Mining.-The longwall system contemplates the extrac- 
 tion of the entire seam or bed, and the original significance of the term 
 “longwall” was a continuous line of breast. No portion of the seam is 
 allowed to remain after leaving the vicinity of the shaft. The method 
 depends on producing a uniform and gradual settlement of the root a tew 
 yards behind the working face. Pack walls are built on each side of the 
 roadways, and at regular intervals in the gob or waste area, and the root 
 settles firmly on these packs, pressing them into the bottom, or compressing 
 them until the roof subsidence is complete. The heignt of the main road- 
 way is maintained by “brushing” the roof or lifting the bottom. Longwall 
 may be advancing or retreating . In longwall advancing, mining begins at oi 
 near the foot of the shaft and advances outwards forming a gradually 
 widening and increasing length of face to the boundary. The passages are 
 made through the excavated portions of the mine, and are maintained b> 
 pack walls built either of the refuse secured in mining or sometimes from 
 material brought in from the surface. In longwall ^fating or withdraw- 
 ing, entries, gangways, or headings are first driven to the boundary or to 
 other convenient distances inbye, and the pillars between these entries are 
 then drawn back toward the shaft; this is also called working home. 
 
 Fig. 2 shows a plan of combined longwall advancing and retreating, in 
 the upper arrangement, or Scotch longwall, the face is semicircular and the 
 roadsTe turned off at angles of 45° This plan issuitablefor seams up to 
 3 ft. thick with a weak top, and which pitch less than 20 , and situated at 
 almost any depth. It is the one from which most of the longwall practice 
 in the central coal basins of the United States is taken. In the lower ^por- 
 tion, which shows one method of longwall retreating, narrow Parallel 
 headings are driven in pairs to the boundary, being from 200 to 300 ft. apart. 
 Such a combination of longwall advancing and retreating insures an 
 unvarying supply of coal, for while one side continually leaves the shaft, the 
 
 Longwall is specially adapted to flat seams or those having a regular and 
 moderate pitch, and which are free from faults; to hard rather than to soft 
 coal; and to the use of undercutting machines. It us also easy of venti- 
 lation and economical in timber, explosives, and trank. 
 
 In all longwall work, the w r eight of the roof is made to act upon the coal 
 face, which is undercut, according to the ability of the miner and the con- 
 ditions of the seam, from 2£ to 3 ft. deep. This weighing action of the roof 
 
282 
 
 METHODS OF WORKING. 
 
 performs the work of the powder and breaks the coal in a few hours or less, 
 after the sprags have been removed. The time required to break the coal 
 will be greater or less, according to conditions. The success of the entire 
 system, as will be readily seen, depends on a uniformly regular advance of 
 the line of the face or breast, and a uniform system of setting and drawing 
 timber at the face; also, uniformity in the pack walls along the roads, and in 
 the amount of gob packing. This system does not permit of long idle spells 
 induced by strikes or other causes. The coal does not break well, either, 
 when some of the men are out a portion of the time, and their places lie idle. 
 The life of the whole system consists in maintaining what miners call a 
 
 Fro. 2. 
 
 traveling weight upon the coal face, which can only be accomplished satis- 
 factorily by uniformity in every part of the work. 
 
 Longwall advancing is better suited to thin seams than to thick ones, to 
 flat rather than pitching, and to good roofs and hard floors. 
 
 Longwall retreating is better adapted to thicker beds; to those liable to gob 
 fires; to seams of hard coal having a considerable pitch; to pockety, or 
 irregular seams; and to a soft and treacherous top. The air-course is also less 
 broken along the face, and better haulage installations can be made. Its 
 chief disadvantage is the large amount of dead work necessitated before 
 returns are received. With this system there is no expense in keeping up 
 the haulage road so far as creep or falling roof is concerned, as the roads are 
 all in solid coal, nor is there any trouble from gob fires or water; and little 
 
STARTING LONGWALL. 
 
 283 
 
 detriment to the working face is caused by the mine having to stand idle 
 for a time. If the seam is high enough for the mules . hors es> jo rock 
 whatever will need to be taken down. The coal seam will be proved before 
 
 ^The Ventilation ^n the retreating plan is as near perfect as it is possible to 
 get it in practice. All the airways are tight, a thing impossible to get in the 
 advancing plan; and it is a comparatively easy matter to shut off hre or to 
 allow a portion of the working face to remain idle. o ,. e 
 
 Longwall retreating is frequently used for working quite limited sections 
 of a mine in which the seam of coal is 16 to 20 tt. thick, aQ d inclined not 
 more than 10°. A series of 8 or 10 pairs of headings are turned off the butt 
 entry and driven a distance, dependent on local conditions where the 
 working face is formed by driving cross-cuts from one to th e° tb er* 
 bice is carried back on the retreating plan, allowing the r ^ of ^9 or 
 
 settle on the gob as the work approaches the butt entry. J 1 * way, any 
 extra weight that would crush and rum the adjacent coal is avoided. This 
 method is also used in lower seams in which the coal is soft, <3r ^eroof, or 
 bottom, or both, are of such a nature as to give trouble m working the ^oom- 
 and-pillar system. Sometimes, instead of driving pairs of headings at 
 considerable 7 distances apart, a number of single headmgs are dri ven com- 
 paratively close together, and connected by cross-cuts froni 1C (to 20 yd. 
 apart When the limit of the section is reached, the working face is formed 
 and carried back, as in the other plan. This latter method is more suita- 
 ble for tender roof, or a coal in which the face and butt cleats are not 
 
 Pr °S?artine Longwall. — There are two methods of starting longwall workings. 
 In the first, the work of extraction begins at the shaft itself, the coal being 
 taken out all around and its place filled with solid packs, leaving only space 
 for the roadways. In the second method, a pillar of solid coal is 
 port the shaft, cut only by the roadways. The longwall work is then 
 started uniformly all around this pillar. Great care is n ee( i e( i iri build, 
 the first pack walls around the shaft pillar, to see that they are 
 and well rammed, in order to break the roof over the coal. The system will 
 not work rightly, however, until the breast has been advanced some dis- 
 tance from the pillar, so as to secure _ the benefit from the weighing action of 
 the roof upon the coal face. The mining will be more difficult in the start, 
 and in some exceptional cases it may even be necessary to place some light 
 shots; which, however, should be avoided, if possible. . . 
 
 The panel system divides a mine into districts or panels by driving entries 
 and cross-entries so as taintersect one another at regular intervals of, usually, 
 about 100 yd. Large pmars are left surrounding the workings within each 
 panel, and any method of development may be used for each panel. This 
 system presents the following advantages: (1) Better control of the ventila- 
 
 tion, since the air in any panel may be temporarily increased or decreased, 
 as required. An explosion occurring in one panel is less liable to afreet the 
 other workings. (2) Coal may be extracted, pillars drawn, and the panels 
 closed and sealed off independently of each other. (8) Greater security is 
 afforded against creep and squeeze. (4) Coal that disintegrates on standing 
 can be quickly worked out. , _ x 
 
 Bearing In, or Undercutting.— In any method of mining where the coal is 
 undermined, advantage should be taken of the roof pressure to assist m 
 both breaking down the coal and also in bearing in. The fact is often 
 overlooked that the roof pressure upon the face coal makes it brittle and 
 more susceptible to the pick, and the good miner starts a shallow mining in 
 the under clay, or lower coal, and carries it the entire width of the face. 
 Bv the time he returns to the side of the breast at which he started, the roof 
 pressure has made the coal more tender and susceptible to the pick, buch a 
 gradual system of mining throws the pressure on the coal face gradually, 
 and the coal breaks in larger pieces. The depth of the undercut depends 
 on the thickness of the seam and the other conditions. Undercutting by 
 mining machines is rapidly replacing hand work wherever these machines 
 
 Ca ^Bunffing^.’Pack Walls, and Stowing.— Pack walls should be built large enough 
 at first and kept well up to the face, to prevent the weight coming upon the 
 timber and also to permit the roof to settle rapidly when the timber is 
 taken out of the face. Often the roof will not stand this 
 without breaking, and possibly closing in the entire face. The face should 
 therefore be kept in shape, and just as soon as there is room for a prop or 
 
284 
 
 METHODS OF WORKING. 
 
 chock, it should be put in immediately, and the pack walls likewise should 
 be extended after each cut or web is loaded out. 
 
 As a general thing, the pack walls in the gob are not so wide as the road- 
 side ones, particularly when the seam produces enough waste material 
 to stow the “marches,” “ cundies,” or “gobs,” between these pack walls. 
 Usually about 50/c of the cubical contents of the solid seam taken out will 
 stow the spaces between the pack walls in thick pitching seams, where the 
 entire gob must be completely filled or nearly so. No waste material, except 
 such as will hasten spontaneous combustion, should be taken out of the 
 mine to the surface. 
 
 Timbering a Longwall Face.— The method of timbering the working face 
 depends on the nature of the roof, floor, coal, etc. The action of the roof on 
 the coal face is regulated almost entirely by timber; consequently, when the 
 coal is of such a nature as to require little weight to make it mine easily, the 
 roof must be timbered with rows of chocks and, if necessary, a few props. 
 
 Control of Roof Pressure.— The working face of a longwall working should 
 advance up grade, but this face cannot always be kept parallel with the 
 strike. When the angle at the line of face, made with the line of strike, is 
 less than 90°, the greater pressure of the covering rocks is thrown on the 
 gob, and, when this angle is more than 90°, the greater pressure comes on 
 the coal. The angle made by the working face with the line of pitch varies 
 inversely as the vertical angle of pitch, or for a high pitch this angle is 
 small and for a low pitch it is large. Where longwall is worked in adjacent 
 sections, care must be taken to prevent the advancing of one section throw- 
 ing a crushing weight on any of the others, and thus producing a crush or 
 an uncontrollable cave. Where the rocks are pitching* and a greater portion 
 of the cracks that cut them run in lines parallel to the strike, neither stone 
 nor timber can efficiently support the roof, which frequently breaks off close 
 to the working face. 
 
 The ends of all stone packs nearest the face of the coal should be in line, 
 and the ends of these pack walls should form a line parallel to the face of 
 the coal. Timbers set at equal distances and in 
 line along a longwall face are much more efficient 
 in supporting the roof than irregularly set tim- 
 bers. Fig. 3 shows the proper way of locating 
 the pack walls and the face timber. 
 
 Number of Entries.— The entries in a mine may 
 be driven single , double , triple, etc. 
 
 The single-entry system is only advisable under 
 certain conditions and for short distances, since 
 the ventilation must be maintained along the 
 face of the rooms, and there is but one haUlage- 
 way, which may easily be closed by a fall or creep. Rooms are turned off 
 one or both sides of the entry. 
 
 The double-entry system is most commonly used. Two parallel entries 
 are driven, separated by an entry pillar whose thickness varies with the 
 depth of the seam, and connected at intervals of about 20 yd. by cross-cuts 
 or breakthroughs to maintain ventilation. 
 
 The triple-entry system is used particularly in very gaseous seams requir- 
 ing separate return airways; or, at times, in mines where the large output 
 requires ample haulage roads. It is usually applied to the main entries only, 
 but sometimes, also, to the cross-entries. In gaseous mines, the middle 
 entry is usually made the haulage road and intake airway, and the outside 
 entries the return air-courses for either side of the mine, respectively. 
 
 A still larger number of entries even has been suggested for deep 
 .'workings where it is difficult to keep open broad passages, but these have 
 not been generally adopted or tried experimentally to any great extent. 
 
 Direction of the Face. — The typical room-andqflllar plan, Fig. 1, shows the 
 main headings and the rooms driven parallel to the direction of the dip, and 
 the cross-headings parallel to the strike, but in most coal seams there are 
 vertical cleavages, called cleats, which cross the coal in two directions about 
 at right angles to each other. Face cleats, as they are called, are the more 
 pronounced, while the end or butt cleats are the shorter, less pronounced 
 joints. The direction of the face with respect to the cleats is of prime 
 importance as greatly facilitating or retarding the mining of the coal. 
 
 Fig. 4 shows the different positions that the face may occupy with respect 
 to the direction of the cleats. The angle of the breast depends on the hard- 
 ness of the coal and freedom of the cleats, and each method has its peculiar 
 
 Fig. 3. 
 
PILLARS. 
 
 285 
 
 adaptation to the varying conditions of a coal seam. When the face cleats 
 are working free and the coal is very soft , it may be necessary to drive “ end 
 on.” The end-on method is best adapted to a very heavy roof pressure, 
 while for a light roof pressure the short-horn method assists in breaking the 
 coal. If the “ face ” cleats are free and the coal breaks readily along them, 
 and it is reasonably hard , the long-horn method is adopted, for when the 
 coal is undercut it needs more support than it gets from the cleats, and its 
 weight must be thrown somewhat upon the end cleats. “ Face on” is 
 adopted when the face cleats are not as free or numerous as the butt cleats. 
 Unless the coal at the face re- 
 ceives sufficient support, the 
 undercutting or bearing in can- 
 not be thoroughly done, or else 
 the amount of spragging and 
 the risk to the miner are in- 
 creased. When the end cleats 
 are less pronounced and nu- 
 merous, and the roof pressure 
 reat, the coal will probably 
 reak better by carrying wide 
 breasts upon the ends of the 
 coal, and it is then an advan- 
 tage to drive double rooms with 
 large pillars between them. In pitching seams, the pillar should have very 
 long sides perpendicular to the strike, if the principal cleats in the coal 
 are parallel to the strike, or nearly so. , 
 
 The short-horn method is adapted to heavy roof pressure and wide room 
 pillars, as the face cleats are here quite pronounced, and the pillars between 
 the rooms thereby weakened to a large extent; hence, wide pillars are more 
 often employed when working on the ends of the coal. When the face 
 cleats are less pronounced, and the end cleats are working freely, a good 
 breast of coal is carried on the face, and, unless other conditions require it, 
 a great width of room pillars is not needed. If this can be done consistently, 
 and good lump coal secured at the same time, the room should cross the pitch 
 as little as possible, as a side pressure upon the pillars having very long sides 
 running diagonally across the pitch is destructive. 
 
 PILLARS. 
 
 Size of Pillars.— It is impossible to give exact rules or formulas for deter- 
 mining the proper size of pillars. Each case in practice requires special 
 consideration, and in laying out the pillars in a virgin field it is well to find 
 out what the current practice is in similar fields. In general, the thicker 
 the seam and the greater its depth from the surface, the greater should be 
 the thickness of the pillar. Some coal deteriorates rapidly when subject to 
 weight and to the disintegrating effect of the atmosphere, and pillars of 
 such coal must be larger than when composed of a hard, compact coal. 
 Permanent pillars, or those that are to remain for a considerable length of 
 time, must be larger than those that are to be promptly removed. Pillars 
 about the bottom of a shaft, or along main haulage roads, should be left large 
 enough to provide for increasing developments, for when landings are 
 enlarged or when haulage systems are introduced, the original pillars fre- 
 quently have to be reduced in size by taking a skip off of them or by taking 
 out chambers for engines and pumps. . 
 
 Shaft Pillars.— Various formulas have been given to determine the size of 
 shaft pillars, and the results given by these several formulas are very diverse. 
 
 Merivale. — S = X 22, where S equals the length of the side of the 
 
 pillar in yards, and D equals depth of shaft in fathoms. 
 
 Andre— Up to 150 yd. depth, have the pillar 35 yd. square, and for greater 
 depths increase 5 yd. on each side for every 25 yd. of increased depth. 
 
 Dron. — Draw lines enclosing all surface buildings that it is necessary to 
 erect about the head of the shaft, and make the shaft pillar so that solid 
 coal will be left outside these lines all around for a distance equal to one- 
 third the depth of the shaft. 
 
 Wardle.— Shaft pillars should not be less than 40 yd. square down to a 
 depth of 60 fathoms, and should increase 10 yd. on a side for every 20 
 fathoms increase in depth. 
 
 Fig. 4. 
 
286 
 
 METHODS OF WORKING. 
 
 Hughes. — Leave 1 yd. in width of pillar for every yard in depth of shaft. 
 
 Pa mely.— Allow a pillar 40 yd. square for any depth up to 100 yd.; for 
 greater depths, increase the pillar 5 yd. for every 20 yd. in depth. 
 
 Calculating the size of pillar from each of these authorities, we find the 
 
 Authority. 
 
 For Shaft 300 Ft. Deep. 
 
 For Shaft 600 Ft. Deep. 
 
 Merivale 
 
 4ndre 
 
 22 yd. square. 
 35 yd. square. 
 
 31 yd. square. 
 45 yd. square. 
 
 Wardle 
 
 Pamely 
 
 Dron 
 
 Hughes 
 
 40 yd. square. 
 
 40 yd. square. 
 33i yd. square.* 
 100 yd. diameter. 
 
 60 yd. square. 
 
 65 yd. square. 
 66f yd. square.* 
 200 yd. diameter. 
 
 •Outside of buildings. 
 
 None of these formulas takes account of the thickness of the seam, and 
 the following formula, which takes account of this veiy important element, 
 was suggested by Mr. R. J. Foster, in “ Mine s and Minerals . 
 
 Radius of pillar = Z\/ D X U 
 in which D = depth of shaft; t = thickness of seam. 
 
 Pitching seams require smaller pillars on the low side than on the rising 
 
 Sid RooV^PM Tars.— The relative width of pillar and breast is dependent on 
 the weight of cover, as compared with the character of the roof and floor, 
 and the crushing strength of the coal. These relative widths are deter- 
 mined largelv by practice. Speaking generally, the narrower the rooms 
 or chambers. ‘the higher the cost in yardage, the greater the production of 
 slack and nut coal, the greater the consumption of powder, track iron, ties, 
 etc., and the greater the cost of dead work. , 
 
 For bituminous coal of medium hardness and good roof and floor, a rale 
 often used is to make the thickness of room pillars equal to 1* of the 
 depth of cover for each foot of thickness of the seam, according to the 
 
 expression W p = ^ X A in which W p = pillar width; t = thickness of 
 
 seam- D = depth of cover, and then make the width of breast or opening 
 equal’ to the depth of cover divided by the width of pillar thus tound, 
 
 according to the expression W 0 = where W 0 = width of room. 
 
 Frail coal and coal that disintegrates readily when exposed to the air, and 
 a soft bottom, may increase the width of pillar required as much as o0* of 
 the amount found above; also, a hard roof may increase the same as much 
 as 254; while on the other hand, a frail roof or a hard coal or fl(x>r may 
 reduce the width of pillar required 25*. The hardness of the roof affects 
 both the width of pillar and width of opemng alike, which is not the case 
 with any of the other factors. 
 
 Dunn’s Tables of Size of Room Pillars for Various Depths. 
 
 The following table is for first working, with the design of afterwards 
 taking out the pillars, the width of the principal workings being 5 yd., and 
 
 Depth. 
 
 Feet. 
 
 Size of 
 Pillars. 
 Yards. 
 
 Proportion 
 
 in 
 
 Pillars. 
 
 Depth. 
 
 Feet. 
 
 Size of 
 Pillars. 
 Yards. 
 
 Proportion 
 
 in 
 
 Pillars. 
 
 120 
 
 20 X 5 
 
 .41 
 
 1,080 
 
 26 X 14 
 
 .69 
 
 240 
 
 20 X 6 
 
 .50 
 
 1,200 
 
 26 X 16 
 
 .71 
 
 360 
 
 22 X 7 
 
 .52 
 
 1,320 
 
 28 X 18 
 
 .73 
 
 480 
 
 22 X 8 
 
 .57 
 
 1,440 
 
 28 X 20 
 
 .75 
 
 600 
 
 22 X 9 
 
 .59 
 
 1,560 
 
 30 X 21 
 
 .77 
 
 720 
 
 22 X 12 
 
 .61 
 
 1,680 
 
 30 X 22i 
 
 .78 
 
 840 
 
 26 X 15 
 
 .63 
 
 1,800 
 
 30 X 24 
 
 .79 
 
 960 
 
 28 X 16 
 
 .66 
 
 
 
 
ROOM PILLARS. 
 
 287 
 
 Extremely large pillars must often be left as a precautionary measure to 
 protect permanent haulageways and surface buildings, or to avoid any pos- 
 sibility of a break in the roof that would cause an inflow of water. 
 
 Table Showing Distance From Center to Center of Breasts or 
 Chambers Measured on the Entry or Gangway, for 
 Different Angles. 
 
 A I to 
 
 I B 
 
 © © 
 0 ^© 
 
 Distance Measured on the Entry in Feet, When Width of 
 Breast + Width of Chamber Is: 
 
 03 r ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h q; f-i . 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 65 
 
 60 
 
 65 
 
 70 
 
 75 
 
 90 
 
 20.0 
 
 25.0 
 
 30.0 
 
 35.0 
 
 40.0 
 
 45.0 
 
 50.0 
 
 ■ 
 
 55.0 
 
 60.0 
 
 65.0 
 
 70.0 
 
 75.0 
 
 85 
 
 20.0 
 
 25.1 
 
 30.1 
 
 35.1 
 
 40.2 
 
 45.2 
 
 50.1 
 
 55.2 
 
 60.2 
 
 65.3 
 
 70.3 
 
 75.3 
 
 80 
 
 20.3 
 
 25.4 
 
 30.5 
 
 35.5 
 
 40.6 
 
 45.7 
 
 50.6 
 
 55.8 
 
 60.9 
 
 66.0 
 
 71.1 
 
 76.2 
 
 75 
 
 20.7 
 
 25.9 
 
 31.1 
 
 36.2 
 
 41.4 
 
 46.6 
 
 51.2 
 
 56.9 
 
 62.1 
 
 67.3 
 
 72.5 
 
 77.7 
 
 70 
 
 21.2 
 
 26.6 
 
 31.9 
 
 37.2 
 
 42.6 
 
 47.9 
 
 53.1 
 
 58.5 
 
 63.9 
 
 69.2 
 
 74.5 
 
 79.8 
 
 65 
 
 22.0 
 
 27.6 
 
 33.1 
 
 38.6 
 
 44.1 
 
 49.6 
 
 55.1 
 
 60.7 
 
 66.2 
 
 7L7 
 
 77.2 
 
 82.8 
 
 60 
 
 23.0 
 
 28.9 
 
 34.6 
 
 40.4 
 
 46.2 
 
 52.0 
 
 57.6 
 
 63.5 
 
 69.3 
 
 75.1 
 
 80.8 
 
 86.6 
 
 55 
 
 24.4 
 
 30.5 
 
 36.6 
 
 42.7 
 
 48.8 
 
 54.9 
 
 60.9 
 
 67.1 
 
 73.3 
 
 79.4 
 
 85.5 
 
 91.6 
 
 50 
 
 25.8 
 
 32.6 
 
 39.2 
 
 45.7 
 
 52.2 
 
 58.7 
 
 65.1 
 
 71.8 
 
 78.3 
 
 84.9 
 
 91.4 
 
 97.9 
 
 45 
 
 28.2 
 
 35.4 
 
 42.4 
 
 49.5 
 
 56.6 
 
 63.6 
 
 ^0.6 
 
 77.8 
 
 84.9 
 
 91.9 
 
 99.0 
 
 106.1 
 
 40 
 
 31.1 
 
 38.9 
 
 46.7 
 
 54.5 
 
 62.2 
 
 70.0 
 
 77.6 
 
 85.6 
 
 93.4 
 
 101.2 
 
 109.0 
 
 116.7 
 
 35 
 
 34.9 
 
 43.6 
 
 52.3 
 
 61.0 
 
 69.7 
 
 78.5 
 
 87.0 
 
 95.9 
 
 104.6 
 
 113.4 
 
 122.1 
 
 130.8 
 
 30 
 
 40.0 
 
 50.0 
 
 60.0 
 
 70.0 
 
 80.0 
 
 90.0 
 
 100.0 
 
 110.0 
 
 120.0 
 
 130.0 
 
 140.0 
 
 150.0 
 
 25 
 
 47.3 
 
 59.2 
 
 71.0 
 
 82.8 
 
 94.6 
 
 106.5 
 
 118.1 
 
 127.2 
 
 142.0 
 
 153.8 
 
 165.7 
 
 177.5 
 
 20 
 
 58.5 
 
 73.1 
 
 87.7 
 
 102.4 
 
 117.0 
 
 131.6 
 
 145.9 
 
 160.8 
 
 175.5 
 
 190.1 
 
 204.7 
 
 219.3 
 
 15 
 
 77.4 
 
 96.6 
 
 115.9 
 
 135.3 
 
 154.5 
 
 ]73.9 
 
 192.8 
 
 212.5 
 
 231.9 
 
 251.2 
 
 270.5 
 
 289.8 
 
 10 
 
 115.2 
 
 144.0 
 
 172.8 
 
 201.6 
 
 230.4 
 
 259.2 
 
 287.3 
 
 316.7 
 
 345.6 
 
 374.4 
 
 403.1 
 
 432.0 
 
 5 
 
 229.5 
 
 286.9 
 
 344.2 
 
 401.6 
 
 459.0 
 
 516.3 
 
 572.5 
 
 631.1 
 
 688.5 
 
 745.8 
 
 803.2 
 
 860.5 
 
 In the following table, the weight thrown upon pillars at different depths 
 by the removal of different proportions of coal is given: 
 
 Weight on Pillars in Pounds per Square Inch. 
 
 Percentage of Coal Left in Pillars. 
 
 Depth 
 
 Sean 
 
 Feel 
 
 90/. 
 
 80/ 
 
 70/ 
 
 60/ 
 
 50/ 
 
 40/ 
 
 30/ 
 
 20/ 
 
 10/ 
 
 100 
 
 Ill 
 
 125 
 
 142 
 
 166 
 
 200 
 
 250 
 
 333 
 
 500 
 
 1,000 
 
 500 
 
 555 
 
 625 
 
 710 
 
 830 
 
 1,000 
 
 1,250 
 
 1,665 
 
 2,500 
 
 5,000 
 
 1,000 
 
 1,111 
 
 1,250 
 
 1,428 
 
 1,666 
 
 2,000 
 
 2,500 
 
 3,333 
 
 5,000 
 
 10,000 
 
 1,500 
 
 2,000 
 
 3.000 
 
 4.000 
 
 5.000 
 10,000 
 
 1,666 
 
 2,222 
 
 3,333 
 
 4,444 
 
 5,555 
 
 11,110 
 
 1,875 
 
 2,500 
 
 3,750 
 
 5,000 
 
 6,250 
 
 12,500 
 
 2,138 
 
 2,956 
 
 4,384 
 
 5,912 
 
 7,340 
 
 2,496 
 
 3,333 
 
 4,999 
 
 6,666 
 
 3.000 
 
 4.000 
 
 6.000 
 8,000 
 
 3,750 
 
 5,000 
 
 7,500 
 
 4,998 
 
 6,666 
 
 7,500 
 
 15,000 
 
 Chain and barrier pillars vary in size even more than shaft pillars, and their 
 widths are almost entirely determined by local considerations. In some 
 States, the minimum width of barrier pillars is regulated by law. 
 
 Barrier Pillars. — For finding the width of barrier pillars in anthracite 
 seams, the following formula, adopted conjointly by the chief mining engi- 
 neers of the Lehigh & Wilkes-Barre Coal Co., Susquehanna Coal Co., D., L. & 
 W. R. R. Co., Delaware & Hudson Canal Co., and the State mine inspectors 
 of Eastern Pennsylvania, is recommended: 
 
 Formula for width of barrier pillars: (Thickness of workings X 1 / of depth 
 below drainage level) + (thickness of workings X 5). 
 
 Thus, for a seam 6 ft. thick, 400 ft. below drainage level, the barrier pillar 
 should be (6 X 4) + (6 X 5) — 54 ft. 
 
Depth Below Water Level — All Dimensions in Feet. 
 
 Formula. — (Thickness of workings X 1* of depth below drainage level) + (thickness of workings X 5) = width of barrier pillar. 
 
 288 
 
 METHODS OF WORKING. 
 
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 Each adjoining owner is to leave one-half of the pillar thickness required. 
 
PILLAR DR A WING. 
 
 289 
 
 Compressive Strength of Anthracite.— Attention has recently been called by 
 Mr. William Griffith, of Scranton, Pa., to the advisability of testing the 
 strength of the different coals and of using this data as a basis for the proper 
 proportioning of the pillars and for determining the probability of a squeeze. 
 Jn some crude experiments, which Mr. Griffith carried on, he found that 
 different coals from even the same locality varied greatly in their strengths. 
 If attention were given to this matter, probably the sizes of pillars could 
 be calculated on a much more certain basis than is possible at present, 
 and the liability to squeeze lessened. 
 
 The table on page 290 gives the results of some preliminary and crude 
 tests made by Mr. Griffith, which supply the only data available as to the 
 crushing strength of anthracite coal. 
 
 Drawing pillars is about the most dangerous work the miner has to perform, 
 but the met of its being so is no doubt the reason why, comparatively 
 speaking, so few serious accidents happen in it. It is not so much that the 
 best, most skilled workmen are chosen to perform pillar drawing, as that 
 the men, being alive to the dangers, are more on the alert and careful to 
 protect themselves. 
 
 Sometimes, if not very often, in chamber or room-and-pillar working it is 
 the custom to work out the rooms or chambers and leave pillars all the way 
 from the shaft to the boundary line over large areas; in other words, the 
 portion of the roof left standing on pillars is very extensive. Mines so 
 worked have sometimes been spoken of as mines on stilts. To this mode of 
 proceeding there are several serious objections. By leaving the pillars until 
 the boundary has been reached, a large number of airways and roadways 
 have to be kept open and in repair, and this number is constantly increasing 
 until the limits of the workings have been reached. This circumstance 
 renders the ventilation more difficult, and thereby increases risks of 
 accident. Moreover, the length of time during which the old rooms and 
 pillars are left open and standing increases the danger of squeeze and creep 
 
 Fig. 5. Fig. 6. 
 
 setting in, by which a large area may in a short time be overrun. Also, by 
 this method, the pillars first formed are last removed, and hence it happens 
 that a large number of them crack and give way under the combined action 
 of atmospheric agencies and great pressure. Even if they resist these actions 
 well, the quality of the coal greatly deteriorates by the long exposure. 
 
 For the above reasons, it is the best practice to carry on the two workings 
 (working the rooms and drawing the ribs and pillars) simultaneously. By 
 so doing, the length, mean duration of the roadways, etc. are reduced, and 
 the pillar coal obtained in much better condition; and, in order to concen- 
 trate the workings as much as possible, the two operations should go on as 
 closely together as practicable. With fairly thick and very soft coals, the 
 rapid working up of the rooms and equally quick drawing of the ribs, as 
 soon as the rooms are driven their full distance, is essential to economical 
 working; for delay in extracting ribs and pillars in such circumstances 
 results in their getting crushed and the coal lost or largely ground to slack, 
 waste of props and material, disordered ventilation, and shortened life of 
 the mine. 
 
 Methods of drawing pillars vary according to the inclinations of the 
 seams, the nature of the roof and floor, and the character of the coal. Figs. 
 5 and 6 show the common methods. In Fig. 5, A, B, and C, the drawing 
 begins by cross-cutting the fast ends of the pillars to obtain a retreating 
 face. A shows a method for soft coal and narrowing pillars, B for wide 
 
Tests of Compressive Strength of Sawed Pieces of anthracite Coal. 
 
 290 
 
 METHODS OF WORKING. 
 
 Remarks. 
 
 Checker coal from Exeter. 
 
 Checker coal M. W. coal pile. 
 
 Checker coal bottom of Clear Spring colliery. 
 
 Clear Spring Pittston vein glassy coal. 
 
 Clear Spring Marcey checker coal. 
 
 Clear Spring Marcey checker coal. 
 
 Clear Spring Marcey checker coal. 
 
 Clear Spring Marcey. Very good test. Checker coal. 
 Clear Spring Marcey. Knot in block. Not fair test. 
 Clear Spring reddish glassy coal. 
 
 Clear Spring reddish. Not fair average test. 
 
 
 Crushing 
 Weight. Tons. 
 Deduced. 
 
 Per 
 Cu. In. 
 
 CO © I— J O') © r-J I-J H<NOiP 
 cir^T^^COiOCOiOT^CO' — 
 
 4.67 
 
 Per 
 Sq. In. 
 
 SaOlOO^<Mr-4lOCOC<l^-N 
 -<^0©COCOI>'^'^COi-HC- 
 <^c4r-5©fHr-3iHCOCli— i' 
 
 
 Breaking 
 Weight. Tons. 
 Deduced. 
 
 Per 
 Cu. In. 
 
 ONClONOJN . 
 
 Oioooco^o 
 — 'GO <N <N <M rH CO 
 
 2.70 
 
 Per 
 Sq. In. 
 
 co 
 
 o io i> ao © © 
 
 O'-, c^. C^. lO © i“ 1 OO lO '-H 
 
 ^ ^ ' H H H rH 
 
 Average 
 
 Weight 
 
 Sustained. 
 
 Crush- 
 
 ing. 
 
 Tons. 
 
 t'-»o>ia©ioioi>-©G 003 
 
 CO(MCOCOl>*CQ'MI>iOCO w 
 
 Break- 
 
 ing. 
 
 Tons. 
 
 28 (?) 
 17 (?) 
 30 (?) 
 28 (?) 
 30 
 20 
 22 
 38 
 32 
 22 
 30 
 
 
 Dimensions of Test 
 Piece. 
 
 Height. 
 
 r-tC4C<liO>COCO<Mi-lT-ICOC^ 
 
 
 Width. 
 
 
 
 Length. 
 
 •«*CC©CO0OTt<''jHTt<Tt<i>lO> 
 
 
 *1S9X jo 
 jaqxnnK 
 
 i-HMCO''*iiO©t>CC©©^| 
 
 1 
 
 pillars, the end being 
 taken in two lifts, while 
 C is for harder coal and 
 shows it taken in three 
 lifts. D and E show the 
 pillars cut into stocks to 
 be drawn by side or end 
 lifts, according to the 
 character of the coal, the 
 inclination of the seam, 
 thickness of the cover, 
 and the strength or weak- 
 ness of the roof and floor. 
 Fig. 6 shows some of the 
 methods used in robbing 
 the pillars in steep pitch- 
 ing, thick beds of anthra- 
 cite. To get the coal out 
 of the pillar at the left 
 of A, a skip is taken off 
 the side, as shown. Suc- 
 cessive skips are thus 
 taken off until the whole 
 is removed, the miner 
 keeping the manway 
 open to the heading be- 
 low as a means of retreat. 
 The pillar between A 
 and B is very similarly 
 worked. To remove that 
 between B and C, a nar- 
 row chute or heading is 
 driven up the middle, 
 and cross-cuts put to the 
 right and left a few yards 
 from the upper end. 
 Shots are placed in the 
 four blocks of coal thus 
 formed, as shown, and 
 they are fired simultane- 
 ously by battery. This 
 operation is repeated in 
 each descending portion 
 unless the pillar begins 
 to run. A pillar from 
 which the coal has 
 started to run is shown 
 to the right of C. 
 
 To secure the highest 
 percentage of pillar coal, 
 a method should be 
 adopted that will pre- 
 vent squeezing or crush- 
 ing, if possible. All the 
 pillars in a panel may be 
 taken out at the same 
 time by end lifts in such 
 a way as to keep the face 
 of all the lifts in line and 
 perpendicular to the 
 sides of the pillars, or 
 the pillars are drawn in 
 lifts of three or more 
 pillars each, the centers 
 of the face of the lifts 
 lying in a straight line 
 that makes an angle of 
 about 40° with the sides 
 
R 0 OM-AND-PILLA R METHODS. 
 
 291 
 
 of the pillars. (See also “ Flushing of Culm,” which is described fully on 
 page 314.) 
 
 Gob fires are due to the spontaneous ignition of coal, and are most likely 
 to occur in pack walls and gobs where there is an insufficiency of air. Ample 
 ventilation is the best preventive. 
 
 Spontaneous Combustion.— According to Prof. Able, Dr. Percy, and Prof. 
 Lewes, the causes of the spontaneous ignition of coal are: First , and chiefly, 
 the condensation and absorption of oxygen from the air by the coal, which 
 of itself causes heating, and this promotes the chemical combination of the 
 volatile hydrocarbons in the coal and some of the carbon itself with the 
 condensed oxygen. This process may be described as self-stimulating, so 
 that, with conditions favorable, sufficient heat may be generated to cause 
 the ignition of portions of the coal. The favorable conditions are: A mod- 
 erately high external temperature; a broken condition of the coal, affording 
 the fresh surfaces for absorbing oxygen; a supply of air sufficient for the 
 
 P urpose, but not in the* nature of a strong current adequate to remove the 
 eat; a considerable percentage of volatile combustible matter or an 
 extremely divided condition. Second , moisture acting on sulphur in the 
 form of iron pyrites. The heating effect of this second cause is very small, 
 and it acts rather by breaking the coal and presenting fresh surfaces for the 
 absorption of oxygen. 
 
 Coal Storage.— Prof. Lewes gives the following recommendations for the 
 storage of coal: “ The coal store should be well roofed in, and have an iron 
 floor bedded in cement; all supports passing through and in contact with 
 the coal should be of iron or brick; if hollow iron supports are used, they 
 should be cast solid with cement. The coal must never be loaded or stored 
 during wet weather, and the depth of coal in the store should not exceed 
 8 ft., and should only be 6 ft. where possible. Under no condition must a 
 steam or exhaust pipe or flue be allowed in or near any wall of the store, 
 nor must the store be within 20 ft. of any boiler, furnace, or bench of retorts. 
 No coal should be stored or shipped to distant ports until at least a month 
 has elapsed since it was brought to the surface. Every care should be taken 
 during loading or storing to prevent breaking or crushing of the coal, and 
 on no account must a large accumulation of small coal be allowed. These 
 precautions, if properly carried out, would amply suffice to entirely do 
 away with spontaneous ignition in stored coal on land.” 
 
 When the coal pile has ignited, the best way to extinguish the fire is to 
 remove the coal, spread it out, and then use water on the burned part. The 
 incandescent portion is invariably in the interior, and when the fire has 
 gained any headway usually forms a crust that effectually prevents the 
 water from acting efficiently. 
 
 MODIFICATIONS OF ROOM-AN D-PI LLAR METHODS. 
 
 Some modifications of the room-and-pillar plan shown in Fig. 1 can 
 usually be applied to seams whose dip does not exceed 3°. When the pitch 
 is greater, rooms are often turned off toward the rise only, and the cross- 
 entries driven correspondingly closer together. When the pitch is from 
 5° to 10°, the cars may still be taken to the face if the rooms are driven 
 across the pitch, thus making an oblique angle with an entry or gangway, 
 the rooms being known as room breasts. 
 
 Buggy Breasts.— For inclinations between 10° and 18°, that is, after mule 
 haulage becomes impossible and until the coal will slide in chutes, buggies 
 are often used. Fig. 8 shows a buggy breast in plan and section. Coal is 
 loaded into a small car or buggy c, which runs to the lower end of the breast 
 and there delivers the coal upon a platform l , from which it is loaded into 
 the mine car. The refuse from the seam is used in building up the track, 
 and if there is not sufficient refuse for this, a timber trestle is used. 
 
 Another form of buggy breast is shown in Fig. 7. Here the coal is 
 dumped directly into the mine car from the buggy. If the breast pitches 
 less than 6°, the buggy can be pushed to the face by hand, but in rooms of 
 a greater pitch, a windlass is permanently fastened to timbers at the bottom 
 of the breast, while the pulleys at the face are temporarily attached to the 
 props by chains, so that they can be advanced as the face advances. The 
 rope used is from £ in. to f in. in diameter, and any form of ordinary 
 horizontal windlass can be used. With the windlass properly geared, one 
 man can easily haul a buggy to the face of a breast in a few minutes time. 
 The buggy runs upon 20-lb. T rails spiked with 2£" X 1" spikes upon 2" X 4" 
 
292 
 
 METHODS OF WORKING . 
 
 hemlock studding sawed into lengths of 14 ft. This system has been 
 thoroughly tested by the Delaware & Hudson Canal Co., Scranton, Pa., 
 and has proved a very successful and economical one. _ _ 
 
 Chute Breasts.— Seams pitching more than 15° are usually worked by 
 chutes, or self-acting inclines. When the pitch is between 15° and 30°, sheet 
 iron is laid to furnish a good sliding surface for the coal. On inclinations 
 of less than 18° to 20°, it is usually necessary to push the coal down the 
 chute Sheet iron is not required on pitches above 30°. It must be remem- 
 bered that these pitches are only fair averages, as much depends on the 
 character of the coal. Anthracite slides more easily than bituminous. _ To 
 secure the best returns from a coal seam, the slope or shaft should be driven 
 to the basin, and the lowest gangways or levels first driven to the property 
 limits, and the coal then worked retreating toward the slope or shaft. 
 Practice is, however, usually contrary to this, and the upper levels or gang- 
 wavs are turned off first, and working places opened out as rapidly as the 
 gangway is driven. Fig. 9 shows a method of grouping rooms that may be 
 
 Fig. 7. 
 
 used where the pitch is from 8° to 20°, the straight heading being driven on 
 the strike and the other headings at such angles as will give a good grade 
 for haulage purposes. 
 
 The pillar-and-stall system is a modification of the room-and-pillar, to 
 which it is similar in ali respects excepting in the relative size of the pillars 
 and breasts. The stalls are usually opened narrow and widened inside, 
 according to conditions of roof, floor, coal, depth, etc., being from 4 to 6 yd. 
 in the single-stall method, with the pillars about the same width. Fig. 10, 
 A and B , shows single and double stalls. This system is adapted to weak roof 
 and floor, or strong roof and soft bottom, to a fragile coal, or wherever 
 ample support is required, and is particularly useful in deep seams with 
 
CONNELLS VILLE METHOD . 
 
 293 
 
 great roof pressure. Double stalls are often driven from 12 to 15 yd. wide, 
 with an intervening pillar of sometimes 30 yd. A „ 
 
 The following are a few of the applications of the pillar-and-stall method 
 of working as they are carried out in some of the leading coal fields of 
 America i 
 
 Connellsville Region (if. L. Auchmuty) .—Fig. 11 shows the common method 
 used in the Connellsville, Pa., region. The average dip is about 50. The 
 face and butt headings are driven, respectively, at right angles to each 
 other on the face and the butt of the coal. The face headings leave 
 
 the main butts about 1,000 ft. apart, while from these face headings, and 
 400 ft. apart, secondary butts are driven, and again from these butts on 
 
 the face of the coal the 
 rooms or wide work- 
 ings are excavated to a 
 length of 300 ft., this 
 having proved the most 
 convenient length for 
 economical working. 
 Koom pillars have a 
 thickness of 30 to 40 
 ft., while the rooms are 
 12 ft. in width and are 
 spaced 42 to 52 ft. 
 between centers, de- 
 pending on depth o 
 strata over the coal. 
 The headings are 8 ft. 
 wide, and in all main 
 butts and faces the dis- 
 tance between centers 
 Fig. 9. of parallel headings is 
 
 60 ft., leaving a solid 
 
 rib of 52 ft. A solid rib of 60 ft. is also left on the side of each main 
 heading. The average thickness of cover at the Leith mine, which 
 
294 
 
 METHODS OF WORKING. 
 
 is here described and which may be considered as a type of the region, 
 is 250 ft., the overlying measures being alternated layers of soft shale 
 ’ & and coal for 4 ft. The bottom is 
 
 an 18" layer of hard fireclay 
 and slate. These floor and roof 
 materials are soft, and are easily 
 disintegrated by air and water. 
 At some mines, cover will Teach 
 as much as 700 ft., and the dip of 
 5 £ (as at Leith) is much heavier 
 at some points on eastern out- 
 crop, and will run as high as 12<, 
 flattening off as the synclinal line 
 of the basin is reached, until it is 
 almost level. In some localities, 
 the material below coal is hard 
 limestone, requiring blasting to 
 remove it, and at other places 
 the roof slates are much more 
 solid than at Leith, and not read- 
 ily disintegrated. The method 
 of drawing ribs is one of the beauties of the system, since it is i harder to do 
 succe ssfull y in a soft coal like the Connellsville coal than in hard coal. The 
 
 Fig. 10. 
 
 Fig. 1L 
 
CLEARFIELD METHOD. 
 
 295 
 
 coal nself is firm. When necessary to protect the top or bottom, 4 to 6 in. of 
 coal are left covering the soft material. 
 
 The method as given above is often applied to a whole series of butts 
 (4 or 5) at once instead of to butt by butt, as shown in Fig. 11. In this case, 
 work is started at the upper end of the uppermost butt and progresses, as 
 shown in Fig. 11; but, after cutting across the butt heading from which the 
 rooms were driven, the butt heading itself and the upper rooms from the 
 second butt, or that just before, are likewise drawn back by continuous 
 slices being removed from the rooms of the upper butt, and on across the 
 next lower butt, etc., all on an angle to the butts, and so continued as the 
 operations progress, until another butt is reached, etc., thus gradually 
 making a longer and longer line of fracture, which is only limited by the 
 number of butts it is desired to include at one time in the section thus 
 mined. This works very nicely and makes long even lines of fracture, the 
 steps of the face of the workings (in the rib drawing) being about 30 ft. 
 ahead of one another. 
 
 Pittsburg Region {H. L. Auchmuty ) .—The coal is worked in much the same 
 way as in the Connellsville region, except that a different system of drawing 
 ribs is used. The coal is worked on the room-and-pillar system, with double 
 entries, with cut-throughs between for air, and on face and butt, entries are 
 about 9 ft. wide, and the rooms 21 ft. wide and about 250 ft. long; narrow (or 
 neck) part of room, 21 ft. long by 9 ft. wide; room pillars, 15 to 20 ft. wide, 
 depending on depth of strata over the coal, which is from a few feet to 
 several hundred feet. The mining is done largely by machines of various 
 types. Coal is hard, of course, and, in many places, the roof immediately 
 over the coal is also quite hard. There are about 4 ft. of alternate layers of 
 hard slate and coal above the coal seam. Rooms are mined from lower end 
 of butt as fast as butt is driven, the ribs being drawn as mining progresses. 
 As the coal is harder than in the Connellsville region, thickness of coal 
 pillar between parallel entries is somewhat less. 
 
 Clearfield Region ( G . F. Duck ).— The butt and face are not strongly marked 
 in the B or Miller seam, the one chiefly worked in this region. Where 
 possible, these cleavages are followed in laying out the workings, but the 
 rule is to drive to the greatest rise or dip and run headings at right angles to 
 the right and left, regardless of anything else. The main dip or rise heading 
 is usually driven straight, and is raised out of swamps or cut down through 
 rolls— very common here— unless they are too pronounced, when the head- 
 ing is curved around them. The same is true of room headings, except that 
 they are more usually crooked, not being graded except over very minor 
 disturbances. 
 
 As the B seam rarely runs over 4 ft. in thickness, and is worked as low as 
 2 ft. 8 in. in the haulage headings, the roof is taken down to give 5 ft. to 
 5 ft. 2 in. above the rail, or 5 ft. 8 in. to 5 ft. 10 in. in the clear. Where the 
 resulting rock is taken outside, the headings are driven 10 ft. wide with 
 
 24 ft. of pillar, roof taken down in haulage heading but not in air-course. 
 Where the rock is gobbed underground, the haulage heading is 18 to 24 ft. 
 wide, air-course 10 ft., pillar 24 ft., and roof taken down in haulage heading 
 only. The thinner the coal, the wider the heading. It is more economical 
 to haul the rock to daylight. The bottom generally consists of 3 ft. to 5 ft. of 
 hard fireclay, frequently carrying sulphur balls. 
 
 In numerous places, the sand rock is immediately over the coal, but in 
 most cases there is from 3 to 5 ft. of slate before the sand rock is reached. 
 Room headings are driven 280 ft. apart, haul rock to daylight, heading 10 ft. 
 wide with 24 ft. pillar to 10 ft. air-course, in which roof is left up. A 15 ft. to 
 
 25 ft. chain pillar is left between air-course and faces of rooms from the lower 
 heading, every fourth to eighth of which is driven through to the air-course 
 to shorten the travel of the air. The rooms are therefore 180 to 200 ft. 
 long, and the men push the cars to the face, an important economical item 
 in this thin coal. 
 
 Rooms are 21 ft. wide with a 15 ft. pillar, and a 15 ft. chain pillar is left 
 between the first room on any room heading and the main heading, and 
 roof is not taken down in rooms. Main-heading track is usually 30-lb. iron, 
 room heading, 12 lb., and 2" X *" strap iron set on edge is used in the rooms 
 in low coal. Mine cars hold from 600 to 800 lb. in low seams, and 1,500 to 
 2,000 lb. in the so-called thick seams, i. e., 3 ft. 8 in. to 4 ft. thick. 
 
 Reynoldsville Region.— The measures are very regular, and the method 
 employed the typical one shown in Fig. 1. The average thickness of the 
 principal seam is 6£ ft. and the pitch is 3° to 4°. The coal is hard and firm, 
 
 \ 
 
 
296 
 
 METHODS OF WORKING. 
 
 and contains no gas; the cover is light, and on top of the coal there are 
 3 or 4 ft of bony coal; the bottom is fireclay. Drift openings and the double- 
 entry system are used. Both main and cross-entries are 10 it. wide, with a 
 94-ft pillar between. The cross-entries are 600 ft. apart, and a 24 ft, chain 
 pillar is left along the main headings. The rooms are about 24 it. wide 
 and open inbye, the necks being 9 ft. wide and 18 ft. long. The pillars are 
 
 from 18 to 30 ft. thick. . 
 
 West Virginia ( James W. Paul).— The general plan of working the Pitts- 
 burg coal in the northern part of West Virginia is as follows. The coal 
 measures vary from 7 to 8 ft. in thickness, and have a covering varying 
 from 50 to 500 ft. The coal does not dip at any place over 5$. In most places the 
 coal is practicallv level, or has just sufficient dip to afford drainage. The usual 
 method of exploitation is to advance two parallel headings, 30 ft. apart on the 
 face of the coal. At intervals of 500 to 600 ft., cross-headings are turned to right 
 and left, and from these headings rooms are turned off. These cross-headings 
 are driven in pairs about 20 or 30 ft, apart. Between the main headings and 
 
 the first room is left a block of coal about 100 ft., and on the cross-headings 
 there is often left a barrier pillar of 100 ft. after every tenth room. 
 
 The headings are driven from 8 to 12 ft. wide, and the rooms are made 
 24 ft. wide and 250 to 300 ft. long. A pillar is left between the rooms 
 about 15 to 20 ft. wide. These pillars are withdrawn as soon as the panel 
 of rooms has been finished. The rooms are driven in from the entry about 
 10 ft. wide for a distance of 20 ft., and then the room is increased m width on 
 one side. The track usually follows near the rib of the room. Cross-cuts 
 on the main and cross-headings are made every 75 to 100 ft., and m rooms 
 about every 100 ft. for ventilation. . . . ~ ^ 
 
 The double-heading svstem of mining and ventilation is m vogue. Over- 
 casts are largelv used, but a great many doors are used in some of the 
 mines. Rooms' are worked in both directions. This is the general 
 practice when the grades are slight. When the coal dips over 1#, the 
 rooms are driven in one direction only. In this case, the rooms are 
 made longer, as much as 350 ft. It is the custom then to Break about 
 every third room into the cross-heading above (a practice ill advised). 
 The floor of this bed of coal, being composed of shale and fireclay, often 
 
ALABAMA METHODS. 
 
 297 
 
 heaves, especially when it is made wet. Some trouble is at times experi- 
 enced by having the floor heave by reason of the pillars being too small for 
 the weight they support. . . 
 
 The dimensions of rooms and pillars given are for a mine (with 
 covering 300 to 500 ft. thick) having a fairly good and strong roof. Where 
 roof, bottom, and thickness of cover change, these dimensions are altered to 
 suit the requirements. The main-heading pillars may be reduced to 30 or 
 40 ft.; the rooms may he made 15 ft. wide with 12 ft. pillars, and no harrier 
 pillars mav be left on the cross-headings. 
 
 The foregoing plan is very much followed in other parts of the State; at 
 least an attempt is made to do so, but local disturbances often require 
 changes in the plan. This plan is followed on some parts of New River, 
 and also in the Flat Top field. 
 
 Alabama Methods (J. E. Strong).— Fig. 12 shows the common methods used 
 in working the Alabama coals. The seams now working vary from 2 to 
 6 ft. thick, and they pitch from 2° to 40°. Where the seams are thin, the 
 coal is hard, and pillars of about 20 to 30 ft. are used to support the roof. 
 
 Fig. 13. 
 
 The thick seams are soft and easily broken, and much larger pillars are left. 
 The character of bottom and top varies; fireclay bottom and slate roof are 
 usually found with the thick seams, and hard bottom and sandstone rqof 
 with the thin, seams. The general plan of laying out the mine is to drive 
 the slope straight with the pitch of the seam; this is usually on the butts of 
 the coal. A single-track slope is 8 ft. wide, and a double-track slope 16 ft. 
 Cross-headings are driven or turned from the slope water level every 
 300 ft.; air-courses are driven parallel on either side of the slope. Where 
 an 8 ft. slope is driven, 30 ft. of pillar are left between the slope and airway, 
 and for a 16 ft. slope, 30 ft. of pillar. The size of pillar, however, depends 
 largely on the character of the roof and thickness and strength of coal. On 
 the lower side of the headings, pillars from 20 to 60 ft. are left on the 
 entry before turning the first room. The rooms are worked across the pitch 
 on an angle of about 5° on the rail, Fig. 12, A, when the coal does not 
 pitch greater than 20°; where the pitch is greater, chutes are worked and 
 the rooms are driven straight up the pitch (Fig. 12, B). In a few cases, 
 where the pitch is not greater than 15°, double rooms are worked with two 
 roadways in each room (Fig. 12, C). A rope with two pulleys is used, and 
 each track keeps the rib side of the room, the loaded car pulling up the 
 empty on the opposite side of the room; distance between room centers, 
 about 42 ft. Where single rooms are worked, the room is driven narrow 
 (8 ft. wide) for 21 ft., when connections are made with the room outside of 
 it; the room is then widened out to about 25 ft., sloping gradually until this 
 width is at tained; pillars of from 10 to 20 ft. thick are left between the 
 rooms, and cross-cuts fdr ventilation are made about every 50 ft.; every third 
 or fourth room is driven through to the entry above; pillars are then drawn 
 back to the entry stumps or pillars. The average cover over the coal now 
 working is from 100 to 600 ft. Air-courses usually have an area of 30 ft., 
 and sufficient coal is taken out to give this area, the roof and bottom being left. 
 
 George’s Creek District, Md.— Fig. 13 shows the method used in the George’s 
 Creek field, Maryland. The coal shows no indication of cleats, and the 
 butts and headings can be driven in any direction. The main heading is 
 driven to secure a light grade for hauling toward the mouth. Cross- 
 headings making an angle of 35° to 40° are usually driven directly to the 
 
298 
 
 METHODS OF WORKING. 
 
 rise, and of the dimensions shown. Pillars are drawn as soon as the rooms 
 are completed, being attacked on the ends and from the rooms on either 
 side, the coal being shoveled to the mine car on a track in the room. Very- 
 wide pillars are split. No effort is made to hold up the overlying strata, and 
 the entire bed is removed as rapidly as possible. An extraction of 85$ of the 
 bed is considered good work. A section of the seam is as follows: Roof 
 coal, 10 in.; coal, 7 ft.; slate, i in.; coal, 10 in.; slate, 1 in.; coal, 10 in.; fireclay; 
 slate. The top bench is bony and frequently left in place to prevent 
 
 disintegration of the roof by the air. Above this coal is from 8 to 10 ft. of 
 “rashings,” consisting of alternating thin beds of coal and shale, that is very 
 brittle, and requires considerable timber to keep it in place.— (“ Mines and 
 Minerals,” Vol. 19, page 422.) 
 
 Blossburg Coal Region, Pa.— Coal is generally mined from drifts, but in a 
 few cases by slopes. Fig. 14 shows the general method adopted; the breasts 
 are run at right angles to the slips; the breast pillars are split by a center 
 heading and taken out as soon as the breasts are finished. The gangway 
 pillars are taken out retreating from the crop or boundaries of the property. 
 
 Fig. 15. 
 
 The general average of the coal seams is not over 31 ft., accompanied by 
 fireclay and some iron ore. The dip of the veins is about 3$. — (‘‘Mines and 
 Minerals,” Vol. 19, page 126.) 
 
 Indiana Coal Mining.— Fig. 15 shows the double-entry room-and-pillar 
 method as used in Indiana. The entries are generally 6 ft. high, 8 ft. broad, 
 
IOWA METHOD. 
 
 299 
 
 the minimum height required by law being 4 ft. 6 in. The rooms are from 
 21 to 40 ft. in width. The mines are generally shallow. The rooms m Fig. 15 
 are shown as widened on both ribs, but a more usual method in this locality 
 is to widen the room on the inbye rib, leaving one straight rib for the protec- 
 tion of the road in the room.— (“ Mines and Minerals,” Yol. 20, page 202.) 
 
 Iowa Coal Mining.— The coal lies at a depth of 200 ft. below the surface, and 
 is geologically similar to that of the Missouri and Illinois fields. It lies in 
 lenticular basinsextending northwest and southeast and outcropping in the 
 larger river beds. The seams are practically level, non-gaseous, and gen- 
 erally underlaid by fireclay and overlaid by a succession of shales, sand- 
 stones, and limestones, which are generally of a yielding nature, giving a 
 strong, good roof for mining. There are three distinct seams, the lower one, 
 which varies from 4 to 7 ft. in thickness, being the only one worked. The 
 coal is a hard, brittle, bituminous coal that shoots with difficulty, but is 
 excellent for steam and domestic uses. About Centerville, the coal has a 
 distinct cleat, but elsewhere in the State this is lacking. 
 
 The entry pillars along the main roads are 6 to 8 yd. thick, for the 
 cross-entries 5 to 6 yd., and for the rooms 3 to 5 yd. Room pillars are 
 drawn in when approaching a cross-cut. Both room-and-pillar and longwall 
 methods are in use, with modifications of each. In the ro 9 m-and-pillar 
 system, the double-entry system is almost invariably used in the larger 
 mines. Rooms are driven off each entry of each pair of cross-entries at 
 distances of 30 to 40 ft., center to center; the rooms are 8 to 10 yd. m width, 
 and pillars 3 to 4 yd. The rooms are narrow for a distance of 3 yd., and then 
 widened inbye at an angle of 45° to their full width. They vary from 50 to 
 100 yd. in length, and the road is carried along the straight rib. 
 
 When double rooms are driven, the mouths of the rooms are 40 to 50 ft. 
 apart, and they are driven narrow from the entry a distance of 4 or 5 yd. 
 
 Fia 16. 
 
 A cross-cut is then made connecting them, and a breast 16 yd. wide is driven 
 up 50 to 60 yd. The pillar between each pair of rooms is 12 to 15 yd. 
 
 In pillar-and-stall work, the stalls are usually turned off narrow and 
 widened inside, the pillar varying from 5 to 8 yd. The stalls are 30 to 40 yd. 
 in length, and the pillars are drawn back. When the stalls are driven m 
 pairs, the pillar 8 to 10 yd. in width is carried between them. 
 
 Longwall . — The main haulage road runs in each direction from the foot of 
 the shaft, and on both sides of this diagonal roads are turned at an angle of 
 45°, or parallel to the main haulageway. These are spaced 10 yd. apart and 
 driven 50 to 60 yd., when they are cut off by another diagonal road. Panel 
 breasts are used where the conditions are such as to induce a squeeze. 
 Rooms are turned narrow off entries and are arranged in sets of 6 to 12 
 rooms, with a pillar 10 to 20 yd. wide between the sets of rooms. When the 
 rooms have progressed a short distance from the entry, they are connected 
 by cross-cuts, and the longwall face is carried forward from this point. 
 Packs are built and the roof allowed to settle, as in longwall. The wide 
 pillars are taken out after the roof has settled. 
 
300 
 
 METHODS OF WORKING. 
 
 Ventilation :— The system of ventilating the workings usually employed is 
 that of conducting the air to the inside workings by means of an air-course 
 forming the back entry of each haulage road. From this point it is carried 
 along the face of the rooms, through the breakthroughs or cross-cuts in the 
 room pillars, returning thence to the haulage road, which is usually made 
 the return airway. When, however, the mine is ventilated by means of a 
 furnace or an exhaust fan, the intake airway is usually made the haulage 
 road, in order to avoid doors at the shaft bottom. 
 
 The Tesla, California, method is shown in Fig. 16. The coal seam averages 
 7 ft, of clear coal, and pitches 60°. This system was adopted m a portion of 
 the mine to get coal rapidly: for, at this point, a short-grained, slate cap 
 rock came in over the coal, making it difficult to keep props in place. The 
 floor is a close blue slate and has a decided heaving tendency. The roof is 
 an excellent sandstone. There is a small but troublesome amount of gas. 
 Two double chutes are driven up the pitch at a distance of 36 ft. apart, con- 
 nected every 40 ft. by cross-cuts. One side of each chute is used for a coal 
 chute and the other for a manway and air-course. At a distance of 12 yd. 
 apart small gangways are driven parallel with the main mine gangways. 
 These are continued from each chute a distance of 300 ft., if the conditions 
 warrant it. The top line is then attacked from the back end and the coal is 
 worked on the cleavage planes; the breast, or room, therefore consists of 
 a 12-yd. face, including the drift or gangway through which the coal is 
 carried to the chutes; a rib of coal (2 or 3 ft.) is left between the breasts to 
 keep the rock from falling on the breast below. Thus in each breast the 
 
 Fig. 17. 
 
 miners have a working face of about 15 or 16 yd., and as the coal is directed 
 to the car by a light chute, moved along as the face advances, the coal is 
 delivered into the cars at small cost, and but little loss results from the 
 falling coal, as a minimum of handling is thus obtained. Immediately 
 above each gangway, and starting from these main chutes, an angle chute is 
 driven at about 45°, connecting with the breast gangway above it, and into 
 these chutes the coal from that breast is delivered, runs into the mam chute, 
 and from it is loaded into the mine cars in the main gangway. These angle 
 chutes serve as a means of keeping the main chute full, and at the same 
 time giving each breast an opportunity to send out coal continuously. 
 Thev also serve the purposes primarily intended, of saving the coal from 
 breakage, by giving it a more gradual descent into the full chute, me 
 breast gangways are driven 5 ft. wide. No timbers are needed in these 
 gangways, as they are driven iu the coal, except ou the foot-waU or floor 
 
TESLA METHOD. 
 
 301 
 
 side, which, as before stated, is a firm sandstone. It is found safest to leave a 
 rib of coal on the top of the breast 2 or 3 ft. thick, until the working- face has 
 passed on 12 or 15 ft., when this rib is cut out and thus all the coal extracted, 
 the roof caving behind and filling in the opening. As cross-cuts are driven 
 every 36 ft., ventilation is kept along the working faces, and a safe and 
 effectual means of securing all the coal in the seam is thus attained. 
 
 Fig. 17 shows another system used in No. 7 vein at the same place. The 
 seam averages 7 ft. of coal. The roof is shelly and breaks quickly, hence 
 the coal must be mined rapidly. 
 
 In this system the gangway chutes are driven at right angles with the 
 strike of the seam, 40 ft. up the pitch; a cross-cut 5 ft. X 6 ft. is then driven 
 
 Fig. 18. 
 
 parallel with the gangway. From this cross-cut, chutes are driven at same 
 distance apart as the gangway chutes (30 ft.), at an angle of 35°, and cross- 
 cuts are driven every 40 ft. between chutes, for ventilation. After a panel 
 of five or more chutes is driven up the required distance, work is com- 
 menced on the upper outside pillar and the pillars on that line are drawn 
 and the next line is attacked, and this is continued until the panel or block 
 is worked down to the cross-cut over the gangway. About every 80 ft. in 
 this level it is found advantageous to build a row of cogs parallel with the 
 strike of the seam as the pillars are drawn. This serves to save the crushing 
 of the pillars, and prevents any accidents from falls of rock. But few 
 timbers are required by this system.— (“Mines and Minerals, ” Yol. 19, 
 page 145.) 
 
302 
 
 METHODS OF WORKING. 
 
 New Castle Colorado, Method.— The following method as described by Mr. 
 R M Hosea * Chief Engineer of the Colorado Fuel and Iron Co., is used 
 at New Castle Colo for highly inclined bituminous seams. The coals mined 
 are only feirly hard, contain considerable gas, and make much waste in 
 mining Fig - 18 shows the method used for extracting the Wheeler or 
 thicker* veinto its ful 1 width of 45 ft., and the E seam 18 ft. thick, excepting 
 that left for pillars. Rooms and pillars are laid out under each other m 
 the two seams whenever practicable. Entries are along the foot-wall, 30 it. 
 un the pitch is an air-course. Rooms and breasts are laid out as shown 
 in B and C Fig. 18. In the Wheeler vein, the manways go through the 
 entrv pillars to the air-course and thence along the ribs each side of 
 the room one manway to the main entry serving for two double rooms. A 
 lower bench of 6 ft. is first mined the full length of the rooms, 120 ft., side 
 man wavs being protected by vertical or leaning props, bordered with 3 
 planks outside and the chute or battery is then put in. At the top the rooms 
 arrconnected by cross-cuts, and, occasionally, intermediate cross-cuts are 
 reauired The room is kept full of loose coal, only sufficient being drawn to 
 keen the’ working floor at the proper height for the mining. When driven 
 to the limit and^ with cross-cuts connected, the coal is all dra\vn out at 
 the chutes which have receptacles for rock and waste at their sides, to be 
 picked out by the loaders. The next operation is to dnve across the seam at 
 
 Fig. 19. 
 
 the air-course until the hanging wall is reached, manways, called back wan- 
 t 4“ be°ng mrnteined as before. A triangular section of coal is mined off, 
 as shown in A, Fig. 18, and the room filled with loose coal. The full thick 
 ness of the seam is now taken off*, shots being first placed at S S, coa | 
 drawn out at the bottom as required. Section D, Fig. 18, shows a method of 
 robbing a pillar. In doing this, the manways are moved back into the 
 uillar each side 10 ft. or so, by mining on the lower bench as before, and 
 E 0 ies are drilled into the roof with long drills, which bring down as much of 
 ^eo^erhanging 1 ^^ as can be reached.-(“ Mines and Minerals, 1 Yol. 17, 
 
 page 377.) 
 
 MODIFICATIONS OF LONGWALL METHOD. 
 
 Fig 19 shows a good arrangement of the main and temporary haulage- 
 wavs in a fiat seam The chief object in any plan of longwall workings is to 
 w! the permanent roadways the arteries of the system, providing the most 
 toect route^rom 11 aU° sections of the mine to the shaft. The temporary 
 
LONG WALL METHOD . 
 
 303 
 
 roads or working places are only maintained for a distance of 60 to 100 yd., 
 until cut off by subroads branching at regular intervals from the main 
 roads. In the figure, full heavy lines indicate the permanent haulageways, 
 
 except only the main 
 intake airway (12 ft. 
 wide), running west 
 from the downcast shaft 
 D , and the main return 
 air-course (12 ft. wide) 
 leading from the face on 
 the east side to the man- 
 way around the upcast U, 
 which is the hoisting- 
 shaft. The full light 
 lines indicate the diag- 
 onal subroads, driven to 
 cut off the working 
 
 S laces, shown by the 
 otted lines. The stables 
 are located as shown in 
 the shaft pillar, between 
 the two shafts, where 
 they will not contami- 
 nate the air going into 
 the mine, but will receive 
 air fresh from the down- 
 cast and discharge it at 
 once into the upcast cur- 
 rent. This position also 
 affords ready access from 
 either shaft in case of 
 accident, and for the 
 handling of feed and 
 refuse. The pumps may 
 be located in any con- 
 venient position at the 
 foot of the upcast. The 
 shaft bottoms are driven 
 14 ft. wide nearly through 
 the shaft pillar, and are 
 
 r A __ 
 
 [Three ', ! Sp/iff'' '• 
 
 L1/7& 
 
 Fig. 20. 
 
 Wor/rec/. Oaf 
 
 MB 
 
 continued 10 ft. wide north 
 and south through the gob. 
 The width of all other roads 
 and subroads is made 8 ft. The 
 Fig. 21. extra width of the straight 
 road through the hoisting shaft 
 is to provide for the future need, when the 
 size of the workings will demand that the 
 mine be ventilated in four sections or splits; and these two roads will then 
 each form the return of two sections. This will be accomplished by over- 
 casting the main road forming the shaft bottom, and carrying half the cur- 
 rent by this means to the east face, where it is again divided. The same 
 thing is done on the west side. The divided currents, after traversing the 
 faces of their own respective sections, unite and return to the hoisting shaft 
 by the main haulage road. 
 
 When the roof is very solid, the gob roads turn off the entries at 45°, and 
 
304 
 
 METHODS OF WORKING. 
 
 maybe a considerable distance apart, so that the tracks can be turned in 
 along the working face and the mine cars loaded at the face. Whenthe 
 roof is tender, making it impossible to maintain sufficient room for the mine 
 cars to pass along the face, gob roads are turned off near together, and the 
 mine cars run to the road heads, to which points the coal is shoveled or 
 hauled in buckets. When the working face has reached such a distance 
 from the bottom of the shaft that it becomes impossible to work rapidly 
 enough to avoid the destructive weighting action of the roof, the mining 
 must be divided into panels or sections, the working face of each of which 
 can be advanced at the proper rate. . . . ,, , 
 
 Figs 20 and 21 show a plan and section, respectively, of two methods 
 largely* used in Europe for working thick pitching, contiguous seams of 
 hard long-grained coal. From the foot of the shafts, levels are driven on 
 the strike, and jig roads turned off these in the top seam at right angles 
 up the pitch. The working faces are advanced in the same horizontal 
 plane, the lower one being always ahead. The coal from the two lower 
 seams is run through horizontal passages to the upper seam, where it is 
 lowered to the levels below by means of jigs or gravity planes The slates 
 between the seams and the refuse obtained in mining are used to fill m as 
 the faces advance. This gobbing must be done quite thoroughly, m order to 
 prevent excessive settling of the roof and consequent crushing of the coal at 
 the faces. Where spontaneous combustion is liable to occur, it is not 
 advisable to use this method, but rather that shown in the lower portion of 
 the figures. Slopes, or inclined planes, are driven down the pitch from the 
 levels to the basin, or, if possible, to the boundary line, where the working 
 faces are formed by driving levels to the right and left of the ends of the 
 slopes. The working faces are here also kept in a horizontal plane with 
 the lower one farthest up the pitch. The coal from the two upper seams is 
 taken through tunnels or flats to the slopes in the lower seam, and hoisted to 
 the shaft bottom. Here all the inclined planes or other passages are mthe 
 solid coal, and the worked-out places are left behind. A very small amount 
 of coal is left in the mine when worked from the basin upwards, and the 
 effects of squeezes are not felt to any great extent, as the weight of the roof 
 is thrown on the gob. Where there is not sufficient refuse material to fill in 
 with, it is taken into the mine from the surface, or from another adjacent 
 mine having an extra amount of stowing material. . _ 
 
 It is not necessary that the slopes be sunk to the boundary line, m which 
 case the main-slope pillars should be large and left in so that the dip 
 workings , as they are called, can be continued downwards when desired. 
 In this way, the first cost of opening up is greatly reduced. The ventila- 
 tion of these workings is quite simple, the intake being split at the ends of the 
 main entries, or slopes, and the air forced along the different working faces 
 to the right and left, and thence to the upcast by way of the main-return 
 airways. If at all possible, it is advisable to provide an outlet near the 
 faces of the rise workings that are advancing upwards, because the lighter 
 gases cannot be forced down-hill with satisfaction unless an excessive 
 velocity of the air-current be maintained. These systems are well adapted 
 to deep or shallow mines, and to give maximum outputs for minimum 
 development, provided the work is carried on quickly and steadily. 
 
 Overhand-Stoping Method.— Where several thick and heavily pitching 
 seams, in which considerable firedamp is given off and the roof falls 
 freely, are to be worked, a shaft is sometimes sunk m the adjacent strata, 
 and at certain distances horizontal tunnels are driven to the coal seams. 
 From these tunnels, levels or haulage roads are driven m each seam to 
 the right and left, provided the seams are not so close together as to make 
 it more profitable to use rock chutes or tunnels, through which the coal 
 is run from one seam into the other. At certain intervals, depending on the 
 length of the lifts horizontally, pairs of headings, usually called dtps, are 
 driven up the pitch until they intersect the levels and tunnels above. 
 Headings are turned off these dips to the right and left, parallel to the mam 
 levels or haulage roads, and when they meet or have reached their limit 
 horizontally they are holed, or cut through, by cross-cuts driven on the 
 pitch. The working faces thus formed are then carried back, as shown in 
 Fig. 22. Skips are taken off the face and the roof allowed to cave in after 
 each operation and fill up the gob behind. The order of working is such 
 that the top faces are worked in advance of the lower ones. The cars, 
 which are taken to the working faces, are handled in the dips by balance 
 carriages, or back balances , as they are termed in some localities,. I he 
 
ANTHRACITE MINING. 
 
 305 
 
 barney, or balance, runs on a narrow track in the middle of the track for the 
 carriage on which the car is placed. The barney will raise the carnage with 
 the empty car on it, and the carriage and loaded car will hoist the barnew 
 These gravity planes are only made one-half the length of the dips, or about 
 150 ft., in order that greater safety may be secured and shorter ropes used. 
 One is placed in the lower half of one of the pairs of dip headings, and 
 another in the upper half of the other, thus necessitating that the cars be 
 changed from one gravity plane to the other midway along the dips. This 
 is done by taking the car off one carriage and pushing it through a break- 
 through or cross-cut to the other. Fig. 22 shows the method of working these 
 lifts in some parts of England and Belgium where the seams are gaseous, and 
 some of them quite thick. The face is stepped more or less deeply, depend- 
 ing on the pitch, in order to protect ea,ch miner from the falling coal of his 
 neighbor. The men reach the higher portion of the working face through 
 timbered manways. The coal is generally run down chutes to the cars 
 below, but in some places it is run to the end of the gangway below by 
 means of inclined chutes, or spouts, laid on the gob. The essential feature 
 for the successful operation of this system is the close and careful stowing of 
 the gob between walls. In Belgium, cord wood and brush wood are very 
 largely used for gobbing material or stowing between the regular timbers. 
 All the coal, except very thin vertical pillars, is taken out. Where there is 
 
 Fig. 22. 
 
 much firedamp, the miners simply nick the coal and leave it stand over 
 night during which time the gas either forces it off the solid or so loosensit 
 that the miner can easily take it off with a pick. (See also Highly Incline 
 Mineral Deposits.) , 
 
 METHODS OF MINING ANTHRACITE. 
 
 A perfectly flat seam of anthracite is seldom found in America, and even 
 where a portion of the seam maybe found lying comparati^ 
 sudden changes in dip must be expected that a system adapted to working 
 on a pitch is almost universally used. A breast may start on a How _ pitch au d 
 the pitch may increase gradually until it beeves vertical, or Pj? 
 may be the case. The cleat is usually lacking m anthracite and the direc- 
 tion of driving the breasts is determined largely by the pitch and by haulage 
 
 COI For^hches up to 30°, the methods shown in Figs. 1,7, 8, 9, 10, 12, and 13 are, 
 in general, applicable, with certain changes due to local considerations. 
 There is considerable difference in the methods of opening rooms m anthra- 
 cite and bituminous mines, owing to variations in thecharac^ 
 coals and to the fact that anthracite will slide on chutes of 1 ^ s o 1 ^ 1 "^ tl a ° T 5 
 than bituminous coal. Where the pitch does not exceed 4Jthewoingm 
 turned off at right angles to the gangway. In moderately thick . & oal i seams, 
 pitching between 4° and 18°, the rooms are generally driven 
 forming room breasts, thus securing a grade that permits the haulage of the 
 cars to the face. 
 
306 
 
 METHODS OF WORKING. 
 
 There are two methods of mining thick coal in breasts when nearly flat. 
 (1) The breasts are opened out and driven to the limit in the lower bench of 
 coal, and the top benches are blown down afterwards, beginning at the face 
 and working back. (2) When the roof is good and there is no danger of its 
 falling and closing up the workings, the upper benches may be worked in 
 the opposite direction, beginning at the gangway and driving towards the 
 limit of the lift, or the working of the upper bench may follow up that of the 
 lower bench. When the seam is less than 12 ft., the top is supported by 
 props; in thicker seams, the expense is so great for propping that but little 
 attempt is made to support the roof. In the thicker anthracite seams 
 (notably the Mammoth), the coal in the breasts is so worked as to make an 
 arch of the upper benches of coal, which acts as a temporary support for the 
 roof, the coal in the arch being extracted when the pillars are robbed. 
 
 When the inclination of anthracite seams is less than 30°, the breasts may 
 be opened with one chute in the center, which ends in a platform projecting 
 into the gangway, off which the coal can be readily loaded into the mine 
 car. When this method is employed, the refuse is thrown to either side of 
 the chute. If the pillars are to be robbed by skipping or slabbing one rib 
 only, it is well to keep most of the refuse on one side. Sometimes, when the 
 top is good, and the breasts are driven wide, two chutes are used, but the 
 cost of making the second chute is considerable and is therefore not 
 advisable unless necessitated by the method of ventilation employed. 
 
 Col. Brown’s Method.— Fig. 23 shows a panel system devised by Col. D. P. 
 Brown, of Lost Creek, Pa., wiiich gives good results in thick seams pitching 
 from 15° to 45°, where the top is brittle, the coal free, and the mine gaseous. 
 Rooms or breasts are turned off the gangway in pairs, at intervals of about 
 
 * Fig. 23. 
 
 60 yd. The breasts are about 8 yd. wide, and the pillar between about 5 yd. 
 wide, which is drawn back as soon as the breasts reach the airway near the 
 level above. In the middle of each large pillar between the several pairs ot 
 breasts, chutes about 4 yd. wide are driven from the gangway up to the 
 airway above. These are provided with a traveling way on one side, giving 
 the miners free access to the workings. Small Headings are driven in the 
 bottom bench of coal, at right angles to these chutes, and about 10 or 20 yd. 
 apart. These headings are continued on either side of the chutes until they 
 intersect the breasts. When the chute and headings are finished, the work 
 of getting the coal in the panel is begun by going to the end of the upper- 
 most heading and widening it out on the rise side until the airway above is 
 reached and a working face oblique to the heading is formed. This face is 
 then drawn back to the chute in the middle of the panel. After the work- 
 ing face in the uppermost section has been drawn back some 10 or 12 yd., 
 work in the next section below is begun, and so on down to the gangway, 
 working the various sections in the descending order. Both sides of the 
 pillar are worked similarly and at the same time toward the chute. 
 
 Small cars, or buggies, are used to convey the coal from the working faces 
 along the headings to the chute, where it is run down to the gangway below 
 and loaded into the regular mine cars. This system affords a great degree of 
 safety to the workmen, because whenever any signs of a fall of roof or coal 
 occur, the men can reach the heading in a very few seconds and be 
 perfectly safe. A great deal of narrow work must be done before any great 
 
anthracite mining. 
 
 307 
 
 Quantity of coal can be produced. The breasts are driven in pairs and at 
 intervals, to get a fair quantity of coal while the narrow work is being done, 
 and they are not an essential part of th,e system It is claimed that the 
 facility and cheapness with which the coal can be mined, ^ 
 
 cleaned in the mine more than counterbalance the extra expense for the 
 
 nal B a tte ry°Work i n g. — Fig. 24 shows a method of opening a breast by two 
 chutes c/c, when there is a great amount of refuse, or when a great amount 
 
 Fig. 24. 
 
 25. 
 
 are 
 
 of £?as is given off. The chutes are extended up along the rib to mthm 
 fewfeet If the working face, either by planking carried ^upright posts, or 
 by building a jugular manway, as shown inthe sections (a) and (5), Fig. 
 These chutes, built of jugulars or inclined props and faced by 2 plank, 
 made as nearly air-tight as possible, to 
 carry the air from the heading a to the 
 working face. Fig. 24 shows a breast 
 on a pitch too steep to enable the miner to 
 keep up to the face. In seams of less than 
 35° the platform / shown near the face ot 
 the breast is unnecessary, and in seams 
 thicker than 12 ft. it cannot be built; 
 hence, this method of working is applica- 
 ble (1) to beds pitching more than 35°, 
 and (2) to thin seams. 
 
 The coal is separated from the refuse 
 on the platform /, and is run down the 
 manway chutes and loaded into the cars 
 from a platform projecting into the gang- 
 way g. The refuse is thrown in the middle 
 of the breast behind the platform. A cer- 
 tain amount of coal is kept on the plat- 
 form to deaden the blow from the falling 
 coal. The chutes are timbered when the 
 character of the coal requires it. This 
 plan can also be employed in thick seams 
 
 refuse’ tcf fin\ InT center^/ the breast so that the miner can work without the 
 
 Pla Fig. I 25 (a) is a section through p,p when jugulars a, a areusedtoform the 
 manways b, b along the sides of the breast; and (5) is a section through the 
 
308 
 
 METHODS OF WORKING. 
 
 same line when upright posts a, a are used to support the plank in forming 
 the manways b, b. The refuse in these eases only partially fills the gob. 
 
 In working very thick seams op heavy dips, where there is not enough 
 refuse to fill the middle of the breast, the miner has nothing to stand on, the 
 
 l 
 
 Plan along ts. Fig. 27. Section on pq. 
 
 platform being impracticable; therefore, it is necessary to leave the loose 
 coal in the breast. Loose coal occupies from 50 $ to 90$ more space than coal 
 in the solid. This surplus is drawn out through a central chute. If the roof 
 
ANTHRACITE MINING. 
 
 309 
 
 is poor, the movement of the coal will not in this way cause it to fall ano 
 mix with the coal; and if the floor is soft, the jugulars, which are stepped 
 into the floor, are not so liable to be unseated, closing the manway and 
 blocking the ventilation. The surplus is sometimes sent down the manways, 
 leaving the loose coal in the center of the breast undisturbed, until the limit 
 is reached. 
 
 Single-Chute Battery.— To prevent the coal from running out through the 
 chutes, the opening into the breast is closed by a battery constructed by 
 laying heavy logs across the openings, as shown at b, Fig. 26, or else built on 
 props, as shown at b , Fig. 27; a hole is left in the center, or at one side of the 
 battery, through which the coal may be drawn. The battery closes all the 
 openings into the breast, except the space occupied by the jugular manways, 
 and is made air-tight, or as nearly so as possible, by a covering of plank. 
 
 Fig. 26 is a plan and section of a breast opened up by a single chute. The 
 plan A is taken on the line m n shown on the section B , which section is 
 taken on the line / 1 shown on the plan A. The pitch is great and the seam 
 is so thick that the breast must be kept full of loose coal for the men to work 
 upon, the surplus being drawn off at the battery b and run into the car 
 standing on the gangway g through the chute c. A manway w is made 
 along each side of the breast, for 
 the purpose of ventilation and 
 affording a passage for the men 
 to reach the working face. The 
 heading a is used for an air- 
 course between breasts. The 
 main airway h is driven over 
 the gangway g , where it will be 
 well protected. 
 
 By drawing the surplus coal 
 through a central chute,- the 
 manways are not injured so 
 much as when it is drawn off 
 through side chutes, as the coal 
 will move principally along the 
 middle of the breast. When the 
 breast is worked up to its limit, 
 all the loose coal is run out of 
 the breast and the drawing back 
 of the pillars is commenced, 
 unless for some purpose they 
 are allowed to stand for a time. 
 
 Double-Chute Battery.— Fig. 27 
 shows a plan and section of 
 double-chute breasts used in 
 very thick seams having a heavy 
 dip. The breasts are entered by 
 two main coal chutes c, c, each 
 of which is provided with a battery 6, through which the coal is drawn. 
 A manway chute rri is driven up through the middle of the pillar for a few 
 yards and is then branched in both directions until each branch (slant 
 chute) intersects the foot of a breast near the battery b, as shown. The jugu- 
 lar manways n, n are started at this point and continued up each side 
 of the breast. The main airway h is driven in the solid through the 
 stump A above the gangway. By driving the main gangway g against the 
 roof, as shown, the pitch of the chute is lessened, and the loading chute c 
 is more readily controlled. 
 
 When the main gangway is not driven against the roof, a gate is placed 
 in the chute below the check-battery, which enables the loader to properly 
 handle the coal. Coal in excess of the amount necessary to keep the miner 
 up to the face may be drawn through the main battery, or sent down the 
 manway chute, from which it is loaded through an air-tight check-battery. 
 
 The main chutes are usually 8 or 9 ft. wide, but sometimes only for the 
 first 6 or 8 ft.; above this they are driven about 6 ft. square. The manway 
 and slant chutes are also about 6 ft. square. 
 
 When the seam is not thick enough to carry the return airway h (Fig, 27) 
 over the gangway, the chutes are driven up in the same manner as in 
 Fig. 27, for a distance of about 30 ft., where they intersect the airway. The 
 breast is opened out just above the airway, a battery being built in the airway 
 
 Fig. 28. 
 
310 
 
 METHODS OF WORKING. 
 
 immediately above each chute. A manway is driven from the gangway 
 up through the middle of the stump until it intersects the airway, and a 
 trap door is placed at this point to confine the air. This manway is made 
 about 4 ft. X 6 ft., or smaller. . . . 
 
 Fig. 28 shows a less complicated plan than Fig. 27. The main chutes n, n 
 are driven up to the heading c, from which the breast is opened out; a log 
 
 battery is built at the top 
 
 Fig. 29. 
 
 of each chute at the points 
 marked a, a. The chutes 
 are used for drawing the 
 battery coal, and for re- 
 ceiving the manway coal, 
 and are also used for trav- 
 eling ways. A check-bat- 
 tery b is placed in the chute 
 to prevent the air-current 
 from taking a short cut 
 from the gangway through 
 the chute to the breast air- 
 ways. This check-battery 
 is of great assistance to the 
 loader when the chute has 
 a very steep pitch, as he 
 can readily control the flow 
 of coal through the draw- 
 hole. 
 
 All these methods are 
 open to the objection that 
 in case of any accident to 
 the breast manway, by 
 which the flow of air, 
 
 shown by the arrows, is obstructed, there is no means of isolating the breast 
 in which the accident 
 occurs, and the venti- 
 lation of all the breasts 
 beyond it is entirely 
 stopped. To overcome 
 this, sometimes the pil- 
 lar A, shown in left- 
 hand breast, Fig. 28, is 
 left in each breast to 
 protect the airway. 
 
 Rock-Chute Mining. 
 
 Fig. 29 shows a section 
 of two seams, sepa- 
 rated by a few yards 
 of rock. Chutes from 
 4i to 7 ft. high and 7 to 
 12 ft. wide are driven 
 in the rock from the 
 gangway or level g to 
 the level l in the seam 
 above, at such an 
 angle that the coal 
 will gravitate from the 
 upper seam into the 
 gangway g. The work- 
 ing, otherwise, is sim- 
 ilar to that previously 
 described. 
 
 Rock-chute mining 
 contemplates the fol- 
 lowing sequence of 
 operation; 
 
 1. The opening of 
 all gangways and air- 
 ways in the lower seam, 
 lying a few feet above it. 
 
 Fig. 30. 
 
 to develop coal as yet untouched in a thick seam 
 
ANTHRACITE MINING. 
 
 311 
 
 2. Developing the thick bed by a regular series of rock chutes driven 
 from the gangway below; workings being opened out from chutes as in 
 ordinary pillar-and-breast working — the panel system or some other plan 
 may be found better than pillar-and-breast workings 
 
 3 Driving the breasts to the limit of the lift and robbing out the pillars 
 from a group of breasts as soon as possible, even if a localized crush is induced. 
 
 4 After one group of breasts is taken out and the roof has settled, open- 
 ing a second series of chutes for the recovery of coal from any large pillars 
 that were not taken out when the crush closed the workings. 
 
 5. While the work of recovering the pillar coal is in progress, a second 
 group of breasts may be worked, and the process continued until all the 
 area to be worked from that gangway has been exhausted. The same 
 process is employed in opening lower lifts. , , 
 
 6 When all the upper bed of coal has been exhausted, the lower seam 
 may be worked by the ordinary method. Workings in this seam may be 
 
 Fig. 31. 
 
 carried on simultaneously with the upper bed, but to avoid the possibility 
 of a squeeze destroying these workings, very large pillars must be left. 
 After exhausting the upper seam, these pillars may be advantageously 
 worked by opening one or two breasts in the center of each, and when these 
 are worked to the upper limit, attacking the thin rib on each side, com- 
 mencing at the top and drawing back. . , 
 
 When the roof of the lower bed is good, the cost of timbering and keeping 
 open the gangways and airways will be considerably less than if these were 
 driven in the upper seam, and this difference, in some cases, may be suf- 
 ficient to pay for driving all the rock chutes. . 
 
 There are three undetermined points in this connection, viz.: (1) The 
 
312 
 
 METHODS OF WORKING. 
 
 maximum distance between the two beds, or the length of rock chute that 
 can be driven with satisfactory financial results. (2) The maximum dip on 
 which such working can be successfully opened. (3) The maximum thick- 
 ness of the upper and also of the lower seam, which w T ill yield results war- 
 ranting the additional outlay w'hen rock chutes are of considerable length. 
 
 Fig. 30 show's how one or more seams are worked by connecting them by 
 a “stone drift,” or “tunnel,” driven horizontally across the measures, 
 through w hich the coal from the adjacent seams is taken to the haulage- 
 way leading to the landing at the foot of the slope or shaft. Tunnels are 
 sometimes driven horizontally through the measures from the surface, so as 
 to cut one or more seams above water level. 
 
 The low r er seam of coal is w r orked from a gangw ay or level l, connected 
 by a tunnel, or stone drift t, to the level or gangway g, in the thick 
 seam, The stone drift may be extended right and left to open seams above 
 and below' the thick seam* This tunnel, or stone drift, is never driven under 
 a breast in the upper seam, but directly under the middle of the pillar. 
 
 In the upper and thicker seam, when the coal is very hard, a breast 5 is 
 worked to the limit and the loose coal nearly all run out through the chute s 
 into the gangw ay g. The “ monkey gangway ” m is driven near the top as a 
 return airway, and is connected to the upper end of the chute s by a level 
 heading n, and to the main gangw ay g by a heading v. These headings are 
 driven for the purpose of ventilation and to provide access to the battery in 
 case the chute s should be closed. In the lower seam, the breast is still being 
 w orked upwards in the ordinary way. 
 
 The J. L. Williams method of working anthracite, Fig. 31, has been applied 
 successfully by the originator at the Richards Mine, Mt. Carmel, Pa. ; and by 
 it 90£ of the available coal is said to have been obtained. The method is a pil- 
 lar-and-stall method with the folio w T ing distinguishing points: (a) Timbering 
 the gob with props set not more than 6 ft. apart, to keep up the roof during 
 the extraction of the pillars. (6) Making holes from the crop, for the 
 delivery of timber into the workings, (c) Removing the pillars in shorter 
 lifts than is possible w hen the roof is supported with culm pillars. ( d ) Keep- 
 ing the gob open with timber for dumping the fallen rock, that would have 
 to be sent to the surface if the breasts were flushed. 
 
 Both the floor and the roof of the mine were weak, so that it w'as not 
 possible to make either the breasts or the pillars wide. In some cases, the 
 floor consisted of 3 ft. of clod, and to prevent its lifting and sliding, every 
 alternate pronwas put through the clod and its foot set in the slate beneath, 
 while the other props were set on pieces of 2" plank 2 ft. in length to keep 
 down the bottom. A small gangway X is driven to take out the chain pillar, 
 and Y is a small gangw ay for taking in timber. 
 
 Running of Coal.— In large seams, when the coal is soft and shelly 91* slip- 
 pery and lies at an angle of more than 50°, and generates large quantities of 
 firedamp a danger to be guarded against is the sudden liberation of gas 
 should a breast run; that is, should the coal at the face loosen and run out 
 bv its own gravity, only stopping w'hen it chokes or fills up the open space 
 below To meet these conditions, the air-course maybe driven above the 
 gangway and used as a return, the fan being attached as an exhaust, and 
 the working breasts ventilated in pairs. The inside manway of one of a pair 
 of breasts is connected with the gangway for the intake, and the outside 
 manwav of the other breast with the return airway, giving each pair of 
 breasts a separate split of the current. In collieries where this system of 
 working is folio w T ed, the coal is soft. A new T breast is worked up a few ^ ards, 
 but as soon as it is opened out, the coal runs freely and the manways are 
 pushed up on each side as rapidly as possible, to keep up with the face. Two 
 miners one on either side, sometimes finish a breast without being able to 
 cross to each other. The work is done exclusively with safety lamps, and 
 when a breast “runs” the gas is liberated in such quantities that it 
 frequently fills breasts from the top to the airway before the men can get 
 down the'manwav on the return side. When the gas reaches the cross-hole, 
 it passes into the ‘return airway without reaching any part where men are 
 working. Should a run of coal block a breast by closing the manway, it 
 affects the current of one pair of breasts alone. As the gangway is the 
 intake, leakage at the batteries passes into the breasts, as the cross-holes are 
 above their level and the gas is thus kept above the starter when at the 
 draw-hole. The gangway, chutes, and airway are supplied by w-ooden 
 pipes, w r hich connect with a door behind the inside chute. If a breast runs 
 up to the surface, it does not affect the return airway, as it is in the solid,. 
 
ANTHRACITE MINING . 
 
 313 
 
 Among the disadvantages urged against this system of working are the 
 
 f<)1 ItTncfeases the friction, as the air must pass in the airway all the distance 
 from the breast to the fan, the area of the airway being small in comparison 
 
 t0 ilis t^faces of the breasts are so much higher than the return airway, 
 the lighter gas must be forced down into the return against the buoyant 
 power of its smaller specific- gravity. 
 
 The reduction of friction obtained by splitting is neutralized by each split 
 running up one small manway and down another; the advantage of running 
 through several pillar headings and thus securing a shorter course being 
 lost. This can be partly obviated by ventilating the breasts in groups, but 
 the dangers avoided in splitting are increased. 
 
 Blackdamp, which accumulates in the empty or partly empty breasts, 
 works its way down and mixes with the intake current, as there is no return 
 current in the breast strong enough to carry it away, the return being closed 
 in the airway. . ^ , 
 
 All things considered, when the seam is soft and has a pitch of 40° and 
 upwards, and emits large quantities of gas in sudden outbursts, as in running 
 breasts, this system is the best that can be adopted. 
 
 When the Coal Is Hard and Gas Is Not Freely Evolved.— The reverse of the 
 svstem just described is followed at some collieries where the coal is hard 
 and but little gas is encountered. The airway is driven over the gangway 
 or against the top, the fan being used to force the air inward to the end of 
 the airway. The air is distributed as it returns, being held up at intervals 
 by distributing doors placed along the gangway. 
 
 Among the advantages claimed for this plan are the following: ■ 
 
 As the pressure is outward, it forces smoke and gas out at any openings 
 that may exist from crop-hole falls or other causes. 
 
 The warm air from the interior of the mine returning up the hoisting 
 slope or shaft prevents it from freezing. 
 
 As the current is carried from the fan to the end of each lift without pass- 
 ing through working places, the opening of doors as cars are passing, etc. 
 does not interfere with the current. . 
 
 If a locomotive is used, the smoke and gases generated by it are carried 
 away from the men toward the bottom. Locomotives are generally used 
 only from the main turnout to the bottom. 
 
 An objection to this svstem is that the gangway, as the return, is apt to be 
 smoky. Starters and loaders are forced to work in more or less smoke, and 
 even the mules work to disadvantage, while if gas is given off, it is passed 
 out over the lights of those working in the gangway. 
 
 However, in places where there is but little gas, and airways of large area 
 can be driven, this plan works very satisfactorily, and some of the best ven- 
 tilated collieries are worked upon it. . 
 
 An objection advanced by some against forcing fans is that they increase 
 the pressure, thus damming the gas back in the strata. In case the speed of 
 the fan is slacked off, the accumulated gas may respond to the lessened pres- 
 sure and spring out in large volumes from its pent-up state. This argument, 
 however, works both wavs. An exhaust fan running at a given speed is 
 taking off pressure, and if anything occurs to block the intake, the pressure 
 is diminished, and the gas responds to the decrease on the same principle. 
 
 Hints for the Smaller Seams When They Are Small and Lie From Horizontal to 
 About 10°. — ' Two gangways may be driven, the lower or main gangway being 
 the intake. Branch gangways should then be driven diagonally or at a slant, 
 with a panel or group of working places on each slant gangway. Large 
 headings should connect the panels. In this system, the air is carried 
 directly to the face of the gangway and up into the breasts, returning back 
 through the working places. The intake and return are separated by a solid 
 pillar, the only openings being the slant gangways on which are the panels. 
 
 The advantages of this plan are several: 
 
 The main gangway is solid, with the exception of the small cross-holes 
 connecting with the gangway above; these furnish air to the gangway and 
 are small and easily kept tight. These stoppings* should be built of brick, 
 and made strong enough to withstand concussion. 
 
 A full trip of wagons can be loaded and coupled in each panel or section 
 without interfering with, or detaining the traffic on, the main road; one trip 
 can be loaded while another is run out to the main gangway for transpor- 
 tation to the bottom. 
 
314 
 
 METHODS OF WORKING. 
 
 The only break in the intake current is when a trip of cars is taken out 
 from, or returns to, a panel or section; this can be partially provided against 
 by double doors, set far enough apart to permit one to close after the trip 
 before the other is opened. This distance can be secured by opening the 
 first three breasts on a back switch above the road through the gangway 
 pillar, or by running each branch over the other far enough to obtain the 
 distance for the double doors. 
 
 If it is not desired to carry the whole volume of air to the end of the air- 
 way, a split can be made at each branch road. These will act as unequal 
 splits in reducing friction, and, although not theoretically correct, are prefer- 
 able to dragging the whole current the full length of the workings. 
 
 The objections urged to this plan are that it involves too much expense 
 in the large amount of narrow work at high prices necessary to open out a 
 colliery, that it necessitates a double track the whole length of the lift, and 
 that the grade ascends into each panel or section. But the latter criticism 
 falls, because the loss of power hauling the empty wagons up a slight grade 
 is more than made up by the loaded wagons running down while the mules 
 are away putting a trip into another panel or section. 
 
 For a large colliery this is without doubt the best and cheapest system. 
 
 When the Seam is Small and Lies at an Angle of More Than 10°.— In small seams 
 lying at an angle of more than 10°, and too small to permit an airway over 
 the chutes, it is more difficult to maintain ventilation. If air holes are put 
 through every few breasts, and a fresh start obtained by closing the back 
 holes, or if an opening can be gotten through to the last lift as often as 
 the current becomes weak, an adequate amount of air can be maintained, 
 because the lift worked can be used as the intake, and the abandoned lift 
 above as the return. To ventilate fresh ground, the filling of the chutes 
 with coal will have to be depended on, or a brattice must be carried along 
 the gangway. This can be done for a limited distance only, as a brattice 
 leaks too much air. As a rule, collieries worked on this plan are run along 
 until the smoke accumulates and the ventilation becomes poor; then a new 
 hole is run through and the brattice removed and used as before for the 
 next section. This operation is repeated until the lift is worked out. Some- 
 times, to make the chutes tight, canvas covers are put on the draw' holes, 
 but as they are usually left to the loaders to adjust, they are often very 
 imperfectly applied. Then, as the coal is frequently very large, the air will 
 leak through the batteries. 
 
 This plan works very satisfactorily if the openings are made at short 
 intervals, say as frequent as every fifth breast, but the distance is usually 
 much greater to save expense. As the power of the current decreases as the 
 distance between the air holes is increased, good ventilation is entirely a 
 question of how often a cut-off is obtained. 
 
 An effective ventilation could be maintained in a small seam at a heavy 
 angle by working with short lifts, say two lifts of 50 yd. instead of one of 
 100 yd., as at present. The gangways should be frequently connected, and 
 one used as an intake and the other as a return. This would necessitate 
 driving two gangways where one is now made to do, but the additional 
 expense would be made up in the greater proportion of coal won. 
 
 FLUSHING CF CULM. 
 
 From 15 $ to 20$ of the coal taken out of an anthracite mine, according 
 to the methods used in the past, became so fine in the course of preparation 
 through the breaker that it could not be used or sold, and had to be piled 
 away as refuse. Recently, the coarser portions of these culm piles have 
 been screened out and sold for use as steam sizes, while the finer part, 
 together with the fine material from the breaker, has been carried back into 
 the mines with water to fill the abandoned portions of the underground 
 workings. 
 
 This culm is carried through a system of conveyors to the hopper, usually 
 an old oil barrel, and the stream of water is conducted into the same hopper 
 by a 3" pipe. The culm is then carried by the water through a pipe from 
 4 to 6 in. in diameter, which passes into the mine through the shaft, bore 
 hole, or other opening, thence alo-ng the gangways to the chambers through 
 the cross-cuts, and to the point where it is desired to deposit the culm. The 
 bottoms or outlets of the chambers to be filled are closed by board partitions 
 fitted closely, or by walls of slate or mine rubbish. The culm, as it issues 
 
CULM FLUSHING. 
 
 315 
 
 from the end of the pipe, takes a very flat slope, and it is carried a long dis- 
 tance by the water, which ultimately filters through the deposited culm to 
 the lower portion of the mine, to be pumped to the surface. When the 
 chamber is filled to the roof, the pipe is withdrawn and extended to the 
 next place to be filled, and so on. Wrought-iron pipe is said to be the best, 
 and the life of the pipe depends on the nature of the water used and the 
 material treated. With fresh water and small culm from the buckwheat 
 screen, it lasts 18 months; when carrying culm from the bank, ranging from 
 dust to pea'coal and some chestnut, 9 months; and when mixed with ashes, 
 6 months. The smaller the material the better. 
 
 The amount of water used depends on the distance to which the culm is 
 carried and the slope of the pipe. , i 
 
 From li to If lb. of water is required to flush 1 lb. of culm to level and 
 down-hill places; 3 to 6 lb. of water to 1 lb. of culm to flush up-hill for heights 
 varying from 10 to 100 ft. above the level of the shaft bottom. Any ele- 
 vation of the pipe very materially increases the amount of water necessary. 
 Mr James B. Davis, superintendent of the Dodson and Black Diamond 
 mines, has ascertained by experiment that 1 cu. ft. of anthracite coal ground 
 to culm can be flushed into a space of nearly 1£ cu. ft., and it is therefore 
 impossible to compress the culm more than one-third. In addition to acting 
 as a filling and a support, to prevent squeezes and crushing, flushing has 
 been advantageously used for fighting and sealing off mine fires. No 
 instance has been recorded where spontaneous combustion has taken place 
 
 in the flushed culm. 
 
 The Dodson culm* plant, which was a pioneer, cost $7,473.42, with the 
 capacity of flushing 119 tons per day, while the Black Diamond culm plant 
 is capable of flushing 287 tons per day and cost $6,280.12, but plants can 
 probably be put up much more cheaply than this. 
 
 The saving from the flushing of culm over depositing it on the surface 
 varies for the ordinary anthracite colliery from $5 to $15 per day. The 
 average cost of putting in stoppings in a 9' vein is given by Mr. Davis as 
 $9.50, including material. 
 
 To remove the pillars after the intervening breasts have been filled with 
 culm, the face of the pillar along the gangway is attacked, and a road driven 
 up through the pillar, splitting it (z, Fig. 32). This road may be the full 
 width of the pillar, but in general it is necessary to leave a narrow stump of 
 coal on either side to keep up the fine flushed material in the adjoining 
 breasts. The thickness of this supporting coal 
 depends entirely on the condition of the 
 flushed material behind it. If that is fine, 
 it will set firmly and form a compact mass 
 that will not run. In such a case, the pillar 
 may be entirely taken out, leaving a vertical 
 wall of solidly packed flushed culm. When 
 the flushed material is of a size larger than 
 buckwheat, it will not set compactly, but will 
 run when it is opened up, and when such 
 material fills the adjoining breasts, the thin 
 pillar of coal must be left to keep back the 
 culm. Timbers are placed flush up against 
 the culm or the coal stumps, as the case may 
 be, and if there is a tendency for the culm 
 to run, lagging is placed behind the timbers. 
 
 In some cases, as much as 700 ft. of timber 
 have been used per 100 ft. of pillar taken 
 out. As the pillar is removed, the top settles 
 until it finally rests upon the flushed culm, 
 and as the weight from the top and the pres- 
 sure from the sides comes upon these props, 
 they are broken, while the coal that has been 
 left will also be crushed. At the Black Dia- 
 mond colliery, the props used are 16 ft. long, 
 
 and at this colliery the top settles about 4 ft. if the flushed material is 
 packed tightly before the roof pressure comes on it. After this settling, 
 new props 12 ft. in length are put in close up against the culm and the 
 broken stump of the original pillar, and they serve to keep the road open 
 up to the working face. 
 
 Fig. 32. 
 
316 
 
 MINING MINERAL DEPOSITS. 
 
 METHODS OF MINING MINERAL DEPOSITS. 
 
 Much of what has already been given under the heading of Coal Mining 
 applies equally to the mining of mineral deposits. It will therefore not be 
 repeated under this heading, and the only methods here given will be those 
 that have not already been covered. 
 
 Highly inclined deposits are mined out as follows: Horizontal passages 
 called drifts , levels , or galleries are driven through the ore at regular intervals, 
 and connected by openings at right angles to the levels, which, in the case 
 of perpendicular or highly inclined deposits, are called winzes or raises, 
 according as they are sunk from above or raised from below. These parallel 
 openings divide the ore body into a series of rectangles, thus serving to test 
 its value. (See also Overhand-Stoping Method, page 304.) 
 
 Levels. — The distance between the individual levels depends on the 
 material being mined. They are placed nearer together in high-grade ore 
 than in low-grade material. The width of the vein also has considerable 
 influence on the distance between the levels. In veins where it is necessary 
 to break into walls to afford working room in the levels, they are usually 
 placed as far apart as is consistent with the economical handling of the 
 material in chutes and convenient access to the working faces of the stopes. 
 The distance between the levels varies from 60 to 100 ft., and should be 
 measured on the dip and not perpendicularly. 
 
 Winzes or Raises.— The distance between the raises or winzes varies from 
 30 to 250 ft., depending largely on the character of the material and the 
 
 method of getting it into the chutes or winzes. Where the material at the 
 working faces is shoveled or thrown directly into the chutes, they are often 
 placed as close as 30 ft., while if the material is carried from the working 
 face to the chute or winze in a wheelbarrow, the chutes may be much 
 farther apart. 
 
 Stoping.— For narrow deposits, there are two general styles of stoping in 
 regular use, called, respectively, underhand and overhand stoping. There are 
 several minor divisions under each. 
 
 Underhand stoping may be conveniently divided into underhand regular 
 and underhand Cornish. 
 
 The regular method of underhand stoping is illustrated in (a), Fig. 33, and 
 may be described as follows: The miner selects a place in any given level 
 or on the surface of the ore deposit from which to commence stoping. A 
 cut 6 or 7 ft. in depth and from 6 to 8 ft. in length is made. This forms the 
 first, or No. 1, bench in the stope. After this, he continues the work in each 
 direction, supporting the track, if any exists above, upon stulls or timbering. 
 After this No. 1 bench has proceeded a sufficient distance, he starts a similar 
 cut in the bottom of it, which forms No. 2 bench, and is driven in 
 both directions as before. At first the ore can be shoveled to the level 
 above, but after considerable depth has been attained, it will be necessary 
 to provide a winze, as shown at /, through which the ore from the lower 
 benches can be hoisted. Stulls covered with lagging are placed across the 
 stope behind each bench as platforms, to support the waste material. Under- 
 hand stoping by the Cornish system is illustrated in (6), Fig. 33, and differs 
 from the system just described only in that the level below has to be driven 
 first, and a winze sunk to it. a is the lower level, b the winze, and c the 
 upper level. The work is then carried on in successive benches, as described. 
 
OVERHAND STORING. 
 
 317 
 
 The advantages of the Cornish method are that any water that collects 
 in the stope flows to the lower level and does not have to be taken care ot in 
 each individual stope. Also, the ore can be tumbled down through the raise 
 to the lower level, thus avoiding the extra hoisting with a windlass, 
 or small hoist. 
 
 The advantages that apply to any system of underhand stoping are as 
 follows: The ore can be extracted at once; while the stope is new, the miner 
 is protected from the roof by stulls and stagings; the loss of fine and valuable 
 mineral is reduced, owing to the opportunity for sorting afforded during the 
 handling of the broken ore. 
 
 The disadvantages are as follows: The manner in which the ore must be 
 handled is expensive; an individual pumping plant will be necessary in each 
 stope of a wet mine with the regular system; should the mine be abandoned 
 for any length of time, the stulls become loose and allow the rock stowed 
 upon them to fall on the face of the ore, rendering the mine unsafe, and 
 burying the ore so as to require a large expenditure of time and money to 
 reopen the workings; in a wet stope, the water flows down over the working 
 faces, interfering with the workmen and forcing them to stand continually 
 
 in water. . , _ _ 
 
 Overhand Stoping.— In this system of stoping, the ore is broken down from 
 above as the work progresses. Work is usually started from the bottom of a 
 raise, as B, Fig. 34. After the lower level A has been driven, the miner 
 stands on top of the lagging over the caps and works out a slice C 5 or 6 ft. 
 high, this being followed by succeeding slices, as D and E. Chutes are 
 timbered or cribbed at intervals, through which the material may be thrown 
 down and any waste packed in the space between the chutes, as at F. In 
 cases where the entire deposit is of value, a portion of the broken ore is 
 allowed to accumulate as a platform upon which the men stand while 
 
 Fig. 34. 
 
 working, only enough being sent through the chutes to provide working 
 room. After overhand stoping is started, the work may be carried on by 
 means of breast holes, as shown at E. The force of gravity assists in break- 
 ing the rock, and reduces the powder necessary for blasting. 
 
 Where rich ore is broken, platforms of planks, or sheets of canvas or bull 
 hide covered with plank, may be placed over the filling to receive the broken 
 ore, thus preventing the loss of fine and valuable material in the filling. 
 One argument which is usually presented against overhand stoping is that 
 the roof is not secured by timbering, but this is offset by the fact that the 
 workmen are always close to the roof and thus examine its condition and 
 break off any dangerous portions or give them such support as may be 
 needed with temporary timbers. 
 
 Overhand stoping may be carried on in a number of modified forms, all of 
 which involve the principle of breaking down the material in such a manner 
 that thq work is aided by gravitation. Sometimes, where practically the 
 entire deposit is removed, temporary platforms supported on stulls are con- 
 structed close to the working face for the workmen. 
 
 The advantages of overhand stoping are that no hoisting or pumping is 
 required in the block of ore being worked, as with underhand stoping with- 
 out a winze; water gives no trouble in the stopes; less timbering is required 
 than in the underhand stoping, because no platforms are required to store 
 waste, and the timbering in working stagings is usually recovered; where 
 the mine is abandoned for a time, the working face is usually left in better 
 
318 
 
 MINING MINERAL DEPOSITS. 
 
 shape with overhand than with underhand stoping. In the overhand system, 
 gravity assists in the breaking of the ore. 
 
 The disadvantages are that the miner is forced to work under an unsup- 
 ported roof, though the fact that he is close to it enables him to examine it 
 and take care of any dangerous portions. There may be greater loss of the 
 fine and valuable material that becomes mixed with the waste than in 
 underhand stoping, though this may be largely prevented by the use of 
 boards or canvas. 
 
 Flat or Slightly Inclined Deposits.— Where flat or slightly inclined ore bodies 
 are being worked, the working drifts (corresponding to levels in steeper 
 deposits) are driven comparatively close together (about 30 ft. apart) and 
 the material between them removed in successive steps, as in underhand or 
 overhand stoping, the space behind the miner being packed with waste mate- 
 rial to support the roof, or the roof being supported by timbering until the ore 
 is removed. Sometimes pillars of ore have to be left to aid in the support 
 of the roof, and when this is the case, the miners try to leave the pillars 
 where the ore is low grade. When a deposit of ore is of uniform value 
 throughout, and the roof of a somewhat flexible character, it may be let 
 down without much, if any, stowing, as in the longwall system of coal 
 mining. In other cases, the material is removed like square work or by pillars 
 and rooms, the pillars in either case being robbed as closely as possible 
 before leaving the workings. 
 
 LARGE DEPOSITS OVER 8 FEET THICK. 
 
 With a deposit much over 8 ft. in 
 
 Fig. 35. 
 
 The advantages are that it 
 
 thickness, it is impossible to keep the 
 walls in place by stulls or single sticks 
 of timber. Large masses of mineral 
 frequently contain very valuable ma- 
 terial, and engineers have developed 
 a number of methods for the removal 
 of their valuable contents. The 
 method depends largely on the value 
 per ton of the material being removed, 
 and local conditions as to the cost of 
 labor, timber, filling material, char- 
 acter of wall rock, etc. The methods 
 used for these deposits are square work , 
 filling , caving , and square-set timbering 
 systems. 
 
 Square work, also called the cham - 
 ber-and-pillar system , is illustrated in 
 Fig. 35. Galleries are driven through 
 the ore as shown, the deeper galleries 
 being smaller than the upper ones, 
 the object being to leave larger pillars 
 for the support of the material above 
 the workings. Galleries are then 
 driven at right angles to these, to leave 
 square pillars, as shown. When this 
 system is applied to a bed that is only 
 30 or 40 ft. thick, from three-fourths to 
 eight-ninths of all the material in the 
 deposit can be removed, the remainder 
 being left as pillars; but where it be- 
 comes necessary to leave floors be- 
 tween the succeeding levels, as shown 
 in (5), Fig. 35, scarcely one-half of the 
 deposit can be removed, even when it 
 is of such a firm nature that the gal- 
 leries can be driven considerably 
 wider than the thickness of the pil- 
 lars. This system of mining is applied 
 to the removal of salt, gypsum, build- 
 ing stone, and various low-grade ores, 
 and is very similar to the room-and- 
 pillar system (see page 280). # 
 
 ;s no timbering, and that, owing to the 
 
MINING THICK DEPOSITS. 
 
 319 
 
 Fig. 36. Fig. 37. 
 
 depending principally on the cost of the stowing material compared with 
 the value per ton of the deposit being removed. In this process, the filling 
 * material should be composed entirely of large pieces, so that it can be 
 packed closely. 
 
 Transverse Rooming With Filling.— In other cases, a filling system is used 
 in which rooms or chambers are driven across the deposit and then con- 
 tinued upwards by overhand stoping, the ore being thrown to a lower level 
 through a chute cribbed up as the work progresses, and the excavated 
 space filled up with broken rock brought down through a chute from above, 
 as showm in Fig. 36. After the rooms are worked out between two levels, 
 the pillars are removed in the same manner. 
 
 Longitudinal Back Stoping With Filling.— In this case, the deposit is worked as 
 a series of overhand stopes, Fig. 37, the space below the workmen being filled 
 with broken rock a brought down through raises b from above, the ore being 
 thrown to a level c, which has been timbered through the filling material on 
 
 larger size of the chambers, the material can be removed at a low cost per 
 ton. The disadvantages are that a large portion of the deposit has to be left 
 untouched, and that where the formation being mined is at all soft, it is not 
 safe to work these large chambers. 
 
 Filling Methods.— Sometimes a filling of worthless material is substituted 
 for the worked-out ore. This system may be carried on by any one of a 
 number of different plans. 
 
 Slicing Method. — In some cases, comparatively small drifts or chambers 
 (from 6 ft. X 6 ft. to 10 ft. X 10 ft.) are driven through the ore across the 
 deposit, and then tightly packed with broken rock, after which other drifts 
 or chambers are driven beside the first ones and also packed or filled. This 
 
 S rocess is continued until a slice has been removed from under the entire 
 eposit. The process is then repeated on top of the filling, taking out suc- 
 cessive chambers and filling them, until another slice has been removed. 
 This method has been used in the copper mines of Spain, and has also been 
 tried at some mines in the United States with varying degrees of success, 
 
320 
 
 MIKING MINERAL DEPOSITS. 
 
 the first or lower floor of the stope. This method has been very successfully 
 applied to some of the large iron mines of the Lake Superior region of the 
 United States. . 
 
 The tilling material used in any one of the various filling methods may be 
 obtained at the surface, may be partially or wholly obtained from the waste 
 rock associated with the vein material and from drifts or passages that have 
 to be driven in barren ground, or it may be obtained by driving drifts into 
 the hanging wall, and opening chambers there, from which the waste may 
 be obtained. (See also Flushing of Culm, page 314.) 
 
 Caving Methods.— The longwall method of mining coal is really a caving 
 system, but where this system is applied to the mining of large masses, it 
 becomes necessary to introduce some special features. There are two 
 general systems in use, caving a back of ore and caving the gob or waste only. 
 
 Caving a Back of Ore. — In this system, drifts or levels are run through the 
 ore a few feet below the top of the deposit, as though the material above were 
 to be removed by overhand stoping, but in place of breaking the material 
 down, it is allowed to cave by gravity. When a back of ore is thick (20 ft. 
 or more), the entire stope is sometimes allowed to cave full and then the 
 broken ore removed by driving heavily timbered drifts through to the 
 farther side and drawing the crushed material into the face of the drift. 
 
 When the overlying worthless material appears, op- 
 erations are continued by removing the last set in 
 the drift and drawing the ore Horn nearer the shaft. 
 This method is continued until practically all the 
 broken ore has been removed. Where the back of 
 the ore is comparatively thin (less than 20 ft.), the 
 caving is usually accomplished at the face of the 
 drift only, the drift being driven a short distance 
 beyond the timbering without support. The ore 
 above this unsupported portion will cave in and can 
 be removed. When the waste rock and old timber 
 from above appear, the operator retreats, removing 
 one set of timber from the drift, caving and remov- 
 ing the ore over it. In this 
 manner, operations are contin- 
 ued until all the ore over the 
 drift or stope has been caved, 
 when another drift or stope is 
 driven beside the first and the 
 ore over it caved. In this 
 method, blasting has to be re- 
 sorted to only in driving the 
 drifts, from one-half to three- 
 fourths of the ore being obtained 
 without the use of powder. 
 
 The advantages of this system 
 are that little blasting is re- 
 quired; practically the entire 
 deposit is recovered; the mining 
 cost per ton is very low. 
 
 The disadvantages are that 
 the ore is liable to become 
 mixed with more or less dirt, which caves down with it; only one level of the 
 Sine can be operated at a time, and the surface of the ground is allowed to 
 cave into the openings, thus rendering it unfit for ordinary surface uses. 
 
 Caving the Waste Only.— In this system, Fig. 38, drifts A and galleries B 
 are drivin through the top of the ore body immediately under the waste 
 rock. After one of these drifts or galleries is completed, the floor is covered 
 with a lagging of plank or poles, and the waste material allowed to cav eon 
 to this platform. Subsequently, other drifts are driven beside the first one, 
 the floor covered with lagging, and the waste allowed to cave. This P r ocess 
 is continued until a slice has been removed over the entire surface of the 
 ore deposit when more drifts are driven lower down and another slice 
 removed After the first slice has been removed, the broken ot waste mate- 
 rial is supported on the lagginglaid on the floors of the first d rifts, and hence 
 the miners have only to support this lagging m order to supigrt the waste. 
 The caving of any individual drift crushes the ore on either side to a con- 
 siderable extent, thus materially reducing the blasting expense. 
 
 Fig. 38. 
 
IRREGULAR DEPOSITS. 
 
 321 
 
 The advantages of this system are that the entire deposit is recovered*, 
 little blasting is required; the ore obtained is clean; the mining expense is 
 comparatively low per ton. , „ . _ , . „ 
 
 The disadvantages are that only one level of a mine can be producing at a 
 time; the surface is allowed to cave, thus rendering it unfit for surface uses. 
 
 SQUARE-SET SYSTEM. 
 
 Frequently, large masses of material are encountered, which it is neces- 
 sary to remove, and at the same time support the surrounding material. 
 At times, it is not desirable to fill the 
 stope while the ore is being removed, 
 and, at the samq time, it is impossible 
 to support the walls by single sticks 
 or stulls. To overcome these diffi- 
 culties, the square-set system has been 
 evolved, which consists in the sup- 
 porting of the walls by means of a 
 series of square frames, from 6 to 9 ft. 
 square, which are placed in position 
 as fast as the ore is removed. The use 
 of these frames reduces the length of 
 the individual sticks, and so produces 
 a firm structure. The timbers may 
 be square-sawed material or round 
 logs. If the walls are soft, the sides 
 and top may require lagging, and if 
 the floor is soft or composed of ore, 
 sills will be necessary under the posts. 
 
 The mining ;s carried on by overhand- 
 stoping system, removing one block at 
 a time and replacing it with the square 
 set. Fig. 39 represents a stope, the 
 walls and roof of which are supported 
 by square sets that are lagged from the outside, 
 sets are made from round timber. 
 
 Fig. 39. 
 
 In this case, the square 
 
 IRREGULAR DEPOSITS. 
 
 Covoting or Gophering.— Bodies of valuable material frequently occur that 
 cannot be mined by any regular system. These are recovered by simply 
 following the ore throughout its irregularities and removing it with the use 
 of as little supporting timber or other material as possible. Owing to the 
 crooked and irregular passages that occur in such mines, the work has been 
 called coyoting , or gophering. Sometimes regular levels are driven at stated 
 intervals, and the coyoting, or gophering carried on from them Many of 
 the small gold and silver mines of the West, the mines of Mexico and South 
 America, and the Missouri lead and zinc deposits are worked by this system, 
 the object being to remove as much of the ore as possible without the use of 
 
 timbering or the driving of unnecessary passages. . 
 
 Probably one of the best examples of working irregular deposits is the 
 mining practice in the Joplin zinc district, Missouri. The deposits of zinc 
 blende are irregularly distributed through a limestone rock, and the mining 
 is carried on in a very crude and irregular fashion. 
 
 After an ore bodv has been found by drilling, a shaft 5 ft. X 5 jt. to 6 it X 
 9 ft. in the clear is sunk by the contractor, the price being $4 per foot for soft 
 and $9 per foot for hard ground for the first 50 to 80 ft., the contractor doing 
 all the work in sinking and timbering. Through the soft ground, the shaft is 
 timbered by 4" round poles or by 2" X 4" or 2" X 6" timbers laid flatwise, 
 skin to skin. The mines are divided into four kinds. (1) Very hard mines 
 that require all the ore to be drilled with air drills and blasted out, and 
 require no timbering. (2) Mines that are hard but have open crevices 
 between the strata where a hand drill can be driven and a charge of dyna- 
 mite lodged and exploded, throwing down a large amount of dirt and so 
 iarring the surrounding ground that it may be easily cut down with the 
 miner’s pick. This kind of ground needs no timber. (3) Mines that are 
 moderately soft and where the miner can place a blast anywhere by driving 
 
822 
 
 COSTS OF MINING ANTHRACITE, 
 
 a spud, throwing down a large amount of ore. The drifts are carried 
 10 ft. X 12 ft. in the clear, and are cut ahead from 6 to 10 ft. before putting 
 in the sets of timber and laggings to hold the roof. (4) Mines that are very 
 soft and where a drift cannot be carried over 8 ft. X 10 ft. in size, where the 
 getting of the ore is all performed by pick and shovel, and where it is neces- 
 sary to timber close and drive spiling overhead as well as along the sides 
 and to resort to mud -sills in the floor of the drift. 
 
 When the shaft reaches the ore and the drift is extended for some dis- 
 tance to prove the ore bodv, underhand stoping is used and 15' holes 
 are drilled bv hand in the bottom. A charge of 50 lb. of 40* dynamite lifts a 
 stope 10 ft. X 10 ft. The cost of 75 tons of ore, hoisting it and dumping it on 
 the mill platform during a shift of 9 hours in the two classes of hard mines 
 mentioned, is, according to Mr. E. Hedburg, as follows: 
 
 1 ground boss - 
 
 2 miners at §2.00 
 
 2 miners at §1.75 
 
 2 shovelers at §1.75 
 
 1 hoister 
 
 1 engineer, who also sharpens picks and drills 
 
 1 engineer 
 
 Dynamite - 
 
 Fuel 
 
 Oil and supplies 
 
 Superintendent - 
 
 Total - 
 
 § 2.50 
 
 4.00 
 3.50 
 
 3.50 
 1.75 
 2.25 
 
 1.50 
 
 6.00 
 
 2.50 
 
 2.50 
 
 3.50 
 $33.40 
 
 Or 44.5 cents per ton of rough ore; this includes pumping the mine. 
 
 In verv soft ground, a drift 8 ft. to 10 ft. high is driven, a spiling put in 
 the top and sides. When one level is worked out, the whole drift is then 
 caved from the surface and allowed to settle down on the floor of timbers. 
 The cost of mining in soft ground is about the same as in hard ore, as the 
 saving of labor and dynamite is expended in timber and time. A typical 
 primitive mining plant in this region, which has a shaft 150 ft. deep, with 
 pump, hoisting engines, and boilers, and including hand jigs, screens, and 
 tools, costs from §2,000 to §3,000; more modern plants are however now 
 being erected, costing §8.000 to §10,000. 
 
 SPECIAL METHODS. 
 
 Frozen Ground.— When the material of placer deposits is frozen, as in 
 Alaska and Siberia, it is mined by building a fire on the surface, which 
 thaws the earth to a depth of from 1 ft. to 14 in. The embers are then 
 scraped awav and the thawed material removed. By repeating this 
 operation, a shaft can be sunk, and then, by building a fire against one side, 
 a drift can be started and continued by thawing the face. 1 ft. at a time. 
 It has been found that 1 ft. of timber piled against the face of a drift will 
 thaw to a depth of about 1 ft. The latest practice thaws the frozen grounc 
 by means of a steam jet instead of by fire. The openings have to be securely 
 but not heavily, timbered. 
 
 Leaching Methods.— Salt, copper, and sulphur have been mined by leaching 
 methods. In the case of salt, a hole is drilled into the salt formation, water 
 allowed to flow down and dissolve the salt, and is then pumped out as a con- 
 centrated brine. For excavating upward in salt, a jet of water is made to 
 play upon the roof of the level to be raised, and the resulting brine is carried 
 off 'in launders. „ _ 
 
 When old workings containing the sulphides of copper are left exposed 
 to the action of air and to percolating waters, part of the copper is converted 
 into soluble sulphate. Water pumped from such mines may be a profitable 
 source of the metal, for bv passing it over iron bars or scrap iron the copper 
 will be separated and deposited as cement copper in the bottom of the 
 vessel containing the iron. . 
 
 In the case of sulphur, superheated steam is forced down to melt the 
 sulphur, which is then pumped out. 
 
LEHIGH REGION. 
 
 323 
 
 COSTS QF MINING ANTHRACITE. 
 
 The following costs include only labor and supplies, and do not include, 
 in general, improvements, royalties, taxes, and other similar fixed charges 
 that are independent of the method of mining. 
 
 LEHIGH REGION (PENNA.). 
 
 The costs for the Lehigh region, though based on the results of a single 
 company, are believed to be very fairly representative of the entire region. 
 They are the mean costs of two collieries where about 2,000 men were 
 employed inside and outside, and apply to the year 1897, when the con- 
 dition at all anthracite mines was very unfavorable to economical working, 
 as the mines were then working on very short time. 
 
 The tonnage at these collieries for the year was as follows: 
 
 January, 29,775.04 
 February, 30,872.97 
 March, 42,827.04 
 
 April, 38,553.08 
 
 May, 34,090.02 
 June, 35,761.89 
 July, 44,409.13 
 August, 37,500.97 
 
 September, 27,406.94 1 
 October, 56,710.04 ! Total, 
 November, 48,177.94 [ 463,672.08. 
 
 December, 37,587.02 J 
 
 The following tables show the distribution of this output by sizes during 
 the year, and the costs per ton itemized under the several headings given: 
 
 Percentages of Different Sizes. 
 
 Month. 
 
 Lump. 
 
 Broken. 
 
 Egg. 
 
 Stove. 
 
 Chestnut. 
 
 Pea. 
 
 January 
 
 10.56 
 
 23.59 
 
 18.27 
 
 18.69 
 
 14.21 
 
 14.68 
 
 February 
 
 12.34 
 
 22.54 
 
 18.11 
 
 17.98 
 
 14.50 
 
 14.53 
 
 March 
 
 11.85 
 
 19.26 
 
 18.81 
 
 19.69 
 
 13.18 
 
 17.21 
 
 April 
 
 12.81 
 
 19.21 
 
 18.35 
 
 19.48 
 
 11.14 
 
 19.01 
 
 May 
 
 13.81 
 
 19.11 
 
 18.35 
 
 19.01 
 
 11.32 
 
 18.40 
 
 June 
 
 15.29 
 
 18.60 
 
 17.73 
 
 19.08 
 
 11.23 
 
 18.07 
 
 July 
 
 14.26 
 
 19.89 
 
 17.41 
 
 18.30 
 
 11.25 
 
 18.89 
 
 August 
 
 13.56 
 
 20.26 
 
 18.10 
 
 17.72 
 
 11.49 
 
 18.87 
 
 September 
 
 12.31 
 
 20.41 
 
 18.27 
 
 18.28 
 
 11.61 
 
 19.12 
 
 October 
 
 12.21 
 
 18.01 
 
 19.15 
 
 18.89 
 
 12.28 
 
 19.46 
 
 November 
 
 11.40 
 
 18.53 
 
 19.78 
 
 19.79 
 
 12.01 
 
 18.49 
 
 December 
 
 10.78 
 
 19.56 
 
 20.09 
 
 20.32 
 
 12.07 
 
 17.18 
 
 Year 
 
 12.58 
 
 19.71 
 
 18.59 
 
 18.99 
 
 12.15 
 
 17.98 
 
 Costs of Mining and Preparation. 
 
 Month. 
 
 Outside. 
 
 Inside. 
 
 Total 
 
 Cost. 
 
 Credits. 
 
 Net Cost 
 
 Labor. 
 
 Supplies. 
 
 Total. 
 
 Labor. 
 
 Supplies. 
 
 Total. 
 
 January... 
 
 .300 
 
 .109 
 
 .409 
 
 .951 
 
 .196 
 
 1.147 
 
 1.595 
 
 .100 
 
 1.495 
 
 February.. 
 
 .297 
 
 .085 
 
 .382 
 
 .909 
 
 .190 
 
 1.099 
 
 1.519 
 
 .063 
 
 1.456 
 
 March 
 
 .243 
 
 .047 
 
 .290 
 
 .844 
 
 .151 
 
 .995 
 
 1.311 
 
 .088 
 
 1.223 
 
 April 
 
 .242 
 
 .071 
 
 .313 
 
 .822 
 
 .137 
 
 .959 
 
 1.303 
 
 .075 
 
 1.228 
 
 May 
 
 .251 
 
 .100 
 
 .351 
 
 .852 
 
 .166 
 
 1.018 
 
 1.397 
 
 .103 
 
 1.294 
 
 June 
 
 .300 
 
 .079 
 
 .379 
 
 .500 
 
 .203 
 
 .703 
 
 1.576 
 
 .103 
 
 1.473 
 
 July 
 
 .240 
 
 .063 
 
 .303 
 
 .487 
 
 .162 
 
 .649 
 
 1.485 
 
 .084 
 
 1.401 
 
 August .... 
 
 .248 
 
 .095 
 
 .343 
 
 .709 
 
 .182 
 
 .891 
 
 1.579 
 
 .085 
 
 1.494 
 
 September 
 
 .278 
 
 .096 
 
 .374 
 
 .682 
 
 .158 
 
 .840 
 
 1.588 
 
 .054 
 
 1.534 
 
 October.... 
 
 .228 
 
 .093 
 
 .321 
 
 .721 
 
 .129 
 
 .850 
 
 1.580 
 
 .072 
 
 1.508 
 
 N ovember 
 
 .247 
 
 .093 
 
 .340 
 
 .806 
 
 .210 
 
 1.016 
 
 1.846 
 
 .091 
 
 1.755 
 
 December. 
 
 .290 
 
 .061 
 
 .351 
 
 .833 
 
 .220 
 
 1.053 
 
 1.883 
 
 .090 
 
 1.793 
 
 Year 
 
 .271 
 
 .092 
 
 .363 
 
 1.109 
 
 .162 
 
 1.271 
 
 1.634 
 
 .088 
 
 1.546 
 
324 
 
 COSTS OF MINING ANTHRACITE. 
 
 Cost per Ton of Supplies Used Inside. 
 
 Distribution. 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May. 1 
 
 Jun. 
 
 JuJ. 
 
 Oils 
 
 .010 
 
 .011 
 
 .007 
 
 .010 
 
 .007 
 
 .009 
 
 .006 
 
 Powder 
 
 .033 
 
 .030 
 
 .031 
 
 .026 
 
 .030 
 
 .032 
 
 .029 
 
 Lumber 
 
 .007 
 
 .025 
 
 .011 
 
 .004 
 
 .002 
 
 .008 
 
 .007 
 
 Props 
 
 Feed 
 
 .040 
 
 .027 
 
 .021 
 
 .015 
 
 .022 
 
 .027 
 
 .019 
 
 .039 
 
 .018 
 
 .019 
 
 .017 
 
 .023 
 
 .022 
 
 .022 
 
 Mules killed, etc 
 
 .010 
 
 .005 
 
 .014 
 
 .019 
 
 .013 
 
 .009 
 
 .015 
 
 T rails, frogs, etc. 
 
 Wire ropes 
 
 .005 
 
 .012 
 
 .018 
 
 .007 
 
 .016 
 
 .009 
 
 .018 
 
 .010 
 
 .025 
 
 .010 
 
 .013 
 
 General supplies , — - 
 
 .019 
 
 .021 
 
 .014 
 
 .009 
 
 .018 
 
 .021 
 
 .013 
 
 Total general supplies 
 
 .166 
 
 .167 
 
 .140 
 
 .109 
 
 .133 
 
 .163 
 
 .134 
 
 Pumping machinery 
 
 Hoisting machinery 
 
 Ventilating machinery 
 
 Boilers 
 
 .009 
 
 .005 
 
 .004 
 
 .012 
 
 .016 
 
 .023 
 
 .004 
 
 Mine cars 
 
 Engines 
 
 .021 
 
 .018 
 
 .007 
 
 .016 
 
 .017 
 
 .,017 
 
 .024 
 
 Total repairs 
 
 .030 
 
 .023 
 
 .011 
 
 .028 
 
 .033 
 
 .040 
 
 .028 
 
 Total cost inside ' 
 
 .196 
 
 .190 
 
 .151 
 
 .137 
 
 .166 
 
 .203 
 
 .162 
 
 Credits 
 
 .100 
 
 .063 
 
 .088 
 
 .075 
 
 .103 
 
 .103 
 
 .084 
 
 Net cost inside - 
 
 .096 
 
 .127 
 
 .063 
 
 .062 
 
 .063 
 
 .100 
 
 .078 
 
 Cost per Ton of Supplies Used Outside. 
 
 Distribution. 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May. 
 
 Jun. 
 
 Jul= 
 
 QJ]g 
 
 .010 
 
 .008 
 
 .007 
 
 .006 
 
 .009 
 
 .009 
 
 .008 
 
 T.nmhpr 
 
 .022 
 
 .025 
 
 .003 
 
 .016 
 
 .015 
 
 .005 
 
 .012 
 
 Feed - 
 
 .008 
 
 .004 
 
 .005 
 
 .004 
 
 .005 
 
 .005 
 
 .006 
 
 Mules killed, etc 
 
 T rails, frogs, etc 
 
 
 .001 
 
 .003 
 
 .004 
 
 .005 
 
 .002 
 
 
 Wire ropes 
 
 General supplies 
 
 .020 
 
 .021 
 
 .015 
 
 .008 
 
 .040 
 
 .016 
 
 .019 
 
 Total general supplies 
 
 .060 
 
 .058 
 
 .031 
 
 I .037 
 
 .073 
 
 .040 
 
 .047 
 
 Pumping machinery 
 
 Hoisting machinery 
 
 .001 
 
 .008 
 
 .002 
 
 .001 
 
 ! .001 
 
 .003 
 
 .008 
 
 .001 
 
 Ventilating machinery 
 
 .028 
 
 .012 
 
 .008 
 
 .014 
 
 .017 
 
 .003 
 
 .008 
 
 Breaker machinery 
 
 .011 
 
 Boilers 
 
 .006 
 
 .005 
 
 .002 
 
 | .013 
 
 .001 
 
 .011 
 
 .005 
 
 RroalrPT 
 
 .006 
 
 .008 
 
 .005 
 
 | .006 
 
 .005 
 
 .006 
 
 .002 
 
 Tracks and sidings 
 
 Miscellaneous 
 
 1 
 
 
 
 .001 
 
 
 
 
 r Vt\tnl wnnirs 
 
 .049 
 
 .027 
 
 .016 
 
 .034 
 
 .027 
 
 .039 
 
 .016 
 
 I Uttlo 1 G 1 O ----- 
 
 
 
 
 
 Total cost dutside 
 
 
 
 1 .109 
 
 .085 
 
 .047 
 
 1 .071 
 
 .100 
 
 .079 
 
 .063 
 
 Net cost outside 
 
 .109 
 
 .085 
 
 1 
 
 .047 
 
 ! .071 
 
 .100 
 
 .079 
 
 .063 
 
 In the two tables above and the one following, the figures were available 
 for seven months of the year only, but an average for these months gives a 
 fair average for the year. 
 
WYOMING REGION. 
 
 32S 
 
 Itemized Cost of Outside Labor. 
 
 Occupations. 
 
 Jan. 
 
 Feb. 
 
 March. 
 
 April* 
 
 May. 
 
 June. 
 
 July. 
 
 Foreman and assistants 
 
 .013 
 
 .012 
 
 .007 
 
 .006 
 
 .006 
 
 .006 
 
 .003 
 
 Clerks, shipper and supply 
 
 .004 
 
 .004 
 
 .003 
 
 .003 
 
 .002 
 
 .002 
 
 .002 
 
 Hoisting engineers 
 
 .009 
 
 .022 
 
 .018 
 
 .021 
 
 .019 
 
 .026 
 
 .021 
 
 Pump and fan engineers 
 
 .003 
 
 .003 
 
 .003 
 
 .002 
 
 .003 
 
 .003 
 
 
 Locomotive engineers and helpers 
 
 .014 
 
 .014 
 
 .011 
 
 .009 
 
 .011 
 
 .013 
 
 .011 
 
 Firemen and ashmen 
 
 .078 
 
 .071 
 
 .056 
 
 .060 
 
 .063 
 
 .069 
 
 .057 
 
 Stablemen 
 
 .005 
 
 .005 
 
 .003 
 
 .003 
 
 .004 
 
 .005 
 
 .003 
 
 Watchmen 
 
 .007 
 
 .006 
 
 .004 
 
 .005 
 
 .006 
 
 .006 
 
 .004 
 
 Total miscellaneous 
 
 .133 
 
 .137 
 
 .105 
 
 .109 
 
 .114 
 
 .130 
 
 .101 
 
 Topmen and footmen 
 
 .004 
 
 .004 
 
 .002 
 
 .003 
 
 .003 
 
 .004 
 
 .003 
 
 Top drivers and oilers 
 
 .007 
 
 .007 
 
 .006 
 
 .006 
 
 .007 
 
 .008 
 
 .007 
 
 Dumpmen 
 
 .002 
 
 .003 
 
 .002 
 
 .002 
 
 .002 
 
 .003 
 
 .002 
 
 Platform and docking boss 
 
 .014 
 
 .013 
 
 .013 
 
 .012 
 
 .012 
 
 .012 
 
 .012 
 
 Chute bosses 
 
 .006 
 
 .006 
 
 .005 
 
 .006 
 
 .005 
 
 .004 
 
 .006 
 
 SI are pickers 
 
 .059 
 
 .063 
 
 .058 
 
 .053 
 
 .048 
 
 .056 
 
 .057 
 
 Car loaders 
 
 .007 
 
 .007 
 
 .007 
 
 .006 
 
 .006 
 
 .008 
 
 .008 
 
 Breaker engineer 
 
 .003 
 
 .003 
 
 .002 
 
 .002 
 
 .002 
 
 .001 
 
 .002 
 
 Dirt and plane engineer 
 
 .007 
 
 .001 
 
 .001 
 
 .001 
 
 .001 
 
 .001 
 
 .001 
 
 Rock and dirt men 
 
 .007 
 
 .009 
 
 .009 
 
 .006 
 
 .004 
 
 .005 
 
 .005 
 
 General laborers 
 
 .012 
 
 .013 
 
 .009 
 
 .011 
 
 .020 
 
 .022 
 
 .012 
 
 Total breaker 
 
 .128 
 
 .129 
 
 .114 
 
 .108 
 
 .110 
 
 .124 
 
 .115 
 
 Pumping machinery 
 
 .001 
 
 .001 
 
 
 
 
 
 
 Hoisting machinery 
 
 .010 
 
 .005 
 
 .005 
 
 .005 
 
 .006 
 
 .009 
 
 .005 
 
 Ventilating machinery 
 
 
 
 
 
 
 .003 
 
 
 Breaker machinery 
 
 .005 
 
 .004 
 
 .004 
 
 .006 
 
 .003 
 
 .002 
 
 .002 
 
 Boilers 
 
 .004 
 
 .004 
 
 .001 
 
 
 
 .005 
 
 .002 
 
 Breakers 
 
 .007 
 
 .009 
 
 .007 
 
 .005 
 
 .011 
 
 .019 
 
 .006 
 
 Tracks and sidings 
 
 .008 
 
 .007 
 
 .007 
 
 .009 
 
 .007 
 
 .008 
 
 .007 
 
 Miscellaneous 
 
 .004 
 
 .001 
 
 
 
 
 
 .002 
 
 Total repairs 
 
 .039 
 
 .031 
 
 .024 
 
 .025 
 
 .027 
 
 .046 
 
 .024 
 
 Total cost outside labor 
 
 .300 
 
 .297 
 
 .243 
 
 .242 
 
 .251 
 
 .300 
 
 .240 
 
 WYOMING REGION (PENNA.). 
 
 The following tables of costs for the Wyoming region give mean results 
 from a number of different collieries which are quite widely separated 
 in location and at which the conditions of working are so different that 
 the mean results given are thought to represent average results for the 
 entire region. They also apply, approximately, to the Lackawanna Valley, 
 where the general conditions are the same, although the seams are much 
 nearer the surface than in the Wyoming region, and the amount of gas 
 present in the coal is much less. These same figures are probably also fairly 
 representative of the Schuylkill and Shamokin fields. 
 
 The collieries for which the following figures are averages are all operated 
 through shafts, varying in depth from 350 to 1,100 ft., and many of the mines 
 are extremely gaseous. The number includes several entirely new and 
 modern surface and underground plants, and the others, though not new, 
 have been overhauled and modernized as much as possible. At these 
 collieries 10,000 men were employed during the year 1895, for which the 
 data are given, and during the same year the output was 1,862,144 tons, 
 distributed during the year as follows: 
 
 Month. 
 
 Ton- 
 
 nage. 
 
 Days 
 
 Worked. 
 
 Month. 
 
 Ton- 
 
 nage. 
 
 Days 
 
 Worked. 
 
 Month. 
 
 Ton- 
 
 nage. 
 
 Days 
 
 Worked. 
 
 January. 
 
 107,952 
 
 7.94 
 
 May 
 
 179,752 
 
 12.84 
 
 September 
 
 161,213 
 
 11.52 
 
 February 
 
 98,109 
 
 7.37 
 
 June 
 
 164,062 
 
 11.92 
 
 October.... 
 
 198,161 
 
 13.90 
 
 March.... 
 
 141,991 
 
 9.95 
 
 July 
 
 145,445 
 
 10.59 
 
 November 
 
 228,433 
 
 17.15 
 
 April .... 
 
 136,375 
 
 9.69 
 
 August .. 
 
 177,241 
 
 12.96 
 
 December 
 
 123,406 
 
 8.87 
 
326 
 
 COSTS OF MINING ANTHRACITE. 
 
 Percentages of Different Sizes. 
 
 Month. 
 
 Lump. 
 
 Steamer. 
 
 Broken. 
 
 Egg. 
 
 Stove. 
 
 Chestnut. 
 
 Pea. 
 
 January 
 
 8.21 
 
 .02 
 
 17.53 
 
 20.31 
 
 21.46 
 
 18.04 
 
 14.43 
 
 February 
 
 8.29 
 
 .12 
 
 17.75 
 
 20.41 
 
 20.85 
 
 17.44 I 
 
 15.14 
 
 March 
 
 6.20 
 
 .55 
 
 17.64 
 
 20.04 
 
 20.65 
 
 18.00 
 
 16.92 
 
 April 
 
 7.01 
 
 .38 
 
 16.76 
 
 20.17 
 
 20.92 
 
 18.12 
 
 16.64 
 
 May 
 
 4.79 
 
 .27 
 
 18.63 
 
 20.33 
 
 21.42 
 
 18.23 
 
 16.33 
 
 June 
 
 3.29 
 
 .21 
 
 22.43 
 
 19.72 
 
 20.21 
 
 18.57 
 
 15.57 
 
 July 
 
 August 
 
 September 
 
 October 
 
 7.84 
 
 .42 
 
 19.42 
 
 19.54 
 
 19.46 
 
 18.58- ! 
 
 14.74 
 
 5.05 
 
 .57 
 
 19.84 
 
 20.69 
 
 18.92 
 
 18.62 
 
 16.31 
 
 4.25 
 
 .27 
 
 18.81 
 
 21.98 
 
 19.98 
 
 19.32 
 
 15.39 
 
 4.72 
 
 .01 
 
 16.77 
 
 22.00 
 
 21.27 
 
 19.88 
 
 15.35 
 
 November 
 
 2.69 
 
 .16 
 
 15.56 
 
 22.42 
 
 22.66 
 
 20.71 
 
 15.80 
 
 December 
 
 4.40 
 
 .57 
 
 14.03 
 
 22.62 
 
 21.27 
 
 21.41 
 
 15.70 
 
 Year 
 
 5.23 
 
 .29 
 
 17.96 
 
 20.95 
 
 20.80 
 
 • 19.03 
 
 15.74 
 
 Costs of Mining and Preparation per Ton. 
 
 ! 
 
 Months. 
 
 Outside. 
 
 Inside. 
 
 Total Cost. 
 
 Credits. 
 
 Net Cost. 
 
 Labor. 
 
 Supplies, j 
 
 Repairs. 
 
 Total. 
 
 Labor. 
 
 Supplies. 
 
 Repairs. 
 
 Total. 
 
 January 
 
 .363 
 
 .042 
 
 .014 
 
 .419 
 
 .934 
 
 .249 
 
 .028 
 
 1.211 
 
 1.630 
 
 .120 
 
 1.510 
 
 February 
 
 .376 
 
 .042 
 
 .014 
 
 .432 
 
 .047 
 
 .273 
 
 .030 
 
 1.250 
 
 1.682 
 
 .104 
 
 1.578 
 
 March 
 
 .297 
 
 .031 
 
 .010 
 
 .338 
 
 .872 
 
 .182 
 
 .022 
 
 1.076 
 
 1.414 
 
 .096 
 
 1.318 
 
 April 
 
 .305 
 
 .034 
 
 .023 
 
 .362 
 
 .870 
 
 .203 
 
 .020 
 
 1.093 
 
 1.455 
 
 .103 
 
 1.352 
 
 May 
 
 .270 
 
 .022 
 
 .011 
 
 .303 
 
 .839 
 
 .164 
 
 .015 
 
 1.018 
 
 1.321 
 
 .101 
 
 1.220 
 
 June 
 
 .290 
 
 .032 
 
 .011 
 
 .333 
 
 .874 
 
 .206 
 
 .018 
 
 1.098 
 
 1.431 
 
 .105 
 
 1.326 
 
 July 
 
 .309 
 
 .046 
 
 .019 
 
 .374 
 
 .879 
 
 .266 
 
 .033 
 
 1.178 
 
 1.552 
 
 .098 
 
 1.454 
 
 August 
 
 .286 
 
 .030 
 
 .017 
 
 .333 
 
 .873 
 
 .194 
 
 .026 
 
 1.093 
 
 1.426 
 
 .102 
 
 1.324 
 
 September — 
 
 .284 
 
 .039 
 
 .012 
 
 .335 
 
 .890 
 
 .201 
 
 .024 
 
 1.115 
 
 1 1.450 
 
 .105 
 
 1.345 
 
 October 
 
 .267 
 
 .036 
 
 .013 
 
 .316 
 
 .856 
 
 .188 
 
 .020 
 
 1.064 
 
 1.380 
 
 .100 
 
 i 1.280 
 
 November .... 
 
 .262 
 
 .029 
 
 .010 
 
 j .301 
 
 .860 
 
 .214 
 
 .018 
 
 1.092 
 
 1.393 
 
 .104 
 
 1.289 
 
 December .... 
 
 .344 
 
 .045 
 
 .018 
 
 ! .407 
 
 .954 
 
 .307 
 
 .028 
 
 1.289 
 
 1.696 
 
 .120 
 
 1.576 
 
 Year 
 
 .297 
 
 .034 
 
 .014 
 
 | .345 
 
 .881 
 
 .214 
 
 .023 
 
 1.118 
 
 1.463 
 
 .104 
 
 1.359 
 
 Coal Production of United States. 
 
 Year. 
 
 Bituminous. 
 
 Anthracite. 
 
 Tons of 2,000 Lb. 
 
 Value. 
 
 Tons of 2,240 Lb. 
 
 Value. 
 
 1890 
 
 1895 
 
 1897 
 
 1898 
 
 1899 
 
 111,302,322 
 
 135,118.193 
 
 147,609.985 
 
 166,592,023 
 
 193,321,987 
 
 $110,420,801 
 
 115,779,771 
 
 119,567,224 
 
 132,586,313 
 
 167,935,304 
 
 46,468,641 
 
 57.999,937 
 
 52,611,680 
 
 53,382,644 
 
 53,944,647 
 
 $66,383,772 
 
 82,019,272 
 
 79,301,954 
 
 75,414,537 
 
 88,142,130 
 
 PRICES OF COAL. 
 
 The table on page 327, given by the U. S. Geological Survey ,will be of interest 
 as showing the fluctuations in the average prices ruling in each State since 188t. 
 Prior to that vear. the statistics were not collected with sufficient accuracy to 
 make a statement of average prices of any practical value. These averages are 
 obtained bv dividing the total value by the total product, except for the years 
 1886, 1887, and 1888, when the item of colliery consumption was not considered 
 
Average Prices per Short Ton for Coal at the Mines Since 1886. 
 
 PRICES OF COAL. 327 
 
 1901. 
 
 05 CO O CO O CO rH rH 05 ©1 lO 0© rH CO CO ©!©00©©la0©>©l©©©«0© 
 
 OHOHCj © © so CO ©J 05 05 t}J ©J TjH T(H©l©lO5©©rHt''-©J00©QOCO 
 
 rH rH ©4 rH >—1 HHHHH H H H Hr- In ©4 rH rH rH rH rH 
 
 1.04 
 
 1.67 
 
 1.18 
 
 •0061 
 
 N-^iOWNOHCOiOGOMCJOOCOHCO t^©)©l©l">HI>Tt<CO<0©©rH© 
 
 HHOHHOOO^CONOOHCjO C0CC^ONOJH<O(N00 0500M 
 
 HHCOHH^HHHHri tHVh rH iHi—irHrHCO H H H rH rH 
 
 1.04 
 
 1.49 
 
 1.14 
 
 1899. 
 
 05 1> 00 <N © O iO 00 CO T*KO 05 CO 05 O O © 00 rH I> <M 00 CO 
 
 ©rH©rH©©OOOOHJC^rHt^I>-CO<NiO C0C0rH00Ol>C0i0<N©l>«0<N 
 
 rH rH ©4 rH rH © HHH rH rH rH rH rH rH CO HH rH rH 
 
 rO 
 
 .87 
 
 1.46 
 
 1.01 
 
 1898. 
 
 iCC0C0iOHN0OH(N^OiO5<ONNN © © rH CO © IH t'- © © 00 rH CO 
 
 t^©U0rH00u0lH0qc0rH©lHlH'^©lO COC^rHOO©©l^©<NlOlH©(N 
 
 © rHCirH c4 HHH HHH rH rH rH CO H H rH rH 
 
 .80 
 
 1.41 
 
 £ 
 
 1897. 
 
 00©iCt^<MC0!NTtlTflC0a0O5©©a0© OOT^OOOO©©rH(M©lH'HCOrH 
 
 00©uOrHl^COl^OOCOr-irHt-l> : T}|©t^ COC0©l>©©00©rH©O5©©J 
 
 © rH ©1 rH CO rHrHrH rH rH rH rHrHrH CO rH rH rH rH 
 
 ^ © <0 
 
 .81 
 
 1.51 
 
 © 
 
 © 
 
 1896. 
 
 ©rH©<D© © Ttl © © 00 © <N GO 1H ©©©©© rH ©©© GO ©© 
 
 05 rH CO rH 00G0r^rHrHt>-Q0CO©T^ ^LOOi>05I^GO©©;©©©CO 
 
 © rH ©4 r-i HrHn rHrHrH rHrHrH ©4 rHrH ©4 rH 
 
 HO O 
 
 .83 
 
 .1.50 
 
 1.02 
 
 1895. 
 
 ©UOCO©CO © rH CO ©©© rH © ©3 © ©© 1H ©<0 <N CO 00 rH CO © 00 CO 
 
 05 ©j CO ©J 00 GO 05 ©} ©4 00 GO © rH go T^©©l>;C0t>O5 00 CO©rH©CO 
 
 © rH©)rH rHrHrH rHrHrH rHrHrH CO rHrH ©l rH 
 
 .86 
 
 1.41 
 
 1.02 
 
 1894. 
 
 CO©JrH-^lO © © © © CO CO t > » !>• fr © tr © ©1 CO t" |>. ©J © © CO © rH 
 
 05 ©1 CO ©| CO <»O5©©|©3(»l^TtfrH©rH©|>.rHa5<»^O5C0'>HX>COI> ; C0 
 © rH ©Q rH rHrHrH rHrH©icOrHrHrH CO ©l rH ©4 rH 
 
 .91 
 
 1.52 
 
 1.09 
 
 CO 
 
 05 
 
 GO 
 
 rH 
 
 ©rfrHTfOO © © GO © CO © tr © CO ©^ © 00 00 00 ^ rH 1> lO 
 
 05 CO CO ©| 05 CO © CO ©} GO GO ©J 05 © H 05 © GO © ©1 hJH GO CO I> CO 
 
 rH ©4 rH rH rH rH rH rHrHrH t— 1 H rH CO rH©4rH ©4 rH 
 
 .96 
 
 1.59 
 
 1.14 
 
 1892. 
 
 r|« 
 
 © ^ © ©} © rH 00 rH ©1 H ©J 05 © CO © ©|Tf©r^©^HCO©l©©GO©rr 
 
 rH©JHJ©05 05 © l> CO CO 05 GO © ©} CO © 05 05 ©q 00 rH CO © GO ©J GO ©] 
 
 rH rH©i^H rHrHrHrH rH rH ©5 rHrH tJ* rH ©l rH ©4 rH 
 
 .99 
 
 1.57 
 
 1.16 
 
 1891. 
 
 t" © © t''- © HCOHt^HCOH©CON QOCO©^©tr rH ©© CO rH © CO 
 
 © rH ©1 CO © 05©!>©qc0 0500©©^©q ©05TjH05©OqrHT^GOGOCOQO© 
 
 H H ©j H H HHHrH rH rH ©4 rHrHrH CO t— 1 ©1 r-i ©4 rH 
 
 .99 
 
 1.46 
 
 1.13 
 
 1890. 
 
 C0O5©©TjH CO © ©J rfH © ©| © as ©J © rf © CO © — 1 ^ © 
 
 ©©4©t 1H© 0505GO©aC0 0500 05©Jrt< CO tr TfJ 05 GO GO rH © tr !>. |>. 00 t". 
 
 rH rH ©4 rH rH V "rHrHrH " ‘ rH rH ©4 rHrHrH ‘©4 * rH ©4 rH ’©4 ’ rH 
 
 .99 
 
 1.43 
 
 1.12 
 
 1889. 
 
 rH©J©rH© X'- ©J © CO GO ©© rH © ©1 © COCO rH © © CO ©1 ©J © 
 
 rH Tf CO © © ©<©t>-COT^©QOI>.CO'rtl t'r rj<© !>©1©©©COOO©I 
 
 H rl ©4rH H rlnHH rH rH ©4 rH rH rH ©4 rH ©4 rH 
 
 w 
 
 1.00 
 
 1.44 
 
 1.13 
 
 1888. 
 
 ©©©©© ©1 © GO © © © © © rH © © ©CO©©©©©©©©© 
 
 rH©©©{© rHTjJ 00 CO © ©J © © ©J © © ©05©©rH©rH©©rH© 
 
 HHT(i©iH HhHHHH rH ©4 CO CO CO CO rH ©4 ©4 rH CO rH CO 
 
 a 1.00 
 a 1.95 
 
 a 1.42 
 
 1887. 
 
 ©GO©©© ©HfrrHtl©©©©T^© © ©00©©©©©^©©© 
 
 CO©©©j© ©COOOCOTHrH05©CO© © ©GO©l©CO©©©©l©© 
 
 rH rH CO ©4 rH rH rH rH —5 rH rH * rH rH CO CO rH "©4 " rH ©j ©j '©4 ‘ CO 
 
 W 
 
 a 1.12 
 a 2.01 
 
 a 1.45 
 
 1886. 
 
 CO©©©© rH©©©©©©0©© © © © © © © © © © © rH © 
 
 Tf © © CO © rHrH©©}©XrH©©CO© © © © © GO rH 00 rH © ©l © © 
 
 rH rH CO ©< l-H HHHHHH H H CO CO rH ©4 rH rH ©4 rH ©J CO 
 
 a 1.06 
 a 1.95 
 
 a 1.30 
 
 State or Territory. 
 
 Alabama 
 
 Arkansas 
 
 California 
 
 Colorado 
 
 Georgia 
 
 Idaho 
 
 Illinois 
 
 Indiana 
 
 Indian Territory 
 
 Iowa 
 
 Kansas 
 
 Kentucky 
 
 Maryland 
 
 Michigan 
 
 Missouri 
 
 Montana 
 
 Nevada 
 
 New Mexico 
 
 North Carolina 
 
 North Dakota 
 
 Ohio 
 
 Oregon 
 
 Pennsylvania bituminous 
 
 Tennessee 
 
 Texas 
 
 Utah 
 
 Virginia 
 
 Washington 
 
 West Virginia 
 
 Wyoming 
 
 Total bituminous 
 
 Pennsylvania anthracite.... 
 
 General average 
 
 a Exclusive of colliery consumption, b Includes Alaska, c Includes Nebraska. 
 
328 
 
 COST OF COKING COAL. 
 
 COST OF COKING COAL 
 
 The cost for labor alone of coking coal has been given by a number of 
 companies in the Connellsville district as 61 cents per ton of coke produced, 
 or 40j cents per ton of coal coked, exclusive of royalties, taxes, rents, and 
 such fixed charges. 
 
 In the “American Manufacturer” for July 27, 1899, Mr. F. C. Keighley 
 gave the following as the proportional costs of the several items of mining 
 and coking Connellsville coal: 
 
 Coke Yard. 
 
 Per 
 
 Cent. 
 
 Coke Yard. 
 
 Per 
 
 Cent. 
 
 Drawing 
 
 70.01 
 
 Shifting cars 
 
 1.28 
 
 Leveling 
 
 8.96 
 
 Y"ard bosses 
 
 1.12 
 
 Charging 
 
 3.48 
 
 Masons on repairs 
 
 6.12 
 
 Carters 
 
 Bookkeeper and superin- 
 
 2.48 
 
 Forking 
 
 Individual cars 
 
 1.60 
 
 .52 
 
 tendent, £ of total for 
 
 
 Sundry 
 
 .51 
 
 TYviTTA q nn vfi rn 
 
 2.04 
 
 1.20 
 
 Yard pumps 
 
 .76 
 
 III! 1 1 tv allU J Ql vA 
 
 Cleaning tracks 
 
 Total - 
 
 100.08 
 
 Mine. 
 
 Per 
 
 Cent. 
 
 Mine. 
 
 Per 
 
 Cent. 
 
 Room coal 
 
 52.15 
 
 Machinist 
 
 .49 
 
 Drivers - 
 
 8.07 
 
 Bookkeeping, £ of total for 
 
 
 TTpudin? oorI i 
 
 11.15 
 
 mine and yard..: 
 
 .49 
 
 Rope haulage 
 
 2.81 
 
 Outside labor 1 
 
 2.03 
 
 Roads 
 
 3.03 
 
 Stable boss 
 
 .96 
 
 Mine bosses 
 
 1.31 
 
 Teams 
 
 .65 
 
 Fire boss 
 
 1.44 
 
 Blacksmith 
 
 .98 
 
 Timber 
 
 2.83 
 
 Carpenters 
 
 f 1.01 
 
 
 .43 
 
 Lamp cleaners 
 
 .82 
 
 irdpptib - 
 
 Superintendence, £ of total 
 
 Inside pumps 
 
 .59 
 
 for mine and yard 
 
 .49 
 
 Steam pumps 
 
 .55 
 
 Cagers 
 
 .66 
 
 Surveys 
 
 .41 
 
 Runners and oilers 
 
 .80 
 
 Extra men 
 
 .51 
 
 Engineers 
 
 1.01 
 
 Supplies - 
 
 .92 
 
 Vi rPTTIPTI 
 
 1.13 
 
 Betterments - 
 
 1.05 
 
 Dumpers 
 
 1.25 
 
 Total 
 
 100.02 
 
 The mine labor is 67.20£ of the total labor cost, and the coke-yard labor is 
 32.804 of the total labor cost. 
 
 The cost of equipping a coke plant and opening a mine to furnish the 
 coal in the Connellsville region is from 8500 to $1,000 per oven, dependent on 
 the kind of opening for the mine and local considerations. 8500 per oven is 
 a fair price for a plant when the conditions are favorable and the mine is a 
 drift mine, and 81.000 is a fair price for a shaft mine about 300 ft. deep, 
 under rather unfavorable conditions. 
 
 Fulton gives the cost of the various types of coke ovens as follows: 
 
 Not saving by-products: Beehive, 8300; Thomas, 8800; McLanahan, 8800; 
 
 Belgian, 81,000; Cop pee. 81.000; Bernard, 81,000. „ 
 
 Saving by-products: Simon Carves, 81.300; Semet-Solvay. $1,600; Huessner, 
 81,400; G. Seibel. 81.300; Otto-Hoffman. 81.600; Festner-Hoffinan, 81.500. 
 
 The usual quantitv of coal required to make 1 ton of coke is 1.4 to 1.6 tons. 
 The lo<5s in loading coke at the ovens and again unloading it at the 
 furnaces or steel works is 24 to 34. During the winter and in wet seasons 
 coke takes on 2d' to 3* of moisture in transit between the ovens and the 
 furnaces. 
 
EXPLOSIVES. 
 
 329 
 
 EXPLOSIVES. 
 
 The characteristics of a good blasting explosive are: (1) sufficient stability 
 and strength; (2) difficulty of detonating by mechanical shock; (3) handy 
 form; (4) absence of injurious effects on the user. 
 
 Explosives are divided into two general classes: (1) low explosives or 
 ^.irect-exploding materials; (2) high explosives or indirect-exploding mate- 
 rials that require a detonator. 
 
 Low Explosives.— Gunpowder or black powder is the type of this group. 
 Its composition varies, depending on the purpose for which it is to be used, 
 but the ingredients commonly used in its manufacture are saltpeter, sulphur, 
 and charcoal. 
 
 The following table gives the composition of blasting powder in different 
 countries: 
 
 Composition of Blasting Powder ( Guttmann ). 
 
 Ingredients. 
 
 Great 
 
 Britain. 
 
 Germany. 
 
 Austria- 
 
 Hungary. 
 
 France. 
 
 Russia. 
 
 Italy. 
 
 United 
 
 States. 
 
 Saltpeter 
 
 75 
 
 66.0 
 
 64 
 
 62 
 
 66.6 
 
 70 
 
 64 
 
 Sulphur 
 
 10 
 
 12.5 
 
 16 
 
 20 
 
 16.7 
 
 18 
 
 16 
 
 Charcoal 
 
 15 
 
 21.5 
 
 20 
 
 18 
 
 16.7 
 
 12 
 
 20 
 
 High Explosives.— These are a mixture of nitroglycerine with an absorbing 
 dope, the composition of which is such that, in addition to thoroughly and 
 permanently absorbing the nitroglycerine, it is itself a gas-producing com- 
 pound. Nitroglycerine at 60° F. has a specific gravity of 1.6. It is odorless, 
 nearly or quite colorless, has a sweetish burny taste, is poisonous even in 
 very small quantities, and is insoluble in water. All nitroglycerine com- 
 pounds freeze at 42° F., and explode when confined at 360° F. It takes fire 
 at 306° F., and, if unconfined, burns harmlessly unless in large quantities, so 
 that a part of it, before coming in contact with the air, becomes heated to 
 the exploding *point. 
 
 Thawing Dynamite. — All frozen cartridges should be thawed, as, when 
 frozen, cartridges are very hard to explode, and even if they do explode, 
 the results are not nearly as satisfactory as when properly thawed. When 
 cartridges are frozen, do not expose to a direct heat, but thaw by one of 
 the following methods: First, place the number of cartridges needed for 
 
 a day’s work on shelves in a room heated by steam pipes (not live steam) 
 or a stove. Where regular blasting is done, a small house can be built for 
 this purpose, fitted with a small steam radiator. Exhaust steam through 
 these pipes gives all heat necessary. Bank your house around with earth, 
 or, preferably, fresh* manure. Second, use two water-tight kettles, one 
 smaller than the other, put cartridges to be thawed in smaller kettle, and 
 place it in larger kettle, filling space between the kettles with hot water at, 
 say, 130° to 140° F., or so that it can be borne by the hand. To keep water 
 warm, do not try to heat it in the kettle, but add fresh warm water. Cover 
 kettles to retain heat. In thawing do not allow the temperature to get 
 above 212° F. Third, where the number of cartridges to be thawed is small, 
 they may be placed about the person of the blaster until ready for use, the 
 heat of the body thawing the cartridges. Keep cartridges away from all 
 fires— this applies to all explosives. Do not be in a hurry, but thaw slowly. 
 Do not thaw before an open fire. Do not put cartridges in an oven, on a hot 
 stove, against hot iron plates, or against brick casing of a boiler. Do not put 
 cartridges in hot water, or expose them to live steam. And do not take any 
 kind of powder, fuse, or caps near a blacksmith shop. 
 
 A large number of high explosives are made that vary but little in their 
 composition, the main difference being in the character of the dope and in 
 the percentage of nitroglycerine. The trade name is usually determined 
 by the percentage of nitroglycerine, thus 10/« dynamite means that the dyna- 
 mite contains 10 Jo of nitroglycerine, etc. 
 
 Safety explosives, or as they are called in England, -permitted explosives, are 
 compounds intended for use in gaseous mines, and they are so constituted 
 
330 
 
 EXPLOSIVES. 
 
 that they will ignite without producing the extremely high temperature 
 given by ordinary explosives. The term flameless explosives was formerly 
 used, but it has been replaced by safety explosives , as the absence of a flame 
 is not now necessary to a permitted explosive. 
 
 Common Blasting Explosives. 
 
 Atlas. 
 
 A 
 
 B+ 
 
 B 
 
 C + 
 
 c 
 
 D+ 
 
 D 
 
 E + 
 E 
 
 <x> 
 
 M <V 
 <D O 
 
 ft be 
 CD o 
 
 Ph H 
 
 75 
 
 60 
 
 50 
 
 45 
 
 40 
 
 33 
 
 30 
 
 27 
 
 20 
 
 Brands Equivalent in Strength to Atlas. 
 
 a> 
 
 0 
 
 o 
 
 a 
 
 3 
 
 p. 
 
 0> 
 
 P4 
 
 A 
 
 B + 
 
 B 
 
 C + 
 
 c 
 
 No. 1 XX 
 No. 1 
 No. 2 SS 
 No. 2 S 
 No. 2 
 No. 2 C 
 No. 3 
 No. 3B 
 No. 4 B 
 
 & 
 
 £ 
 
 (2 
 
 1 
 
 0 
 
 No. 1 XX 
 No. 1 
 No.2SS 
 No. 2S 
 No. 2 
 
 Old No. 1 
 No. 1 A 
 New No. 1 
 No. 2 Extrs 
 No. 2 
 No. 2C 
 No. 3 
 XXX 
 
 xxxx 
 
 Giant Gelatine. 
 
 Hecla Powder. 
 
 9 
 
 cD 
 
 £ 
 
 o 
 
 Ph 
 
 o3 
 
 % 
 
 No. 1 A 
 
 No. 1 XX 
 
 
 No. 1 
 
 No. 1 XS 
 
 No. 1 
 
 No. 2 
 
 i 
 
 No. IX 
 
 No. 2 XX 
 No. 2 X 
 
 No. 3 C 
 
 No. 1 
 
 No. 2 
 
 No. 2X 
 
 No. 3 A 
 
 
 No. 2 
 
 No. 3 
 
 
 No. 3 X 
 
 No. 3B 
 
 
 No. 3 
 
 No. 4 
 
 Drilling -Adapt the size and depth of the hole to the work to be accom- 
 plished As a rule for ordinary rock blasting, the distance between the boles 
 shouM be equa?to from one-half to the total depth of the holes, the holes 
 set back from the face twice as far for dynamite as for common black 
 powder sav a distance equal to the depths of the holes or slightly less, and 
 Cd oSe-third 1 thl lenlth of the hole^ These directions are only general, 
 and do not apply to very deep holes. de Pe^ 
 
 ness of the rock, also on size of drill holes. In all cases, the experience 
 
 shafts, experience shows 
 
 in weak rock. All the holes in the heading or shaft should have the same 
 
 “iff SSSSSS-t 
 
 the free faces formed by firing the previous holes , . 
 
 l"°ho!e should^ever'hS’e^ a^harge™ f S more than 12 in. of explosive placed 
 iu it Where several holes are fired together, this rule is sometimes slight y 
 deviated from It fs usually best to employ a length of charge between these 
 two limits as for instance, about 10 times the diameter of the hole* 
 
 Chambering orsquibbing is the blasting out of a cavity at the bottom of a 
 drill hole to allow of a larger charge of explosive being used. . , 
 
 Rniiino a drill hole is the working of clay into any cracks op^hing into 
 a drill "lofe to p?eveit the power g of the "blast being scattered through 
 
 the char C g r h.| k -The charge must fit and fill the bot^m of bore and be packed 
 wise with 
 
 . -The charge must fit and fill the bottom of bore and be pacK< 
 a'knife, a^^^achH^S’oppe^ into^the^hole, C strike g a wooch 
 
 h* 
 
 len 
 
BLASTING. 
 
 331 
 
 rammer on it with sufficient force to make the powder completely fill the 
 bottom and diameter of the bore. Where water is not present, a more per- 
 fect loading is made by taking powder out of cartridge and dropping it in 
 loosely, ram each 6 or 8 in. of the charge, using the paper of each cartridge 
 as a wad, to take down any powder that may have stuck to the sides of the 
 hole. If water is standing in the hole, do not break the paper of the car- 
 tridges and avoid ramming more than enough to settle the charge on the 
 bottom, using cartridges of as large diameter as will readily run into the bore. 
 
 When cartridges are used, the last cartridge placed in the hole should 
 contain an electric exploder, or cap with fuse attached. When loose powder 
 is used, a piece of cartridge 2 or 3 in. in length, with exploder or cap attached, 
 should be pressed firmly on top of charge. Some blasters put an exploder or 
 cap in the first cartridge put in the hole, placing remainder of charge on top. 
 The charge should be placed in a solid part of the material to be broken. If 
 possible, the face should be undercut and then the overhanging material 
 shot down. Best results are obtained when the bore holes cross the faces or 
 layers of the material at right angles. The charges should be placed so as to 
 disturb the sides and roof of a tunnel through material of medium hardness 
 as little as possible. The charge at the bottom of the tunnel snould be 
 placed from 6 to 12 in. below the permanent level. 
 
 Amount of Charge. — No invariable rule can be laid down as to the diameter 
 and length of cartridges to be used under any and all circumstances, nor the 
 amount or grade of powder required for all kinds of work. Much depends 
 on the good sense and judgment of the persons using the explosive. 
 Guttmann, in his well-known handbook on blasting, says: “There is no lack of 
 theories for tne determination of blasting charges, ‘but their application 
 depends on empirical facts determined by practical work. I therefore 
 advise that the calculation of charges under ordinary conditions be neg- 
 lected, and recommend watching actual operations for some weeks, asking 
 for explanation from the most expert miners. In this way experience will 
 be gotten in a short time that will enable one to estimate with some precision 
 the proper charge to use after inspecting the spot to be blasted." 
 
 A good rule by which to determine the weight of black powder to use in 
 any given hole in bituminous workings is the following: Find the distance 
 in feet from the charge out in the line of least resistance. Multiply the 
 fourth power of this distance by ^ the diameter of the hole in inches, and 
 divide this product by the thickness of the seam in inches. The result will 
 be the weight of the charge in pounds. Thus, for a 2\ rt hole in a seam of 
 bituminous coal 6 ft. thick, where the charge is placed 4£ ft. deep from the 
 face of the coal, or cutting, we have for the weight of charge to be used, 
 
 9 X 9 - 9 - 9 ^ - 4X2 - 5 5 71 b 
 
 2 X 2 X 2 X 2 X 6X 12 6 
 
 Tamping.— -In deep holes, water makes a good tamping, but fine sand, 
 clay, etc. are generally used. Fill in for the first 5 or 6 in. carefully, so as 
 not to displace cap and primer; then with a hardwood rammer pack bal- 
 ance of material as solid as possible, ramming with the hand alone, and not 
 using any form of hammer. Never use a metal tamping rod. 
 
 Firing.— If the work is wet, or the charge used under water, use water- 
 proof fuse, and protect the end of the fuse by applying bar soap, pitch, or 
 tallow around the edge of the cap. Water must not be allowed to reach the 
 powder in the fuse or the fulminate in the cap. Exploding by electricity is 
 best under water at great depth, as the pressure of water is so great on the 
 fuse that it is forced through and dampens it so as to prevent firing. 
 
 Seam Blasting.— If a seam is found in the rock, remove the powder from 
 the cartridges and push it into the seam and fire a cap beside it. This will 
 open the seam so that a larger quantity of explosive can be used, and the 
 rock broken without drilling. In blasting coal, slate, marble, granite, free- 
 stone, or any other material that it is desirable to obtain in large blocks, 
 cartridges of small diameter should be used in wide bore holes, the charge 
 being rolled in several folds of paper, to prevent its touching the sides of the 
 bore holes. The intensity of action and the crushing effect of the explosive 
 are thus lessened. 
 
 Firing by Detonation.— Nitroglycerine explosives always require detonation 
 by a cap or exploder in order to develop their full force. Fig. 1 illustrates 
 the method of attaching such an exploder to the end of a fuse and the pla- 
 cing of it in the cartridge. The exploders are loaded with fulminate of mer- 
 cury and are slipped over the end of the fuse, after which the upper end is 
 
332 
 
 EXPLOSIVES. 
 
 crimped tightly against the end of the fuse, as shown. (Miners sometimes 
 bite the caps on to the fuse w ith their teeth. This is a very dangerous pro- 
 ceeding and should never be allowed, as, with strong caps, one of them 
 exploding in a man’s mouth would prove fatal.) In placing the cap or 
 
 Fig. 1. 
 
 Fig. 2. 
 
 Fig. 3. 
 
 exploder into the dynamite or giant-powder cartridge, care should be taken 
 that only about two-thirds of the cap be embedded in the material of the 
 cartridge, for if the fuse had to pass through a portion of the material before 
 
 reaching the cap. there w ould be danger of its igniting the material, thus 
 causing deflagration of the cartridge in place of detonation. The fumes 
 given off by high explosives are much w^orse in the case of del 
 cartridge. 
 
 The electric exploder, Fig. 2, has wires A and B , which carry the current to 
 the exploder. D is a cement (usually sulphur) that protects the explosive 
 compound C (usually mercury fulminate) and the w^hole is contained m a 
 Conner shell. A small platinum wire E is heated by the passage of a current 
 ^ and ignites the explosive. Fig. 3 
 
 shows the method of placing a cap 
 or an electric exploder in a cartridge 
 of powder. 
 
 When a number of holes are ex- 
 ploded at one time, the electric 
 exploders are connected in series, as 
 shown in Fig. 4, for a small number 
 of holes, and as in Fig. 5 for a larger 
 number. 
 
 The battery for furnishing the 
 current of electricity is a magneto 
 machine that is worked by either pulling up or by depressing a handle or 
 rack bar, or else by turning a crank. _ . , . . 
 
 Directions for Blasting by Electricity.— Drill the number of holes desired to be 
 
 Fig. 4. 
 
 fired at one time; depth and spacing of holes depending on character of 
 rock size of drill holes, etc., the blaster, of course, using his judgment in this 
 matter Load the hole in the usual manner, fitting one cartridge with a fuse 
 
ARRANGEMENT OF DRILL HOLES. 
 
 333 
 
 (electric exploder) instead of cap and fuse. The fuse head is fitted into 
 the bottom end of the cartridge, or into the middle of one side of the 
 cartridge, where a hole has been punched with a pencil or small sharp stick 
 to receive it; push the powder close around the fuse head. The fuse can then 
 be held in position by tying a string around the cartridge and the fuse wires, 
 binding the wires to the cartridge, as shown in Fig. 3. A shows head of fuse, £ 
 the two fuse wires, C string used to tie wires to cartridge. Avoid taking 
 hitches in fuse wires, as by this very common practice, the insulation of 
 the wires may be injured and the current of electricity may pass from one 
 wire to the other, without passing through the cap, hazarding a misfire. 
 
 The cartridge containing the fuse is put in on top of the charge by some 
 blasters; by others, at bottom of the charge. The best place for it is in the 
 center of the charge, having part of the charge above and part below it. 
 In tamping the hole, great care must be taken not to cut the wires, or injure the 
 cotton covering of fuse wires, or to pull the fuse out of the cartridge. Allow 
 at least 8 in. of the fuse wire to project above the hole, to make connections. 
 
 When all the holes to be fired at one time are tamped, separate the ends 
 of the two wires in each hole, and, by the use of connecting wire, join one 
 wire of the first hole with one of the second, the other or free wire of the 
 second with one of the third, and so on to the last hole, leaving a free wire 
 at each end hole. 
 
 All connections of wires should be made by twisting together the bare 
 and clean ends; it is best to have the joined parts bright. Scrape off the cotton 
 
 covering at the ends of the wires to be connected, say for 2 in., then rub the 
 wire with a small hard stone. This makes a bright fresh wire. Be sure that 
 all connections are clean, bright, and well twisted. Do not hook or loop wires 
 in making connections. Bare joints in wire should never be allowed to touch 
 the ground, particularly so if the ground is wet. This can be prevented by 
 putting dry stones under the joints. The charges having all been connected, 
 as directed above, the free wire of the first hole should be joined to one of the 
 leading wires, and the free wire of the last hole to the other of the two 
 leading wires. The leading wires should be long enough to reach a point at 
 a safe distance from the blast, say 250 ft. at least. All being ready, and not 
 till the men are at a safe distance, connect the leading wires, one to each of 
 the projecting screws on the front side or top of the battery, through each 
 of which a hole is bored for the purpose, and bring the nuts down firmly on 
 the wires. Now, to fire, take hold of the handle for the purpose, lift the 
 rack bar (or square rod, toothed on one side) to its full length, and press it 
 down, for the first inch of its stroke with moderate speed, but finishing the 
 stroke with all force, bringing rack bar to the bottom of the box with a solid 
 thud, and the blast will be made. Do not churn rack bar up and down. It 
 
334 
 
 EXPLOSIVES. 
 
 is unnecessary and harmful to the machine. One quick stroke of the rack 
 bar is sufficient to make the blast. Never use fuses (exploders) made by 
 different manufacturers in the same blast. Connecting wire should be of 
 
 Fig. 8. Fig. 9. 
 
 same size as the fuse wire; leading wire should be at least twice as large. 
 Covering on wire should not “ strip ” or come off easily. 
 
 The power of an explosive cannot be exactly calculated from the quantity 
 and temperature of the gas resulting from its detonation, as it is impossible 
 to determine the exact composition of gas at the moment of explosion and 
 during the subsequent cooling period. Tables that give the relative strength 
 
 Fig. 10. Fig. 11. 
 
 of explosives are apt to be misleading, as so much depends on the compo- 
 sition of the explosive, and since there are so many explosives of varying 
 compositions that are sold under the same name. 
 
 Pressures Developed by Explosives.— According to experiments conducted 
 
ARRANGEMENT OF DRILL HOLES. 
 
 335 
 
 by Sarrau, Vielle, Nonle, and Abel, the following approximate maximum 
 pressures, in tons per square inch, developed by various explosives, have 
 been arrived at: Mercury fulminate, 193; nitroglycerine, 86; guncotton, 71; 
 blasting powder, 43. _ , . 
 
 Values of Explosives.— Taking gunpowder (containing 61# saltpeter) as a 
 standard, and calling its value 1, the following are the comparative values 
 
 Fig. 12. Fig. 13. 
 
 of the other explosives: Dynamite, containing 75# nitroglycerine, 2.2; blasting 
 gelatine, containing 92 # nitroglycerine, 3.2; nitroglycerine, 3.3. 
 
 The arrangement of aril I holes for driving and sinking should be such as to 
 permit the easy handling of the drills and also to minimize the number of 
 holes and the weight of explosive. Two distinct systems are m use: (1) the 
 center cut , by which a center core or key is first removed, and after that 
 concentric layers about this core; (2) the square cut , in which the lines of 
 
336 
 
 MACHINE MIXING. 
 
 holes are parallel to the sides of the excavation, the rock being removed in 
 wedges instead of in concentric circles. 
 
 The center-cut method is shown in Figs. 6, 7, 8, and 9, Fig. 6 showing the 
 face of a heading, Fig. 7 an elevation or vertical section, and Figs. 8 and 9 
 plans. The numbers of the holes correspond in the several views. The 
 holes are supposed to be drilled by rock drills, and they are so placed that 
 all except the breaking-in holes have an equal line of resistance. The num- 
 ber of holes given is supposed to take out a clean cut of the whole section 
 abed to the extent of 3 ft. 6 in. The order of firing the holes is: (1) break- 
 ing-in shots 1, 2, 8 , and 4 simultaneously; (2) 5, 6, 7, 8 ; (3) 9, 10, 11, 12; 
 (4) IS, U, 15, 16; (5) 17, 18, 19, 20. 
 
 The square-cut arrangement is shown in Figs. 12 (face), 10, 11 (plans), 
 and 13 (vertical elevation). The entering wedge, Fig. 11, is best removed in 
 two stages: First, the part egh by the shots 1, 2, 8, and 4; and second, part 
 efh by shots 5, 6, 7, and 8. The other shots are then fired: (1) 9, 10, 11, 12: 
 (2) 13, lit, 15, 16; (3) 17, 18, 19, 20, each volley being fired either simultaneously 
 or consecutively. Where there is a natural parting in the heading, advan- 
 tage is, of course, taken of this- in the location of the shots. 
 
 Figs. 14 and 15 show two arrangements of drill holes used in sinking the 
 Parker shaft at Franklin Furnace, N. J. The size of the shaft was 10 ft. X 
 20 ft. in the rock. At first, only 6' cuts were put in, but these were gradually 
 increased until IF and 12' cuts were pulled. The best average obtained was 
 66 ft. of shaft from 6 consecutive cuts. 
 
 MACHINE MINING. 
 
 The number of coal-mining machines in use has increased rapidly within 
 a very short time. In 1896 there were 1,446 in use in the United States. 
 During 1897 there was an increase of 542, or 37.5$, while the average yearly 
 gain from 1891 to 1896 was only about 22$. The total tonnage won by 
 machines in 20 States in 1897 was 22,649,220 short tons, or 16.17$ of the total 
 product of these States, and 15.3$ of the total bituminous product of the 
 United States. A universal mining machine has not yet been brought out, 
 and one of the principal reasons for the failure of mining machines in a 
 number of instances has been the attempt to use a machine under condi- 
 tions to which it was not adapted. When a mining machine is designed and 
 built to suit the conditions under which it is to be operated, it is safe to say 
 that there are but few mines in which they cannot be successfully utilized. 
 They are of particular advantage where there is a long working face and 
 where the coal is over 3 ft. in thickness. Low seams require more under- 
 cutting for the given output than high seams. As a rule it has not been 
 found economical to use machines in seams pitching over 12° to 15°, though 
 pick machines have been used in mines having an inclination of 23°, the 
 difliculty being not so much in the cutting as in moving the machine from 
 place to place. 
 
 There are four general types of mining machines in use; pick machines, 
 chain-cutter machines, cutter-bar machines, and longwall machines. The 
 first two are the types almost universally used in America. Cutter-bar 
 machines have almost entirely disappeared from use excepting one type 
 which is at present used in Iowa. Longwall mining machines have not 
 been very generally adopted in America, as the longwall method of mining 
 is not extensively used. 
 
 Both compressed air and electricity are used for operating mining 
 machines. Pick machines driven by compressed air are made by three 
 separate concerns. Four companies make electric chain machines and one of 
 these four is also making a compressed-air chain machine. One makes a 
 longwall machine, and one has brought out a pick machine for electric 
 power. 
 
 Pick machines work very similarly to a rock drill. They can be used 
 wherever mining machines are applicable, and their particular advantage is 
 that they are more perfectly under the control of the operator, who can cut 
 around pvrites and similar obstructions without cutting them with the 
 machine. ‘ This renders such a machine particularly applicable for seams of 
 
VENTILATION OF MINES. 
 
 337 
 
 coal having rolls in the bottom and containing pyrites or other hard impuri- 
 ties. They are also applicable for working pillars on which there is a 
 saueeze, as they are light and can be easily handled and readily removed. 
 q Chain-cutter machines consist of a. low metal bed frame upon which is 
 mounted a motor that rotates a chain to which suitable cutting teeth are 
 attached. To operate chain machines to the best advantage, the coal should 
 be comparatively free from pyrites. They also require more room than pick 
 machines, and a space from 12 to 15 ft. in width is nece&sa p , „ a Jo n l = t ^ e ^® 
 to work them to advantage. These machines have proved failures m some 
 mines on account of the incessant jarring of the roof by the re & r Jack. 
 Chain-cutter machines cannot be used to undercut coal when a squeeze is 
 upon it. Coal seams that are comparatively level and free from pyrites and 
 have a comparatively good roof can undoubtedly be more satisfactorily and 
 economically cut with chain-cutter machines than with any other type. 
 
 The average height of cut is to 5 in., and at this height the chain- 
 cutter machines makes only about 60/c as much small coal as a pick machine. 
 This is not always an advantage, as it does not always allow sufficient height 
 for the coal to fall down after the cut is made. In a 3£' seam, 3 men are 
 required to handle the machine to advantage. .. , . _ . . „ 
 
 Shearing —All the pick machines can be converted into shearing machines 
 and can be used for longwall work by using a longer striking arm and a 
 longer supply hose. The chain machines are used to do shearing work by 
 having the cutting parts turned vertically. . . .. „ , 
 
 Capacity.— The average producing capacity of each mining machine used 
 in the United States was 11,398 tons in 1891, 11,373 tons m 1896, and 11,393 
 tons in 1897. So much depends on the local conditions that it is almost 
 impossible to give specific data of rates of working and costs, but the 
 
 following are fair working approximations. 
 
 A good pick machine will undercut 450 sq. ft. m 10 hours, while an ordi- 
 nary miner will undercut 120 sq. ft. in the same time. In a seam varying from 
 41 to 6 ft in thickness, the machine will undercut from 50 to 100 tons of coal 
 in 10 hours. The cost of undercutting under these conditions has been giv en 
 as approximately 10 cents per ton. Extraordinary records show 1,400 sq. it. 
 to have been cut in 9 hours in Western Pennsylvania, and in an 8' seam, 
 
 240 tons have been undercut in a shift of 10 hours. 
 
 A good chain cutter will make from 30 to 45 cuts, 44 m. wide and 6 it. 
 deep in 10 hours under moderately fair conditions, while m high seams 
 two men handling the same machine under ordinary conditions can make 
 60 cuts per shift, and under particularly favorable conditions, 80 to 120 cuts 
 per shift. 
 
 VENTILATION OF MINES. 
 
 This subject is divided naturally into (a) gases occurring m workings, 
 explosive conditions, quantity of air, distribution of air, and (o) ventilating 
 methods and machinery. 
 
 THE ATMOSPHERE. 
 
 Composition.— Air consists chiefly of oxygen and nitrogen, with small and 
 varying amounts of carbonic-acid gas, ammonia gas, and aqueous vapor. 
 These gases are not chemically combined, but exist in a free state m umtorm 
 
 proportion, as follows: ■ ■ r ,,r • 
 
 * ^ By Volume. By \\ eight. 
 
 100.0 
 
 100.0 
 
 Wherever air is found, its composition is practically the same. 
 
 Weight —The weight of 1 cu. ft. of air at 32° F. and under a barometric 
 pressure of 30 in. is .080975 lb. Air decreases in weight per cubic foot as we 
 ascend in the atmosphere, and increases as we descend below the surface 
 of the earth. 
 
338 
 
 VENTILATION OF MINES. 
 
 The weight of 1 cu. ft. of dry air at any temperature and barometric 
 pressure is found by means of the formula 
 
 1.3253 X B 
 W 459 + t * 
 
 in which w = weight of 1 cu. ft. of dry air; B = barometric pressure 
 (inches of mercury); t = temperature (degrees F.). 
 
 The constant 1.3253 is the weight in pounds avoirdupois of 459 cu. ft. of 
 dry air at a temperature of 1° F. and 1 in. barometric pressure. 
 
 Example.— Find the weight of 1 cu. ft. of dry air at a temperature of 
 60° F. and a barometric pressure of 30 in. 
 
 w 
 
 1.3253 X 30 
 “459 + 60 
 
 = .07661b. 
 
 Table of Weight of Dry Air. 
 
 Weight of 1 cu. ft. of dry air at different temperatures and barometric 
 
 1 3253 X B 
 
 pressures, as calculated by the formula w = + . 
 
 Temperature. 
 Degrees F. 
 t 
 
 Weight of 1 Cu. Ft. of Dry Air (Lb. Avoirdcpois). 
 
 Barometer (In.). 
 B — 27. 
 
 Barometer (In.). 
 B = 28. 
 
 Barometer (In.). 
 B = 29. 
 
 Barometer (In.). 
 B = 30. 
 
 0 
 
 .07796 
 
 .08085 
 
 .08373 
 
 .08662 
 
 5 
 
 .07718 
 
 .08002 
 
 .08285 
 
 .08569 
 
 10 
 
 .07631 
 
 .07914 
 
 .08196 
 
 .08478 
 
 15 
 
 .07550 
 
 .07830 
 
 .08109 
 
 .08388 
 
 20 
 
 .07470 
 
 .07747 
 
 .08023 
 
 .08300 
 
 25 
 
 .07393 
 
 .07667 
 
 .07041 
 
 .08215 
 
 30 
 
 .07318 
 
 .07589 
 
 .07860 
 
 .08131 
 
 32 
 
 .07288 
 
 .07558 
 
 .07828 
 
 .08098 
 
 35 
 
 .07244 
 
 .07512 
 
 .07780 
 
 .08048 
 
 40 
 
 .07171 
 
 .07435 
 
 .07701 
 
 .07967 
 
 45 
 
 .07099 
 
 .07362 
 
 .07625 
 
 .07888 
 
 50 
 
 .07031 
 
 .07291 
 
 .07551 
 
 .07811 
 
 55 
 
 .06961 
 
 .07219 
 
 .07477 
 
 .07735 
 
 60 
 
 .06895 
 
 .07150 
 
 .07405 
 
 .07660 
 
 65 
 
 .06828 
 
 .07081 
 
 .07324 
 
 .07587 
 
 70 
 
 .06766 
 
 .07016 
 
 .07266 
 
 .07516 
 
 75 
 
 .06701 
 
 .06949 
 
 .07197 
 
 .07445 
 
 80 
 
 .06648 
 
 .06884 
 
 .07130 
 
 .07376 
 
 85 
 
 .06576 
 
 .06820 
 
 .07064 
 
 .07308 
 
 90 
 
 .06519 
 
 .06760 
 
 .07001 
 
 .07242 
 
 95 
 
 .06490 
 
 .06699 
 
 .06938 
 
 .07177 
 
 100 
 
 .06401 
 
 .06638 
 
 .06875 
 
 .07112 
 
 110 
 
 .06288 
 
 .06521 
 
 .06754 
 
 .06987 
 
 120 
 
 .06180 
 
 .06409 
 
 .06638 
 
 .06867 
 
 130 
 
 .06075 
 
 .06300 
 
 .06525 
 
 .06750 
 
 140 
 
 .05974 
 
 .06195 
 
 .06416 
 
 .06637 
 
 150 
 
 .05874 
 
 .06092 
 
 .06310 
 
 .06528 
 
 160 
 
 .05781 
 
 .05995 
 
 .06209 
 
 .06423 
 
 170 
 
 . .05688 
 
 .05899 
 
 .06110 
 
 .06321 
 
 180 
 
 .05601 
 
 .05808 
 
 .06015 
 
 .06222 
 
 190 
 
 .05514 
 
 .05718 
 
 .05922 
 
 .06126 
 
 200 
 
 .05430 
 
 .05631 
 
 .05832 
 
 .06033 
 
 220 
 
 .05271 
 
 .05466 
 
 .05661 
 
 .05856 
 
 240 
 
 .05119 
 
 .05309 
 
 .05498 
 
 .05688 
 
 260 
 
 .04978 
 
 .05162 
 
 .05346 
 
 .05530 
 
 280 
 
 .04840 
 
 .05020 
 
 .05200 
 
 .05380 
 
 300 
 
 .04714 
 
 .04888 
 
 .05063 
 
 .05238 
 
 350 
 
 .04423 
 
 .04587 
 
 .04751 
 
 .04915 
 
 400 
 
 .04166 
 
 .04321 
 
 .04475 
 
 .04629 
 
THE BAROMETER. 
 
 339 
 
 Atmospheric Pressure.— The term barometric pressure is the pressure caused 
 bv the weight of the atmosphere above a given point. It is measured by the 
 barometer, and this gives rise to the synonymous term barometric pressure. 
 Atmospheric pressure is usually stated in pounds per square inch, while 
 barometric pressure is stated in inches of mercury. Thus, at sea level, the 
 atmospheric pressure under normal conditions of the atmosphere is 14./ lb. 
 per sq. in., while the barometric pressure at the same level is 30 in. of 
 mercury column, or simply 30 in. , . ' , . 
 
 Barometric Variations.— The pressure of the atmosphere is not constant, but 
 is subject to fluctuations depending on the condition of the atmosphere. 
 Besides these, there are fluctuations that are more or less regular and are 
 called barometric variations. There is both a yearly and a diurnal, or daily, 
 variation. Of these two, the more important and the. more regular is the 
 daily variation, in which the barometer attains a maximum height from 9 to 
 10 o’clock a. m., and a minimum about 4 o’clock p. m. Other maximum and 
 minimum readings are obtained at 10 p. m. and 3 a. m., respectively; but 
 these are not as pronounced as those occurring in the daytime. The daily 
 barometric variations range from .01 to .08 in. 
 
 Mercurial Barometer.— This barometer is often called the cistern barometer; 
 or when the lower end of the tube is bent upwards instead of the mouth of 
 the tube being submerged in a basin, it is known as the siphon barometer. 
 The instrument is constructed by filling a glass tube 3 ft. long, and having a 
 bore of £ in. diameter, with mercury, which is boiled to drive off the air. 
 The thumb is now placed tightly over the open end, the tube inverted, and 
 its mouth submerged in a basin of mercury. When the thumb is withdrawn, 
 the mercury sinks in the tube, flowing out into the basin, until the top of 
 the mercury column is about 30 in. above the surface of the mercury m the 
 basin, and after a few oscillations above and below this point, comes to rest. 
 The vacuum thus left in the tube above the mercury column is as perfect 
 a vacuum as it is possible to form, and is called a Torricelli vacuum, after its 
 discoverer. There being evidently no pressure in the tube above the 
 mercury column, and as the weight of this column standing above the sur- 
 face of the mercury in the basin is supported by the pressure of the atmos- 
 phere, it is the exact measure of the pressure of the atmosphere on the 
 surface of the mercury in the basin. If the experiment is performed at sea 
 level, the height of the mercury will be found to average about 30 in., at 
 higher elevations it is less, while if we descend deep shafts below this level, 
 it Is greater. Roughly speaking, an allowance of 1 in. of barometric height 
 is made for each 900 ft. of ascent or descent from sea level (see calculation of 
 barometric elevations). A thermometer is attached to each mercurial 
 barometer to note the temperature of the reading, as it is customary m all 
 accurate work with this instrument to reduce each reading to an equivalent 
 reading at 32° F., which is the standard temperature for barometric readings. 
 A scale is provided at the top of the mercury column with its inches so 
 marked upon it as to make due allowance for what is called the error of 
 capacity. In other words, the inches of the scale are longer than real 
 inches, since the level of the mercury in the basin rises as it sinks m the tube, 
 and vice versa. The top of the mercury column is always oval, convex 
 upwards, owing to capillary attraction, and the scale is read where it is 
 tangent to this convex surface. , , „ ., . , 
 
 Aneroid Barometer.— This is a more portable form than the mercurial 
 barometer. It consists of a brass box resembling a steam-pressure gauge, 
 having a similar dial and pointer, the dial, however, being graduated to 
 read inches, corresponding to inches of mercury column, instead of reading 
 pounds, as in a pressure gauge. Within the outer case is a delicate brass 
 box having its upper and lower sides corrugated, which causes it to act as a 
 bellows, moving in and out as the atmospheric pressure on it changes. The 
 air within the box has been partially exhausted, to render it sensitive to 
 atmospheric changes. The movement of the upper surface of the box is 
 communicated to the pointer by a series of levers^ and the dial is graduated 
 to correspond with the mercurial barometer. 
 
 Calculation of Atmospheric Pressure.— The weight of the mercury column 
 of the barometer is the exact measure of the pressure of the atmosphere, 
 since it is the downward pressure of the atmosphere that supports the 
 mercury column, area for area; that is to say, the pressure of the atmosphere 
 on 1 sq. in. supports a column of mercury whose area is 1 sq. in., and whose 
 height is such that the weight of the mercury column is equal to the weight 
 of the atmospheric column. Hence, since 1 cu. in. of mercury weighs .49 lb., 
 
340 
 
 VENTILATION OF MINES. 
 
 the atmospheric pressure that supports 30 in. of mercury column is .49 X 30 
 = 14.7 lb. per sq. in. In like manner, the atmospheric pressure correspond- 
 ing to any height of mercury column may be calculated. It will be observed 
 that the sectional size of the mercury column is not important, since it is 
 supported by the atmospheric pressure on an equal area, but the calculation 
 of pressure is based on 1 sq. in. 
 
 Water Column Corresponding to Any Mercury Column.— The density of mercury 
 
 referred to water is practically 13.6; hence, the height of a water column 
 corresponding to a given mercury column is 13.6 times the height of the 
 mercury column. For example, at sea level, where the average barometric 
 pressure is 30 in. of mercury, the height of water column that the atmos- 
 pheric pressure will support is 13.6 X = 34 ft. This is the theoretical 
 height to which it is possible to raise water by means of a suction pump, but 
 the length of the suction pipe should not exceed 75 $ or 80$ of the theoretical 
 water column. . , , , 
 
 Calculation of Barometric Elevations.— Such elevations, although approxi- 
 mate, are useful for many purposes. The barometric readings are reduced 
 to equivalent readings at the standard temperature of 32° F., and for deter- 
 mining differences in elevation, the readings of two barometers should be 
 taken, if possible, at the same time. It must not be supposed, however, that 
 the barometer alwavs reads the same for the same elevation at this tempera- 
 ture. The temperature of the atmosphere has indeed very little effect on 
 the atmospheric pressure, which is due to the weight of air above the point 
 of observation, aerial currents, and other phenomena. 
 
 In the more accurate determinations of vertical height or elevation by 
 means of the barometer, the following formula is usually employed: 
 
 R = reading of barometer (inches) at lower station; 
 r — reading of barometer (inches) at higher station; 
 
 T = temperature (F.) at lower station; 
 t = temperature (F.) at higher station; 
 
 H = difference of level in feet between the two stations. 
 
 H= 56,300 (log B- log 0(l + ^)- 
 
 log R 
 
 H 
 
 56,300 1 + 
 
 ( i +— -1 
 
 \ + 900 I 
 
 + log r. 
 
 More simply: H = 49,000 (§^) (l + ) . 
 
 R = 
 
 ’49,000 (900 + T+ t) -f 90017] 
 .49,000 (900 + T+ t) - 900 id * 
 
 Correction for Temperature.— Mercury expands about .0001 of its volume 
 for each degree Fahrenheit. To reduce, therefore, a reading at any tem- 
 perature to the corresponding reading at the standard temperature of 32° F., 
 subtract of the observed height for each degree above 32°; or, if the 
 temperature is below 32°, add TT ,^ for each degree. 
 
 Thus, 30.667 in. at 62° F. is equivalent to a reading of 30.555 in. at 32° F., 
 
 since 30.667 -^-^=(30.667) - 30.667 -.092 = 30.555 in. 
 
 Depth of Shafts.— The barometer is often employed to determine the 
 depth of a shaft or the depth of any point in a mine below a corresponding 
 point on the surface. The aneroid is employed for this work, being more 
 portable. Allowance must always be made in such cases for the venti- 
 lating pressure of the mine. A simple formula often used for such calcu- 
 lations is the following: • 
 
 H= 55,OOo(l-^), 
 
 in which the letters stand for the same factors as designated above. 
 
 The most important use of the barometer in mining practice, however, is 
 found in the warning that it gives of the decrease of atmospheric pressure, 
 and the expansion of mine gases that always follows. 
 
CHEMISTRY OF GASES. 
 
 341 
 
 CHEMISTRY OF GASES. 
 
 All matter exists in one of three forms, solid , liquid , or gaseous, according to 
 the predominance of the attractive or the repulsive forces existing between the 
 molecules For example, water exists as ice, or in a solid form, when the 
 attractive force exceeds the repulsive force between its molecules As 
 the^enmeratu^ is raised or heat is applied, the ice assumes the liquid form 
 due to^hl more rapid vibration of the molecules of which it is composed. 
 In other words, the repulsive force existing between the molecules is 
 increased aid the result is a liquid. If we still further raise the tempera- 
 S% bv applying more heat, the vibration of the molecules becomes yet 
 more rapid, the repulsive force is increased between the molecules, and a 
 gas or vapor called steam is formed. . . • 
 
 An atom is the smallest conceivable division of matter 
 A molecule is a collection of two or more atoms, united by affinity. 
 
 The atom cannot consist of more than one element. The molecule may 
 be either simple or compound. If compound, it is a chemical compound, 
 
 itS Che^afcomp^nTs^ compound is one formed by the union 
 
 of two or more atoms chemically, such atoms uniting always in fixed or 
 definite°proportions. The properties of a chemical compound are always 
 
 Mechanical Mixture. — A mechanical mixture is_ composed of different sub- 
 stances that are not chemically united, and which are mixed in no fixed 
 Smmrtion The properties of a mechanical mixture present a regular 
 proportion, ine p p iTnllTn to a minimum state. Thus, a solution of 
 gradation from a is not a chemical compound of salt and water, 
 
 bTCplv a mechanical mixture of the salt in the water. If more salt is 
 
 Slo t^tPr thp strength of the mixture or the brine is increased; 
 added to the water the strengm onu«mt . g legg> Qn the other ha nd, 
 
 an it -^if?s ! chemffial compound formed by the union of 1 atom of sodium 
 withTatoin of chlorine, the two atoms being bound together by chemical 
 Affinity, and always uniting in the same proportion, 1 atom of each, to form 
 
 Sal The air that we breathe is a mechanical mixture of nitrogen and oxygen 
 gases with small amounts of other ingredients. The nitrogen and oxygen 
 gases’ are in a free state; that is to say, they are not combined as in a chem- 
 fca? compound. This Is true, although the proportion of these two gases, 
 oxvgen and nitrogen, in the atmosphere, is uniformly in the ratio of say, 
 1 volume of oxygen to 4 of nitrogen. Firedamp is another example of true 
 mechanical mixture, consisting chiefly of a mixture of marsh gas CH± and 
 Sfwith smaU amounts of other hydrocarbons and a varying amount of 
 carbonic-acid gas, which is always present in firedamp. These gases are not 
 combined chemically, but are mixed in varying proportions. 
 
 Atomic volume, or specific volume , means simply relative volume. These 
 terms refer to the relative volume of gases resulting from any particular 
 reaction. By means of the laws of atomic volume, we can ascertain the 
 volumes of the different gases resulting from any particular reaction. 1 he 
 chemical reaction that takes place between the elements constituting 
 the different gases is expressed by means of a chemical equation. When 
 we have expressed such reaction by a chemical equation, we can then 
 calculate the volumes of the gases formed, with respect to the original 
 volumes of the gases entering into the reaction. It must be observed, how- 
 ever, that the atomic volumes express merely the relative^ volumes of gases; 
 or, in other words, the ratio of the volumes of gases before and after the 
 
 rea LawToTvollme a — The following laws of volume refer to gases only, and 
 
 ^First— EquaWoVum^of all gases, under the same conditions of tempera- 
 ture and pressure, contain the same number of molecules. Hence, the 
 molecules of all simple gases are of the same size. - 
 
 Second.— The molecules of compound gases, under like con ditions of tem- 
 perature and pressure, occupy twice the volume of an atom of hydrogen gas. 
 
 There are very few exceptions to these two laws of gaseous volume, and 
 the exceptions are unimportant so far as mining practice concerned. _ 
 An element is a form of matter that is composed wholly of like atoms. 
 Thus hvdrogen, oxygen, iron, copper, gold, and silver are elements. 
 
 Chemical Symbols and Equations.— To facilitate the writing of chemical 
 
342 
 
 VENTILATION OF MINES. 
 
 equations expressing the reaction that takes place between elements under 
 certain conditions, it is usual to express the elements by letters called 
 symbols. These symbols stand for the elements that they represent, and are 
 written as capital letters, except where two letters are used to express a 
 symbol, in which case the first letter only is a capital. Thus, C is the 
 symbol for the element carbon, but Cu is the symbol for copper (cuprum) 
 and Co is the symbol for cobalt. It is important that these symbols be 
 written exactly in this manner; otherwise they are liable to be frequently 
 misconstrued. For example, Co stands for cobalt, while the symbol CO 
 
 Table of Elements. 
 
 Element. 
 
 Symbol. 
 
 Atomic 
 
 Weight. 
 
 Element. 
 
 Symbol. 
 
 Atomic 
 
 Weight. 
 
 Aluminum 
 
 Al 
 
 27.5 
 
 Manganese 
 
 Mn 
 
 55.0 
 
 Antimony (stibium) 
 
 Sb 
 
 120.0 
 
 Molybdenum 
 
 Mo 
 
 96.0 
 
 Argon(?) 
 
 A 
 
 
 Neodymium(?) 
 
 Nd 
 
 
 Arsenic 
 
 As 
 
 75.0 
 
 Nickel 
 
 Ni 
 
 58.8 
 
 Barium 
 
 P'7- 
 
 137 0 
 
 Niobium . 
 
 Nb 
 
 94.0 
 
 Beryllium 
 
 Be 
 
 0 4 
 
 Nitrogen 
 
 N 
 
 14.0 
 
 Bismuth 
 
 Bi 
 
 208.0 
 
 Osmium 
 
 Os 
 
 191.0 
 
 Boron 
 
 B 
 
 11.0 
 
 Oxygen 
 
 O 
 
 16.0 
 
 Bromine 
 
 Br 
 
 80.0 
 
 Palladium 
 
 Pd 
 
 106.5 
 
 Cadmium 
 
 Cd 
 
 112.0 
 
 Phosphorus. 
 
 P 
 
 3L0 
 
 Caesium 
 
 Cs 
 
 133.0 
 
 platinum 
 
 Pt 
 
 197.0 
 
 Calcium 
 
 Ca 
 
 4o!o 
 
 Potassium (kalium) 
 
 K 
 
 39!o 
 
 Carbon 
 
 C 
 
 12.0 
 
 Praseodymium(?) 
 
 Pr 
 
 
 Cerium 
 
 Ce 
 
 138.0 
 
 Rhodium 
 
 Eh 
 
 104.0 
 
 Chlorine 
 
 Cl 
 
 35.5 
 
 Rubidium 
 
 Eb 
 
 85.0 
 
 Chromium 
 
 Cr 
 
 52^5 
 
 Ruthenium 
 
 Eu 
 
 10L0 
 
 Cobalt 
 
 Co 
 
 59.0 
 
 Samarium(?) 
 
 Sa 
 
 
 Columbium 
 
 Cb 
 
 93.7 
 
 Scandium 
 
 Sc 
 
 
 Copper (cuprum) 
 
 Cu 
 
 63.0 
 
 Selenium 
 
 Se 
 
 79.0 
 
 Didymium 
 
 D 
 
 147.0 • 
 
 Silicon 
 
 Si 
 
 28.0 
 
 Erbium(?) 
 
 Er 
 
 169.0 
 
 Silver (argentum) 
 
 Ag 
 
 108.0 
 
 Fluorine 
 
 F 
 
 19.0 
 
 Sodium (natrium) 
 
 Na 
 
 23.0 
 
 Gallium 
 
 Ga 
 
 69.0 
 
 Strontium 
 
 Sr 
 
 87.5 
 
 Germanium 
 
 Ge 
 
 
 Sulphur 
 
 S 
 
 32.0 
 
 Gold (aurum) 
 
 Au 
 
 196.7 
 
 Tantalum 
 
 Ta 
 
 182.0 
 
 Helium(?) 
 
 He 
 
 
 Tellurium 
 
 Te 
 
 127.0 
 
 TTvrlrn^pn 
 
 H 
 
 1.0 
 
 Thallium 
 
 Tl 
 
 205.0 
 
 Xlj 
 
 Tnrlinm 
 
 In 
 
 113.4 
 
 Thorium 
 
 Th 
 
 231.5 
 
 TodinP! 
 
 I 
 
 127.0 
 
 Tin (stannum) 
 
 Sn 
 
 108.0 
 
 Iridium 
 
 Ir 
 
 193.0 
 
 Titanium 
 
 Ti 
 
 48.0 
 
 Iron (ferrum) 
 
 Fe 
 
 56.0 
 
 Tungsten (wolfram) ... 
 
 W 
 
 184.0 
 
 Lanthanum 
 
 La 
 
 139.0 
 
 Uranium 
 
 U 
 
 240.0 
 
 Lead (plumbum) 
 
 Pb 
 
 207.0 
 
 Vanadium 
 
 V 
 
 51.2 
 
 Lithium 
 
 Li 
 
 7.0 
 
 Ytterbium 
 
 Yb 
 
 
 Magnesium 
 
 Mg 
 
 24.0 
 
 Yttrium 
 
 Y 
 
 89.0 
 
 Mercury (hydrargy- 
 
 
 
 Zinc 
 
 Zn 
 
 65.0 
 
 rum) 
 
 Hg 
 
 200.0 
 
 Zirconium 
 
 Zr 
 
 90.0 
 
 stands for carbonic-oxide gas, which is a chemical compound composed ot 
 two elements, carbon and oxygen. ...... 
 
 A molecule is expressed by writing the symbols of its elementary atoms. 
 Where more than 1 atom of a substance or element enters into the compo- 
 sition of a molecule, the number of atoms of such element is expressed by a 
 small subscript letter written immediately after the symbol of the element. 
 Thus carbonic-acid gas is composed of 1 atom of carbon chemically united 
 with’ 2 atoms of oxygen, and is expressed by the symbol C0 2 . Where the 
 symbol is written without such subscript figure, 1 atom only is meant. 
 Thus carbonic-oxide gas being composed of 1 atom of carbon chemically 
 united to 1 atom of oxygen, is expressed by the symbol CO. 
 
CHEMISTRY OF GASES. 
 
 343 
 
 A large figure written before the symbols expressing the molecule indi- 
 cates the number of molecules entering into the reaction. A large figure is 
 sometimes used before the symbol of a single element to indicate the number 
 of atoms of that element that enter the reaction. In any reaction occurring 
 between atoms of matter, no matter is destroyed. In any reaction, there 
 are always the same number of atoms after the reaction as before the 
 reaction took place. A chemical equation is therefore an expression of 
 equality between the atoms before and after a reaction takes place. The 
 first member of the equation contains the substances that act upon each 
 other while the second member of the equation contains the substances 
 that are formed by the reaction. The number of atoms is the same in each 
 member of the equation. „ , . , . 
 
 Example.— T o express the reaction that takes place when carbonic- 
 oxide gas burns in the air to produce carbonic-acid gas, we write 
 CO -+• 0 -H 4 N — C0 2 -f- 4 N 
 
 In this equation, each molecule of carbonic-oxide gas CO takes up 1 atom 
 of the free oxygen of the atmosphere to form carbonic-acid gas C0 2 . The 
 nitrogen in the atmosphere being 4 times the volume of oxygen, is expressed 
 as 4 atoms in the equation. This nitrogen, however, remains inactive, and 
 takes no part in the reaction. It is written on both sides of the equation for 
 the purpose of determining the atomic volumes of the gases before and after 
 the reaction, as explained below. 
 
 The reaction for an explosion of firedamp is 
 
 + 40 + 16N = 00 2 + 2 H,0 + 16 N m 
 
 In this equation, each molecule of marsh gas CH* is dissociated; that is 
 to say, its atoms are separated. The atom of carbon in the molecule unites 
 with 2 atoms of the oxygen of the air to form carbonic-acid gas C0 2 . The 
 4 atoms of hydrogen, in like manner, combine with two atoms of oxygen in 
 the air to form 2 molecules of water or steam 2 (HO), or 2 H 2 0. The nitro- 
 gen in this equation is equal to 4 times the volume of the oxygen consumed, 
 and is therefore written as 1 6N, since a total of 4 atoms of oxygen have 
 been used. The nitrogen is however inert, and plays no part in the reac- 
 tion itself, but is written here on both sides of the equation, as in the 
 previous equation, in order to properly represent the atomic volumes of 
 the gases or their relative volumes before and after the reaction takes place. 
 
 Calculation of the Relative Volumes of Gases— To calculate the relative 
 volumes of the gases before and after the reactions expressed in each of the 
 equations given in the preceding paragraphs, write beneath the symbol 
 of each molecule or atom its atomic volume. In the chemical equation 
 expressing the reaction that takes place when carbonic-oxide gas CO burns 
 to carbonic-acid gas C0 2 , we have as follows: 
 
 CO + O + 4N = C0 2 + 4 N 
 
 Atomic volumes, 2+1+4= 2+ 4 
 
 or in this reaction, 7 volumes have been reduced to 6 volumes. Such a 
 change of volume often takes place in chemical reactions. All attempts to 
 explain the cause of this change of volume, however, have thus far failed; 
 but that the change of volume does take place has been demonstrated by a 
 large number of experiments. . ' _ . . . , 
 
 To calculate the volume of air consumed in the complete explosion of 
 100 cu. ft. of carbonic-oxide gas CO, we write the ratio of the relative volumes 
 of carbonic-oxide gas and air, which is 2 : (1 + 4), or 2 : 5; and to obtain the 
 actual volume of air consumed in the explosion of 100 cu. ft. of carbonic- 
 
 oxide gas, we write the proportion 2 : 5 : : 100 : x, or x = — ^ — = 250 cu. ft. 
 
 To find the volume of carbonic-acid gas C0 2 produced in the complete 
 explosion of 100 cu. ft. of earbonic-oxide gas CO, write the ratio of the atomic 
 volumes of these two gases 2 : 2, which shows no change of volume, and. 
 therefore, the. volume of carbonic-acid gas C0 2 produced will be the same as 
 the volume of carbonic-oxide gas CO burned. 
 
 To find the volume of air consumed m the complete explosion of 100 
 cu. ft. of marsh gas CH h write the equation expressing the reaction that 
 takes place in this explosion as given above, 
 
 CH± + 40 + 16 N = C0 2 -f 2 H 2 0 + 1 6N 
 
 Atomic volumes, 2 + 4 + 16 = 2 + 4 + 16 
 
 There is no change of volume caused by the explosion, since 22 volumes 
 on one side of the equation produce, likewise, 22 volumes on the other side; 
 or 22 volumes before the explosion produce 22 volumes after the explosion. 
 
344 
 
 VENTILATION OF MINES. 
 
 To find the volume of air consumed, we write the ratio of the atomic 
 volumes of marsh gas and air 2 : (4 + 16), or 2 : 20, or 1 : 10; that is to 
 say, roughly speaking, the amount of air consumed in the complete explo- 
 sion of marsh gas is 10 times the volume of the marsh gas. This is not exact, 
 however, as the volume of nitrogen in the air is 3.83 times the volume of 
 oxygen. Making this correction, the volume of air consumed in the complete 
 explosion of marsh gas is 9.66 times the volume of the gas. 
 
 To determine the percentage of pure marsh gas in the above firedamp 
 mixture (marsh gas and air), we write the ratio of the atomic volumes of 
 
 these two, 2 : (2 + 4 + 15.32), or 1 : 10.66; and — X 100 = 9.38$ of CH±. 
 
 1 U* DO 
 
 The volume of carbonic-acid gas formed in this reaction is equal to the 
 volume of marsh gas consumed, and the volume of watery vapor is double 
 the volume of marsh gas consumed; the total volume of gas and vapor 
 formed by the reaction is the same as the original volume of marsh gas and 
 air, or firedamp mixture, since the sum of the atomic volumes on each side 
 of the equation is the same. 
 
 Atomic weight is the relative weight of an atom of an element compared 
 with an atom of hydrogen. Atomic weight is, then, not an absolute weight 
 to be expressed in pounds, ounces, or any other denomination, but is simply 
 relative weight. The atomic weight of each of the elements is given in the 
 table on page 342. 
 
 Molecular weight is the sum of the atomic weights of the elements forming 
 the molecule, taking the atomic weight of each element as many times as 
 there are atoms of that element in the molecule. A molecule of water is 
 composed of 2 atoms of hydrogen and 1 atom of oxygen, and as the atomic 
 weight of hydrogen is 1 and that of oxygen 16, the molecular weight of 
 water is (2 X 1) +16 = 18. In the same manner, since a molecule of marsh 
 gas CH4 is composed of 4 atoms of hydrogen and 1 of carbon, and the atomic 
 weight of hydrogen is 1 and that of carbon 12, the molecular weight of marsh 
 gas is (4 X 1) + 12 = 16. 
 
 The density of a gas is the weight of any volume compared with the 
 weight of the same volume of hydrogen or some other standard. The 
 density of a gas is constant at all temperatures and pressures, the change 
 of temperature and pressure affecting the gas in question and the standard 
 alike. The density of air referred to hydrogen is 14.38. 
 
 (a) The density of any simple gas, referred to hydrogen as unity, is equal to 
 its atomic weight. ( b ) The density of any compound gas , referred to hydrogen 
 as unity, is one-half of its molecular weight. 
 
 Specific Gravity of Gases.— The specific gravity of a gas is the weight of 
 that gas referred, to the weight of a like volume of air as a standard. It is, 
 in other words, the ratio between the weight of like volumes of any gas and 
 air, both the air and gas being subject to the same temperature and pressure. 
 Thus, since the weight of 1 cu. ft. of air at a temperature of 60° F. and 30 in. 
 barometric pressure is .0766 lb., and the weight of 1 cu. ft. of carbonic-acid 
 gas C0 2 is .11712 lb. at the same temperature and pressure, the specific 
 
 .11712 
 
 gravity of carbonic-acid gas is --y — = 1.529. 
 
 Weight of Gases.— The weight of 1 cu. ft. of any gas at any given tempera- 
 ture and pressure is found by first calculating the weight of 1 cu. ft. of dry 
 air at the same temperature and pressure by means of the formula given on 
 page 338 for air, and then multiplying this weight by the specific gravity of 
 the gas referred to air as a standard. 
 
 Example.— To determine the weight of 1 cu. ft. of carbonic-acid gas at a 
 temperature of 60° F., and 30 in. barometric pressure, we multiply the 
 weight of 1 cu. ft. of dry air, at this temperature and pressure, as found 
 above (.0766 lb.), by the specific gravity of carbonic-acid gas (1.529). Thus, 
 .0766 X 1.529 = .11712 lb. 
 
 The table on page 349 gives the specific gravity of the gases common in 
 mining practice, referred to air as a standard. 
 
 Expansion of Air and Gases.— All air and gases expand uniformly at the 
 same rate. The expansion and contraction of air and gases follow two 
 simple laws that we will consider under the heads (a) Ratio of volume and 
 absolute temperature and ( b ) Ratio of volume and absolute pressure. 
 
 Absolute temperature means the temperature as reckoned from absolute 
 zero, which is the point on the temperature scale below which it is assumed 
 that no substance can exist in a gaseous state. * The absolute zero of the 
 
PRESSURE. 
 
 345 
 
 » 
 
 Fahrenheit scale is assumed in mining practice as 459° below zero. Hence, 
 the absolute temperature corresponding to any common temperature is 
 found by adding 459° to the common temperature. Thus, the absolute 
 temperature corresponding to 60° F. is 459 + 60 = 519°. 
 
 Absolute Pressure.— The term absolute pressure refers to the total pressure 
 supported by air or gas; i. e., the pressure above a vacuum. Gauge pressure 
 is the pressure above the atmosphere. Absolute pressure is always the 
 atmospheric pressure plus the gauge pressure. If a gauge pressure on a 
 boiler indicates 60 lb. per sq. in., the absolute pressure supported by the 
 steam in the boiler will be 60 + 14.7 = 74.7 lb. per sq. in. Or, if the ventila- 
 ting pressure in a mine is equal to 13 lb. per sq. ft., the absolute pressure 
 supported by the air in the airways will be 13 + (14.7 X 144) = 2,129.8 lb. 
 ner so ft 
 
 Relation of Volume and Absolute Temperature ( Charles ’ or Gay Lussac's law). 
 The volume of any air or gas varies directly as its absolute temperature. 
 
 Relation of Volume and Absolute Pressure ( Boyle's or Mariotte’s law).— The 
 volume of any air or gas varies inversely as the absolute pressure it supports. 
 For example, if we double the absolute pressure supported by air or gas, 
 the volume of the air or gas will be reduced to one-half its original volume; 
 if we multiply the absolute pressure 3 times, we reduce the volume to one- 
 third the original volume; etc. „ „ . 
 
 Example.— The intake current of a mine is 50,000 cu. ft. of air per minute; 
 the ventilating pressure is 13 lb. per sq. ft. The temperature of the intake is 
 20° F.; the temperature of the return air is 70° F. Calculate the volume of 
 the return air-current per minute, according to the rules of expansion of air, 
 due to the increase of temperature and decrease of pressure, in the return 
 current. 
 
 The increased volume of the return air, due to the decrease of pressure 
 and increase of temperature, is found by writing a compound proportion, 
 the first member of which consists of two ratios, viz., the direct ratio of the 
 absolute temperatures, and the inverse ratio of the absolute pressures, accord- 
 ing to the two laws stated above. That is, we write 
 
 Or, 
 
 (459 + 20) : (459 + 70) 
 2,116.8 : 2,129.8 
 
 529 X 2,129.8 X 50,000 
 479X2,116.8 
 
 50,000 : x. 
 
 55,558 cu. ft. 
 
 Example— In a compressed-air plant, the gauge shows a pressure of 80 lb. 
 per sq. in. ; the area of the piston is 20 sq. in., and its stroke 10 in. The pump 
 makes 100 strokes per minute. Assuming there is no leakage of air past the 
 piston, what will be the volume of air discharged from the pump into the 
 mine per minute? , „ , 
 
 The volume of air discharged from the pump cylinder per minute 
 
 = 20 X 10 X 100 _ 11<57 cu ft (cylinder pressure). The absolute pressure 
 
 on the air in the cylinder is 80 + 14.7 = 94.7 lb. The absolute pressure on 
 the discharged air is simply the atmospheric pressure (14.7 lb.). Hence, we 
 
 write the proportion 14.7 : 94.7 : : 11.57 : x ; or, x = — ; X 11.57 = 74.54 cu. ft. 
 
 per minute, nearly. , . . , 
 
 In calculating the expanded volume of air or gas, it will be observed that 
 the ratio of the original volume to the expanded volume is always equal to 
 the product of the direct ratio of the absolute temperatures and the inverse 
 ratio of the absolute pressures, which gives a compound proportion, the first 
 member of which consists of two ratios, the one a direct ratio and the other 
 an inverse ratio. r A . . - 
 
 Weight Produces Pressure.— In the study of the barometer as a means of 
 measuring atmospheric pressure, we observe that the weight of the atmos- 
 
 g here produced the atmospheric pressure. In like manner, the weight of all 
 uids produces pressure, and this pressure acts equally in all directions. 
 This is an important consideration in the study of mine ventilation, since it 
 has given rise to the measurement of pressure by air or motive columns. 
 
 Calculation of Pressure.— An air column, or motive column, in ventilation, 
 is a column of air having a base of 1 sq. ft., and of such height that its weight 
 shall be equal to any given pressure. To calculate the height of air column 
 corresponding to any given pressure, divide the pressure in pounds per 
 square foot by the weight of 1 cu. ft. of the air. Mine pressure is also 
 
346 
 
 VENTILATION OF MINES. 
 
 measured by the water column that it will support, as in the water gauge, or by 
 the mercury column, as in the barometer. In the measurement of pressure 
 by means of the water column, the weight of the water column must be 
 equal to the pressure, ( ireafor area. 
 
 Since the weights of these columns are proportional to their sectional areas, 
 it makes no difference what this area may be, the weight of the column 
 calculated for a sectional area of 1 sq. in. will 
 equal the pressure per square inch that supports 
 the same. 
 
 Hence, since 1 cu. in. of mercury weighs .49 lb., 
 .49 X 30 = 14.7 lb. is the atmospheric pressure 
 corresponding to a height of 30 in. of mercury, or, 
 as we say, 30 in. of barometer. If we consider a 
 cubical box, as shown in the accompanying 
 figure, holding exactly 1 cu. ft. of water, and 
 assume the weight of the water to be 62.5 lb., 
 as is usual in practice, and divide the bottom of 
 the box into 144 sq. in., as show T n in Fig. 1, we 
 observe: 
 
 (a) The pressure of the water on the bottom 
 of the box is equal to the weight of the water, 
 
 62.5 lb.; that is to say, the pressure per square foot due to lft. of water column 
 is 62.5 lb. 
 
 ( b ) The pressure on the bottom of the box, when the water is only 
 1 in. deep, is equal to the weight of a layer of water 1 in. thick, or 
 62 5 
 
 — = 5.2 lb.; or, the pressure per square foot due to 1 in. of water column 
 12 
 
 is 5.2 lb. 
 
 (c) The pressure per square inch on the bottom of the box is equal to 
 the weight of a prism of water 1 ft. high, and having a base of 1 sq. in. 
 
 = .434; or, the pressure per square inch due to 1 ft. of water column is .454 lb. 
 
 These principles relating to the pressure of fluids are important to the 
 student of mining, of which the following are examples: 
 
 1. In a mountainous country, several thousand feet above sea level, 
 where the barometer registers, say, only 21 in., it is desired to know the 
 theoretical height a pump will draw. .49 X 21 = 10.29 lb. atmospheric 
 
 pressure, and = 24 ft., nearly. The theoretical suction, in this instance, 
 
 is 24 ft., nearly, but the actual draft or suction would vary from f to $ of 
 this, according to the perfection of the pump. 
 
 2. The water-gauge reading between the intake and return airways of a 
 certain mine is 2.5 in.; to determine the pressure per square foot, we have, 
 
 2.5 X 5.2 = 13 lb. per sq. ft. 
 
 3. To determine the pressure per square foot on a mine dam, due to a 
 vertical head of 200 ft., 62.5 X 200 = 12,500 lb. 
 
 4. To express in air column or motive column, a mine pressure equivalent 
 to a water-gauge reading of 3 in., assuming the temperature of the air to be 
 60° F. and the barometric pressure 30 in., we have for the weight of 1 cu. ft. 
 
 1 3253 X 30 
 
 of air at this temperature and pressure w = — — y = .0766 lb. The 
 
 pressure per square foot due to 3 in. of water gauge is 3 X 5.2 = 15.6. Then, 
 
 15 6 
 
 we have for motive column, m = = 204 ft. 
 
 .0766 
 
 Diffusion and Transpiration of Gases.— Diffusion of gases means the mixing 
 of the gaseous volumes. Graham took several glass tubes, and inserting in 
 one end of each a plug of plaster of Paris that was porous, he filled each 
 tube with a different gas; as for example, oxygen, hydrogen, nitrogen, etc. 
 He then placed the open end of each inverted tube in a basin of mercury, 
 supporting the tubes in an erect position. The gas in each tube immedi- 
 ately began to diffuse through the porous plaster plug into the atmosphere, 
 and it was observed that the mercury rose in each tube to take the place of 
 the gas that passed into the atmosphere. The mercury rose in the hydrogen 
 tube 4 times as fast as in the oxygen tube, and in the other tubes the mer- 
 cury rose at different rates. 
 
 Rate of Diffusion (Graham’s Law).— The rate of diffusion of gases into air is 
 
 
 ~~7 
 
 % 
 
 '\ 
 
 m 
 
 62 
 
 S /As 
 
 
 — jg . 
 
 
 Fig. 1. 
 
DIFFUSION OF GASES. 
 
 347 
 
 in the inverse ratio of the square roots of their densities. The density of 
 oxvaen being 16 and hydrogen 1, the rate of diffusion of oxygen as compared 
 with hytogfn is 1 to 4; that is to say, the rate of diffusion of oxygen is only 
 one-fourth that of hydrogen. 
 
 Table Showing the Corresponding Mercury and Air Columns, and 
 Pressure per Square Foot for Each Inch of Water Column. 
 
 0) 
 
 bo 
 
 03 <D 
 
 © P 
 C 3 1 ”" 1 
 £ 
 
 P 
 
 a 
 
 p . 
 
 7? M 
 
 M P 
 
 O 
 
 © 
 
 .0735 
 
 .1471 
 
 .2206 
 
 .2941 
 
 .3676 
 
 aw 
 
 £o~ 
 
 8% 
 
 © 
 
 © 
 
 68 
 
 136 
 
 204 
 
 272 
 
 340 
 
 © ■ 
 M a* 
 PGO 
 
 8§J 
 
 rQ 
 
 w 
 
 5.2 
 
 10.4 
 
 15.6 
 
 20.8 
 
 26.0 
 
 © 
 
 bo 
 
 a °° 
 
 03 n; 
 
 O.P 
 © 
 © P 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 P 
 
 a 
 
 p . 
 
 02 
 
 83 
 
 frS 
 
 P H 
 
 O 
 
 u 
 
 © 
 
 .4412 
 
 .5147 
 
 .5882 
 
 .6618 
 
 .7353 
 
 aw 
 
 .JhH 
 
 © 
 
 © 
 
 407 
 
 475 
 
 543 
 
 611 
 
 679 
 
 © • 
 U U* 
 p CQ 
 
 *3 
 
 31.2 
 
 36.4 
 
 41.6 
 
 46.8 
 
 52.0 
 
 Table Showing the Corresponding Water Column, and Pressure per 
 
 Barometer. 
 
 Inches. 
 
 Water Column J 
 Feet. 
 
 Pressure. 
 
 Lb. per Sq. In. 
 
 Barometer. 
 
 Inches. 
 
 Water Column. 
 Feet. 
 
 Pressure. 
 
 Lb. per Sq. In. 
 
 1 
 
 1.13 
 
 .49 
 
 . 16 
 
 18.13 
 
 7.84 
 
 2 
 
 2.27 
 
 .98 
 
 17 
 
 19.27 
 
 8.33 
 
 3 
 
 3.40 
 
 1.47 
 
 18 
 
 20.40 
 
 8.82 
 
 4 
 
 4.54 
 
 1.96 ! 
 
 19 
 
 21.53 
 
 9.31 
 
 5 
 
 5.67 
 
 2.45 
 
 20 
 
 22.67 
 
 9.80 
 
 5 
 
 6.80 
 
 2.94 
 
 21 
 
 23.80 
 
 10.29 
 
 7 
 
 7.93 
 
 3.43 
 
 22 
 
 24.93 
 
 10.78 
 
 8 
 
 9.06 
 
 3.92 
 
 23 
 
 26.07 
 
 11.27 
 
 9 
 
 10.20 
 
 4.41 
 
 24 
 
 27.20 
 
 11.76 
 
 10 
 
 11.33 
 
 4.90 
 
 25 
 
 28.33 
 
 12.25 
 
 11 
 
 12.46 
 
 5.39 
 
 26 
 
 29.47 
 
 12.74 
 
 12 
 
 13.60 
 
 5.88 
 
 27 
 
 30.60 
 
 13.23 
 
 13 
 
 14.73 
 
 6.37 
 
 28 
 
 31.73 
 
 13.72 
 
 14 
 
 15.87 
 
 6.86 
 
 29 
 
 32.87 
 
 14.21 
 
 15 
 
 17.00 
 
 7.35 
 
 30 
 
 34.00 
 
 14.70 
 
 t a picDBuiovi iu. 
 
 30 in., is *h; the weighty of 
 
 w q ater hutVeTato of til to watTr'i^oftSfaiumed ’aJl.2:Y,Udo forVick 
 calculation. The ; Speciflc.gjavity ofm 
 
 calculation. j.ixc opci/im/ 
 
 ture for barometric readings) is 13.62; and a nVrpn ft 
 
 temperature weighs 849 lb. For ordinary calculation tne weight of 1 cu. ft. 
 of water is taken as 62.5 lb. The exact weight of 1 cu. ft. of pure water, at a 
 temperature of 32° F. t is, however, 62.418 lb. 
 
 The diffusion of marsh gas (Sp. Gr. .559) is much more rapid than that of 
 carbonic-acid gas (Sp. Gr. 1.529). The diffusion of gases, however, is greatly 
 assisted by the movement of the air-current; or by the movement of the gas 
 as it tends to rise or fall, according to its relative densityand Positionmthe 
 airway. For example, suppose a gas feeder to be located in .the * floor : of an 
 airway. The marsh gas given off from the feeder, being lighter than air, 
 
348 
 
 VENTILATION OF MINES. 
 
 tends to rise toward the roof. The action of rising helps a diffusion of this 
 gas very much. On the other hand, a feeder located in the roof gives off the 
 same gas, which tends to accumulate along the roof, and if "the air-current 
 is at all sluggish at this point, the diffusion of the marsh gas will be compar- 
 atively slow. It often happens that a feeder in the roof or other high point 
 of the workings gives off gas more quickly than diffusion can take place, 
 where the air-current is sluggish. This results in the accumulation of a 
 body of pure marsh gas at this point. In like manner, we often have an 
 accumulation of a large body of carbonic-acid gas, or blackdamp, near the 
 floor or other low place in the mine workings, where the air-current is 
 sluggish and where the blackdamp is formed quicker than diffusion takes 
 place. 
 
 Limit of Diffusion. — The diffusion of gases continues to take place until the 
 mixture of the gases is uniform. It is a curious fact that this takes place 
 earlier or quicker in the case of gases whose densities differ widely, than where 
 the densities of the two gases are nearly alike. Thus, saturation will take 
 place more quickly in the diffusion of carbonic-acid gas into air than in the 
 diffusion of firedamp into air, although the rate of diffusion of the latter is 
 greater than of the former, firedamp being lighter than carbonic-acid gas. 
 
 The property of diffusion is of the greatest importance in the ventilation 
 of mines, since it is owing to this that the air-current is- enabled to sweep 
 away these gases from their lurking places in the workings more rapidly 
 and effectually than it otherwise could. 
 
 Transpiration of gases is the exuding of the gases from the pores of the coal 
 in which they are contained. It is a well-known fact that transpiration 
 takes place more rapidly from a newly exposed face of coal. This is owing 
 to the fact that the gas pent up in the coal, or occluded in the seam, tends 
 to escape at the first opportunity, when the seam is exposed to the atmos- 
 phere. The gas is under a certain pressure, as we have previously observed, 
 and, as the mine workings penetrate the coal seam, the gases are forced 
 outward from the coal by their own pressure, thus expanding into the air of 
 the mine. 
 
 The transpiration of gas from coal seams differs very widely, in some 
 seams it being so rapid and violent as to splinter and break the coal in its 
 effort to escape. It frequently causes a crackling sound peculiar to a very 
 gaseous seam, and in some cases, causes fine coal to be thrown into the face 
 of the miner. 
 
 GASES FOUND IN MINES. 
 
 Oxygen 0 is a colorless, odorless, tasteless, non-poisonous gas. It is heavier 
 than air, having a specific gravity of 1.1056. It is the great supporter of life 
 and combustion. Oxidation , or the union of any of the elements with 
 oxygen, is simply another term for combustion in its broadest sense. Most 
 forms of matter containing carbon are easily decomposed at certain temper- 
 atures, through carbon seeking to combine with the oxygen of the air to 
 form carbonic-acid gas C0 2 . This union of the oxygen with other elements, 
 or oxidation, takes place at all temperatures. It is less active when the 
 temperature is low, and is then known as slow combustion. An example of 
 this is found in the gob fires that occur so frequently in mine workings. 
 The fine coal that is so often thrown back into the gob is acted on first by 
 moisture, and as its temperature rises, carbonic-oxide gas is formed in small 
 quantities by the union of the carbon of the coal with the oxygen of the air; 
 as the temperature rises, more gas is formed. Heat is caused by the chemical 
 action due to the interchange of the atoms, this heat being often sufficient to 
 ignite the gas formed, spontaneous or active combustion resulting. Oxygen 
 is the element in the atmosphere on which all life depends. 
 
 Nitrogen A is a colorless, odorless, and tasteless gas: it is neither combus- 
 tible nor a supporter of combustion; it is not poisonous, and is lighter than 
 air, having a specific gravity of .9713. Nitrogen is a particularly inert gas; it 
 takes no active part in any combustion, in the sense of causing such com- 
 bustion. Its province is to dilute oxygen of the atmosphere, on which 
 life depends. Were it not for this dilution, oxidation would be too rapid, 
 and not as completely under control as at present. The effect of nitrogen 
 on human life would be to suffocate, if breathed pure, inasmuch as it would 
 exclude oxygen from the lungs. Nitrogen itself has no life-giving power. 
 
 Marsh gas CH h often called light carbureted hydrogen , or methane, is a 
 chemical compound, consisting of 4 atoms of hydrogen to 1 atom of carbon. 
 
GASES FO UND IN MINES. 
 
 349 
 
 It is one of the chief gases occluded in coal seams, and results from the 
 metamorphism of the carbonaceous matter from which coal is formed, when 
 such metamorphism has taken place with the exclusion of air, and m presence 
 of water. Pure marsh gas is colorless, odorless, and tasteless, and. is iignter 
 than air. Its specific gravity is .559, and it diffuses rapidly in the air, forming 
 a firedamp mixture. Marsh gas burns with a blue flame, but it will not 
 support combustion, and a lamp placed in it is immediately extinguished. 
 In the mine, it is a difficult matter to place a lamp m a body of pure marsh 
 gas, since the gas diffuses so rapidly that a firedamp mixture always sur- 
 rounds a body of pure gas, which may exist high up m some cavity of the 
 roof, or at the face of a steep pitch where the circulation is slow and the 
 feeder at the face is giving off a large amount of gas. The flaming of 
 the lamp in passing through a fir edamp mixture would at once cause the 
 withdrawal of the lamp before reaching the body of pure marsh gas. But 
 could a lamp be placed in a body of pure marsh gas, it would be extinguished 
 at once. Marsh gas is not poisonous, and when mixed with air m sufficient 
 proportion, it may be breathed for a considerable time with impunity (see 
 Firedamp). Pure marsh gas does not support life, but suffocates by exclu- 
 ding oxygen from the lungs. , , , . , 
 
 Other Hydrocarbons.— All gases that are compounds of carbon and hydrogen 
 are called hydrocarbons. Qf these, the chief member is marsh gas, or light 
 carbureted hydrogen, described in the preceding paragraph; the other 
 hydrocarbons are called heavy hydrocarbons. The chief of these are olefiant 
 gas CM f 4 , and ethane C 2 H Q . Both of these gases, like marsh gas, are the 
 result of the metamorphism of carbonaceous matter, during the formation of 
 the coal, but unlike marsh gas, they have been produced in the absence of 
 water, and as a result they contain a larger percentage of carbon than 
 marsh gas. They always exist in common with marsh gas, as occluded gases 
 in coal seams, but to a far less extent. Each of these gases possesses a higher 
 illuminating power, burning with a brighter flame than marsh gas. This is 
 due to the larger percentage of carbon present in their composition. Their 
 remaining properties are very similar to the properties given for marsh gas; 
 thev, however, when present in a firedamp mixture, lower the temperature 
 of ignition, and render the mixture more dangerous than it would be 
 otherwise. , , 
 
 Constants for Mine Gases— The following table shows the symbols, specific 
 gravities, and relative velocities of diffusion and transpiration of the principal 
 mine gases, arranged in the order of their specific gravities, air being taken asl. 
 The values given in the next to the last column of this table were 
 obtained by experimenting with the gases, and agree quite closely with 
 the calculated values given in the preceding column. From this column 
 we see that 1,344 volumes of marsh gas will diffuse in the same time as 1,000 
 volumes of air, or 812 volumes of carbonic-acid gas. 
 
 Table of Mine Gases. 
 
 Name of Gas. 
 
 Symbol. 
 
 Specific 
 
 l 
 
 Relative Velocity [Relative Velocity 
 r»f Diffnsirn. of Transniration. 
 
 Gravity. 
 
 V Sp. Gr- 
 
 (Air = 1.) 
 
 (Air = 1.) 
 
 Air 
 
 
 1.00000 
 
 1.0000 
 
 1.000 
 
 1.0000 
 
 Carbonic acid 
 
 co 2 
 
 1.529 
 
 .8087 
 
 .812 
 
 1.2371 
 
 Sulphureted hy- 
 drogen 
 
 'MS 
 
 1.1912 
 
 .9162 
 
 .95 
 
 .903 
 
 Oxygen 
 
 0 
 
 1.1056 
 
 .9510 
 
 .9487 
 
 Olefiant 
 
 c 2 h 4l 
 
 .978 
 
 1.0112 
 
 1.0191 
 
 1.788 
 
 Nitrogen 
 
 N 
 
 .9713 
 
 1.0147 
 
 1.0143 
 
 1.0303 
 
 Carbonic oxide ... 
 
 CO 
 
 .967 
 
 1.0169 
 
 1.0149 
 
 1.034 
 
 Steam 
 
 Marsh gas 
 
 MO 
 
 CHt 
 
 .6235 
 
 .559 
 
 1.2664 
 
 1.3375 
 
 1.344 
 
 1.639 
 
 Hvdrogen 
 
 M 
 
 .06926 
 
 3.7794 
 
 3.83 
 
 2.066 
 
 Carbonic-oxide gas CO, often called whitedamp , is a chemical compound 
 consisting of 1 atom of carbon united to 1 atom of oxygen. To a certain 
 
350 
 
 VENTILATION OF MINES. 
 
 extent it occurs as an occluded gas in coal. It is chiefly formed, however, 
 in coal mines, by the slow combustion of carbonaceous matter in the gobs or 
 waste places of the mine, where the supply of air is limited. It is always the 
 product of the slow combustion of carbon in a limited supply of air. It is 
 therefore one of the chief products of gob fires, and is also a product of the 
 explosion of powder in blasting. This gas often fills the crevice made 
 behind a standing shot, and causes the flash that takes place when the 
 miner puts his lamp behind such shot to examine the same. This gas is 
 formed in large quantities whenever the flame of a blast or explosion is 
 projected into an atmosphere in which coal dust is suspended. The force of 
 a blast often blows the dust into the air, and the flame acting on it distils 
 carbonic-oxide gas. 
 
 Carbonic-oxide gas is lighter than air, having a specific gravity of .967, 
 and it thereicre accumulates near the roof and in the higher working- 
 places. It is colorless, odorless, and tasteless. It is combustible, burning 
 with a light-blue flame. This is the flame often seen over a freshly fed 
 anthracite fire. It is also a supporter of combustion, being the only mine 
 gas that burns and also supports combustion. This property leads to very 
 important results in mines, inasmuch as it lengthens the flame of a lamp or 
 the flame of a blast. Any flame is fed by this gas, and is thereby transmitted 
 through the mine airways, from one point to another. The same property 
 extends very widely an otherwise local explosion. This gas has the widest 
 explosive range of any gas known to mining, except hydrogen. The effect 
 of its presence in firedamp mixtures is always to widen the explosive range 
 of the firedamp, causing it to become explosive in larger and smaller 
 proportions than it otherwise would. Carbonic-oxide gas is a very poison- 
 ous gas, and acts on the human system as a narcotic, producing drowsiness 
 or stupor, followed by acute pains in the head, back, and limbs, and after- 
 ward by delirium. It acts, when breathed into the lungs, to absorb the 
 oxygen from the blood, or, in other words, poisons the blood. 
 
 Carbonic-oxide gas is detected in mine workings by its effect on the flame 
 of a lamp, w T hich burns more brightly in the presence of the gas, and reaches 
 upwards as a slim, quivering taper, having often a pale-blue tip that, 
 however, is not readily observed. 
 
 Carbonic-acid gas CO 2 , often called blackdamp or chokedamp , is a chemical 
 compound consisting of 1 atom of carbon united to 2 atoms of oxygen. It is 
 heavier than air, having a specific gravity of 1.529. It therefore accumulates 
 near the floor or in the low places of the mine workings. It is always the 
 result of the complete combustion of carbon in a plentiful supply of air, and 
 is a product of the breathing of men and animals, burning of lamps, or any 
 other complete combustion. It is always present in occluded gases. 
 
 Carbonic-acid gas is a colorless, odorless gas, but possesses a peculiarly 
 sweet taste, which may be detected in the mouth when it is inhaled in large 
 quantities. It is not combustible, nor is it a supporter of combustion. 
 Lamps are at once extinguished by it. It diffuses slowly into the atmosphere, 
 and is a difficult gas to remove in ventilating. It is not poisonous, but acts 
 to suffocate by excluding oxygen from the lungs. Its effect, when breathed 
 for any length of time, is to cause headache and nausea, followed by weak- 
 ness and pains in the back and limbs; when present in larger quantities, it 
 causes death by suffocation. This gas, when present in firedamp mixtures, 
 has the opposite effect from that of carbonic-oxide gas, inasmuch as it 
 narrows the explosive range of the firedamp, and renders such mixtures 
 inexplosive, which would otherwise be explosive (see Firedamp). 
 
 Carbonic-acid gas is detected in the mine air by the dimness of the lamps 
 and by their extinguishment when present in larger quantities. It should 
 always be looked for at the floor, and in low places of the mine workings. 
 
 Sulphureted hydrogen H 2 S occurs at times as an occluded gas in coal seams, 
 but more often exudes from the strata immediately underlying or over- 
 lying those seams. It is generally supposed to be formed by the disintegra- 
 tion of pyrites in the presence of moisture. It is heavier than air, having a 
 specific gravity of 1.1912. It is a colorless gas, having a very disagreeable 
 odor resembling that of rotten eggs, and is known to the miners as stinkdamp. 
 It is an exceedingly dangerous gas when occurring in considerable quan- 
 tities. When mixed with 7 times its volume of air, it is violently explosive. 
 It is extremely poisonous, acting to derange the system when breathed 
 in small quantities, and, when inhaled in larger quantities, it produces 
 unconsciousness and prostration. Its smell serves as the best means for its 
 detection. 
 
GASES FOUND IN MINES. 
 
 351 
 
 Firedamp. — The general term firedamp relates to any explosive mixture of 
 marsh gas and air, although in some localities this term is understood as 
 referring to any mixture of marsh gas and air whatever, whether explosive 
 or otherwise. Many persons speak of pure marsh gas as firedamp. The first 
 meaning given above, however, is the general acceptation of the term. 
 
 Pure marsh gas when present in small quantities in the air burns m the 
 flame of the lamp without explosion. As the quantity of the gas is 
 increased, the effect on the flame of the lamp is at once noticeable. As the 
 proportion of gas in the air is further increased, and approaches the lower 
 explosive limit, the lamp flame enlarges, snaps, and crackles. When the 
 proportion of gas to air is 1 to IB, slight explosions occur within the lamp, 
 the flame of the lamp jumping violently. As the proportion of gas is 
 increased, the violence of the explosion is augmented until it reaches a 
 maximum, when the proportion of gas to air is 1 to 9i (exactly, 1: 9.38). 
 This is the proportion of gas and air in firedamp, when at its maximum 
 explosive violence. From this point, as the quantity of gas is still further 
 increased, the violence of the explosion decreases, until it becomes very 
 feeble when the proportion of gas and air is 1 to 5i, and ceases altogether 
 beyond this point. The explosive limits of marsh gas, or the limits of fire- 
 damp mixtures, are then as follows: Lower limit, 1 volume of gas to 13 
 volumes of air; higher limit, 1 volume of gas to 5£ volumes of air. These 
 limits refer to pure firedamp, or, in other words, a firedamp mixture con- 
 sisting of pure marsh gas and air. , . . , . . , 
 
 It rarely happens that firedamp, as found m mine workings, is pure, but 
 contains admixtures of other gases, such as carbonic-acid gas CO 2 , carbonic- 
 oxide gas CO, and heavy hydrocarbons. . j A 
 
 Afterdamp. — The term afterdamp relates to the gaseous mixture that exists 
 in mine workings after an explosion of gas. The composition of afterdamp 
 is exceedingly variable, and admits of no general analysis that can be 
 applied with certainty to any one explosion. The conditions that obtain 
 in an explosion are so manifold, and control so completely the character of 
 the gases formed, that it is impossible to give more than a general analysis 
 of afterdamp. . „ , 
 
 The chief products of the complete explosion of pure firedamp are car- 
 bonic-acid gas, watery vapor, and nitrogen (see page 343). The explosion of 
 firedamp is seldom, however, complete. Where a large body of gas has 
 exploded, the air-current in the mine workings does not furnish sufficient 
 air for the complete combustion of the firedamp, and as a result, a large 
 amount of carbonic-oxide gas is formed, and is present in the afterdamp of 
 the explosion. The presence of this gas (CO) renders the afterdamp far 
 more dangerous than it would otherwise be, for two reasons: The gas itself 
 is very poisonous, and its presence is not at once detected by the zealous 
 men that are working to rescue their fellow workmen. The lamps burn 
 very brightly in this gas, and the rescuers press forward unconscious of their 
 real danger until overcome by the effects of the gas. The presence of coal 
 dust in suspension in the mine air at the time of the explosion, or thrown 
 into the air by the force of the explosion, results at once in the production 
 of a large amount of carbonic-oxide gas. It is a well-known fact, also, that 
 carbonic-acid gas CO 2 , formed as a direct product of the explosion, or which 
 may be present in the mine air before the explosion, coming in contact 
 with the incandescent carbon of the coal dust, is converted by it, at the high 
 temperature of the flame, into carbonic-oxide gas CO. We observe, there- 
 fore, that an explosion, in all its effects, tends to the rapid and abundant 
 production of this most dangerous gas. 
 
 The other products of an explosion are numerous, but for the larger part, 
 unimportant, except as they do not support life. We have mentioned and 
 described only those gases forming the larger portion of the afterdamp, or 
 constituting the active agents in an explosion. 
 
 Occurrence of Oases in Mines.— Most of the gases occurring m mines are 
 occluded in the coal or the strata adjacent to the coal seam. These occluded 
 gases differ widely in different coals, but are chiefly marsh gas with varying 
 quantities of heavy hydrocarbons (olefiant gas, ethane, etc.) and carbonic- 
 acid gas to a limited extent. In some coals, a large percentage of nitrogen 
 gas is occluded, which, however, transpires very slowly into the atmosphere. 
 Sulphureted hydrogen, when present, usually exudes from the underlying 
 or overlying strata of the coal seam. These occluded gases are the result of 
 coal formation, according to the best authorities and evidence. They exist 
 in the pores of the coal under considerable pressure due to the weight of the 
 
352 
 
 VEX TIL A TIOX OF MIXFX. 
 
 superincumbent strata. When the strata adjacent to the coal seam are 
 impervious to gases, the occluded gases remain pent up, and we have what 
 is called a gaseous seam. The tendency of the gas is alwavs to escape to the 
 surface or into the mine workings, at the first opportunity. 
 
 These gases have each their separate effects, and their combined effect is 
 sometimes very complicated. We can only study to become familiar with 
 the separate characteristics of each of these gases, and judge of their com- 
 bined effect when present in firedamp mixtures. For example, one effect of 
 all these gases when present in firedamp mixtures is to dilute the firedamp, 
 and to that extent weaken its explosive force. Dilution of the firedamp bv 
 carbonic-acid gas CO* decreases very rapidly the explosiveness of the fire- 
 damp. When carbonic-acid gas is present in firedamp to the extent of one- 
 seventh its volume, explosion ceases altogether; in other words, the 
 firedamp is rendered inexplosive. The effect of smaller quantities of this 
 gas is to contract the explosive limits of the firedamp as well as to weaken 
 the explosion. The effect of carbonic-oxide gas CO when present in fire- 
 damp mixtures is likewise to dilute the firedamp. The flame, however, is 
 lengthened by this gas. and the explosive limits of the fire dam p mixture are 
 widened. In other words, mixtures of marsh gas and air. which were not 
 explosive mixtures, are rendered explosive by the presence of carbonic- 
 oxide gas. 
 
 The chief source of the other mine gases lies in the slow combustion 
 of carbonaceous matter in the gob. gob fires, burning of lamps, breathing 
 of men and animals, etc. The table on page 353 shows the percentage of 
 occluded gases in a number of coals and their volume at normal temper- 
 ature and pressure. 
 
 Gas Feeders i Pockets!. — The occluded gas of a coal seam escapes when- 
 ever opportunity is offered, and accumulates in the pockets and crevices 
 of the adjoining strata, forming what are called gas feeders. These consti- 
 tute a very dangerous element in the mining of gaseous seams, inasmuch as 
 when such a crevice or feeder is tapped by the miner’s drill, the gas, which 
 is usually under heavy pressure, blows out in a large volume, at times even 
 blowing the drill from the hole. 
 
 Pressure of Occluded Gases. — Occluded gases exist under a pressure that is 
 proportionate to the weight of the overlying strata. Numerous experiments 
 in England. France, and Belgium show* that the pressure of gases occluded 
 in coal seams frequently amounts to from 10 to 16 atmospheres, and in some 
 cases has reached 32 atmospheres. These high pressures of occluded gases 
 manifest themselves frequently in the boring of gas wells, where the tools 
 are at times blown from the bore hole. 
 
 Pressure of Occluded Gas. 
 
 Name of Mine. 
 
 Depth of Hole. 
 Feet. 
 
 Pressure. 
 
 Pounds. 
 
 Elmore mine, main bed 
 
 8.53 
 
 4.36 
 
 Hetton mine, Hutton bed 
 
 8.98 
 
 6.96 
 
 Eppleton mine, Hutton bed... 
 
 46.90 
 
 36.14 
 
 Balden mine, Bensham bed... 
 
 31.85 
 
 71.41 
 
 Harris Navigation mine 
 
 32.80 
 
 22.04 
 
 Merthvr Vale mine 
 
 49.20 
 
 39.67 
 
 Celvnen mine 
 
 54.48 
 
 68.32 
 
 Harton mine (1,214 ft. deep) 
 
 16.24 
 
 196.30 
 
 Harton mine 
 
 27.55 
 
 230.44 
 
 Harton mine 
 
 37.13 
 
 294.45 
 
 Amount of Gas. — Experiments made by the Prussian Firedamp Commission 
 have given results varying from 357 to 2,400 cu. ft. of gas liberated per ton of 
 coal mined. Mr. Chesneau gives 1,377 cu. ft. at the Herin mine, Anzin. 
 Experiments at the Ronchamp mines give 883 cil ft. 
 
 Outbursts of gas are frequent occurrences in some coal seams. They are 
 caused by the occluded gas finding its way to a vertical crevice or cleat in the 
 
gases found in mines. 
 
 353 
 
 Gases Enclosed in the Pores of Coal and Evolved in Vacuo 
 at 212° F.— (Thomas.) 
 
 Name of Colliery. 
 
 Quality. 
 
 C0 2 
 
 0 
 
 CHt 
 
 N 
 
 Quar 
 
 . <* 
 o> 3 
 
 60 
 
 68 
 
 rH 
 
 itity. 
 
 d 
 
 -M O 
 
 fa u H 
 U £ 
 
 CC 
 
 Navigation 
 
 Steam. 
 
 13.21 
 
 .49 
 
 81.64 
 
 4.66 
 
 250.0 
 
 80 
 
 Dunraven 
 
 Steam. 
 
 5.46 
 
 .44 
 
 84.22 
 
 9.88 
 
 218.0 
 
 70 
 
 Cyfarthfa 
 
 Steam. 
 
 18.90 
 
 1.02 
 
 67.47 
 
 12.61 
 
 147.0 
 
 47 
 
 Bute 
 
 Steam. 
 
 9.25 
 
 .34 
 
 86.92 
 
 3.49 
 
 375.0 
 
 120 
 
 Bonville’s Court 
 
 Anthracite. 
 
 2.62 
 
 
 93.13 
 
 4.25 
 
 555.0 
 
 178 
 
 Watney’s 
 
 Anthracite. 
 
 14.72 
 
 
 84.18 
 
 1.10 
 
 600.0 
 
 192 
 
 Plymouth Iron Works 
 
 Bituminous. 
 
 36.42 
 
 .80 
 
 
 62.78 
 
 55.9 
 
 18 
 
 Cwm Clydach 
 
 Bituminous. 
 
 5.44 
 
 1.05 
 
 63.76 
 
 29.75 
 
 55.1 
 
 18 
 
 Bettwys 
 
 Bituminous. 
 
 22.16 
 
 6.09 
 
 2.68 
 
 69.07 
 
 1 
 
 24.0 
 
 8 
 
 Gases Enclosed in the Pores of Coal and Evolved in Vacuo 
 at 212° F.— (If. LeChatellier.) 
 
 Locality. 
 
 CH 4 
 
 C0 2 
 
 N 
 
 0 
 
 Analyst. 
 
 Dunraven mine (blowers) 
 
 96.70 
 
 .47 
 
 2.79 
 
 
 J. W. Thomas. 
 
 Dunraven mine (bore hole) 
 
 96.50 
 
 .44 
 
 3.02 
 
 
 J. W. Thomas. 
 
 Garswood mine 
 
 84.16 
 
 .86 
 
 12.30 
 
 2.65 
 
 W. Kellner. 
 
 Garswood mine (blowers) 
 
 88.86 
 
 .41 
 
 8.90 
 
 1.83 
 
 W. Kellner. 
 
 Glamorgan mine (blowers) 
 
 93.01 
 
 .27 
 
 5.94 
 
 .78 
 
 W. Kellner. 
 
 Dombran mine (blowers) 
 
 95.11 
 
 .48 
 
 4.07 
 
 .34 
 
 f Austrian Firedamp 
 
 Karwin mine 
 
 94.59 
 
 .18 
 
 4.48 
 
 .75 
 
 | Commission. 
 
 Karwin mine (blowers) 
 
 99.10 
 
 .20 
 
 .70 
 
 
 
 Hruschau mine 
 
 79.16 
 
 .19 
 
 17.04 
 
 .61 
 
 
 Hruschau mine (blowers) 
 
 87.93 
 
 .83 
 
 10.25 
 
 .99 
 
 
 Peters wald mine (blowers) 
 
 90.00 
 
 .15 
 
 9.25 
 
 .60 
 
 
 Segen Gottes mine 
 
 83.51 
 
 1.17 
 
 15.02 
 
 .30 
 
 Sauer. 
 
 Segen Gottes mine (borehole) 
 
 87.16 
 
 1.11 
 
 11.73 
 
 
 Sauer. 
 
 Liebe Gottes mine (borehole) 
 
 77.69 
 
 3.77 
 
 18.48 
 
 .06 
 
 Sauer. 
 
 Gases Enclosed in the Pores of Coal and Evolved in Vacuo 
 at 212° F .—{Schondorfr.) 
 
 Locality. 
 
 Clh 
 
 CiH, 
 
 II 
 
 C0 2 
 
 N+O 
 
 Blowers. 
 
 Bonifacius mine at Kray (Essen) 
 
 Consolidation mine at Schalk ( Westphalia 1 
 Konig mine at Neunkirchen (Saarbruck) 
 
 Oberkirchen mine at Schaumburg 
 
 Cavities in the roof, Lothringen mine at 
 
 Castrop (Westphalia) 
 
 New Iserlohn mine at Lawgendren (West- 
 phalia) 
 
 90.94 
 
 89.88 
 
 84.89 
 (60.46 
 1 93.66 
 
 27.95 
 ( 4.75 
 \ 4.00 
 
 1.62 
 
 37.64 
 
 .88 
 
 .06 
 
 1.40 
 
 5.84 
 
 2.11 
 
 1.35 
 
 .09 
 
 .30 
 
 .67 
 
 .65 
 
 2.56 
 
 .63 
 
 .45 
 
 1.34 
 
 .40 
 
 7.36 
 
 3.61 
 
 12.84 
 
 4.80 
 
 70.25 
 
 65.00 
 
 95.00 
 
354 
 
 VENTILATION OF MINES. 
 
 Fig. 2. 
 
 coal seam, as illustrated in Fig. 2, and the pressure of the gas thus becomes 
 distributed over a large area. Thus, a pressure of 10 atmospheres of a gas 
 feeder becoming distributed over an area of 200 sq. ft. results in a total pres- 
 sure of upwards of 2,000 tons, upon a comparatively small area of coal. 
 
 As mine openings approach proximity 
 to such a locality, this pressure man- 
 ifests itself by bursting the coal from 
 its position in the face, and throwing 
 it into the entries, in some cases com- 
 pletely blocking the openings or pas- 
 sageways. Such an occurrence is 
 termed an outburst. It is frequently 
 accompanied by thunderings and 
 poundings, which manifest themselves 
 for several days previous to the actual 
 outburst of gas. These poundings are 
 _ taken as a warning by the miners 
 
 S • experienced in such regions. The 
 ' s ? poundings are probably the result of 
 
 the gas working its way from one 
 crevice to another, always advancing 
 closer and closer to the mine openings, where they finally burst forth with 
 extreme violence. 
 
 Testing for Gas’ by Lamp Flame.— Marsh gas and firedamp are detected in 
 mine workings by the small flame cap that envelopes and surmounts the 
 flame of the lamp in a firedamp mixture. This flame cap is caused by the 
 gaseous mixture, which burns as it comes in contact with the flame. 
 The proportion of gas in the mixture determines the height of the flame cap. 
 When testing for gas, the lamp flame is first reduced to a small, uniform 
 size, and although this is not a universal practice, it has the advantage of 
 giving uniform results. The lamp is held in an upright position, m one 
 hand, while the eyes are carefully screened by the other hand from the 
 glare of the light, the lamp being slowly raised toward the root where gas is 
 suspected. The flame is carefully watched for the first appearance of a cap, 
 and the height of the cap is carefully noted. Many lamps are provided with 
 a graduated scale set opposite to the flame, so that the height of a cap may 
 be estimated with accuracy. After the observation, the lamp is quietly and 
 promptly withdrawn from the gas. Should flaming occur within the lamp, 
 as sometimes happens when it is raised too quickly, or when the gaseous 
 mixture is strong, the lamp should be withdrawn carefully and not with 
 undue haste, as there is danger of the flame of the gases burning within the 
 lamp being forced through the gauze by a rapid movement. This requires 
 great presence of mind on the part of the person using the lamp. 
 
 In Fig. 3 the heights of flame cap due to the presence of different proportions 
 of marsh gas are shown. These heights, as given, refer to the experimental 
 heights of flame cap ob- 
 tained with pure marsh 
 gas. It should be ob- 
 served, however, that the 
 presence of other gases 
 in the firedamp will vary 
 its explosive character, 
 and this fact very mate- 
 rially modifies the explo- 
 siveness of certain caps. 
 
 For example, in the ex- 
 periments on pure marsh 
 gas, a 2" flame cap w as 
 found to be inexplosive; 
 while, in the mine, and 
 with the variable char- „ . 
 
 acter of the firedamp mixtures usually found there, a flame ot 1 T % in. is 
 often found to indicate explosive conditions. Again, flames of even less 
 height than this often indicate dangerous conditions, especially where 
 the coal is inflammable and there is much fine dust present m the 
 atmosphere. These conditions account readily for the various statements 
 that we commonly see in regard to the explosiveness of certain flames. In 
 fact, each fire boss learns, after years of experience, to depend wholly on his 
 
 1:50 LAO 
 
 1:30 1:25 
 
 Fig. 3. 
 
 1:20 1:18 1:16 
 
SAFETY LAMPS. 
 
 355 
 
 own knowledge, guided by tbe conditions tbat exist in tbe workings and 
 with which he has become familiar. 
 
 SAFETY LAMPS. 
 
 The safety lamp is designed to give light in gaseous workings without the 
 danger of igniting the gases present in the atmosphere. The principle of 
 the safety lamp depends on the cooling effect that an iron-wire gauze exerts 
 on flame. It is a well-known fact that all gases ignite at certain fixed 
 temperatures, and if this temperature is decreased from any cause, the flame 
 is extinguished. , „ . 
 
 Use of Safety Lamps.— Safety lamps are used for two general purposes in 
 the mine, and may be classified under two heads: (a) lamps. for general use; 
 (b) lamps for testing for gas. , „ , . , 
 
 Safety Lamps for General Work.— The essential features of a lamp designed 
 for general mine work are: (1) safety in strong currents; (2) good illuminating 
 power; (3) security of lock fastening; (4) freedom from flaming; (5) security 
 against accident; (6) simplicity of construction. The conditions under 
 which a lamp is placed at the working face differ from those that attend 
 the testing for gas. The illuminating power of the lamp must be good, so 
 that the workman can see clearly what he is doing. The lamp must not 
 be too sensitive to gas, or its tendency to flame will necessitate that a care- 
 ful watch be kept of it, and this would interfere with the prosecution of the 
 miner’s work. Again, the miner is too often careless or neglectful of his 
 lamp, and would fail to give it the required attention The lamp is often 
 upset, and is apt to be broken by flying coal, or by a fall, unless carefully 
 protected. The lamp should be so securely locked as not to permit of any 
 tampering on the part of the miner without its being detected in the lamp 
 room. In order that the lamp may be thoroughly and rapidly cleaned, its 
 construction should be simple. The lamp should be easily taken apart and 
 put together again after it is cleaned. _ . . . 
 
 Lamps for Testing.— The essential features of a lamp for testing purposes 
 are* (1) free admission of air below the flame; (2) no reflecting surface 
 behind the flame; (3) ability to test for a thin layer of gas at the roof. 
 When testing for gas, it is important to have a free admission of the air 
 below and around the flame, as the flame cap is very sensitive and is inter- 
 fered with seriously by the conflicting ascending and descending currents 
 in a lamp in which the air enters above the flame. A more uniform cap will 
 be given where the currents ascend quietly around the flame. # This feature 
 is very important to the production of a good flame cap, and it is this feature 
 that makes the Davy lamp such a favorite among fire bosses. In order that 
 a flame cap shall be readily observed, there should be no reflection behind 
 it, as the eye is easily deceived under these conditions. A scale by which, 
 the height of the flame cap may be accurately measured, is a convenient 
 feature in many lamps for testing purposes. 
 
 In the use of the common Davy lamp in testing for gases, it is a common, 
 although dangerous, practice to turn the lamp on its side and place it close 
 up against the roof. In this position, the flame is very apt to pass through the 
 gauze, from two causes: The gauze is readily heated, because the flame cap 
 is close against it, and when heated, affords no protection against the 
 passage of the flame and the ignition of the gas outside the lamp. Again, 
 in this position, small particles from the roof are apt to fall upon the gauze, 
 and this may often assist in the passage of the flame through the gauze. A 
 dirty gauze is unsafe. When the lamp is turned sideways, the gauze may 
 become smoked by contact with the flame, and this smoke, or deposit of 
 carbon, assists greatly the passage of flame through the gauze. Another 
 common and dangerous practice on the part of the fire boss is to brush the 
 gas down on the lamp with his cap. By so doing, there is great danger of 
 the flame being blown through the gauze and igniting the gas that may be 
 present. On these accounts, it is essential that a good lamp for testing pur- 
 poses shall be able to draw its air from a point close to the roof, in cases 
 where it is necessary to do so. This is often accomplished by an extra tube, 
 which is supplied with the lamp, and which may be taken off the lamp 
 
 __i j. nrui ~ nrv Ck rvn t Girl A nf t.np IRTT1T) TO tllft TOT). 
 
 its flame. A more uniform 
 the lard oil commonly used in the safety lamp. 
 
356 
 
 SAFETY LAMPS. 
 
 Detection of Small Percentages of Gas.— The Davy lamp in the hands of a 
 careful person may be made to detect the presence of gas in quantities as 
 low as 3f*. It is claimed by some tire bosses that 2 $ of gas may be detected 
 with a good Davy. For the detection of small quantities of gas, specially 
 constructed lamps have been used. These lamps are designed to burn 
 alcohol or hydrogen, giving a non-luminous flame. Among these may be 
 mentioned the Pieler lamp, burning alcohol, which it is claimed will detect 
 as small a quantity of gas as A device known as the Clowes gas tester 
 has been invented, and may be attached to many safety lamps. It consists 
 of a hydrogen tube that is designed to furnish a small stream of hydrogen 
 to the lamp flame when testing for gas. Surrounding the wick of the lamp 
 is a closely fitting cone, to which the hydrogen from the tube is supplied. 
 When the lamp is to be used for testing for gas, the wick is lowered, 
 extinguishing the oil flame after the hydrogen is turned on. It is claimed 
 that gas may be detected in as small quantities as iJo by this apparatus 
 attached to any good safety lamp admitting its air below the flame. 
 
 The Shaw gas tester is useful for determining the percentage of marsh 
 gas in the mine air, but it cannot be applied at the face, and samples of gas 
 must be taken to the surface for analysis. . 
 
 Oils for Safety Lamps.— Most safety lamps burn vegetable oils, which are 
 considered the safest for mining use, and so reported by the English Mine 
 Commission. Such oils are rape-seed oil and colza oil, made from cabbage 
 seed. Seal oil is also largely used, and was regarded as a safe oil by the 
 English Mine Commission. Seal oil affords a better light than vegetable oils, 
 and in its use there is less charring of the wick. A mixture of 1 part of coal 
 oil to 2 parts of rape or seal oil is often used, and improves the light, but the 
 smoke from the flame is increased. The Ashworth-Hepplewhite-Gray lamp 
 is constructed to burn coal oil, or a mixture of coal and lard oil. The Wolf 
 lamp is especially designed for burning naphtha or benzine. Special tests 
 have been made to prove the safety of using such a fluid in this lamp, and 
 resulted in demonstrating the fact that the lamp was safe under any condi- 
 tions that might arise. A thorough test was made, the oil vessel of the 
 burning lamp being heated to 180° F., at which point the lamp was 
 extinguished without ma nifesting any dangerous results. 
 
 Types of Safety Lamps.- In the year 1815, Sir Humphrey Davy and George 
 Stevenson, the latter a poor miner, discovered, simultaneously, that flame 
 would not pass through small openings in a perforated iron plate. This led 
 to the construction of what are known as the Davy, and the Stevenson or 
 “ Geordy,” lamps. The Davy lamp is still a great favorite among fire bosses 
 for the detection of gas in mine air. Inasmuch as all safety lamps, of which 
 there are a large number, depend on the same principle, we will only 
 describe such lamps as possess essential features, and which show important 
 improvements and the gradual developments in safety -lamp construction. 
 
 Davy Lamp.— Fig. 4 (a) shows a wire gauze cylinder about 5 in. in height 
 and 1£ in. in diameter, surmounted by a gauze cap 2 in. in depth. The 
 gauze, which has 28 wires to the inch, or 784 apertures to the square inch, is 
 fastened to a brass standard, which secures it to the oil cup or lamp below. 
 The gauze at the top of the lamp is doubled by the cap, which gives greater 
 security at this point, where the flame tends to pass through the gauze more 
 quickly, and where the gauze is more readily burned out. The mixture of 
 gas and air enters the lamp in the lower part of the gauze, and burns within 
 the lamp, the products of combustion passing out through the upper portion 
 of the gauze cylinder. This lamp gives a good flame cap, on account of the 
 free access of the air below the flame, which prevents smoking and increases 
 the illuminating power of the lamp. As a lamp for general use, the Davy 
 lamp, however, is unsafe, on account of its liability to flame. In many 
 mining localities the use of this lamp is prohibited by law, except for 
 purposes of examining for gas, when it must be used solely by properly 
 authorized fire bosses. The flame of the lamp is also unprotected from the 
 force of rapid air-currents, and is not safe when the velocity of the current 
 exceeds 6 ft. per second. The illuminating power of the lamp is also not 
 sufficient for general work. , . , , , 
 
 Clanny Lamp.— The unbonneted Clanny lamp, Fig. 4 (6), is constructed 
 according to the same principles as the Davy lamp, differing only in the 
 fact that the lower part of the wire gauze surrounding the flame is replaced 
 by a strong glass cylinder or chimney. The purpose of this is to increase 
 the illuminating power of the lamp. The lamp, when clean, gives a good 
 light, but the entrance of the air at a point above the flame, and its descent 
 
(dJ (e) 
 
 Fig. 4. 
 
 (f> 
 
358 
 
 SAFETY LAMPS. 
 
 within the lamp to the flame, causes the lamp to smoke, due to the conflict 
 of the ascending and descending air-currents within the lamp. The smoke 
 becomes deposited on the glass chimney, which interferes greatly with the 
 light. This lamp is not a good one for gas testing, and in fact cannot be 
 used for that purpose to any advantage. The unbonneted Clanny is not 
 safe in an air-current having a velocity greater than 8 ft. per second. The 
 bonneted Clanny obviates this difficulty to a large extent, but increases the 
 tendency of the lamp to smoke. 
 
 Mueseler Lamp.— This lamp, Fig. 4 (c), in all respects resembles the Clanny 
 lamp just described, except that the tendency in the Clanny lamp to smoke 
 is overcome in the Mueseler by increasing the draft by means of an interior 
 wrought-iron chimney or tube, supported within the lamp, and reaching 
 down to within an inch of the base of the flame. The air enters the lamp 
 as in the Clanny, above the flame, but is deflected downwards by the central 
 tube, and passes under the edge of this tube, ascending through it to the top 
 of the lamp, where it escapes. The Mueseler lamp is a better lamp for 
 illuminating purposes than the Clanny, and presents more security, when 
 bonneted, against explosions within the lamp. This lamp will withstand a 
 current of very much higher velocity than the Clanny lamp, and is 
 reputed to be safe in a current having a velocity of 100 ft. per second. The 
 lamp is not a good lamp for the detection of gas. It does not flame, 
 however, as quickly as the Clanny lamp. 
 
 Marsaut Lamp— This lamp, Fig. 4 (d), is built after the Clanny lamp in 
 every respect, but is supplied with multiple-gauze chimneys, one within the 
 other, the effect of which is to increase the security against explosion of gas 
 within the lamp. The bonneted Marsaut lamp is a peculiarly strong lamp 
 in this respect. The gauze used in the caps of this lamp has 984 apertures to 
 the square inch. This lamp is often extinguished in an explosive mixture 
 by the force of the explosion within itself. It gives a good light and is a good 
 lamp for general work; it is not, however, a good lamp for testing for gas. 
 
 Ashworth-Hepplewhite-Gray Lamp.— This lamp, Fig. 4 (e), combines a number 
 of characteristic features. It is designed for general work, as well as for 
 testing for gas. It often happens that gas accumulates in a thin layer along 
 the roof of an entry or working place, and is not detected by the use of the 
 Davy lamp or any ordinary lamp. The Gray lamp is so arranged that it 
 can be made to draw its air from the top of the lamp, by means of openings 
 in the top of the four standards of the lamp, the air passing down 
 through the standards, and into the lamp, below the flame. When not 
 in use for testing, openings can be made in the lower part of the stand- 
 ards by moving a slide, and air enters at these openings. The lamp is essen- 
 tially a bonneted Clanny. The glass chimney, however, as well as the gauze 
 that surmounts it, is made in a conical form, the purpose of this being to 
 diffuse the light upward for examination of the roof of the mine. The 
 conical form given to the gauze also strengthens the lamp against explo- 
 sions of gas within. This lamp is a very good all-around lamp, and possesses 
 good illuminating power. . , . . 
 
 Wolf Lamp.— The Wolf lamp, Fig. 4 (/), is rapidly growing m popularity 
 having been already introduced in a large number of mines in America ana 
 England, and on the Continent. This lamp is essentially a Clanny lamp 
 with a free admission of air. It is compact and efficient, and has good 
 illuminating power, and is also constructed in different forms, combining, 
 as desired, any or all of the features of previous lamps. Two of its charac- 
 teristic features, however, consist in a self-lighting arrangement accom- 
 plished by means of a percussive device, which ignites a wax taper within 
 the lamp, and a locking device, which can be opened only with a powerful 
 magnet. This relighting device is an important feature in any safety lamp 
 for general use, inasmuch as the most dangerous conditions exist immedi- 
 ately after an explosion, and the miners are always left to grope their way 
 in the dark. A large number of lives are lost, owing to the confusion that 
 ensues, the men becoming bewildered and losing their wav, when they are 
 shortly overcome by the afterdamp of the explosion. This lamp permits of 
 immediate relighting with safety to the men. „ , _ . , _ 
 
 Locking Lamps.— The ordinary lock consists of a lead plug, which, when 
 inserted in the lamp, will show the least tampering on the part of the 
 miner. Other locks consist of an ordinary turnbolt operated by a peculiar 
 key. Magnetic locks allow of the opening of the lamp only by means of a 
 
 Strong magnet kept in the lamp room. 
 
 Cleaning Safety Lamps.— Safety lamps should be thoroughly and regularly 
 
CARE OF SAFETY LAMPS. 
 
 359 
 
 cleaned and filled between each shift. Each lamp should then be lighted 
 and inspected by a competent person before being given to the miner. A 
 careful inspection of the gauze of the lamp is necessary, as well as of all the 
 joints by which air may enter the lamp. It should be known to a certainty 
 that each lamp is securely locked before leaving the lamp room. 
 
 Relighting Stations.— These stations are located at certain places in gaseous 
 mines where they can be supplied with a current of fresh air, and where 
 there is no danger from the gases of the mine. The lamp is apt to be 
 overturned, or to fall, and is often extinguished thereby; and if these 
 stations were not provided, the man would have to return with his lamp to 
 the surface in order to have it relighted. Such a station is always located at 
 the entrance of the gaseous portion of a mine, in cases where the entire 
 mine does not liberate gas. 
 
 Illuminating Power of Safety Lamps.— The following table gives the illumi- 
 nating power or candlepower of some of the principal lamps. The light of a 
 sperm candle is taken as 1, or unity. 
 
 f 
 
 Name of Lamp. 
 
 Illuminating Power 
 of Lamp. 
 
 Davy 
 
 Geordy 
 
 Clanny •. 
 
 Mueseler 
 
 Evan Thomas. 
 
 Marsaut, 3 gauzes 
 
 Marsaut, 2 gauzes 
 
 Marsaut, with Ho wat’s deflector 
 
 Ashworth-Hepplewhite-Gray 
 
 Wolf 
 
 .16 
 
 .10 
 
 .20 
 
 .35 
 
 .45 
 
 .45 
 
 .55 
 
 .65 
 
 .65 
 
 .90 
 
 EXPLOSIVE CONDITIONS IN MINES. 
 
 In the ventilation of gaseous seams, the air-current may be rendered 
 explosive by the sudden occurrence of any one of a number of circum- 
 stances that cannot be anticipated. Among these are the following: (1) 
 Derangement of the ventilating current. (2) Sudden increase of gas due to 
 outburts, falls of roof, feeders, fall of barometric pressure, etc. (3) Presence 
 of coal dust thrown into suspension in the air, in the ordinary working of 
 the mine, or by the force of blasting at the working face, or by a blown-out, 
 or windy, shot. (4) Pressure due to a heavy blast, or any concussion of the 
 air caused by closing of doors, etc. (5) Rapid succession of shots in close 
 workings. (6) Accidental discharges of an explosive in a dirty atmosphere. 
 Any or all of these causes may precipitate an explosion at any moment. 
 Hence, the condition of the air-current should be maintained far within the 
 explosive limit. The explosive conditions vary considerably in different 
 coal seams. The nature of the coal and its enclosing strata, its friability 
 and inflammability, together with the character of its occluded gases, deter- 
 mine, to a large extent, the explosive conditions in the seam. Experience 
 in any particular seam or district must always be the best guide, and furnish 
 the best standard for determining the explosiveness of any given lamp 
 flame. For example, a 2" flame may be comparatively safe in a small mine 
 where the coal is hard and not particularly flammable, while a li" flame 
 cap would be considered unsafe in mines where the conditions are more 
 favorable to the generation of gas and formation of coal dust. The daily 
 output of the mine and the general care that is enforced upon the miners 
 at the working face are factors that should always be considered and taken 
 into serious account in determining explosive conditions (see Testing for 
 Gas by Lamp Flame). 
 
 Derangement of Ventilating Current.— The flow of the air-current must be 
 uniform and continuous. Doors must be kept closed, since the mere setting 
 open of a door, for a short period of time, is sufficient to precipitate a serious 
 explosion. Any contemplated change in the current, by the erection of 
 brattices, air bridges, stoppings, etc., should be carefully considered before 
 the work is begun, and every precaution adopted to secure the safety of the 
 
360 
 
 VENTILATION OF MINES. 
 
 men. Derangement of the current may occur through a fall of roof upon 
 the main airway, by which the area of the airway is reduced, which results 
 in the reduction of the quantity of air passing in the mine. If this fall is 
 not noticed at once, serious results may happen. The utmost vigilance is 
 therefore required on the part of fire bosses and all connected with mine 
 workings. The failure of the ventilating apparatus is another source that 
 gives rise to the derangement of the current. As a rule, furnaces are not 
 now employed for the ventilation of gaseous seams. There are, however, 
 some furnaces in use in such seams, and these require constant attention 
 lest the fire should burn low. Upon any accident occurring to the ventila- 
 ting machinery, notice should at once be given to the inside foreman, and 
 the men withdrawn as rapidly as possible. 
 
 A sudden increase of gas may occur at any time in a gaseous seam, owing 
 to an outburst, which suddenly yields a large volume of gas and may 
 render the mine air in that section extremely explosive. The men working 
 on the return of such a current must be hastily withdrawn, and all lights 
 extinguished. A heavy fall of coal in the mine workings or in the airways, 
 or the tapping of a large gas feeder, produces the same effect in a less degree. 
 The nearer the fall of roof takes place to the face of the workings, the more 
 liable it is to be followed with a large flow of gas, inasmuch as the gas near 
 the face has not had time to drain off, as in the case of old workings. This 
 fact is always true in reference to new workings in a gaseous seam. The gas 
 continues to flow freely for a considerable period, w r hen its flow gradually 
 decreases until it about ceases. When a large feeder has been tapped, it 
 may be plugged for a time, if necessary, but the better practice is to allow it 
 to flow freely and diffuse into the air-current, which should be sufficiently 
 increased to dilute the quantity of gas given off and to render it inexplosive. 
 The men upon the return air should be notified. It is dangerous practice to 
 light these feeders. 
 
 When there is a large area of abandoned workings m the mine, any 
 considerable fall of barometric pressure is usually followed by a large 
 outflow of gas from the gobs or waste places of the mine. A fall of 1 in. m 
 5 hours represents a very rapid decrease of barometric pressure. At all large 
 collieries there is, or should be, a good standard barometer located upon the 
 surface near the shaft. In many cases, these barometers are self-recording, 
 and are often provided with an automatic alarm that gives warning when- 
 ever a fall of barometric pressure occurs. This warning should at once be 
 conveyed to the men in the workings, and every precaution adopted to 
 avoid evil results. The fact is fairly well established that a fall of atmos- 
 pheric pressure is not followed by an outflow of gas from the mine workings 
 for the space of, say, 3 hours after such fall occurs. This statement must be 
 regarded with caution, however, as it largely depends on the condition and 
 extent of the abandoned workings. Where these are full of gas, its expan- 
 sion affects the condition of the airways much more quickly than m cases 
 where these working places are partly ventilated. 
 
 Effect of Coal Dust in Mine Workings —According to the greater or less flam- 
 mability of the coal, the presence of fine dust in the airways and workings of 
 the mine becomes a dangerous factor. Certain coals are extremely friable 
 and are reduced readily to fine dust, which is thrown into suspension m the 
 air-current by the ordinary operations of the miners in their work, as well 
 as by the concussion of the air from numerous causes, and by the movement 
 of cars and the traveling of men and animals upon the various haul- 
 ways and passageways. For a long time it was questioned whether the 
 presence of dust was a dangerous factor, except where there was also a 
 small percentage of gas in the air. Evidence, however, has well established 
 the fact that coal dust of itself is a dangerous element, and may often be the 
 sole cause of an explosion, when acted upon by a flame of sufficient intensity 
 and magnitude. The action of the flame is to distil carbonic-oxide gas (70 
 from the fine particles of dust suspended in the air. The explosion of the 
 
 § as thus formed causes a further disturbance and raises a larger supply oi 
 ust, which likewise contributes to the liberation of fresh quantities of gas, 
 and thus an explosion is generated and transmitted. Small quantities of 
 marsh gas greatly increase the violence of this action, but explosions in 
 flouring mills and w 7 ell-ventilated coal bins establish the fact that such 
 occurrences are not dependent on the presence of marsh gas. .... 
 
 Too much faith must not be placed in the use of water by sprinkling lor 
 laying the dust. This has a beneficial effect in the immediate vicinity, but 
 a large amount of water is required to render an untidy working place safe 
 
MINE EXPLOSIONS. 
 
 361 
 
 at firing time. Better practice is to allow no accumulations of dust at the 
 face. This should be regularly loaded out with the coal. 
 
 Pressure as Affecting Explosive Conditions.— Gaseous mixtures that are not 
 explosive in the ordinary condition of a mine, often become explosive 
 under the momentary pressure to which they are subjected by heavy 
 blasting, and, in some instances, this may occur from the concussion 
 of the air caused by the quick shutting of a door. In the latter case, how- 
 ever, the explosive condition of the air would necessarily have to be close to 
 the limit, in order for such a slight occurrence to precipitate an explosion. 
 The factor of pressure as increasing the explosiveness of gaseous mixtures 
 should be considered and constantly borne in mind. 
 
 Rapid Succession of Shots in Close Workings.— It constantly happens that 
 two, three, or more shots are fired by means of fuse or touch squibs in a 
 single chamber or heading, where the circulation of air is not always the 
 best. The practical effect is that a considerable quantity of carbonic-oxide 
 gas CO is produced by the firing of the first shot, and this gas does not have 
 time to diffuse or become diluted by the air-current before it is fired by the 
 flame of the following shots. An explosion may often be precipitated by such 
 an occurrence, if the workings are at all dusty. Two shots at the most are all 
 that should be fired at one time in a close chamber or heading. 
 
 Mine Explosions.— The explosion of gas in a mine usually arises from the 
 ignition of an explosive mixture of gas and air called firedamp , which has 
 accumulated in some unused chamber or cavity of the roof, or in the waste 
 places of the mine, and has been ignited by a naked light, by the flame of a 
 shot, or by a mine fire. The initial force of an explosion is generally 
 expended locally, but the flame continues to feed upon the carbonic-oxide 
 gas generated by the incomplete combustion of the firedamp mixture, 
 and distilled also from the coal dust thrown into the air by the agitation. 
 Air is required to burn this carbonic-oxide gas; this causes the flame to 
 travel against the air-current, or in the direction in which fresh air is found. 
 In the other direction, or behind the explosion, the flame is soon extin- 
 guished in its own trail when the initial force of the explosion is expended. 
 The explosion continues to travel along the airways against the current as 
 long as there is sufficient gas or coal dust for it to feed upon, or until its 
 temperature is cooled below the point of ignition, by some cause such as, for 
 example, the rapid expansion of the area of the workings. We observe the 
 chief factor in transmitting an explosion is the presence of carbonic-oxide 
 gas, which lengthens the flame and extends the effect. 
 
 The recoil of an explosion is the return of the flame along the path that it 
 has just traversed. In the recoil, the flame burns more quietly, advances 
 more slowly, and travels close to the ropf. The evidence found at the point 
 where a recoil took place, or an explosion turned back, has been sufficient 
 to establish the fact that the recoil is caused primarily by a cooling of the tem- 
 perature, probably caused very largely by an expansion of the area of the 
 airway. Soot is often deposited at this point in considerable quantity, if the 
 action of the flame is not such as to consume it. This fact alone shows the 
 combustion at this point to have been incomplete. Immediately in the rear 
 of the flame is a mixture of carbonic-oxide gas CO, which bursts into flame 
 at the sudden stoppage of the advancing explosion. This is rendered 
 possible by the flow of cold air from the adjacent chambers and workings 
 along the floor of the airway. The flame now retreats, burning the trail of 
 carbonic-oxide gas along the roof, fed by the cold air along the floor. 
 
 To Explore Workings After a Serious Explosion.— The shafts or slopes and the 
 ventilating machinery should claim the first attention of those on the 
 surface, and an effort should be made to reach the bottom as expeditiously 
 as possible. Assistance from neighboring collieries, both in the way of 
 skilled labor and advice, should also be requested. Should the shaft or slope 
 need repairs before communication between top and bottom is restored, the 
 person in charge on the surface should, in the meantime, see that props of 
 the lengths in ordinary use, brattice boards, brattice cloth, and nails are 
 brought to a convenient place for putting on the cage or car, and he ought 
 also to collect all the tools likely to be required, such as axes, saws, ham- 
 mers, etc. It is also important that rough tracings of the workings be 
 prepared for the use of the leader of each squad of explorers. Officials will 
 understand how useful these will be to those that are penetrating into work- 
 ings about which every man of his squad may have been heretofore ignorant. 
 
 When the explorers have arrived at the bottom and are ready to proceed, 
 there should be for each section, if more than one is operated upon, two 
 
362 
 
 VENTILATION OF MINES. 
 
 managers, each, having his own squad of men, and his own particular duty 
 to do. One may take charge of restoring the ventilation, the inspection of 
 the workings, and the clearing of the roads; the other may appoint and 
 have charge of the bottom man, the conveying of material, and the detailing 
 of stretcher companies where required. They can consult and help each 
 other in every difficulty, but system is necessary if the work is to be done in 
 the shortest possible time. . A . _ ... 
 
 The manager who has charge of the men m front should appoint two 
 experienced men with good nerves to act as foremen, instructing them to 
 inspect and report to him the condition of the workings within a short 
 radius. He should then form the rest of his men into, say, three squads of 
 three each, who will work together at stoppings or falls until separated by 
 him or until the end of the shift. Being near the bottom, it will probably 
 be found that all is clear for three or four breast or stoop lengths, and stop- 
 pings are required to be put up. Material will be required for this, and 
 when the cage is first sent to the top for it, it should not be kept there to 
 enable the top man to put on a big load, but it should be sent down with all 
 despatch, loaded with a half dozen each of props and brattice boards, with 
 one piece of cloth and nails. This will allow a start to be made, and will 
 prevent the anxious men from worrying over what to them is an unac- 
 countable delay. Larger loads can be sent down in subsequent trips. For 
 convenience in carrying, the brattice cloth may be cut in lengths to suit the 
 gangways or headings with 2 or 3 ft. to spare. Squad No. 1 should be 
 detailed to the first stopping. This may be put up with boards at top and 
 bottom and cloth between. If the air-current is strong, a few of the follow- 
 ing stoppings may be put up by squads No. 2 and No 3, with cloth only 
 stretched between two props. These can be very rapidly put up and will 
 drive the ventilation forwards, thus allowing the firemen to extend rapidly 
 the area of inspection. These stoppings can be completed by No. 1 detail. 
 In a short time it may become impossible to proceed in this manner, lhe 
 foul air will in all probability become more difficult to dislodge, and eventu- 
 ally one detail may be able to put up stoppings as quickly as the firedamp or 
 chokedamp can be carried off. Part of what may be called the ventilaung 
 detail can now be transferred to the bearer detail, the duties of the latter 
 having become heavier as the stoppings advanced. It is not an easy task to 
 carry props long distances in a stooping posture, and when to that is added, 
 it may be, the carrying out of the living or dead bodies, the men begin to 
 fag very soon. But the person in charge here must see that the forward 
 party is kept in material for stoppings so that no delay may occur on that 
 account. A system of staging gives relief to the carrying parties. 
 
 To conclude with a few general remarks Let those that nave never yet 
 assisted to explore a mine after an explosion be assured of this, that the 
 chief requisites in a leader are a capacity for hard work and the ability to 
 organize his men into a system, however roughly, whereby work will be best 
 forwarded. It mil not speed the work to say to a dozen or more men, 
 generally, do this or that, neither is it beneficial to allow all the workmen 
 to discuss matters and suggest plans. Those in charge ought to arrange what 
 is to be done. Anything else results in noise and confusion. And let men 
 that are sent from other collieries take with them their own toefis and lamps. 
 Those in charge ought to take note of the position, etc. of bodies found, and 
 of every point which is likely to throw light on the cause or origin of the 
 explosion. This can be more correctly done before the roads are disturbed 
 by oust and travel. These notes might not only be the means of ascertain- 
 ing the cause of explosion, but also of pointing out a way of prevention m 
 
 In no case after an explosion should the air-current of the mine be 
 reversed from its usual course, except only after careful consideration, 
 because of the reliance placed by the entombed workmen on their knowl- 
 edge of the direction in which the air should be moving- and the reversal 
 of the current may drive the gases of the explosion upon them with disas- 
 trous results. Conditions must be allowed to remain as they exist, and the 
 rescuers conform themselves to such conditions in the best manner possible. 
 
 QUANTITY OF AIR REQUIRED FOR VENTILATION. 
 
 The quantity of air required for the adequate ventilation of a mine can- 
 not be stated as a rule applicable in every case. Regulations tlmt would 
 supply a proper amount of air for the ventilation of a tlnck seam would be 
 
ELEMENTS IN VENTILATION. 
 
 363 
 
 found to cause great inconvenience if applied without modification to the 
 workings in a thin seam. Likewise, the ventilation of an old mine with 
 extended workings, a large area of which has been abandoned, and m many 
 cases not properly sealed off, will require, naturally, a larger quantity of air 
 per capita than a newly opened mine or shaft. The natural conditions 
 existing in rise and dip workings, with respect to the gases that may be 
 liberated or generated in those workings, call for the modification of the 
 quantity of air required in each case. For example, dip workings, where 
 much blackdamp is generated, will require a larger quantity of air, or 
 higher velocity at the working face, to carry off such damps; and rise work- 
 ings, liberating a large amount of marsh gas, will likewise require a higher 
 velocity at the working face. On the other hand, a reversal of these 
 conditions, such as a large quantity of marsh gas being liberated m dip 
 workings, or a similar amount of blackdamp being generated in rise work- 
 ings, will require a comparatively low velocity of the air at each respective 
 
 Quantity Required by State Laws.— The quantity of air required by the 
 laws of the several States is generally specified as 100 cu ft. per man 
 per minute, and in many cases an additional amount of 500 cu. it. per 
 animal per minute is stated. This quantity is m no case stated as the 
 actual amount of air required for the use of each man or animal, but is only 
 the result of experience, as showing the quantity of air required for the 
 proper ventilation of the average mine, based on the number of men and 
 animals employed. The number of men employed m a mine is an indica- 
 tion of the extent of the working face, while the number of animals 
 employed is an indication likewise of the extent of the haulage roads, or the 
 development of the mine. These amounts refer particularly to non-gaseous 
 
 Sea The Bituminous Mine Law of Pennsylvania specifies that there shall be 
 not less than 100 cu. ft. per minute per person m any mine, while 150 cu. it. 
 are required in a mine where firedamp has been detected. . > 
 
 The Anthracite Mine Law of Pennsylvania specifies a minimum quantity 
 of 200 cu. ft. per minute per person. Each of these laws contains modifying 
 clauses, which specify that the amount of air in circulation shall be sufficient 
 to “dilute, render harmless, and sweep away” smoke and noxious or 
 
 da Quantity of Air Required for Dilution of Mine G.*...-To determiiie > this 
 requires a knowledge of the quantity of gas generated °^ llb ^ted m the 
 workings. The quantity of air for dilution should be ample, and should be 
 such as not to permit the condition of the current to approach the explosive 
 point. The ventilation should be ample at the face. - r . 
 
 Quantity of Air Required to Produce the Necessary Velocity of Current at the 
 Face — This consideration modifies considerably the quantity of air required 
 for the ventilation of thick and thin seams. The velocity of the current is 
 dependent not only on the quantity of air in circulation, but on the area of the 
 air passage. This area is quite small in thin seams, and often very large in 
 thick seams. As a result, the velocity is often low at the face of thick 
 seams, and insufficient for the proper ventilation of the face, although the 
 quantity of air passing into such a mine may be very large. A certain 
 velocity of the current is always required in order to sweep away the gases. 
 This velocity depends on the character of the gases and the position ot the 
 workings. Heavy damps are hard to move from dip workings where they 
 have accumulated; and, likewise, lighter damps accumulating ’at 'the: face 
 of steep pitches are hard to brush away, and the velocity of the current in 
 these cases must be equal to the task of driving out these gases. 
 
 ELEMENTS IN VENTILATION. 
 
 The elements in any circulation of air are (a) horsepower or power 
 applied; (5) resistance of the airways, or mine resistance, which gives rise 
 to the total pressure in the airway; (c) velocity generated by the power 
 applied against the mine resistance. ,. , . 
 
 Horsepower or Power of the Cur rent. -The power applied is often ^spoken 
 of as the power upon the air. It is the effective power of the ventilating 
 motor, whatever this may be, including all the ventilating , ^£ e J^cies, 
 whether natural or otherwise. The power upon the air may be the power 
 exerted by a motive column due to natural causes, or to a furnace, or may 
 
364 
 
 VENTILATION OF MINES. 
 
 be the power of a mechanical motor. The power upon the air is always 
 measured in foot-pounds per minute, which expresses the units of work 
 accomplished in the circulation. 
 
 Mine Resistance.— The resistance offered by a mine to the passage of an 
 air-current, or the mine resistance, is due to the friction of the air rubbing 
 along the sides, top, and bottom of the air passages. This friction causes the 
 total ventilating pressure in the airway, and is equal to it. Calling the 
 resistance E, the unit of ventilating pressure (pressure per square foot) p, 
 and the sectional area of the airway a, we have, E = pa; that is to say, the 
 total pressure is equal to the mine resistance. 
 
 Velocity of the Air-Current.— Whenever a given power is applied against a 
 given resistance, a certain velocity results. For example, if the power u 
 (foot-pounds per minute) is applied against the resistance pa, a velocity v 
 (feet per minute) is the result; and since the total pressure pa moves at 
 the velocity v, the work performed each minute by the power applied is the 
 product of the total pressure by the space through which it moves per 
 minute, or the velocity. Thus, u = (pa) v. 
 
 Relation of Power, Pressure, and Velocity.— The relation of these elements of 
 ventilation is not a simple relation. For example, a given power applied to 
 move air through an airway establishes a certain resistance and velocity in 
 the airway. The resistance of the airway is not an independent factor; that 
 is to say, it does not exist as a factor of the airway independent of the 
 velocity, but bears a certain relation to the velocity. Power always produces 
 resistance and velocity, and these two factors always sustain a fixed relation. 
 
 This relation is expressed as follows: The total pressure or resistance varies 
 as the square of the velocity; i. e., if the power is sufficient to double the 
 velocity, the pressure will be increased 4 times; if the power is sufficient to 
 multiplv the velocity B times, the pressure will be increased 9 times. Thus, 
 we observe that a change o*f power applied to any airway means both a 
 change of pressure and a change of velocity. 
 
 Again, since the power is expressed by the equation u = (pa) v, and since 
 p a, or the total pressure, varies as v 2 , the work varies as vK From this it 
 follows that, if the velocity is multiplied by 2, and, consequently, the total 
 pressure by 4, the work performed (pa) v will be multiplied by 2 3 = 8. We 
 thus learn that the power applied varies as the cube of the velocity. 
 
 MEASUREMENT OF VENTILATING CURRENTS. 
 
 The measurement and calculation of any circulation in a mine airway 
 includes the measurement of (a) the velocity of the air-current, (b) of pres- 
 sure, (c) of temperature, (d) calculation of pressure, quantity, and horse- 
 power of the circulation. 
 
 These measurements should be made at a point in the airway where the 
 airway has a uniform section for some distance, and not far from the foot of 
 the downcast shaft or the fan drift. 
 
 Measurement of Velocity.— For the purpose of mine inspection, the velocity 
 of the air-current should be measured at the foot of the downcast, at the 
 mouth of each split of the air-current, and at each inside breakthrough, in 
 each split. These measurements are necessary in order to show that all the 
 air designed for each split passes around the face of the workings. 
 
 The measurement of the velocity of a current is best made by means of 
 the anemometer. This instrument consists of a vane placed in a circular 
 frame and having its blades so inclined to the direction of its motion that 
 1 ft. of lineal velocity in the passing air-current will produce 1 revolution of 
 the vane. These revolutions are recorded by means of several pointers, 
 each having a separate dial upon the face of the instrument, the motion 
 being communicated by a series of gear-wheels arranged decimally to each 
 other. Most anemometers are provided with a large central pointer that 
 makes 1 revolution for each 100 revolutions of the vane. The dial for this 
 pointer is marked by 100 divisions, which record the number of lineal feet 
 of velocity. In very accurate work with the anemometer, certain constants 
 are used as suggested by the instrument maker, but these constants are of 
 little value in ordinary practice and are of doubtful' value even in more 
 accurate observations. 
 
 The measurement of the velocity of an air-current must necessarily 
 represent only approximately the true velocity in the airway. The air 
 travels with a greater velocity in the center of the airway, and is retarded at 
 
MEASUREMENT OF PRESSURE. 
 
 385 
 
 the sides, top, and bottom by the friction of these surfaces. Hence, the 
 air to a large extent rolls upon these surfaces, which naturally generates an 
 eddy at the sides of airways. When measuring the air, the anemometer 
 should be held in a position exactly perpendicular to the direction of the 
 current, and moved to occupy different positions in the airway, being 
 held an equal time in each position, or it may be moved continuously 
 around the margin of the airway, and through the central portion. The 
 person taking the observation should observe the caution of not obstruct- 
 ing the area of the airway by his body, as the area is thereby reduced, and 
 the velocity of the current increased. The area of the airway is accurately 
 measured at the point where the observations are taken. 
 
 To obtain the quantity of air passing (cubic feet per minute), multiply 
 the area of the airway, at the point where the velocity is measured, by the 
 velocity. 
 
 Example.— The anemometer gives a reading of 1,320 ft. in 2 minutes, the 
 height of the airway is 6 ft. 6 in., and its average width 8 ft. 8 in. What 
 volume of air is passing in the airway per minute? 
 
 1 320 
 
 X 8f X — 37,180 cu. ft. per min. 
 
 The measurement of the ventilating pressure is made by means of a water 
 column in the form of a water gauge. 
 
 Water Gauge.— The water gauge is simply a glass U tube open at both ends. 
 Water is placed in the bent portion of the tube, and stands at the same 
 height in both arms of the tube when each end of the tube is subjected to 
 the same pressure. If, 
 however, one end of the 
 tube is subjected to a 
 greater pressure than 
 the other end, the water 
 will be forced down in 
 that arm of the tube, 
 and will rise a corre- 
 sponding height in the 
 other arm, the differ- 
 ence of level in the two 
 arms of the tube repre- 
 senting the water col- 
 umn balanced by the 
 excess of pressure to 
 which the water in the 
 first .arm is subjected. 
 
 An adjustable scale 
 graduated in inches 
 measures the height of 
 the water column. The 
 zero of the scale is ad- 
 justed to the lower 
 water level, and the Fig. 6. 
 
 upper water level will 
 then give the reading of the water gauge. One end of the glass tube is 
 drawn to a narrow opening to exclude dust, while the other end is bent to 
 a right angle, and passing back through the standard to which the tube is 
 attached, is cemented into the brass tube that passes through a hole in- the 
 partition or brattice, when the water gauge is in use. The bend of the tube 
 is contracted to reduce the tendency to oscillation in the height of water 
 column. (See Fig. 5.) 
 
 When in use, the water gauge must be in a perpendicular position. It is 
 placed upon a brattice occupying a position between two airways, as shown 
 at A, Fig. 6. The brass tube forming one end of the water gauge is 
 inserted m a cork, and passes through a hole bored in the brattice. The 
 water gauge must not be subjected to the direct force of the air-current, as 
 m this case the true pressure will not be given. Fig. 6 shows the instrument 
 as occupying a position in the breakthrough, between two entries. It will 
 be observed that the water gauge records a difference of pressure, each end 
 of the water gauge being subject to atmospheric pressure, but one end in 
 addition being subject to the ventilating pressure, which is the difference of 
 
 Fig. 5. 
 
366 
 
 VENTILATION OF MINES, 
 
 pressure between the two entries. The water gauge thus enables us to 
 measure the resistance of the mine inbye from its position between tw o 
 airways. If placed in the first breakthrough, at the foot of the shaft, it 
 measures the entire resistance of the mine, but if placed at the mouth ot a 
 split it measures only the resistance of that split. It never measures the 
 resistance outbye from its position in the mine, but always inbye (see Calcula- 
 tion of Pressure). ,« , . 
 
 Measurement of Temperature.— It is important to measure the temperature 
 of the air-current at the point where the velocity is measured, as the tem- 
 perature is an important factor of the volume of air passing (see Expan- 
 sion of Air and Gases, etc.). , __ . . _ 
 
 Thermometers.— Thermometers measure changes m the temperature ot the 
 atmosphere bv the contraction and expansion of mercury or spirits; or they 
 may be made entirely of metal, and the changes of temperature are then 
 measured by the expansion and contraction of the sensitive metallic 
 portion These latter are known as aneroid thermometers. The Fahren- 
 heit thermometer is the one most commonly used in America. By this 
 scale, the freezing point of water at the sea level is placed at 32° above zero; 
 the boiling point of water at sea level is placed at 212° above zero, so that the 
 space between these two points is divided into 180°. ^ . 
 
 Reaumur and Centigrade thermometers are used on the continent ot 
 Europe. Of these two, the first is generally used in Germany, and the 
 second in France, but the latter is almost exclusively used by the scientists 
 
 of all nations. . _ ^ . . , , 
 
 In the Reaumur thermometer, the freezing and boiling points are placed 
 at 0° and 80°, respectively. In the Centigrade, the freezing and boiling 
 points are placed at 0° and 100°, respectively. _ ,. ., .. 
 
 To Convert Fahrenheit Into Centigrade— (1) Subtract 32 and divide the 
 remainder by 1.8, or multiply by f . A ... ... . , , 
 
 If a Fahrenheit thermometer registers 16/°, what will be the register by a 
 Centigrade thermometer ? 
 
 167 - 32 (167 - 32)5 
 
 1.8 
 
 = 75° Centigrade. 
 
 75° Centigrade. 
 
 To Convert Centigrade Into Fahrenheit. — (1) Multiply by 1.8, or f, and add 32. 
 
 • Ars 75°, what will be the register by a 
 
 If the Centigrade thermometer registers 
 Fahrenheit thermometer? 
 
 75 X 1.8 + 32 = 167° Fahrenheit. + 32 
 
 167° Fahrenheit. 
 
 To Convert Fahrenheit Into Reaumur.— (1) Subtract 32, and divide by 2.25, 
 
 ° r mhe'Fahrenheit thermometer registers 113°, what will be the register by 
 the Reaumur thermometer? 
 
 113 — 32 
 2.25 
 
 = 36° Reaumur. 
 
 I 113 — 32 ^ = 36° Reaumur. 
 9 
 
 To Convert Reaumur Into Fahrenheit— (1) Multiply by 2.25, or multiply by f, 
 
 If the Reaumur thermometer registers 36°, what will be the register by 
 the Fahrenheit thermometer? 
 
 36 X 2.25 + 32 = 113° Fahrenheit. + 32 = 113° Fahrenheit. 
 
 To Convert Reaumur Into Centigrade.— Multiply by 1-25. . . 
 
 If a Reaumur thermometer registers 32°, what will be the register by 
 a Centigrade thermometer? 
 
 32 X 1.25 = 40° Centigrade. 
 
 To Convert Centigrade Into Reaumur.— Multiply by .8. . . 
 
 If a Centigrade thermometer registers 40°, what will be the register by 
 a Reaumur thermometer? 
 
 40 x .8 = 32° Reaumur. 
 
 Calculation of Mine Resistance— The mine resistance is equal to the total 
 pressure p a that it causes. This mine resistance is dependent upon tnree 
 factors: (a) The resistance k offered by 1 sq. ft. of rubbing surface to 
 
 a current naving a velocity of 1 ft. per minute. The coefficient of friction k, 
 or the unit of resistance , is the resistance offered by the unit of rubbing sur- 
 face to a current of a unit velocity. This unit resistance has been variously 
 estimated by different authorities (see following table). The value most 
 universally accepted, however, is that known as the Atkinson coefficient 
 
THE EQ UIVALENT ORIFICE. 
 
 367 
 
 (.0000000217). (b) The mine resistance, which varies as the square of the 
 velocity. ( c ) The rubbing surface. Hence, if we multiply the unit resist- 
 ance by the square of the velocity, and by the rubbing surface, we will 
 obtain the total mine resistance as expressed by the formula pa = ksv *. 
 
 Table of Various Coefficients of Friction of Air in Mines. 
 
 Pressure per 
 Sq. Ft. Decimals 
 of a Pound. 
 
 J. J. Atkinson’s treatise 
 
 A. Devillez in Ventilation des Mines: 
 
 Forchies - 
 
 Crachet-Picquery 
 
 Grand Baisson 
 
 Average of 2, 3, and 4 
 
 Used in Ventilation des Mines 
 
 Arched Tunnels 
 
 Along a working face of coal 
 
 G. G. Andre, Atmosphere of Coal Mines 
 
 Peclet, Cheminee (Devillez, p. 112) 
 
 j) Clark 
 
 According to Goupiiliere’s Cours d’ Exploitation des Mines, 
 Vol. II, p. 389: 
 
 D’Aubuisson - 
 
 Navier 
 
 W. Fairley 
 
 J. Stanley James 
 
 D. Murgue 
 
 .0000000217 
 
 .000000008211 
 
 .000000008928 
 
 .000000008611 
 
 .000000008585 
 
 .000000009511 
 
 .000000002113 
 
 .000000014266 
 
 .000000022424 
 
 .000000003697 
 
 .000000002272 
 
 .000000001955 
 
 .000000001872 
 
 .00000001 
 
 .00000000929 
 
 .000000008242 
 
 It will be observed that J. J. Atkinson’s coefficient is greatly in excess of 
 any other, with the exception of Andre’s. Fairley’s is derived from an 
 average taken between Atkinson, Devillez, and Clark, and, undoubtedly, it 
 is an exceedingly simple coefficient to work out calculations with, as it will 
 save a great mass of figures. James, in his work on colliery ventilation, 
 reduces the coefficient still further on the authority of the Belgian Mine 
 Commission, but he gives a most unwieldy figure to use. 
 
 Atkinson’s figure is the one most in use, and if it is too high, it errs on the 
 side of safety, and it is always advisable to have plenty of spare ventilating 
 power at a mine. For this reason, and until a regular and thorough investi- 
 gation, made by a commission of competent men, provides a standard coef- 
 ficient, we prefer to abide by Atkinson’s coefficient, and it is used in all our 
 calculations 
 
 Calculation of Power, or Units of Work per Minute. — If we multiply the total 
 pressure by the velocity (feet per minute) with which it moves, we obtain the 
 unite of work per minute, or the power upon the air. Hence, u = pav = 
 ksv* which isrthe fundamental expression for work per minute , or power. 
 
 The Equivalent Orifice.— This term, often used in regard to ventilation, 
 evaluates the mine resistance, or, as will be seen from the equation given 
 below for its value, it expresses the ratio that exists between the quantity 
 of air passing in an airway and the pressure or water gauge that is produced 
 by the circulation. This term was suggested by M. Daniel Murgue, and 
 refers to the flow of a fluid through an orifice in a thin plate, under a given 
 head. The formula expressing the velocity of flow through such an orifice 
 is v = 1 / 2 gh] multiplying both members of this equation by A, and substi- 
 tuting for the first member A v, its value q , we have, after transposing and 
 
 correcting for vena contracta, A = - , in which .62 is the coefficient for 
 
 .62y 2 g h 
 
 the vena contracta of the flow. Reducing this to cubic feet per minute and 
 inches of water gauge represented by i , we have, finally, the equation 
 
 A = .0004 X —7=. By this formula, Murgue has suggested assimilating the 
 
 V i 
 
 flow of air through a mine to the flow of a fluid through a thin plate; since, 
 in each case, the quantity and the head or pressure vary in the same ratio. 
 Thus, applying this formula to a mine, Murgue multiplies the ratio of the 
 quantity of air passing (cubic feet per minute) and the square root of the 
 water gauge (inches) by .0004, and obtains an area A , which he calls 
 the equivalent orifice of the mine. . , „ 
 
 Potential Factor of a Mine. ( Proposed by J. T. Beard.) — Equations 8 and 27, 
 
368 
 
 VENTILATION OF MIXES. 
 
 pages 370-371, give, respectively, the pressure and the power that will circu- 
 late a given quantitv of air per minute in a given airway. These equations 
 may be written as equal ratios, expressed in factors of the current and the 
 * T) Jc s u k s 
 
 airway, respectively; thus, — = — ^ , and — = which show that the ratio 
 
 between the pressure and the square of the quantity it circulates in any 
 given airwav is equal to the ratio between the power and the cube of the 
 quantity it circulates. Solving each of these equations with respect to q, 
 we have the following: 
 
 With respect to pressure, 
 
 *- ( a \i rs)^- 
 
 With respect to power, 
 
 *“ (rny*- 
 
 Hence, we observe that, in any airway, for a constant pressure , the quan- 
 
 tity of air in circulation is proportional to the expression a-y and, for a 
 
 constant power , the quantity is proportional to the expression which 
 
 terms are called the potentials of the mine with respect to pressure and 
 power, respectively; and their values — — and are the potentials of the 
 
 Vp 
 
 current with respect to pressure and power, respectively. These factors, it 
 will be observed, evaluate the airway, as they determine the quantity of air 
 a given pressure or power will circulate in that airway (cubic feet per 
 minute). Bv their use. the relative quantities of air any given pressure or 
 power will circulate in different airways are readily determined. The rule 
 mav be stated as follows: . . . 
 
 Rule. — For any given pressure or power, the quantity of air in circulation is 
 always ' proportional to the potential for pressure, or the potential for power , as 
 the case may he. 
 
 This rule finds important application in splitting (see Calculation of 
 Natural Splitting) . In all cases where the potential is used as a ratio, the 
 relative potential maybe employed by omitting the factor k; or it may be 
 employed to obtain the pressure and power, in several splits by multiplying 
 the final result by k (see Formulas 46, 47, etc., page 378). 
 
 Example.— 20,000 cu. ft. of air is passing in a mine in which the airway 
 is 6 ft. X 8 ft., and 10.000 ft. long, under a certain pressure; it is required to 
 find what quantity of air this same pressure will circulate in a mine in which 
 the airwav is 6 ft. X 12 ft., and 8,000 ft. long. 
 
 Calculating the potential X p with respect to the pressure for each 
 of these mines, or airways, we have, using the relative potential, 
 
 Xi= 6X8< 
 
 6X8 
 
 6X12 
 
 — .62845, and X 2 — 6 X 12^ 2 (6 + J2) x 8,000 
 
 \ 2(6 + 8) X 10,000 
 
 = 1.1384. Since the ratio of the quantities is e^ual to the ratio of the 
 potentials with respect to pressure, in these two mines, we write the propor- 
 
 20,000 X 1.1384 
 .62845 
 
 = 36,229 cu. ft. per 
 
 tion 20,000 : q 2 : : .62845 : 1.1384, and q 2 = 
 
 min. . . . . . ,, 
 
 Example.— 20,000 cu. ft. of air is passing in a mine in which the airway 
 is 6 ft. X 8 ft., and 10,000 ft. long, under a certain power; it is required to find 
 what quantity of air will be circulated by this same power m a mi ne in 
 which the airway is 6 ft. X 12 ft., and 8,000 ft. long. 
 
 We calculate the potential X u with respect to power for each of these 
 
 6X8 
 
 mines, using, as before, the relative potential. Thus, X\ = ^ io 000 
 
 -- .7337, and X 2 
 
 6X 12 
 
 f 2(6 
 
 1.0905. Then, in this case, since the 
 
 12)X 8,000 
 
 ratio of the quantities is equal to the ratio of the potentials with 
 
POTENTIAL FACTOR. 
 
 369 
 
 respect to power, we write the proportion, 20,000 : q% : : .7337 : 1.0905, and 
 
 a = 20,000 X 1^0905 _ 29 726 cu. ft. per min. 
 v2 7337 
 
 The following table will serve to illustrate the use of the formulas 
 employed in these calculations. It will be observed that there are 
 several formulas for quantity, and for velocity, and for work or horse- 
 power, but in each respective case the several formulas are derived by 
 simple transposition of the terms of the original formula, and are tabulated 
 here for convenience. Choice must be made in the use of any of these for- 
 mulas, according to the known terms in each example. . Thus, an example 
 mav ask- What pressure will be produced in passing a given quantity of air 
 through 'a certain mine, the size and length of the airways being given? 
 
 We then use the formula p = if t ^ ie question asks what quan- 
 
 tity of air a given pressure will produce in this same mine, we use the 
 
 formula q 
 
 -4 
 
 pa 
 
 ks 
 
 Xfl. 
 
 It will be observed that this second formula is a 
 
 simple transposition of the first. , „ , x ... , 
 
 In like manner the question maybe asked, what power will produce a 
 certain quantity of air in a certain airway; and the expression used, in this 
 
 case, is u = k S y~. Or, the question may be asked, what quantity of air will 
 
 be produced in a given airway by means of a certain power^or work applied 
 
 to the airway. In this case, the formula used is q = If the fic- 
 
 tion asks for the power required to produce a given velocity in a fiven air- 
 way the formula employed is u = ksv s . All of these formulas are derived 
 
 by combining the simple formulas p = — — , q = av, and u = qp. 
 
 To illustrate the use of the formulas, we take as an example an under- 
 ground road, 5 ft. wide by 4 ft. high, and 2,000 ft. in length, and calculate the 
 value of each symbol or letter, assuming a velocity of 500 ft. per minute. 
 
 
 Symbol. 
 
 Value of Symbol for this 
 Particular Example. 
 
 Area of airway (5 ft. X 4 ft.) 
 
 tinrconowpr ♦ 
 
 a 
 
 h 
 
 20 sq. ft. 
 2.959 H. P. 
 
 Pnoffipipnt of friction + 
 
 k 
 
 .0000000217 lb. 
 
 VUCUILIV/Ilt VII llibtlL/ll | 
 
 T nn rrfVi rtf* QirWflV 
 
 l 
 
 2,000 ft. 
 
 ±J0IlgXIl U1 dll WdJ • • • 
 
 Perimeter of airway, 2(5 ft. + 4 ft.) 
 
 Pressure (lb. per sq. ft.) 
 
 nna-nfitxr r»f nir fpn ft, ner mm. ) 
 
 0 
 
 P 
 
 q 
 
 18 ft. 
 
 9.765 lb. 
 10,000 cu. ft. 
 
 quantity ui an it. 
 
 A rna nf rnhhiTiP 1, SlirftinP 
 
 s 
 
 36,000 sq. ft. 
 
 AltJd UI lUUUIllg ouiiavyt 
 
 Units of work per minute (power) 
 
 Volnpitv (ft ripr min ^ 
 
 u 
 
 V 
 
 97,650 ft.-lb. 
 500 ft. 
 
 V clUtltj \ A o* pci min. ; 
 
 Wotpr an n orp 
 
 i 
 
 1.87788 ip. 
 
 vv dtei gdugu - 
 
 TTmiiirolorit nrifipp nf flip TTVlTlfi 
 
 A 
 
 2.919 sq. ft. 
 
 JjU UlValcIll UllliUC ui tiiv 
 
 ■prktori+ml fnr nowpr 
 
 X u 
 
 217.16 units. 
 
 i UtUlltldl lUi pu vv Cl 
 
 ■pz-ktcntial fnr* tytpwii re 
 
 X v 
 
 3,200 units. 
 
 I Utt/Utldl 1U1 piCOOUlC 
 
 Weight of 1 cu. ft. of downcast air 
 
 Motive column (downcast air) 
 
 moYvfVi nf fnrnflPP shaft, 
 
 P 
 
 W 
 
 M 
 
 D 
 
 .08098 lb. 
 120.5 ft. 
 306.77 ft. 
 
 Average temperature of the upcast column.. 
 Average temperature of the downcast column 
 
 T 
 
 t 
 
 350° F. 
 32° F. 
 
 *A horsepower is equal to 33,000 units of work. . , , . 
 
 t This coefficient of friction is an invariable quantity, and is the same in every calculation 
 relating to the friction of air in mines. 
 
 Note.— The water gauge is calculated to five decimal places to enable all 
 the other values to be accurately arrived at. In practice, it is only read to 
 one decimal place. 
 
370 
 
 VENTILATION OF MINES. 
 
 FORMULAS. 
 
 On the right side of each formula, the various calculations, based on the 
 example given, are worked out in figures. 
 
 To Find: 
 
 No. 
 
 Formula. 
 
 • 
 
 Specimen Calculation. 
 
 Rubbing sur- 
 face of an air- 
 way. (Sq.ft.) 
 
 1 
 
 s — l 0 
 
 2,000 X 18 = 36,000 sq. ft. 
 
 Area of an 
 airway. (Sq.ft.) 
 
 2 
 
 a = — 
 
 V 
 
 20 sq. ft. of area. 
 
 ouU 
 
 Velocity. 
 (Ft. per min.) 
 
 3 
 
 II 
 
 ^^ = 500 ft. 
 
 
 4 
 
 3 / U 
 
 V ” \ks 
 
 si 97,650 
 
 \ .0000000217 X 36,000 
 
 
 5 
 
 v J Va 
 
 1 9.765X20 
 
 
 
 \ ks 
 
 \ .0000000217 X 36,000 
 
 
 6 
 
 u 
 
 97,650 cqq «■ 
 
 9.765 X 20 500 * 
 
 
 ~ pa 
 
 Pressure. 
 
 7 
 
 ksv 2 
 
 .0000000217 X 36,000 X 500 2 n 
 
 (Lb. persq. ft.) 
 
 P 
 
 a 
 
 20 ~ 9./6D1D. 
 
 
 8 
 
 k sq 2 
 
 J 
 
 .0000000217 X 36,000 X 10,000 2 
 20 s 
 
 = 9.765 lb. 
 
 
 9 
 
 II 
 
 |g 
 
 
 
 10 
 
 p = Mw 
 
 120.58 X .08098 = 9.765 lb. 
 
 
 11 
 
 p — 5.2 i 
 
 5.2 X 1.87788 = 9.765 lb. 
 
 
 12 
 
 II 
 
 hk 
 
 cel 
 
 
 
 13 
 
 II 
 
 Sc 
 
 
 Water gauge. 
 (Inches.) 
 
 14 
 
 i - 
 
 5.2 
 
 9 '™ 5 = 1.87788 in. 
 
 Resistance of 
 an airway. 
 (Total pressure, 
 
 15 
 
 16 
 
 pa = ksv 2 
 u 
 
 pa = - 
 
 * V 
 
 .0000000217 X 36,000 X 500 2 = 195.31b. 
 9 ^° = 195.3 n>. 
 
FORMULAS. 
 
 371 
 
 To Find: 1 
 
 No. 
 
 Formula. | 
 
 1 
 
 Specimen Calculation. 
 
 Quantity. 
 
 17 
 
 q == av 
 
 20 X 500 = 10,000 cu. ft. 
 
 (Cu.ft.per min) 
 
 
 ►c 
 
 It 
 
 •e 18 
 
 97 650 
 
 
 18 
 
 y/,oou = 1Q 000 cu ft 
 9.765 
 
 
 19 
 
 
 1 9.765 X 20 
 
 \ .0000000217 X 36,000 
 
 
 
 = 10,000 cu. ft. 
 
 
 20 
 
 9 = ^ Xa 
 
 3 1 97,650 ^ 0 q 
 
 \ .0000000217 X 36,000 
 = 10,000 cu. ft. 
 
 
 21 
 
 q = X u fu 
 
 217.16 X ^ 97,650 = 10,000 cu. ft. 
 
 
 22 
 
 q = fX/u 
 
 ^3,2002 X 97,650 = 10,000 Cu. ft. 
 
 
 23 
 
 q = X p Vp 
 
 3,200 X V 9-765 = 10,000 cu. ft. 
 
 Units of work 
 
 24 
 
 u = avp 
 
 20 X 500 X 9.765 = 97,650 ft.-lb. 
 
 per minute, or 
 power on the 
 
 25 
 
 II 
 
 8 
 
 10,000 X 9.765 = 97,650 ft.-lb. 
 
 air. 
 
 (Ft.-lb.permin) 
 
 26 
 
 u = ksv* 
 
 .0000000217 X 36,000 X 500 3 
 = 97,650 ft.-lb. 
 
 
 
 ks q 3 
 
 .0000000217 X 36,000 X 10,000 3 
 
 
 27 
 
 U ~~ a 3 
 
 20 3 
 
 
 
 = 97,650 ft.-lb. 
 
 
 28 
 
 u = h 33,000 
 
 2.959 X 33,000 = 97,650 ft.-lb. 
 
 
 29 
 
 q z 
 
 u = x~z 
 
 ■A-u 
 
 = 97,650 ft.-lb. 
 
 
 30 
 
 q 3 
 
 - 97 - 65oft - ib - 
 
 Horsepower. 
 
 31 
 
 h U 
 
 “ 33,000 
 
 = 2.959 H. P. 
 
 33,000 
 
 Power poten- 
 tial. 
 (Units.) 
 
 32 
 
 | w 
 
 ° £ 
 II 
 
 20 — 217.16 units. 
 
 ^ .0000000217 X 36,000 
 
 
 33 
 
 1 * 1 *. 
 
 II 
 
 Hi 
 
 j/lO’OOO 2 - 217.16 units. 
 \ 9.765 
 
 
 
 ■y q 
 
 10,000 _ 2^7 i6 units. 
 
 
 34 
 
 ■A-u — O/— 
 
 fn 
 
 ^97,650 
 
372 
 
 VENTILATION OF MINES. 
 
 To Find: 
 
 No. 
 
 Formula. 
 
 Specimen Calculation. 
 
 Pressure poten- 
 tial. 
 (Units.) 
 
 35 
 
 r 
 
 36 
 
 
 
 H & 
 
 f " 
 
 11 S 
 
 •ell *|a 
 
 Co 1 
 
 1 20 
 
 „ 20 \ .0000000217 X 36,000 
 
 = 3,200 units. 
 
 - Q - Q ° = 3,200 Units. 
 
 V 9.765 
 
 Equivalent 
 orifice. 
 (Sq. ft.) 
 
 37 
 
 .0004 q 
 
 Vi 
 
 .0 004 X 10 ,°°0 — 2.919 sq. ft. 
 1/1.87788 
 
 Motive column, 
 downcast air. 
 (Feet.) 
 
 38 
 
 39 
 
 K - 1>x 459+r 
 
 M = - P - 
 w 
 
 30M7x m + ^ - m5ft - 
 
 Motivecolumn, 
 upcast air. 
 (Feet.) 
 
 40 
 
 39 
 
 M = 2 - 
 
 w 
 
 306 - 77x !“;32- m7ft ; 
 
 mZ = m7ft - 
 
 Variation of the Elements.— In the illustration of the foregoing table, we 
 have assumed fixed conditions of motive column, as well as fixed conditions 
 in the mine airways. It is often convenient, however, to know how the 
 different elements, as velocity v, quantity q, pressure p, power u, etc., will 
 vary in different circulations; since we may, by this means, compare the 
 circulations in different airways, or the results obtained by applying different 
 pressures and powers to the same airway. These laws of variation must 
 always be applied with great care. For example, before we can ascertain 
 how the quantity in circulation will vary in different airways, we must 
 -mow whether the pressure or the power is constant or the same for each 
 hrway. The following rules may always be applied: 
 
 For a constant pressure: v varies as q varies as (relative poten- 
 tial for pressure). 
 
 For a constant power: varies as 
 
 q varies as — ^ (relative potential 
 
 for power). 
 
 For a constant velocity: q varies as a; p varies as — ; u varies as lo. 
 
 For a constant quantity: v varies inversely as a; p varies inversely as X u 3 
 (potential for power); u varies inversely as X M 3 (potential for power) or 
 directly as p. 
 
 For the same airway: The following terms vary as each other: v , q, 
 
 Vp, fu. 
 
 Similar Airways. 
 
 v = length of similar side, or similar dimension. 
 
 For a constant pressure: v varies as ^ j ; q varies as r 2 X rvariesas 
 
 lv 2 , or V lq 2 . 
 
DISTRIBUTION OF AIR. 
 
 373 
 
 1. 3 jr 2 
 
 For a constant power: v varies as q varies as r X yjj ; r varies as 
 
 r ^ or ^« 3 
 
 For a constant velocity: q varies as r 2 ; p varies as u varies as Ir; 
 
 /-l u 
 r varies as y g, or 
 
 For a constant quantity: v varies inversely as r 2 ; p and u vary inversely as 
 or 
 
 Furnace Ventilation. 
 
 p (motive column) varies as D; q varies as j/ D. 
 
 Fan Ventilation. 
 
 r b . l 
 
 , ; r varies as — — 
 
 * V*' 
 
 It has been customary in calculations pertaining to the yield of centrif- 
 ugal ventilators to assume as follows: q varies as n; p varies as ?i 2 ; 
 u varies as n 3 . 
 
 More recent investigation, however, shows that when we double the 
 speed we do not obtain double the quantity of air in circulation; or, in other 
 words, the quantity does not vary exactly as the number of revolutions of 
 the fan. Investigation also points to the fact that the efficiency of centrif- 
 ugal ventilators decreases as the speed increases. To what extent this is the 
 case has not been thoroughly established. The variation between the speed 
 of a fan and the quantity, pressure, power, and efficiency, as calculated from 
 a large number of reliable fan tests, may be stated as follows: 
 
 Forthesamefan, discharging against a constant potential: q varies as n- 91 . 
 p varies as it 1 * 94 . Complement of efficiency (1 — K) varies as w 425 . 
 
 The efficiency here referred to is the mechanical efficiency , or the ratio 
 between the effective work qp and the theoretical work of the fan. 
 
 DISTRIBUTION OF AIR IN IWNE VENTILATION. 
 
 When a mine is first opened, the air is conducted in a single current 
 around the face of all the headings and workings, and returns again to the 
 upcast shaft, where it is discharged into the atmosphere. As the develop- 
 ment of the mine advances, however, it becomes necessary to divide the air 
 into two or more splits or currents. This division or splitting of the air- 
 current is usually accomplished at the foot of the downcast, or as soon as 
 possible after the current enters the mine. There are several reasons why 
 the air-current should be thus divided. The most important reason is that 
 the mine is thereby divided into separate districts, each of which has its 
 own ventilating current, which may be increased or decreased at will. 
 Fresh air is thus obtained at the face of the workings, and the ventilation is 
 under more perfect control. It often happens that certain portions of a mine 
 are more gaseous than others, and it is necessary to increase the volume of 
 air in these portions, which can be readily accomplished when each district 
 has its own separate circulation. Again, the gases and foul air are not 
 conducted from one district to another, but each district is supplied with 
 fresh air direct from the main intake. Should an explosion occur in any 
 part of the mine, it is more apt to be confined to one locality when a mine is 
 thus divided into separate districts. Another consideration is the reduced 
 power necessary to accomplish the same circulation in the mine; or the 
 increased circulation obtained by the use of the same power. 
 
 Requirements of Law in Regard to Splitting.— The Anthracite Mine Law of 
 Pennsylvania specifies that every mine employing more than 75 persons 
 must be divided into two or more ventilating districts, thus limiting the 
 number that are allowed to work on one air-current to 75 persons. The 
 Bituminous Mine Law of Pennsylvania limits the number allowed to work 
 upon one current to 65 persons, except in special cases, where this number 
 may be increased to 100 persons at the discretion of the mine inspector. 
 
 Practical Splitting of the Air-Current— When the air-current is divided into 
 two or more branches, it is said to be split. The current may be divided one 
 or more times; when split or divided once, the current is said to be traveling 
 
374 
 
 VENTILATION OF MINES . 
 
 in two splits, each branch being termed a split. The number of splits in 
 which a current is made to travel is understood as the number of separate 
 currents in the mine, and not as the number of divisions of the current. 
 
 Primary Splits.— When the main air-current is divided into two or more 
 splits, each of these is called a primary split. . 
 
 Secondary Splits. — Secondary splits are the divisions of a primary split. 
 
 Tertiary Splits.— Tertiary splits result from the division of a secondary split. 
 
 Equal Splits of Air.— When a mine is spoken of as having two or more equal 
 splits, it is understood to mean that the length and the size of the separate 
 airways forming those splits are equal in each case. It follows, of course, 
 from this that the ventilating current traveling in each split will be the 
 same, inasmuch as they are all subject to the same ventilating pressure. 
 When an equal circulation is obtained in two or more splits by the use of 
 regulators, these splits cannot be spoken of as equal splits. * 
 
 Unequal Splits of Air.— By this is meant that the airways forming the splits 
 are of unequal size or length. Under this head we will consider (a) Natural 
 Division of the Air- Current; ( b ) Proportionate Division of the Air- Current. 
 
 Natural Division of the Air-Current.— By natural division of air is meant any 
 division of the air that is accomplished without the use of regulators; or, in 
 other words, such division of the air-current as results from natural means. 
 If the main air-current at any given point in a mine is free to traverse two 
 separate airways in passing to the foot of the upcast shaft, and each of these 
 airways is free or an open split, i. e., contains no regulator, the division of 
 the air will be a natural division. In such a case, the larger quantity of air 
 will always traverse the shorter split of airway. In other words, an air-cur- 
 rent always seeks the shortest way out of a mine. A comparatively small 
 current, however, will always traverse the long split or airway. 
 
 Calculation of Natural Splitting.— It is always assumed, in the calculation 
 of the splitting of air-currents, that the pressure at the mouth of each split, 
 starting from any given point, is the same. Since this is the case, in order 
 to find the quantity of air passing in each of several splits starting from a 
 common point, the rule given under Potential Factor of a Mine is apphed. 
 This rule may be stated as follows: 
 
 The ratio between the quantity of air passing in any split and the pressure 
 potential of that split is the same for all splits starting from a common point. 
 Also the ratio between the. entire quantity of air in circulation in the several 
 splits and the sum of the pressure potentials of those splits is the same as the 
 above ratio, and is equal to the square root of the pressure. 
 
 Expressed as a formula, indicating the sum of the pressure potentials 
 
 (Xi + X 2 + etc.) by the expression 2I P , this rule is = Jr = V P- 
 
 Hence, p = 
 
 and u = 
 
 express the pressure and power, 
 
 (2J P ) 2 ~ (2X P ) 2 . . _ 
 
 respectively, absorbed by the circulation of the splits. These are the basal 
 formulas for splitting, from which any of the factors may be calculated by 
 transposition. They will be found illustrated in the table at the end of this 
 section. We will give here two examples only, showing the calculation of 
 the natural division of an air-current between several splits. \\ e have, from 
 
 Xi ^ 
 
 the above formulas, q\ = ^ ^ Q. 
 
 Example. — In a certain mine, an air-current of 60,000 cu. ft. perminute 
 is traveling in two splits as follows: Split A, 6 ft. X 8ft., 5,000 ft. long- 
 split B, 5 ft. X 8 ft., 10,000 ft. long. It is required to find the natural division 
 
 of this air-current. . , , . , r+ 
 
 Calculating the relative potentials for pressure in each split, we have 
 
 for split A 
 
 for split B, X 2 
 
 .*1 -«Vs 
 
 = 4 °^ 
 
 48 
 
 2(6 + 8)5,000 
 40 
 
 2(5 + 8)10,000 
 
 = .4961 
 
 and 2 X p = 1.3849; 
 
 and substituting these values, we have, 
 
 Qi 
 
 Q2 
 
 - v 60,000 = 38,506 cu. ft. per min.; 
 1.3849 
 
 x 60.000 = 21,494 cu. ft. per min. 
 
 1.3849 
 
 and 
 
DISTRIBUTION OF AIR. 
 
 375 
 
 Example.— In a certain mine, there is an air-current of 100,000 cu. ft. per 
 minute traveling in three splits as follows: Split A, 6 ft. X 10 ft., 8,000 ft. long; 
 split B , 6 ft. X 12 ft., 15,000 ft. long; split C, 5 ft. X 10 ft., 6,000 ft. long. Find 
 the natural division of this current of air. 
 
 Calculating the respective relative potentials with respect to pressure, 
 we have 
 
 for split A X, = 60 y 2(6 + 1 ff x8 , 000 - -9185; 
 
 for split B, X t = 72 ^ + - .8314; 
 
 for split C, X 3 = 50 V 2(5 + 10 rx 6 , - 60b = ' 8833 ' 
 
 Adding these potentials, we have 2 X p = .9185 + .8314 + .8333 = 2.5832. 
 Then, applying the foregoing rule, we have 
 
 qi = X 100,000 = 35,556 cu. ft. per min.; 
 
 « 2 = ~~ X 100,000 = 32,184 cu. ft. per min.; 
 
 QQOO 
 
 and x 100 ’ 000 = 32,260 cu - ft - P er min - 
 
 Total, 100,000 
 
 Proportional Division of the Air-Current.— It continually happens that differ- 
 ent proportions of air are required in the several splits of a mine than would 
 be obtained by the natural division of the air-current. It is usually the case 
 that the longer splits employ a larger number of men, and require a larger 
 quantity of air passing through them. They, moreover, liberate a larger 
 quantity of mine gases, for which they require a larger quantity of air than 
 is passing in the smaller splits. The natural division of the air-current 
 would give to these longer splits less air, and to the shorter ones a larger 
 amount of air, which is directly the reverse of what is needed. On this 
 account, recourse must be had to some means of dividing this air pro- 
 portionately, as required. This is accomplished by the use of regulators, of 
 which there are two general types, the box regulator and the door regulator. 
 
 Box Regulator. — This is simply an obstruction placed in those airways that 
 would naturally take more air than the amount required. It consists of a 
 brattice or door placed in the entry, and having a small shutter that can be 
 opened 1 to a greater or less amount. The shutter is so arranged as to allow 
 the passage of more or less air, according to the requirements. The box 
 regulator is, as a rule, placed at the end or near the end of the return air- 
 way of a split. It is usually placed at this point as a matter of convenience, 
 because, in this position, it obstructs the roads to a less extent, the haulage 
 from the back entry in this split being carried over to the main haulway, 
 through a cross-cut, before this point is reached. The difficulty, however, 
 can be avoided, in most cases, by proper consideration in the planning of 
 the mine with respect to haulage and ventilation. The objection to this 
 form of regulator is that, in effect, it lengthens the airway, or increases its 
 resistance, making the resistance of all the airways, per foot of area, the 
 same. It is readily observed that, by thus increasing the resistance of the 
 mine, the horsepower of the ventilation is largely increased, for the same 
 circulation. This is an important point, as it will be found that the power 
 required for ventilation is thus increased anywhere from 50$ to 100$ over 
 the power required when the other form of regulator can be adopted. 
 
 Door Regulator.— In this form of regulator, which was first introduced by 
 Beard, the division of the air is made at the mouth of the split. The regu- 
 lator consists of a door hung from a point of the rib between two entries, 
 and swung into the current so as to cut the air like a knife. The door is 
 provided with a set lock, so that it may be secured in any position, to give 
 more or less air to the one or the other of the splits, as required. The posi- 
 tion of this regulator door, as well as the position of the shutter in the box 
 regulator, is always ascertained practically by trial. The door is set so as 
 to divide the area of the airway proportionate to the work absorbed in the 
 
376 
 
 VENTILATION OF MINES. 
 
 respective splits. The pressure in any split is not increased, each split 
 retaining its natural pressure. . . . , 
 
 Calculation of Pressure for Box Regulators— When any required division of 
 the air-current is to be obtained by the use of box regulators, these are 
 placed in all the splits, save one. This split is called the open, or free, split, 
 
 and its pressure is calculated in the usual way by the formula p = ~as~' 
 
 The natural pressure in this open split determines the pressure of the entire 
 mine, since all the splits are subject to the same pressure in this form of 
 
 ^ First* determine in which splits regulators will have to be placed, in order 
 to accomplish the required division of the air. Calculate the natural pres- 
 sure, or pressure due to the circulation of the air-current, for each split, 
 
 when passing its required amount of air, using the formula p = ^ The 
 
 split showing the greatest natural pressure is taken as the free split. In each 
 of the other splits, box regulators must be placed, to increase the pressure 
 in those splits; or, in other words, to increase the resistance of those splits 
 per unit of area. . , . . 
 
 Example. — The ventilation required m a certain mine is: 
 
 split A , 6 ft. X 9 ft., 8,000 ft. long; 40,000 cu. ft. per min. 
 
 split B , 5 ft. X 8 ft., 6,000 ft. long; 40,000 cu. ft. per min. 
 
 split a 9 ft. X 9 ft., 8,000 ft. long; 10,000 cu. ft. per min. 
 
 split D, 6 ft. X 8 ft., 10,000 ft. long; 30,000 cu. ft. per mm. 
 
 In which of these splits should regulators be placed, to accomplish the 
 required division of air, and what will be the mine pressure? . 
 
 Calculating the pressure due to friction in each split when passing its 
 required amount of air, we find, 
 
 .0000000217 X 2(6 + 9)8,000 X 40,0002 
 ' 5# ~ 
 
 .0000000217 X 2(5 + 8)6,000 X 40.0002 
 ~40* 
 
 .0000000217 X 2(9 + 9)8,000 X 10 . 000 2 
 81 3 
 
 .0000000217 X 2(6 + 8 )10,000 X 30,0 002 = 49 45 lb per sq ft 
 
 for split A, p = 
 for split B, p = 
 
 for split C, p = 
 
 = 52.92 lb. per sq. ft.; 
 = 84.63 lb. per sq. ft.; 
 = 1.176 lb. per sq. ft.; 
 
 for split D, p = 
 
 48 * 
 
 Split B has the greatest pressure, and is therefore the free split. Box 
 regulators are placed in each of the other splits to increase their respective 
 pressures to the pressure of the free split or the mine pressure. Therefore, 
 the mine pressure in this circulation is 84.63 lb. per sq. ft. _ 
 
 The Size of opening in a box regulator is calculated by the formula for 
 determining the flow of air through an orifice in a thin plate under a certain 
 head or pressure. The difference in pressure between the two sides of a box 
 regulator is the pressure establishing the flow through the opening, which 
 corresponds to the head h in the formula v = \Z%9 h -. This regulator is 
 usually placed at the end of a split or airway, and since the regulator 
 increases the pressure in the lesser split so as to make it equal to the pressure 
 in the other split, the pressure due to the regulator will be equal to the 
 ventilating pressure at the mouth of the split, less the natural . pressure or 
 the pressure due to friction in this split. Hence, when the position of the 
 regulator is at the end of the split, the pressure due to friction m the split is 
 
 first calculated by the formula p = ^- 2 , and this pressure is deducted from 
 
 the ventilating pressure of the free or open split, which gives the pressure 
 due to the regulator. This is then reduced to inches of water gauge, and 
 
 .OCKHq Tbe va ] ue 0 f ^ thus obtained is 
 
 substituted for i in the formula A — 
 
 V K 
 
 the area (square feet) of the opening in the regulator. 
 
 Example.— 50,000 cu. ft. of air is passing per minute m a certain mine, 
 in two equal splits, under a pressure equal to 2 in. of water gauge, and it is 
 required to reduce the quantity of air passing m one of these splits, by a box 
 regulator placed at the end of the split, so as to pass but 15,000 cu. ft. per 
 
DISTRIBUTION OF AIR. 
 
 377 
 
 minute in this split. Find the area of the opening in the regulator, assu- 
 ming that the ventilating power is decreased to maintain the pressure con- 
 stant at the mouth of the splits after placing the regulator. The size and 
 length of each split is 6 ft. X 10 ft. and 10,000 ft. long. . , „ ^ 
 
 The natural pressure for the split in which the regulator is placed will be 
 
 .0000000217 X 2(6 + 10) X 10, 000 X 15, OOP 2 = ? m lb per sq> 
 
 _ ksq 2 __ 
 
 a 3 
 
 Then, 
 
 7.233 
 
 5.2 
 
 (6 X 10) 3 
 
 1.4 in. of water gauge (nearly), due to friction of the air- 
 
 current in this split. 
 
 „ . .0004 q 
 
 Finally, A = — j=- - 
 
 V 
 
 And, 2 — 1,4 = .6 in. water gauge due to regulator. 
 
 .0004 X 15,000 
 
 V -6 
 
 = 7.746 sq. ft., area of opening. 
 
 Size of Opening for a Door Regulator.— The sectional area at the regulator is 
 divided proportionately to the work to be performed in the respective splits 
 according to the proportion A\ : A 2 :: U\ : u 2 . Or since A\-t A 2 = a, we have 
 
 A\ :a :: U \ : u x + u 2 , and A x = — X a. This furnishes a method of pro- 
 portionate splitting in which each split is ventilated under its own natural 
 pressure. The same result would be obtained by the placing of the box 
 regulator at the intake of any split, thereby regulating the amount of air 
 passing into that split, but the door regulator presents less resistance to the 
 flow of the air-current. The practical difference between these two forms 
 of regulators is that in the use of the box regulator each split is ventilated 
 under a pressure equal to the natural pressure of the open or free split 
 which very largely increases the horsepower required for the ventilation of 
 the mine- while in the use of the door regulator each split is ventilated 
 under its own natural pressure, and the proportionate division of the air is 
 accomplished without any increase of horsepower. This is more clearly 
 . , x — — -• — — h s , and the table showing the corn- 
 
 explained in the two following parai 
 parative horsepowers of the two methods. 
 
 Calculation of Horsepower for Box Regulators.— By the use of the box regu- 
 lator, the pressure in all the splits is made equal to the greatest natural 
 pressure in any one. This split is made the open or free split, and its natural 
 pressure becomes the pressure for all the splits, or the mine pressure. This 
 mine pressure, multiplied by the total quantity of air in circulation (the sum 
 of the quantities passing in the several splits), and divided by 33,000, gives 
 the horsepower upon the air, or the horsepower of the circulation. Thus, 
 in the first example given on page 376, in which for split B the pressure 
 p = 84.63 lb. per sq. ft. and the total quantity of air passing per minute 
 is 120,000 cu. ft., we have 
 
 Calculation of Horsepower for Door Regulators. — In the use of the door 
 regulator, each split is ventilated under its own natural pressure, and, 
 hence, in the calculation of the horsepower of such a circulation, the power 
 of each split must be calculated separately, and the sum of these several 
 powers will be the entire power of the circulation. For the purpose of com- 
 parison, we tabulate below the results obtained in the application of these 
 two methods of dividing the air in the above example. 
 
 Splits. 
 
 Natural 
 
 Division. 
 
 Required 
 
 Division. 
 
 Horsepower. 
 
 Door 
 
 Regulator. 
 
 Box 
 
 Regulator. 
 
 Split A, 6 ft. X 9 ft., 8,000 ft. long 
 Split B, 5 ft. X 8 ft., 6,000 ft. long 
 Split C, 9 ft. X 9 ft,, 8,000 ft. long 
 Split D, 6 ft. X 8 ft., 10, 000 ft. long 
 
 Totals 
 
 28,277 
 
 22,360 
 
 47,423 
 
 21,940 
 
 40,000 
 
 40.000 
 
 10.000 
 30,000 
 
 64.145 
 102.582 
 .356 
 1 44.955 
 
 102.582 
 
 102.582 
 
 25.645 
 
 76.936 
 
 120,000 
 
 1 
 
 120,000 
 
 212.038 
 
 307.745 
 
 
378 
 
 VENTILATION OF MINES 
 
 SPLITTING FORMULAS 
 
 The following table of formulas will serve to illustrate the methods of 
 calculation in splitting. The example assumes the same airway as that given 
 on page 369 and used to illustrate the table of formulas, page 370, but the air- 
 current is divided, as specified in the table: 
 
 Natural Division. 
 
 Primary Splits— Split (1) = 4 ft. X 5 ft., 800 ft. long. Split (2) = 4 ft. X 5 ft., 
 1,200 ft. long. 
 
 To Find: 
 
 No. 
 
 Formula. 
 
 Specimen Calculation. 
 
 Potential for 
 pressure. 
 
 35 
 
 Y n 1 a 
 
 x *~ a \lks' 
 
 U ? = (Xx + X 2 
 + etc.). 
 
 (1) 20 ^ 00000 oo 217x14 , 4 oo 5 ’ 060 ' 
 
 / 20 _ . 
 
 (2) 20 0000000217 X 2b 600 
 
 5,060 + 4,131 = 9191. 
 
 Natural divi- 
 sion. 
 
 41 
 
 X v 
 
 q ~ 2 X p X Q ' 
 
 (1) X 10,000 = 5,505 cu. ft. 
 
 (2) ^lli x 10,000 = 4,495 cu ‘ ft ' 
 
 Or the natural division may be calculated from the pressure at the mouth 
 of the several splits by using formula (23); thus, 
 
 
 23 
 
 q^X p \/ p. 
 
 (1) 5,060 ]/ 1.1838 = 5,505 cu. ft. 
 
 (2) 4,131 V 1.1838 = 4,495 cu. ft. 
 See formula (42). 
 
 Pressure. 
 
 42 
 
 Q 2 
 
 P (2 X p )* 
 
 
 Power. 
 
 43 
 
 Q 3 
 
 (zx„r 
 
 
 Quantity. 
 
 44 
 
 45 
 
 Q = S.X p y'p. 
 
 <2 = X p )-w. 
 
 9,191 V 1.1838 = 10,000 cu. ft. 
 1^9, 191 2 X 11,838 = 10,000 cu. ft. 
 
 Increase of 
 quantity due 
 to splitting. 
 (Pressure con- 
 stant. ) 
 
 46 
 
 2 X p 
 
 Q = -X 
 
 •A-p — O 
 
 Q 191 
 
 i~j~ X 10,000 = 28,722 cu. ft. 
 
 Increase in 
 quantity due 
 to splitting. 
 (Power con- 
 stant.) 
 
 47 
 
 
 
 & 
 
 It 
 
 10,00 °-V(|^) 2 = 20,205 CU,ft 
 
SPLITTING FORMULAS 
 
 379 
 
 500 ft. long. 
 
 Secondary Splits. — (1) 4 ft. X 5 ft., 800 ft. Jong. (2) 4 ft. X 5ft 
 (3) 4 ft. X 5 ft., 400 ft. long. (4) 4 ft. X 5 ft., 300 ft. long. 
 
 The calculation is often shortened, when many splits are concerned, by 
 using the relative potential , omitting the factor A;; but the final result must 
 then be multiplied by k to obtain the pressure or power; or, these factors 
 must be divided by Ac, when finding the quantity, as in formulas (49) to (51). 
 
 tH 
 
 Oi 
 
 O 
 
 rH 
 
 
 
 iO 
 
 iO 
 
 o3 jq 
 
 
 
 
 fH O 
 
 P*5S 
 
 fH 
 
 3 
 
 t « 
 
 fH* 
 
 <D 
 
 >3 
 
 p 
 
 55 
 
 09 
 
 0> 
 
 M 
 
 £ 
 
 o 
 
 I 
 
 to* 
 
 P. 
 
 P 
 
 <? 
 
 > a 5 2 
 *3 o.o3 2 
 
 pj* Ph’43 Pi 
 
380 
 
 VENTILA TIOX OF MIXES. 
 
 Proportionate Division. 
 
 Primary Splits (only).— (1) 4 ft. X 5 ft., 800 ft. long = 3,500 cu. ft. (2] 
 4 ft. X 5 ft., 1,200 ft. long = 6,500 cu. ft. 
 
 To Find: 
 
 No. 
 
 Formula. 
 
 Specimen Calculation. 
 
 Pressure due to 
 friction. 
 
 13 
 
 (f 
 
 P X 2* 
 
 (1) S--"- 
 
 (2) IS = 2 - 4757,b - 
 
 To accomplish this division of air, the pressure in split (1) must be 
 increased by means of a regulator to make it equal to the pressure in the 
 free or open split (2), and, hence, the pressure due to the regulator is 
 equal to the difference between the natural pressures in these splits. 
 
 Pressure due to 
 the regulator 
 in split (1). 
 
 53 
 
 P = P-2 — Pi- 
 
 2.4757 — .47845 = 1.99725 lb. 
 
 Area of the 
 opening in 
 regulator. 
 
 37 
 
 , _ .0004 q 
 ^ /— * 
 V * 
 
 .0004 X 3,500 _ 2 259 gq ft 
 1.99725 
 \ 5.2 
 
 Secondary Splits.— (1) 4 ft, X 5 ft., 800 ft. — 3,500 cu. ft. (2) 4 ft. X 5 ft., 
 500 ft. — 6,500 cu. ft. (3) 4 ft. X 5 ft., 400 ft. — 4,000 cu. ft. (4) 4 ft. X 5 ft., 
 300 ft. — 2,500 cu. ft. 
 
 Note— When using the relative potential, multiply the result by k, to 
 obtain the pressure, or the power. 
 
 Pressure due to 
 friction. Free 
 split-second- 
 ary pressure. 
 
 13 
 
 *8 
 
 11 J 
 
 (1) .0000000217 2 =- -47848 lb. 
 
 (2) .0000000217 = 1.0314 lb. 
 
 (3) -0000000217 = -31248 lb - 
 
 (4) .0000000217 (£^)* = .0915461b. 
 
 V 1.2172 / 
 
 Since the natural pressure in (3) is greater than that in (4), (3) is the free 
 split, and its natural pressure is the pressure for the secondary splits. The 
 pressure for the primary splits is then found by first adding the pressures in 
 (2) and (3), and if their sum is greater than the natural pressure for (1), it 
 becomes the pressure for the primary splits, or the mine pressure. If the 
 natural pressure for (1) is the greater, this is made the free split, and its 
 natural pressure becomes the primary or mine pressure. In this case, the 
 secondary pressure must be increased by placing a regulator in split (3). 
 
 Primary or 
 mine pressure. 
 
 
 P 2 +P 3 . 
 
 1.0314 + .31248 = 1.34388. 
 
 Pressure due to 
 
 
 P3 — Pi- 
 
 (4) .31248 — .091546 = .220934 lb. 
 
 the regula- 1 
 tors. 
 
 
 (Pa + Pa)—Pi- 
 
 (1) (1.0314 + .31248) — .47848 
 
 = .86540 lb. 
 
 Areas of open- 
 ings in the 
 regulators. 
 
 37 
 
 i _ .0004 q 
 
 Vi 
 
 (4) • 0004 X 2 ' 500 - 4.85 1 4sa.ft. 
 
 /.220934 
 
 M 5.2 
 
 (1) '° 004 X 3,500 = 3.4328 sq. ft. 
 
 1.8654 
 \ 5.2 
 
METHODS AND APPLIANCES. 
 
 381 
 
 METHODS AND APPLIANCES IN THE VENTILATION 
 
 OF MINES. 
 
 Ascensional Ventilation.— Every mine, as far as practicable, should be venti- 
 lated upon the plan known as ascensional ventilation. This term refers 
 particularly to the ventilation of inclined seams. The air should enter the 
 mine at its lowest point, as nearly as possible, and from thence be conducted 
 through the mine to the higher points, and there escape by a separate 
 shaft, if such an arrangement is practicable. Where the seam is dipping 
 considerably and is mined through a vertical shaft, the upcast shaft should 
 be located as far to the rise of the downcast shaft as possible. The intake air 
 is then first conducted to the lowest point of the dip workings, which it 
 traverses upon its way to the higher workings. In the case of a slope 
 working where a pair of entries is driven to the dip, one being used as the 
 intake and the other the return, there being cross-entries or levels driven at 
 regular intervals along the slope, the air should be conducted at once to the 
 inside workings, from which point it returns, ventilating each pair of cross- 
 entries from the inside, outwards. Where the development of the cross- 
 entries or levels is considerable, their circulation is considered separately, 
 and a fresh air split is made in the intake at each pair of levels. In all 
 ventilation, the main point to be observed is to conduct the air-current first 
 to the inside workings, from whence it is distributed along the working face 
 as it returns toward the upcast. 
 
 General Arrangement of Mine Plan.— Every mine should be planned with 
 respect to three main requirements, viz.: (a) haulage; ( b ) drainage; (c) 
 ventilation. These requirements are so closely connected with one another 
 that the consideration of one of them necessitates a reference to all. The 
 mine should be planned so that the coal and the water will gravitate toward 
 the opening, as far as possible. There are many reasons, in the consideration 
 of non-gaseous mines, why the haulage should be effected upon the return 
 airways. The haulage road is always a dusty road, caused by the traveling 
 of men and mules, as well as by the loss of coal in transit, which becomes 
 reduced to fine slack and powder. If the haulage is accomplished upon the 
 intake entry or air-course, this dust is carried continually into the mine and 
 working places, which should be avoided whenever possible. When the 
 loaded cars move in the same direction as the return air, the ventilation of 
 the mine is not as seriously impeded. It is often the case that fewer doors are 
 required upon the return airway than upon the intake, which is a feature 
 favorable to haulage roads. Again, in this arrangement, the hoisting shaft 
 is made the upcast shaft, which prevents the formation of ice, and conse- 
 quent delay in hoisting in the winter season. The arrangement, however, 
 presupposes the use of the force fan or blower, since if a furnace or exhaust 
 fan is employed, a door, or probably double doors, would have to be placed 
 upon the main haulage road at the shaft bottom, which would be a great 
 hindrance. 
 
 In the ventilation of gaseous mines; however, other and more important 
 considerations demand attention. The gaseous character of the return 
 current prevents making the return airway a haulageway. In such mines, 
 the haulage should always be accomplished upon the intake air, as any other 
 system would often result in serious consequences. In such gaseous mine, 
 men and animals must be kept off the return airways as far as this is 
 possible. . 
 
 As far as practicable, ventilation should be accomplished m sections 
 or districts, each district having its own split of air from the main intake, and 
 its own return connecting with the main return of the mine. Reference 
 has been made to this under Distribution of the Air in Mine Ventilation. 
 This splitting of the air-current is accomplished preferably by means of an 
 air bridge, either an under crossing or an over crossing. There are, in 
 general, three systems of ventilation, with respect to the ventilating motor 
 employed: (a) natural ventilation; (b) furnace ventilation; (c) mechanical 
 / VC71/t'i/l/CLt'i071/ 
 
 Natural ventilation means such ventilation as is secured by natural means, 
 or without the intervention of artificial appliances, such as the furnace, 
 or any mechanical appliances by which the circulation of air is maintained. 
 In natural ventilation, the ventilating motor or air motor is an air column 
 that exists in the downcast shaft by virtue of the greater weight of the 
 downcast air. This air column acts to force the air through the airways 
 
382 
 
 VENTILATION OF MINES. 
 
 of the mine. An air column always exists where the 
 currents of air pass through a certain vertical height, and Afferent 
 
 temperatures This is the case whether the opening is a shaft or a slope, 
 since? in 'either case, there is a vertical height which ^ part determines 
 the height of air column. The other factor determining the hei § h ^ of mr 
 
 column is the difference of temperature between rfu^de^ticai 
 
 The calculation of the ventilating pressure m natural ventilation is identical 
 with that of furnace ventilation, which is described later. , lirnri 
 
 ventilation of Rise and Dio Workings.— We have referred to the air column 
 existing either in vertical shafts or slopes as the motive column or venti- 
 Mng motor Such an air column will be readily seen to exist m any rise 
 or din workings within the mine, and may assist or retard the circulation 
 of the air-current through the mine. It is this air column that renders the 
 ventilation of dip workings easy, and that of rise workings correspondingly 
 difficult depending, however, on the relative temperature of the intekeaim 
 return currents; the latter usually is the warmer of the two, i\hich gpwes 
 rise to the air column. The influence of such air columns m ^{. al ^ a * jbe 
 taken into account in the calculation of any ventilation. This is often 
 
 neglected • • 
 
 The influence of air columns in rise or dip workings, within the unne, 
 becomes very manifest where, from any reason, the mam intake current is 
 increased or decreased. For example, a mine is ventilated in two sphts, a 
 rise and a dip split; a current of 50,000 cu. ft. of air is passing m the mam 
 airwav, 30,000 cu. ft. passing into the dip workings, and 20,000 mto the roe 
 workings. A fall of roof in the main intake airway, or other cause,reduces 
 the main current from 50,000 to 35,000 cu. ft. Instead, now, of 21 , 
 going to the dip workings and 14,000 to the nse workings, we find that this 
 proportion no longer exists, but that the dip workings are taking more than 
 their proportion of air, and the rise workings less. Thus the emulation 
 being decreased to 35,000 cu. ft., the dip workings will ^ k h e Q ?°Vh^ 
 
 cu. ft., and the rise workings 10,000 cu. ft. On the other hand, had the 
 intake current been increased instead of decreased, the workings ^ would 
 
 then take more than their proportion, while the dip workings would take 
 less The reason for this distribution is evident; suppose, for example, the 
 intake or mine pressure is 3 in. of water gauge, and m 
 there is k in of water gauge acting to assist ventilation, whilea like water 
 gauge ofi in in the ri^ workings lets to retard ventilation The effective 
 water auge in the dip workings is therefore 3, in., while the effective 
 w a nue in the rise workings is 21 in., or they are to each other as 7 . 5. 
 If, now, the mine pressure is decreased to, say, 2 in., the^ effective nse and 
 diD pressures will be, respectivelv, 2£ in. and 11 in., or as o : 3. e observe, 
 before the decrease, the dip pressure was h or 1.4, times the i ^ P^^f e ’ 
 while after the decrear iok place m the mine pressure, the dip pressure 
 became for 1 M times the rise pressure. The relative quanUtiespa^ng m 
 the dip split before and after the decrease took placets compared with the 
 
 quantities passing in the rise split, will be as the V 1.4 : j/l.66, ?bowing an 
 increase of proportion. Now, instead of a decrease taking piacemthemme 
 pressure, let us suppose it is increased , say, from 3 in. to 4 m. The eff^tive 
 pressures in t 1 lip and rise workings will then be, re ^ c k vel ^Jj “■ 
 and 3£ in., or they will be to each other as 9:7, instead of 7. 5. Her e we 
 observe that the dip pressure is If, or 1.15, times the nse pr^sure, instead 
 of 1.4. The relative quantities, therefore, passing in the dip split, before 
 and after the increase of the mine pressure, as compared with the quantities 
 
 passing in the rise split, will be in the ratio of V 1.4 : l/l.lo. 
 decrease of proportion. We observe that any alteration of the mm e 
 sure bv which it is increased or decreased does not affect the inside dip or 
 rise columns, and hence the disproportion obtains. In case of a dec^^ of 
 the mine pressure, the dip workings receive more than p^^rtion 
 of air, and in case of an increase of the mine pressure, they receive less 
 
 tha i nfl uence of^Msons^-In any ventilation, air columns are always established 
 in slopes and shafts, owing to the relative temperatures of the outade and 
 inside 1 air. The temperature of the upcast, or return column, may always be 
 assumed to be the same as that of the inside air The of the 
 
 downcast, or intake column, generally approximates the tempw^^eof the 
 outside air. although, in deep shafts or long slopes, this temperature imy be 
 Changed considerably before the bottom of the shaft or slope is reached, and 
 
METHODS AND APPLIANCES 
 
 383 
 
 consequently the average temperature of the downcast or intake, is often 
 different from that of the outside air. The difference of temperatures will 
 also vary with the season of the year. In winter the outside temperature is 
 below that of the mine, and the circulation in shafts and slopes is assisted, 
 since the return columns are warmer and lighter than the intake columns 
 for the same circulation. In the summer season, however, the reverse of 
 this is the case. The course of the air-current will thus often be changed. 
 When the outside temperature approaches the average temperature of the 
 mine, there will be no ventilation at all in such mines, except such as is 
 caused bv accidental wind pressure. „ . . , 
 
 In furnace ventilation the temperature of the upcast column is increased 
 above that of the downcast column by means of a furnace. The chief 
 points to be considered in furnace ventilation are in regard arrange- 
 
 ment and size of the furnace. Furnace ventilation should n<?t be applied to 
 gaseous seams, and in some cases is prohibited by law. It is, however, in 
 use in manv mines liberating gas. In such cases the furnace fire is fed by a 
 current of air taken directly from the air-course sufficient to maintain the 
 fire, and the return current from the mine is conducted by means of a dumb 
 drift, or an inclined passageway, into the shaft at a point from 50 to 100 ft. 
 above the seam. At this point, the heat of the furnace gases is not sufficient 
 for the ignition of the mine gases. The presence of carbonic-acid gas in the 
 furnace gases also renders the mine gases inexplosive. In other cases 
 where the dumb drift is not used, a sufficient amount of fresh air is allowed 
 to pass into the return current to insure its dilution below the explosive 
 point before it reaches the furnace. . _ „ 
 
 Construction of a Mine Furnace.— In the construction of a mine furnace, a 
 sufficient area of passage must be maintained over the fire and around the 
 furnace to allow the passage of the air-current circulating in i the mine. The 
 velocity of the current at the furnace should be estimated not to exceed 
 20 ft. per second, and the entire area of passage calculated from this velocity. 
 Thus for a current of 50,000 cu. ft. of air per minute, the area of passage 
 through and around the furnace should be not less than 
 
 50,QQ - = 41f sq. ft. 
 
 60 X 20 3 H 
 
 This is a safe method of calculation, notwithstanding the fact that the 
 velocity of the air is often much more than 20 ft. per second, yet the volume 
 of the air is largelv increased owing to the increase of temperature. 
 
 The length of the furnace bars is limited to the distance in which good 
 firing can be accomplished, and should not exceed 5 ft. 1 he width of the grate 
 will therefore determine the grate area. The grate area must, m every case, 
 be sufficient for the heating of the air of the current to a temperature such 
 as to maintain the average temperature of the furnace shaft high enough to 
 produce the required air column, or ventilating pressure, in the mine. 
 The area A of the grate of the furnace is best determined by the formula 
 
 A = x H. P., in which A = grate area in square feet; H. P. = horse- 
 •\[ D 
 
 power of the circulation; and D = depth of shaft in feet. The horsepower 
 for any proposed circulation may always be determined by divining the 
 quantity of air (cubic feet per minute) by the mine potential Xu, and cubing 
 and dividing the result by 33,000; thus, 
 
 3 1 
 
 H. P 
 
 -m 
 
 X 
 
 33,000' 
 
 The furnace should have proper cooling spaces above and at each side; 
 upon one side, at least, should be a passageway or manway. The furnace 
 should be located at a point from 10 to 15 yd. back from the f 90 t of the shafh 
 at a place in the airway where the roof is strong. This is well secured 
 by railroad iron immediately over the furnace. A good foundation is 
 obtained in the floor, and the walls of the furnace carried up above the 
 level of the grate bars, when the furnace arch is sprung. If possible, a 
 full semicircle should be used in preference to a flat arch. The sides ana 
 arch of the furnace should be carried backwards to the shaft; this is 
 necessary in order to prevent ignition of the coal. The walls and arcn 
 are constructed of firebrick a sufficient distance from the furnace, and after- 
 wards of a good quality of hard brick; the shaft is also lined with brick 
 or protected by sheet iron a sufficient height to prevent the ignition ot 
 the curbing. 
 
38a 
 
 VENTILATION OF MINES. 
 
 Air Columns in Furnace Ventilation.— As previously stated, natural ventilation 
 and furnace ventilation are identical, m so far as in each the ventilating 
 motor is an air column. This air column is an imaginary column of air 
 whose weight is equal to the difference between the weights of the upcast 
 and downcast columns. The upcast and downcast columns in furnace 
 ventilation are sometimes referred to as the primary and secondary 
 columns, respectively. The primary or furnace column is, in nearly every 
 case, a vertical column, and consists of a single air column whose average 
 temperature is easily approximated. According to the manner of opening 
 the mine, whether by shaft, slope, or drift, the secondary column may be 
 a vertical column in the shaft, an inclined column in the slope, or an outside 
 air column in case of a drift opening. Again, it is to be observed that in 
 case of a slope opening where the top of the furnace shaft is much higher 
 than the mouth of the slope, and the dip of the slope is considerable, the 
 secondary column consists of two columns of different temperatures, an 
 outside air column and the slope column. These two parts of the secondary 
 column must be calculated separately, and their sum taken for the weight of 
 the secondary column. The level of the top of the furnace shaft determines 
 the top of both the primary and secondary columns, whether these columns 
 are in the outer air or in the mine. The weight of the upcast or primary 
 column is largely affected by its gaseous condition. For example, if the 
 return current from the mine is laden with blackdamp C0 2 , its weight will 
 be much increased, since this gas is practically 1£ times as heavy as air, 
 while, if laden with marsh gas, or firedamp mix- 
 ture, its weight will be considerably reduced. 
 These causes decrease and increase, respectively, 
 the ventilating pressure in the mine. 
 
 Inclined Air Columns.— In a slope opening, the 
 air column is inclined; it is none the less, how- 
 ever, an air column, and must be calculated in 
 the same manner as a vertical column whose ver- 
 tical height corresponds to the amount of dip of 
 the slope. Fig. 7 shows a vertical shaft and a 
 slope, the air column in each of these being the same for the same tem- 
 perature. The air column in all dips and rises must be estimated in like 
 manner, by ascertaining the vertical height of the dip. 
 
 Calculation of Ventilating Pressure in Furnace Ventilation.— The ventilating 
 pressure in the mine airways, in natural or in furnace ventilation, is caused 
 by the difference of the weights of the primary and secondary columns. Air 
 always moves from a point of higher pressure toward a point of lower 
 pressure, and this movement of the air is caused by the difference between 
 these two pressures. In this calculation each column is supposed to have 
 an area of base of 1 sq. ft. Hence, if we multiply the weight of 1 cu. ft. of 
 air at a given barometric pressure, and having a temperature equal to the 
 average temperature of the column, by the vertical height D of the column, 
 we obtain not only the weight of the column but the pressure at its base due 
 to its weight. Now, since the ventilating pressure per square foot in the 
 airway is equal to the difference of the weights of the primary and secondary 
 columns, we write 
 
 1.3253 X B 1.3253 X B 
 
 Fig. 7. 
 
 P=( 
 
 X D. 
 
 459 At 459+ T 
 
 Example. — Find the ventilating pressure in a mine ventilated by a 
 furnace, the temperatures of the upcast and downcast columns being, 
 respectively, 350° F. and 40° F., the depth of the upcast and downcast shafts 
 being each* 600 ft., and the barometer 30 in. # 
 
 Substituting the given values in the above equation, we have 
 
 p = 1.3253 X30X600 lb ' P« ft ' 
 
 Calculation of Motive Column or Air Column. — It is often convenient to 
 express the ventilating pressure p (lb. per sq. ft.) in terms of air column or 
 motive column M, in feet. The height of the air column M is equal to the 
 
 pressure p divided by the weight w of 1 cu. ft. of air, or M = The expres- 
 sion for motive column may be written either in terms of the upcast air or 
 of the downcast air, the former giving a higher motive column than the 
 latter for the same pressure, since the upcast air is lighter than that of the 
 
MECHANICAL VENTILATORS. 
 
 385 
 
 downcast. As the surplus weight of the downcast column of air produces 
 the ventilating pressure, it is preferable to write the air column in terms of 
 the downcast air, or, in other words, to consider the air column as beiijg 
 located in the downcast shaft, and pressing the air downwards and through 
 the airways of the mine. If we divide the expression previously given for the 
 
 which is the expression for motive column in terms of the downcast air. 
 
 If, on the other hand, we divide the expression for the ventilating 
 
 the upcast air. 
 
 Influence of Furnace Stack.— To increase the height of the primary or 
 furnace column, a stack is often erected over the mouth of the furnace shaft. 
 The effect of this is to increase the ventilating pressure in the mine in 
 proportion to the increased height of the primary column, and to increase 
 the quantity of air passing in the mine in proportion to the square root ot 
 this height. Thus, the square root of the ratio of the heights of the primary 
 column, before and after the stack is erected, is equal to the ratio of the 
 quantities of air passing before and after the erection of the stack. Or, 
 calling these quantities qi and q 2 , and the height of stack d, we have 
 
 A large number of mechanical ventilators have been invented and applied, 
 with more or less success, to the ventilation of mines. The earliest type of 
 ventilator was the wind cowl, by which the pressure of the wind at the sur- 
 face was brought to bear effectively upon the mine airways by the action of 
 a cowl whose mouth could be turned toward the wind; this was naturally 
 very unreliable. The waterfall was also extensively applied at one time, but 
 its application could only be made where there was a reliable source of 
 water supply, and where the drainage of the mine could be effected through 
 a tunnel, or where the mine opening could be placed in connection with 
 such a waterfall outside of the mine. Where these conditions are obtained, 
 as is the case in some mountainous districts, the waterfall is still in use, as 
 it is an effective means of ventilation, and is economical. Its application, 
 however, must be limited to the ventilation of small mines. The steam jet 
 is another mechanical device for producing an air-current in the mine. The 
 steam is allowed to issue from a jet at the bottom of an upcast shaft, and, 
 by the force of its discharge, causes an upward current in the shaft. Its use, 
 however, is very limited, and is practically restricted to the ventilation of 
 shafts while sinking. In this connection it may be mentioned, however, 
 that the discharged steam from the mine pumps, where practicable, may be 
 conducted into the upcast shaft; or the discharge pipe from the pumps may 
 be carried up the upcast shaft, its heat increasing the temperature of the 
 shaft, and thereby increasing the motive column and the ventilation. 
 
 Fan Ventilation.— Mechanical motors of this type present two distinct 
 modes of action in producing an air-current: (a) by propulsion of the air; 
 and (b) by establishing a pressure due to the centrifugal force incident to 
 the revolution of the fan. Fans have been constructed to act wholly on 
 one or the other of these principles, while others have been constructed to 
 act on both of these principles combined. 
 
 Disk Fans. — The action of this type of fan resembles that of a windmill, 
 except that in the latter the wind drives the mill, while in the former the 
 fan propels the air or produces the wind. This type of fan consists of a 
 number of vanes radiating from a central shaft, and inclined to the plane of 
 revolution. The fan is set up in the passageway between the outer air and 
 the mine airways. Power being applied to the shaft, the revolution of the 
 
 MECHANICAL VENTILATORS. 
 
386 
 
 VENTILATION OF MINES. 
 
 vanes propels the air, and produces a current in the airways. The fan may 
 force the air through, or exhaust the air from, the airways, according to the 
 direction of its revolution. This type of fan is most efficient under light 
 pressures. It has found an extensive application in mining practice, and 
 has a large number of devotees, but has been replaced to a large degree 
 in the ventilation of extensive mines. This type of fan acts wholly by 
 
 PJ °(Sntrifugal fans include all fans that act solely on the centrifugal principle, 
 and those that combine the centrifugal and propulsion principles. The 
 action of the fan, whether by centrifugal force alone, or combined with 
 propulsion, depends on the form of the fan blades. In this type of fan, the 
 blades are all set at right angles to the plane of revolution, and not inclined, 
 as in the disk fan just described. The blades may, however, be either radial 
 blades, sometimes spoken of as paddle blades, or they may be inclined to the 
 radius either forward in the direction of revolution, or backward. When 
 the blades are radial, the action of the fan is centrifugal only. The inclina- 
 tion of the blades backward from the direction of motion gives rise to an 
 action of propulsion, in addition to the centrifugal action of the fan. Ihe 
 blades in this position may be either straight blades in an inclined position^ 
 as in the original Guibal fan, or they may be curved backward in the form 
 of a spiral, as in the Schiele and Waddle fans. - 
 
 Centrifugal fans may be (a) exhaust fans or (b) force fans of blowers. In 
 each, the action of the fan is essentially the same; 1. e., to create a difference 
 of pressure between its intake or central opening, and its discharge at the 
 circumference. The centrifugal force developed by the revolution of the air . 
 between the blades of the fan causes the air within the fan to crowd toward 
 the circumference; as a result, a depression is caused at the center and a 
 compression at the circumference, giving rise to a difference of pressure 
 between the intake and the discharge of the fan 
 
 Exhaust Fans.— If the intake opening of the fan be placed m connection 
 with the mine airways, and the discharge be open to the atmosphere, the 
 fan will act to create a depression in the fan drift leading to the mine, which 
 will cause a flow of air through the mine airways and into and through the 
 fan. In this case, the fan is exhausting, its position being ahead of the 
 current that it produces in the airway. The atmospheric pressure at 
 the intake of the mine forces the air or propels the current toward the 
 depression in the fan drift caused by the fan’s action. , 
 
 Force Fans and Blowers.-If the discharge opening of the fan he placed m 
 connection with the mine airways, a compression will result m the fan anit 
 owing to the fan’s action, and the air will flow from this point of compres- 
 sion through the airways of the mine, and be discharged into the upcast, 
 and thence into the atmosphere. The ventilating pressure m the case ot 
 either the exhaust fan or the force fan is equal to the difference of pressure 
 created by the fan’s action. In the former case, when the fan is exhausting, 
 the absolute pressure in the fan drift is equal to the atmospheric pressure 
 less the ventilating pressure, while in the latter case, when a fan is forcing, 
 the absolute pressure in the fan drift is equal to the atmospheric pressure 
 increased by the ventilating pressure. This gives rise to two distinct systems 
 of ventilation, known as (a) vacuum system and (b) plenum system. 
 
 Vacuum System of Ventilation.-In this system, the ventilation of the mine 
 is accomplished by creating a depression m the return airway of the rome. 
 This depression may be created by the action of an exhaust tan, as just 
 described, or by the action of a furnace. In either case, the absolute pres- 
 sure in the mine is below that of the atmosphere, or, we may say, the mine 
 is ventilated under a pressure below the atmospheric pressure. This system 
 has many points of advantage over the plenum system, and for years was 
 considered by many the only practicable system of ventilation. Its appli- 
 cation, however, is controlled by conditions in the mine with respect to 
 the gases liberated, the arrangement of the haulage system, etc. # 
 
 Plenum System of Ventilation.-In this system, the air-current is 
 through the mine airways by means of the compression or ventilating 
 pressure created at the intake opening of the mine. This ventilating pres- 
 sure may be established by a fan, waterfall, wind cowl, or . 
 mechanical means at hand . In this system, the absolute pressure m the mine 
 is above that of the atmosphere; or, as we say, the mine is ventilated under 
 a pressure above the atmospheric pressure. , 
 
 Comparison of Vacuum and Plenum Systems.— No hard-and-fast rule can be 
 made to apply in every case, as each system has its particular advantages. 
 
TYPES OF FANS. 
 
 387 
 
 In case of a sudden stoppage of the ventilating motor ,at a mine, there is 
 i-n vapiinm svstem a rise of mine pressure, instead of a tall, ana. tne 
 ^ IJe lack ’into the workings for a while, while, in the plenum 
 
 Inv stoDDaee of the ventilating motor is followed at once by a fall of 
 pressure iii the^ine, and mine gases expand more freely into _the passage - 
 the verv moment when their presence is most dangerous. This 
 3 must be caremily considered in the ventilation of deep workings In 
 shallow workings, the plenum system is often advantageous, especially if 
 there is a lar^e area of abandoned workings that have a vent or opening to 
 thfatmosnhCTe^ either Through an old shift or through crevices extending 
 to the surface 'Every crevice or other vent becomes a discharge opening by 
 whfchThe Sne gaS find their way to the surface, and the gases accum.i- 
 latine in the old workings are driven back into the workings, and find their 
 way'to'tte surface instead of being drawn into th « ;mlne amwaj as wotdd be 
 th please in an exhaust system. Any given fall of the barometer altects tne 
 expansion 1 of mine gases to a less extent in the plenum system than in 
 the vacuum system, but this small advantage would not give it consider- 
 ation^ dSerhiining between the adoption of the one or the other of these 
 two^vS^s^S^ard must be had, however, to other conditions more vital 
 than this In the ventilation of gaseous seams, owing to the necessity of 
 making the intake airway the haulage road, the exhaust system has usualiy 
 been adopted, as the main road is thereby left unobstructed by doors. 
 
 TYPES OF CENTRIFUGAL FANS. 
 
 We shall only mention the more prominent types of fans that have been 
 or are still in use, giving the 
 characteristic features, as nearly 
 as possible, of each fan. Many 
 fans have been built, however, 
 combining many of the features 
 that originally characterized a 
 single type of fan. . 
 
 Nasmyth Fan. — Fig. 8 is the 
 original type of fan representing 
 straight paddle blades radiating 
 from the center, which is its 
 characteristic feature. This was 
 probably the earliest attempt to 
 apply the centrifugal principle 
 to a mine ventilator, and al- 
 though not recognized at the 
 
 . • j-1- _ DArn a AT 
 
 ‘some of the most essential principles in centrifugal 
 
 Illation It possessed certain disadvantages, 
 
 _ — j. — /.nntvni /Yr intake opening. The blades, also, were siraigni 
 
 throughout their entire length being normal 
 both to the inner and outer circles of the tan. 
 
 UOLIl IU LHC lliii ci a iici r . - . 
 
 and thus did not provide for receiving the air 
 without shock at the throat of the fan. The 
 depth of Nasmyth’s blades equaled one-half the 
 radius of the fan, which was, under ordinary 
 conditions of mine practice, far too great, and 
 gave the fan a low efficiency. 
 
 Biram’s Ventilator. - About 1850, Biram at- 
 tempted to improve upon the Nasmyth ventilator 
 bv reducing the depth of blade so that it was 
 but one-tenth of the radius. The blades were 
 straight, as in Nasmyth’s ventilator, but inclined 
 backwards from the direction of motion at a 
 considerable angle. A large number of these 
 blades were employed. This fan was run at a 
 considerable speed, but proved very inefficient. It depended ^o^e on the 
 effort of propulsion given to the air than on the 
 
 the depth of the blade was as much too small as that of Nasmyth s was too 
 great. The intake or central opening in this fan was as contracted as in 
 
 the Wad(He^ Venti?ator.— Intis’ fan, Fig. 10, the invents attempted to reenforce 
 the discharge pressure at the circumference against the pressure of the 
 
388 
 
 VENTILATION OF MINES. 
 
 atmosphere. The discharge took place all around the entire circumference 
 of the fan, which was entirely opened to the atmosphere. The blades were 
 curved backward from the direction of motion in spiral form. The width 
 of the blade decreased from the throat toward the circumference, so as to 
 present an inverse ratio to fhe length of radius. Thus, the area of passage 
 
 between the fan blades was 
 maintained constant from 
 the throat to the circumfer- 
 ence of the fan. The pur- 
 pose of this was to maintain 
 the velocity of the air 
 through the fan constant, 
 and to fortify the pressure 
 due to the fan against the 
 atmospheric pressure at the 
 point of discharge. The es- 
 sential features of the Wad- 
 dle ventilator were, there- 
 fore, curved blades tapered 
 toward the circumference, 
 and a free discharge into 
 the atmosphere all around 
 the circumference. This 
 type is the best type of the 
 open-running fan having no peripheral casing, and discharging air into 
 the atmosphere all around the circumference. 
 
 Schiele Ventilator.— This ventilator, Fig. 11, was constructed on the same 
 principles as the Waddle ventilator just described, but differed from the 
 latter, as the discharge was made into a spiral chamber surrounding the 
 fan and leading to an expanding or 6vase chimney. There was some advan- 
 tage in this feature, as it protected the fan against the direct influence of 
 the atmosphere, and reduced the velocity of discharge; but, in each of 
 these fans, the intake opening was contracted, and the depth of blade was 
 very great, yielding a comparatively low efficiency. 
 
 Guibal Ventilator.— The next important step in the improvement of centrif- 
 ugal ventilators was introduced by M. Guibal, who constructed a fan, 
 Fig. 12, embodying the features of the Nasmyth ventilator, with the addition 
 of a casing built over the fan to protect its circumference. This casing was, 
 however, a tight-fitting casing, and as such, differed very materially from the 
 Schiele casing. In the Guibal fan the blades were arranged upon a series of 
 parallel bars passing upon each side of the center and at some distance from 
 it. By this construction, the blades were not radial at their inner edge or 
 the throat of the fan. They were curved, however, as they approached the 
 circumference of the fan, so as to be normal or radial at the circumference. 
 
 The advantage of this construction was to give a strong skeleton or frame- 
 work to the revolving parts, and, further, each blade was inclined to the 
 radius at its inner extremity, the effect of which was to receive tbe air upon 
 the blade with less shock than was the case in the Nasmyth ventilator. The 
 intake or central opening, however, was very contracted, and the tight-fitting 
 
EFFICIENCY OF FANS. 
 
 389 
 
 casing about the circumference prevented the effective action of the fan 
 during a considerable portion of its revolution. The fan was supplied 
 with an 6vase chimney, which was a feature of the Schiele fen, but 
 vibration was so strong that a shutter was required at the cut-off below 
 
 the chimney, to prevent it. This shutter was made adjustable, and is known 
 as the Walker shutter , having been applied to the fan later , . , 
 
 The Guibal ventilator presents some important and valuable features 
 in the protecting cover, and in the blades meeting the outer circumference 
 radially, and in the air being received with less shock than before. On the 
 whole it has proved a very efficient ventilator, although much work is lost 
 by reason of its contracted central orifice and tight casing, where the same 
 
 is used. - . _ _ , , ,, 
 
 Murphy Ventilator.— Fig. 13 consists of twin fans supported on the same 
 shaft and set a few feet apart. Each fen receives its air on one side only, 
 the openings being turned toward each other. This ventilator is built with 
 a small diameter, and is run at a high speed. The blades are curved back- 
 wards from the direction of motion. The intake opening is considerably 
 enlarged; a spiral casing generally surrounds the fan, and in every respect 
 this fan makes an efficient high-speed motor. It has received considerable 
 favor in the United States, where it has been introduced into a large number 
 
 of mines. A ., , , , 
 
 Capell Ventilator.— Perhaps none of the centrifugal ventilators have been 
 as little understood in regard to their principle of action as the Capell fen. 
 The fan is constructed along the lines of the Schiele ventilator, but diners 
 from that ventilator in the manner of receiving its intake air and delivering 
 the same into the main body of the fen. Here, and revolving with it, is a set 
 of smaller supernumerary blades. These blades occupy a cylindrical space 
 within the main body of the fan, and are inclined to the plane of revolu- 
 tion so as to assist in deflecting the entering air through small ports or 
 openings into the main body of the fan, where it is revolved and discharged 
 at the circumference into a spiral space resembling that surrounding the 
 
 Schiele fan. The larger blades of this 
 fan are curved backwards as the Schiele 
 blades, but are not tapered toward the 
 circumference. The fan is capable of giv- 
 ing a high water gauge, and is efficient 
 as a mine ventilator. The space surround- 
 ing the fan is extended to form an ex- 
 panding chimney. The fan may be used 
 either as an exhaust fan or a blower. 
 The best results in the United States have 
 been obtained by blowers. In Germany, 
 where this fen is in general use, there are 
 no blowers. 
 
 The position of the fan, whether used as 
 an exhaust or blower, should be suffi- 
 ciently removed from the fan shaft to 
 avoid damage to the fan in case of ex- 
 plosion in the mine. Even in non-gaseous 
 mines, the fan should be located a short distance back from the shaft 
 mouth, to avoid damage due to settlement. Connection should be made 
 with the fan shaft by means of an ample drift, which should be deflected 
 into the shaft so as to produce as little shock to the current as possible. In 
 
390 
 
 VEXTILA TIOX OF MIXES. 
 
 ' — ** f 
 
 u~~\ 
 
 U] 
 
 F* 
 
 K] 
 
 
 „ , 
 
 A 
 
 1 
 
 case of gaseous seams, explosion doors should be provided at the shaft 
 mouth. The ventilator at every large mine should be arranged so that & 
 mav be converted from an exhaust to a blow-down fan at short notice. 
 This is managed bv housing the central onhees or intake of the fan m 
 such a manner as to connect them directly with the fan drift. A noor 
 ab. Fig. 15, is arranged at the foot of the expanding chimney, the latter 
 - ^ ^ ° hwng placed between the tan 
 
 and the shaft. This door, 
 when the fan is exhausting, 
 is in the lower position a b. 
 and then forms a portion, of 
 the spiral casing leading to 
 the chimney. When the fan 
 is blowing, however, the door 
 is swung upwards so as to oc- 
 cupy the position ae . being 
 tangent to the cut-off at c, 
 thereby closing the discharge 
 into the chimney and causing 
 it to enter the fan drift behind 
 the door. At the same time, 
 the positions of the two doors, 
 e d and/d, in the fan drift, are 
 changed to c t and / s, respec- 
 tively, to open the fan drift 
 to the discharge from the fan, 
 and to close the openings lead- 
 ing from the fan drift to the 
 housing upon each side of the 
 fan. while another set of doors 
 A A upon each side of the fan. 
 in the housing, which were 
 -Prr- u previously cl:sed rigidly .are 
 
 now set wide open to admit 
 the outside air to the intake openings of the fan. The fan is thus made to 
 draw it- air from the atmosphere, and discharge it into the fan drift, instead oof 
 drawing its air from th e fan drift and discharging into the chimnev. as before. 
 
 The manometrical efficiency of a fan is the ratio between its effective and 
 theoretical pressures. It has-been assumed that the theoretical PJ^iredue 
 
 to the fan’s action is given by the equation h = — , or i = pXlooo » 
 
 u hpincr a « before the tangential speed feet per second), and g the force of 
 ™?iP?V3AlCd: h = head of air column in feet: i = water gauge m inches. 
 ~ The term mechanical efficiency, as applied to the ventilator, ^btie ratm 
 between its effective and theoretical powers. In estimating the e ^ c J^ncv of 
 a ventilator, it is customary, though incorrect, to estimate the theoretical 
 power of the fan from an engine card taken from the steam 
 to en°ine The efficiency of the steam engine is thus confused with the 
 efficiencyof the ventilat6r. Mr. Beard gives the following formula for 
 
 the theoretical work of the to per minute: U = .001699^- 1 V&b 
 
 in which m = ratio between outer and inner diameters of { ^(D = md). 
 
 y _ velocitv i feet per minute) of air m fan drift: R — outer radim of 
 fan blades (feet) r b = breadth of fan blades 1 feet); n == number of reyolu- 
 of fan per minute. If we divide the power upon the air. as determined 
 hv^hfexprSdoTqp bv the theoretical work given in the last eqnanon we 
 cffitaffi thev^lue of the ’coefficient of efficiency According to tins formula 
 the efficiencv of the ventilator changes with the speed. pnt 
 
 speed increases, but not in the same ratio. An expression for the^coe^ient 
 
 of efficiency of a ventilator is given by Beard as follows: K = - ^ + 163>6 o^ 
 TbP factor c is a constant of design whose value may vary from 2 to 7. but 
 en^o Stfc res^Jmw its e^^yforpa.^ 
 
 the^mor P a£dwith straight radial blades having only a forward 
 
FAN CONSTRUCTION. 
 
 391 
 
 curve at the lip of the blade to avoid the shock of the entry air against the 
 revolving blades, and the spiral casing starting a short distance upon the 
 cut-off and extending uniformly around the circumference of the fan, 
 the value of this constant may be 2 or 3. Where none of these accessories 
 to the efficiency of the fan is employed, the value of c may be as high as 7. 
 
 FAN CONSTRUCTION. 
 
 Size of Central Orifice.— The velocity of the intake should vary between 
 1,000 ft. and 1,500 ft. per minute, while 1,200 ft. may be used for fan calcula- 
 tions. If d = dia meter of opening, and q = quantity of air passing per 
 
 minute, d = ^ ^0 x 7 ? 854 fOT sin S le - intake faIls ’ and d = V^400 X .7854 
 for double-intake fans. 
 
 Upon entering the fan the air travels in a radial direction; this change of 
 direction is accompanied by a slight reduction of the velocity, hence the 
 throat area of the fan must be slightly in excess of the intake area. The 
 throat is the surface of the imaginary cylinder that has for its two bases the 
 two intake openings of the fan, and for its length the width of the fan, = 
 ndb. [The throat area is commonly made 1.25 times the total area of the 
 intake orifices, which gives for breadth of blade b = f d for double intake, 
 and b = T % d for single intake. — Beard.] 
 
 Diameter of Fan.— Murgue assumes the tangential velocity of the blade 
 tips (it) to create a depression double that due to the velocity as expressed 
 
 by the equation H = — , or if the manometrical efficiency = K , and the* 
 
 rfi la h 
 
 effective head produced = h, h = K H = K — , or m= ^ If' From this 
 
 equation, the tangential velocity (feet per second) may be calculated for 
 any given effective head h. This effective head h is the head of air column 
 effective in producing the circulation in the airway. To convert the 
 effective head of air column into inches of water gauge (£), we have 
 
 h = ^ 2 )<°i2 Hav ^ 11 8' f° un( l the tangential speed required in feet per 
 
 second, this is multiplied by 60, to obtain the speed in feet per minute, 
 and dividing this result by the desired number of revolutions per minute, or 
 the desired speed of the ventilator, the outer circumference of the fan 
 blades is obtained. No reference is made in the equation to the quantity 
 of air in circulation, which is determined from the equivalent orifice of 
 
 the mine and of the fan by the equation V — an * w hich 
 
 V = volume of air (cu. ft. per sec.); a = equivalent orifice of the mine: 
 o = the equivalent orifice of the fan. M. Murgue also uses the equation 
 
 h = — — , and suggests that the value of K for any particular type of 
 
 machine should be first decided, after which the tangential speed required 
 to produce any given effective head of air column ( h ) is easily calculated 
 
 from the formula u 
 
 4 
 
 Igh 
 
 method 
 
 The breadth of the blade is left largely to 
 
 judgment, while this method of calculation gives the same size of fan 
 for any given effective head desired, regardless of the quantity of air to 
 be circulated, which is the same as saying that the ventilator will present 
 the same efficiency when a large amount of air is crowded through its 
 orifice of passage as when a smaller amount of air is necessary. 
 
 Mr. Beard uses the following formulas for determining the several dimen- 
 sions of a ventilating fan: 
 
 m 
 
 = a/_J 
 
 \ nV V ' 
 
 c + 
 
 163, 600 2 \ 
 X 3 j 
 
 Q 
 
 n 2 i/ V 
 
 385,000,000 p 
 (m 3 — l)n 2 K V^IV 
 
 + i; 
 
 D = 
 
 e = 
 
 m 
 
 |/ 77Z 3 — 1 
 
 i/m 3 — 1 
 170 m 
 
 3,770 V p . 
 n i/K*v' 
 4,'X 3 K* V 
 
392 
 
 VENTILATION OF MINES. 
 
 in which m = — , which is the ratio between the outer diameter of the fan 
 a 
 
 blades D and the inner diameter of the blade d , which equals the diameter of 
 the intake orifice; b = width of fan blade; e = expansion of spiral casing at 
 
 point of cut-olf. , . , . ,. . , 
 
 The other symbols stand for the same quantities as previously indicated. 
 
 Curvature of Blades.— It was at one time supposed that the curvature of 
 the blades should be such that the radial passage of the air-current would 
 be undisturbed by the revolution of the fan; but fans constructed on this 
 principle gave no adequate results, and the theoretical spiral thus developed 
 was entirely abandoned. A certain curvature of the blade backward, 
 however, is assumed by many to increase the efficiency of the fan. This 
 has not been proved in practice, but the effect of the backward curvature 
 appears simply to necessitate a higher speed of revolution in the fan, in 
 order to obtain the same results as are obtained with radial blades. 
 
 The Guibal blade, radial at its outer extremity, or normal to the outer 
 circumference, and curved forward in the direction of motion, atits inner 
 extremity, so that the lip of the blade approaches tangency to the throat 
 circle, seems the most effective blade in centrifugal ventilation. 
 
 Tapered Blades.— The object of the taper is to produce a constant area of 
 passage from the throat to the circumference of the fan, and thus prevent 
 the reduction of the velocity of the current in its passage through the 
 fan This feature presents an attempt similar to that attempted by the 
 curvature of the blades, to hasten the passage of the air through the fan. 
 It has not been proved, however, to have produced any beneficial result, 
 except in the strengthening of the discharge pressure against the atmos. 
 pheric pressure, in open-running fans. On the other hand, the slowing up 
 of the air in its passage through a covered fan has by no means been 
 proved a detriment, but is assumed by many to be an advantage, inasmuch 
 as the air thus remains longer within the influence of the fan blades. 
 
 The number of blades depends on the size of the fan. An increased number 
 strengthens the fan’s action at the circumference, or supports the air at that 
 point, and thus prevents the backlash or the reentry of air into the fan. due 
 to the eddies occurring at the circumference when the blades are too iar 
 apart. To a certain extent, the number of blades is modified by the speed 
 of revolution, high-speed motors requiring a somewhat lesser number, 
 while low-speed motors require more. In any case, the number of blades 
 should not be so great as to abnormally increase the resistance to the air- 
 current. In general, the distance upon the outer circumference from tip 
 to tip of the fan blades should be from 2 to 3 times the depth of the blade. 
 
 The spiral casing gradually reduces the velocity of the air and reduces the 
 shock incident to the discharge of the air into the ati^sphere. The spiral 
 casing should be so proportioned that the velocity of the flow from the fan 
 blades will be maintained constant around the entire circumference, and 
 this should not be less than the velocity of the blade tips. The expansion e 
 of the casing at the cut-off should be such as to provide a velocity of the 
 air at this point equal to the velocity of the blade tips, according to the 
 
 equation e = -|-v. in which T> = diameter of fan; n = number revolu- 
 
 1 it Dnb 
 
 tions per minute; b = breadth of fan blade. , . . 
 
 The evase chimnev reduces the velocity of the air, as it is discharged into 
 the atmosphere to The chimney should be sufficiently high 
 
 to protect toe fan from the effect of high winds, but should not extend too 
 far above the fan casing, the point of cut-off b ® i ^ 1 ® 1 ^ t ®ij Delow thlS ’ a 
 about the level of a tangent to the throat circle at its lower side. 
 
 High-Speed and Low-Speed Motors— The question of speed of the jrenti 
 latin! motor is largely an open one, inasmuch as the same work maybe 
 performed by a small ventilator running at a high speed as is performed by 
 a large ventilator running at a low speed. . 
 
 It is important to design a mine ventilator at a speed &™h for* his teen 
 its being- increased in case of emergency. If the ventilator has oeen 
 designed at a high speed, a demand for an increase of speed cannot be met 
 as rladily as whin the ventilator is designed at a ^^m^ 
 other words, the exigencies of mine ventilation demand that a ventilator 
 shall be capable of greatly increased speed. 
 
 Fan Tests.— A large number of fan tests have been made, from time t 
 
CONDUCTING AIR-CURRENTS. 
 
 393 
 
 time, on different types of fans and under different conditions, with respect 
 to the resistance against which the fan is operated, and the quantity of air 
 required, and the speed of the ventilator. The experiments have resulted, 
 to a large extent, in tabulating a mass of contradictory data. The condi- 
 tions that affect the yield of the centrifugal ventilator are so numerous, and 
 the tabulation of the necessary data has been so often neglected m these 
 experiments, as to render them practically useless for the purpose of 
 scientific investigation. In conducting a reliable fan test, the following 
 points should be observed: (1) Take the velocity, pressure, and temperature 
 of the air at the same point in the airway, as nearly as practicable. This 
 point should be selected near the foot of the downcast shaft, or m the fan 
 drift at a suitable distance from the fan, to avoid oscillations of pressure and 
 velocity. (2) The area of the fan drift should be uniform for a suitable 
 distance in each direction from the point of observation, and tins area 
 should be carefully measured. (3) Take the anemometer readings at differ- 
 ent positions in the airway, so as to obtain an average reading over the entire 
 sectional area. Do not interpose the body in this area so as to decrease the 
 sectional area of the airway. (4) Take outside temperature of the air and 
 the barometric pressure at the time of making the test. (5) The intake and 
 discharge openings of the fan should be protected against wind pressure. 
 (6) At least three observations should be made, at as many different speeds 
 of the ventilator, and the number of revolutions of the fan carefully observed 
 and recorded for each observation. . 
 
 Mr. R. Van A. Norris (Trans. A. I. M. E., Yol. XX, page 637) gives the 
 results of a large number of experiments performed upon different mine 
 ventilating fans. This table, like all other tabulated fan tests, shows a large 
 amount of contradictory data. The conclusions drawn by Mr. Norris from 
 these tests are interesting and would be given here excepting that they 
 might be misleading if considered apart from the description of the experi- 
 ments and the discussion leading up to the conclusions. 
 
 CONDUCTING AIR-CURRENTS. 
 
 Doors.— A mine door is used for the purpose of deflecting the air-current 
 from its course in one entry so as to cause it to traverse another entry, at the 
 same time permitting the passage of mine cars through the first entry. The 
 essential points in the construction of a mine door are that it shall be bung 
 from a strong door frame in such a manner as to close with the current. The 
 door should be hung so as to have a slight fall. If necessary, canvas flaps 
 may be supplied to prevent leakage around the door, and particularly at the 
 bottom. Double doors are used on main entries at the shaft bottom, or at 
 any point where the opening of the door causes a stoppage of the entire cir- 
 culation of the mine. Such doors should be placed a sufficient distance 
 apart to allow an entire trip of mine cars to stand between them, so that one 
 of the doors will always be closed while the other is open. 
 
 Stoppings.— Stoppings are used to close break-throughs that have been 
 made through two entries, or rooms, for the purpose of maintaining the cir- 
 culation as the workings advance; also to close or seal off abandoned rooms 
 or working places. Stoppings must be air-tight and substantially built. A 
 good form of stopping is constructed by laying up a double wall of slate, 
 having about 8 or 10 in. of space between the two walls. This space is filled, 
 as the building progresses, with dirt taken from the roadways, or other fine 
 material. In the building of stoppings to seal off* mine fires, it is important 
 to begin the work at the end nearest the return air, and work toward the 
 intake end, which should be sealed off last. This method avoids the danger 
 of an explosion occurring within the workings that are being sealed off, as 
 the necessary dilution of the gases within is accomplished by the fresh air- 
 current, until the intake is finally sealed. Where the intake is sealed first, 
 an explosion is almost inevitable, as has been proved in many instances. 
 
 Air Bridges.— An air bridge is a bridge constructed for the passage of air 
 across and over another airway, this being called an overcast; or, the cross- 
 ing may be made to pass under the airway, this being called an unaercast. 
 In almost every instance, overcasts are preferable to undercasts for several 
 reasons. An undercast is liable to be filled with water accumulating from 
 mine drainage; it is also liable to fill with heavy damps from the mine, when 
 the ventilation is sluggish, and to offer considerable resistance to the free 
 passage of the air-current. An undercast can never be maintained as air- 
 tight as an overcast, on account of the continual travel through the 
 
394 
 
 HOISTING AND HAULAGE. 
 
 haulageway or passageway leading over it. This continual passing over 
 the bridge causes a fine dust to sift into the airway and mingle with the 
 air-current. All these objections are overcome in the construction of the 
 overcast 
 
 An air brattice is any partition erected in an airway for the purpose of 
 deflecting the current. A thin board stopping is sometimes spoken of as a 
 brattice; but the term applies more particularly to a thin board or canvas 
 partition running the length of an entry or room and dividing it into two 
 airways, so that the air will be obliged to pass up one side of the partition 
 and return on the other side of the partition, thus sweeping the face of the 
 heading or chamber. Such a temporary brattice is often constructed by 
 nailing brattice cloth or heavy duck canvas to upright posts set from 4 to 6 ft. 
 apart along one side of the entry a short distance from the rib. 
 
 Curtains. — These are sometimes called canvas doors. Heavy duck, or 
 canvas, is hung from the roof of the entry to divide the air or deflect a 
 portion of it into another chamber or entry. Curtains are thus used very 
 often previous to setting a permanent door frame. They are of much use 
 in longwall work, or where there is a continued settlement of the roof, 
 which would prevent the construction of a permanent door; also, in tempo- 
 rary openings where a door is not required. 
 
 HOISTING AND HAULAGE. 
 
 HOISTING. 
 
 There are two general systems of hoisting in use: (a) Hoisting without 
 attempting to balance the load. In this system, the cage and its load are hoisted 
 by an engine and lowered by gravity. (b) Hoisting in balance. In this 
 system, the descending cage or a special counterbalance assists the engine to 
 hoist the loaded ascending cage. Hoisting in balance is usually effected by 
 the use of (1) double cylindrical drums; (2) flat ropes winding on reels; 
 (3) conical drums; (4) the Koepe system; (5) the Whiting system 
 
 1. Double cylindrical drums are widely used: they consist essentially of an 
 
 engine coupled directly or else geared to the common axis of the drums. 
 The drums are usually provided with friction or positive clutches, and 
 brakes, so that they can be run singly if desired, or the load can be lowered 
 by gravity and the brake. , _ , , , 
 
 2. Flat ropes wound on reels are sometimes used either for unbalanced 
 hoisting with a single reel or for balanced hoisting with a double reel. With 
 
 & the double reels, the load on 
 
 the engine is balanced 
 throughout the entire hoist, 
 for, as the rope is wound on 
 the reel, the diameter of the 
 reel is increased, and the 
 lever arm through which 
 the power of the engine is 
 applied is also increased and 
 the mechanical efficiency of 
 the hoisting system de- 
 creased. Thus, when the 
 cage is at the bottom of the 
 shaft and the entire weight 
 of the rope is out, giving the 
 maximum load to be hoisted, 
 the drum is of a minimum 
 diameter and the engine has, 
 therefore, its greatest lever- 
 age to start the load. A flat 
 rope has the advantage of 
 preventing fleeting, but its first cost, extra weight, wear, and difficulty of 
 repairing have prevented its very general adoption. 
 
 3. Conical Drums— A conical drum, Fig. 1, equalizes the load on an engine 
 just as a flat rope on a reel does. On account of the fleeting of the rope, 
 however, the drum must be set at a considerable distance from the shaft to 
 prevent the rope leaving the head-sheave. A tail-rope gives the most 
 
 Fig. 1. 
 
HOISTING. 
 
 395 
 
 perfect counterbalance, the weight of the cage and rope on each side being 
 exactly equal. 
 
 4. In the Koepe system. Fig. 2, one rope runs over and the other under 
 driving sheaves S. A tail-rope R is used, and the head-sheaves x , x' are 
 placed vertically and at such an 
 
 angle to each other that their 
 grooves and the groove in the dri- 
 ving sheave are in line. As the 
 main driving shaft is short, the en- 
 gines can be placed close together, 
 thus requiring a smaller foundation 
 and engine house than for a drum 
 hoist. The objection to the system 
 is the liability of the rope to slipping 
 about the driving sheave, and for 
 this reason a hoisting indicator can- 
 not be depended on. The system is 
 also inconvenient for hoisting from different 
 levels in the same shaft, and, in case of the rope 
 breaking, both cages fall to the bottom. 
 
 5. The Whiting system, Fig. 3, uses two narrow- 
 grooved drums placed tandem instead of a single- 
 driving sheave as is used in the Koepe system. 
 
 The rope passes from the cage A over a head-sheave, 
 under the guide sheave T and around the sheaves 
 if, F three times, then out to the fleet sheave C, 
 back under another guide sheave, and up over 
 another head-sheave to the cage B. The sheave M 
 is driven by a motor either coupled direct to its 
 shaft, or geared. The drums F and M are coupled 
 together by a pair of connecting-rods, like the 
 drivers of a locomotive, and this arrangement 
 makes it possible to utilize all the friction of both 
 drums to drive the rope. Thus a tail-rope is not depended on to produce 
 more friction, though one is generally used as a balance to the loads. 
 
 It is best to incline the follower sheave F from the vertical an amount 
 
 Fig. 2. 
 
 Q 
 
 n 
 
 equal in its diameter to the distance between 
 the centers of two adjacent grooves, the object 
 being to eliminate chafing between the ropes 
 around the drums and to prevent them from 
 running off by enabling the rope to run from 
 each groove in one drum straight to the proper 
 groove in the other. This throws the shaft and 
 crankpins out of parallel with those of the main 
 drum, but this difficulty is overcome by the 
 connections in the ends of the parallel rods. 
 
 The fleet sheave C is arranged to travel back- 
 wards and forwards, as shown by the dotted 
 lines, in order to change the working length of 
 
 the rope, whereby hoisting can be done from different levels in the 
 The power used for hoisting is generally steam for the main hoists, 
 tricity is, however, coming rapidly into use, particularly for smaller 
 
 shaft. 
 
 Elec- 
 
 hoists 
 
396 
 
 HOISTING AND HAULAGE . 
 
 and local Installations, and for main hoists in locations where fue* in- 
 expensive and water-power available. Gasoline engines are also being used 
 to an increasing degree, particularly for smaller hoists and in local installa- 
 tions, and they are said to give very satisfactory results. 
 
 PROBLEMS IN HOISTING. 
 
 To Balance a Conical Dram.— Having given the diameter of one end of a 
 conical drum, to determine the diameter of the other end that will equalize 
 the load on the engines. In Fig. 1. call total load at bottom A, empty cage 
 at top B, loaded cage at top C, empty cage plus rope at bottom!), small 
 diameter of dram z, and large diameter y; then. Ax — By = Cy — Vx. 
 
 Example.— In a shaft, the cage weighs 2 tons, the empty car 1 ton. the 
 loaded car 3 tons, and the rope 2 tons. What should be the small diameter of 
 a conical drum whose large diameter is 30 ft.? 
 
 (2 + 2 + 3)x - (2 + 1)30 = (2 + 3)30 - (2+ 2 + l)z, 
 or 7 x — 90 = loO — 5 z. 
 
 12z = 240, 
 x = 20 ft. 
 
 To Find the Size of the Hoisting Engine.— Let D = diameter of cylinder. 
 
 P = mean effective steam pressure in cylinders, r = ratio of stroke to diam- 
 eter of cylinder and i c = work per revolution required to be done: then, by 
 making one cylinder capable of doing the work, n = number of strokes, 
 u = work per minute (ft.-lb.). 
 
 D = 1®^. or 2) = 
 
 Example —What should be the size of the cylinders of a hoisting engine 
 that is to perform 152.580 ft.-lb. of work per revolution, if the mean effective 
 pressure is 45 lb. per sq. in. and the stroke of the piston is twice its diam eter ? 
 
 D = = 23 - 5in ' 
 
 To get up speed in a few seconds, more power than would be represented 
 by the load to be lifted is required. Mr. Percy gives the following rule for 
 t his case: In a properly balanced winding arrangement, with uniform load, 
 multiplv the weight of coal in pounds by the average speed of the cage m 
 feet per minute: add one-half to cover the frictional resistances, and call 
 ihat the load. Then the power that must equal this must be the average 
 effective pressure of steam in pounds per square inch on the piston, multi- 
 plied by the area of one cylinder in square inches, and multiplied again by 
 the average speed of the piston in feet per minute. 
 
 Approximately, the average effective pressure of steam will be two-thirds 
 of the pressure shown on the gauge near the engines. A good average piston 
 speed is 400 ft. per minute. . , „ . _ _ , , n . , c . 
 
 To Find the Actual Horsepower of an Engine for Hoisting Any Load Out of a Shaft 
 at a Given Rate of Speed — To the weight of the loaded car add the weight of 
 the rope and cage. This will give the gross weight. 
 
 Then, H. P. 
 
 gross weight in lb. X speed in ft. per minute . L ^ 
 “ 33.000 
 
 contingencies, friction, etc. . , . , . , ^ ~ ^ 
 
 Example. — H aving a shaft 600 ft. deep, gross weight of load 20,000 Id., to 
 be hoisted in II minutes, what horsepower is required? 
 
 H . p. _ = 243 H. P„ nearly . 
 
 To which add £ for contingen- 
 
 cies, and we have 324 H. P. . ^ _ ... , 
 
 In a shaft with two hoistways, use the net weight -j- the weight of one 
 rope, instead of the gross weight. . 
 
 The following rules regarding winding engines are given by Percy: 
 
 1. To Find the Load That a Given Pair of Direct-Acting Engines Will Start. 
 Multiply the area of one cylinder by the average pressure of the steam 
 per square inch in the cylinder, and twice the length ot the stroke. Divide 
 this by the circumference of the drum, and deduct i for friction, etc. 
 
HEAD-FRAMES. 
 
 397 
 
 Example.— Given a pair of engines, cylinders 20 in. diameter by 40 in. 
 stroke, the drum 12 ft. diameter, and the pressure at steam gauge 50 lb., 
 steam cut-off at average pressure of steam in cylinder 48.2 lb. 
 
 Then, area of cylinder = 314.16 sq. in. 314.16 X 48.2 X 80 = 1,211,400.96. 
 
 The circumference of the drum = 452.4 in. 1,211,400.96 -4- 452.4 = 2,677. 
 f of 2,677 = 1,784 lb., or the net load. 
 
 The gross load would include the weight of rope, cage, and car, but as 
 these are balanced by the descending rope, cage, and car, the net load only 
 is found. The drum mentioned is cylindrical. 
 
 2. Knowing the Load and the Diameter of a Cylindrical Drum, and the Length of 
 Stroke, the Cut-ofF and Pressure of Steam at Steam Gauge, to Find the Area and 
 Diameter of Cylinders of a Pair of Direct-Acting Engines. — Multiply the load by 
 the circumference of the drum, and add one-half for friction, etc. Divide 
 this by the mean average steam pressure, multiplied by twice the length of 
 the stroke. 
 
 Example. — Having the drum 10 ft. in diameter, the stroke 6 ft., the steam 
 pressure at gauge 60 lb., the cut-off at £ of stroke, and the load 5 tons, or 
 
 11,1 Then’ 11,200 X 31.416 (circumference of drum) = 351,859. 351,859 + £ of 
 351,859 (or 175,930) = 527,789. , _ 
 
 The mean average pressure = 56.2 lb. 56.2 X (6 X 2) = 674.4. 527,789 
 - 4 - 674.4 = 782.6 sq. in., area of piston. 
 
 782.6 .7854 = 996. V 996 — 31.56 in., or diameter of cylinder. 
 
 3 To Find the Approximate Period of Winding on a Cylindrical Drum With a Pair 
 of Direct-Acting Engines.— Assume the piston to travel at an average velocity of 
 400 ft. per minute, and divide this by twice the length of the stroke, and 
 multiply by the circumference of the drum. This gives the speed of cage in 
 feet per minute. Divide the depth of shaft by this, and the result will be the 
 period of winding. , „ ^ _ . _ . , 
 
 Example. — Drum, 31.416 ft. circumference; stroke, 6 ft.; depth of shaft, 
 
 1 500 ft 
 
 ’ Then, 400 - 4 - 12 = 33.33. 33.33 X 31.416 = 1,047.1. 1,500 -4- 1,047.1 = 1.43 
 
 min., or about 1 min. 26 sec. . „ _ A . , _ .. , 
 
 4. To Find the Useful Horsepower During a Winding. — Multiply the depth of 
 shaft by net weight raised; divide this by number of minutes occupied in 
 winding, and divide again by 33,000. . , _ . , 
 
 Example. — Net weight, 2 tons = 4,4801b.; depth, 1,500 ft.; period of wmd- 
 
 mg Then, m 4^0 e x 1,500 = 6,720,000. 6,720,000 - 4 - 1.43 = 4,699,301. 4,699,301 
 
 -4- 33,000 = 142+ H. P. 
 
 HEAD-FRAMES. 
 
 Head-frames are built of wood or steel, and some of the typical forms are 
 shown on pages 275 and 276. They vary in 
 height from 30 to 100 ft., depending on local 
 conditions. , , „ _ . , , 
 
 The inclined leg of a head-frame should be 
 placed so as to take up the resultant strain due 
 to the load hanging down the shaft and the 
 pull of an engine. Fig. 4 shows the graphical 
 method of determining the direction and mag- 
 nitude of this resultant force. Produce the 
 direction of the two portions of the rope lead- 
 ing to the drum and down the shaft until 
 they intersect at G, measure off a distance G K 
 to scale to represent the load hanging down the 
 shaft; similarly, measure off GH to the same 
 scale to represent the pull of the engine, com- 
 plete the parallelogram GHLK; the direction 
 of the line G L represents the direction of the 
 resultant force, and its length represents the 
 amount of this force. The inclined leg of 
 the head-frame should be placed as nearly as 
 possible parallel to this resultant line, and 
 should be designed to withstand a compressive 
 strain equal to this resultant. _ . , .. 
 
 Head-sheaves are made of iron, being sometimes entirely cast, or else tne 
 
 Fig. a. 
 
 ✓ 
 
398 
 
 HOISTING AND HAULAGE. 
 
 rim and hub are cast separately and wrought-iron spokes are used. The 
 former are cheaper and quite satisfactory, but the latter are lighter and 
 stronger, and therefore usually better. The diameter of the sheave depends 
 on the diameter of the rope, and the table giving this will be found on 
 page 120. The groove in the sheave should be wood-lined, to reduce wear 
 on the rope. Wrought-iron spokes should be staggered in the hub and not 
 placed radially. _ _ A A _ 
 
 Guides and conductors are usually of timber rigidly attached to the sides of 
 a shaft. In England and certain parts of Europe, wire ropes are used for guides 
 and are strongly advocated, but they have never found favor in America. 
 These ropes when used are weighted at the bottom, and Percy gives 1 ton 
 for each 600 ft. in depth for each wire as a good weight to be used. When 
 not thus weighted, the ropes are fastened at the bottom and attached to 
 levers at the top, the levers being weighted to produce the requisite tension. 
 
 Safety catches usually consist of a pair of toothed cams placed on either 
 side of the cages and enclosing the guides. When the load is on the hoisting 
 rope, these cams are kept away from the guides by suitable springs; but if 
 the rope breaks, the springs come into action and throw the catcnes or 
 dogs so that they grip the guides, and the tendency to fall increases the 
 
 ^Detaching hooks are devices that automatically disconnect the rope from 
 the cage in case of overwinding. 
 
 HAULAGE. 
 
 The magnitude of modern mines and the practice of loading or of treating 
 the coal or ore at a large central station makes the underground haulage of 
 the material one of the most important problems in connection with 
 mining. A good haulage system is now essential to make most mines a 
 commercial success. Haulage may be considered under the following heads: 
 
 1. Inclined Roads— Gravity planes, engine planes. , 
 
 2. Level Roads— Mule haulage, rope haulage (tail-rope and endless rope), 
 motor haulage (steam, electricity, compressed air, or gasoline). 
 
 Gravity Planes.— The loaded car or trip hauls the empty car up the grade. 
 Two ropes are attached to a drum so that the rope attached to the loaded 
 
 car unwinds from the drum as the car de- 
 - scends, while the rope attached to the empty 
 
 4 ^ car is wound on the drum and the car thus 
 
 — hauled up the plane. The natural slope of 
 
 Horizontal. the ground, in a large measure, determines 
 
 Fig. 5. the grade of the incline, but where it is pos- 
 
 sible to alter the direction of the incline, 
 the grade may be lessened by constructing the incline across the slope of 
 the ground. The grade of the incline may be increased by carrying the 
 upper landing forwards till a point is reached from which the required 
 grade is obtained. „ A _ , 
 
 The following rule gives suggestions based on practice that has been 
 successful: For lengths not exceeding 500 ft., the minimum grade for the 
 incline should be 5 $ when the weight of the descending load is 8,000 lb. and 
 that of the ascending load 2,8001b. Or the inclination should not be less 
 than 5 H if the respective descending and ascending loads are one-half of 
 those just given. When the length of the plane is from 500 to 2,000 It., the 
 grade should be increased from 5$ to 10$, according to the loads. A load 
 of 4,000 lb. on a 10$ grade 2,000 ft. long will hoist a weight of 1,400 lb. 
 
 The angle of inertia is that angle or inclination at which a car will start to 
 move down the slope or plane. The car, when it has once started on this 
 
 S rade, will continue to accelerate its speed as it descends the plane A B , 
 ig. 5 . If we decrease the angle of inclination until the plane A B occupies 
 the position A C, such that the moving car will continue to move, at a 
 uniform velocity instead of accelerating its speed, the angle EGA will be 
 the angle of rolling friction, and the tangent of this angle will be the coefficient 
 of rolling friction for the car. . _ .. 
 
 The upper portion of a plane is made steeper than the lower portion so 
 that the trip may start quickly at the head and afterwards maintain a 
 uniform velocity. With a good brake to control the cars, the uniform grade 
 of a central portion of a gravity plane should not fall much below 3 , which 
 corresponds practically to a 5£$ grade. 
 
HAULAGE. 
 
 ^99 
 
 The acceleration f of the haulage system is given by the formula 
 
 / = x g X sin a 
 
 Pl + P2 
 
 where p, and p, are the descending and ascending pulls, respectively. 
 
 The length of steep pitch is given by the formula- 
 
 i = |L, 
 
 to haul the loaded trip up, is 
 
 T = ( W + w l) sin a + ( W + w l) cos a X /*> 
 
 where IF = weight of loaded trip; wl = weight of rope; a = slope angle; 
 
 ** "example 6 — 1 Mndthe°possible tension of a rope used to lower a loaded trip 
 nf two cars upon a plane 800 ft. long, having a uniform grade of 5 <fo at a 
 speed of 20 miles per hour, using a factor of _10, and l 
 
 To find the number of cars that must run in a trip on a self-acting incline, use 
 the formula 
 
 N 
 
 (40 sin a + cos a) TT 3 
 
 (40 sin a — cos a) W\ — (40 sin a + cos a) W 2 
 
 in which N = number of cars; a = angle of inclination of plane; 
 
 = weight in pounds of one loaded car; W 2 = weight m pounds of one 
 empty g car; W 3 = weight in pounds of haulage rope; — coefficient of 
 
 ^ Example — A gravity plane has an inclination of 8°; it is 2,000 ft. ^ng, the 
 rope weighs 4,000 lb., a loaded car weighs 3,000 lb., and an empty car weighs 
 1 800 lb. What number of cars must be m the trip to start it? 
 
 ’ Substituting values in the above formula, we have 
 
 N 
 
 (40 X .13917 -f .99027)4,000 
 
 = 13.6. 
 
 (40 X .13917 — .99027) 3,000 — (40 X .13917 + .99027)1,800 
 
 Fnffinp Planes —With an engine plane, the load is delivered at the foot of 
 the nlane and has to be hoisted. The engine may be either at the top or the 
 bottom. The grade of the plane is usually uniform from top to bottomland 
 there mav be a single track, a double track, or three rails with a turnout. 
 
 S^EngmeTRequired for Engine-Plane Haulage.-(a) ^gineat Hef of 
 Plane Sinale Track. — Calling the load on the engine or the tension of the 
 rone at the winding drum T, the weight of the ascending loaded trip TF,1 the 
 weight of the rope per lineal foot w, and the length of the plane l, the ang 
 of inclination or the slope angle being a, as before, we have 
 T= {W +wl) {sin a + cos a). 
 
 Assume an approximate value for w, and determine T approximately. 
 The size of rone required for this load is then obtained from the table fo 
 haul&J ^ropes?and with this new value of w, the correct load on the engine 
 
 is ^XAMPLK— What size of rope will be required to haul up an incline a 
 loaded trip of 10 mine cars weighing 1,000 lb. each, and carrying a load of 
 2,000 lb. each, the inclination of the plane or the slope angle being 16 and 
 its length 500 yd., assuming for the coefficient of frictmn ^ - ■ • 500) 
 
 jy _ 30 ( 000 lb., and assuming w = .89 lb., W + wl — 30,000 + Co;* X 
 
 31,335 lb. Sin a + 
 
 = .27564 + -fP- = .29967. 
 40 40 
 
 Hence, T = 31,320 X 
 
 .29967 = 9,394 lb. To provide against shock, we double th e loa^orimllon 
 the rone in calculating the size of rope required; thus, 9,394 X 2 ^ 18,788 lb.^ 
 and uSng a factor of safety of 6, we have for the breaking strain of the rope 
 
 18,788 X 6 = 57 tong In the table of w i re ropes, a If" plow-steel rope 
 
 2,000 
 
400 
 
 HOISTING AND HAULAGE. 
 
 presents a breaking strain of 56 tons. Since a 1|" rope weighs 2 lb. per 
 lineal foot, we have IF 4- wl = 30.000 + (2 X 1,500) = 33.000 lb. Then T = 
 33,000 X .29967 = 9,889 lb. , 
 
 (6) Engine at Head of Incline , Double Track.— The load on the engine 
 equals the difference between the gravity pulls of the ascending and 
 descending trips, including the rope, plus the friction pull of both the trips 
 and one rope, since there is only one rope on the plane at any time. Calling 
 the weight of the ascending trip TF, as before, and that of the descending 
 trip TFi, we have for the difference of the gravity pulls when the loaded trip 
 is at the foot of the incline, ( TF— TFi + w l) sin a. and for the friction pull of 
 the entire moving system ( TF + TFi + w l) /x cos a, and L = ( W — W\ + w l) 
 sin a + ( TF -f W\ -I- w l) n cos a. 
 
 Assuming the same conditions as given in the example of the preceding 
 paragraph, we have for the load L on the engine, L = [10 X 2,000 -f 2 x 
 
 1,500] .27564 + [10 X (3,000 + 1,000) + 2 X 1,500] X = 23,000 X .27564 
 
 + 43,000 X .02403 = 7,373 lb. instead of 9,394, the unbalanced load for single 
 track. ... 
 
 (c) Engine at Foot of Incline.— The load on the engine is the same as m 
 (a), except that the gravity pull is the pull due to the weight of the loaded 
 cars only, the weight of the ascending rope being balanced by the descend- 
 ing rope, while the friction pull is increased by the friction of the descend- 
 ing rope. C alling the load on the engine X, as before, we have, in this case, 
 
 L = TF sina-i(lF-j-2wZ)M- cos a. 
 
 Assuming the conditions of the previous example and calculating the 
 load on the engine for this case, we have L = 30,000 X .27564 H- [30,000 + 
 2(2X1,500)].02403 = 9,1341b. 
 
 To Find the Horsepower of an Engine Required to Hoist a Given Load Up a Single- 
 Track Incline in a Given Time.— Multiply the length of the incline in feet by the 
 natural sine of the angle of inclination, which will give you the vertical lift. 
 Divide the vertical lift by the given time in minutes. Multiply this by the 
 gross load, including weight of rope, and divide the product by 33,000. 
 
 Example.— Length of incline, 600 ft.; angle of inclination, 35°; weight of 
 loaded car and 600 ft. of rope, 5,000 lb. ; time of hoisting, 2 minutes. Required, 
 the horsepower. 
 
 Sine of 35° = .573576. .573576 X 600 = 344.1456. 344.1456 -s- 2 = 172.728. 
 
 172.728 X 5,000 ~ 
 
 ^ooo — = h - p - 
 
 Add from 25^ to 50^ for contingencies, friction, etc. In mine practice. 50£ 
 is not any too much to add. because the condition of track, cars, etc., is not 
 as good, as a general rule, as on railroad planes. 
 
 To Find the Horsepower of an Engine Required to Hoist a Given Load Up a Double- 
 Track Incline in a Given Time.— Proceed as above, using the net load, to which 
 should be added the weight of one rope, instead of the gross load. 
 
 ROPE HAULAGE. 
 
 The tail-rope system of haulage uses two ropes and a pair of drums on the 
 same shaft. The ma in rope passes from one drum directly to the front of the 
 loaded trip, and the tail-rope passes from the other drum to the large sheave 
 wheel at the end of the road and back to the rear of the loaded trip. While 
 hauling the loaded trip, the drum on which the tail-rope is wound is allowed 
 to turn freelv on its journal by throwing its clutch out, while the engine 
 turns the other drum. When the empty trip is being hauled, the clutch on 
 the main-rope drum is thrown out and the one on the tail-rope drum is 
 thrown in. The engine then turns the tail-rope drum and allows the other 
 one to pav out rope as the trip advances. 
 
 The tail-rope svstem is suitable for steep, circuitous, and undulating 
 roads. The trip can be kept stretched at all points, and thus the cars will 
 be prevented from bumping together or from being jerked apart as the trip is 
 passing over changes in the grade. It is undoubtedly the most satisfactory 
 system of rope haulage under the natural conditions of most haulage roads 
 in mines, and especially so where but one road is available for haulage 
 purposes. 
 
ROPE HA ULAGE. 
 
 401 
 
 CALCULATION OF THE TENSION OF HAULING ROPE. 
 
 T = tension or pull upon rope (lb.). 
 
 W = weight of loaded trip (lb.). 
 
 w = weight of rope per lineal foot (lb.). - . jj-w 
 
 l = length of two ropes; equals 2 times the distance from winding 
 drum to tail-sheave (ft.). 
 d = vertical drop of rope (ft.). 
 a = slope angle of maximum grade. 
 
 T = TF(sin a + /u. cos a)+w(d-jrn l). 
 
 Example.— What size of steel wire rope will be required to haul a trip of 
 20 mine cars the weight of the loaded cars being 3,000 lb. each, the depth of 
 the shaft 300* ft., and the distance from the loot of the shaft to the t^^g| 
 900 vd. the maximum grade in this haulage being 10 , n — 4 o • Assumi g 
 
 ^ on IVv ■nor linnol ft 
 
 '' rope, weighing .89 lb. per lineal ft. 
 , .9848 \ 
 
 r = 
 
 60,000 (.17365 + 
 what over 6 tons. 
 
 40 ) 
 
 .89 (soo+^gr) = sa -y 12 ' 300 lb - or some ‘ 
 
 ^eferringTo^the tables for steel haulage ropes with 6 strands i of ’1 'wire: 3 
 each we find the breaking strain of a f" rope, weighing .89 lb. per hneal it., 
 is 1 8 6 tons which will give a factor of safety of about 3. We would, however, 
 use a or evenal'' rope, as a change of ropes would then be required less 
 often. 8 Making the necessary corrections for 1" rope weighing 1.58 lb. per 
 
 ^Tliee n dTe s s - r o p e°s y s t e m uses an endless rope, which is kept m nning con- 
 tinuously by a pair of drums geared together and set tandem. The drums 
 are comparatively 
 narrow and provided 
 with grooves for the 
 rope to run in. Two 
 drums are necessary 
 to get sufficient fric- 
 tion to drive the rope 
 when the trip is at- 
 tached to it. The 
 rope is passed around 
 both drums a num- 
 ber of times, depend- 
 ing on the amount of 
 friction desired, 
 without completely 
 encircling either. It 
 then passes to a ten- 
 sion wheel at the rear 
 of the drums and 
 thence to the sheave 
 wheel at the far end 
 of the road and back 
 to the drums. To be 
 used to best advan- 
 
 qmres^hatthe grade be in one direction and that it he 
 
 from a number of places en route. The cars are attached to the rop y 
 friction grips in a manner quite similar to the way in which street cars 
 are attached to cable lines. It is evident, therefore, that any jerking due 
 to the cars bumping together or stretching the 
 
 injure the rope where the grip takes hold. A double road is an essential 
 
 fCa The endless-rope system of haulage is best adapted to roads Ffairlvrel^ 
 fairly uniform grade, particularly when the trips are not sp^ed at fairly reg 
 lar intervals along the road. Owing to delays m the dehvery of the cars by the 
 drivers and to irregularity in unloading at the Uppte, it 
 possible to have the several trips regularly spaced, and m 
 load on the engine varies greatly. In order to take up any elongation of the 
 rope due either to change in temperature or to stretching some form of 
 balance car or balance weight is used This weight should be sufecient to 
 keep the empty rope taut, and any tendency of the rope to slip on tne 
 
 Fig. 6. 
 
402 
 
 HOISTING AND HA ULAGE. 
 
 winding drum may be overcome by increasing the weight in the bal- 
 ance car. 
 
 Fig. 6 shows a device for working a district haulage by connecting it with 
 the main haulage. The main rope makes one or two complete turns around a 
 fleet wheel located at the mouth of each district, and then continues on its 
 course. This fleet wheel d is directly connected with the driving sheave m 
 for the district by means of beveled gears g , h, as shown. The driving 
 sheave m is thrown in or out of gear by levers o and q. 
 
 To Determine the Friction Pull on an Endless-Rope Haulage. — Let 
 
 0 = output (lb. per min.); tcx = weight of mine car (lb.); 
 
 = speed of winding (ft. per min.); w = weight of rope (lb.); 
 
 = length of haulage road (ft.); T = load on the rope (lb.); 
 
 = capacity of mine car (lb.); /u. = coefficient of friction. 
 
 v 
 
 l 
 
 c 
 
 Then, 
 
 — = weight of material in transit; 
 v 
 
 2 ^ W\ = weight of moving cars, loaded and empty; 
 2 1 w = weight of rope; 
 
 ^ ^ 1 + + 2 w l = entire moving load. 
 
 And if the coefficient of friction equals 
 
 [t( 
 
 
 ] 
 
 H. P. = 
 
 til 
 
 33,000 
 
 [•( 
 
 1 + 
 
 *-?> 
 
 + 2w 
 
 ’] 
 
 Example.— F ind the horsepower for an endless-rope system 5,000 ft. long 
 for an output of 1,000 tons per day of 10 hours in a flat seam, the mine cars 
 having a capacity of 2.000 lb. each and weighing 1,200 lb. each. 
 
 Assuming a speed of winding of 8 miles per hour or 704 ft. per minute, and 
 for the coefficient of friction /m = 
 
 T = a [ 5,00 °^^ ( 1 + + 2 X 1-58 X 5,OOo] = 1,697, say 1,700 lb. 
 
 H. P. = 
 
 704 
 5,000 
 
 40 X 33,000 
 
 [3,333 ( 
 
 2 X 1,200 \ 
 
 + 2,000 
 
 X 1.58 X 704 J = 36.2 H. P. or, 
 
 36.2 
 
 assuming an efficiency for the engine of 60^,—^ = 60 H. P. 
 
 Inclined Roads.— The calculation of power for inclined roads is the same as 
 that just given, excepting that the work due to lifting the coal through a 
 height h must be added to that found by the previous formulas. If h equals 
 the elevation due to the grade of the incline, the additional work of the 
 engine due to hoisting the load from this elevation will be OA and the 
 total work per minute u will be 
 
 u = fi i[o ^1 + + 2 w v J + O h. 
 
 Example. — Assuming the same conditions as given above, and, in addi- 
 tion, a rise or elevation of 100 ft. in the entire length of the haulage way, we 
 
 hare u = ^[ 3 , 333(1 + 2 ^^°° ) + 2 X 1.58 X 70l] + 3,333 X 100 
 
 = 1,528,050 ft.-lb. per minute = 46.3 H. P., or assuming an efficiency of 60£ 
 
 for the engine, ^ = 77 H. P. 
 
 MOTOR HAULAGE. 
 
 Locomotive Haulage. — Wire-rope haulage is very efficient in headings, on 
 heavy grades, and against large loads, but in crooked passages it entails great 
 costs for renewals and repairs. When the grades do not exceed 5£ for short 
 distances and average 3# against, or for short distances 84 and 54 average in 
 favor of loads, locomotives have been found the most economical form ot 
 haulage. 
 
MOTOR HAULAGE. 
 
 403 
 
 M EK^'iSS® s““m l-omoaves are 
 
 K^J3^S3!»^^"y5aa 
 
 a?r and electricity of ^ whfchTnumber of types have been designed in recent 
 yearf and whTch have been very successful and have shown a marked 
 efficiency over the mule . _ 194 , compressed air is particularly 
 
 auDlicable irf gaseolis mines, as it improves ventilation and is perfectly safe 
 under all conditions. The great disadvantage in compressed-air haulage is 
 
 ^M^H^MyeXof^he Baldwin Locomotive Works, gives the following 
 
 in In 8 oMer th^co^pre^ed^locomotives may be able to make a fair 
 lenffth° of run, the tanks for storage purposes must necessarily be rather 
 cumbersome^ and constructed to carry high-storage pressures lnorder that 
 theTmay be designed correctly and get a minimum of storage for the 
 maximum work expected, it is necessary to have a complete profile of the 
 nrnnosed haulage road and to make a tabulated statement of the air con 
 Siwfon oSTe Various grades, noting the “ cut-off” necessary to produce the 
 reamsite tractive effort. By making a summation of these varmus amounts, 
 an^addtng^O^y we will have the possible amount of air used in doing cer- 
 
 tai IMrLcesL e ry lfi therefore, to provide storage on the loepmotive for this 
 amount of air at a much greater pressure than that used m the cylinders. 
 Tn order that the locomotive may receive a quick charge at the stations 
 specially provided for the purpose, it is necessary to have stationary storage 
 
 to 800 lb and to have the volume of this storage at least double the tank 
 capacity of the locomotives comprising the system. This allows an equalized 
 nressure in the locomotive storage of approximately 600 lb. . . 
 
 ^ The following formula is useful in determining the capacity of stationary 
 
 storage: P' = in which V = volume of storage on locomotive; 
 
 X = volume of stationary storage desired; p = cylinder pressure; P = 
 stationary storage pressure; and if = locomotive storage P res ®^- . , 
 
 If the average time for each trip is 30 minutes, the compressor must be 
 able to compress in that time to pressure P, the calculated amount of air 
 required for one trip or series of trips for the various locomotives included 
 in the haulage. In general, it is customary to extend extra-strong pipeinto 
 the mine and of such length and diameter as to bave the required volume 
 for the stationary storage. There are times however when it would be found 
 more economical to arrange for tank storage either inside or putsitm the 
 mine, but in general, especially when the mine is advancing, it is the better 
 practice to install pipe storage since it increases the range of the locomotive 
 as the workings advance. . ^ ^ - 
 
 The following table gives the various tractive efforts of different sizes of 
 compressed-air locomotives, when working at 100 lb. cylinder pressure, and 
 various cut-offs. If other pressures or strokes are used, the tractive efforts 
 are directly proportionate. This table is calculated by means of the formula, 
 
 tractive effort = 
 
 d 2 lxP' 
 
 D ’ 
 
 in which d = diameter of cylinder; D = diameter of driver; Z = length of 
 stroke; p — working pressure of the cylinders; and x — variable due to the 
 various cut-offs. 
 
404 
 
 HOISTING AND HAULAGE. 
 
 Tractive Efforts of Compressed-Air Locomotives. 
 
 Cylinder. 
 
 Diam- 
 eter of 
 Driver. 
 
 Inches. 
 
 Weight 
 
 on 
 
 Driver. 
 
 Pounds 
 
 Tractive Effort for Each 100-Lb. Cylinder 
 Pressure at Various Cut-Offs. 
 
 Diam. 
 
 Inches. 
 
 Stroke. 
 
 Inches. 
 
 7 
 
 e 
 
 3 
 
 3F 
 
 5. 
 
 8 
 
 ■ 1 
 
 * 
 
 3. 
 
 8 
 
 4 
 
 1 
 
 8 
 
 5 
 
 10 
 
 24 
 
 6,000 
 
 1,020 
 
 990 
 
 920 
 
 835 
 
 710 
 
 530 
 
 325 
 
 6 
 
 10 
 
 24 
 
 8,500 
 
 1,470 
 
 1,425 
 
 1,320 
 
 1,200 
 
 1,020 
 
 760 
 
 445 
 
 7 
 
 12 
 
 26 
 
 13,000 
 
 2,200 
 
 2,150 
 
 1,990 
 
 1,810 
 
 1,540 
 
 1,140 
 
 700 
 
 8 
 
 12 
 
 26 
 
 18,000 
 
 2,880 
 
 2,750 
 
 2,600 
 
 2,360 
 
 2,000 
 
 1,510 
 
 900 
 
 9 
 
 14 
 
 26 
 
 25,000 
 
 4,340 
 
 4,140 
 
 3,840 
 
 3,490 
 
 2,960 
 
 2,220 
 
 1,350 
 
 10 
 
 14 
 
 26 
 
 32,000 
 
 5,280 
 
 5,150 
 
 4,740 
 
 4,310 
 
 3,660 
 
 2,630 
 
 1,670 
 
 11 
 
 16 
 
 28 
 
 42,000 
 
 6,770 
 
 6,450 
 
 5,980 
 
 5,440 
 
 4,620 
 
 3,470 
 
 2,140 
 
 12 
 
 16 
 
 28 
 
 52,000 
 
 8,050 
 
 7,800 
 
 7,200 
 
 6,550 
 
 5,580 
 
 4,150 
 
 2,550 
 
 On account of certain losses due to radiation, etc. for cut-off at full length 
 of stroke in steam practice, x is taken as .85. While cylinder surface acts as 
 a detriment to the use of steam, it acts entirely opposite in the use of air, 
 for the reason that, in the expansion of the air, very low temperatures are 
 produced, and, with a maximum of cylinder surface exposed, we absorb a 
 maximum of heat from the surrounding air, which virtually adds new 
 energy to the air, thus acting as a reheater. Therefore in air practice, x is 
 made .98 for full-stroke cut-off, with the others proportionately high. If 
 simple-expansion cylinders are used, the working pressure should not 
 exceed 130 lb., while, with compounds, one can easily use from 180 to 
 225 lb. with great economy. Where it is imperative to have a minimum- 
 sized locomotive storage with a maximum run, this can be accomplished 
 with compound locomotives. Originally, it was the custom to lag the cylin- 
 ders as in steam practice, but now it is found advantageous to leave them 
 bare and to corrugate both sides and ends so as to present a maximum 
 surface to the surrounding atmosphere while running, thus absorbing 
 new energy. 
 
 Example.— It is desired to haul trips of 60 cars, empties weighing 2,000 lb. 
 and loads 6,000 lb. each, over a track having the following profile, and with 
 one charge of air. All grades are in favor of loads. (The following calcu- 
 lations have been made with the slide rule.) 
 
 Profile of Road. 
 
 Grade. 
 
 Distance. 
 
 Grade. 
 
 Distance. 
 
 Grade. 
 
 Distance. 
 
 1.3# 
 
 800 ft. 
 
 0.30# 
 
 700 ft. 
 
 2.4# 
 
 400 ft. 
 
 2.0# 
 
 600 ft. 
 
 1.77# 
 
 1,025 ft. 
 
 3.5# 
 
 425 ft. 
 
 1.3# 
 
 800 ft. 
 
 0.90# 
 
 300 ft. 
 
 1.2# 
 
 320 ft. 
 
 The maximum grade being 3.5#, and the car friction in this case being 1 #, 
 the total resistance when ascending a 3.5# grade due to cars is, hence, 
 3.5# -HI# = 4.5#. Since it is desired to haul 60-car trips, and all grades are 
 in favor of loads, it is only necessary to provide a locomotive capable of 
 hauling 60 empties weighing 120,000 lb. up the above-mentioned grade. The 
 drawbar pull necessary to do this is 4.5# of 120,000 lb. = 5,400 lb. In general, 
 it will require a locomotive having a weight on drivers of 5 times the 
 tractive effort desired if steel tires are used, as is the practice in the con- 
 struction of air locomotives, and 6 times the tractive effort if cast-iron chilled 
 wheels are used, as is the practice in electric locomotives. We will there- 
 fore assume the necessary weight of the locomotive to give the proper 
 adhesion as 32,000 lb., and we calculate that the tractive effort necessary to 
 haul itself up the 3.5# grade would be 3.5# + .5# = 4# (.5# covering the fric- 
 tion of the locomotive on the level) of 32,000 lb., or 1,2801b., to which we 
 add the necessary drawbar pull to haul the desired load, 1,280 + 5,400, and 
 have a total tractive effort of 6,680, which is about the limit of a locomo 
 live on dry rail with sand. 
 
MOTOR HAULAGE. 
 
 405 
 
 Bv consulting the table of tractive efforts of compressed-air locomotives, 
 we see that, at 100 lb. working pressure, a 10" X 14", 26" driver. locomotive has 
 a maximum tractive effort at £ cut-off, which is practically full stroke, of 
 5,280 lb., and by dividing - 5 T % 8 <£ into our necessary tractive effort, we find that 
 
 the necessary working pressure would be about 130 lb. # 
 
 On this basis, we then make up the following table in order to ascertain 
 the necessary air consumption: 
 
 Grade. 
 
 Distance. 
 
 Going in 
 T.E. 
 
 1.3 
 
 800' 
 
 3,375 
 
 2.0 
 
 600' 
 
 4,450 
 
 3,375 
 
 1.3 
 
 800' 
 
 0.3 
 
 700' 
 
 1,830 
 
 1.77 
 
 1,025' 
 
 4,100 
 
 0.9 
 
 300' 
 
 2,750 
 
 2.4 
 
 400' 
 
 5,075 
 
 3.5 
 
 425' 
 
 6,760 
 
 1.2 
 
 320' 
 
 2,220 
 
 0.3 
 
 700' 
 
 Coming 
 
 2,600 
 
 Strokes. 
 
 120 
 
 80 
 
 120 
 
 105 
 
 150 
 
 40 
 
 60 
 
 65 
 
 50 
 
 Cut-Off. 
 
 Cu. In. Air Used. 
 126,000 
 
 176.000 
 
 126.000 
 55,125 
 
 315.000 
 42,000 
 
 157,500 
 
 239.000 
 52,500 
 
 110,000 
 
 20 l additional . 
 
 1,399,125 
 
 279,825 
 
 Total - 1,678,950 cu. m. 
 
 This equals 975 cu. ft. at 130 lb. pressure used in hauling the required loads 
 on a single round trip. Since we should return to the starting point with 
 130 lb. in the locomotive storage, it is evident that the volume of the tanks 
 shall allow for the use of 975 cu. ft. in addition to 1 volume at 130 lb. 
 
 Let V = volume of storage on locomotive; P' = pressure of storage on 
 locomotive; p — working pressure; V' = volume at working pressure nec- 
 essary to do the work required. _ ., „„„ 
 
 Then the product of the volume of the locomotive storage by its pressure 
 must equal the sum of the volume necessary to do the work required 
 multiplied by the working pressure, and the locomotive storage volume by 
 the working pressure, thus, 
 
 p’v = pv + pv\ or,r= v przr p - 
 
 130 
 
 If P' = 650, then V = 975 X _ f 3 Q = 244 cu - ft - 
 
 With one locomotive, making trips every 30 minutes, we must arrange for 
 a compressor capable of compressing 975 cu. ft. at 130 lb. m this time, bince 
 it is customary to rate compressors at their capacity in free air per minute, 
 
 the above is equivalent to 
 
 975 X 130 
 
 = 288 cu. ft. free air per minute. 
 
 14.7 X 30 
 
 This must be compressed to 800 lb. in the compressor, and stored in 
 stationary storage. If X is the volume of the stationary storage, 
 
 pV+PX 
 VAX * 
 
 P' = 
 
 P'-P 
 
 or X = 244 X 
 
 650-130 
 800 — 650 
 
 = 846 cu. ft. 
 
 “ r P-P" 
 
 The length of the haulage is 5,370 ft., hence the cross-section of the pipe 
 
 846 
 
 necessary to furnish the requisite storage is = .157 sq. ft. 
 
 From the following table, this would require a 5£" pipe, but for practical 
 purposes it is possible that a 5" pipe would be selected. 
 
 # Returning with loads, it is possible that there is only one grade that the trip will hare 
 to be hauled. 
 
406 
 
 HOISTING AND HAULAGE. 
 
 Standard Steam and Extra-Strong Pipe Used for Compressed-Air 
 Haulage Plants. 
 
 Trade 
 
 Diam- 
 
 eter. 
 
 In. 
 
 Cu. Ft. in 
 1 Lineal 
 Ft. 
 
 Lineal Ft. 
 Necessary 
 to Make 
 1 Cu. Ft. 
 
 Steam. 
 
 Extra Strong. 
 
 Trade 
 
 Diam- 
 
 eter. 
 
 In. 
 
 Thick- 
 
 ness. 
 
 Weight 
 per Ft. 
 
 Thick- 
 
 ness. 
 
 Weight 
 per Ft. 
 
 2 
 
 .0218 
 
 45.41 
 
 .15 
 
 3.61 
 
 .22 
 
 5.02 
 
 2 
 
 2i 
 
 .0341 
 
 29.32 
 
 .20 
 
 5.74 
 
 .28 
 
 7.67 
 
 2£ 
 
 3 
 
 .0491 
 
 20.36 
 
 .21 
 
 7.54 
 
 .30 
 
 10.20 
 
 3 
 
 3£ 
 
 .0668 
 
 15.00 
 
 .22 
 
 9.00 
 
 .32 
 
 12.50 
 
 3£ 
 
 4 
 
 .0873 
 
 11.52 
 
 .23 
 
 10.70 
 
 .34 
 
 15.00 
 
 4 
 
 4£ 
 
 .1105 
 
 9.05 
 
 .24 
 
 12.30 
 
 .35 
 
 17.60 
 
 4£ 
 
 5 
 
 .1364 
 
 7.33 
 
 .25 
 
 14.50 
 
 .37 
 
 20.50 
 
 5 
 
 5£ 
 
 .1650 
 
 6.06 
 
 .26 
 
 16.40 
 
 .40 
 
 24.50 
 
 
 6 
 
 .1963 
 
 5.10 
 
 .28 
 
 18.80 
 
 .43 
 
 28.60 
 
 6 
 
 From the following table we see that it would require 2.88 X 32.5 — 
 93.6 H. P.; hence, we would be compelled to arrange for a boiler capacity 
 of practically 100 H. P., provided we used a three-stage compressor, as is the 
 general custom. 
 
 Horsepower Necessary to Compress 100 Cu. Ft. of Free Air to Various 
 
 Pressures and With Two-, Three-, and Four-Stage Compressors. 
 
 Gauge 
 
 Pres- 
 
 sure. 
 
 Horsepower Necessary. 
 
 Gauge 
 
 Pressure. 
 
 Horsepower Necessary. 
 
 Two- 
 
 Stage. 
 
 Three- 
 
 Stage. 
 
 Four- 
 
 Stage. 
 
 Two- 
 
 Stage. 
 
 Three- 
 
 Stage. 
 
 Four- 
 
 Stage. 
 
 100 
 
 15.7 
 
 15.2 
 
 14.2 
 
 900 
 
 36.3 
 
 33.7 
 
 31.0 
 
 200 
 
 21.2 
 
 20.3 
 
 18.8 
 
 1,000 
 
 37.8 
 
 34.9 
 
 31.8 
 
 300 
 
 24.5 
 
 23.1 
 
 21.8 
 
 1,200 
 
 39.7 
 
 36.5 
 
 33.4 
 
 400 
 
 27.7 
 
 25.9 
 
 24.0 
 
 1,400 
 
 41.3 
 
 37.9 
 
 34.5 
 
 500 
 
 29.4 
 
 27.7 
 
 25.9 
 
 1,600 
 
 43.0 
 
 39.4 
 
 35.6 
 
 600 
 
 31.6 
 
 29.5 
 
 27.4 
 
 1,800 
 
 44.5 
 
 40.5 
 
 36.7 
 
 700 
 
 33.4 
 
 31.2 
 
 28.9 
 
 2,000 
 
 45.4 
 
 41.6 
 
 37.8 
 
 800 
 
 34.9 
 
 32.5 
 
 30.1 
 
 2,500 
 
 
 43.0 
 
 39.0 
 
 Electric Haulage.— Mr. H. K. Myers says in regard to mine haulage by 
 electricity: In general, it costs from 6 to 10 cents per ton to deliver coal 
 from face of workings to shaft, slope, or tipple, where the haul is 1 mile and 
 the tracks approximately level; yet I know three mines that at present haul 
 from parting with the trolley system, the miner delivering from face of 
 room, making an average round trip of 9,000 ft., at a total cost of 1 cent per 
 ton. These mines have never had a mule in them, and it would be almost 
 an impossibility to introduce them, for the reason that the seam is of such 
 thickness that the clearance between tie and roof is only about 4 ft. Since 
 the advent of the electric-mining locomotive, there has been a change in 
 the mine wagons universally used. Formerly it was customary to find as 
 much as 60 lb. per ton car resistance on the level, while at present it is as 
 low as 15 lb. . _ . . , . 
 
 In dimensioning mining locomotives, it is customary to make the weight 
 from 6 to 8 times the necessary tractive effort, dependent entirely on the 
 nature of the work. If the work is constant and a maximum, then the 
 weight will be only 6 times the torque of the motors, while if the work is 
 intermittent with a short-time maximum tractive effort, then the factor 
 will be 8. The weight of an electric locomotive running at a speed of 6 to 
 
MOTOR HAULAGE. 
 
 407 
 
 8 miles per hour with intermittent load may also be mM be 
 
 400 lb for each rated horsepower of the motor, and the weight should be 
 Himes the rated drawbar pull, regardless of speed. For continuous work, 
 these weights should be decreased 25/*. 
 
 Drawbar Pull on Various Grades for Different Sized Locomotives. 
 
 Horse- 
 
 Weight. 
 
 Grades. 
 
 power. 
 
 Level. 
 
 1/* 
 
 2 J 
 
 3/ 
 
 4/ 
 
 5 J 
 
 6/* 
 
 10 
 
 20 
 
 30 
 
 50 
 
 70 
 
 100 
 
 4.000 
 
 8.000 
 12,000 
 20,000 
 28,000 
 40,000 
 
 500 
 
 1,000 
 
 1.500 
 
 2.500 
 
 3.500 
 5,000 
 
 460 
 
 920 
 
 1,380 
 
 2,300 
 
 3,220 
 
 4,600 
 
 420 
 
 840 
 
 1,260 
 
 2,100 
 
 2,940 
 
 4,200 
 
 380 
 
 760 
 
 1,140 
 
 1,900 
 
 2,660 
 
 3,800 
 
 340 
 
 680 
 
 1,020 
 
 1,700 
 
 2,380 
 
 3,400 
 
 300 
 
 600 
 
 900 
 
 1,500 
 
 2,100 
 
 3,000 
 
 260 
 
 520 
 
 780 
 
 1,300 
 
 1,820 
 
 2,600 
 
 In mines it is found that the friction between wheel and rail is less than 
 on the surface, due to dampness and powdered coal on the rail. The tractive 
 efforts with chilled wheels is usually considered ¥ of the weight, lhe 
 table on page 408 and diagram on page 409 give hauling capacities of 
 
 locomotives in tons of 2,000 lb. . , . 
 
 For maximum continuous work, it is necessary to have a grade such 
 that the efforts to haul the same number of empty wagons as loaded 
 are equal. With the car resistance considered 1/ and the loaded cars weigh- 
 ing 3 times as much as the empties, this is found to be * of 1/. The most 
 critical point in the designing of mining locomotives is to make the limiting 
 dimensions a minimum. The demands for various dimensions are wonder- 
 ful. The headings in mines are never of more generous proportions than 
 really necessary, and all clearances a minimum. The minimum dimen^ons 
 for mining locomotives are as small as 2 ft. for wheel base, 8 ft. for length 
 over all, and 3 ft. width. Scarcely two orders carry the same dimensions, 
 and it is impossible to have any kind of a standard. In consequence of this, 
 it is necessary to have a great variety of motors suitable for gauges as nar- 
 row as 18 in. and for wheels as small as 20 in. m diameter. With such a 
 variety, it becomes possible to construct a locomotive weighing 40,000 lb. 
 on 3' gauge, having the width over all 62 in., height 35 in., and length 
 12 ft. In construction, it is necessary to have the most modern form of 
 motors and the most rigid mechanical construction. 
 
 The motors now used are of the best possible construction and efficiency. 
 They are of the slow-speed street-car type, 6 to 8 miles per hour winding, 
 and range in size from 4 to 50 H. P. It is customary to use the rheostatic 
 type of controller for mining locomotives, on account of its small dimensions 
 and apparent efficiency for this class of work, but it is doubtless but a short 
 time until a very compact form of series-parallel type will be devised. On 
 account of the use of this rheostatic controller, it becomes necessary to Pro- 
 vide for large diverter capacity, and since the locomotive is designed for the 
 maximum tractive effort, it is hardly ever possible to run without resist- 
 ance and, hence, a large amount of current must be dispeped with the 
 consequent heating. If the motors are overloaded, they heat rapidly, this 
 heating varying as the square of the current. A motor that has a rating of 
 40 amperes for regular work, if worked for 3 minutes at 100 amperes, should 
 not be subjected to such a strain oftener than once in 18f minutes, as shown 
 by the following equation: 
 
 402 x x = 3 X 100 2 ; x = 18£ minutes. 
 
 Using the same problem given under compressed-air locomotives, in 
 which the maximum tractive effort was 6,760 lb., we find from the table of 
 drawbar pulls that a locomotive equipped with two 50 H. P. motors (equals 
 100 H. P.) will carry the load with an overload, these motors being rated for 
 continuous work at approximately 32 amperes of 500 volts. 
 
 Using the formula = which t — various times at which 
 
408 
 
 HOISTING AND HAULAGE. 
 
 various amounts a of current are used on the corresponding grades, 2 the 
 summation of the items t a 2 calculated for each section or grade, and T = 
 total time that should be taken for each trip, we calculate the following 
 table: 
 
 Grade. 
 
 Dist. 
 
 T. E. 
 
 Time. 
 
 Minutes. 
 
 Amperes. 
 
 to? 
 
 1.30 
 
 800 
 
 3,375 
 
 1.5 
 
 114 
 
 Empties. 
 
 19,600 
 
 2.00 
 
 600 
 
 4,450 
 
 1.1 
 
 134 
 
 19,900 
 
 1.30 
 
 800 
 
 3,375 
 
 1.5 
 
 114 
 
 19,600 
 
 .30 
 
 700 
 
 1,830 
 
 1.3 
 
 74 
 
 7,100 
 
 1.77 
 
 1,025 
 
 4,100 
 
 2.0 
 
 128 
 
 32,800 
 
 .90 
 
 300 
 
 2,750 
 
 .6 
 
 100 
 
 6,000 
 
 2.40 
 
 400 
 
 5,075 
 
 .8 
 
 148 
 
 17,300 
 
 3.50 
 
 425 
 
 6,760 
 
 .8 
 
 182 
 
 26,500 
 
 1.20 
 
 320 
 
 2,220 
 
 .6 
 
 86 
 
 4,400 
 
 .30 
 
 700 
 
 2,600 
 
 1.3 
 
 96 
 
 Loads. 
 
 13,900 
 
 
 
 
 
 
 167,100 = 2 t a 2 
 
 ^Jl6U00 = 64; 4 096 T = 167100; T = 4a 
 
 By this means we can make 60-car trips every 40 minutes without injury 
 to the motors, based upon a speed of 6 miles per hour. (See also page 215.) 
 
 Speed of haulage depends on the system of haulage used and on the con- 
 dition of the haulage road. The law in Pennsylvania provides for a speed of 
 haulage not over 6 miles per hour, and this is the speed at which electric and 
 
 Hauling Capacity of Electric Locomotives. 
 
 Horsepower. 
 
 Weight. 
 
 Drawbar Pull 
 on Level. 
 
 Frictional Car 
 Resistance per 
 Ton on Level. 
 
 Grades. 
 
 Level. 
 
 
 1 Jo 
 
 W 
 
 2{ 
 
 m 
 
 3 jo 
 
 3 if 
 
 4 f 
 
 5 f 
 
 6 i 
 
 10 
 
 4,000 
 
 500 
 
 20 
 
 23 
 
 15 
 
 10 
 
 8 
 
 6.3 
 
 5.2 
 
 4.2 
 
 3.5 
 
 3.0 
 
 2.2 
 
 1.6 
 
 
 
 
 30 
 
 15 
 
 11 
 
 8.4 
 
 6.7 
 
 5.4 
 
 4.5 
 
 3.8 
 
 3.2 
 
 2.7 
 
 2.0 
 
 1.5 
 
 
 
 
 40 
 
 12 
 
 9 
 
 7 
 
 5.7 
 
 4.7 
 
 4.0 
 
 3.4 
 
 3.0 
 
 2.5 
 
 1.8 
 
 1.4 
 
 20 
 
 8,000 
 
 1,000 
 
 20 
 
 46 
 
 29 
 
 21 
 
 16 
 
 13 
 
 10.3 
 
 8.4 
 
 7.1 
 
 6.0 
 
 4.3 
 
 3.1 
 
 
 
 
 30 
 
 31 
 
 22 
 
 17 
 
 13 
 
 11 
 
 9 
 
 7.5 
 
 6.4 
 
 5.4 
 
 4.0 
 
 3.0 
 
 
 
 
 40 
 
 23 
 
 18 
 
 14 
 
 11 
 
 9.5 
 
 8 
 
 6.8 
 
 5.9 
 
 5.0 
 
 3.7 
 
 2.8 
 
 30 
 
 12,000 
 
 1,500 
 
 20 
 
 69 
 
 44 
 
 32 
 
 24 
 
 19 
 
 15 
 
 13 
 
 10.7 
 
 9.0 
 
 6.5 
 
 4.7 
 
 
 
 
 30 
 
 43 
 
 33 
 
 25 
 
 20 
 
 16 
 
 13 
 
 11 
 
 9.6 
 
 8.2 
 
 6.0 
 
 4.4 
 
 
 
 
 40 
 
 34 
 
 26 
 
 21 
 
 17 
 
 14 
 
 12 
 
 10 
 
 8.8 
 
 7.5 
 
 5.6 
 
 4.1 
 
 50 
 
 20,000 
 
 2,500 
 
 20 
 
 115 
 
 73 
 
 52 
 
 40 
 
 32 
 
 26 
 
 21 
 
 18 
 
 15 
 
 11 
 
 7.9 
 
 
 
 
 30 
 
 77 
 
 55 
 
 42 
 
 33 
 
 27 
 
 22 
 
 19 
 
 16 
 
 14 
 
 10 
 
 7.3 
 
 
 
 
 40 
 
 58 
 
 44 
 
 35 
 
 29 
 
 24 
 
 20 
 
 17 
 
 15 
 
 13 
 
 9.3 
 
 6.8 
 
 70 
 
 28,000 
 
 3,500 
 
 20 
 
 161 
 
 103 
 
 74 
 
 56 
 
 44 
 
 36 
 
 30 
 
 25 
 
 21 
 
 15 
 
 11 
 
 
 
 
 30 
 
 107 
 
 77 
 
 59 
 
 47 
 
 38 
 
 31 
 
 26 
 
 22 
 
 19 
 
 14 
 
 10 
 
 
 
 
 40 
 
 81 
 
 61 
 
 50 
 
 40 
 
 33 
 
 28 
 
 24 
 
 20 
 
 18 
 
 13 
 
 9.6 
 
 100 
 
 40,000 
 
 5,000 
 
 20 
 
 230 
 
 147 
 
 105 
 
 80 
 
 63 
 
 52 
 
 42 
 
 36 
 
 30 
 
 22 
 
 16 
 
 
 
 
 30 
 
 153 
 
 110 
 
 84 
 
 67 
 
 54 
 
 45 
 
 38 
 
 32 
 
 27 
 
 20 
 
 15 
 
 
 
 
 40 
 
 115 
 
 88 
 
 70 
 
 57 
 
 47 
 
 40 
 
 34 
 
 29 
 
 25 
 
 19 
 
 14 
 
MOTOR HAULAGE. 
 
 409 
 
 compressed-air haulages are usually calculated and at which loaded trips 
 are usually run. Empty trips are usually run at a slightly higher speed. 
 
 The speed for tail-rope haulage is given by three prominent makers of 
 such plants, as follows: (a) 600 to 700 ft. per minute; (6) 8 to 10 miles per 
 hour; (c) 6 to 8 miles per hour. 
 
 The speed for endless-rope haulage is given by the same makers as (a) 
 140 to 150 ft. per minute; (5) 1 to 2 miles per hour; (c) 150 to 200 ft. per 
 minute. A slow speed for endless rope is to be preferred as being much 
 more economical in the wear of the rope and cars, and many prefer a single- 
 car system to a trip system, thus doing away with the trip rider By han- 
 dling: the cars singly or even in trains of two and at a slow speed, the load 
 can be picked up without any slippage of the rope through the grips; while 
 if trains of from 12 to 25 cars are used, with the rope traveling 3 to 3* miles 
 per hour, it is impossible to pick up the load without having the rope slip 
 
 through the grip, thus heating the rope and cutting it. The slow-speed, 
 single-car or small-train, system requires more cars, but this is counterbal- 
 anced by the life of the cars and rope. Those that have tried both systems 
 prefer the slow-speed small trip to the high-speed large trip. 
 
 It has been found in general practice that the maximum pulling power 
 of a mule as well as a locomotive is, approximately, one-fifth its weight, or, 
 in other words, a locomotive will pull as much as the same weight of mules 
 will pull, and at a speed about three times as great. ...... 
 
 Cost of Haulage.— So much depends on local considerations that it is 
 difficult to give costs of haulage that will be of service. Mule haulage 
 has been given as costing, under different conditions, 5.74 cents and 7.92 
 cents per ton-mile, and in other locations 2.35 cents, 2.95 cents, and 7.15 cents 
 per ton of coal hauled. „ . ... 
 
 The Berwind-White Coal Mining Co., at Windber, Pa., uses 30 electric loco- 
 motives at various mines, which average, approximately, 400 tons per day ol 
 9 hours, per locomotive, over an average haul of 2 miles for the round trip. 
 The approximate cost for operating one of these locomotives, including the 
 wages of motorman, trip rider, and proportion of power-house expense, is 
 about $6.00 ner day, or cents per ton of coal haul per mile. If the total 
 load, including weight of cars, is considered, it figures f of 1 cent per ton per 
 mile. These figures do not, however, include grades, which is an important 
 factor in equating costs per ton per mile. In these mines there are no mules 
 
410 
 
 HOISTING AND HAULAGE. 
 
 whatever, the locomotives distributing the empty cars to room partings, for 
 the men to push to the face. If the haul is done between side tracks and 
 under similar grade conditions, the same locomotives could easily handle 
 1,000 to 1,200 cars per day. . , , 
 
 Mr. F. J. Platt, of Scranton, Pa., gives the following comparative costs of 
 electric and mule haulage per ton of coal hauled and under approximately 
 the same conditions in the same mine: 
 
 Name of Mine. 
 
 Mule Haulage. 
 Cents. 
 
 Electric Haulage. 
 „ Cents. 
 
 Green Ridge Colliery 
 
 New York & Scranton Coal Co 
 
 New York & Scranton Coal Co 
 
 Mt. Pleasant Colliery 
 
 Hillside Coal & Iron Co 
 
 Hillside Coal & Iron Co 
 
 7.15 
 
 6.58 
 
 2.35 
 
 2.95 
 
 10.77 
 
 9.10 
 
 2.76 
 
 2.62 
 
 1.07 
 
 1.27 
 
 4.56 
 
 4.65 
 
 The following costs of electric haulage, per ton of material hauled, are 
 given in the catalogue of the General Electric Co.: 
 
 Name of Mine. 
 
 Mule Haulage. 
 Cents. 
 
 Electric Haulage. 
 Cents. 
 
 Wythe Lead & Zinc Co 
 
 Blossburg Coal Co. 
 
 Cleveland-Cliffs Iron Co. (’94, ’95, ’96) 
 
 10 
 
 2.56 
 
 7.9 
 
 3.9, 4.5, 4.8 
 
 At Carbondale, Pa., compressed-air locomotives have hauled coal for 
 1 5 cents per ton-mile, at Mill Creek, Pa., for 3.77 cents per ton-mile, and 
 at Glen Lyon, Pa., for 1.89 to 1.93 cents per ton-mile. 
 
 THIRD-RAIL MINE LOCOMOTIVES. 
 
 By W. L. Affelder* 
 
 Traction locomotives have overcome practically every obstacle that has 
 appeared in their path except that of grade. No conservative manufacturer 
 will recommend a traction locomotive for a haulage in which the grade 
 against the loaded trips exceeds 5 per cent., and the more conservative 
 Diace 4 ner cent, as- the practical limit. A traction locomotive will work 
 successfully on considerably steeper grades when the grades are short and 
 all of a large trip will not be on the grade at the same time, but where a 
 grade of over 4 per cent, is continuous over a considerable distance, some 
 other svstem of mechanical haulage should be adopted. This fact led 
 several companies into experimenting on electric haulage m which tractive 
 force due to the weight of the locomotive would not be a factor, and in 
 which friction and gravity alone would have to be overcome. 
 
 In 1899, the Morgan Electric Machine Co. placed their first third- 
 and-traction-rail locomotive, or so-called ‘‘sprocket locomoUve, in the Star 
 City Ind., mine of the Harder & Hafer Co., of Chicago. This system, as 
 developed, combines the flexibility of the trolley-traction system and the 
 advantages of the wire-rope systems in surmounting grades Ihe ihndiaA 
 is generally placed 5 inches to the right of the center of the track. There 
 
 are three sizes of third rail— standard, heavy, and special, and the compo- 
 nent parts of < 
 
 spike^to^he* tie's, which are first trimmed, if necessary, to receive it. On 
 this stringer are spiked, at intervals of about 18 inches, pme blocks E of 
 
 nent parts of each are made in 16-foot lengths. The standard size will 
 be described. A 6*"xH" white : pine bottom _stnnger^ r securely 
 
 *See 
 
 ‘Mines and Minerals,” March, 1904. 
 
THIRD-RAIL MINE LOCOMOTIVES. 
 
 410a 
 
 sufficient thickness to bring the height of the completed fluid rail 4 inches 
 above the height of the steel rails. Two tongitudinal pme strips D, each 
 in. X 1£ in., are spiked to the blocks, leaving a H-m. slot between them. 
 They are trimmed on top in such a way as to allow the iron track C , which 
 is 4 in. wide and 1-in. thick, to be partially countersunk. This track con- 
 sists of a flat bar of iron with lf-in. square holes punched through it at 
 intervals of the same distance. It is made continuous by means of perforated 
 fish plates J that are securely bolted to the ends of the two bars at _ a joint. 
 Two pine strips DD, which are similar to strips I) except that they are 
 trimmed on the lower side to cover the iron rail, are laid on the strips I) 
 and fastened to them with 4-in. spikes. By means of an insulated c0 PPer 
 cable, which is connected with the iron bar, the positive electric current is 
 
 introduced into the third rail. The wooden portion of the rail acts both as 
 a carrier of the iron bar or rack and as a medium for insulating it. The 
 negative current is carried by the bonded steel rails. 
 
 The locomotives are made in two standard sizes: one having an oU-id.r. 
 motor and the other two motors of the same kind. 
 
 On each of the two axles B are two track wheels H and one steel sprocket 
 A with its accompanying gear. The track wheels are tight on the axles, but 
 the sprocket and gear are loose, the sprocket being insulated both from the 
 axle and from the gear by means of maple blocks, shown m solid black m the 
 figure. The teeth of the two sprocket wheels, which are geared to run in 
 unison, run in the slot between the two wooden strips D D and D of the 
 
410b 
 
 HOISTING AND HAULAGE. 
 
 third rail and engage the iron rail C. In coming in contact with the charged 
 iron rail, they take up the current, and through the agency of copper con- 
 tact springs that rub against them, impart the current to the motor or 
 motors. In revolving, the motor drives the sprockets, and as the construc- 
 tion of the third rail is such as to ma k e it absolutely rigid, the movement of 
 the sprockets in the perforated iron track produces motion of the locomo- 
 tive. As all transmission of power is through the cut-steel gear-wheels, loss 
 of power is entirely eliminated. 
 
 The following advantages are claimed for this system: The full power of 
 the third-rail locomotive — weighing only 6.000 lb.— is available at all times 
 regardless of the grades, within limits, or slippery character of the track sur- 
 face. The third rail can be extended easily and cheaply by the track layers 
 as the regular track is extended. There is little liability of explosions being 
 caused by the electric current, as the conductor is close to the floor; and 
 from this* same line power can be taken off at any point to light the mine 
 and to run machinery. No sand, trolley pole, or trolley wire are required: 
 and men and animals are safe, as it is practically impossible for them to 
 accidentally come in contact with the electric current. Also, heavy fells of 
 roof will not injure the third rail. Moreover, in this system, only a com- 
 paratively small weight has to be moved, thus saving in power and wear. 
 
 A modification of the above-described system is manufactured by the 
 Morgan Electric Machine Co., consisting of a combination of a complete 
 trolley traction locomotive with the third-rail feature for use on grades. 
 
 Gathering locomotives are used to take the cars from the rooms. They are 
 similar in their general construction to the ordinary traction locomotive bnt 
 are shorter and lower. In traveling along the entries the locomotive obtains 
 its power by means of a regular trolley attachment, but when leaving an 
 entrv to go into a room the trolley pole is fastened down and a flexible 
 insulated cable is hooked npon the trolley wire and upon the track. The 
 current returns through the bonded rails in rooms where steel rails are used, 
 and when wooden rails are used in the rooms a double cable like that on a 
 mining machine is used, one cable being attached to the trolley wire upon 
 the entrv and the other to the ground wire or entry rail. The reel upon 
 which the cable winds acts automatically to keep the cable taut in winding 
 and unwinding. It is operated by chains and sprocket wheels or by friction 
 plates. Several makes of gathering locomotives are now being operated 
 successfully, both in anthracite and bituminous mines. 
 
 MINE ROADS AND TRACKS. 
 
 Underground or mine-ear tracks should be solidly laid on good sills, rest- 
 ing on the solid floor of the mine. They should be well ballasted, and 
 should have good clean gutters on the lower side of the entry, so that the 
 rails mav be protected as much as possible from the action .of the mine 
 water. Much of the following data, on mine roads is based on an article on 
 Mine Roads.’’ by Mr. H. L. Auchnmty. “Mines and Minerals." March. 
 1900. 
 
 " Grade.— The trades depend entirely on oirotimstanees. V.::. when possible, 
 the grade should be in favor of the load, and should be at least 5 in. in 100 ft. 
 to insure flow in the gutters alongside the track. On main roads, where 
 wagons having a capaeitv of 15 to 2.5 tons are hauled by anim a l power, the 
 grades should not exceed 14 to 24 in flavor of the loaded wagon. Such a rate 
 of grade provides for an easy return haul of the empty trip without wearing 
 out the stock, and likewise insures good drainage. With grades under 14, 
 unless the ditches are kept perfectly clean, the drainage is apt to be sluggish, 
 and then, in low places, we are sure to find a wet and muddy track, which 
 is a great source of waste energy. . 
 
 Where banting is done by locomotives, whether by compressed air or 
 steam, the adverse grades should not be over 1.54 to 2.54 if it can possibly be 
 avoided. When gradients are heavy, too great a percentage of the tractive 
 power of the locomotive is consumed in drawing itself up the grade. 
 
 Ties should be spaced about 2 ft. apart, center to center, m ak i n g 15 to a 
 30' rail. The rail should be well spiked to the ties with four spikes to each 
 tie. the joint between two rails on one side of the track being located about 
 midway between two joints on the opposite rail. Care should be taken in 
 locating the spikes that they are not all in the center of the tie. thereby 
 
MINE ROADS AND TRACKS. 
 
 411 
 
 causing a tendency to split the same. It is best to place them each side of 
 the center with two spikes between the rails, on one side, and the two spikes 
 on the outside of the rail on the other side of the center of the tie. With tne 
 
 spikes so located, there is no tendency totototo pp^oVtlfe tif Tte” 
 outside and inside spike are on the same side of the center oi tne lie. lies 
 having a 5 in. face and 4 in. deep hy 5£ ft. in length should be 
 ordinary sizes of rail, i. e., 16 lb. to 20 lb., and, m general, the thickness 
 should be sufficiently great that the spike does not pass e ^ re ^^|^?racks 
 tie as then its holding power is greatly diminished. On haulage tracks 
 where 35-lb. to 40-lb. rail is used, the ties should be at least 5 in. deep 
 have a face of 6 in., the ties ordinarily used for lighter sizes of rails being 
 entirely too thin for rails of this weight, as a larger spike than the ordinary 
 3 in. X I in. is required to securely hold the vails to place. The ends of the 
 ties should be lined up along one side of the track, so that they are all the 
 same (Stancl from the rail and, with each tie placed at right angles to 
 the rail as it should be, we have a well-spaced, neat-looking track, which, 
 when we!l tamped with the ballast, is perfectly solid. On curves, the ties 
 should be laid so as to form radii of the curves of the track. , 
 
 Rails.— The weight of rail to be chosen in any individual case depends 
 entirely on the weight of wagons used, and the motive power. For wagons 
 whose ^Lpacity is alout 1.5 tons, the weight of rail, when the motxve power is 
 live stock should not be less than 16 lb. per yd., while for wagons having a 
 capacity of 2 tons or over, a 20-lb. rail should be used. There is no economy 
 in P using a very light rail, as the base is gradually eaten away by the mine 
 water when it comes in contact with the metal, and m the case of a heav} 
 section of rail it will be much longer before the rail becomes weakened. 
 
 On main roads, where haulage machinery of one kind or another is used, 
 the weight of rail for 2-ton wagons should be from 25 lb. to 35 lb. per yd., 
 and on steep slopes as high as 40 lb. per yd. . ~ 
 
 In the case of locomotive haulage, authorities claim that the weighty 
 should be regulated by allowing 1 ton for each driver for each 10 lb. weight 
 of rail per yd . 
 
 Gaupe —The gauge of the track in coal mines should not be less than 30 in. 
 nor more than 48 in. A mean between these two, or a gauge 
 42 in. is desirable, because it combines, to a certain extent, advantages 
 claimed for the extremes. The advocates of broad gauges believe that the 
 greater stability of the track and the consequent reduction ^ ^ulage expen- 
 ses, the increased capacity of the broad-gauged mine cars » the reduction m 
 the outlay for rolling stock, and for repairs to the same, more than equal 
 the disadvantages of broad as compared to the narrow gauges. 
 
 Advocates of the narrow gauges think that the ease of hauling around 
 sharp curves, the reduction in cost of construction, and the use of mine cars 
 with inside wheels, are advantages greater than those advanced bythe 
 advocates of the broad gauges. An allowance about im should alwaysbe 
 made between the wheel gauge and the track gauge. By so < *°iug, tne : resist 
 ance to hauling is greatly overcome, and there is no binding of the wagons 
 on the track, hence a less likelihood of having derailed wagons. With an 
 average running wagon, there is a resistance of 15 to 20 lb. per ton tractive 
 force on a level track, which would be equal to the resistance occasioned by 
 a gmde of .75^ to H, and with wagons that bind on the track, this resistance 
 is greatly increased. . f 
 
 Curves should be of as large a radius as possible, and never if possible, of 
 less radius than 25 ft. The resistance of curves is very considerable. - The 
 less the radius of the curve, and the greater the length of thee urvedtrack 
 occupied bythe trip, or train, the greater the resistance. The length of 
 wheel bases of the cars, the condition of rolling stock and of the track and 
 the rate of speed, all influence the resistance, and there is no formula 
 that will apply to all cases. In practice on surface railroads, engineers 
 compensate for curves on grades at the rate of ijh* ft. in each hundred feet 
 for each degree of curvature, the grade being stated in feet per hundred. 
 In mine work, this compensation is not made, as the gam will not pay for 
 the labor that must necessarily be employed to do work m a thoroughly 
 scientific manner. , 
 
 Sharper curves can be used on narrow-gauge roads than on broad-gauge 
 roads, because the difference in length of the inner and ^terrailsoncurves 
 on the same degree is not quite so great, and also because the wheel bases 
 of cars are less. The track should be spread about * in. on easy curves, and 
 
412 
 
 HOISTIXG AND HAULAGE. 
 
 on very short curves about 1 in., or as much as the tread of the wheels will 
 permit’. A good rule is to widen the track ^ in. for each 2£° of curvature. 
 Short and irregular curves are to be avoided whenever possible, as they 
 increase the load and are destructive to rails and rolling stock. When a sharp 
 curve is necessary, the rail should be bent to the right curvature by a 
 portable rail bender, or by a jack and clamps. 
 
 To Bend Rails to Proper Arc for Any Radius.— Rails are usually 30 ft. long, and 
 the most convenient chord to use in bending mine rails is 10 ft. 
 
 Then, having the radius and chord, we find the rise of middle ordinate by 
 squaring the radius, and from it take the square of £ the chord. Extract the 
 square root of the remainder and subtract it from the radius; the result will 
 be the rise of the middle ordinate. Thus, having a radius of 30 ft. and a 
 chord of 10 ft., the middle ordinate will be 
 
 30 — y'&E — 5-, or 0.42 ft. 
 
 Rail Elevation.— In elevating rails on curves, consider whether the hauling 
 is to be done bv a rope, or by a locomotive, or electric motor. For either of 
 the latter, elevate the rail on the outside of the curve; but for the first, 
 elevate the inner rail, since as the power is applied by a long flexible rope, 
 there is alwavs a tendency for both rope and wagons to take the long chord 
 of the curve as soon as the point of curve is reached. On slope haulages, 
 operated bv a single rope, when the weight of the wagons traveling on the 
 grade of the slope is sufficient to draw the rope off the hoisting drum, the 
 rails on curves should be elevated on the outside, the effect then being 
 similar to that of a locomotive, i. e.. the centrifugal force tends to throw the 
 wagon to the outside of the track. In such cases, the elevation should be 
 moderate so as not to interfere with the trip when drawn out again by the 
 rope—the opposite effect being then experienced. On an 18° curve (319 ft. 
 radius 1. an elevation of 2 in. or 3 in. in the outer rail, where the haulage was 
 by slope rope, has never given any trouble in operating. In general, the 
 elevation of rail necessary for different degrees of curvature for a 42" track 
 gauge should be made in accordance with the following table: 
 
 Table of Elevations. 
 
 For outer rail of curves for a speed of 10 to 15 miles per hour and a gauge 
 of track of 42 in. for locomotives; or for slope haulages where cars run down 
 grade by gravity. 
 
 Degree 
 
 of 
 
 Curve. 
 
 Radius 
 
 of 
 
 Curve (Ft.). 
 
 Elevation 
 of Outer 
 Rail (In.). 
 
 Degree 
 
 of 
 
 Curve. 
 
 Radius 
 
 of 
 
 Curve (Ft.). 
 
 Elevation 
 of Outer 
 Rail (In.). 
 
 1 
 
 5,729.6 
 
 i 
 
 I 
 
 10.0 
 
 573.7 
 
 H 
 
 2 
 
 2.864.9 
 
 12.0 
 
 478.3 
 
 
 3 
 
 1,910.1 
 
 5 
 
 T6 
 
 TS 
 
 15.0 
 
 383.1 
 
 if 
 
 4 
 
 1,432.7 
 
 18.0 
 
 319.6 
 
 in 
 
 5 
 
 1.146.3 
 
 T6 
 
 20.0 
 
 287.9 
 
 
 6 
 
 955.4 
 
 H 
 
 57.3 
 
 100.0 
 
 
 7 
 
 819.0 
 
 a 
 
 95.5 
 
 60.0 
 
 4£ 
 
 8 
 
 9 
 
 716.8 
 
 637.3 
 
 _ 
 
 e 
 
 l 
 
 114.6 
 
 50.0 
 
 4£ 
 
 I 
 
 No elevation should be over 4£ in., which would be equivalent to an 
 elevation of 6 in. for standard track gauge of 4 ft. 9 in., the latter being con- 
 sidered as the maximum for standard gauge. A ^ 
 
 Rollers.— The rollers on level tracks should not be more than about 20 ft. 
 apart to properly carry the rope, and on gravity slopes where the lower end 
 of the slope gradually flattens off. the distance between rollers should not 
 be more than 12 to 15* ft., as this spacing allows the trip of wagons to run 
 much farther, by keeping the rope well off the ties, than if they are farther 
 apart, thereby not supporting the rope, and causing a great amount of 
 friction between the rope and the ties. With tracks in fair shape and rollers 
 12 to 15 ft. apart, the resistance, due to the rope in running empty wagons 
 down grades varying from 3.8* to 6.2*, varied from 6* to 15* of the weight of 
 ropes by actual trial. 
 
MINE ROADS AND TRACKS. 
 
 413 
 
 Switches.— The switch, or latch , most commonly used in mines is shown 
 in Fig. 7. When the branch or siding is in constant use, an ordmar\ railway 
 lrog is substituted for the bar b. The latches a, a : are wedge-shapeS .bars of 
 iron (made as high as the rail) withan^ “nne^ldtoletoW^/a rod 
 
 Fig. 7. 
 
 attached to a lever so that they may 
 both be moved at once from the side of 
 the track, or by a person situated some 
 distance away. This switch is made self- 
 closing or automatic whenever it is 
 necessary to run all the cars off at the 
 branch (the switch then being used 
 only to admit cars to the main track) 
 by attaching the latches through a bar 
 or lever to a metallic spring, a stick of 
 some elastic wood, or a counter weight, 
 to pull them back into a certain position 
 whenever they have been pushed to one 
 side or the other by the passage of a car 
 
 -I ^ ^ n -t-i-tVl i oofi rtn c nf 
 
 on the main track. Figs. 10, 11, 12, and 13 show some of the applications of 
 
 these spring latches or automatic switches. , 
 
 A modification of this switch is shown m Fig. 8, which represents a form 
 of double switch. These latches are set by the drivers, who kick them over 
 and drop a small square of plate iron between them to hold them m place 
 This switch costs more than the other style and is better adapted to outside 
 
 r0a The^ordinary movable rail switch in common use on all surface railways 
 is sometimes used in mine roads. It is commonly used m slopes arranged as • 
 shown by Fig. 12, to replace latches set by the 
 car, and is also largely used in outside roads. 
 
 For crossings, ordinary railway frogs and 
 grade crossings are sometimes used, as is also 
 a small turntable, which then answers two 
 purposes. More frequently the plan shown 
 in Fig. 9, in which four movable bars are 
 thrown across the main track whenever the 
 other road is to be used, is adopted. 
 
 The subordinate road is built from li to 2 
 in. higher than the main road, to allow the 
 
 bars to clear the main-track rails. . . , . 
 
 Turnouts.— On gangways or headings used as mam haulage roads, turnouts 
 should be constructed at convenient intervals to allow the loaded and 
 emptv trips to pass. These turnouts should be long enough to accommodate 
 from 5 or 6 up to 15 or 20 cars. The switches at each end may be made seli- 
 acting so that the empty trip, coming in, is thrown on the turnout, and in 
 running out on the main track at the otner end, the loaded cars open the 
 switch, which immediately closes. 
 
 As there is constant trouble with self-setting switches, either from small 
 
 fragments of coal or slate clogging them 
 up, or from insufficient power of the spring 
 to move them, they are viewed with dis- 
 favor by many mine managers, who do 
 not care to use them under any conditions. 
 
 Slope Bottoms.— At the foot of a slope, 
 or at the landing on any lift, the gang- 
 way is widened out to accommodate at 
 least two tracks— one for the empty and 
 one for the loaded cars. The empty track 
 should be on the upper side of the gang- 
 way, or that side nearer the floor of the 
 seam, and the loaded track on that side 
 of the gangway nearer the roof of the seam. 
 
 An arrangement of tracks often used 
 is shown in Fig. 10. At a distance of 40 
 
 IS SIlOWll 111 -Tig. -L V. Al/ Cl uioiauuv 
 
 or 50 ft. above the gangway, the slope is widened out to accommodate the 
 branch leading into the gangway loaded track. This branch descends 
 with a gradually lessening inclination until nearly at the level of the gang- 
 way it turns into the main loaded track. A short distance abov e the gangway , 
 
414 
 
 HOISTING AND HAULAGE. 
 
 a bridge or door is placed, which, when closed, forms a latch by which the 
 empty cars are taken off the slope. The empty track is about 6 ft. higher 
 than the loaded track, and is carried over it on a trestle. The illustration in 
 Fig. 10 shows the plan as arranged for a single slope, or one side only of a 
 
 slope taking the coal from both directions. 
 When coal is being raised from this lift, 
 the bridge is closed; the empty car comes 
 down and is run off over the bridge; the 
 car is unhooked from the rope, and the 
 chain and hook attached to the rope are 
 thrown down to the branch below on 
 which a loaded car is standing; the loaded 
 car is attached, the signal given, the car 
 ascends to the main track on the slope, 
 opening the switch— or the switch may be 
 set each time by the bottom men, by a lever 
 at the bottom of the branch. This plan can 
 only be economically applied in thick 
 seams, as the height necessary to allow one 
 track to cross the other on a trestle cannot 
 be obtained in seams of moderate thick- 
 ness without taking down a large amount 
 of top. 
 
 A more simple plan, which dispenses 
 with the bridge, is often used. The branch 
 is laid off, as shown by Fig. 10, but, near 
 the point where it enters the gangway, a switch opening into the empty 
 track is placed. By this arrangement, the. tracks cannot be as well arranged 
 for handling the cars by gravity as in the former plan, in which the empty 
 cars when detached from the rope run by 
 gravity into the empty siding, and the loaded 
 cars descend by gravity around the curve to 
 the foot of the branch, where they lie ready to 
 be attached to the rope. 
 
 When the pitch of the slope is so steep that 
 the coal or ore falls out of the cars, during 
 hoisting a gunboat is used or the cars are 
 raised on a slope carriage — in either case, the 
 arrangement of the tracks ^at lift landings is 
 entirely different. With either a gunboat or 
 a slope carriage, the arrangement of tracks 
 on the slope is the same; but, in the former 
 case, a connection between the slope and 
 gangway tracks is often advisable. When a , . , ™ 
 
 gunboat is used, the gangway tracks run direct to the slope, and a tipple, or 
 dump, is placed on each side to dump the mine cars f 
 but when the cars are raised on a slope carriage, the gangway tracks 
 run direct (at right angles) to the slope, to carry 
 the car to the cage or carriage. The fl 9 or of the 
 cage is horizontal, and has a track on it that tits 
 on the end of the gangway track when the car- 
 riage is at the bottom, and this track is arranged 
 with stops similar to those on cages used in shafts. 
 
 Another common arrangement of tracks at 
 the bottom of a slope is shown in Fig. 11. A 
 branch is made by widening the slope out near 
 the bottom, and this, being a few feet higher than 
 the main track, is used to run off the empties by 
 gravity. The loaded cars run in by gravity 
 around the curve to the foot of the slope in 
 position to be attached to the rope. 
 
 In ascending, the loaded car forces its way 
 Fig. 12. through the switch, or the switch may be set by a 
 
 lever located at the foot of the slope. When the 
 empty car descends, it runs in on the branch, where the chain is unhooked 
 and thrown over in front of the loaded car, and runs around the curve into 
 
 ll^olSSved that in this plan the loaded car (and consequently the 
 
 Fig. 11. 
 
 the 
 
MINE ROADS AND TRACKS. 
 
 415 
 
 bottom men) stands on the track in line with the slope, and is in danger from 
 any objects falling down the slope, or from the breakage of the rope or coup- 
 lings; but this can be obviated by making the bottom on the curve. The 
 illustration in Fig. 11 shows only one side of the slope; the other side is, of 
 course, similar. 
 
 All these plans necessitate the location of that part of the gangway near 
 the slope, in the upper benches of the coal or near the top rock. The gang- 
 way is then curved gently around toward the floor, so that, when it has been 
 driven far enough to leave a sufficiently thick pillar, the bottom bench is 
 reached and the gangway is then driven along the bottom rock. 
 
 A very different bottom arrangement is shown by Fig. 12, which also 
 represents a plan frequently adopted on surface planes. The two slope 
 tracks are merged into one a short distance from the bottom of the slope, and 
 on the opposite sides of the bottom two tracks curve around into the gang- 
 way on opposite sides of the slope. As these branches curve into the main 
 gangway tracks, a switch sends off a side track for the empty cars. The 
 switch on the slope is either set by the car— and this can be done because 
 the next loaded goes up on the same side on which the last empty descended 
 —or by a lever located at the bottom. . . 
 
 It will at once be seen that in this plan no opportunity is afforded oi 
 handling the cars by gravity. The curved branches are made nearly level, 
 and the momentum of the descending car, if quickly detached, is often 
 sufficient to carry it partly or wholly around the curve, even against a slight 
 
 adverse grade. The disadvantage above noted of having the bottom in 
 direct line with the slope (where there is danger from breakage and falling 
 material) also obtains in this plan. 
 
 In the plan shown by Fig. 13, the grades may be so arranged that the cars 
 can be entirely handled by gravity. The latches on the main-slope track 
 may be closed automatically by a spring or weight, the loaded car running 
 through them in its ascent on the slope, or both sets may be operated by a 
 single lever at the bottom. The switch at the upper end of the central track 
 (loaded) is set by a hand lever. All three sets may be linked together, so 
 that they can all be properly set by a single lever. Reference to Fig. 11 will 
 show that this is only a modification of that method. It requires space at 
 the bottom for only three tracks, while Fig. 13 requires width to accom- 
 modate four tracks, and is objectionable because it is more complicated. 
 The extra set of latches at the top of the central track, and the curvature of 
 both main tracks into this central one, must inevitably cause much trouble 
 and delay from cars jumping the track at this point. 
 
 The plan shown in Fig. 14 is open to many of the objections pertaining to 
 some of those already described, and which need not be reiterated here. It 
 can only be employed in thick seams, or in seams of moderate thickness 
 lving at a slight angle or dip. 
 
 In planning the arrangement of tracks on a slope, it is advisable to place 
 as few switches as possible on the slope itself, to keep the main track 
 
 Fig. 13. 
 
 Fig. 14. 
 
416 
 
 HOISTING AND HAULAGE. 
 
 'ESEzZEB a 
 
 LOADED*—: 
 
 unbroken, to make the tracks as straight as possible, to have nothing stand- 
 ing at the bottom in direct line with the slope tracks, and to arrange the 
 tracks so that cars are handled by gravity. . , 
 
 The arrangement of tracks near the top of the slope, and on the surface, 
 is often very similar to the bottom arrangements, as already described; but 
 as all loaded cars (except rock and slate cars, which are run off on a separate 
 switch) are to be sent off on one track, and all the empties come in on the 
 same track to the head of the slope, and as there is usually abundance of 
 room for tracks and sidings, these top arrangements are, in a measure, much 
 more easily designed. In some instances, the two mam-slope tracks run 
 into a single track near the head of the slope— a plan somewhat similar to 
 the bottom arrangement shown by Fig. 12— and the cars are then brought to 
 the surface on one track, which, after passing the knuckle, bifurcates into a 
 loaded and empty track. A similar arrangement is frequently adopted at 
 slopes on which a carriage or gunboat is used. When the two mam-slope 
 tracks are continued up over the knuckle to the surface— the most common 
 and best plan— the arrangement of tracks and switches may be planned 
 entirely with a view to the quickest and most economical method of 
 handling the cars. „ , _ . 
 
 Vertical Curves.— The vertical curves at the knuckle and bottom of a slope 
 or plane should have a sufficiently large radius, so that when passing over 
 them the car will rest on the rail with both front and back wheels. The 
 wheel base of the car must be considered in adopting the radius for these 
 curves, for if the curve is of too short a radius, there is danger of the car 
 jumping the track every time it passes over the curve. 
 
 Tracks for Bottom of Shaft.— Fig. 15 shows the arrangement of tracks at the 
 foot of a shaft, with one of the cages at surface. The grades should be so 
 arranged that from the inside latches of the crossings the empty track 
 should have a slight down grade from 
 the shaft, and the loaded track a slight 
 down grade toward the shaft. The cross- __ y __ 
 ings and the short straight piece of road 
 close to the shaft should be level. “=* 
 
 As it is often desired to move empty Fig. 15. 
 
 cars from one side of the shaft to the 
 
 other, without stopping the hoisting, a narrow branch road should be cut 
 through the shaft pillar, and used for this purpose. Where the pitch of the 
 seam prevents this, a road should be laid alongside the shaft, room to 
 accommodate it being cut out of the rock on the side most desirable. (See 
 also Shaft Bottom, page 276.) _ _ „ , 
 
 In arranging tracks for shaft bottoms, at tops and bottoms of slopes, on 
 coal bins, for mechanical-haulage landings, at foot of slopes or shafts, or m 
 the body of the mine, it is customary to provide double tracks of sufficient 
 length to hold the requisite number of wagons for economically operating 
 the plant and with sufficient distance from center to center of tracks, and 
 from centers of tracks to sides of entries, to easily pass around the wagons 
 where it may be necessary, either in handling them, or in lubricating the 
 wheels. For wagons with a capacity of from 1£ to 2 tons, it generally 
 requires an entry to be about 15 to 17 ft. wide in the clear for ordinary land- 
 ings in the body ’of the mine, while at shaft bottoms the necessary width 
 may attain 17 to 18 ft. in the clear, owing largely to location and local 
 requirements. The curved crossovers connecting the tracks at shaft bot- 
 toms should be designed with radii of as great length as can be introduced, 
 thereby giving an easy running track. They should not be less than from 
 20 to 50 ft. on center lines for ordinary gauge of tracks, i. e., 36 to 44 in. 
 
 On landings constructed in the body of the mine for the reception ot 
 empty and full wagons handled by mechanical haulage from shaft or slope, 
 and from this point transported by animal power to the various working 
 places in the mine, a grade of about 1 f> in favor of the loaded wagons to be 
 handled by the stock will be found quite an assistance in delivering the 
 wagons to the haulage. The frogs and switches for these landings, as well 
 as those required at the shaft or slope, should be formed of regular track 
 rails, and can generally be arranged to be thrown by a spring or a con- 
 veniently located hand lever, as has been described, instead of being kicked 
 to position, as was the custom at one time. 
 
 Besides these usual arrangements of shaft-bottom landings, at many 
 plants the natural grades of the entries can be taken advantage of in 
 designing convenient and economical methods for handling the mine cars. 
 
MINE ROADS AND TRACKS. 
 
 417 
 
 For instance, where the coal is to be hauled from the dip workings of a 
 mine by some form of mechanical haulage, and a summit can conveniently 
 be arranged for in the track on the same side of the hoisting shaft, at the 
 proper distance therefrom, to accommodate the requisite number of loaded 
 wagons to be hauled, thus allowing them to run by gravity over, say, a 
 1 <f 0 grade to the shaft, several varieties of empty-track arrangements can be 
 made. The most simple form is to have the empty wagon descend a short 
 grade of from ApJo to 5$ when pushed from the cage by the succeeding full 
 one. The momentum thus secured is quite sufficient to carry the car up an 
 opposing grade of about 1.5 Jo. It again descends on the same track, and 
 passing through an automatic switch, continues to the empty-car siding. 
 From this latter point it is handled by the regular haulage machinery, and 
 in its route passes around the shaft through an entry especially prepared for 
 this arrangement. A shaft bottom so constructed is very economical to 
 operate, requiring but few men to handle the wagons. 
 
 Occasionally, it becomes more expedient to have a separate short haulage 
 to draw 'the empty wagons to the main haulage when it cannot be easily 
 arranged to construct a complete gravity landing. Several other modifica- 
 tions of such a general design can be made. All the different devices, 
 however, depend largely on the local requirements of the particular mine 
 under consideration. , . . „ ^ , , , . 
 
 When endless-rope haulage is employed, it is generally found to be most 
 convenient to have the landings for full and empty wagons, m the body of 
 the mine, reached by switches off of the main-haulage track, the cars 
 coming on and leaving the main track at slight knuckles introduced in the 
 track, in order to allow a place for the passing of the rope, which then 
 moves along through a short cut or channel through the switch rails. 
 The flanges of the wagons pass over the rope in this manner without any 
 injury to it. • 
 
 Surface Tracks for Slopes and Shafts. — The arrangement of the tracks on the 
 surface naturally differs at every mine, owing to the different existing 
 conditions. All surface roads should be so arranged that the loaded cars can 
 be moved with the least possible power, always looking out for the return of 
 the empties with as little expenditure of power as possible. To secure the 
 running of the loaded cars from the mouth of the shaft or slope by gravity , a 
 slight grade is necessary, the amount of which depends on the friction of 
 the cars, which varies greatly. Care should be take that an excessive grade 
 is not constructed, or there will be trouble in returning the empties from the 
 dump to the head of the shaft or slope. . , 
 
 The tracks connecting the top of the shaft and the tipple may be very 
 short, or of considerable length, depending on the conditions at each mine. 
 Usually from 20 to 60 ft. will be sufficient, although no definite rule can be 
 given for this. . , ’ x , , _ . , 
 
 There are two general arrangements of tracks about the head of a shait: 
 First, where the loaded cars are removed from the cage and the empty cars 
 placed upon it from the same side of the shaft; second, where the loaded 
 cars are removed from one side of the shaft and the empty cars returned to 
 the cages from the opposite side of the shaft. 
 
 In either case there are usually several empty cars on the platform ready 
 to be put on the cages when the loaded cars have been removed. 
 
 Where the conditions are such that the loaded cars can be run by gravity 
 to the dump, a good plan is to have a short incline, equipped with an endless 
 chain, in the empty track. The empty cars can be run to the foot of this, 
 hoisted by machinery to the top, and thus gain height enough to run them 
 back to the shaft or slope by gravity. 
 
 At the Philadelphia & Reading Coal & Iron Co.’s Ellango wan colliery, 
 where the tipple at the head of the breaker is above the level of the head 
 of the shaft, the following plan is used: The loaded cars are taken off 
 the east side of the cages, and run by gravity to the foot of an incline, 
 where the axles of the car are grasped by hooks on an endless chain 
 and the car pulled up to the tipple. After being dumped, the car is run 
 back from the tipple to the head of the incline, and is carried to the 
 foot of the empty track of the incline by an endless chain. The foot of 
 the empty track is several feet higher than that of the loaded track, and the 
 cars are run by gravity around to the west side of the cages, and are put on 
 from that side. The empty cars, as they run on the cage, have momentum 
 enough to start the loaded car off the cage and on toward the foot of the 
 incline. There are a number of hooks attached to both the empty ana 
 
418 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 loaded chain on the incline, and there are often several loaded and several 
 empty cars on different parts of the plane at once. This arrangement 
 permits of the hoisting of from 700 to 800 cars per day out of a shaft 110 yd. 
 deep, with single-deck cages. # * 
 
 Another excellent arrangement for handling coal on the surface is the 
 invention of Mr. Robert Ramsey, and has been adopted by the H. C. Frick 
 Coke Co. and a number of other prominent operators. A description of 
 this arrangement as applied at the H. C. Frick Coke Co.’s Standard Shaft 
 is as follows: The landing of the shaft is made slightly higher than the level 
 of the tipple, which is north of the shaft. South of the shaft is located a 
 double steam ram, one ram being directly in line with the track on each 
 cage. Directly in front of the rams is a transfer truck, worked east and 
 west by wire rope. The loaded car on the cage is run by gravity to the 
 tipple, where it is dumped by means of a nicely balanced dumping arrange- 
 ment. As soon as it is empty it rights itself and runs by gravity alongside 
 the shaft to the transfer truck, which carries it up a grade to a point directly 
 in line with the cage that is at the landing, and one of the steam rams 
 pushes it on the cage, and at the same time starts the loaded car off toward 
 the tipple. This second loaded car is then returned by the same means to 
 the opposite cage. The whole mechanism is operated by one man, by means 
 of conveniently arranged levers, each of which is automatically locked, 
 except when the proper time to use it arrives. It is therefore impossible 
 for the topman to work the wrong lever and put an empty car into the 
 wrong compartment of the shaft. Besides the one man at the levers, there 
 is but one other man employed at the tipple, and his work is solely to look 
 after the cars when dumping. All switches are worked automatically, and 
 the average hoisting at this shaft is at the rate of 3 wagons per minute. The 
 shaft is about 250 ft. deep, and single-deck cages are used. 
 
 The Lehigh & Wilkes-Barre Coal Co. has a system in use at a number of 
 collieries that has also proven very effective. In this system the loaded cars 
 are run by gravity from the cage to the dump, and the empties are hauled 
 from the dump back to a transfer truck by a system of endless-rope haulage. 
 The transfer truck carries the car to a point opposite the back of the 
 cage. The empty car runs by gravity to the cage, and its momentum starts 
 the loaded car on the cage on its way to the dump. This system necessi- 
 tates the employment of more topmen, but is a very good one. At the Not- 
 tingham shaft, which is 470 ft. from landing to landing, from 140 to 150 cars 
 per hour are hoisted on single-deck cages. 
 
 ORE DRESSING AND THE PREPARATION 
 
 OF COAL. 
 
 CRUSHING MACHINERY. 
 
 The object of crushing ore or coal is: first, to free the mineral or other 
 valuable constituents from the gangue, slate, pyrites (sulphur), or other 
 worthless or objectionable constituents so that they can be subsequently 
 separated; or, second, simply to reduce the size of the individual pieces and 
 so get the material into a more salable or convenient condition for use. 
 
 Selection of a Crusher.— The style of crusher employed is influenced by the 
 following conditions: (a) The amount of material to be crushed in a given 
 time, (b) The size of the material as it goes to the crusher, (c) The 
 physical characteristics of the material to be crushed; that is, whether it is 
 hard or soft, tough or brittle, clayey or sticky. ( d ) The object of the crush- 
 ing; that is, whether it is to free the mineral constituents or simply to reduce 
 the size of the individual pieces, (e) The character of the product desired; 
 that is, whether an approximately sized product is desirable and whether 
 dust or fine material is objectionable. 
 
 All crushing machinery may be divided into the following classes: Jaw 
 crushers, gyratory crushers, cracking rolls, disintegrating rolls, crushing 
 rolls, roller mills, ball mills, stamp mills, hammers, and miscellaneous 
 forms of crushers. 
 
CRUSHING MACHINERY. 
 
 419 
 
 JAW CRUSHERS. 
 
 With iaw crushers, the material is crushed between two jaws, one or 
 both being movable. All jaw crushers have the common detect of imparting 
 a considerable amount of vibration or shake to the framework of the build- 
 ins containing them, owing to the reciprocating 
 
 ing parts. There are three 
 styles of jaw crushers in 
 common use. 
 
 The Blake crusher is 
 shown in Fig. 1, a being a 
 fixed jaw and b a movable 
 jaw that is operated by a 
 toggle joint and the pitman 
 d from a suitable crank- 
 shaft. The jaw b is hung 
 or pivoted at the top. The 
 advantages of this style are 
 as follows: The large pieces 
 of rock to be crushed are 
 received between the 
 upper part of the jaws, 
 where the motion is least 
 and the purchase or lever- 
 age greatest, so that they 
 are broken with the small- 
 est possible expenditure of 
 
 Fig. 1. 
 
 energy. The movement of the jaws is greatest at the discharge opening, 
 thus 8 affording a free and rapid discharge of the material crushed, and 
 insuring a large capacity for the machine. The principal disadvantage is 
 that the great variation in the discharge opening results m a considerable 
 range in the size of the material delivered. , . . 
 
 This style of crusher has found a wide field for breaking down material 
 
 Table of Blake Crushers. 
 
 Size of Re- 
 ceiving Ca- 
 pacity. 
 
 Approximate Prod- 
 uct per Hour, 
 Cubic Yards, to 
 2 Inches. 
 
 Weight of 
 Heaviest 
 Piece. 
 
 ■s 
 
 •pH 
 
 > 
 
 > 
 
 . O 
 
 H 
 
 Inches. 
 
 Lb. 
 
 Lb. 
 
 3X li 
 
 Laboratory 
 
 40 
 
 100 
 
 6 X 2 
 
 One 
 
 560 
 
 1,200 
 
 10 X 4 
 
 Three 
 
 1,800 
 
 4,900 
 
 10 X 7 
 
 Five 
 
 3,800 
 
 8,000 
 
 15 X 9 
 
 Eight 
 
 7,400 
 
 15,500 
 
 15X10 
 
 Nine 
 
 7,800 
 
 16,000 
 
 20 X 6 
 
 Ten 
 
 5,300 
 
 11,200 
 
 20 X 10 
 
 Ten 
 
 8,100 
 
 18,300 
 
 12X30 
 
 Sixteen 
 
 14,200 
 
 33,000 
 
 15X30 
 
 Twenty 
 
 14,200 
 
 35,000 
 
 Extreme Dimensions. 
 
 Length. Breadth. 
 
 Ft. In. 
 
 1 
 
 10 
 
 0 
 
 1 
 
 6 
 
 6 
 
 3 
 
 10 
 
 10 
 
 10 
 
 Height. 
 
 Ft. In. Ft. In. 
 
 6 
 
 1 
 
 3 
 9 
 0 
 5 
 
 11 
 
 9 
 
 4 
 4 
 
 0 10 
 2 3 
 
 9 
 
 5 
 
 11 
 
 11 
 
 6 
 11 
 
 4 
 
 4 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 250 
 
 s-< . 
 
 ^ <X> 
 
 ft 0 
 a> 
 
 g © 
 
 H 
 
 4 
 
 6 
 
 8 
 
 15 
 
 15 
 
 15 
 
 20 
 
 30 
 
 30 
 
 and preparing it for other crushers, or for breaking large quantities of any 
 material where an approximate sizing is not essential. 
 
 The Dodge crusher, Fig. 2, has a fixed jaw a and a movable jaw b, operated 
 by a cam on the shaft g. The movable jaw is pivoted at the bottom, so that 
 the minimum movement between the jaws is at the discharge opening. The 
 advantage of this is that the least movement occurs at the discharge opening, 
 
420 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 and hence the product is of a fairly uniform size, so that the crusher may 
 be used as a rough sizing apparatus. The disadvantages are that the large 
 pieces of rock have to be crushed 
 in the upper part of the space ^ 
 between the jaws, where the 
 motion is greatest and the pur- 
 chase or leverage least, thus re- 
 quiring an excessive amount of 
 power, especially when dealing 
 with hard material. The move- 
 ment of the jaw at the discharge 
 opening is so much less than 
 that above that there is danger 
 of clogging or blocking the ma- 
 chine, especially when working 
 upon tough or sticky material. 
 
 The capacity of the Dodge style , , , 
 
 of machine is less than that of the Blake. It is used largely as a secondary 
 crusher, or for crushing comparatively small amounts of material where an 
 approximately sized product is desired. 
 
 Fig. 2. 
 
 The Dodge Crusher. 
 
 No. 
 
 Size of Jaw 
 Opening. 
 
 Diameter of 
 Pulleys. 
 
 Width of Belt 
 Used. 
 
 Horsepower 
 
 Required. 
 
 No. Tons per 
 Hour, Nut Size. 
 
 Revolutions per 
 Minute. 
 
 Weight Complete. 
 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 
 • 
 
 
 
 1 
 
 4 X 6 
 
 20 
 
 4 
 
 2 to 4 
 
 i to 1 
 
 275 
 
 1,200 
 
 2 
 
 7 X 9 
 
 24 
 
 5 
 
 4 to 8 
 
 1 to 3 
 
 235 
 
 4,300 
 
 3 
 
 8X12 
 
 30 
 
 6 
 
 8 to 12 
 
 2 to 5 
 
 220 
 
 5,600 
 
 4 
 
 10X16 
 
 36 
 
 8 
 
 12 to 18 
 
 5 to 8 
 
 200 
 
 12,000 
 
 Roll* Jaw Crushers.— Fig. 3 is a sectional view of a Sturtevant roll-jaw 
 crusher. The rolling motion of the jaw subjects the material to a rolling 
 and squeezing action, instead of a direct squeeze. The product of this 
 
 crusher is approximately sized and 
 there is no greater producer of sized 
 output. The adjustment of the 
 machine for fine crushing neces- 
 sarily contracts the space for dis- 
 charge, and thereby lessens its 
 capacity. When set wide, or for 
 material from 1 to H inches in size, 
 the discharge is very free and the 
 capacity is claimed to be greater 
 than that of any other jaw and 
 toggle machine. 
 
 Gyratory Crushers.— These crush- 
 ers, Fig. 4, are all large capacity, con- 
 tinuous-action crushers, a is a ring 
 or hopper against which the material 
 is crushed by a conical head c, which 
 Fig. 3. fits on a shaft g, the bottom of which 
 
 is placed in an eccentric bearing 
 so that the amount of space between a and c varies as the head rotates. 
 The material to be crushed is dumped into the receiving hopper h, and the 
 machine is thus automatically fed. 
 
ROLLS. 
 
 421 
 
 The advantages of this style are that the large pieces of material are 
 received at the top of the jaws, where the motion is least and the leverage 
 or purchase greatest, thus reducing the work necessary m this heavy 
 y preliminary crushing. The relative move- 
 
 ment between the crushing members is a 
 maximum at the discharge opening, but 
 the amount of this movement is so small 
 that the product is approximately sized. 
 The fact that the maximum movement is 
 at the point of discharge assures a free 
 discharge. There is practically no shaking 
 imparted to the building by gyratory 
 crushers. Their capacity is very great, and 
 with a large size, material may be dumped 
 into the hopper h directly from the cars. 
 For small capacity a gyratory crusher is 
 more expensive than a jaw crusher. 
 
 Frequently, where very great amounts 
 of material are to be crushed, large gyra- 
 tory crushers are used as secondary crush- 
 ers after jaw crushers of the Blake pattern, 
 the discharge from the jaw crushers ran- 
 ging from 6" to 12" cubes, and that from the 
 gyratory crushers from 1£" to 2£" cubes. 
 (See table on page 422). 
 
 ROLLS. 
 
 Cracking Rolls.— This is a general name 
 applied to rolls having teeth, which are 
 usually made separate and inserted. 
 These rolls, Fig. 5, are employed for 
 Fig 4 breaking coal, phosphate rock, etc., the 
 
 object being to break the material into 
 angular pieces with the smallest possible production of very fine material. 
 The principal field for cracking rolls is in the preparation of anthracite 
 coal, and the exact style or design of the roll depends largely on the 
 physical condition of the coal under treatment. In most cases, the rolls 
 are constructed with an iron cylinder having steel teeth inserted, the 
 size, spacing, and form of the 
 teeth depending on the size 
 and physical condition of 
 the material to be broken. 
 
 Cracking rolls vary from 12 
 to 48 in. in diameter and 
 from 24 to 36 in. in face 
 width. The teeth of the 
 larger sizes are from 3 to 3| 
 in. high, and of the smaller 
 1 in. or less. 
 
 The average practice in 
 the anthracite regions of 
 Pennsylvania is to give the 
 points of the teeth a speed of 
 about 1,000 ft. per minute, 
 though the speed in different 
 cases varies from 750 to 1,200 
 ft. per minute. One of the 
 largest anthracite companies 
 has a standard roll speed of 
 97.5 R. P. M. for the main 
 rolls and 124.5 R. P. M. for 
 the pony rolls. The harder 
 the coal, the faster the rolls , 
 
 can be run. If run slow and overcrowded, the rolls will make more culm 
 than when driven at a proper speed. One advantage of comparatively 
 fast driven rolls is that the higher speed has a tendency to free the rolls 
 by throwing out, by centrifugal force, any material lodged between the 
 
 
 
 
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 O© 
 
 tIt- 
 
 
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 |o© 
 
 -~r§T 
 
 
 
 
 
 
 nx 
 
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 r§r 
 
 
 T§I 
 
 OO 
 
 in 
 
 
 
 
 
 
 si 
 
 Fig. 5. 
 
422 
 
 ORE DRESSING AND PREPARATION OF COAL . 
 
 teeth. In one test 
 it was found that 
 less fine coal was 
 produced at 800 ft. 
 per minute, but 
 that the rolls 
 blocked at this 
 speed and hence 
 had to be driven 
 1,000 ft. per minute. 
 
 In one case a 
 pair of main rolls 
 24 in. in diameter, 
 30 in. face, running 
 at 1,000 ft. per min- 
 ute, handled 2,500 
 tons of coal in 24 
 hours. A pair of 
 19" X 24'' main 
 rolls run at 1,000 ft. 
 per minute handled 
 300 tons mine run 
 in 10 hours. 
 
 A well-known 
 maker of rolls for 
 crushing bitumi- 
 nous coal gives a 
 speed of 100 to 150 
 li. P. M., according 
 to the output re- 
 quired, for rolls 24 
 in. in diameter and 
 33 in. long. As a 
 rule, cracking rolls 
 are never run up to 
 their full capacity, 
 as is the case with 
 crushing rolls. 
 
 The form of the 
 teeth varies greatly, 
 but, as a rule, the 
 larger rolls have 
 straight pointed 
 teeth of the spar- 
 row-bill or s o m e 
 similar form, Fig. 6 
 a. The old curved, 
 or hawk-billed, 
 teeth, Fig. 6 b, have 
 now gone almost 
 wholly out of use. 
 
 On* small sized 
 rolls, rectangular 
 teeth with a height 
 equal to one side of 
 the square base are 
 frequently em- 
 ployed, and these 
 may be cast in seg- 
 ments of manga- 
 nese or chrome 
 steel. 
 
 Corrugated rolls 
 
 have teeth or cor- 
 rugations extend- 
 ing their entire 
 length. They were 
 first introduced by 
 
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ROLLS. 
 
 423 
 
 Fig 
 
 Mr E. B. Coxe, at Drifton, Pa M but they have not come into general use 
 owing to the fact that, while they break some coal fairly well, in most 
 cases it has been found that a continuous edge causes too much disinte- 
 gration along its length, while a point splits the coal into three or four 
 pieces only, all the cracks radiating from the place where the point strikes, 
 thus producing very much less culm. Another advantage possessed by the 
 toothed rolls is that if anything hard passes through the corrugated roll 
 and breaks out a piece of the corrugation, the entire roll 
 is ruined, while, in the case of the toothed rolls, any one 
 of the teeth may be replaced. 
 
 Disintegrating rolls and pulverizers are sometimes used to 
 reduce coking coal to the size of corn or rice before intro- 
 ducing it into the ovens. One roll is driven at double the 
 speed of the other, the slower roll acting as a feed-roll, and 
 the other as a disintegrator. The slower roll is commonly 
 driven at from 1,800 to 2,000 ft. per minute peripheral 
 speed, and the faster roll at from 3,600 to 4,000 ft. per mi- 
 nute. The teeth are always fine, rarely being over f m. 
 high. In some cases, the inner roll is provided with a 
 series of saw teeth from £ in. to | in. high and having 
 about f in. pitch, the individual teeth being set so as to 
 form a slight spiral about the body of the roll. The other 
 roll is provided with teeth having their greatest dimension 
 in the direction of rotation, so that they tend to cross the teeth on the opposite 
 roll. These teeth are also set so as to form a slight spiral, and thus prevent 
 blocking. In other cases, the teeth on both rolls are set in the form of 
 quite a steep spiral. , _ . , , 
 
 Hammers— For the reduction of coal, crushers employing hammers have 
 been used, Fig. 7. The crushing chamber is usually of a circular or barrel form, 
 and the crushing is done by means of hammers pivoted about a central shaft. 
 These swing out by centrifugal force and strike blows upon the coal to be 
 broken. When it is reduced sufficiently fine, it is discharged through bars 
 or gratings at the lower portion of the machine. This style of machinery is 
 usually employed in preparing coal for coke ovens, thus occupying the same 
 J field as the disintegrating 
 
 rolls. A No. 3 pulverizer of 
 this type will crush 50 to 75 
 tons per hour run of mine, 
 down to £ in., or it will crush 
 100 tons per hour of slack. 
 Such a machine occupies 
 about 8 sq. ft. of floor space 
 and requires 25 to 30 H. P. to 
 run it. 
 
 Crushing Rolls.— The prin- 
 cipal representative of this 
 type of machine is the ordi- 
 nary Cornish roll having a 
 fairly wide face and rather 
 small diameter. The diam- 
 eter of these rolls was kept 
 down for a great many years 
 on account of the fact that 
 the chilled cast-iron shells 
 could not be obtained in 
 
 large sizes and were expen- 
 
 sive and hard to handle. 
 With the advent of the rolled- 
 steel shells, it became possible to employ larger diameters and higher 
 speeds. Rolls of the Cornish type vary from 4" face and 9" diameter to 
 
 16" face and 42" diameter. The distinctive feature of the Cornish roll 
 
 is a comparatively wide face compared with the diameter, and a 
 rather slow peripheral speed. Many of the modern Cornish rolls are 
 provided with rolled-steel shells, especially when employed for very fine 
 crushing, owing to the fact that these shells are of a more uniform 
 texture, work more evenly, can be worn much thinner before being 
 discarded, and can be trued up with less difficulty than is the case 
 when chilled iron is employed. To guard against the bending of the rou 
 
 Fig. 7. 
 
424 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 shaft or breaking of the machine in case any hard material (such as a pick 
 or hammer) gets between the rolls, one roll is mounted in a movable 
 bearing and kept in place by a compressed spring washer. This washer is 
 composed of two plates between which are placed one or more steel springs. 
 The plates are kept together by several small bolts, which are screwed 
 up so as to compress the springs to a certain degree. Then the entire 
 arrangement is employed as a washer on the rod that keeps the rolls 
 together. Should the pressure exerted on the rolls exceed that already 
 exerted in the spring, the plates would be brought nearer together and the 
 roll allowed to move back and pass the hard substance, but at any pressure 
 below this, the roll acts as if placed in a fixed bearing. 
 
 Cracking, corrugated, and disintegrating rolls are usually provided with 
 breaking pieces back of one of the rolls, so that in case any extra hard piece 
 passes through the rolls, the breaking piece will give way, allowing the 
 rolls to move back and thus prevent the bending of the shaft or breaking 
 of the machine itself. Compressed spring washers have never come into 
 general use in connection with this style of machinery. 
 
 Amount Crushed.— The amount of material that 
 can pass between any pair of rolls is proportionate 
 to the number of square feet of roll surface passing 
 per minute; hence, the capacity may be increased 
 by keeping the face width the same and in- 
 creasing the speed, or the same capacity may 
 be obtained by reducing the face and increasing 
 the speed. 
 
 According to Stutz (A. I. M. E. IX, page 464), 
 if the distance between the contact points of 
 the material with the rolls be t, Fig. 8, the dis- 
 tance between the crushing face of the rolls w, 
 the angle a, as shown in the figure, and R the radius of the roll, then 
 22 _ t — w _ t — w 
 ~ 2 vers, sin a ~~ 2(1 — cos a)’ 
 
 According to Pernolet, the amount of material that may be crushed 
 by a pair of rolls in a given time is equal to one-fourth or one-fifth of a 
 band or layer whose length is the circumference of the roll multiplied 
 by the number of revolutions; whose width is the length of the rolls, 
 and whose thickness is equal to the space or distance between the rolls. 
 
 
 
 
 1 
 
 1 \ 
 
 
 Fig. 8. 
 
 Or, Q = 
 
 dnnlw 
 
 , where d = diameter of rolls; ir = 3.14; n = number of revo- 
 
 lutions in the given time: l = length of rolls; w = space between rolls; 
 and £ = coefficient, to allow for the irregular feeding of the material and 
 the space between the pieces. . _ _ 
 
 The Denver Engineering Works gives the following formulas for the 
 capacity of crushing rolls: 
 
 T = tons per hour; R = rev. per min.; S = mesh (inches). 
 
 For 14" X 27" rolls, T = 7.725 R S. 
 
 For 16" X 36" rolls, T = 11.775 RS. 
 
 For 12" X 20" rolls, T = 4.9 R S. 
 
 Speeds. — The pressure on the bearings necessary to crush ore depends 
 directly on the face width, and hence if the capacity can be kept the same 
 and the face width decreased, it is evident that there will be less pressure 
 on the bearings and less loss in friction. The difficulty of keeping the 
 bearings cool when crushing hard rock with the old Cornish rolls has led 
 to the adoption of high-speed, narrow-faced rolls for certain classes of work. 
 One objection to running the small diameter rolls fast is that the larger 
 pieces of ore have a tendency to dance on the face of the rolls rather than 
 to be crushed, while the bite is better when the speed is slower. 
 
 The advantages of high-speed, narrow-faced rolls are: greater capacity 
 for a given bearing pressure; less loss of power from friction; less dancing 
 of the ore on the roll face, owing to the fact that the angle of approach 
 between the surfaces of large rolls is more acute than with rolls of a small 
 diameter. High-speed, large-diameter rolls will handle C 9 arser material and 
 hence make a greater range of reduction than small-diameter rolls. The 
 disadvantage of high-speed rolls is that they tend to hammer and pulverize 
 the ore, so that with verv brittle minerals a high speed may be detrimental. 
 In general, it may be stated that for crushing to any definite size with the 
 lowest possible production of very fine material, rolls are the best form of 
 
ROLLS. 
 
 425 
 
 machinery on the market. For fine crushing of brittle material, quite slow 
 speeds may give the best results. . , . ,. ,, 
 
 The accompanying table gives some facts m regard to the crushing-roll 
 practice of several manufacturers, the data having been taken from their 
 catalogues or other information furnished by them. 
 
 Crushing Rolls. 
 
 Name. 
 
 Size. 
 
 Inches. 
 
 Peripheral 
 Speed in 
 Ft. per Min. 
 
 Spring Pres- 
 sure in Lb. per 
 In. of Face 
 Width. 
 
 Character of 
 Rolls. 
 
 Frazer & Chalmers... 
 
 24X8 
 
 36X16 
 
 600-1,500 
 
 4,000 for hard 
 quartz. 
 
 Cornish. 
 
 Frazer & Chalmers,.. 
 
 44X5 
 
 56X8 
 
 2,200-2,300 
 
 
 Narrow face, 
 high speed. 
 
 Earle C. Bacon 
 
 
 1,000 
 
 
 Cornish. 
 
 Sturtevant Mill Co. 
 
 16X3 
 
 27X5 
 
 3,000 
 
 • 
 
 Special cen- 
 trifugal. 
 
 E. P. Allis Co 
 
 20X12 
 
 26X14 
 
 30X14 
 
 36X14 
 
 i 
 
 800 
 
 
 Cornish. 
 
 E. P. Allis Co 
 
 
 1,885 
 
 
 Narrow face, 
 high speed. 
 
 Colorado Ironworks 
 
 20X12 
 27X14 
 36 X 16 
 40X16 
 
 600 
 
 4,000 for hard 
 rock. 
 
 4,800 for very 
 hard rock. 
 
 Cornish. 
 
 Colorado Iron Works 
 
 36X6 
 
 42X6 
 
 54X8 
 
 2,100-2,800 
 
 
 Narrow face, 
 high speed. 
 
 Denver Engineering 
 Works Co 
 
 20 X 12 
 to 
 
 36X16 
 
 350-100 
 
 3,500-4,500 
 
 Cornish. 
 
 
 
 
 
 Gates Iron Works ... 
 
 9X4 
 26X15 
 36 X 15 
 
 470-850 
 
 2,266-3,333 
 
 Cornish. 
 
 The Gates Iron Works has furnished the following formulas relating to 
 crushing rolls, in which D = diameter of roll m inches; N == number of 
 R. P. M.; S = maximum size of ore cube in inches fed to the rolls, 
 S' = maximum size of cube for a given diameter of roll. 
 
 oqo s 
 
 N = f x isd- s ' = M76XZ> - 
 
 It will be seen from the first of these formulas than N is an inverse 
 function of S, which agrees with the results shown in the previous diagram. 
 As a rule, it is best not to try to run rolls up to the maximum size that they 
 
 will crush, but to feed smaller material to them. . ... 
 
 The Denver Engineering Works Company has furnished the diagram, 
 Fig. 9, and formulas relating to rolls. This diagram serves very well to 
 illustrate the fact that small rolls do not grip or crush large pieces as well 
 when running at comparatively high peripherial speeds as when running 
 at slow speeds. In the case of the 10" X 16" roll, a difference of from 
 1" to cube size made a difference of 20 R. P. M. in order to obtain the 
 
426 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 most effective crushing speed, and the difference between £" and £" cube 
 sizes made a difference almost as great. It will also be noticed that the 
 larger diameters, as, for instance, the 42" roll, are not so greatly affected by 
 this cause, owing to the fact that the effective or crushing angle between the 
 rolls is much more acute than in the case of the smaller diameters. 
 
 CRUSHING MILLS. 
 
 Radial Roller Mills.— In this type of mill, the crushing is performed on a 
 ring or die by a series of heavy rolls pressing on it by gravity. In some 
 cases, the rolls travel around on the die and in others the die travels in 
 relation to the rolls. Fig. 10 represents one form of Chilian mill that is the 
 leading type of this class. , , ,,, . ,, . .. 
 
 The peculiarity of the grinding action of the radial rolling mills is that it 
 is not a pure crushing action, but a triturating or grinding action as well, 
 owing to the fact that while the different portions the face of the roll are 
 all traveling at the same speed, the outer portions have to travel over 
 a greater length of ring than the inner po tions, so that there is only one 
 line along which true crushing action occurs. Some manufacturers have 
 made the crushing ring and the rollers both with coning faces, the vertices 
 of both cones meeting at a common point. This has resulted in a true 
 crushing action, but for so'me classes of work the triturating action is to be 
 preferred, as, for instance, in the grinding of silver ores for the patio process 
 of amalgamation. , . 
 
 Centrifugal Roller Mills.— In centrifugal roller nulls the crushing is accom- 
 plished between rapidlv moving rolls and the inside of a stationary die or 
 ring. The Huntington ‘mill. Fig. 11, is one of the principal representatives 
 of this class of machinery. The rollers c are supported from bearings e and 
 are carried rapidly around by means of the frame a and the ^Ihe 
 
 ore is crushed against the ring d. In order to prevent the accumulation 
 
CRUSHING MILLS. 
 
 427 
 
 Fig. 10. 
 
 of ore below the rollers, and to throw it out for crushing, scrapers / are pro- 
 vided. The crushed ore discharges through screens, as shown in the illus- 
 tration. There are many styles of this class of machinery having different 
 numbers of rollers, varying from 1 up, and some machines have been intro- 
 duced combining a portion 
 of the action of radjal and 
 centrifugal machines, the 
 faces of the die or ring being 
 at an angle and the rollers 
 being mounted in inclined 
 bearings so that they tend to 
 crowd out and down upon 
 the ring. Centrifugal roller 
 mills have found two espe- 
 cial fields in concentration 
 works, one for crushing clay 
 or soft ores containing free 
 gold, and the other for re- 
 grinding middlings for fur- 
 ther concentration. Rolls of 
 this type are also extensively 
 employed in grinding cement 
 and phosphate rocks. 
 
 Ball Mills.— There are two 
 types of ball mills: (1) those 
 in which the crushing is per- 
 formed by balls traveling in 
 a fixed path, and (2) those 
 in which the crushing is per- 
 formed by a large mass of balls of various sizes rolling over one another. 
 In the first type the balls travel in a fixed path, track, or race that 
 may be either vertical or horizontal. Where it is vertical, the balls 
 must be driven at such a rapid rate that their centrifugal force will 
 keep them in contact with the crushing ring or track. This form may be 
 likened to a bicycle ball bearing on a large scale, the crushing being 
 accomplished between the balls and the race or track. The serious objec- 
 tion to this class of ball mills is found in the uneven wear of both the balls 
 and the race, so that the work soon becomes unevenly distributed, and also 
 in the fact that the balls cannot be used after they have been worn to a 
 slight extent. , . 
 
 In the second class of machines the balls are introduced into a large 
 barrel or chamber, where they roll over one another, the ore being crushed 
 
 between the different balls and 
 between the balls and the lin- 
 ing of the chamber. In this 
 style of machine the crushed 
 material may be discharged 
 through openings in the per- 
 iphery or through openings in 
 one end of the barrel. One 
 great advantage with this style 
 of mill is that the balls can be 
 entirely worn out and it is 
 only necessary to charge a 
 sufficient number of new balls 
 with the ore each day to make 
 up for the wear of those in 
 the mill. 
 
 STAMPS. 
 
 Gravity stamps are especially 
 well suited for material the 
 Fig. 11. valuable portion of which does 
 
 not have a tendency to slime. 
 The fact that these stamps are very simple in construction, easy to transport 
 and erect, as well as to operate, gives them a decided advantage over other 
 forms of crushers. Fig. 12 illustrates a 10-stamp battery of the gravity type. 
 
428 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 Fie 13 is a detail of the mortar stamp heads and dies. The mortar a is placed 
 on a suitable foundation of timbers b and the ore crushed on dies £ by the 
 stamps s, which are secured by means of tapered joints to the he^s or bosses 
 h The stems e are attached to the heads h and the whole lifted by the cams 
 (shown in detail in Fig. 12). The cams operate under tappets on the stems 
 as shown in Fig. 12. As the cam operates under the -edge of the tappet, it 
 not only lifts the stamp, but gives a partial rotation, thus equalizing the wear 
 on both the stamp and die. The ore is fed in at the back of the mortar and 
 the crushed material discharged through the screen, as shown in Figs. 12 and 
 13 Usually a single screen at the front is employed, but sometimes two or 
 more upon different sides of the mortar may be introduced. For treating 
 free-milling gold ores in which the gold occurs in rather large grains free 
 from iron pyrites, the California style of battery was developed, the charac- 
 teristics of Which are a small drop (4 in. to 6 in.), low discharge ^m.U 
 heavy stamp (750 to 1,000 lb.), and a high speed or number of drops permin- 
 ute (90 to 105). The advantage of this style is rapid crushing, but the 
 majority of the gold had to be saved on apron plates outside the mortar. 
 
 For working ores that contain large quantities of iron pyrites with the 
 g-old values occurring in the cleavage planes of the pyrites, the Gilpin 
 County . Colo., style of battery was developed. This is characterized by a 
 Loumy, y high drQp (lg tQ 20 in v a high discharge 
 
 (14 in.), a light stamp (550 to 600 lb.), and 
 a comparatively slow rate of drop (30 per 
 minute) . With this style of battery, most 
 of the gold was obtained on amalgamated 
 plates in the battery, but its use was 
 accompanied by excessive sliming on 
 
 Fig. 12. 
 
 Fig. 13. 
 
 account of the fact that the high discharge kept the material m the mortar 
 for a long time, and subjected it to repeated treatment. w 
 
 Modern practice tends toward the use of rather heavy 
 1 000 lb ) quick drop (90 to 105 per minute), and low discharge (4 to 6 m.). 
 The advantages are P tU the capacity of the hatte^ 
 
 sliming - reduced to a minimum. If the ore contains sulphides carrying go , 
 
 they afe separated by concentration upon manners or 1 amon nlates do 
 subsea uentlv treated by chlorination or smelting. If the apron plates ao 
 not catch the major portion of the values, the tailings may treated by t e 
 cyanfdfproceT This last method is that employed at many large gold 
 mines, especially those of the Transvaal in South Africa. T . . 
 
 Order of Drop.— There is much diversity of practice m this respert. . It is 
 desirable to drop the stamps in such rotation as to insure an even disti ^b 
 tion of the pulp on the several dies. Adjacent stamps should not drop co ^ 
 secutivelv, as this occasions accumulation of the pulp at one end of 
 mortar in conseauence of which the efficiency of the stamps at that end is 
 reduced by having a decreased height of drop and a cu shion 
 pulverization of the ore. The stamps at the other themortarhave 
 
 too little work, and are liable to “ pound iron.” The order of drop 1, 4, 2, 5, 3 
 
STAMPS. 
 
 429 
 
 seems to best fulfil the requirements. It gives a good 
 
 but the two just 
 
 m 6 ?n i large & ^lfs ^lie stendanl^rop is gtven as 1 7, 3, 9, 5, 2, 8 , A 1CK 
 1 s, 4, 10,2, 7, 5, 9, 3 , 6 as a close favorite; while 1, 5, 9, 7, 3, 2, 6 , 10, 8 , 4 and 
 
 1,6 SDee 3 d of 'SUmps— Heav^stainps and stamps having high drops should 
 
 KKrS.7S| 6 doK r 
 
 dislodged! and breakage is imminent. A fast drop produces a good splash, 
 which is very desirable for battery amalgamation. 
 
 Shoes and Dies. — Shoes and dies are either of iron or steel. In most mills, 
 remote from foundries where transportation is an important item m the cost 
 nf^hops and dies steel shoes and dies have replaced those of iron. Chrome 
 ° t f p S ( shot? and ^ dies have been introduced and have proved superior. In 
 feme S S, steel shoesVd iron dies ; are used L The ^ £SS 
 
 evenly with steel shoes than the steel dies do. The life is about 2. to 3 times 
 that of iron shoes and dies, and the cost about twice as great as those of iron. 
 The mixture of steel ( from the old chrome steel shoes and dies) with iron 
 nrodS Xes and dVes Sat wear considerably longer than those of pure 
 Son* and may be advantageously introduced l where 
 
 r»n<?«iiblp for the old steel, because of want of local facilities ior me 
 utilization of this residue. In many districts, the old iron shoes and dies are 
 sold to local foundries for from 1 £ to 2 cents per lb. • of the 
 
 The weights of the shoes bear a certain relation to the weights of tne 
 tnoopts stems and bosses. Chrome steel shoes made for stamps of 850 to 
 9% P lb weiSi ’from 150 to 155 lb., and measure about 9 in. m diameter by 
 71 to 8 in long. The neck is from 4£ to 5 in. long, with a taper to correspond 
 to the socketof the boss or stamp head. Iron shoes are usually from 15 to 
 20 lb. lighter than the above weights. The chrome steel dies weigh from 
 110 to 125 lb., and measure (where shoes of the above dimensions are used ) 
 
 Q in in diameter bv 4 to 41 in. in height, with a rectangular foot-plate 10 , »in. 
 by 91 in. by i in. thick. Iron shoes usually weigh from 20 to 25 lb. less than 
 
 ^ e Lif?°of e the^l^oes^and Dies. — There are many conditions that affect the 
 durability of shoes and dies as, for instance, the hardness of the rock the 
 wPiXt sneed and height of drop of the stamp, the manner of feeding 
 the ore’ etc Iron shoes of good quality last from 30 to 47 days. Old shoes 
 wea? usualiv down to U in. or 1 in. in thickness, and weigh about 25 or 40 lb. 
 Old dies usually wear down to about 11 in. in thickness, and weigh from 
 20 to 50 lb. The consumption of iron or steel in shoes and d ies depends < on 
 the character of the ore crushed. Other conditions being the same, it will 
 depeM onlhe coarseness of the stamjang and the heigh t of discharge 
 Dies wear less rapidly than the shoes, as they are Protected 
 and the pulp, which covers them to the depth of from .1* to 3 in. BUt while 
 the actual wear of dies is less than that of the shoes, the life of the dies 
 is shorter than that of the shoes, owing to the fact that tbe shoes have 
 several inches greater length of wearing part than the dies. The con 
 sumption of iron for shoes and dies per ton of ore crushed &hfornia, 
 froni H to 3 lb. To obtain the maximum crushing capacity of the 
 battery the dies must be kept as high (with reference to the lower edge 
 of the 7 screens) as is compatible with the safety of the sheens 
 successful amalgamation in the battery. To prevent the JPO^nding of iron, 
 it is necessary to preserve more or less uniformity m the level of the dies. 
 Should one die in the battery project much above the others, little or no 
 pulp would remain upon it, and the shoe would consequently drop upon 
 
 ^^am^Stamo Heads and Stems.— Cams and stamp heads ought to last 
 several years. P They are usually broken through carelessness, The stems 
 break at the socket of the stamp head. Stems are reversible; w h ^i ^ r oken, 
 they may be swedged or planed down and additional lengths welde 
 
 Wh Tappets.— When there is much grease on the tappet or cam or when the 
 
430 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 tappets have so worn that the face of the cam strikes a grooved instead of a 
 level face on a tappet, the rotary motion is greatly impaired. Tappets last 
 for several years, from 4 to 5 years being their usual life. Sometimes they 
 are broken by being too tightly keyed. When their faces are worn, they 
 are planed down. They are reversible, so that when one face has been 
 worn as far as possible, the other face is placed downwards. They are 
 usually of steel, and weigh about 112 lb. when 900-lb., stamps are used. 
 
 Battery Water. — The amount of water fed to the battery depends on the 
 character of the ore and the size of the screen. Clayey arid highly sulphu- 
 reted ores require the maximum amount of water. The amount of water 
 used per ton of ore stamped varies from 1,000 to 2,400 gallons. The mean 
 amount used per ton of ore stamped is about 1,800 gallons. From I to 14 
 miner’s inches per battery should be provided. Tn winter, when the battery 
 water is chilly, it should, when possible, be heated to tepidity, as this pro- 
 motes amalgamation. A high temperature should be avoided, as it renders 
 the quicksilver too lively. 
 
 Duty of Stamps.— The capacity of gravity stamps varies from a little over 
 1 ton per stamp for 24 hours to as high as 4 tons per stamp for 24 hours, 
 depending on the quality of the ore. Usually, an average of from 1.7 to 2 
 tons per stamp for 24 hours in a combination mill would be good practice, 
 while where the ore is crushed to a rather coarse screen and the tailings 
 treated with the cyanide process, a larger capacity is usually obtained. 
 
 The number of tons of ore crushed per stamp depends chiefly on the 
 weight of the stamp, the number of drops per minute, the height of drop, 
 the height of discharge, the size of the screens, the width of the mortar, and 
 chiefly on the character of the ore. Hard ores and ores of a clayey nature 
 (from the difficulty experienced in discharging the clayey pulp) decrease 
 the duty of the stamps. About 2£ tons per stamp in 24 hours is the average 
 duty of the stamp in California. The discharging capacity of a mortar 
 depends on the height and size of the discharge opening, the character of 
 the screen, and the width of the mortar discharge, as will be illustrated 
 from two well-known mills. 
 
 The Homestake Mill uses an 850-lb. stamp dropped 9 in., 85 times per 
 minute, developing 78,030,000 ft.-lb. in 24 hours, and crushing 44 tons of rock, 
 or 1 ton for every 17,340,000 ft.-lb. developed. 
 
 The Caledonia Mill uses an 850-lb. stamp, dropping 12 in., 74 times per 
 minute, crushing 3.3 tons, of rock and developing 90,576,000 ft.-lb. in 24 
 hours, or 1 ton to every 24,447,272 ft.-lb. developed. Although developing 
 more foot-pounds in 24 hours, and therefore seemingly more efficient, yet it 
 crushes less rock than the former. The reasons for this are (1) that the rock 
 is harder than that of the Homestake; (2) the width of mortar is 16 in. 
 against 134 in.; and (3) the 2" recess for the 8" copper plate below the feed. 
 On the other hand, the Caledonia has a lower discharge from the mortar, 
 using 6 in. against 10 in. in the Homestake; but this advantage is again 
 neutralized by a smaller screen, the Caledonia using 258 sq. in. against 
 376 sq. in. of the Homestake. 
 
 Horsepower of Stamps.— The H. P. of a stamp battery = 
 No.ofstampsXwgt.ofeach stamp X No. of drops per min. X drop of each in in. 
 
 12 X 33,000 
 
 The weight of each stamp is equal to the sum of the weights of the stem, 
 tappet, stamp head, and shoe. To the nominal H. P. add 25$ for friction of 
 machinery in calculating driving H. P. 
 
 Cost of Stamping.— The cost of stamping varies from a little over $1.00 
 per ton up. The Montana Co., Limited, operating a 60-stamp combination 
 mill, in 1888 treated 40,530 tons of ore at $1.13 per ton. In Australia, stamp- 
 mill costs have been reported varying from $1.30 to $2.50 per ton where 
 fairly favorable conditions for working could be obtained. Figures from 
 other districts compare favorably with these, but it would be impossible to 
 give any absolute rule by means of which the cost can be determined in 
 advance, without an intimate knowledge of the character of the ore and 
 the local conditions. 
 
 Pneumatic Stamps.— This is a name given to a form of large capacity 
 power stamp, the head of which is connected to a piston in an air cylinder. 
 The cylinder is raised and lowered by power, the air forming an elastic 
 connection by means of which the stamp is operated. They are quite 
 extensively employed in crushing tin ore, but have never come into general 
 use for other purposes. The capacity is as high as 30 tons per 24 hours. 
 
SIZING AND CLASSIFYING APPARATUS. 
 
 431 
 
 Power Stamps —Various forms of stamps have been brought out at differ- 
 pnt times intended to operate by power like a trip hammer, or in which 
 the stamp's were connected directly to the cranks operating them by means 
 MS Neariy all of these forms have failed on account of exces- 
 2 ? ve wear small capacity and the large amount of power consumed . 
 
 lt«T’st^D 3 .-The Targe capacity steam stamp, which was evolved in 
 connection with the concentration of the Lake S u P e J™ r copper “es, °on- 
 sists of a steam cylinder in which °P er at es ^.Piston, to the stem of which the 
 otomn head is directly connected. Machines of this style are usually 
 made very large and heavy, frequently extending throug wo or ree 
 stories of the mill, and having a capacity eqmvaient to from 60 to 100 ordi 
 nary gravity stamps. In most forms, live steam is admitted on top of the 
 piston during the (lescent of the stamp, thus increasing the force of the 
 blow For lifting the stamp, the steam is throttled so that a lower pressure 
 is employed. The discharge is usually through a coarso screen, * to 8 
 mesh not being uncommon. One interesting fact connected with the large 
 steam stamps is that their heavy blows do not cause as excessive sliming as 
 th^Ughte? gravity Stamps, and on this account this form of stamp has been 
 introduced m some cases for crushing free-milling gold ores. 
 
 For prospecting work, for testing properties, or foroperatmgsmall prop- 
 erties a number of forms of portable or semiportable steam stamps have 
 come ’ out* 1 during ?he last few years. One of these (the Tremam) is illus- 
 trated in Fig. 14. In this form, two pistons work m cylin- 
 ders side by side and strike alternate blows in a common 
 mortar. The steam is introduced at full boiler pressure on 
 the lower side of the cylinder, which, owing to the large 
 diameter of the piston rod, has a small area. This high pres- 
 sure steam is then allowed to expand on to the top ot the 
 piston, thus urging it down with greater force than its own 
 weight would. These steam stamps can be run at a much 
 higher speed than gravity stamps, and hence have a greater 
 capacity. In the figure shown, three screens are employed, 
 one in front and one at each end of the mortar. There are 
 several other forms of portable steam stamps manufactured. 
 
 They all have the advantage that for a small property they 
 can be installed with much less trouble than any other form 
 of crusher, owing to the fact that no steam engine is required 
 and the steam necessary to drive them can usually be 
 obtained from the boiler operating the hoisting engine, 
 
 ^Mfscetlaneous Forms of Crushers.-Most crushers can be 
 classed under one of the previous heads, but there are spme . 
 forms that depend on the material itself to do the crushing: £or mstance, 
 in the preparation of coal for coke ovens, there has been a combined crusner 
 and separator invented that may be described as follows: A Ja r ge h^izontal 
 drum or cylinder, provided with screen openings around 
 mounted in a horizontal position The coal to be separated is fed into one 
 end and is caught by shelves or plates projecting radially into the cylinder. 
 These lift the material to the upper side, from which it falls by gravity and 
 strikes the bottom, thus crushing the softer parts The sulphur 
 being harder than the coal, are not crushed by the same height ; of fall, and 
 hence, by a proper adjustment of the diameter of the cylinder, the °oal may 
 be crushed and discharged through the screen while the slate and sulphur 
 will pass out at the opposite end of the cylinder. 
 
 SIZING AND CLASSIFYING APPARATUS. 
 
 Stationary Screens, Grizzlies, Head-Bars, or Platform Bars.—' These {*F® 
 various names given to an inclined screen employed for removing tne 
 fine material from the run of mine so that only the coarse portion will be 
 passed to the crushers. At concentrating works, the term 
 employed, and a common form is shown in Fig .15 This is composed of flat 
 bars held apart by cast-iron washers through which the bar bolts are passed 
 to hold the entire frame together. Grizzlies are usually Placed at an angle 
 of from 45° to 55°, and ordinarily, for the head of a large concentrating 
 works, they are from 3 to 6 ft. wide and from 8 to 12 ft. long, the amount 
 of space between the bars depending on the size of the run-of-mme material 
 and on its subsequent treatment. 
 
 Fig 
 
432 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 In the anthracite coal breakers, the terms platform bars or head-bars are 
 usually employed, and these bars are made of 1£" to 2" round iron placed at 
 an inclination of 5 in. to 1 ft., the spacing depending on the size of coal it is 
 desired to make in the breaker. 
 
 At the present time, in accordance with an agreement between the oper- 
 ators and the miners’ officials, the standard size for a bituminous lump 
 
 screen (the bars are called a screen) for Ohio, 
 _ _ Pennsylvania, Indiana, and Illinois is 12 ft. 
 
 long and 6 ft. wide over the screen surface. The 
 screen consists of 6 bearing bars 4 in. by $ in. 
 of soft steel and 39 steel screen bars, Fig. 16, 
 with li in. clear space between bars. . In Iowa, 
 the same sized bar is used, but the space be- 
 tween the bars is If in. In the other Western 
 and Southern States there is at present no 
 standard. 
 
 The standard nut-coal screen for Pennsyl- 
 vania and Ohio is £ in. space, but f in. is 
 Fig. 15. sometimes used and the nut screen is often 
 
 varied to suit the special trade. At present 
 very few pea screens are used, but if placed under a nut screen, the 
 space is from f in. to f in. In the Pittsburg region of Pennsylvania, all 
 coal passing through f" screen is called slack , while, on the Monongaheuo 
 River, coal passing through 11" screen is called slack and is used in stokers. 
 Many companies are at present crushing their run-of-mine coal to make 
 slack suitable for stokers. 
 
 It is difficult to classify bituminous coal by sizes, but as nearly as possible 
 the following seem to be the standard: Lump , all coal passing over 11" screen; 
 nut, all coal passing through 11" openings and over openings; slack, all 
 coal passing through %" screen. If a pea-coal screen is used, all coal passing 
 over 1", 1", and f" would be pea coal, and that passing through |", 1", or f" 
 would be slack. 
 
 Adjustable Bars .- The top of the bar is cylindrical and projects beyond the 
 web which supports it, so that any lump which passes through the upper 
 part will fall freely without jamming. The two ends of the bar are V shaped 
 and fit into similarly shaped grooves, so that the bars can be set at distances 
 from each other varying with the sum of the width of the bases of the 
 triangles, the usual opening being about 4 in. These bars are generally 
 4 ft. long, but they can be of any size. 
 
 Finger bars are screen bars that are fixed at one end only, and the bars 
 are narrower at the lower end than at the top, so that the spaces between 
 them are wider at the bottom than at the top, thus giving less tendency for 
 pieces of material to become wedged between the bars. 
 
 Movable or oscillating bars are screen bars that are attached to eccentrics 
 at their lower ends, the eccentrics of adjoining bars being 
 placed 180° apart. This movement throws the material for- 
 wards and the bars do not therefore require nearly the same 
 inclination as fixed bars. 
 
 Shaking screens have an advantage in that the entire area 
 of the screen is available for sizing, and hence a greater 
 capacity can be obtained from a given area of screening 
 surface. They also occupy less vertical height than a revolv- 
 ing screen. In coal breakers they are particularly applicable 
 where the coal is wet and has a tendency to stick together. 
 
 The principal disadvantage of the shaking screen is that 
 the reciprocating motion imparts a vibration to the framing 
 of the building. For anthracite coal, the screens usually 
 have an angle or pitch of from £ in. to 2 in. per foot, the 
 average being about £ in. per foot. These screens are run at 
 from 90 to 280 shakes per minute, the average being about 200 
 shakes per minute, or 100 revolutions per minute for the 
 cam-shaft. The throw of the eccentric or cam varies from 
 2 in. to 5 in. Fig. 16. 
 
 Similar screens are employed for sizing salt, but are usually 
 placed at a much steeper incline and are frequently so hung that they have a 
 combined rocking and swinging motion. Shaking screens are rarely employed 
 in concentrating works on account of the fact that revolving screens can 
 be hung in the upper part of the mill where they will not interfere with 
 
Screens. 
 
 433 
 
 oyecti?nablT^ile a ^e^do C &e\izmg a satisfactorily a wirBout°imparting jar 
 
 fOI For brokena n’d egg coal, i sq. ft. per ton for lOhoura. 
 
 For stove and chestnut coal, T sq. ft f t PF r ^“ h f “ “ h 
 For pea and buckwheat coal, * sq. ft. for 10 hours 
 
 fi^on^tSt lasttag 6 ? hcmrs^/OW) tons of coal were passed over screens having 
 
 nerforated plates of the following dimensions: 
 perioraxea ^ ^ perforations for making slack. 
 
 56 sq. ft. with 1£" perforations for making pea coal. 
 
 28 sq ft. with 2 i" perforations for making nut coal. 
 
 28 sa ft with 4i" perforations for making egg coal. 
 
 SliiSHSSSS 
 
 Mesh for Shaking Screens. 
 
 Kind of Coal. 
 
 Steamboat... 
 
 Lump 
 
 Broken 
 
 Egg 
 
 Stove 
 
 Chestnut ... 
 
 Pea 
 
 Buckw r heat 
 Rice 
 
 Lehigh Valley 
 Coal Co. 
 
 Phila. & Reading 
 Coal & Iron Co. 
 
 Round. 
 
 Round. 
 
 Square. 
 
 
 5f" 
 
 5" 
 
 4i" 
 
 4i" 
 
 4" 
 
 3i" 
 
 3i" 
 
 24" 
 
 2tV' 
 
 21" 
 
 2" 
 
 ^*16 
 
 It" 
 
 if 
 
 1" 
 
 li" 
 
 7// 
 
 V 
 
 H" 
 
 4" 
 
 i rr 
 
 3 
 
 H" 
 
 *" 
 
 c 
 
 16 
 
 i // 
 
 4 
 
 \tf 
 
 e 
 
 Kind of Coal. 
 
 Steamboat. 
 Large broken. 
 Small broken. 
 Egg. 
 
 Stove. 
 
 Chestnut. 
 
 P6£l 
 
 Buckwheat. 
 
 Rice. 
 
 s^ssi5fia.'S 
 
 many 1 mills ‘where^h^^achi^ry ^mu^of necessity be crowded into a 
 minimum space and be hard to get at. number of flat sides, 
 
 or dashed against the screen surfaces to break it or to loosen aanering cmy 
 
434 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 or dirt. The shaft is sometimes hollow, and streams of water from this 
 hollow shaft wash the material as it is being screened. 
 
 Revolving screens are frequently jacketed, that is, two or more screens 
 are placed concentrically about the same shaft, the inmost one being the 
 coarsest, and each succeeding screen serving to make additional separations. 
 This method reduces the space necessary for a given amount of sizing 
 machinery. In other cases, a long cylindrical screen has a coarse mesh near 
 its discharge end and finer mesh near the entrance end, thus making two or 
 more through products as well as the overproduct. The disadvantage of 
 jacketed screens is that the necessarily slow speed of the inmost screen 
 reduces the capacity of the entire combination, so that if rapid work is 
 essential, it is better to use fairly large-diameter screens placed one after the 
 other in place of jacketed screens. Another disadvantage is that, to renew 
 the inner jackets, it is often necessary to remove the outer ones. 
 
 The disadvantages of having two or more sizes of wire cloth on one screen 
 are that the fine-meshed screen near the head is worn out rapidly, as all the 
 material both coarse and fine passes over it, while, when separate screens 
 are employed, each screen has to deal only with its through or oversized 
 product, all coarser material having been removed. 
 
 Speed. — The periphery of a revolving screen should travel about 200 ft. 
 per minute. In the case of very fine material, screens are sometimes run 
 faster than this. 
 
 The following have been adopted as standard speeds for screens by one 
 of the largest anthracite coal companies: 
 
 Speed of Screens. 
 
 Rev. per Minute. Rev. per Minute. 
 
 Mud screens 8.87 Big screens 8.52 
 
 Counter mud screens 15.49 Pony screens 10.87 
 
 Cast-iron screens 11.25 Buckwheat screens 15.30 
 
 Duty of Anthracite Screens.— The following table gives the number of 
 square feet of screen surface required for a given duty in the case of 
 revolving screens working upon anthracite coal: 
 
 Egg coal, 1 ton per 1 sq. ft. per 10 hours. 
 
 Stove coal, 1 ton per 1£ sq. ft. per 10 hours. 
 
 Chestnut coal, 1 ton per If sq. ft. per 10 hours. 
 
 Pea coal, 1 ton per 2 sq. ft. per 10 hours. 
 
 Buckwheat coal, 1 ton per 2f sq. ft. per 10 hours. 
 
 Rice coal, 1 ton per 3f sq. ft. per 10 hours. 
 
 Culm, 1 ton per 5 sq. ft. per 10 hour:,. 
 
 These figures may be reduced from 20$ to 30$ for very dry or wash coal. 
 Revolving Screen Mesh for Anthracite.— A standard mesh for revolving 
 screens for sizing anthracite coal was adopted some years ago, but it is only 
 approximately adhered to and a considerable variation from the standard is 
 found throughout the anthracite region. 
 
 The following are probably as nearly standard meshes for revolving 
 screens for sizing anthracite coal as can be given: 
 
 Mesh for Sizing Coal. 
 
 passes through 3 y' mesh, 
 passes over mesh, and through mesh, 
 passes over £" mesh, and through £" mesh, 
 passes over mesh, and through f" mesh, 
 passes over f" mesh, and through If" mesh, 
 passes over If" mesh, and through 2" mesh, 
 passes over 2" mesh, and through 2f" mesh, 
 passes over 2f" mesh, and out end of screen, 
 over 3" mesh, and out end of screen. 
 
 Culm 
 Birdseye 
 Buckwheat 
 Pea 
 
 Chestnut 
 Stove 
 Egg 
 
 * Grate 
 
 * Special grate 
 
 * Special steamboat passes over 3" bars, and through 6" bars. 
 
 Hydraulic Classifiers.— The separation of materials by this class of 
 machinery depends upon the law of equally falling bodies, which may be 
 stated as follows: Bodies falling free in a fluid , fall at a speed proportional 
 to their weight divided by the resistance. From this it will be seen that small 
 masses of a heavy mineral will fall as rapidly as large masses of a light 
 mineral, owing to the fact that the weight increases as the volume and the 
 
 * These sizes and “lump” size are seldom made, and there is no uniformity whatever in the 
 sizes called by these names. 
 
HYDRA XJLIC CLASSIFIERS. 
 
 435 
 
 resistance only as the area, so that if a quantity of galena and quartz of 
 various sizes were introduced into water, it would settle into approximate 
 layers, each composed of relatively large pieces of quartz and relatively 
 small pieces of galena. This same action would be true in the case ot any 
 
 minerals differing in specific gravity. The principal representatives of the 
 hydraulic classifying machines are the Spitzkasten and Spitzlutten. . 
 
 The Spitzkasten consists of a series of pyramidal boxes, one of which is 
 shown in Fig. 17. The material enters the box at a , passes down under the 
 diving board b , and discharges into the next box through the trough c. At 
 the bottom of the box, water is introduced through the pipe d from the 
 launder g in such quantity as to more than supply the opening or spigot e. 
 The heavy particles of mineral settle against this rising stream of water into 
 the elbow/, from which they are washed out through e. Each succeeding 
 box is made larger than the preceding, and the rising current is so regulated 
 that a different product will settle out in each. 
 
 The Spitzlutten is a V-shaped box, inside of which is set another V having 
 the same slope, the material flowing down between the two V’s on one side 
 and up and out on the other. The distance between the V’s can be regulated, 
 as can also the rising current of water, thus obtaining the separation desired. 
 
 Many other forms of separators, all depending on this same principle, 
 have been brought out, some having a conical form, some being arranged in 
 the form of troughs, and others as boxes of various shapes. 
 
 The Calumet classifier, Fig. 18, consists of a series of boxes or pockets in the 
 bottom of a gradually widening 
 trough. Wash water enters 
 through a pipe a and discharges 
 directly against the discharge 
 spigot d , which is however not 
 large enough to carry all the 
 water off directly, hence it twirls 
 and eddies in the bottom of the 
 box so that only the heaviest 
 particles having weight enough 
 to settle in this disturbed water 
 pass out through the spigot. A 
 shield c reflects upward currents 
 and confines the agitation to the 
 bottom. The pr 1 - -- 
 
 the direction of 
 
 
 s. 
 
 . . .r* — 
 
 ■ M 
 
 s 
 
 
 HrH* 
 
 
 k 
 
 flowing in 
 e arrows is 
 downwards by the 
 
 Fig. 18. 
 
 deflected 
 
 boards s. . 
 
 The settling boxes employed for mills are really a form of hydraulic classi- 
 fier. They are usually very large V-shaped boxes provided with a diving 
 board similar to that shown at b in Fig. 17, but no current of water is 
 
436 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 introduced. In some cases a small stream of the heavy muds or concen- 
 trates is kept continually flowing from the bottom, while in others they 
 are drawn off inteimittentlv. One very important point to be observed 
 in settling boxes is that the settling action depends on the arresting ot 
 the current, and with a given amount of floor space, very much more 
 efficient settling can be obtained from two boxes placed side by side and 
 having half the material pass through each than from two boxes placed 
 in series, and that the width of the box is of vastly more importance 
 than the length. The depth is also fairly important, and a diving board 
 mast always be introduced to prevent surface currents. If the boxes 
 are properly arranged, nearly all the solid material will be settled out ot 
 
 the water. ,, . . . - 
 
 The Jeffrey-Robinson coal washer, Fig. 19, which operates on the principle of 
 the Spitzkasten, c^sists of a steel chamber B in the form of an inverted 
 cone, inside of which are 
 projecting arms and stir- 
 ring plates <7, C revolved 
 by a driving gear A. The 
 water supply enters at 
 the bottom from the 
 water pipe P through 
 perforations 31. The coal 
 is introduced through a 
 chute S and is kept in a 
 continual state of agita- 
 tion by the current of 
 water, and being lighter 
 than the impurities, it 
 passes out through the 
 overflow K onto the con- 
 veyors E, F and through 
 the chutes X, X, while 
 the water and sludge 
 drain through the hop- 
 per into the sludge tank 
 G . whence, if necessary, 
 the same water can be 
 again pumped by the pul- 
 
 someter H back into the . . , ... „ , 
 
 washer. (As mentioned elsewhere, it is poor practice to use this water over 
 again when it is desired to decrease the percentage of sulphur m the 
 washed product as greatly as possible. ) . ^ _ , T , „ 
 
 The heavy impurities sink to the bottom into the chamber J and when 
 this is full the upper of the two valves shown is closed and the lower valve is 
 
 opened to discharge the refuse. . . , 
 
 The following data in regard to one of these washers is given by 
 Air J. J. Ormsbee in the Transactions of the A. I. M. E. These results were 
 obtained at the Pratt Alines, Alabama, with a plant having a nominal 
 capacity of 400 tons per day. By washing slack that passed between screen 
 bars spaced f in. in the clear, the washed coal contained 42# less ash than 
 the unwashed coal, the reduction in sulphur was 15#, while the volatile 
 matter was increased 4$, and the fixed carbon 5#. W ith coal passing over 8 
 perforations, the results were a reduction of 48# m ash, 1 a# in sulphur and a 
 £ain of 5* in volatile matter and fri fixed carbon. These results indicate that 
 the washer is better adapted to large sizes than to fines. The amount of 
 water used per ton of washed coal was 85.1 gallons and the cost \\as 
 2.25 cents per ton for washing 400 tons, itemized as follows: Labor at washer, 
 82 00; labor at boiler, fuel, etc., 84.00; repairs and supplies, 83.00; total, 89.00. 
 
 Loe Washer.— For removing clay from ores or other material, the log 
 washer illustrated in Fig. 20 has proved itself to be efficient. Lither 
 single or double logs are employed, the form shown being double- 
 log washer. The logs work over troughs which have a slight inclination, 
 so that the water will flow from one end to the other. W ater is introduced 
 at the upper end and discharged at the lower end. The material to be washed 
 is introduced near the lower end and is fed up against the water by spiral 
 arms or plates fixed about the logs, as shown in the illustration. As the 
 material is advanced, the clay or other sticky substance is broken up, 
 washed away, and discharged at the lower end with the wash water. 
 
LOG AND TROUGH WASHERS. 
 
 437 
 
 Fig. 20. 
 
 These washers have "been extensively employed for cleaning iron ores 
 occurring as rather hard masses in clay. , , , - 
 
 There is no general standard size of these washers, hut most of the double- 
 log washers for both steel and wood logs are the same, except in length oi 
 logs, the washer box being 7 ft. 
 
 4 in. wide, 2 ft. deep at discharge 
 end, and 4 ft. deep at receiving 
 end. The length of logs varies 
 from 20 to 30 ft. The logs are 
 generally given an elevation of 
 1 in. in 1 ft., and sometimes li in- 
 in 1 ft. 
 
 The capacity of an ore washer 
 depends very much on the 
 quality of the material, avera- 
 ging for one pair of logs from 100 
 tons per day, when the matrix 
 is of a clayey nature, to 350 tons 
 with loose sandy material. The 
 capacity of a washer is based 
 on the amount of material from 
 the mines it will put through 
 more than the tonnage of clean 
 
 ore, and this amount varies from . , 
 
 500 to 1,000 yd. per 10 hours. The amount of water used varies from 300 to 
 500 gal. per minute. The total expense for labor and fuel, including the 
 water supply, varies from 5 cents to 25 cents per ton of ore, averaging 
 possibly 10 cents per ton. , A 
 
 The Scaife trough washer consists of a semicircular iron trough 2 ft. 
 in diameter and 24 ft. long. Inside is a series of fixed dams or partitions 
 that can be made higher or lower, as required, by means of plates. A shaft 
 running the entire length of the trough and turning in babbitted journals 
 carries a number of stirring arms or forks and is given a reciprocating 
 motion by a connecting-rod attached to a driving pulley at its center. The 
 coal is fed with water at the upper end of the trough, and by the action of 
 the flowing water and the agitation of the arms, the slate, pyrites, and other 
 impurities settle at the bottom and are caught behind the dams, while the 
 clean coal passes over the dams and out at the lower end of the trough. 
 When the spaces behind the dams are filled, feeding is stopped and the 
 refuse in the dams quickly dumped. This form of washer is particularly 
 successful with coal mixed with fireclay. One washer handles from 75 to 
 100 tons of coal per day, and one man can attend to six washers. Each 
 washer requires less than 1 H. P. to operate it. The larger the coal, the 
 greater must be the slope and the quantity of water used. 
 
 Jigs. — This is a general term applied to that class of concentrating 
 machines in which the separation of the mineral from the gangue takes 
 place on a screen or bed of material and is effected by pulsating up-and- 
 down currents of a fluid medium. 
 
 There are a number of different methods in use for driving the pistons 
 that cause the pulsations of the water in jigs. Some of these use plain 
 eccentrics, giving the same time to both the up and the down strokes of the 
 pistons, while others employ special arrangements of parts, which give a 
 quick down stroke and a slow up stroke, thus allowing the water ample 
 time to Work its way back through the bed without any sucking action 
 from the piston. This tends to make a better separation in some cases than 
 the use of the plain eccentrics. , . , , 
 
 Stationary Screen Jigs.— This class is illustrated by Fig. 21, which shows a 
 3-compartment jig. The separation takes place on screens supported on 
 wooden frames g, and is effected by moving the water in each compartment 
 so that it ascends through the screen, lifting the mineral and allowing it to 
 settle again, thus giving the material an opportunity to arrange itself accord- 
 ing to the law of equally falling particles. Each compartment is composed 
 of two separate parts, one containing the screen on the support g and the 
 other adjoining it and arranged so that the piston in it may impart the 
 necessary pulsations to the water. These pistons are usually loose fitting and 
 are operated by the eccentrics e on the shaft s. Jigs operating on coarse ore 
 should be fed with approximately sized material, when the ore will accumu- 
 late near the bottom on the screen and the barren portion or gangue will 
 
438 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 be carried over the discharge. Formerly, the concentrates were discharged 
 from jigs intermittently by digging out the tailings first, then the middlings, 
 which are composed of pieces containing some ore and some gangue, and 
 then the concentrates, but it has been found that the concentrates will 
 flow over the screen like a liquid of comparatively heavy specific gravity, 
 
 Fig. 21. 
 
 Advantage has been taken of this fact in the design of several forms of jig 
 discharges. , , 
 
 The Heberle gate, Fig. 22, acts as follows: a is a U-shaped shield fastened 
 against the inside of the jig and held in place by a band b, the ends of 
 
 which are drawn down into the form of 
 bolts and pass through the sides of the 
 jig, where they are secured with suitable 
 nuts. The shield a may be raised or 
 lowered by loosening the band b. The 
 discharge takes place through the open- 
 ing / in the side of the jig, the size and 
 position of the opening being regulated 
 by slides c. The concentrates k rest on 
 the screen e supported by a grating d, 
 while the tailings i occupy a higher posi- 
 tion. The shield a prevents the tailings 
 from flowing out through the opening /, 
 while the concentrates flow along the 
 screen and rise to a height somewhat 
 lower than the top of the tailings in the 
 jig, when they are discharged through the 
 opening / over the spout h, as shown at 
 p. The tailings are usually discharged 
 over the dam at the end of the jig, and 
 in some cases a third discharge is provided of the Heberle-gate pattern 
 and so arranged that the middlings will flow out through it and discharge 
 separately from the tailings and the concentrates. 
 
 In the case of jigs handling fine material, the material may be sorted by 
 hydraulic classifiers and then introduced on to the jigs. In this class the 
 mineral will be in the form of relatively small pieces, while the gangue will 
 occur as relatively large pieces. Advantage may be taken of this fact by 
 regulating the mesh of the jig screen so that the concentrates will pass 
 through into the space below the screen, commonly called the hutch, while 
 the tailings will pass over the tailing dam. In some cases the gate 
 discharge is employed on the side to remove the middlings. The middlings 
 are recrushed and treated on other machines. This form of concentration 
 has been used very largely in connection with the Lake Superior copper 
 ores, the values of which occur as metallic copper in a relatively light 
 gangue, and also in concentrating tin ores that occur in the light gangue 
 containing considerable mica. 
 
THEOR Y OF JIGGING. 
 
 439 
 
 Theory of Jigging.-By far the most exhaustive investigations on the theory 
 of jigging carried on in America are those of Prof. Robert H. Richards, the 
 Massachusetts Institute of Technology, and the greater part of ^following 
 theoretical discussion is based on his several papers published m the Trans 
 
 actions of the American Institute of Mining Fmgineers. 
 
 Four laws of jigging are given by the several ®f 
 
 equal settling particles, under free settling conditions, (2) the law ot 
 interstitial currents, or settling under hindered settling conditions; (3) the 
 
 la ^hl a firet le of a these is 4 the mosTim^rtant.'but the others are elements that 
 
 Can E^ t a,^^p 1 a rtl^t^U„ t re? W»owin g formulas to repre- 
 senAhe relation between diameter of grains and rate of falling in water for 
 irregularly shaped grains: _______ 
 
 V = 2.73 / — 1). for roundish grains; 
 
 V = 2.44/ J)[S — 1), for average grains; 
 
 V = 2.37j /W(8 — 1), for long grains; 
 
 V = 1.92i/ D(8 — 1), for flat grains, _ 
 
 in which V = velocity in meters per second; D = diameter of particles in 
 
 me By S means ofXes^^fferenffomSas. the ratios of 
 different particles that will be equal settling in water can be computed. 
 Professor Richards has not found these formu]as to ho d in a]l c^es in 
 practice, and, as the result of elaborate experiments, he gives the following 
 table * 
 
 Equal Settling Factors or Multipliers. 
 
 Table of eaual settling factors or multipliers for obtaining the diameter of 
 a quartz grain that will be equal settling under free settling conditions with 
 the mineral specified. 
 
 
 Specific Gravity. 
 
 Velocity in Inches per Second. 
 
 Rittinger’s 
 
 Multipliers. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Author’s Multipliers. 
 
 Anthracite 
 
 Epidote 
 
 Sphalerite 
 
 Pyrrhotite 
 
 Chalcocite 
 
 Arsenopyrite ... 
 
 Cassiterite 
 
 Anitmony 
 
 Wolframite 
 
 Galena 
 
 Copper 
 
 Quartz 
 
 1.473 
 
 3.380 
 
 4.046 
 
 4.508 
 
 5.334 
 
 5.627 
 
 6.261 
 
 6.706 
 
 6.937 
 
 7.856 
 
 8.479 
 
 2.640 
 
 .500 
 
 1.57 
 
 .352 
 
 1.35 
 
 1.46 
 
 1.73 
 
 1.90 
 
 1.90 
 
 2.11 
 
 2.71 
 
 2.71 
 
 2.71' 
 
 2.71 
 
 .225 
 
 1.05 
 
 1.05 
 
 1.29 
 
 1.47 
 
 1.57 
 
 1.79 
 
 2.00 
 
 1.83 
 
 1.83 
 
 2.00 
 
 .213 
 
 1.13 
 
 1.17 
 
 1.48 
 
 1.62 
 
 1.89 
 
 2.00 
 
 2.00 
 
 2.07 
 
 2.26 
 
 2.36 
 
 1.50 
 
 1.62 
 
 2.00 
 
 2.07 
 
 2.42 
 
 2.73 
 
 2.73 
 
 2.86 
 
 3.00 
 
 3.00 
 
 1.61 
 
 1.64 
 
 2.22 
 
 2.28 
 
 2.56 
 
 2.93 
 
 2.93 
 
 3.04 
 
 3.42 
 
 3.20 
 
 1.56 
 
 1.68 
 
 2.26 
 
 2.41 
 
 2.72 
 
 3.03 
 
 3.03 
 
 3.21 
 
 3.65 
 
 3.58 
 
 1.56 
 
 1.66 
 
 2.13 
 
 2.44 
 
 2.84 
 
 3.05 
 
 2.98 
 
 3.28 
 
 3.76 
 
 3.76 
 
 1.47 
 
 1.56 
 
 2.08 
 
 2.17 
 
 2.94 
 
 3.12 
 
 3.00 
 
 3.26 
 
 3.75 
 
 3.75 
 
 .288 
 
 1.45 
 
 1.85 
 
 2.14 
 
 2.64 
 2.82 
 3.32 
 3.48 
 
 3.64 
 4.01 
 4.56 
 
 The significance of the above table is as follows: If a piece o- anthracite 
 of a certain size falls in water with a velocity of 4m. per second a piece of 
 quartz 0.213 times the diameter of the anthracite will fall with the same 
 velocity. If a piece of copper of a certain size falls with a velocity of 7 in. 
 per second, a piece of quartz 3.58 times as large as the copper will fall with 
 
 ^InterstitTal^Currents, or Law of Settling Under Hindered Settling Conditions 
 
 If d equals the diameter of a falling particle, and D that of the tube m which 
 
 it falls, the larger the fraction the greater will be the retardation or loss 
 
440 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 ODD 
 
 © 
 
 Pyrrhotite 2.808 
 
 Sphalerite 2.127 
 
 Epidote 1.628 
 
 Anthracite 1782 
 
 of velocity by the particle. When this fraction equals 1, the particle' stops. 
 If in Fig. 23 (a), the larger circles represent particles of quartz and the 
 smaller circles equal settling particles of galena, then if these mixed parti- 
 cles are settling together or are held in suspension by a rising current of 
 water, each particle may be considered to be falling in a tube, the walls of 
 which consist of the surrounding particles. Substituting a circle in each 
 
 case for the imaginary tube, we have 
 Fig. 23 (6) representing the condi- 
 f ) ( ) tions for galena and quartz, the 
 
 v, ^ outer circle in each case represent- 
 
 ing the imaginary tube. Evidently, 
 
 is much smaller for the galena 
 
 ^ ^ than for the quartz, and it will 
 
 V. J V J {0t therefore be much less impeded in 
 
 v o. its fall than the quartz; hence, the 
 
 (a) Fig. 23. particles of galena found adjacent 
 
 to the particles of quartz will be 
 smaller than the ratio that the law of equal settling particles under 
 free settling conditions would indicate. Application of this principle is 
 found when a mass of- grains is subjected to a rising current of sufficient 
 force to rearrange the grains according to their settling power and the 
 grains are said to be treated under hindered settling conditions, as on the 
 
 6 Interstdfal factors, or multipliers for obtaining the diameter of the particle 
 of quartz that under hindered settling conditions will be found adjacent 
 to and in equilibrium with the particle of the mineral specified, are the 
 following; 
 
 Copper 8.598 Cassiterite 4.698 
 
 Galena 5.842 Arsenopyrite . .3.737 
 
 Wolframite... 5.155 Chalcocite 3.115 
 
 Antimony ...4.897 Magnetite 2.808 
 
 These signify that, after pulsion has done its work on a jig bed, for exam- 
 ple, where quartz and anthracite are being jigged, the grains will be so 
 arranged that the grains of quartz are .1782 times the diameter of the grains 
 
 of anthracite that are adjacent to and in equilibrium with them. 
 
 Acceleration. — A particle of galena that is equal settling to the particle of 
 quartz reaches its maximum velocity in perhaps T V the time required by the 
 quartz. The oft-repeated pulsations of a jig, therefore, give the galena par- 
 ticles a decided advantage over the quartz, placing beside the quartz, when 
 equilibrium is reached, a much smaller particle of galena than we should 
 expect according to the law of equal settling particles. 
 
 Suction acts to draw down through the screen small grains, mainly oi tne 
 heavier mineral, which are distributed among large grains. It increases as 
 the length of plunger stroke, with the difference in specific gravity of the 
 two minerals, and with the diminishing of the thickness of the bed on the 
 sieve, whether of the heavier mineral only or of both minerals. The law 
 of suction seems to be that jigging is greatly hindered by strong suction 
 where the two minerals are nearly of the same size, the quickest and best 
 work then being done with no suction; but when the two minerals differ 
 much in size of particles, the quartz being the larger, strong suction is not 
 only a great advantage, but may be necessary to get any separation at all. 
 Experiments have indicated an approximate boundary between grains that 
 are helped and those that are hindered by suction; namely, if the diameter 
 of the quartz particles is equal to or greater than 3.52 times the diameter 
 of the other mineral particles, then separation is helped by suction; it less, 
 separation is hindered. This value 3.52 (obtained by dividing .0683 by .0195) 
 is approximate only, and it will differ with the fracture of the quartz under 
 consideration; if the quartz grains are much flattened, it will have a large 
 
 Val Eccentric jigs invariably spend more time on pulsion than accelerated 
 jigs. Is it not fair to conclude that the eccentric jigs are better adapted tor 
 treating sands that require the most pulsion? Such sands are the sized 
 products from the trommel and the first spigot of the hydraulic classifier. 
 On the other hand, mav not the long-protracted mild suction of the acceler- 
 ated jig be best adapted to the treatment of such products as require 
 primary suction for their separation; fox example, the second spigot and 
 
THEORY OF JIGGING. 
 
 44 J 
 
 the following spigots of the hydraulic classifier? This may he the reason 
 
 Sfoft^e i n e C ra "ecSre 
 
 the condition as presented in this particular by experiments 
 
 Two extreme suggestions arise from a contemplation o t ^be^per* 5: 
 we have carried on: (1) On closely sized products, an accelerated Jigshould 
 hf^ rim back wards to lengthen out the pulsion period, which is the only 
 neriod that does any work; and (2) the accelerated jig should be run for- 
 wards on the spigot products of the hydraulic separator, to increase t e 
 
 period of suctiom suggestion, there are two difficulties, either of 
 
 which mlv cancel thl advantage: the violent downward motion of he 
 
 quick return will tend to “ blind up ’’the sieve; and, second, the same action 
 will tend to pulverize a soft mineral like galena. with 
 
 Professor Richards summarizes his experiments m connection with 
 iigging as follows- The two chief reactions of jigging are pulsion and suc- 
 tion. g The effect of pulsion depends on the laws of interstitial currents, or 
 of ooual settling particles, under hindered settling conditions. The c bi®£ 
 function of pulsfon is to save the larger grains of the hea ^ e ^P 1 ^ e ^ 1 ’^tion 
 grains that settle faster and farther than the waste. The effect of suction 
 depends on the interstitial factor of the minerals to be separated. It this 
 factor is greater than 3.70, suction will be efficient and rapid If the factor 
 is less than 3.70, suction will be much pampered and leered. function of 
 lone- stroke will help to overcome this difficulty. I he cni ci luncuon oi 
 suction is to save the particles that are too small tobesayed by 
 
 of interstitial currents, acting through the P ul ^°? th & n J r^Wffing cfo^elv 
 mixed sizes, pulsion with full suction should be used For Jigging closely 
 <5i7Pd Tvroduots nulsion with a minimum of suction should be usea. 
 
 The degree of sizing needed as preparation for jigging, if perfect work is 
 desired depends S <m the interstitial ffictor of the minerals to be separated. 
 If the factor is above 3.70 (assuming this value to 1 
 
 thpn sizino- is simplv a matter of convenience. The fine slimes snouia oi 
 course be removed- and if it is more convenient to send egg size, nut size, 
 pea Jfze ^ anTsand size, each to its own jig, the suitable screens should be 
 Provided for this purpose and a hydraulic separator for grading the finest 
 sizes. But if on the other hand, the factor is below 3.70, then the jigging of 
 mixed sizes cannot give perfectly clean work, and the separation will be 
 flooroximate onlv. To effect the most perfect separation, close sizing must 
 be ^adopted and the closer the sizes are to one another, the more rapid and 
 nerfpetPwili the iigging be. There may be conditions where the jigging oi 
 mixed sizes of tins cfass will be considered sufficiently satisfactory , as an 
 
 GX Removal o^Siflp hur^F rom C C^f ^The object of washing coal is to remove the 
 slate and pyrites, thus reducing the amount of ash and sulphur. Manyforms 
 of wash er s easily and cheaply reduce the slate from 20$ in the coal to 8$ of 
 nsh fn the coke but it is much more difficult to reduce 4$ of sulphur m the 
 ronl to K or less of sulphur in the coke. Sulphur occurs in the coal m 
 three forms as hydrogen sulphide, calcium sulphate, and pyrite. The first 
 is volatile and is removed in coking, the second cannot usually be J^moved 
 and it is the removal of the third form with which 
 washing has to do. The presence of water in the coke ovens apparently 
 assists the removal of the sulphur; but wet coals require a longer time for 
 
 coking than dry and therefore, pyrite should be removed as far as practicable 
 
 before* charging the coal into the coke ovens. The pyrite m coal it c 5“ t 
 from the mfne seems to be in particles even fine^ 
 
 This irrmalnable nowder or flour pyrites floats m air or water, inis uemg 
 F£ 1S the cram™ practice of using the water over and over agam in a 
 washerv caLot Mve tfie best results In the removal 9 f sulphur, as some 
 flour Dvrites will he carried hack each time and remain with the washed 
 coal P ExDerimentsmade by Mr. C. C. Upham, of New York Citv, show that 
 the critical size at which an almost complete division of the coal and pyrites 
 takes place varies with coals from different districts and beds, and m laying 
 out coahwashing plants, the proper fineness of crushing should he deter- 
 mined beforehand by careful experiment. 
 
442 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 Preparation of Anthracite.— The method of preparing anthracite coal is 
 clearly shown, graphically, by the diagram below. This consists in screen- 
 ing the coal over bars and through revolving or over shaking screens, 
 together with breaking it with rolls to produce the required market size. 
 
 BARS 3" APART 
 BARS 4 f APART 
 n\un SCREE H 
 
 Diagram Showing Method of Preparing Anthracite Coal. 
 
 The large lumps of slate and other impurities are separated by hand on the 
 platform near the dump, while the smaller portions are picked out by auto- 
 matic pickers or by hand by boys or old men seated along the chutes leading 
 to the shipping pockets or bins. The smaller sizes are cleaned by jigging. 
 
HANDLING OF COAL. 
 
 443 
 
 HANDLING OF MATERIAL. 
 
 Anthracite Coal.— The following may be taken as average figures for the 
 angle or grade of chutes for anthracite coal, to be used where the chutes are 
 lined with sheet steel: For broken or egg coal, 2* in. per ft.; for stove or 
 chestnut coal, 3£ in. per ft.; for pea coal, 4* in. per ft.; for buckwheat coal, 
 6 in. per ft.; for rice coal, 7 in. per ft.; for culm, 8 in. per ft. 
 
 If the coal is to start on the chute, 1 in. per ft. should be added to each of 
 the above figures; while if the chutes are lined with manganese bronze m 
 place of steel, the above figures can be reduced 1 in. per ft. for coal in 
 motion, or would remain as in the table to start the coal. When the run of 
 mine is to be handled, as in the main chute, at the head of the breaker, the 
 angle should be not less than 5 in. per ft., or practically 22£° from the hori- 
 zontal. If chutes for hard coal are lined with glass, the angle can be 
 reduced from 30$ to 50$, depending somewhat on the nature of the coal. 
 In all cases, the flatter the coal, the steeper the angle must be, on account of 
 the large friction surfaces exposed, compared with the weight ot the piece. 
 If chutes are lined with cast iron, the angle should be about the same as 
 that emploved for steel, though sometimes a slightly greater angle is allowed. 
 
 The following table is printed through the courtesy of the Link-Belt 
 Engineering Co., Philadelphia, Pa.: 
 
 Pitch at Which Anthracite Coal Will Run, in Inches per Foot. 
 
 Kind of Coal. 
 
 Sheet Iron. 
 
 Cast 
 
 Iron. 
 
 Glass. 
 
 Glass. 
 
 Start 
 
 on. 
 
 Con- 
 
 tinue 
 
 on. 
 
 Start 
 
 on. 
 
 Start 
 
 on. 
 
 Con- 
 
 tinue 
 
 on. 
 
 Start 
 
 on. 
 
 Con- 
 
 tinue 
 
 on. 
 
 Dry. 
 
 Wet. 
 
 Broken slate 
 
 5 £ 
 
 4f 
 
 5 * 
 
 3£ 
 
 3 
 
 
 
 Dry egg slate 
 
 5 £ 
 
 4* 
 
 5 £ 
 
 3£ 
 
 3 
 
 
 
 Dry stove slate 
 
 5 £ 
 
 4» 
 
 5 £ 
 
 3£ 
 
 3 
 
 
 
 Dry chestnut slate 
 
 L ' 
 
 4» 
 
 5 £ 
 
 3£ 
 
 3 
 
 
 
 Broken coal 
 
 
 
 
 2* 
 
 2£ 
 
 
 
 Egg coal 
 
 3| 
 
 3 
 
 
 2| 
 
 2£ 
 
 2£ 
 
 
 Stove coal 
 
 4£ 
 
 4£ 
 
 4£ 
 
 3 
 
 2£ 
 
 2£ 
 
 2 £ 
 
 Chestnut coal 
 
 4f 
 
 4£ 
 
 4£ 
 
 3 
 
 ■ 2£ 
 
 3 
 
 2£ 
 
 Pea coal 
 
 5 £ 
 
 5 
 
 51 
 
 3£ 
 
 2£ 
 
 3 
 
 
 Buckwheat No.l ... 
 
 
 
 
 3f * 
 
 3£ 
 
 3£ 
 
 3£ 
 
 O 1 
 
 Buckwheat No.2... 
 
 
 
 
 3£ 
 
 3£ 
 
 3£ 
 
 3£ 
 
 A 1 
 
 Buckwheat No.3... 
 
 
 
 
 4f 
 
 3f 
 
 4? 
 
 
 Buckwheat No.4.... 
 
 
 
 
 4£ 
 
 4* 
 
 4t 
 
 4£ 
 
 Bituminous Coal —When the run of mine is to be handled, the angle ot 
 the chutes should be from 35° to 45° from the horizontal, or from 8* m. to 
 12 in per ft. If the coal is wet, the angle should always be steeper, and 
 coarse coal will slide on a flatter angle than slack or fine coaL 
 
 Ore, Rock, Etc.— For coarse fairly dry ore, i. e., from 2 in. or 3 in. size up, 
 chutes may have an angle of 45°, or if the material is always to be in motion, 
 the ore will sometimes slide on 40°. For fine ore or run of mine, the chutes 
 should have an angle of 50° from the horizontal or practically 14 m. vertical 
 to 1 ft. horizontal. . , „ , r x_ 
 
 Flumes and Launders.— Water flumes are given grades varying from 4 or 5 to 
 20 or 30 ft. per mile, depending on the surface of the ground and the amount 
 of water to be carried. Practical results have demonstrated the fact that m 
 ordinary ground the water should travel at the rate of from 180 to 200 ft. per 
 
 mil Where rather coarse stuff is to be carried through launders in a mill with 
 a comparatively small amount of water, an angle of 2 m. per ft. should be 
 used. With an excess of water, 1 in. per ft. will be ample. The spouting tor 
 vanners or launders from trommels carrying rather fine material should 
 have a grade of about 2 in. per ft. In placer mining, the minimum grade 
 
444 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 for the sluice should not be less than i in. per ft., or about 4j in. per rod. 
 Experiments made in river gravel have shown that with a grade of from 1 m 
 20 to 1 in 25, 60 cu. ft. per min. will wash 140 to 175 cu. yd. per day of 24 hours. 
 
 No absolutely definite figures can be given on this subject, owing to the 
 fact that the nature of the ore plays an important part, and while angular 
 quartzose ore can be transported at a comparatively flat angle, it may be 
 necessary to use quite a steep angle if the material is of a clayey nature or 
 contains many large flat plates, exposing large friction surfaces when com- 
 pared to the mass of the pieces. „ x . _ . _ _ . , 
 
 The following tables are printed through the courtesy of the Link-Belt 
 Engineering Co., Philadelphia, Pa.: 
 
 Horizontal Pressure Exerted by Bituminous Coal Against Vertical 
 Retaining Walls per Foot of Length. 
 
 97 so. Y ' 
 
 / m 
 
 Surface horizontal { {^^0 lowest ft. 
 
 Surface sloping { preslurefowest ft. 
 Angle of repose 
 
 d — height of wall in feet or a b 
 
 = 6.37 d 2 
 
 = 6.37(2 ci — 1) 
 = 10 d 2 
 
 = 10(2 d — 1) 
 
 = 35° 
 
 BITUMINOUS. 
 
 4-j 
 
 Horizontal 
 Surface. 
 b m. 
 
 Sloping 
 Surface. 
 b e. 
 
 rt 
 
 Horizontal 
 Surface. 
 b m. 
 
 Sloping 
 
 Surface. 
 
 be. 
 
 S 
 
 rO 
 
 A 
 
 <D 
 
 A 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 e 
 
 rO 
 
 •g 
 
 A 
 
 © 
 
 A 
 
 . j 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 1 
 
 6.4 
 
 6.4 
 
 10 
 
 10 
 
 26 
 
 4,305 
 
 325 
 
 6,760 
 
 510 
 
 2 
 
 25.0 
 
 19.0 
 
 40 
 
 30 
 
 27 
 
 4,641 
 
 338 
 
 7,290 
 
 530 
 
 3 
 
 57.0 
 
 32.0 
 
 - 90 
 
 50 
 
 28 
 
 4,993 
 
 350 
 
 7,840 
 
 550 
 
 4 
 
 102.0 
 
 45.0 
 
 160 
 
 70 
 
 29 
 
 5,358 
 
 363 
 
 8,410 
 
 570 
 
 5 
 
 159.0 
 
 57.0 
 
 250 
 
 90 
 
 30 
 
 5,733 
 
 376 
 
 9,000 
 
 590 
 
 6 
 
 229.0 
 
 70.0 
 
 360 
 
 110 
 
 31 
 
 6,122 
 
 389 
 
 9,610 
 
 610 
 
 7 
 
 312.0 
 
 83.0 
 
 490 
 
 130 
 
 32 
 
 6,523 
 
 401 
 
 10,240 
 
 630 
 
 8 
 
 407.0 
 
 96.0 
 
 640 
 
 150 
 
 33 
 
 6,935 
 
 414 
 
 10,890 
 
 650 
 
 9 
 
 516.0 
 
 108.0 
 
 810 
 
 170 
 
 34 
 
 7,362 
 
 427 
 
 11,560 
 
 670 
 
 10 
 
 637.0 
 
 121.0 
 
 1,000 
 
 190 
 
 35 
 
 7,778 
 
 440 
 
 12,250 
 
 690 
 
 11 
 
 770.0 
 
 134.0 
 
 1,210 
 
 210 
 
 36 
 
 8,253 
 
 452 
 
 12,960 
 
 710 
 
 12 
 
 917.0 
 
 146.0 
 
 1,440 
 
 230 
 
 37 
 
 8,754 
 
 465 
 
 13,690 
 
 730 
 
 13 
 
 1,076.0 
 
 159.0 
 
 1,690 
 
 250 
 
 38 
 
 9,193 
 
 478 
 
 14,440 
 
 750 
 
 14 
 
 1,248.0 
 
 172.0 
 
 1,960 
 
 270 
 
 39 
 
 9,682 
 
 490 
 
 15,210 
 
 770 
 
 15 
 
 1,433.0 
 
 185.0 
 
 2,250 
 
 290 
 
 40 
 
 10,192 
 
 503 
 
 16,000 
 
 790 
 
 16 
 
 1,630.0 
 
 197.0 
 
 2.560 
 
 310 
 
 41 
 
 10,669 
 
 516 
 
 16,810 
 
 810 
 
 17 
 
 1,840.0 
 
 210.0 
 
 2,890 
 
 330 
 
 42 
 
 11,236 
 
 529 
 
 17,640 
 
 830 
 
 18 
 
 2,063.0 
 
 223.0 
 
 3,240 
 
 350 
 
 43 
 
 11,797 
 
 541 
 
 18,490 
 
 850 
 
 19 
 
 2,298.0 
 
 236.0 
 
 3,610 
 
 370 
 
 44 
 
 12,331 
 
 554 
 
 19,360 
 
 870 
 
 20 
 
 2,548.0 
 
 248.0 
 
 4,000 
 
 390 
 
 45 
 
 12,968 
 
 567 
 
 20,250 
 
 890 
 
 21 
 
 2,809.0 
 
 261.0 
 
 4,410 
 
 410 
 
 46 
 
 13,478 
 
 580 
 
 21,160 
 
 910 
 
 22 
 
 3^083.0 
 
 274.0 
 
 4,840 
 
 430 
 
 47 
 
 14,100 
 
 592 
 
 22,090 
 
 930 
 
 23 
 
 3,369.0 
 
 287.0 
 
 5,290 
 
 450 
 
 48 
 
 14,679 
 
 605 
 
 23,040 
 
 950 
 
 24 
 
 3,669.0 
 
 299.0 
 
 5,760 
 
 470 
 
 49 
 
 15,275 
 
 618 
 
 24,010 
 
 970 
 
 25 
 
 3*981.0 
 
 312.0 
 
 6,250 
 
 490 
 
 50 
 
 15,925 
 
 631 
 
 j 25,000 
 
 990 
 
 Weight of coal = 47 lb. per cu. ft. 
 
HANDLING OF COAL. 
 
 445 
 
 Horizontal Pressure Exerted by Anthracite Coal Against Vertical 
 Retaining Walls per Foot of Length. 
 
 total pressure = 
 pressure lowest ft. = 
 
 Surface horizontal 
 
 Surface sloping { pVessure lowest ft. 
 Angle of repose 
 d = height of wall in feet. 
 
 9.78 d 2 
 9.78(2 d- 1) 
 14.22 d 2 
 = 14.22(2 d-1) 
 = 27° 
 
 ANTHRACITE. 
 
 4-a 
 
 q 
 
 Horizontal 
 
 Surface. 
 
 bm. 
 
 Sloping 
 
 Surface. 
 
 be. 
 
 8 
 
 rO 
 
 6 
 
 6 
 
 r-H fH 
 
 03 3 
 
 
 g 
 
 'S p 
 
 02 02 
 
 & 
 
 n 
 
 02 
 O 02 
 
 Ph 
 
 xn 0> 
 
 Z* 
 
 Qj O 
 
 "O 02 
 
 £ 
 
 02 
 
 <u 
 
 u 1> 
 Q. O 
 
 1 
 
 9.78 
 
 9.78 
 
 14.22 
 
 14.22 
 
 2 
 
 39.12 
 
 29.34 
 
 56.88 
 
 42.66 
 
 3 
 
 88.02 
 
 48.90 
 
 127.98 
 
 71.10 
 
 4 
 
 156.48 
 
 68.46 
 
 227.52 
 
 99.54 
 
 5 
 
 244.50 
 
 88.02 
 
 355.50 
 
 127.98 
 
 6 
 
 352.08 
 
 107.58 
 
 511.92 
 
 156.42 
 
 7 
 
 479.22 
 
 127.14 
 
 696.78 
 
 184.86 
 
 8 
 
 625.92 
 
 146.70 
 
 910.08 
 
 213.30 
 
 9 
 
 792.18 
 
 166.26 
 
 1,151.82 
 
 241.74 
 
 10 
 
 978.00 
 
 185.82 
 
 1,422.00 
 
 270.18 
 
 11 
 
 1,183.38 
 
 205.38 
 
 1,720.62 
 
 298.62 
 
 12 
 
 1,408.32 
 
 224.94 
 
 2,047.68 
 
 327.06 
 
 13 
 
 1,652.82 
 
 244.50 
 
 2,403.18 
 
 355.50 
 
 14 
 
 1,916.88 
 
 264.06 
 
 2,787.12 
 
 383.94 
 
 15 
 
 2,200.50 
 
 283.62 
 
 3,199.50 
 
 412.38 
 
 16 
 
 2,503.68 
 
 303.18 
 
 3,640.32 
 
 440.82 
 
 17 
 
 2,826.42 
 
 322.74 
 
 4,109.56 
 
 469.26 
 
 18 
 
 3,168.72 
 
 342.30 
 
 4,607.28 
 
 497.70 
 
 19 
 
 3,530.58 
 
 361.86 
 
 5,133.42 
 
 526.14 
 
 20 
 
 3,912.00 
 
 381.42 
 
 5,688.00 
 
 554.58 
 
 21 
 
 4,313.00 
 
 400.98 
 
 6,271.00 
 
 583.26 
 
 22 
 
 4,733.50 
 
 420.54 
 
 6,882.50 
 
 611.46 
 
 23 
 
 5,173.70 
 
 440.10 
 
 7,522.50 
 
 639.90 
 
 24 
 
 5,633.30 
 
 459.67 
 
 8,190.70 
 
 668.35 
 
 25 
 
 6,112.60 
 
 479.22 
 
 8,887.50 
 
 696.79 
 
 4-S 
 
 P 
 
 Horizontal 
 
 Surface. 
 
 bm. 
 
 Sloping 
 
 Surface. 
 
 be. 
 
 e 
 
 rO 
 
 A 
 
 ■P 
 
 ft 
 
 o» 
 
 p 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 Total 
 
 Pressure. 
 
 Pressure 
 Lowest Ft. 
 
 26 
 
 6,611.1 
 
 498.78 
 
 9,612.8 
 
 725.21 
 
 27 
 
 7,129.5 
 
 518.35 
 
 10,366.0 
 
 753.67 
 
 28 
 
 7,667.6 
 
 537.90 
 
 11,149.0 
 
 782.10 
 
 29 
 
 8,225.0 
 
 557.46 
 
 11,988.0 
 
 810.54 
 
 30 
 
 8,802.0 
 
 577.01 
 
 12,797.0 
 
 839.00 
 
 31 
 
 9,398.5 
 
 596.59 
 
 13,665.0 
 
 867.41 
 
 32 
 
 10,015.0 
 
 616.14 
 
 14,561.0 
 
 895.86 
 
 33 
 
 10,650.0 
 
 635.70 
 
 15,486.0 
 
 924.30 
 
 34 
 
 11,306.0 
 
 655.26 
 
 16,439.0 
 
 952.70 
 
 35 
 
 11,980.0 
 
 674.81 
 
 17,420.0 
 
 981.19 
 
 36 
 
 12,675.0 
 
 694.39 
 
 18,429.0 
 
 1,009.60 
 
 37 
 
 13,389.0 
 
 713.94 
 
 19,467.0 
 
 1,038.10 
 
 38 
 
 14,123.0 
 
 733.50 
 
 20,533.0 
 
 1,066.50 
 
 39 
 
 14,875.0 
 
 753.07 
 
 21,629.0 
 
 1,095.00 
 
 40 
 
 15,648.0 
 
 772.63 
 
 22,752.0 
 
 1,123.40 
 
 41 
 
 16,440.0 
 
 792.20 
 
 23,904.0 
 
 1,151.80 
 
 42 
 
 17,252.0 
 
 811.74 
 
 25,084.0 
 
 1,180.30 
 
 43 
 
 18,083.0 
 
 830.73 
 
 26,293.0 
 
 1,208.70 
 
 44 
 
 18,934.0 
 
 850.86 
 
 27,530.0 
 
 1,237.20 
 
 45 
 
 19,804.0 
 
 870.41 
 
 28,793.0 
 
 1,265.60 
 
 46 
 
 20,695.0 
 
 889.99 
 
 30,090.0 
 
 1,294.00 
 
 47 
 
 21,605.0 
 
 909.54 
 
 31,412.0 
 
 1,322.30 
 
 48 
 
 22,533.0 
 
 929.10 
 
 32,763.0 
 
 1,350.90 
 
 49 
 
 23,482.0 
 
 948.66 
 
 34,143.0 
 
 1,379.40 
 
 50 
 
 24,450.0 
 
 968.21 
 
 35,550.0 
 
 1,407.90 
 
 Horsepowers for Coal Conveyors (Coal Included). 
 
 Speed, 100 ft. per minute. Conveyors, 100 ft. long. Standard steel troughs. 
 
 ■"I® 
 
 °.o 
 
 flP 
 
 Size of 
 Flights 
 In. 
 
 4 X 10 
 4X12 
 5X12 
 5X15 
 6X18 
 
 Horizontal . 
 
 Inclined. 
 
 12 In. 
 
 Between 
 
 Flights. 
 
 2 } 
 
 3 
 
 3* 
 
 4* 
 
 54 
 
 18 In. 
 
 Between 
 
 Flights. 
 
 2 
 
 24 
 
 3 
 
 34 
 
 44 
 
 12 In. 
 Between 
 Flights. 
 
 3 
 34 
 
 4 
 54 
 64 
 
 o .g 
 
 N rd 
 
 5° 
 
 Size of 
 Flights. 
 In. 
 
 5X15 
 6X18 
 8X 18 
 8 X 20 
 8 X 24 
 10 X 24 
 
 Horizontal. 
 
 16 In. 
 
 Between 
 
 Flights. 
 
 4 
 
 5 
 
 7 
 
 8 
 94 
 
 124 
 
 24 In. 
 
 Between 
 
 Flights. 
 
 34 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 Inclined. 
 
 16 In. 
 
 Between 
 
 Flights. 
 
 44 
 
 54 
 
 8 
 
 10 
 
 114 
 
 14 
 
446 
 
 ORE DRESSING AND PREPARATION OF COAL. 
 
 Weights and Capacities of Standard Steel Buckets. 
 
 Chain. 
 
 Size of Bucket. 
 In. 
 
 Weight of 
 Bucket. 
 Lb. 
 
 Capacity of 
 Bucket. 
 Lb. 
 
 Capacity of Elevator. 100' per Min. 
 
 Number of 
 Drawing. 
 
 Lb. per Min. 
 
 Net Tons per Hr. 
 
 a> 
 
 W) 
 
 *d - 
 
 12 X 9 X 11? 
 
 181 
 
 11 
 
 1,100 
 
 33.0 
 
 5,357 
 
 o 
 
 Q 
 
 14 X 9X11? 
 
 221 
 
 121 
 
 1,250 
 
 37.5 
 
 5,357 
 
 0 
 
 18 X 9X11? 
 
 27 
 
 161 
 
 1,650 
 
 49.5 
 
 5,357 
 
 
 24 X 9 X Ilf 
 
 36 
 
 22 
 
 2,200 
 
 66.0 
 
 5,357 
 
 
 12X10 X 161 
 
 20 
 
 19 
 
 1,380 
 
 41.4 
 
 5,357 
 
 tS 
 
 bC 
 
 18 X 10 X 161 
 
 29 
 
 281 
 
 2,072 
 
 62.2 
 
 5,357 
 
 'O 
 
 O 
 
 24X10X161 
 
 38 
 
 38 
 
 2,760 
 
 82.8 
 
 5,357 
 
 ft 
 
 30 X 10 X 161 
 
 461 
 
 471 
 
 3,450 
 
 103.5 
 
 5,357 
 
 a 
 
 18 X 12 X 161 
 
 31 
 
 33 
 
 2,400 
 
 72.0 
 
 5,357 
 
 
 24 X 12 X 161 
 
 40 
 
 44 
 
 3,200 
 
 96.0 
 
 5,357 
 
 
 30 X 12 X 161 
 
 48 
 
 55 
 
 4,000 
 
 120.0 
 
 5,357 
 
 Buckets taken £ full. Buckets continuous. 1 lb. of coal = 34 cu. in. 
 
 Elevating Capacities of Malleable Iron Buckets. 
 
 Table gives tons (2,000 lb.) of pea coal per hour at 100 ft. per minute. 
 
 Buckets. 
 
 Capacities. 
 
 Distance Between Buckets in In. 
 
 Size 
 
 In. 
 
 
 Wt. 
 
 Lb. 
 
 Cu. 
 
 In. 
 
 Lb. 
 
 8 
 
 10 
 
 1 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 2f 
 
 X 
 
 4 
 
 0.75 
 
 15 
 
 0.48 
 
 2.16 
 
 1.73 
 
 1.44 
 
 1.23 
 
 1.08 
 
 1.94 
 
 
 
 
 31 
 
 x 
 
 5 
 
 1.50 
 
 31 
 
 0.97 
 
 4.36 
 
 3.49 
 
 2.91 
 
 2.49 
 
 2.18 
 
 
 
 
 4 
 
 X 
 
 6 
 
 2.00 
 
 51 
 
 1.57 
 
 7.06 
 
 5.65 
 
 4.71 
 
 4.04 
 
 3.53 
 
 3.14 
 
 2.83 
 
 3.81 
 
 
 41 
 
 x 
 
 7 
 
 2.56 
 
 75 
 
 2.33 
 
 10.38 
 
 8.39 
 
 6.99 
 
 5.99 
 
 5.19 
 
 4.66 
 
 4.19 
 
 4.72 
 
 5 
 
 x 
 
 8 
 
 3.56 
 
 102 
 
 3.15 
 
 
 11.34 
 
 9.45 
 
 8.10 
 
 7.09 
 
 6.30 
 
 5.67 
 
 5.15 
 
 6 
 
 x 
 
 10 
 
 5.47 
 
 185 
 
 5.73 
 
 
 
 17.19 
 
 14.73 
 
 12.88 
 
 11.46 
 
 10.31 
 
 9.38 
 
 8.59 
 
 7 
 
 x 
 
 12 
 
 8.97 
 
 287 
 
 8.90 
 
 
 
 
 22.88 
 
 20.02 
 
 17.80 
 
 16.02 
 
 14.56 
 
 13.35 
 
 7 
 
 X 
 
 14 
 
 11.41 
 
 295 
 
 9.14 
 
 
 
 
 
 20.56 
 
 18.28 
 
 16.45 
 
 14.95 
 
 13.71 
 
 10 
 
 X 
 
 18 
 
 
 
 
 
 
 1 
 
 
 
 
 1 
 
 
 
 Weight of 1 cu. ft. of pea coal = 53.5 lb. 32.3 cu. in., or .0187 cu. ft. = 1 lb. 
 
 Conveying Capacities of Flights at 100 Ft. Per Minute. 
 
 (Tons of Pea Coal per Hour.) 
 
 Size 
 
 of 
 
 Flight. 
 
 In. 
 
 
 Horizontal. 
 
 
 
 Inclined. 
 
 
 
 
 
 
 10° 
 
 20° 
 
 30° 
 
 
 
 
 Lb. Coal 
 per 
 
 Flight. 
 
 
 
 Every 
 16 In. 
 
 Every 
 18 In. 
 
 Every 
 24 In. 
 
 Every 
 24 In. 
 
 Every 
 24 In. 
 
 Every 
 24 In. 
 
 4 X 10 
 
 33.75 
 
 30 
 
 22.5 
 
 15 
 
 18.0 
 
 14.25 
 
 10.5 
 
 4 X 12 
 
 42.75 
 
 38 
 
 28.5 
 
 19 
 
 24.0 
 
 18.00 
 
 13.5 
 
 5 X 12 
 
 51.75 
 
 46 
 
 34.5 
 
 23 
 
 28.5 
 
 22.50 
 
 16.5 
 
 5 X 15 
 
 69.75 
 
 62 
 
 46.5 
 
 31 
 
 40.5 
 
 31.50 
 
 22.5 
 
 6 X 18 
 
 
 80 
 
 60.0 
 
 40 
 
 49.5 
 
 40.50 
 
 31.5 
 
 8X18 
 8X 20 
 8 X 24 
 10 X 24 
 
 
 120 
 
 90.0 
 
 105.0 
 
 135.0 
 172.5 
 
 60 
 
 70 
 
 90 
 
 115 
 
 72.0 
 
 84.0 
 120.0 
 150.0 
 
 57.00 
 66.50 
 
 96.00 
 120.00 
 
 48.0 
 
 56.0 
 
 72.0 
 
 90.0 
 
 Note. — T hese ratings are for continuous feed. 2,000 lb. — 1 ton. 
 
HANDLING OF COAL. 
 
 447 
 
 Horsepower for Bucket Elevators, 
 
 N = number taken from table; 
 
 = height of elevatpr in feet; 
 co = weight of material in one bucket; 
 d = distance apart of buckets, in inches. 
 
 Revolu- 
 
 tions 
 
 per 
 
 Minute. 
 
 Diameter of Head-Wheels. 
 
 Revolu- 
 
 tions 
 
 per 
 
 Minute. 
 
 22 In. 
 
 24 In. 
 
 26 In. 
 
 28 In. 
 
 30 In. 
 
 32 In. 
 
 10 
 
 .064 
 
 .070 
 
 .075 
 
 .080 
 
 .087 
 
 .093 
 
 10 
 
 12 
 
 .077 
 
 .083 
 
 .090 
 
 .097 
 
 .104 
 
 .111 
 
 12 
 
 14 
 
 .089 
 
 .096 
 
 .106 
 
 .114 
 
 .121 
 
 .130 
 
 14 
 
 16 
 
 .102 
 
 .111 
 
 .121 
 
 .130 
 
 .140 
 
 .148 
 
 16 
 
 18 
 
 .115 
 
 .125 
 
 .136 
 
 .146 
 
 .157 
 
 .167 
 
 18 
 
 20 
 
 .128 
 
 .139 
 
 .151 
 
 .162 
 
 .174 
 
 .186 
 
 20 
 
 22 
 
 .140 
 
 .153 
 
 .166 
 
 .179 
 
 .191 
 
 .204 
 
 22 
 
 24 
 
 .153 
 
 .167 
 
 .181 
 
 .195 
 
 .209 
 
 .223 
 
 24 
 
 26 
 
 .166 
 
 .181 
 
 .196 
 
 .211 
 
 .226 
 
 .242 
 
 26 
 
 28 
 
 .179 
 
 .195 
 
 .211 
 
 .227 
 
 .244 
 
 .260 
 
 28 
 
 30 
 
 .191 
 
 .209 
 
 .226 
 
 .244 
 
 .261 
 
 .279 
 
 30 
 
 32 
 
 .204 
 
 .223 
 
 .241 
 
 .260 
 
 .278 
 
 .297 
 
 32 
 
 34 
 
 .217 
 
 .237 
 
 .256 
 
 .276 
 
 .296 
 
 .316 
 
 34 
 
 36 
 
 .230 
 
 .251 
 
 .271 
 
 .292 
 
 .313 
 
 .334 
 
 36 
 
 38 
 
 .242 
 
 .265 
 
 .287 
 
 .309 
 
 .331 
 
 .353 
 
 38 
 
 40 
 
 .255 
 
 .279 
 
 .302 
 
 .325 
 
 .348 
 
 .372 
 
 40 
 
 COST OF UNLOADING COAL. 
 
 Coal is generally unloaded from railroad cars into the hold of a vessel by 
 some form of unloader, which usually raises the car bodily and dumps it 
 directly into the hold of the vessel. In this way the cost of unloading has 
 been reduced to a very small figure, and the speed of unloading greatly 
 increased. The cost of unloading is given by the makers of the Brown hoist 
 as varying from 2£ cents per ton up to 4£ cents per ton; deducting in each 
 case 2 cents for trimming the coal in the vessel, the actual cost of loading 
 varies from £ cent to 2£ cents per ton, depending on the conditions. Along 
 the Lakes it is customary to pay a premium of £ cent per ton to all connected 
 with the loading, for all coal loaded in excess of 2,500 tons per day and 1,800 
 tons per night. The Brown hoist has a guaranteed capacity of at least 800 
 tons per hour, but this has been greatly exceeded in practice TheMcMyler 
 end dump has a record of 4.65 tons per minute, and the McMyler side dump 
 of 8.41 tons per minute. These figures apply to the lake cities of the U. b. 
 
 The C. W. Hunt Co., West New Brighton, N. Y., gives the follovvmg 
 figures for handling coal along the Atlantic seaboard: The cost of shoveling 
 coal by hand in the hold of the vessel into ordinary iron buckets is about 6 
 to 7 cents per ton of 2,000 lb.; the cost for iron ore, phosphate rock, or sand, 
 about 10 i less. The cost of shoveling coal and hoisting it out of vessel to the 
 wharf with an ordinary hoist with manila rope is 12 to 13 cents per ton, so 
 that the hoisting costs about the same as the shoveling. The cost for both 
 shoveling and hoisting with a steam engine is 10 to 11 cents per ton. ine 
 cost when using a steam shovel or grab bucket for taking up coal out of the 
 vessel varies greatly in different classes of vessels, but usually runs from 
 about 1£ to 5 cents per ton, averaging about 3 cents. After the coal is 
 hoisted, it can be carried into storage with an automatic railway or other 
 efficient plant, at a cost of about 1 to 1£ cents per ton For great distances a 
 cable railway or a convevor can be used, which handles the material about 
 as cheaply as for short distances, but the cost of plant is greatly increased. 
 
 In unloading anthracite from cars on a trestle into pockets or on the 
 ground, the loss on all sizes is 2 to 3j$ when the coal is not resized; when it 
 is resized the loss is 8£ to 9j&. 
 
448 
 
 BEIQ UETING. 
 
 The cost of stocking and unloading anthracite by the Dodge system is 
 given by Mr. Piez, “ Mines and Minerals/ * June, 1898, page 488, as follows: 
 
 Year. 
 
 Engine Service, 
 Stocking and 
 Lifting, per Ton. 
 Cents. 
 
 Office Expense, 
 per Ton. Cents. 
 
 Steam, Wages 
 and Fuel, per 
 Ton. Cents. 
 
 Labor, Dumping 
 and Lifting, per 
 Ton. Cents. 
 
 Repairs, per 
 Ton. Cents. 
 
 Supplies, per 
 Ton. Cents. 
 
 Total, per Ton. 
 Cents. 
 
 1895 
 
 .87 
 
 .29 
 
 .97 
 
 2.67 
 
 .78 
 
 .25 
 
 5.83 
 
 1896 
 
 .78 
 
 .30 
 
 .82 
 
 2.19 
 
 .90 
 
 .27 
 
 5.26 
 
 1897 
 
 .69 
 
 .32 
 
 .62 
 
 1.88 
 
 .97 
 
 .16 
 
 4.64 
 
 BRIQUETING. 
 
 Machines Employed— Fuel, fuel dust, and other products may be briqueted 
 by a number of different styles of machines, but all these may be divided 
 into two classes, briquet and eggette machines. The eggette machines have 
 a pair of rollers, the faces of which are provided with semispherical or semi- 
 ovoid openings. The material that is fed between these rolls crowds into 
 the openings of the two rolls, thus forming small spheres. The material is 
 mixed with a suitable bond before being fed to the rolls, and the eggettes are 
 received on any suitable form of traveling belt or chute and removed for 
 drying or storage. This style of machine has not been used to any great 
 extent in this country. The briqueting machines all act more or less on the 
 principle of the brick machine, having some kind of a die or mold into 
 which the material is crowded. The material is either pressed as it is being 
 fed into the mold or subsequently by some form of plunger. For some 
 materials, common brick machines, such as are used in the manu- 
 facture of building brick, are employed, while in others special forms are 
 necessary. , . 
 
 Briqueting of Fuel.— Fuel briquets have not come into general use m the 
 United States for two reasons: (1) on account of the great amount of cheap 
 fuel available, which has prevented the utilization of culm, coal dust, etc.; 
 and (2) on account of the lack of or high price of suitable bonding material. 
 This latter condition is now being removed by the introduction of by-product 
 coke ovens, from which supplies of coal tar can be obtained. Aside from peat 
 and certain kinds of brown coal, and possibly some caking coals, it is neces- 
 sary to employ a bond in the making of any fuel briquets. This is especially 
 true in the case of anthracite coal The present tendency is to employ no 
 inorganic bonding materials, as they increase the ash. The material to be 
 briqueted should be as clean and free from dirt or slate as possible, and the 
 particles should be of practically uniform size, the most satisfactory product 
 being from coal crushed to about fin. cube size. The coal must be thor- 
 oughly mixed with bonding material and then subjected to a heavy pressure. 
 One advantage claimed for briquets is that they can be made of such a form 
 as to occupy less space than the original fuel. The French navy has found 
 it possible to store 10^ more briquets than coal in a given space, and also 
 that the loss by breakage and pulverization is very much less. Under favor- 
 able conditions, fuel can be briqueted for 20 cents per ton, and the following 
 are some of the advantages claimed for these briquets: They are sound 
 throughout and will not decrepitate while burning, thus reducing the 
 loss by fine material working through the grates. The bond, if properly 
 selected, renders the briquets practically waterproof, so that they are not 
 injured if kept in storage, do not evolve combustible gases, nor ignite from 
 spontaneous combustion. There is no fine material mixed with the briquets, 
 and hence a more uniform fire can be maintained with them. 
 
 Briqueting of Flue Dust.— Flue dust from iron blast furnaces has been suc- 
 cessfully briqueted in a number of instances. One firm employs a common 
 brick machine, making bricks 2£ in. X 4f in. X 9 in. With this machine, 
 they mix the flue dust with 3$ of lime and 3^ of cement, the lime acting as 
 a flux in the furnace. These machines work with comparatively light pres- 
 sure. When regular briqueting machines, producing round bricks and 
 
TREATMENT OF INJURED PERSONS. 
 
 449 
 
 employing high pressures are employed, no cementne^ 
 being mixed with 4 4 to 
 
 are loaded ^into barrels and taken direct to the blast furnace, with as 
 
 u Another firm, figuring on a basis of 130 tons per 24 hours, and using 3j6 
 lime in the solution, gave the following figures: 
 
 4 tons lime, 13.00 per **r:5 
 
 2 machine tenders (day and night), 12hours, at #2.50... 
 
 2 laborers (day and night), 12 hours, at $1.75 
 
 Oil and waste - - 
 
 Interest on cost of plant 1 . ^ ' 
 
 Thi* is less than 20 cents per ton. This estimate does not take into con- 
 sideration the cost of power, which would be about 35 H. P., nor does it take 
 into consideration hauling of material to plant and removing of briquets. 
 
 Cubic Feet Occupied by 2,000 Pounds of Yabious Coals. 
 
 ( Link-Belt Engineering Co ., Philadelphia , Pa.) 
 
 5.00 
 
 3.50 
 
 2.00 
 
 1.50 
 
 Varieties. 
 
 Lackawanna, anthracite.. 
 
 Garfield red ash, anthracite . 
 Ly kens Valley, anthracite .... 
 
 Shamokin, anthracite : -— 
 
 Plymouth red ash, anthracite. 
 
 Wilkes-Barre, anthracite 
 
 Lehigh, anthracite 
 
 Lorberry, anthracite 
 
 Scranton, anthracite 
 
 Pittston, anthracite 
 
 Broken. 
 
 Egg. 
 
 Stove. 
 
 37.10 
 
 36.65 
 
 34.90 
 
 37.30 
 
 36.95 
 
 36.35 
 
 37.55 
 
 37.25 
 
 37.55 
 
 38.05 
 
 37.70 
 
 37.25 
 
 34.90 
 
 34.85 
 
 34.75 
 
 34.95 
 
 34.35 
 
 33.75 
 
 33.30 
 
 33.80 
 
 33.55 
 
 34.65 
 
 34.20 
 
 33.80 
 
 35.35 
 
 35.20 
 
 34.60 
 
 35.45 
 
 34.95 
 
 34.35 
 
 34.35 
 
 36.35 
 37.25 
 37.25 
 
 34.70 
 34.00 
 
 32.55 
 
 33.55 
 33.30 
 
 33.70 
 
 Pea, 
 
 37.25 
 
 37.50 
 
 38.50 
 
 38.50 
 36.90 
 36.90 
 33.05 
 35.20 
 34.95 
 
 35.50 
 
 Cumberland, bituminous 36.65 
 
 Clearfield, bituminous 33.55 
 
 New River, bituminous. | 40.15 
 
 Pocahontas, bituminous 
 
 American cannel, bituminous 
 English cannel, bituminous... 
 
 34.00 
 
 41.50 
 
 42.30 
 
 TREATMENT OF INJU RED PERSONS. 
 
 The dangers to be feared in case of wounds, are: shock or collapse , loss of 
 
 blood , and unnecessary suffering in the moving qf the P at ™nt. insen- 
 
 Tn shock the iniured person lies pale, faint, and cold, sometimes 
 sible with feeble pulse and superficial breathing. The cause of death in 
 case of a shock is arrest of heart action, produced by the suspension of the 
 functions of the brain and spinal cord. In treatment, the two most import- 
 ant" of the injured person; (2) the application of 
 
 eX ThTinTured^erson should at once be placed in a recumbent portion, his 
 head resting 1 on a plane lower than that of his trunk, legs, and feet. He 
 should be lell wrapped up and protected from the chilling influences of 
 external air When there is danger of immediate death, stimulants should 
 bfliven; in all other conditions of shock, stimulants are injurious 
 
 Sss of Blood. — In case of loss of blood, two conditions P res ^\thems^ 
 m \ The bleeding is arrested spontaneously or otherwise, but the injured 
 tli symptoms of loss .of blood; (2 'the injured pers0 n is 
 actually bleeding and he is, or is not, suffering from loss of blood. 
 
 In the first condition, life is threatened by anemia .of brain spinal 
 
 cord, and all the efforts of treatment are to direct the flow of whatever 
 
450 
 
 TEE A TMENT OF INJURED PERSONS. 
 
 quantity of blood may still remain in the body to the vital centers in the 
 brain and spinal cord. This is most efficiently done by placing the injured 
 person in a recumbent position, with his head resting on a plane somewhat 
 
 Fig. 1. 
 
 Fig. 2. 
 
 lower than that of his trunk and legs. In graver cases, constricting bands 
 should be applied to both arms, as near the shoulders as possible, and to 
 both thighs, as near the abdomen as possible. This last maneuver directs 
 the entire quantity of blood in the body to the suffering centers, the centers 
 of life itself. Stimulants may be sparingly administered. 
 
 If there is bleeding, do not try to stop it by binding up the wound. The 
 current of blood to the part must be checked. To do this, find the artery, by *ts 
 beating; lay a firm and even compress or pad (made of cloth or rags rolled 
 up, or a round stone or piece of wood well wrapped) over the artery 
 ( Fig. 1 ) . Tie a handkerchief around the limb and compress; put a 
 bit of stick through the handkerchief and twist the latter up 
 until it is just tight enough to stop the bleed- 
 ing; then put one end of the stick under 
 the handkerchief, to prevent untwisting, 
 as in Fig. 2. 
 
 The artery in the thigh runs along the 
 inner side of the muscle in front near the 
 bone, as shown by dotted line in Fig. 8. 
 
 A little above the knee it passes to the 
 back of the bone. In injuries at or above 
 the knee, apply the compress higher up. 
 on the inner side of the thigh, at the point 
 P, Fig. 3, with the knot on the outside of 
 the thigh. 
 
 When the leg is injured below the 
 knee, apply the compress at the back of 
 the thigh, just above the knee, at P, Fig. 4, 
 and the knot in front, as in Figs. 1 and 2. 
 
 The artery in the arm runs down the inner side of the large muscle in 
 front, quite close to the bone, as shown by dotted line; low down it is further 
 forwards, towards the bend of the elbow. It is most easily compressed a 
 little above the middle, at P. Fig. 5. Care should be taken to examine the 
 limb from time to time, and to lessen the compression if it becomes cold 
 or purple; tighten up the handkerchief 
 again if the bleeding begins afresh. 
 
 To Transport a Wounded Person Comfortably. 
 
 Make a soft and even bed for the injured 
 part, of straw, folded blankets, quilts, or 
 
 Fig. 3. 
 
 Fig. 4. 
 
 Fig. 5 
 
 Fig. 6. 
 
 pillows, laid on a board with side pieces of board nailed on. when this can 
 be done. If possible, let the patient be laid on a door, shutter, settee, or 
 some firm support, properly covered. Have sufficient force to lift him 
 steadily, and let those that bear him not keep step. 
 
 Should any important arteries be opened, apply the handkerchief, as 
 recommended. Secure the vessel by a surgeon’s dressing forceps, or by a 
 
TREATMENT OF PERSONS OVERCOME BY GAS. 
 
 451 
 
 hook then have a silk ligature put around the vessel, and tighten, should 
 the bleeding be from arterial vessels of small size, apply persulphate of iron, 
 either in tmcture or in powder, by wetting a piece of lmt or sponge with the 
 solution 1 then after bleeding ceases, apply a compress against the parts, to 
 sustain them during the application of the persulphate of iron, and to 
 vent ^ furthe^bleeding, should it occur. The persulphate of iron should be 
 
 ke meedin^rom 1 1 cVl^WoTivc! A %ad or compress is placed immediately 
 before the ear over the region marked by a dotted line, Fig. 6. The com- 
 nress is firmly secured by a handkerchief. If this ^o®s not ^rest bleeding, a 
 similar compress on the opposite side should be applied. Should the bleed- 
 inTissurfmmTwound of the posterior or back part of the head, a compress 
 should be placed behind the ear, over the region marked by the dotted line, 
 Fig. 6, and firmly secured by a handkerchief or bandage. 
 
 TREATMENT OF PERSONS OVERCOME BY GAS. 
 
 Miners are exposed to asphyxia when the circulation of the air is not suf- 
 ficiently active, when the mine exhales a quantity of deleterious gas, when 
 they imprudently penetrate into old and abandoned workings, and when 
 
 th6 The symptoms of asphyxia are sudden cessation of the respiration, of the 
 
 pulsations of the heart, and of the action of the senses; the 
 
 swollen and marked with reddish spots, the eyes are protruded, the features 
 
 are distorted, and the face is often livid, etc. . ^ 
 
 The best and first remedy to employ, and m which the greatest confidence 
 ought to be placed, is the renewal of the air necessary for respiration. 
 
 Pr T ee prompflywithdraw the asphyxiated person from the deleterious place 
 
 and exf^se^him^to roun( j the neck and chest, and dash cold water 
 
 in ‘ke^ace^and s hould e be h made to irritate the mucous membrane with i the 
 feathered end of a quill, which should be gently moved in the nostrils of the 
 insensible person, or to stimulate it with a bottle of volatile alkali placed 
 
 under .tbe ^ U p wa rmth of the body, and apply mustard plasters over the 
 
 heart an pro duce respiration, Doctor Sylvester’s method 
 of producing artificial respiration should be tried as follows: 
 
 Place the patient on the back on a flat surface, inclined a little upwards 
 from the feet; raise and support the head and shoulders on a smaU firm 
 cushion or folded article of dress placed under the shoulder blades. D raw 
 forwards the patient’s tongue and keep it projecting beyond the lips, an 
 elastic band over the tongue and under the chin will answer this purpose, or 
 a piece of string or tape may be tied around them, or by raising the lower 
 iaw the teeth may be made to retain the tongue in that position. Remove 
 all tight clothing from about the neck and chest, especially the suspenders. 
 Then g standing at the patient’s head, grasp the arms just above the elbows, 
 and draw the arms gently and steadily upwards above the head, and keep 
 them stretched upwards for 2 seconds (by this means air is diawn into the 
 lungs'). Then turn down the patient’s arms and press them gently and 
 firmly for 2 seconds against the sides of the chest (by this means air is Pressed 
 out of the lungs) . Repeat these measures alternately , deliberately , and per- 
 severingly about 15 times in a minute, until a spipntaneous effort to respire 
 is perceived, immediately upon which cease to imitate the movements of 
 breathing, and proceed to induce circulation and warmth. 
 
 6 To promote warmth and circulation, rub the limbs upwards with firm, 
 
 grasping pressure and energy, using handkerchiefs flannels, etc. Apply hot 
 flannels! bottles of hot water, heated bricks, etc. to the pit of the stomach, 
 the arm pits, between the thighs, and to the soles of the feet. , . 
 
 7 On the restoration of life, a teaspoonful of warm water should be given, 
 and then, if the power of swallowing has returned, small quantities of wine, 
 warm brandy and water, or coffee should be administered. 
 
 8. These remedies should be promptly applied, and as death does not 
 certainly appear for a long time, they ought only to be discontinued^ when it 
 is clearly confirmed. Absence of the pulsation of the heart is not a sure sign 
 of death, neither is the want of respiration. 
 
452 
 
 COAL DEALER'S TABLE. 
 
 Coal Dealers’ Computing Table, for Ascertaining the Price of Any 
 Number of Pounds, at a Given Price per Ton of 2,000 Pounds. 
 
 Lb . 
 
 $0.75 
 
 $1.00 
 
 $1.25 
 
 $1.50 
 
 $1.75 
 
 $2.00 
 
 $2.25 
 
 $2.50 
 
 $2.75 
 
 10 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 .01 
 
 20 
 
 .01 
 
 .01 
 
 .01 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .03 
 
 .03 
 
 30 
 
 .01 
 
 .02 
 
 .02 
 
 .02 
 
 .03 
 
 .03 
 
 .03 
 
 .04 
 
 .04 
 
 40 
 
 .02 
 
 .02 
 
 .03 
 
 .03 
 
 .04 
 
 .04 
 
 .04 
 
 .05 
 
 .06 
 
 50 
 
 .02 
 
 .02 
 
 .03 
 
 .04 
 
 .04 
 
 .05 
 
 .06 
 
 .06 
 
 .07 
 
 60 
 
 .02 
 
 .03 
 
 .04 
 
 .05 
 
 .05 
 
 .06 
 
 .07 
 
 .08 
 
 .08 
 
 70 
 
 .03 
 
 .03 
 
 .04 
 
 .05 
 
 .06 
 
 .07 
 
 .08 
 
 .09 
 
 .10 
 
 80 
 
 .03 
 
 .04 
 
 .05 
 
 .06 
 
 .07 
 
 .08 
 
 .09 
 
 .10 
 
 .11 
 
 90 
 
 .03 
 
 .04 
 
 .06 
 
 .07 
 
 ..08 
 
 .09 
 
 .10 
 
 .11 
 
 .12 
 
 100 
 
 .04 
 
 .05 
 
 .06 
 
 .08 
 
 .09 
 
 .10 
 
 .11 
 
 .13 
 
 .14 
 
 200 
 
 .08 
 
 .10 
 
 .13 
 
 .15 
 
 .17 
 
 .20 
 
 .23 
 
 .25 
 
 .28 
 
 300 
 
 .11 
 
 .15 
 
 .19 
 
 .23 
 
 .26 
 
 .30 
 
 .34 
 
 .38 
 
 .41 
 
 400 
 
 .15 
 
 .20 
 
 .25 
 
 .30 
 
 .35 
 
 .40 
 
 .45 
 
 .50 
 
 .55 
 
 500 
 
 .19 
 
 .25 
 
 .31 
 
 .38 
 
 44 
 
 .50 
 
 .56 
 
 .63 
 
 .69 
 
 600 
 
 .23 
 
 .30 
 
 .37 
 
 .45 
 
 .53 
 
 .60 
 
 .68 
 
 .75 
 
 .83 
 
 700 
 
 .26 
 
 .35 
 
 .44 
 
 .53 
 
 .61 
 
 .70 
 
 .77 
 
 .88 
 
 .96 
 
 800 
 
 .30 
 
 .40 
 
 .50 
 
 .60 
 
 .70 
 
 .80 
 
 .90 
 
 1.00 
 
 1.10 
 
 900 
 
 .34 
 
 .45 
 
 .56 
 
 .68 
 
 .79 
 
 .90 
 
 1.01 
 
 1.13 
 
 1.24 
 
 1,000 
 
 .38 
 
 .50 
 
 .63 
 
 .75 
 
 .88 
 
 1.00 
 
 1.13 
 
 1.25 
 
 1.38 
 
 1,100 
 
 .41 
 
 .55 
 
 .69 
 
 .83 
 
 .96 
 
 1.10 
 
 1.24 
 
 1.38 
 
 1.51 
 
 1,200 
 
 .45 
 
 .60 
 
 .75 
 
 .90 
 
 1.05 
 
 1.20 
 
 1.35 
 
 1.50 
 
 1.65 
 
 L 300 
 
 .49 
 
 .65 
 
 .81 
 
 .98 
 
 1.14 
 
 1.30 
 
 1.46 
 
 1.63 
 
 1.79 
 
 L 400 
 
 .52 
 
 .70 
 
 .88 
 
 1.05 
 
 1.22 
 
 1.40 
 
 1.58 
 
 1.75 
 
 1.93 
 
 1,500 
 
 .56 
 
 .75 
 
 .94 
 
 1.13 
 
 1.31 
 
 1.50 
 
 1.69 
 
 1.88 
 
 2.06 
 
 1,600 
 
 .60 
 
 .80 
 
 1.00 
 
 1.20 
 
 1.40 
 
 1.60 
 
 1.80 
 
 2.00 
 
 2.20 
 
 1,700 
 
 .64 
 
 .85 
 
 1.06 
 
 1.28 
 
 1.49 
 
 1.70 
 
 1.91 
 
 2.13 
 
 2.34 
 
 1,800 
 
 .68 
 
 .90 
 
 1.13 
 
 1.35 
 
 1.58 
 
 1.80 
 
 2.03 
 
 2.25 
 
 2.48 
 
 1,900 
 
 .71 
 
 .95 
 
 1.19 
 
 1.43 
 
 1.66 
 
 1.90 
 
 2.14 
 
 2.38 
 
 2.61 
 
 Lb . 
 
 $3.00 
 
 $3.25 
 
 $3.50 
 
 $3.75 
 
 $4.00 
 
 $4.25 
 
 $4.50 
 
 $4.75 
 
 $5.00 
 
 10 
 
 .02 
 
 
 .02 
 
 .02 
 
 .02 
 
 .02 
 
 .03 
 
 .03 
 
 .03 
 
 20 
 
 .03 
 
 .03 
 
 .04 
 
 .04 
 
 .04 
 
 .05 
 
 .05 
 
 .05 
 
 .05 
 
 30 
 
 .05 
 
 .05 
 
 .05 
 
 .06 
 
 .06 
 
 .07 
 
 .07 
 
 .07 
 
 .08 
 
 40 
 
 .06 
 
 .07 
 
 .07 
 
 .08 
 
 .08 
 
 .09 
 
 .09 
 
 .10 
 
 .10 
 
 50 
 
 .08 
 
 .08 
 
 .09 
 
 .09 
 
 .10 
 
 .11 
 
 .12 
 
 ,12 
 
 .13 
 
 60 
 
 .09 
 
 .10 
 
 .11 
 
 .11 
 
 .12 
 
 .13 
 
 .14 
 
 .15 
 
 .15 
 
 70 
 
 .11 
 
 .11 
 
 .12 
 
 .13 
 
 .14 
 
 .15 
 
 .16 
 
 .17 
 
 .18 
 
 80 
 
 .12 
 
 .13 
 
 .14 
 
 .15 
 
 .16 
 
 .17 
 
 .18 
 
 .19 
 
 .20 
 
 90 
 
 .14 
 
 .15 
 
 .16 
 
 .17 
 
 .18 
 
 .19 
 
 .20 
 
 .22 
 
 .23 
 
 100 
 
 .15 
 
 .16 
 
 .18 
 
 .19 
 
 .20 
 
 .22 
 
 .23 
 
 .24 
 
 .25 
 
 200 
 
 .30 
 
 .33 
 
 .35 
 
 .38 
 
 .40 
 
 .43 
 
 .45 
 
 .48 
 
 .50 
 
 300 
 
 .45 
 
 .49 
 
 .53 
 
 .56 
 
 .60 
 
 .64 
 
 .68 
 
 .72 
 
 .75 
 
 400 
 
 .60 
 
 .65 
 
 .70 
 
 .75 
 
 .80 
 
 .85 
 
 .90 
 
 .95 
 
 1.00 
 
 500 
 
 .75 
 
 .81 
 
 .88 
 
 .94 
 
 1.00 
 
 1.07 
 
 1.13 
 
 1.19 
 
 1.25 
 
 600 
 
 .90 
 
 .98 
 
 1.05 
 
 1.13 
 
 1.20 
 
 1.28 
 
 1.35 
 
 1.43 
 
 1.50 
 
 700 
 
 1.05 
 
 1.14 
 
 1.23 
 
 1.31 
 
 1.40 
 
 1.49 
 
 1.58 
 
 1.67 
 
 1.75 
 
 800 
 
 1.20 
 
 1.30 
 
 1.40 
 
 1.50 
 
 1.60 
 
 1.70 
 
 1.80 
 
 1.90 
 
 2.00 
 
 900 
 
 1.35 
 
 1.46 
 
 1.58 
 
 1.69 
 
 1.80 
 
 1.92 
 
 2.03 
 
 2.14 
 
 2.25 
 
 1,000 
 
 1.50 
 
 1.63 
 
 1.75 
 
 1.88 
 
 2.00 
 
 2.13 
 
 2.25 
 
 2.38 
 
 2.50 
 
 1,100 
 
 1.65 
 
 1.79 
 
 1.93 
 
 2.06 
 
 2.20 
 
 2.34 
 
 2.48 
 
 2.62 
 
 2.75 
 
 1,200 
 
 1.80 
 
 1.95 
 
 2.10 
 
 2.25 
 
 2.40 
 
 2.55 
 
 2.70 
 
 2.85 
 
 3.00 
 
 1,300 
 
 1.95 
 
 2.11 
 
 2.28 
 
 2.44 
 
 2.60 
 
 2.77 
 
 2.93 
 
 3.09 
 
 3.25 
 
 1,400 
 
 2.10 
 
 2.28 
 
 2.45 
 
 2.63 
 
 2.80 
 
 2.98 
 
 3.15 
 
 3.33 
 
 3.50 
 
 1,500 
 
 2.25 
 
 2.44 
 
 2.63 
 
 2.81 
 
 3.00 
 
 3.19 
 
 3.38 
 
 3.57 
 
 3.75 
 
 1,600 
 
 2.40 
 
 2.60 
 
 2.80 
 
 3.00 
 
 3.20 
 
 3.40 
 
 3.60 
 
 3.80 
 
 4.00 
 
 1,700 
 
 2.55 
 
 2.76 
 
 2.98 
 
 3.19 
 
 3.40 
 
 3.62 
 
 3.83 
 
 4.04 
 
 4.25 
 
 1,800 
 
 2.70 
 
 2.93 
 
 3.15 
 
 3.38 
 
 3.60 
 
 3.83 
 
 4.05 
 
 4 28 
 
 4.50 
 
 1,900 
 
 2.85 
 
 3.09 
 
 3.33 
 
 3.56 
 
 3.80 
 
 4.04 
 
 4.28 
 
 4.52 
 
 4.75 
 
NATURAL SINES AND COSINES, 
 
 453 
 
 TABLE OF NATURAL SINES, COSINES, 
 TANGENTS, AND COTANGENTS. 
 
 EXPLANATION. 
 
 Given an angle, to find its sine, cosine, tangent, and cotangent: 
 
 Example— Find the sine, cosine, tangent, and cotangent of 37° 24 . . 
 Look in the table of natural sines along the tops of the pages, and find 37 . 
 Glancing down the left-hand column marked ('), until 24 is found, find 
 opposite this 24 in the column marked sine and headed 3/°, the number 
 .60738; then .60738 = sin 37° 24'. Similarly, find m the column marked 
 cosine and headed 37°, the number .79441, which corresponds to cos 3/° 24'. 
 So also, find in the column marked tangent and headed 3/°, and opposite 24 , 
 the number .76456; and in the column marked cotangent and headed 37 , ana 
 opposite 24', the number 1.30795. , , , ... . ... 
 
 In most of the tables published, the angles run only from 0° to 45° at the 
 heads of the columns; to find an angle greater than 45°, look at the bottom of 
 the vaqe and glance upwards , using the extreme right-hand column to find 
 minutes , which begin with 0 at the bottom and run upwards, 1, 2, 3, etc. 
 up to 60. 
 
 Example.— Find the sine of 77° 43'. _ _ . , 
 
 Look along the bottom of the tables until the column marked sine and 
 marked 77° is found. Glancing up the column of minutes on the right until 
 43 ' is found, find opposite 43 ' in the column marked sine at the bottom and 
 marked 77°, the number .97711; this is the sine of //°43 . Similarly, the 
 cosine, tangent, and cotangent may be found. 
 
 To find the sine, cosine, tangent, or cotangent Of an angle whose exact 
 
 value is not given in the table: , . 
 
 r u | e —Find in the table the sine , cosine , tangent , or cotangent corresponding 
 
 to the degrees and minutes of the angle. . 
 
 For the seconds, find the difference of the values of the sine, cosine, tangent, or 
 cotangent taken from the table between which the seconds of the angle fall; multiply 
 this difference by a fraction whose numerator is the number of seconds m the given 
 angle and whose denominator is 60 . . . , ^ v 
 
 If sine or tangent, add this correction to the value first found; if cosine or 
 
 cotangent, subtract the correction. 
 
 Example.— Find the sine, cosine, tangent, and cotangent of 56 43 17 . 
 Sin 56° 43' = .83597. Sin 56° 44' = .83613. Since 56° 43 17 is greater than 
 56° 43 ' and less than 56° 44 ', the value of the sine of the angle lies between 
 
 V? * 2 a ’ QQC 1 Q QQWV7 — 0001 mil 1 tlTll VI T1 & 
 
 til 0.11 OO 11 i Hlv V tllUC LAAvy oaaav/ p « . « i • 
 
 “83597 and .83613; the difference equals .83613 — .83597 = .00016; multiplying 
 this by the fraction M, .00016 X ii = 00005. nearly, which is to be added 
 
 .00005, nearly, which is to be added 
 to .83597, the "value first found', or .83597 + .00005 = .83602. Hence, sm 
 56° 43' 1 7" = 83602 
 
 Cos 56° 43' = 54878; cos 56° 44' = .54854; the difference equals .54878 
 _ 54054 _> 00024 and .00024 Y U = .00007, nearly. Now, since the cosine is 
 deshed! we must’ sSak this correction from cos 56° 43' or .54878; subtract- 
 ing, .54878 - .00007 = .54871. Hence, cos 56° 43' 17" = .54871. . 
 
 Given the sine, cosine, tangent, or cotangent, to find the angle corresponding^ 
 
 Example.— The sine of an angle is .47486; what is the angle? ,, 
 
 Consulting the table of natural sines, glance down the columns marked 
 sine until .47486 is found, opposite 21 ' in the ^ft-hand column and under the 
 column headed 28°. Therefore, the angle whose sine = .47486 is 28° 21 , or 
 sin 28° 21' = .47486. 
 
 To find the angle corresponding to a given sine, cosine, tangent, or 
 cotangent whose exact value is not contained in the table: ... 
 
 R U | %—Find the difference of the two numbers in the table between which the 
 given sine , cosine, tangent, or cotangent falls, and use the number of parts m this 
 difference as the denominator of a fraction. 
 
454 
 
 NATURAL SINES AND COSINES. 
 
 Find the difference between the number belonging to the smaller angle and the 
 given sine , cosine, tangent , or cotangent, and use the number of parts in the dif- 
 ference just found, as the numerator of the fraction mentioned above. Multiply 
 this fraction by 60, and the result will be the number of seconds to be added to the 
 smaller angle. . , 
 
 Example. — Find the angle whose sine equals .5/698. . 
 
 Looking in the table of natural sines, in the column marked sine, it is 
 found between .57691 = sin 35° 14' and .57715 = 35° 15'. The difference 
 between them is .57715 — .57691 = .00024, or 24 parts. The difference between 
 the sine of the smaller angle, or sin 35° 14' = .57691, and the given sine, or 
 .57698, is .57698 — .57691 = .00007, or 7 parts. . „ 
 
 Then, ^ X 60 = 17.5", and the angle = 35° 14' 17.5", or sm 3o° 14 17.(/ 
 
 = .57698. „ „ . . _ 
 
 The cosecant of an angle is equal to the reciprocal of its sine, and the 
 secant is equal to the reciprocal of its cosine. Hence, to multiply a quantity 
 bv the cosecant, divide it by the sine; or, to divide it by the cosecant 
 niultiplv it bv the sine. Similarlv, to multiply a quantity by the secant ol 
 an angle, divide it by the cosine; or, to divide it by the secant, multiply it 
 by the cosine. 
 
NATURAL SINES AND COSINES. 
 
 455 
 
 
 0° 
 
 1° 
 
 2° 
 
 3 ° 
 
 4 ° 
 
 / 
 
 / 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .00000 
 
 .00029 
 
 .00058 
 
 .00087 
 
 .00116 
 
 .00145 
 
 .00175 
 
 .00204 
 
 .00233 
 
 .00262 
 
 .00291 
 
 .00320 
 
 .00349 
 
 .00378 
 
 .00407 
 
 .00436 
 
 .00465 
 
 .00495 
 
 .00524 
 
 .00553 
 
 .00582 
 
 .00611 
 
 .00640 
 
 .00669 
 
 .00698 
 
 .00727 
 
 .00756 
 
 .00785 
 
 .00814 
 
 .00844 
 
 .00873 
 
 .00902 
 
 .00931 
 
 .00960 
 
 .00989 
 
 .01018 
 
 .01047 
 
 .01076 
 
 .01105 
 
 .01134 
 
 .01164 
 
 .01193 
 
 .01222 
 
 .01251 
 
 .01280 
 
 .01309 
 
 .01338 
 
 .01367 
 
 .01396 
 
 .01425 
 
 .01454 
 
 .01483 
 
 .01513 
 
 .01542 
 
 .01571 
 
 .01600 
 
 .01629 
 
 .01658 
 
 .01687 
 
 .01716 
 
 .01745 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 1 . 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99999 
 
 .99998 
 
 .99998 
 
 .99998 
 
 .99998 
 
 .99998 
 
 .99998 
 
 .99997 
 
 .99997 
 
 .99997 
 
 .99997 
 
 .99996 
 
 .99996 
 
 .99996 
 
 .99996 
 
 .99995 
 
 .99995 
 
 .99995 
 
 .99995 
 
 .99994 
 
 .99994 
 
 .99994 
 
 .99993 
 
 .99993 
 
 .99993 
 
 .99992 
 
 .99992 
 
 .99991 
 
 .99991 
 
 .99991 
 
 .99990 
 
 .99990 
 
 .99989 
 
 .99989 
 
 .99989 
 
 .99988 
 
 .99988 
 
 .99987 
 
 .99987 
 
 .99986 
 
 .99986 
 
 .99985 
 
 .99985 
 
 .01745 
 
 .01774 
 
 .01803 
 
 .01832 
 
 .01862 
 
 .01891 
 
 .01920 
 
 .01949 
 
 .01978 
 
 .02007 
 
 .02036 
 
 .02065 
 
 .02094 
 
 .02123 
 
 .02152 
 
 .02181 
 
 .02211 
 
 .02240 
 
 .02269 
 
 .02298 
 
 .02327 
 
 .02356 
 
 .02385 
 
 .02414 
 
 .02443 
 
 .02472 
 
 .02501 
 
 .02530 
 
 .02560 
 
 .02589 
 
 .02618 
 
 .02647 
 
 .02676 
 
 .02705 
 
 .02734 
 
 .02763 
 
 .02792 
 
 .02821 
 
 .02850 
 
 .02879 
 
 .02908 
 
 .02938 
 
 .02967 
 
 .02996 
 
 .03025 
 
 .03054 
 
 .03083 
 
 .03112 
 
 .03141 
 
 .03170 
 
 .03199 
 
 .03228 
 
 .03257 
 
 .03286 
 
 .03316 
 
 .03345 
 
 .03374 
 
 .03403 
 
 .03432 
 
 .03461 
 
 .03490 
 
 .99985 
 
 .99984 
 
 .99984 
 
 .99983 
 
 .99983 
 
 .99982 
 
 .99982 
 
 .99981 
 
 .99980 
 
 .99980 
 
 .99979 
 
 .99979 
 
 .99978 
 
 .99977 
 
 .99977 
 
 .99976 
 
 .99976 
 
 .99975 
 
 .99974 
 
 .99974 
 
 .99973 
 
 .99972 
 
 .99972 
 
 .99971 
 
 .99970 
 
 .99969 
 
 .99969 
 
 .99968 
 
 .99967 
 
 .99966 
 
 .99966 
 
 .99965 
 
 .99964 
 
 .99963 
 
 .99963 
 
 .99962 
 
 .99961 
 
 .99960 
 
 .99959 
 
 .99959 
 
 .99958 
 
 .99957 
 
 .99956 
 
 .99955 
 
 .99954 
 
 .99953 
 
 .99952 
 
 .99952 
 
 .99951 
 
 .99950 
 
 .99949 
 
 .99948 
 
 .99947 
 
 .99946 
 
 .99945 
 
 .99944 
 
 .99943 
 
 .99942 
 
 .99941 
 
 ,99940 
 
 .99939 
 
 .03490 
 
 .03519 
 
 .03548 
 
 .03577 
 
 .03606 
 
 .03635 
 
 .03664 
 
 .03693 
 
 .03723 
 
 .03752 
 
 .03781 
 
 .03810 
 
 .03839 
 
 .03868 
 
 .03897 
 
 .03926 
 
 .03955 
 
 .03984 
 
 .04013 
 
 .04042 
 
 .04071 
 
 .04100 
 
 .04129 
 
 .04159 
 
 .04188 
 
 .04217 
 
 .04246 
 
 .04275 
 
 .04304 
 
 .04333 
 
 .04362 
 
 .04391 
 
 .04420 
 
 .04449 
 
 .04478 
 
 .04507 
 
 .04536 
 
 .04565 
 
 .04594 
 
 .04623 
 
 .04653 
 
 .04682 
 
 .04711 
 
 .04740 
 
 .04769 
 
 .04798 
 
 .04827 
 
 .04856 
 
 .04885 
 
 .04914 
 
 .04943 
 
 .04972 
 
 .05001 
 
 .05030 
 
 .05059 
 
 .05088 
 
 .05117 
 
 .05146 
 
 .05175 
 
 .05205 
 
 .05234 
 
 .99939 
 
 .99938 
 
 .99937 
 
 .99936 
 
 .99935 
 
 .99934 
 
 .99933 
 
 .99932 
 
 .99931 
 
 .99930 
 
 .99929 
 
 .99927 
 
 .99926 
 
 .99925 
 
 .99924 
 
 .99923 
 
 .99922 
 
 .99921 
 
 .99919 
 
 .99918 
 
 .99917 
 
 .99916 
 
 .99915 
 
 .99913 
 
 .99912 
 
 .99911 
 
 .99910 
 
 .99909 
 
 .99907 
 
 .99906 
 
 .99905 
 
 .99904 
 
 .99902 
 
 .99901 
 
 .99900 
 
 .99898 
 
 .99897 
 
 .99896 
 
 .99894 
 
 .99893 
 
 .99892 
 
 .99890 
 
 .99889 
 
 .99888 
 
 .99886 
 
 .99885 
 
 .99883 
 
 .99882 
 
 .99881 
 
 .99879 
 
 .99878 
 
 .99876 
 
 .99875 
 
 .99873 
 
 .99872 
 
 .99870 
 
 .99869 
 
 .99867 
 
 .99866 
 
 .99864 
 
 .99863 
 
 .05234 
 
 .05263 
 
 .05292 
 
 .05321 
 
 .05350 
 
 .05379 
 
 .05408 
 
 .05437 
 
 .05466 
 
 .05495 
 
 .05524 
 
 .05553 
 
 .05582 
 
 .05611 
 
 .05640 
 
 .05669 
 
 .05698 
 
 .05727 
 
 .05756 
 
 .05785 
 
 .05814 
 
 .05844 
 
 .05873 
 
 .05902 
 
 .05931 
 
 .05960 
 
 .05989 
 
 .06018 
 
 .06047 
 
 .06076 
 
 .06105 
 
 .06134 
 
 .06163 
 
 .06192 
 
 .06221 
 
 .06250 
 
 .06279 
 
 .06308 
 
 .06337 
 
 .06366 
 
 .06395 
 
 .06424 
 
 .06453 
 
 .06482 
 
 .06511 
 
 .06540 
 
 .06569 
 
 .06598 
 
 .06627 
 
 .06656 
 
 .06685 
 
 .06714 
 
 .06743 
 
 .06773 
 
 .06802 
 
 .06831 
 
 .06860 
 
 .06889 
 
 .06918 
 
 .06947 
 
 .06976 
 
 .99863 
 
 .99861 
 
 .99860 
 
 .99858 
 
 .99857 
 
 .99855 
 
 .99854 
 
 .99852 
 
 .99851 
 
 .99849 
 
 .99847 
 
 .99846 
 
 .99844 
 
 .99842 
 
 .99841 
 
 .99839 
 
 •99838 
 
 .99836 
 
 .99834 
 
 .99833 
 
 .99831 
 
 .99829 
 
 .99827 
 
 .99826 
 
 .99824 
 
 .99822 
 
 .99821 
 
 .99819 
 
 .99817 
 
 .99815 
 
 .99813 
 
 .99812 
 
 .99810 
 
 .99808 
 
 .99806 
 
 .99804 
 
 .99803 
 
 .99801 
 
 .99799 
 
 .99797 
 
 .99795 
 
 .99793 
 
 .99792 
 
 .99790 
 
 .99788 
 
 .99786 
 
 .99784 
 
 .99782 
 
 .99780 
 
 .99778 
 
 .99776 
 
 .99774 
 
 .99772 
 
 .99770 
 
 .99768 
 
 .99766 
 
 .99764 
 
 .99762 
 
 .99760 
 
 .99758 
 
 .99756 
 
 .06976 
 
 .07005 
 
 .07034 
 
 .07063 
 
 .07092 
 
 .07121 
 
 .07150 
 
 .07179 
 
 .07208 
 
 .07237 
 
 .07266 
 
 .07295 
 
 .07324 
 
 .07353 
 
 .07382 
 
 .07411 
 
 .07440 
 
 .07469 
 
 .07498 
 
 .07527 
 
 .07556 
 
 .07585 
 
 .07614 
 
 .07643 
 
 .07672 
 
 .07701 
 
 .07730 
 
 .07759 
 
 .07788 
 
 .07817 
 
 .07846 
 
 .07875 
 
 .07904 
 
 .07933 
 
 .07962 
 
 .07991 
 
 .08020 
 
 .08049 
 
 .08078 
 
 .08107 
 
 .08136 
 
 .08165 
 
 .08194 
 
 .08223 
 
 .08252 
 
 .08281 
 
 .08310 
 
 .08339 
 
 .08368 
 
 .08397 
 
 .08426 
 
 .08455 
 
 .08484 
 
 .08513 
 
 .08542 
 
 .08571 
 
 .08600 
 
 .08629 
 
 .08658 
 
 .08687 
 
 .08716 
 
 .99756 
 
 .99754 
 
 .99752 
 
 .99750 
 
 .99748 
 
 .99746 
 
 .99744 
 
 .99742 
 
 .99740 
 
 .99738 
 
 .99736 
 
 .99734 
 
 .99731 
 
 .99729 
 
 .99727 
 
 .99725 
 
 .99723 
 
 .99721 
 
 .99719 
 
 .99716 
 
 .99714 
 
 .99712 
 
 .99710 
 
 .99708 
 
 .99705 
 
 .99703 
 
 .99701 
 
 .99699 
 
 .99696 
 
 .99694 
 
 .99692 
 
 .99689 
 
 .99687 
 
 .99685 
 
 .99683 
 
 .99680 
 
 .99678 
 
 .99676 
 
 .99673 
 
 .99671 
 
 .99668 
 
 .99666 
 
 .99664 
 
 .99661 
 
 .99659 
 
 .99657 
 
 .99654 
 
 .99652 
 
 .99649 
 
 .99647 
 
 .99644 
 
 .99642 
 
 .99639 
 
 .99637 
 
 .99635 
 
 .99632 
 
 .99630 
 
 .99627 
 
 .99625 
 
 .99622 
 
 .99619 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 .24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 « 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 t 
 
 / 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 ! Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 89 ° 
 
 88 ° 
 
 87 ° 
 
 86° 
 
 85 ° 
 
 
456 
 
 NA TURAL SINES AND COSINES. 
 
 
 5° 
 
 6° 
 
 7° 
 
 8° 
 
 9° 
 
 t 
 
 / 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 o 
 
 08716 
 
 .99619 
 
 .10453 
 
 .99452 
 
 .12187 
 
 .99255 
 
 .13917 
 
 .99027 
 
 .15643 
 
 .98769 
 
 60 
 
 1 
 
 08745 
 
 .99617 
 
 .10482 
 
 .99449 
 
 .12216 
 
 .99251 
 
 .13946 
 
 .99023 
 
 .15672 
 
 .98764 
 
 59 
 
 
 08774 
 
 .99614 
 
 .10511 
 
 .99446 
 
 .12245 
 
 .99248 
 
 .13975 
 
 .99019 
 
 .15701 
 
 .98760 
 
 58 
 
 3 
 
 08803 
 
 99612 
 
 .10540 
 
 .99443 
 
 .12274 
 
 .99244 
 
 .14004 
 
 .99015 
 
 .15730 
 
 .98755 
 
 57 
 
 
 08831 
 
 99609 
 
 .10569 
 
 .99440 
 
 .12302 
 
 .99240 
 
 .14033 
 
 .99011 
 
 .15758 
 
 .98751 
 
 56 
 
 5 
 
 08800 
 
 99607 
 
 .10597 
 
 .99437 
 
 .12331 
 
 .99237 
 
 .14061 
 
 .99006 
 
 .15787 
 
 .98746 
 
 55 
 
 0 
 
 *08889 
 
 .99604 
 
 .10626 
 
 .99434 
 
 .12360 
 
 .99233 
 
 .14090 
 
 .99002 
 
 .15816 
 
 .98741 
 
 54 
 
 7 
 
 08918 
 
 .99602 
 
 .10655 
 
 .99431 
 
 .12389 
 
 .99230 
 
 .14119 
 
 .98998 
 
 .15845 
 
 .98737 
 
 53 
 
 g 
 
 08947 
 
 .99599 
 
 .10684 
 
 .99428 
 
 .12418 
 
 .99226 
 
 .14148 
 
 .98994 
 
 .15873 
 
 .98732 
 
 52 
 
 9 
 
 08976 
 
 .99596 
 
 .10713 
 
 .99424 
 
 .12447 
 
 .99222 
 
 .14177 
 
 .98990 
 
 .15902 
 
 .98728 
 
 51 
 
 10 
 
 !o9005 
 
 .99594 
 
 .10742 
 
 .99421 
 
 .12476 
 
 .99219 
 
 .14205 
 
 .98986 
 
 .15931 
 
 .98723 
 
 50 
 
 
 09034 
 
 99591 
 
 .10771 
 
 .99418 
 
 .12504 
 
 .99215 
 
 .14234 
 
 .98982 
 
 .15959 
 
 .98718 
 
 49 
 
 
 09063 
 
 99588 
 
 .10800 
 
 .99415 
 
 .12533 
 
 .99211 
 
 .14263 
 
 .98978 
 
 .15988 
 
 .98714 
 
 48 
 
 13 
 
 09092 
 
 99586 
 
 .10829 
 
 .99412 
 
 .12562 
 
 .99208 
 
 .14292 
 
 .98973 
 
 .16017 
 
 .98709 
 
 47 
 
 
 09121 
 
 99583 
 
 .10858 
 
 .99409 
 
 .12591 
 
 .99204 
 
 .14320 
 
 .98969 
 
 .16046 
 
 .98704 
 
 46 
 
 15 
 
 09150 
 
 99580 
 
 .10887 
 
 .99406 
 
 .12620 
 
 .99200 
 
 .14349 
 
 .98965 
 
 .16074 
 
 .98700 
 
 45 
 
 
 09179 
 
 99578 
 
 .10916 
 
 .99402 
 
 .12649 
 
 .99197 
 
 .14378 
 
 .98961 
 
 .16103 
 
 .98695 
 
 44 
 
 
 09908 
 
 99575 
 
 .10945 
 
 .99399 
 
 .12678 
 
 .99193 
 
 .14407 
 
 .98957 
 
 .16132 
 
 .98690 
 
 43 
 
 
 09937 
 
 99572 
 
 .10973 
 
 .99396 
 
 .12706 
 
 .99189 
 
 .14436 
 
 .98953 
 
 .16160 
 
 .98686 
 
 42 
 
 19 
 
 09266 
 
 .99570 
 
 .11002 
 
 .99393 
 
 .12735 
 
 .99186 
 
 .14464 
 
 .98948 
 
 .16189 
 
 .98681 
 
 41 
 
 20 
 
 ^09295 
 
 .99567 
 
 .11031 
 
 .99390 
 
 .12764 
 
 .99182 
 
 .14493 
 
 .98944 
 
 .16218 
 
 .98676 
 
 40 
 
 21 
 
 09324 
 
 .99564 
 
 .11060 
 
 .99386 
 
 .12793 
 
 .99178 
 
 .14522 
 
 .98940 
 
 .16246 
 
 .98671 
 
 39 
 
 22 
 
 09353 
 
 .99562 
 
 .11089 
 
 .99383 
 
 .12822 
 
 .99175 
 
 .14551 
 
 .98936 
 
 .16275 
 
 .98667 
 
 38 
 
 23 
 
 09382 
 
 99559 
 
 .11118 
 
 .99380 
 
 .12851 
 
 .99171 
 
 .14580 
 
 .98931 
 
 .16304 
 
 .98662 
 
 37 
 
 94- 
 
 0941 1 
 
 .99556 
 
 .11147 
 
 .99377 
 
 .12880 
 
 .99167 
 
 .14608 
 
 .98927 
 
 .16333 
 
 .98657 
 
 36 
 
 25 
 
 09440 
 
 .99553 
 
 .11176 
 
 .99374 
 
 .12908 
 
 .99163 
 
 .14637 
 
 .98923 
 
 .16361 
 
 .98652 
 
 35 
 
 26 
 
 09469 
 
 .99551 
 
 .11205 
 
 .99370 
 
 .12937 
 
 .99160 
 
 .14666 
 
 .98919 
 
 .16390 
 
 .98648 
 
 34 
 
 27 
 
 *09498 
 
 .99548 
 
 .11234 
 
 .99367 
 
 .12966 
 
 .99156 
 
 .14695 
 
 .98914 
 
 .16419 
 
 .98643 
 
 33 
 
 28 
 
 ]09527 
 
 .99545 
 
 .11263 
 
 .99364 
 
 .12995 
 
 .99152 
 
 .14723 
 
 .98910 
 
 .16447 
 
 .98638 
 
 32 
 
 29 
 
 !o9556 
 
 .99542 
 
 .11291 
 
 .99360 
 
 .13024 
 
 .99148 
 
 .14752 
 
 .98906 
 
 .16476 
 
 .98633 
 
 31 
 
 30 
 
 ]o9585 
 
 .99540 
 
 .11320 
 
 .99357 
 
 .13053 
 
 .99144 
 
 .14781 
 
 .98902 
 
 .16505 
 
 .98629 
 
 30 
 
 31 
 
 .09614 
 
 .99537 
 
 .11349 
 
 .99354 
 
 .13081 
 
 .99141 
 
 .14810 
 
 .98897 
 
 .16533 
 
 .98624 
 
 29 
 
 32 
 
 .09642 
 
 .99534 
 
 .11378 
 
 .99351 
 
 .13110 
 
 .99137 
 
 .14838 
 
 .98893 
 
 .16562 
 
 .98619 
 
 08 
 
 33 
 
 .09671 
 
 .99531 
 
 .11407 
 
 .99347 
 
 .13139 
 
 .99133 
 
 .14867 
 
 .98889 
 
 .16591 
 
 .98614 
 
 27 
 
 34 
 
 .09700 
 
 ^99528 
 
 .11436 
 
 .99344 
 
 .13168 
 
 .99129 
 
 .14896 
 
 .98884 
 
 .16620 
 
 .98609 
 
 | 26 
 
 35 
 
 .09729 
 
 .99526 
 
 .11465 
 
 .99341 
 
 .13197 
 
 .99125 
 
 .14925 
 
 .98880 
 
 .16648 
 
 .98604 
 
 25 
 
 36 
 
 .09758 
 
 .99523 
 
 .11494 
 
 .99337 
 
 .13226 
 
 .99122 
 
 .14954 
 
 .98876 
 
 .16677 
 
 .98600 
 
 24 
 
 37 
 
 .09787 
 
 .99520 
 
 .11523 
 
 .99334 
 
 .13254 
 
 .99118 
 
 .14982 
 
 .98871 
 
 .16706 
 
 .98595 
 
 23 
 
 38 
 
 .09816 
 
 .99517 
 
 .11552 
 
 .99331 
 
 .13283 
 
 .99114 
 
 .15011 
 
 .98867 
 
 .16734 
 
 .98590 
 
 22 
 
 39 
 
 .09845 
 
 .99514 
 
 .11580 
 
 .99327 
 
 .13312 
 
 .99110 
 
 .15040 
 
 .98863 
 
 .16763 
 
 .98585 
 
 21 
 
 40 
 
 !09874 
 
 .99511 
 
 .11609 
 
 .99324 
 
 .13341 
 
 .99106 
 
 .15069 
 
 .98858 
 
 .16792 
 
 .98580 
 
 20 
 
 41 
 
 .09903 
 
 .99508 
 
 .11638 
 
 .99320 
 
 .13370 
 
 .99102 
 
 .15097 
 
 .98854 
 
 .16820 
 
 .98575 
 
 19 
 
 42 
 
 .09932 
 
 .99506 
 
 .11667 
 
 .99317 
 
 .18399 
 
 .99098 
 
 .15126 
 
 .98849 
 
 .16849 
 
 .98570 
 
 18 
 
 43 
 
 !o9961 
 
 .99503 
 
 .11696 
 
 .99314 
 
 .13427 
 
 .99094 
 
 .15155 
 
 .98845 
 
 .16878 
 
 .98565 
 
 17 
 
 44 
 
 !o9990 
 
 .99500 
 
 .11725 
 
 .99310 
 
 .13456 
 
 .99091 
 
 .15184 
 
 .98841 
 
 .16906 
 
 .98561 
 
 16 
 
 45 
 
 J0019 
 
 .99497 
 
 .11754 
 
 .99307 
 
 .13485 
 
 .99087 
 
 .15212 
 
 .98836 
 
 .16935 
 
 .98556 
 
 15 
 
 46 
 
 J0048 
 
 .99494 
 
 .11783 
 
 .99303 
 
 .13514 
 
 .99083 
 
 .15241 
 
 .98832 
 
 .16964 
 
 .98551 
 
 14 
 
 47 
 
 .10077 
 
 .99491 
 
 .11812 
 
 .99300 
 
 .13543 
 
 .99079 
 
 .15270 
 
 .98827 
 
 .16992 
 
 .98546 
 
 13 
 
 48 
 
 .10106 
 
 .99488 
 
 .11840 
 
 .99297 
 
 .13572 
 
 .99075 
 
 .15299 
 
 .98823 
 
 .17021 
 
 .98541 
 
 12 
 
 49 
 
 !l0135 
 
 .99485 
 
 .11869 
 
 .99293 
 
 .13600 
 
 .99071 
 
 .15327 
 
 .98818 
 
 .17050 
 
 .98536 
 
 11 
 
 50 
 
 '.10164 
 
 .99482 
 
 .11898 
 
 .99290 
 
 .13629 
 
 .99067 
 
 .15356 
 
 .98814 
 
 .17078 
 
 .98531 
 
 10 
 
 51 
 
 .10192 
 
 .99479 
 
 .11927 
 
 .99286 
 
 .13658 
 
 .99063 
 
 .15385 
 
 .98809 
 
 .17107 
 
 .98526 
 
 9 
 
 52 
 
 !l0221 
 
 .99476 
 
 .11956 
 
 .99283 
 
 .13687 
 
 .99059 
 
 .15414 
 
 .98805 
 
 .17136 
 
 .98521 
 
 8 
 
 53 
 
 !l0250 
 
 .99473 
 
 .11985 
 
 .99279 
 
 .13716 
 
 .99055 
 
 .15442 
 
 .98800 
 
 .17164 
 
 .98516 
 
 7 
 
 54 
 
 .10279 
 
 .99470 
 
 .12014 
 
 .99276 
 
 .13744 
 
 .99051 
 
 .15471 
 
 .98796 
 
 .17193 
 
 .98511 
 
 6 
 
 55 
 
 !l03G8 
 
 .99467 
 
 .12043 
 
 .99272 
 
 .13773 
 
 .99047 
 
 .15500 
 
 .98791 
 
 .17222 
 
 .98506 
 
 5 
 
 56 
 
 .10337 
 
 .99464 
 
 .12071 
 
 .99269 
 
 .13802 
 
 .99043 
 
 .15529 
 
 .98787 
 
 .17250 
 
 .98501 
 
 4 
 
 57 
 
 .10366 
 
 .99461 
 
 .12100 
 
 .99265 
 
 .13831 
 
 .99039 
 
 .15557 
 
 .98782 
 
 .17279 
 
 .98496 
 
 3 
 
 58 
 
 .10395 
 
 .99458 
 
 .12129 
 
 .99262 
 
 .13860 
 
 .99035 
 
 .15586 
 
 .98778 
 
 .17308 
 
 .98491 
 
 2 
 
 59 
 
 J0424 
 
 .99455 
 
 .12158 
 
 .99258 
 
 .13889 
 
 .99031 
 
 .15615 
 
 .98773 
 
 .17336 
 
 .98486 
 
 1 
 
 60 
 
 ’.10453 
 
 .99452 
 
 .12187 
 
 .99255 
 
 .13917 
 
 .99027 
 
 .15643 
 
 .98769 
 
 .17365 
 
 .98481 
 
 0 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 f 
 
 84° 
 
 83° 
 
 82° 
 
 81° 
 
 80° 
 
 
NATURAL SINES AND COSINES. 
 
 457 
 
 
 10° 
 
 11° 
 
 12° 
 
 13° 
 
 14 c 
 
 > 
 
 r 
 
 t - 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 69 
 80 
 
 .17365 
 
 .17393 
 
 .17422 
 
 .17451 
 
 .17479 
 
 .17508 
 
 .17537 
 
 .17565 
 
 .17594 
 
 .17623 
 
 .17651 
 
 .17680 
 
 .17708 
 
 .17737 
 
 .17766 
 
 .17794 
 
 .17823 
 
 .17852 
 
 .17880 
 
 .17909 
 
 .17937 
 
 .17966 
 
 .17995 
 
 .18023 
 
 .18052 
 
 .18081 
 
 .18109 
 
 .18138 
 
 .18166 
 
 .18195 
 
 .18224 
 
 .18252 
 
 .18281 
 
 .18309 
 
 .18338 
 
 .18367 
 
 .18395 
 
 .18424 
 
 .18452 
 
 .18481 
 
 .18509 
 
 .18538 
 
 .18567 
 
 .18595 
 
 .18624 
 
 .18652 
 
 .18681 
 
 .18710 
 
 .18738 
 
 .18767 
 
 .18795 
 
 .18824 
 
 .18852 
 
 .18881 
 
 .18910 
 
 .18938 
 
 .18967 
 
 .18995 
 
 .19024 
 
 .19052 
 
 .19081 
 
 .98481 
 
 .98476 
 
 .98471 
 
 .98466 
 
 .98461 
 
 .98455 
 
 .98450 
 
 .98445 
 
 .98440 
 
 .98435 
 
 .98430 
 
 .98425 
 
 .98420 
 
 .98414 
 
 .98409 
 
 .98404 
 
 .98399 
 
 .98394 
 
 .98389 
 
 .98383 
 
 .98378 
 
 .98373 
 
 .98368 
 
 .98362 
 
 .98357 
 
 .98352 
 
 .98347 
 
 .98341 
 
 .98336 
 
 .98331 
 
 .98325 
 
 .98320 
 
 .98315 
 
 .98310 
 
 .98304 
 
 .98299 
 
 .98294 
 
 .98288 
 
 .98283 
 
 .98277 
 
 .98272 
 
 .98267 
 
 .98261 
 
 .98256 
 
 .98250 
 
 .98245 
 
 .98240 
 
 .98234 
 
 .98229 
 
 .98223 
 
 .98218 
 
 .98212 
 
 .98207 
 
 .98201 
 
 .98196 
 
 .98190 
 
 .98185 
 
 .98179 
 
 .98174 
 
 .98168 
 
 .98163 
 
 .19081 
 
 .19109 
 
 .19138 
 
 .19167 
 
 .19195 
 
 .19224 
 
 .19252 
 
 .19281 
 
 .19309 
 
 .19338 
 
 .19366 
 
 .19395 
 
 .19423 
 
 .19452 
 
 .19481 
 
 .19509 
 
 .19538 
 
 .19566 
 
 .19595 
 
 .19623 
 
 .19652 
 
 .19680 
 
 .19709 
 
 .19737 
 
 .19766 
 
 .19794 
 
 .19823 
 
 .19851 
 
 .19880 
 
 .19908 
 
 .19937 
 
 .19965 
 
 .19994 
 
 .20022 
 
 .20051 
 
 .20079 
 
 .20108 
 
 .20136 
 
 .20165 
 
 .20193 
 
 .20222 
 
 .20250 
 
 .20279 
 
 .20307 
 
 .20336 
 
 .20364 
 
 .20393 
 
 .20421 
 
 .20450 
 
 .20478 
 
 .20507 
 
 .20535 
 
 .20563 
 
 .20592 
 
 .20620 
 
 .20649 
 
 .20677 
 
 .20706 
 
 .20734 
 
 .20763 
 
 .20791 
 
 .98163 
 
 .98157 
 
 .98152 
 
 .98146 
 
 .98140 
 
 .98135 
 
 .98129 
 
 .98124 
 
 .98118 
 
 .98112 
 
 .98107 
 
 .98101 
 
 .98096 
 
 .98090 
 
 .98084 
 
 .98079 
 
 .98073 
 
 .98067 
 
 .98061 
 
 .98056 
 
 .98050 
 
 .98044 
 
 .98039 
 
 .98033 
 
 .98027 
 
 .98021 
 
 .98016 
 
 .98010 
 
 .98004 
 
 .97998 
 
 .97992 
 
 .97987 
 
 .97981 
 
 .97975 
 
 .97969 
 
 .97963 
 
 .97958 
 
 .97952 
 
 .97946 
 
 .97940 
 
 .97934 
 
 .97928 
 
 .97922 
 
 .97916 
 
 .97910 
 
 .97905 
 
 .97899 
 
 .97893 
 
 .97887 
 
 .97881 
 
 .97875 
 
 .97869 
 
 .97863 
 
 .97857 
 
 .97851 
 
 .97845 
 
 .97839 
 
 .97833 
 
 .97827 
 
 .97821 
 
 .97815 
 
 .20791 
 
 .20820 
 
 .20848 
 
 .20877 
 
 .20905 
 
 .20933 
 
 .20962 
 
 .20990 
 
 .21019 
 
 .21047 
 
 .21076 
 
 .21104 
 
 .21132 
 
 .21161 
 
 .21189 
 
 .21218 
 
 .21246 
 
 .21275 
 
 .21303 
 
 .21331 
 
 .21360 
 
 .21388 
 
 .21417 
 
 .21445 
 
 .21474 
 
 .21502 
 
 .21530 
 
 .21559 
 
 .21587 
 
 .21616 
 
 .21644 
 
 .21672 
 
 .21701 
 
 .21729 
 
 .21758 
 
 .21786 
 
 .21814 
 
 .21843 
 
 .21871 
 
 .21899 
 
 .21928 
 
 .21956 
 
 .21985 
 
 .22013 
 
 .22041 
 
 .22070 
 
 .22098 
 
 .22126 
 
 .22155 
 
 .22183 
 
 .22212 
 
 .22240 
 
 .22268 
 
 .22297 
 
 .22325 
 
 .22353 
 
 .22382 
 
 .22410 
 
 .22438 
 
 .22467 
 
 .22495 
 
 .97815 
 
 .97809 
 
 .97803 
 
 .97797 
 
 .97791 
 
 .97784 
 
 .97778 
 
 .97772 
 
 .97766 
 
 .97760 
 
 .97754 
 
 .97748 
 
 .97742 
 
 .97735 
 
 .97729 
 
 .97723 
 
 .97717 
 
 .97711 
 
 .97705 
 
 .97698 
 
 .97692 
 
 .97686 
 
 .97680 
 
 .97673 
 
 .97667 
 
 .97661 
 
 .97655 
 
 .97648 
 
 .97642 
 
 .97636 
 
 .97630 
 
 .97623 
 
 .97617 
 
 .97611 
 
 .97604 
 
 .97598 
 
 .97592 
 
 .97585 
 
 .97579 
 
 .97573 
 
 .97566 
 
 .97560 
 
 .97553 
 
 .97547 
 
 .97541 
 
 .97534 
 
 .97528 
 
 .97521 
 
 .97515 
 
 .97508 
 
 .97502 
 
 .97496 
 
 .97489 
 
 .97483 
 
 .97476 
 
 .97470 
 
 .97463 
 
 .97457 
 
 .97450 
 
 .97444 
 
 .97437 
 
 .22495 
 
 .22523 
 
 .22552 
 
 .22580 
 
 .22608 
 
 .22637 
 
 .22665 
 
 .22693 
 
 .22722 
 
 .22750 
 
 .22778 
 
 .22807 
 
 .22835 
 
 .22863 
 
 .22892 
 
 .22920 
 
 .22948 
 
 .22977 
 
 .23005 
 
 .23033 
 
 .23062 
 
 .23090 
 
 .23118 
 
 .23146 
 
 .23175 
 
 .23203 
 
 .23231 
 
 .23260 
 
 .23288 
 
 .23316 
 
 .23345 
 
 .23373 
 
 .23401 
 
 .23429 
 
 .23458 
 
 .23486 
 
 .23514 
 
 .23542 
 
 .23571 
 
 .23599 
 
 .23627 
 
 .23656 
 
 .23684 
 
 .23712 
 
 .23740 
 
 .23769 
 
 .23797 
 
 .23825 
 
 .23853 
 
 .23882 
 
 .23910 
 
 .23938 
 
 .23966 
 
 .23995 
 
 .24023 
 
 .24051 
 
 .24079 
 
 .24108 
 
 .24136 
 
 .24164 
 
 .24192 
 
 .97437 
 
 .97430 
 
 .97424 
 
 .97417 
 
 .97411 
 
 .97404 
 
 .97398 
 
 .97391 
 
 .97384 
 
 .97378 
 
 .97371 
 
 .97365 
 
 .97358 
 
 .97351 
 
 .97345 
 
 .97338 
 
 .97331 
 
 .97325 
 
 .97318 
 
 .97311 
 
 .97304 
 
 .97298 
 
 .97291 
 
 .97284 
 
 .97278 
 
 .97271 
 
 .97264 
 
 .97257 
 
 .97251 
 
 .97244 
 
 .97237 
 
 .97230 
 
 .97223 
 
 .97217 
 
 .97210 
 
 .97203 
 
 .97196 
 
 .97189 
 
 .97182 
 
 .97176 
 
 .97169 
 
 .97162 
 
 .97155 
 
 .97148 
 
 .97141 
 
 .97134 
 
 .97127 
 
 .97120 
 
 .97113 
 
 .97106 
 
 .97100 
 
 .97093 
 
 .97086 
 
 .97079 
 
 .97072 
 
 .97065 
 
 .97058 
 
 .97051 
 
 .97044 
 
 .97037 
 
 .97030 
 
 .24192 
 
 .24220 
 
 .24249 
 
 .24277 
 
 .24305 
 
 .24333 
 
 .24362 
 
 .24390 
 
 .24418 
 
 .24446 
 
 .24474 
 
 .24503 
 
 .24531 
 
 .24559 
 
 .24587 
 
 .24615 
 
 .24644 
 
 .24672 
 
 .24700 
 
 .24728 
 
 .24756 
 
 .24784 
 
 .24813 
 
 .24841 
 
 .24869 
 
 .24897 
 
 .24925 
 
 .24954 
 
 .24982 
 
 .25010 
 
 .25038 
 
 .25066 
 
 .25094 
 
 .25122 
 
 .25151 
 
 .25179 
 
 .25207 
 
 .25235 
 
 .25263 
 
 .25291 
 
 .25320 
 
 .25348 
 
 .25376 
 
 .25404 
 
 .25432 
 
 .25460 
 
 .25488 
 
 .25516 
 
 .25545 
 
 .25573 
 
 .25601 
 
 .25629 
 
 .25657 
 
 .25685 
 
 .25713 
 
 .25741 
 
 .25769 
 
 .25798 
 
 .25826 
 
 .25854 
 
 .25882 
 
 .97030 
 
 .97023 
 
 .97015 
 
 .97008 
 
 .97001 
 
 .96994 
 
 .96987 
 
 .96980 
 
 .96973 
 
 .96966 
 
 .96959 
 
 .96952 
 
 .96945 
 
 .96937 
 
 .96930 
 
 .96923 
 
 .96916 
 
 .96909 
 
 .96902 
 
 .96894 
 
 .96887 
 
 .96880 
 
 .96873 
 
 .96866 
 
 .96858 
 
 .96851 
 
 .96844 
 
 .96837 
 
 .96829 
 
 .96822 
 
 .96815 
 
 .96807 
 
 .96800 
 
 .96793 
 
 .96786 
 
 .96778 
 
 .96771 
 
 .96764 
 
 .96756 
 
 .96749 
 
 .96742 
 
 .96734 
 
 .96727 
 
 .96719 
 
 .96712 
 
 .96705 
 
 .96697 
 
 .96690 
 
 .96682 
 
 .96675 
 
 .96667 
 
 .96660 
 
 .96653 
 
 .96645 
 
 .96638 
 
 .96630 
 
 .96623 
 
 .96615 
 
 .96608 
 
 .96600 
 
 .96593 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 t 
 
 Cosine 
 
 > Sine 
 
 Cosine 
 
 ; Sine 
 
 Cosine 
 
 : Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 i Sine 
 
 - t 
 
 1 
 
 79° 
 
 78° 
 
 77° 
 
 76° 
 
 75° 
 
 ... 
 
 
458 
 
 NATURAL SINES AND COSINES , 
 
 
 15 ° 
 
 16 ° 
 
 17 ° 
 
 18 ° | 
 
 19 ° 
 
 / 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 0 
 
 .25882 
 
 .96593 
 
 .27564 
 
 .96126 
 
 .29237 
 
 .95630 
 
 .30902 
 
 .95106 
 
 .32557 
 
 .94552 
 
 60 
 
 1 
 
 .25910 
 
 .96585 
 
 .27592 
 
 .96118 
 
 .29265 
 
 .95622 
 
 .30929 
 
 .95097 
 
 .32584 
 
 .94542 
 
 59 
 
 2 
 
 .25938 
 
 .96578 
 
 .27620 
 
 .96110 
 
 .29293 
 
 .95613 
 
 .30957 
 
 .95088 
 
 .32612 
 
 .94533 
 
 58 
 
 3 
 
 .25966 
 
 .96570 
 
 .27648 
 
 .96102 
 
 .29321 
 
 .95605 
 
 .30985 
 
 .95079 
 
 .32639 
 
 .94523 
 
 57 
 
 4 
 
 .25994 
 
 .96562 
 
 .27676 
 
 .96094 
 
 .29348 
 
 .95596 
 
 .31012 
 
 .95070 
 
 .32667 
 
 .94514 
 
 56 
 
 5 
 
 .26022 
 
 .96555 
 
 .27704 
 
 .96086 
 
 .29376 
 
 .95588 
 
 .31040 
 
 .95061 
 
 .32694 
 
 .94504 
 
 55 
 
 6 
 
 .26050 
 
 .96547 
 
 .27731 
 
 .96078 
 
 .29404 
 
 .95579 
 
 .31068 
 
 .95052 
 
 .32722 
 
 .94495 
 
 54 
 
 7 
 
 .26079 
 
 .96540 
 
 .27759 
 
 .96070 
 
 .29432 
 
 .95571 
 
 .31095 
 
 .95043 
 
 .32749 
 
 .94485 
 
 53 
 
 8 
 
 .26107 
 
 .96532 
 
 .27787 
 
 .96062 
 
 .29460 
 
 .95562 
 
 .31123 
 
 .95033 
 
 .32777 
 
 .94476 
 
 52 
 
 9 
 
 .26135 
 
 .96524 
 
 .27815 
 
 .96054 
 
 .29487 
 
 .95554 
 
 .31151 
 
 .95024 
 
 .32804 
 
 .94466 
 
 51 
 
 10 
 
 .26163 
 
 .96517 
 
 .27843 
 
 .96046 
 
 .29515 
 
 .95545 
 
 .31178 
 
 .95015 
 
 .32832 
 
 .94457 
 
 50 
 
 11 
 
 .26191 
 
 .96509 
 
 .27871 
 
 .96037 
 
 .29543 
 
 .95536 
 
 .31206 
 
 .95006 
 
 .32859 
 
 .94447 
 
 49 
 
 12 
 
 .26219 
 
 .96502 
 
 .27899 
 
 .96029 
 
 .29571 
 
 .95528 
 
 .31233 
 
 .94997 
 
 .32887 
 
 .94438 
 
 48 
 
 13 
 
 .26247 
 
 .96494 
 
 .27927 
 
 .96021 
 
 .29599 
 
 .95519 
 
 .31261 
 
 .94988 
 
 .32914 
 
 .94428 
 
 47 
 
 14 
 
 .26275 
 
 .96486 
 
 .27955 
 
 .96013 
 
 .29626 
 
 .95511 
 
 .31289 
 
 .94979 
 
 .32942 
 
 .94418 
 
 46 
 
 15 
 
 .26303 
 
 .96479 
 
 .27983 
 
 .96005 
 
 .29654 
 
 .95502 
 
 .31316 
 
 .94970 
 
 .32969 
 
 .94409 
 
 45 
 
 16 
 
 .26331 
 
 .96471 
 
 .28011 
 
 .95997 
 
 .29682 
 
 .95493 
 
 .31344 
 
 .94961 
 
 .32997 
 
 .94399 
 
 44 
 
 17 
 
 .26359 
 
 .96463 
 
 .28039 
 
 .95989 
 
 .29710 
 
 .95485 
 
 .31372 
 
 .94952 
 
 .33024 
 
 .94390 
 
 43 
 
 18 
 
 .26387 
 
 .96456 
 
 .28067 
 
 .95981 
 
 .29737 
 
 .95476 
 
 .31399 
 
 .94943 
 
 .33051 
 
 .94380 
 
 42 
 
 19 
 
 .26415 
 
 .96448 
 
 .28095 
 
 .95972 
 
 .29765 
 
 .95467 
 
 .31427 
 
 .94933 
 
 .33079 
 
 .94370 
 
 41 
 
 20 
 
 .26443 
 
 .96440 
 
 .28123 
 
 .95964 
 
 .29793 
 
 .95459 
 
 .31454 
 
 .94924 
 
 .33106 
 
 .94361 
 
 40 
 
 21 
 
 .26471 
 
 .96433 
 
 .28150 
 
 .95956 
 
 .29821 
 
 .95450 
 
 .31482 
 
 .94915 
 
 .33134 
 
 .94351 
 
 39 
 
 22 
 
 .26500 
 
 .96425 
 
 .28178 
 
 .95948 
 
 .29849 
 
 .95441 
 
 .31510 
 
 .94906 
 
 .33161 
 
 .94342 
 
 38 
 
 23 
 
 .26528 
 
 .96417 
 
 .28206 
 
 .95940 
 
 .29876 
 
 .95433 
 
 .31537 
 
 .94897 
 
 .33189 
 
 .94332 
 
 37 
 
 24 
 
 .26556 
 
 .96410 
 
 .28234 
 
 .95931 
 
 .29904 
 
 .95424 
 
 .31565 
 
 .94888 
 
 .33216 
 
 .94322 
 
 36 
 
 25 
 
 .26584 
 
 .96402 
 
 .28262 
 
 .95923 
 
 .29932 
 
 .95415 
 
 .31593 
 
 .94878 
 
 .33244 
 
 .94313 
 
 35 
 
 26 
 
 .26612 
 
 .96394 
 
 .28290 
 
 .95915 
 
 .29960 
 
 .95407 
 
 .31620 
 
 .94869 
 
 .33271 
 
 .94303 
 
 34 
 
 27 
 
 .26640 
 
 .96386 
 
 .28318 
 
 .95907 
 
 .29987 
 
 .95398 
 
 .31648 
 
 .94860 
 
 .33298 
 
 .94293 
 
 33 
 
 28 
 
 .26668 
 
 .96379 
 
 .28346 
 
 .95898 
 
 .30015 
 
 .95389 
 
 .31675 
 
 .94851 
 
 .33326 
 
 .94284 
 
 32 
 
 29 
 
 .26696 
 
 .96371 
 
 .28374 
 
 .95890 
 
 .30043 
 
 .95380 
 
 .31703 
 
 .94842 
 
 .33353 
 
 .94274 
 
 31 
 
 30 
 
 .26724 
 
 .96363 
 
 28402 
 
 .95882 
 
 .30071 
 
 .95372 
 
 .31730 
 
 .94832 
 
 .33381 
 
 .94264 
 
 30 
 
 31 
 
 .26752 
 
 .96355 
 
 .28429 
 
 .95874 
 
 .30098 
 
 .95363 
 
 .31758 
 
 .94823 
 
 .33408 
 
 .94254 
 
 29 
 
 32 
 
 .26780 
 
 .96347 
 
 .28457 
 
 .95865 
 
 .30126 
 
 .95354 
 
 .31786 
 
 .94814 
 
 .33436 
 
 .94245 
 
 28 
 
 33 
 
 .26808 
 
 .96340 
 
 .28485 
 
 .95857 
 
 .30154 
 
 .95345 
 
 .31813 
 
 .94805 
 
 .33463 
 
 .94235 
 
 27 
 
 34 
 
 .26836 
 
 .96332 
 
 .28513 
 
 .95849 
 
 .30182 
 
 .95337 
 
 .31841 
 
 .94795 
 
 .33490 
 
 .94225 
 
 26 
 
 35 
 
 .26864 
 
 .96324 
 
 .28541 
 
 .95841 
 
 .30209 
 
 .95328 
 
 .31868 
 
 .94786 
 
 .33518 
 
 .94215 
 
 25 
 
 36 
 
 .26892 
 
 .96316 
 
 .28569 
 
 .95832 
 
 .30237 
 
 .95319 
 
 .31896 
 
 .94777 
 
 .33545 
 
 .94206 
 
 24 
 
 37 
 
 .26920 
 
 .96308 
 
 .28597 
 
 .95824 
 
 .30265 
 
 .95310 
 
 .31923 
 
 .94768 
 
 .33573 
 
 .94196 
 
 23 
 
 38 
 
 .26948 
 
 .96301 
 
 .28625 
 
 .95816 
 
 .30292 
 
 .95301 
 
 .31951 
 
 .94758 
 
 .33600 
 
 .94186 
 
 22 
 
 39 
 
 .26976 
 
 .96293 
 
 .28652 
 
 .95807 
 
 .30320 
 
 .95293 
 
 .31979 
 
 .94749 
 
 .33627 
 
 .94176 
 
 21 
 
 40 
 
 .27004 
 
 .96285 
 
 .28680 
 
 .95799 
 
 .30348 
 
 .95284 
 
 .32006 
 
 .94740 
 
 .33655 
 
 .94167 
 
 20 
 
 41 
 
 .27032 
 
 .96277 
 
 .28708 
 
 .95791 
 
 .30376 
 
 .95275 
 
 .32034 
 
 .94730 
 
 .33682 
 
 .94157 
 
 19 
 
 42 
 
 .27060 
 
 .96269 
 
 .28736 
 
 .95782 
 
 .30403 
 
 .95266 
 
 .32061 
 
 .94721 
 
 .33710 
 
 .94147 
 
 18 
 
 43 
 
 .27088 
 
 .96261 
 
 .28764 
 
 .95774 
 
 .30431 
 
 .95257 
 
 ' .32089 
 
 .94712 
 
 .33737 
 
 .94137 
 
 17 
 
 44 
 
 .27116 
 
 .96253 
 
 .28792 
 
 .95766 
 
 .30459 
 
 .95248 
 
 .32116 
 
 .94702 
 
 .33764 
 
 .94127 
 
 16 
 
 45 
 
 .27144 
 
 .96246 
 
 .28820 
 
 .95757 
 
 .30486 
 
 .95240 
 
 .32144 
 
 .94693 
 
 .33792 
 
 .94118 
 
 15 
 
 46 
 
 .27172 
 
 .96238 
 
 .28847 
 
 .95749 
 
 .30514 
 
 .95231 
 
 .32171 
 
 .94684 
 
 .33819 
 
 .94108 
 
 14 
 
 47 
 
 .27200 
 
 .96230 
 
 .28875 
 
 .95740 
 
 .30542 
 
 .95222 
 
 .32199 
 
 .94674 
 
 .33846 
 
 .94098 
 
 13 
 
 48 
 
 .27228 
 
 .96222 
 
 .28903 
 
 .95732 
 
 .30570 
 
 .95213 
 
 .32227 
 
 .94665 
 
 .33874 
 
 .94088 
 
 12 
 
 49 
 
 .27256 
 
 .96214 
 
 .28931 
 
 .95724 
 
 .30597 
 
 .95204 
 
 .32254 
 
 .94656 
 
 .33901 
 
 .94078 
 
 11 
 
 50 
 
 .27284 
 
 .96206 
 
 .28959 
 
 .95715 
 
 .30625 
 
 .95195 
 
 .32282 
 
 .94646 
 
 .33929 
 
 .94068 
 
 10 
 
 51 
 
 .27312 
 
 .96198 
 
 .28987 
 
 .95707 
 
 .30653 
 
 .95186 
 
 .32309 
 
 .94637 
 
 .33956 
 
 .94058 
 
 9 
 
 52 
 
 .27340 
 
 .96190 
 
 .29015 
 
 .95698 
 
 .30680 
 
 .95177 
 
 .32337 
 
 .94627 
 
 .33983 
 
 .94049 
 
 8 
 
 53 
 
 .27368 
 
 .96182 
 
 .29042 
 
 .95690 
 
 .30708 
 
 .95168 
 
 .32364 
 
 .94618 
 
 .34011 
 
 .94039 
 
 7 
 
 54 
 
 .27396 
 
 .96174 
 
 .29070 
 
 .95681 
 
 .30736 
 
 .95159 
 
 .32392 
 
 .94609 
 
 .34038 
 
 .94029 
 
 6 
 
 55 
 
 .27424 
 
 .96166 
 
 .29098 
 
 .95673 
 
 .30763 
 
 .95160 
 
 .32419 
 
 .94599 
 
 .34065 
 
 .94019 
 
 5 
 
 56 
 
 .27452 
 
 .96158 
 
 .29126 
 
 .95664 
 
 .30791 
 
 .95142 
 
 .32447 
 
 .94590 
 
 .34093 
 
 .94009 
 
 4 
 
 57 
 
 .27480 
 
 .96150 
 
 .29154 
 
 .95656 
 
 .30819 
 
 .95133 
 
 .32474 
 
 .94580 
 
 .34120 
 
 .93999 
 
 3 
 
 58 
 
 .27508 
 
 .96142 
 
 .29182 
 
 .95647 
 
 .30846 
 
 .95124 
 
 .32502 
 
 .94571 
 
 .34147 
 
 .93989 
 
 2 
 
 59 
 
 .27536 
 
 .96134 
 
 .29209 
 
 .95639 
 
 .30874 
 
 .95115 
 
 .32529 
 
 .94561 
 
 .34175 
 
 .93979 
 
 1 
 
 60 
 
 .27564 
 
 .96126 
 
 .29237 
 
 .95630 
 
 .30902 
 
 .95106 
 
 .32557 
 
 .94552 
 
 .34202 
 
 .93969 
 
 0 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 
 / 
 
 7 
 
 4° 
 
 78 ° 
 
 72 ° 
 
 71 ° 
 
 70 ° 
 
 \ 
 
 / 
 
NATURAL SINES AND COSINES. 
 
 459 
 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 / 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 o 
 
 .34202 
 
 .93969 
 
 .35837 
 
 .93358 
 
 .37461 
 
 .92718 
 
 .39073 
 
 .92050 
 
 .40674 
 
 .91355 
 
 60 
 
 1 
 
 .34229 
 
 .93959 
 
 .35864 
 
 .93348 
 
 .37488 
 
 .92707 
 
 .39100 
 
 .92039 
 
 .40700 
 
 .91343 
 
 59 
 
 2 
 
 .34257 
 
 .93949 
 
 .35891 
 
 .93337 
 
 .37515 
 
 .92697 
 
 .39127 
 
 .92028 
 
 .40727 
 
 .91331 
 
 58 
 
 3 
 
 .34284 
 
 .93939 
 
 .35918 
 
 .93327 
 
 .37542 
 
 .92686 
 
 .39153 
 
 .92016 
 
 .40753 
 
 .91319 
 
 57 
 
 4 
 
 .34311 
 
 .93929 
 
 .35945 
 
 .93316 
 
 .37569 
 
 .92675 
 
 .39180 
 
 .92005 
 
 .40780 
 
 .91307 
 
 56 
 
 5 
 
 .34339 
 
 .93919 
 
 .35973 
 
 .93306 
 
 .37595 
 
 .92664 
 
 .39207 
 
 .91994 
 
 .40806 
 
 .91295 
 
 55 
 
 6 
 
 .34366 
 
 .93909 
 
 .36000 
 
 .93295 
 
 .37622 
 
 .92653 
 
 .39234 
 
 .91982 
 
 .40833 
 
 .91283 
 
 54 
 
 7 
 
 .34393 
 
 .93899 
 
 .36027 
 
 .93285 
 
 .37649 
 
 .92642 
 
 .39260 
 
 .91971 
 
 .40860 
 
 .91272 
 
 53 
 
 g 
 
 .34421 
 
 .93889 
 
 .36054 
 
 .93274 
 
 .37676 
 
 .92631 
 
 .39287 
 
 .91959 
 
 .40886 
 
 .91260 
 
 52 
 
 9 
 
 .34448 
 
 .93879 
 
 .36081 
 
 .93264 
 
 .37703 
 
 .92620 
 
 .39314 
 
 .91948 
 
 .40913 
 
 .91248 
 
 51 
 
 10 
 
 .34475 
 
 .93869 
 
 .36108 
 
 .93253 
 
 .37730 
 
 .92609 
 
 .39341 
 
 .91936 
 
 .40939 
 
 .91236 
 
 50 
 
 11 
 
 .34503 
 
 .93859 
 
 .36135 
 
 .93243 
 
 .37757 
 
 .92598 
 
 .39367 
 
 .91925 
 
 .40966 
 
 .91224 
 
 49 
 
 12 
 
 .34530 
 
 .93849 
 
 .36162 
 
 .93232 
 
 .37784 
 
 .92587 
 
 .39394 
 
 .91914 
 
 .40992 
 
 .91212 
 
 48 
 
 13 
 
 .34557 
 
 .93839 
 
 .36190 
 
 .93222 
 
 .37811 
 
 .92576 
 
 .39421 
 
 .91902 
 
 .41019 
 
 .91200 
 
 47 
 
 14 
 
 .34584 
 
 .93829 
 
 .36217 
 
 .93211 
 
 .37838 
 
 .92565 
 
 .39448 
 
 .91891 
 
 .41045 
 
 .91188 
 
 46 
 
 15 
 
 .34612 
 
 .93819 
 
 .36244 
 
 .93201 
 
 .37865 
 
 .92554 
 
 .39474 
 
 .91879 
 
 .41072 
 
 .91176 
 
 45 
 
 16 
 
 .34639 
 
 .93809 
 
 .36271 
 
 .93190 
 
 .37892 
 
 .92543 
 
 .39501 
 
 .91868 
 
 .41098 
 
 .91164 
 
 44 
 
 17 
 
 .34666 
 
 .93799 
 
 .36298 
 
 .93180 
 
 .37919 
 
 .92532 
 
 .39528 
 
 .91856 
 
 .41125 
 
 .91152 
 
 43 
 
 18 
 
 .34694 
 
 .93789 
 
 .36325 
 
 .93169 
 
 .37946 
 
 .92521 
 
 .39555 
 
 .91845 
 
 .41151 
 
 .91140 
 
 42 
 
 19 
 
 .34721 
 
 .93779 
 
 .36352 
 
 .93159 
 
 .37973 
 
 .92510 
 
 .39581 
 
 .91833 
 
 .41178 
 
 .91128 
 
 41 
 
 20 
 
 .$4748 
 
 .93769 
 
 .36379 
 
 .93148 
 
 .37999 
 
 .92499 
 
 .39608 
 
 .91822 
 
 .41204 
 
 .91116 
 
 40 
 
 21 
 
 .34775 
 
 .93759 
 
 .36406 
 
 .93137 
 
 .38026 
 
 .92488 
 
 .39635 
 
 .91810 
 
 .41231 
 
 .91104 
 
 39 
 
 22 
 
 .34803 
 
 .93748 
 
 .36434 
 
 .93127 
 
 .38053 
 
 .92477 
 
 .39661 
 
 .91799 
 
 .41257 
 
 .91092 
 
 38 
 
 23 
 
 *.34830 
 
 .93738 
 
 .36461 
 
 .93116 
 
 .38080 
 
 .92466 
 
 .39688 
 
 .91787 
 
 .41284 
 
 .91080 
 
 37 
 
 24 
 
 .34857 
 
 .93728 
 
 .36488 
 
 .93106 
 
 .38107 
 
 .92455 
 
 .39715 
 
 .91775 
 
 .41310 
 
 .91068 
 
 36 
 
 25 
 
 .34884 
 
 .93718 
 
 .36515 
 
 .93095 
 
 .38134 
 
 .92444 
 
 .39741 
 
 .91764 
 
 .41337 
 
 .91056 
 
 35 
 
 26 
 
 .34912 
 
 .93708 
 
 .36542 
 
 .93084 
 
 .38161 
 
 .92432 
 
 .39768 
 
 .91752 
 
 .41363 
 
 .91044 
 
 34 
 
 27 
 
 .34939 
 
 .93698 
 
 .36569 
 
 .93074 
 
 .38188 
 
 .92421 
 
 .39795 
 
 .91741 
 
 .41390 
 
 .91032 
 
 33 
 
 28 
 
 .34966 
 
 .93688 
 
 .36596 
 
 .93063 
 
 .38215 
 
 .92410 
 
 .39822 
 
 .91729 
 
 .41416 
 
 .91020 
 
 32 
 
 29 
 
 .34993 
 
 .93677 
 
 .36623 
 
 .93052 
 
 .38241 
 
 .92399 
 
 .39848 
 
 .91718 
 
 .41443 
 
 .91008 
 
 31 
 
 30 
 
 .35021 
 
 .93667 
 
 .36650 
 
 .93042 
 
 .38268 
 
 .92388 
 
 .39875 
 
 .91706 
 
 .41469 
 
 .90996 
 
 30 
 
 31 ' 
 
 .35048 
 
 .93657 
 
 .36677 
 
 .93031 
 
 .38295 
 
 .92377 
 
 .39902 
 
 .91694 
 
 .41496 
 
 .90984 
 
 29 
 
 32 
 
 .35075 
 
 .93647 
 
 .36704 
 
 .93020 
 
 .38322 
 
 .92366 
 
 .39928 
 
 .91683 
 
 .41522 
 
 .90972 
 
 28 
 
 33 
 
 .35102 
 
 .93637 
 
 .36731 
 
 .93010 
 
 .38349 
 
 .92355 
 
 .39955 
 
 .91671 
 
 .41549 
 
 .90960 
 
 27 
 
 34 
 
 .35130 
 
 .93626 
 
 .36758 
 
 .92999 
 
 .38376 
 
 .92343 
 
 .39982 
 
 .91660 
 
 .41575 
 
 .90948 
 
 26 
 
 35 
 
 .35157 
 
 .93616 
 
 .36785 
 
 .92988 
 
 .38403 
 
 .92332 
 
 .40008 
 
 .91648 
 
 .41602 
 
 .90936 
 
 25 
 
 36 
 
 .35184 
 
 .93606 
 
 .36812 
 
 .92978 
 
 .38430 
 
 .92321 
 
 .40035 
 
 .91636 
 
 .41628 
 
 .90924 
 
 24 
 
 37 
 
 .35211 
 
 .93596 
 
 .36839 
 
 .92967 
 
 .38456 
 
 .92310 
 
 .40062 
 
 .91625 
 
 .41655 
 
 .90911 
 
 23 
 
 38 
 
 .35239 
 
 .93585 
 
 .36867 
 
 *92956 
 
 .38483 
 
 .92299 
 
 .40088 
 
 .91613 
 
 .41681 
 
 .90899 
 
 22 
 
 39 
 
 .35266 
 
 .93575 
 
 .36894 
 
 .92945 
 
 .38510 
 
 .92287 
 
 .40115 
 
 .91601 
 
 .41707 
 
 .90887 
 
 21 
 
 40 
 
 .35293 
 
 .93565 
 
 .36921 
 
 .92935 
 
 .38537 
 
 .92276 
 
 .40141 
 
 .91590 
 
 .41734 
 
 .90875 
 
 20 
 
 41 
 
 .35320 
 
 .93555 
 
 .36948 
 
 .92924 
 
 .38564 
 
 .92265 
 
 .40168 
 
 .91578 
 
 .41760 
 
 .90863 
 
 19 
 
 42 
 
 .35347 
 
 .93544 
 
 .36975 
 
 .92913 
 
 .38591 
 
 .92254 
 
 .40195 
 
 .91566 
 
 .41787 
 
 .90851 
 
 18 
 
 43 
 
 .35375 
 
 .93534 
 
 .37002 
 
 .92902 
 
 .38617 
 
 .92243 
 
 .40221 
 
 .91555 
 
 .41813 
 
 .90839 
 
 17 
 
 44 
 
 .35402 
 
 .93524 
 
 .37029 
 
 .92892 
 
 .38644 
 
 .92231 
 
 .40248 
 
 .91543 
 
 .41840 
 
 .90826 
 
 16 
 
 45 
 
 .35429 
 
 .93514 
 
 .37056 
 
 .92881 
 
 .38671 
 
 .92220 
 
 .40275 
 
 .91531 
 
 .41866 
 
 .90814 
 
 15 
 
 46 
 
 .35456 
 
 .93502 
 
 .37083 
 
 .92870 
 
 .38698 
 
 .92209 
 
 .40301 
 
 .91519 
 
 .41892 
 
 .90802 
 
 14 
 
 47 
 
 .35484 
 
 .93493 
 
 .37110 
 
 .92859 
 
 .38725 
 
 .92198 
 
 .40328 
 
 .91508 
 
 .41919 
 
 .90790 
 
 13 
 
 48 
 
 .35511 
 
 .93483 
 
 .37137 
 
 .92849 
 
 .38752 
 
 .92186 
 
 .40355 
 
 .91496 
 
 .41945 
 
 .90778 
 
 12 
 
 49 
 
 .35538 
 
 .93472 
 
 .37164 
 
 .92838 
 
 .38778 
 
 .92175 
 
 .40381 
 
 .91484 
 
 .41972 
 
 .90766 
 
 11 
 
 50 
 
 .35565 
 
 .93462 
 
 .37191 
 
 .92827 
 
 .38805 
 
 .92164 
 
 .40408 
 
 .91472 
 
 .41998 
 
 .90753 
 
 10 
 
 51 
 
 .35592 
 
 .93452 
 
 .37218 
 
 .92816 
 
 .38832 
 
 .92152 
 
 .40434 
 
 .91461 
 
 .42024 
 
 .90741 
 
 9 
 
 52 
 
 .35619 
 
 .93441 
 
 .37245 
 
 .92805 
 
 .38859 
 
 .92141 
 
 .40461 
 
 .91449 
 
 .42051 
 
 .90729 
 
 8 
 
 53 
 
 .35647 
 
 .93431 
 
 .37272 
 
 .92794 
 
 .38886 
 
 .92130 
 
 .40488 
 
 .91437 
 
 .42077 
 
 .90717 
 
 7 
 
 54 
 
 .35674 
 
 .93420 
 
 .37299 
 
 .92784 
 
 .38912 
 
 .92119 
 
 .40514 
 
 .91425 
 
 .42104 
 
 .90704 
 
 6 
 
 55 
 
 .35701 
 
 .93410 
 
 .37326 
 
 .92773 
 
 .38939 
 
 .92107 
 
 .40541 
 
 .91414 
 
 .42130 
 
 .90692 
 
 5 
 
 56 
 
 .35728 
 
 .93400 
 
 .37353 
 
 .92762 
 
 .38966 
 
 .92096 
 
 .40567 
 
 .91402 
 
 .42156 
 
 .90680 
 
 4 
 
 57 
 
 .35755 
 
 .93389 
 
 .37380 
 
 .92751 
 
 .38993 
 
 .92085 
 
 .40594 
 
 .91390 
 
 .42183 
 
 .90668 
 
 3 
 
 58 
 
 .35782 
 
 .93379 
 
 .37407 
 
 .92740 
 
 .39020 
 
 .92073 
 
 .40621 
 
 .91378 
 
 .42209 
 
 .90655 
 
 2 
 
 59 
 
 .35810 
 
 .93368 
 
 .37434 
 
 .92729 
 
 .39046 
 
 .92062 
 
 .40647 
 
 .91366 
 
 .42235 
 
 .90643 
 
 1 
 
 60 
 
 .35837 
 
 .93358 
 
 .37461 
 
 .92718 
 
 .39073 
 
 .92050 
 
 .40674 
 
 .91355 
 
 .42262 
 
 .90631 
 
 0 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 
 I 
 
 69° 
 
 68° 
 
 67° 
 
 66° 
 
 65° 
 
 f 
 
460 
 
 natural sines and cosines. 
 
 
 25 ° 
 
 26 ° 
 
 27 ° 
 
 28 ° 
 
 29 ° 
 
 / 
 
 I - 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 0 
 
 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 
 87 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .42262 
 
 .42288 
 
 .42315 
 
 .42341 
 
 .42367 
 
 .42394 
 
 .42420 
 
 .42446 
 
 .42473 
 
 .42499 
 
 .42525 
 
 .42552 
 
 .42578 
 
 .42604 
 
 .42631 
 
 .42657 
 
 .42683 
 
 .42709 
 
 .42736 
 
 .42762 
 
 .42788 
 
 .42815 
 
 .42841 
 
 .42867 
 
 .42894 
 
 .42920 
 
 .42946 
 
 .42972 
 
 .42999 
 
 .43025 
 
 .43051 
 
 .43077 
 
 .43104 
 
 .43130 
 
 .43156 
 
 .43182 
 
 .43209 
 
 .43235 
 
 .43261 
 
 .43287 
 
 .43313 
 
 .43340 
 
 .43366 
 
 .43392 
 
 .43418 
 
 .43445 
 
 .43471 
 
 .43497 
 
 .43523 
 
 .43549 
 
 .43575 
 
 .43602 
 
 .43628 
 
 .43654 
 
 .43680 
 
 .43706 
 
 .43733 
 
 .43759 
 
 .43785 
 
 .43811 
 
 .43837 
 
 .90631 
 
 .90618 
 
 .90606 
 
 .90594 
 
 .90582 
 
 .90569 
 
 .90557 
 
 .90545 
 
 .90532 
 
 .90520 
 
 .90507 
 
 .90495 
 
 .90483 
 
 .90470 
 
 .90458 
 
 .90446 
 
 .90433 
 
 .90421 
 
 .90408 
 
 .90396 
 
 .90383 
 
 .90371 
 
 .90358 
 
 .90346 
 
 .90334 
 
 .90321 
 
 .90309 
 
 .90296 
 
 .90284 
 
 .90271 
 
 .90259 
 
 .90246 
 
 .90233 
 
 .90221 
 
 .90208 
 
 .90196 
 
 .90183 
 
 .90171 
 
 .90158 
 
 .90146 
 
 .90133 
 
 .90120 
 
 .90108 
 
 .90095 
 
 .90082 
 
 .90070 
 
 .90057 
 
 .90045 
 
 .90032 
 
 .90019 
 
 .90007 
 
 .89994 
 
 .89981 
 
 .89968 
 
 .89956 
 
 .89943 
 
 .89930 
 
 .89918 
 
 .89905 
 
 .89892 
 
 .89879 
 
 .43837 
 
 .43863 
 
 .43889 
 
 .43916 
 
 .43942 
 
 .43968 
 
 .43994 
 
 .44020 
 
 .44046 
 
 .44072 
 
 .44098 
 
 .44124 
 
 .44151 
 
 .44177 
 
 .44203 
 
 .44229 
 
 .44255 
 
 .44281 
 
 .44307 
 
 .44333 
 
 .44359 
 
 .44385 
 
 .44411 
 
 .44437 
 
 .44464 
 
 .44490 
 
 .44516 
 
 .44542 
 
 .44568 
 
 .44594 
 
 .44620 
 
 .44646 
 
 .44672 
 
 .44698 
 
 .44724 
 
 .44750 
 
 .44776 
 
 .44802 
 
 .44828 
 
 .44854 
 
 .44880 
 
 .44906 
 
 .44932 
 
 .44958 
 
 .44984 
 
 .45010 
 
 .45036 
 
 .45062 
 
 .45088 
 
 .45114 
 
 .45140 
 
 .45166 
 
 .45192 
 
 .45218 
 
 .45243 
 
 .45269 
 
 .45295 
 
 .45321 
 
 .45347 
 
 .45373 
 
 .45399 
 
 .89879 
 
 .89867 
 
 .89854 
 
 .89841 
 
 .89828 
 
 .89816 
 
 .89803 
 
 .89790 
 
 .89777 
 
 .89764 
 
 .89752 
 
 .89739 
 
 .89726 
 
 .89713 
 
 .89700 
 
 .89687 
 
 .89674 
 
 .89662 
 
 .89649 
 
 .89636 
 
 .89623 
 
 .89610 
 
 .89597 
 
 .89584 
 
 .89571 
 
 .89558 
 
 .89545 
 
 .89532 
 
 .89519 
 
 .89506 
 
 .89493 
 
 .89480 
 
 .89467 
 
 .89454 
 
 .89441 
 
 .89428 
 
 .89415 
 
 .89402 
 
 .89389 
 
 .89376 
 
 .89363 
 
 .89350 
 
 .89337 
 
 .89324 
 
 .89311 
 
 .89298 
 
 .89285 
 
 .89272 
 
 .89259 
 
 .89245 
 
 .89232 
 
 .89219 
 
 .89206 
 
 .89193 
 
 .89180 
 
 .89167 
 
 .89153 
 
 .89140 
 
 .89127 
 
 .89114 
 
 .89101 
 
 .45399 
 
 .45425 
 
 .45451 
 
 .45477 
 
 .45503 
 
 .45529 
 
 .45554 
 
 .45580 
 
 .45606 
 
 .45632 
 
 .45658 
 
 .45684 
 .45710 
 .45736 
 .45762 | 
 .45787 
 .45813 | 
 .45839 | 
 .45865 
 .45891 : 
 .45917 j 
 
 .45942 
 
 .45968 
 
 .45994 
 
 .46020 
 
 .46046 
 
 .46072 ! 
 
 .46097 
 
 .46123 
 
 .46149 | 
 
 .46175 
 
 .46201 ! 
 
 .46226 
 
 .46252 
 
 .46278 
 
 .46304 
 
 .46330 
 
 .46355 
 
 .46381 
 
 .46407 
 
 .46433 
 
 .46458 
 
 .46484 
 
 .46510 
 
 .46536 
 
 .46561 
 
 .46587 
 
 .46613 
 
 .46639 
 
 .46664 
 
 .46690 
 
 .46716 
 
 .46742 
 
 .46767 
 
 .46793 
 
 .46819 
 
 .46844 
 
 .46870 
 
 .46896 
 
 .46921 
 
 .46947 
 
 .89101 
 
 .89087 
 
 .89074 
 
 .89061 
 
 .89048 
 
 .89035 
 
 .89021 
 
 .89008 
 
 .88995 
 
 .88981 
 
 .88968 
 
 .88955 
 
 .88942 
 
 .88928 
 
 .88915 
 
 .88902 
 
 .88888 
 
 .88875 
 
 .88862 
 
 .88848 
 
 .88835 
 
 .88822 
 
 .88808 
 
 .88795 
 
 .88782 
 
 .88768 
 
 .88755 
 
 .88741 
 
 .88728 
 
 .88715 
 
 .88701 
 
 ! .88688 
 .88674 
 | .88661 
 i .88647 
 ! .88634 
 .88620 
 .88607 
 1 .88593 
 .88580 
 .88566 
 
 .88553 
 .88539 
 ! .88526 
 .88512 
 ! .88499 
 ! .88485 
 1 .88472 
 1 .88458 
 1 .88445 
 : .88431 
 
 ! .88417 
 1 .88404 
 i .88390 
 I .88377 
 1 .88363 
 ' .88349 
 .88336 
 ! .88322 
 | .88308 
 .88295 
 
 .46947 
 
 .46973 
 
 .46999 
 
 .47024 
 
 .47050 
 
 .47076 
 
 .47101 
 
 .47127 
 
 .47153 
 
 .47178 
 
 .47204 
 
 .47229 
 
 .47255 
 
 .47281 
 
 .47306 
 
 .47332 
 
 .47358 
 
 .47383 
 
 .47409 
 
 .47434 
 
 .47460 
 
 .47486 
 
 .47511 
 
 .47537 
 
 .47562 
 
 .47588 
 
 .47614 
 
 .47639 
 
 .47665 
 
 .47690 
 
 .47716 
 
 .47741 
 
 .47767 
 
 .47793 
 
 .47818 
 
 .47844 
 
 .47869 
 
 .47895 
 
 .47920 
 
 .47946 
 
 .47971 
 
 .47997 
 
 .48022 
 
 .48048 
 
 .48073 
 
 .48099 
 
 .48124 
 
 .48150 
 
 .48175 
 
 .48201 
 
 .48226 
 
 .48252 
 
 .48277 
 
 .48303 
 
 .48328 
 
 .48354 
 
 .48379 
 
 .48405 
 
 .48430 
 
 .48456 
 
 .48481 
 
 .88295 
 
 .88281 
 
 .88267 
 
 .88254 
 
 .88240 
 
 .88226 
 
 .88213 
 
 .88199 
 
 .88185 
 
 .88172 
 
 .88158 
 
 .88144 
 
 .88130 
 
 .88117 
 
 .88103 
 
 .88089 
 
 .88075 
 
 .88062 
 
 .88048 
 
 .88034 
 
 .88020 
 
 .88006 
 
 .87993 
 
 .87979 
 
 .87965 
 
 .87951 
 
 .87937 
 
 .87923 
 
 .87909 
 
 .87896 
 
 .87882 
 
 .87868 
 
 .87854 
 
 .87840 
 
 .87826 
 
 .87812 
 
 .87798 
 
 .87784 
 
 .87770 
 
 .87756 
 
 .87743 
 
 .87729 
 
 .87715 
 
 .87701 
 
 .87687 
 
 .87673 
 
 .87659 
 
 .87645 
 
 .87631 
 
 .87617 
 
 .87603 
 
 .87589 
 
 .87575 
 
 .87561 
 
 .87546 
 
 .87532 
 
 .87518 
 
 .87504 
 
 .87490 
 
 .87476 
 
 .87462 
 
 .48481 
 
 .48506 
 
 .48532 
 
 .48557 
 
 .48583 
 
 .48608 
 
 .48634 ! 
 
 .48659 j 
 
 .48684 | 
 
 .48710 
 
 .48735 
 
 .48761 
 
 .48786 
 
 .48811 
 
 .48837 
 
 .48862 
 
 .48888 
 
 .48913 
 
 .48938 
 
 .48964 | 
 
 .48989 
 
 .49014 
 
 .49040 
 
 .49065 
 
 .49090 
 
 .49116 
 
 .49141 
 
 .49166 
 
 .49192 
 
 .49217 
 
 .49242 
 
 .49268 
 
 .49293 
 
 .49318 
 
 .49344 
 
 .49369 
 
 .49394 
 
 .49419 
 
 .49445 
 
 .49470 
 
 .49495 
 
 .49521 
 
 .49546 
 
 .49571 
 
 .49596 
 
 .49622 
 
 .49647 
 
 .49672 
 
 .49697 
 
 .49723 
 
 .49748 
 
 .49773 
 
 .49798 
 
 .49824 
 
 .49849 
 
 .49874 
 
 .49899 
 
 .49924 
 
 .49950 
 
 .49975 
 
 .50000 
 
 .87462 
 
 .87448 
 
 .87434 
 
 .87420 
 
 .87406 
 
 .87391 
 
 .87377 
 
 .87363 
 
 .87349 
 
 .87335 
 
 .87321 
 
 .87306 
 
 .87292 
 
 .87278 
 
 .87264 
 
 .87250 
 
 .87235 
 
 .87221 
 
 .87207 
 
 .87193 
 
 .87178 
 
 .87164 
 
 .87150 
 
 .87136 
 
 .87121 
 
 .87107 
 
 .87093 
 
 .87079 
 
 .87064 
 
 .87050 
 
 .87036 
 
 .87021 
 
 .87007 
 
 .86993 
 
 .86978 
 
 .86964 
 
 .86949 
 
 .86935 
 
 .86921 
 
 .86906 
 
 .86892 
 
 .86878 
 
 .86863 
 
 .86849 
 
 .86834 
 
 .86820 
 
 .86805 
 
 .86791 
 
 .86777 
 
 .86762 
 
 .86748 
 
 ,.86733 
 
 .86719 
 
 .86704 
 
 .86690 
 
 .86675 
 
 .86661 
 
 .86646 
 
 .86632 
 
 .86617 
 
 .86603 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 | Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 / 
 
 / 
 
 64 ° 
 
 63 ° 
 
 62 ° 
 
 61 ° 
 
 60 ° 
 
 
NATURAL SINES AND COSINES. 
 
 461 
 
 
 30° 
 
 31° 
 
 32° 
 
 33° 
 
 34° 
 
 / 
 
 t 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .50000 
 
 .50025 
 
 .50050 
 
 .50076 
 
 .50101 
 
 .50126 
 
 .50151 
 
 .50176 
 
 .50201 
 
 .50227 
 
 .50252 
 
 .50277 
 
 .50302 
 
 .50327 
 
 .50352 
 
 .50377 
 
 .50403 
 
 .50428 
 
 .50453 
 
 .50478 
 
 .50503 
 
 .50528 
 
 .50553 
 
 .50578 
 
 .50603 
 
 .50628 
 
 .50654 
 
 .50679 
 
 .50704 
 
 .50729 
 
 .50754 
 
 .50779 
 
 .50804 
 
 .50829 
 
 .50854 
 
 .50879 
 
 .50904 
 
 .50929 
 
 .50954 
 
 .50979 
 
 .51004 
 
 .51029 
 
 .51054 
 
 .51079 
 
 .51104 
 
 .51129 
 
 .51154 
 
 .51179 
 
 .51204 
 
 .51229 
 
 .51254 
 
 .51279 
 
 .51304 
 
 .51329 
 
 .51354 
 
 .51379 
 
 .51404 
 
 .51429 
 
 .51454 
 
 .51479 
 
 .51504 
 
 .86603 
 
 .86588 
 
 .86573 
 
 .86559 
 
 .86544 
 
 .86530 
 
 .86515 
 
 .86501 
 
 .86486 
 
 .86471 
 
 .86457 
 
 .86442 
 
 .86427 
 
 .86413 
 
 .86398 
 
 .86384 
 
 .86369 
 
 .86354 
 
 .86340 
 
 .86325 
 
 .86310 
 
 .86295 
 
 .86281 
 
 .86266 
 
 .86251 
 
 .86237 
 
 .86222 
 
 .86207 
 
 .86192 
 
 .86178 
 
 .86163 
 
 .86148 
 
 .86133 
 
 .86119 
 
 .86104 
 
 .86089 
 
 .86074 
 
 .86059 
 
 .86045 
 
 .86030 
 
 .86015 
 
 .86000 
 
 .85985 
 
 .85970 
 
 .85956 
 
 .85941 
 
 .85926 
 
 .85911 
 
 .85896 
 
 .85881 
 
 .85866 
 
 .85851 
 
 .85836 
 
 .85821 
 
 .85806 
 
 .85792 
 
 .85777 
 
 .85762 
 
 .85747 
 
 .85732 
 
 .85717 
 
 .51504 
 
 .51529 
 
 .51554 
 
 .51579 
 
 .51604 
 
 .51628 
 
 .51653 
 
 .51678 
 
 .51703 
 
 .51728 
 
 .51753 
 
 .51778 
 
 .51803 
 
 .51828 
 
 .51852 
 
 .51877 
 
 .51902 
 
 .51927 
 
 .51952 
 
 .51977 
 
 .52002 
 
 .52026 
 
 .52051 
 
 .52076 
 
 .52101 
 
 .52126 
 
 .52151 
 
 .52175 
 
 .52200 
 
 .52225 
 
 .52250 
 
 .52275 
 
 .52299 
 
 .52324 
 
 .52349 
 
 .52374 
 
 .52399 
 
 .52423 
 
 .52448 
 
 .52473 
 
 .52498 
 
 .52522 
 
 .52547 
 
 .52572 
 
 .52597 
 
 .52621 
 
 .52646 
 
 .52671 
 
 .52696 
 
 .52720 
 
 .52745 
 
 .52770 
 
 .52794 
 
 .52819 
 
 .52844 
 
 .52869 
 
 .52893 
 
 .52918 
 
 .52943 
 
 .52967 
 
 .52992 
 
 .85717 
 
 .85702 
 
 .85687 
 
 .85672 
 
 .85657 
 
 .85642 
 
 .85627 
 
 .85612 
 
 .85597 
 
 .85582 
 
 .85567 
 
 .85551 
 
 .85536 
 
 .85521 
 
 .85506 
 
 .85491 
 
 .85476 
 
 .85461 
 
 .85446 
 
 .85431 
 
 .85416 
 
 .85401 
 
 .85385 
 
 .85370 
 
 .85355 
 
 .85340 
 
 .85325 
 
 .85310 
 
 .85294 
 
 .85279 
 
 .85264 
 
 .85249 
 
 .85234 
 
 .85218 
 
 .85203 
 
 .85188 
 
 .85173 
 
 .85157 
 
 .85142 
 
 .85127 
 
 .85112 
 
 .85096 
 
 .85081 
 
 .85066 
 
 .85051 
 
 .85035 
 
 .85020 
 
 .85005 
 
 .84989 
 
 .84974 
 
 .84959 
 
 .84943 
 
 .84928 
 
 .84913 
 
 .84897 
 
 .84882 
 
 .84866 
 
 .84851 
 
 .84836 
 
 .84820 
 
 .84805 
 
 .52992 
 
 .53017 
 
 .53041 
 
 .53066 
 
 .53091 
 
 .53115 
 
 .53140 
 
 .53164 
 
 .53189 
 
 .53214 
 
 .53238 
 
 .53263 
 
 .53288 
 
 .53312 
 
 .53337 
 
 .53361 
 
 .53386 
 
 .53411 
 
 .53435 
 
 .53460 
 
 .53484 
 
 .53509 
 
 .53534 
 
 .53558 
 
 .53583 
 
 .53607 
 
 .53632 
 
 .53656 
 
 .53681 
 
 .53705 
 
 .53730 
 
 .53754 
 .53779 
 .53804 
 .53828 
 .53853 
 . .53877 
 .53902 
 .53926 
 .53951 
 .53975 
 
 .54000 
 
 .54024 
 
 .54049 
 
 .54073 
 
 .54097 
 
 .54122 
 
 .54146 
 
 .54171 
 
 .54195 
 
 .54220 
 
 .54244 
 
 .54269 
 
 .54293 
 
 .54317 
 
 .54342 
 
 .54366 
 
 .54391 
 
 .54415 
 
 .54440 
 
 .54464 
 
 .84805 
 
 .84789 
 
 .84774 
 
 .84759 
 
 .84743 
 
 .84728 
 
 .84712 
 
 .84697 
 
 .84681 
 
 .84666 
 
 .84650 
 
 .84635 
 
 .84619 
 
 .84604 
 
 .84588 
 
 .84573 
 
 .84557 
 
 .84542 
 
 .84526 
 
 .84511 
 
 .84495 
 
 .84480 
 
 .84464 
 
 .84448 
 
 .84433 
 
 .84417 
 
 .84402 
 
 .84386 
 
 .84370 
 
 .84355 
 
 .84339 
 
 .84324 
 
 .84308 
 
 .84292 
 
 .84277 
 
 .84261 
 
 .84245 
 
 .84230 
 
 .84214 
 
 .84198 
 
 .84182 
 
 .84167 
 
 .84151 
 
 .84135 
 
 .84120 
 
 .84104 
 
 .84088 
 
 .84072 
 
 .84057 
 
 .84041 
 
 .84025 
 
 .84009 
 
 .83994 
 
 .83978 
 
 .83962 
 
 .83946 
 
 .83930 
 
 .83915 
 
 .83899 
 
 .83883 
 
 .83867 
 
 .54464 
 
 .54488 
 
 .54513 
 
 .54537 
 
 .54561 
 
 .54586 
 
 .54610 
 
 .54635 
 
 .54659 
 
 .54683 
 
 .54708 
 
 .54732 
 
 .54756 
 
 .54781 
 
 .54805 
 
 .54829 
 
 .54854 
 
 .54878 
 
 .54902 
 
 .54927 
 
 .54951 
 
 .54975 
 
 .54999 
 
 .55024 
 
 .55048 
 
 .55072 
 
 .55097 
 
 .55121 
 
 .55145 
 
 .55169 
 
 .55194 
 
 .55218 
 
 .55242 
 
 .55266 
 
 .55291 
 
 .55315 
 
 .55339 
 
 .55363 
 
 .55388 
 
 .55412 
 
 .55436 
 
 .55460 
 
 .55484 
 
 .55509 
 
 .55533 
 
 .55557 
 
 .55581 
 
 .55605 
 
 .55630 
 
 .55654 
 
 .55678 
 
 .55702 
 
 .55726 
 
 .55750 
 
 .55775 
 
 .55799 
 
 .55823 
 
 .55847 
 
 .55871 
 
 .55895 
 
 .55919 
 
 .83867 
 
 .83851 
 
 .83835 
 
 .83819 
 
 .83804 
 
 .83788 
 
 .83772 
 
 .83756 
 
 .83740 
 
 .83724 
 
 .83708 
 
 .83692 
 
 .83676 
 
 .83660 
 
 .83645 
 
 .83629 
 
 .83613 
 
 .83597 
 
 .83581 
 
 .83565 
 
 .83549 
 
 .83533 
 
 .83517 
 
 .83501 
 
 .83485 
 
 .83469 
 
 .83453 
 
 .83437 
 
 .83421 
 
 .83405 
 
 .83389 
 
 .83373 
 
 .83356 
 
 .83340 
 
 .83324 
 
 .83308 
 
 .83292 
 
 .83276 
 
 .83260 
 
 .83244 
 
 .83228 
 
 .83212 
 
 .83195 
 
 .83179 
 
 .83163 
 
 .83147 
 
 .83131 
 
 .83115 
 
 .83098 
 
 .83082 
 
 .83066 
 
 .83050 
 
 .83034 
 
 .83017 
 
 .83001 
 
 .82985 
 
 .82969 
 
 .82953 
 
 .82936 
 
 .82920 
 
 .82904 
 
 .55919 
 
 .55943 
 
 .55968 
 
 .55992 
 
 .56016 
 
 .56040 
 
 .56064 
 
 .56088 
 
 .56112 
 
 .56136 
 
 .56160 
 
 .56184 
 
 .56208 
 
 .56232 
 
 .50256 
 
 .56280 
 
 .56305 
 
 .56329 
 
 .56353 
 
 .56377 
 
 .56401 
 
 .56425 
 
 .56449 
 
 .56473 
 
 .56497 
 
 .56521 
 
 .56545 
 
 .56569 
 
 .56593 
 
 .56617 
 
 .56641 
 
 .56665 
 
 .56689 
 
 .56713 
 
 .56736 
 
 .56760 
 
 .56784 
 
 .56808 
 
 .56832 
 
 .56856 
 
 .56880 
 
 .56904 
 
 .56928 
 
 .56952 
 
 .56976 
 
 .57000 
 
 .57024 
 
 .57047 
 
 .57071 
 
 .57095 
 
 .57119 
 
 .57143 
 
 .57167 
 
 .57191 
 
 .57215 
 
 .57238 
 
 .57262 
 
 .57286 
 
 .57310 
 
 .57334 
 
 .57358 
 
 .82904 
 
 .82887 
 
 .82871 
 
 .82855 
 
 .82839 
 
 .82822 
 
 .82806 
 
 .82790 
 
 .82773 
 
 .82757 
 
 .82741 
 
 .82724 
 
 .82708 
 
 .82692 
 
 .82675 
 
 .82659 
 
 .82643 
 
 .82626 
 
 .82610 
 
 .82593 
 
 .82577 
 
 .82561 
 
 .82544 
 
 .82528 
 
 .82511 
 
 .82495 
 
 .82478 
 
 .82462 
 
 .82446 
 
 .82429 
 
 .82413 
 
 .82396 
 
 .82380 
 
 .82363 
 
 .82347 
 
 .82330 
 
 .82314 
 
 .82297 
 
 .82281 
 
 .82264 
 
 .82248 
 
 .82231 
 
 .82214 
 
 .82198 
 
 .82181 
 
 .82165 
 
 .82148 
 
 .82132 
 
 .82115 
 
 .82098 
 
 .82082 
 
 .82065 
 
 .82048 
 
 .82032 
 
 .82015 
 
 .81999 
 
 .81982 
 
 .81965 
 
 .81949 
 
 .81932 
 
 .81915 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 , 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 / 
 
 Cosine 
 
 i Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 
 59° 
 
 58° 
 
 i 57° 
 
 56° 
 
 55° 
 
 f 
 
462 
 
 NATURAL SINES AND COSINES. 
 
 
 35 ° 
 
 36 ° 
 
 37 ° 
 
 38 ° 
 
 39 ° 
 
 f 
 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 
 0 
 
 .57358 
 
 .81915 
 
 .58779 
 
 .80902 
 
 .60182 
 
 .79864 
 
 .61566 
 
 .78801 
 
 .62932 
 
 .77715 
 
 60 
 
 1 
 
 .57381 
 
 .81899 
 
 .58802 
 
 .80885 
 
 .60205 
 
 .79846 
 
 .61589 
 
 .78783 
 
 .62955 
 
 .77696 
 
 59 
 
 2 
 
 .57405 
 
 .81882 
 
 .58826 
 
 .80867 
 
 .60228 
 
 .79829 
 
 .61612 
 
 .78765 
 
 .62977 
 
 .77678 
 
 58 
 
 3 
 
 .57429 
 
 .81865 
 
 .58849 
 
 .80850 
 
 .60251 
 
 .79811 
 
 .61635 
 
 .78747 
 
 .63000 
 
 .77660 
 
 57 
 
 4 
 
 .57453 
 
 .81848 
 
 .58873 
 
 .80833 
 
 .60274 
 
 .79793 
 
 .61658 
 
 .78729 
 
 .63022 
 
 .77641 
 
 56 
 
 5 
 
 .57477 
 
 .81832 
 
 .58896 
 
 .80816 
 
 .60298 
 
 .79776 
 
 .61681 
 
 .78711 
 
 .63045 
 
 .77623 
 
 55 
 
 6 
 
 .57501 
 
 .81815 
 
 .58920 
 
 .80799 
 
 .60321 
 
 .79758 
 
 .61704 
 
 .78694 
 
 .63068 
 
 .77605 
 
 54 
 
 7 
 
 .57524 
 
 .81798 
 
 .58943 
 
 .80782 
 
 .60344 
 
 .79741 
 
 .61726 
 
 .78676 
 
 .63090 
 
 .77586 
 
 53 
 
 8 
 
 .57548 
 
 .81782 
 
 .58967 
 
 .80765 
 
 .60367 
 
 .79723 
 
 .61749 
 
 .78658 
 
 .63113 
 
 .77568 
 
 52 
 
 9 
 
 .57572 
 
 .81765 
 
 .58990 
 
 .80748 
 
 .60390 
 
 .79706 
 
 .61772 
 
 .78640 
 
 .63135 
 
 .77550 
 
 51 
 
 10 
 
 .57596 
 
 .81748 
 
 .59014 
 
 .80730 
 
 .60414 
 
 .79688 
 
 .61795 
 
 .78622 
 
 .63158 
 
 .77531 
 
 50 
 
 11 
 
 .57619 
 
 .81731 
 
 .59037 
 
 .80713 
 
 .60437 
 
 .79671 
 
 .61818 
 
 .78604 
 
 .63180 
 
 .77513 
 
 49 
 
 12 
 
 .57643 
 
 .81714 
 
 .59061 
 
 .80696 
 
 .60460 
 
 .79653 
 
 .61841 
 
 .78586 
 
 .63203 
 
 .77494 
 
 48 
 
 13 
 
 .57667 
 
 .81698 
 
 .59084 
 
 .80679 
 
 .60483 
 
 .79635 
 
 .61864 
 
 .78568 
 
 .63225 
 
 .77476 
 
 47 
 
 14 
 
 .57691 
 
 .81681 
 
 .59108 
 
 .80662 
 
 .60506 
 
 .79618 
 
 .61887 
 
 .78550 
 
 .63248 
 
 .77458 
 
 46 
 
 15 
 
 .57715 
 
 .81664 
 
 .59131 
 
 .80644 
 
 .60529 
 
 .79600 
 
 .61909 
 
 .78532 
 
 .63271 
 
 .77439 
 
 45 
 
 16 
 
 .57738 
 
 .81647 
 
 .59154 
 
 .80627 
 
 .60553 
 
 .79583 
 
 .61932 
 
 .78514 
 
 .63293 
 
 .77421 
 
 44 
 
 17 
 
 .57762 
 
 .81631 
 
 .59178 
 
 .80610 
 
 .60576 
 
 .79565 
 
 .61955 
 
 .78496 
 
 .63316 
 
 .77402 
 
 43 
 
 18 
 
 .57786 
 
 .81614 
 
 .59201 
 
 .80593 
 
 .60599 
 
 .79547 
 
 .61978 
 
 .78478 
 
 .63338 
 
 .77384 
 
 42 
 
 19 
 
 .57810 
 
 .81597 
 
 .59225 
 
 .80576 
 
 .60622 
 
 .79530 
 
 .62001 
 
 .78460 
 
 .63361 
 
 .77366 
 
 41 
 
 20 
 
 .57833 
 
 .81580 
 
 .59248 
 
 .80558 
 
 .60645 
 
 .79512 
 
 .62024 
 
 .78442 
 
 .63383 
 
 .77347 
 
 40 
 
 21 
 
 .57857 
 
 .81563 
 
 .59272 
 
 .80541 
 
 .60668 
 
 .79494 
 
 .62046 
 
 .78424 
 
 .63406 
 
 .77329 
 
 39 
 
 22 
 
 .57881 
 
 .81546 
 
 .59295 
 
 .80524 
 
 .60691 
 
 .79477 
 
 .62069 
 
 .78405 
 
 .63428 
 
 .77310 
 
 38 
 
 23 
 
 .57904 
 
 .81530 
 
 .59318 
 
 .80507 
 
 .60714 
 
 .79459 
 
 .62092 
 
 .78387 
 
 .63451 
 
 .77292 
 
 37 
 
 24 
 
 .57928 
 
 .81513 
 
 .59342 
 
 .80489 
 
 .60738 
 
 .79441 
 
 .62115 
 
 .78369 
 
 .63473 
 
 .77273 
 
 36 
 
 25 
 
 .57952 
 
 .81496 
 
 .59365 
 
 .80472 
 
 .60761 
 
 .79424 
 
 .62138 
 
 .78351 
 
 .63496 
 
 .77255 
 
 35 
 
 26 
 
 .57976 
 
 .81479 
 
 .59389 
 
 .80455 
 
 .60784 
 
 .79406 
 
 .62160 
 
 .78333 
 
 .63518 
 
 .77236 
 
 34 
 
 27 
 
 .57999 
 
 .81462 
 
 .59412 
 
 .80438 
 
 .60807 
 
 .79388 
 
 .62183 
 
 .78315 
 
 .63540 
 
 .77218 
 
 33 
 
 28 
 
 .58023 
 
 .81445 
 
 .59436 
 
 .80420 
 
 .60830 
 
 .79371 
 
 .62206 
 
 .78297 
 
 .63563 
 
 .77199 
 
 32 
 
 29 
 
 .58047 
 
 .81428 
 
 .59459 
 
 .80403 
 
 .60853 
 
 .79353 
 
 .62229 
 
 .78279 
 
 .63585 
 
 .77181 
 
 31 
 
 30 
 
 .58070 
 
 .81412 
 
 .59482 
 
 .80386 
 
 .60876 
 
 .79335 
 
 .62251 
 
 .78261 
 
 .63608 
 
 .77162 
 
 30 
 
 31 
 
 .58094 
 
 .81395 
 
 .59506 
 
 .80368 
 
 .60899 
 
 .79318 
 
 .62274 
 
 .78243 
 
 .63630 
 
 .77144 
 
 29 
 
 32 
 
 .58118 
 
 .81378 
 
 .59529 
 
 .80351 
 
 .60922 
 
 .79300 
 
 .62297 
 
 .78225 
 
 .63653 
 
 .77125 
 
 28 
 
 33 
 
 .58141 
 
 .81361 
 
 .59552 
 
 .80334 
 
 .60945 
 
 .79282 
 
 .62320 
 
 .78206 
 
 .63675 
 
 .77107 
 
 27 
 
 34 
 
 .58165 
 
 .81344 
 
 .59576 
 
 .80316 
 
 .60968 
 
 .79264 
 
 .62342 
 
 .78188 
 
 .63698 
 
 .77088 
 
 26 
 
 35 
 
 .58189 
 
 .81327 
 
 .59599 
 
 .80299 
 
 .60991 
 
 .79247 
 
 .62365 
 
 .78170 
 
 .63720 
 
 .77070 
 
 25 
 
 36 
 
 .58212 
 
 .81310 
 
 .59622 
 
 .80282 
 
 .61015 
 
 .79229 
 
 .62388 
 
 .78152 
 
 .63742 
 
 .77051 
 
 24 
 
 37 
 
 .58236 
 
 .81293 
 
 .59646 
 
 .80264 
 
 .61038 
 
 .79211 
 
 .62411 
 
 .78134 
 
 .63765 
 
 .77033 
 
 23 
 
 38 
 
 .58260 
 
 .81276 
 
 .59669 
 
 .80247 
 
 .61061 
 
 .79193 
 
 .62433 
 
 .78116 
 
 .63787 
 
 .77014 
 
 22 
 
 39 
 
 .58283 
 
 .81259 
 
 .59693 
 
 .80230 
 
 .61084 
 
 .79176 
 
 .62456 
 
 .78098 
 
 .63810 
 
 .76996 
 
 21 
 
 40 
 
 .58307 
 
 .81242 
 
 .59716 
 
 .80212 
 
 .61107 
 
 .79158 
 
 .62479 
 
 .78079 
 
 .63832 
 
 .76977 
 
 20 
 
 41 
 
 .58330 
 
 .81225 
 
 .59739 
 
 .80195 
 
 .61130 
 
 .79140 
 
 .62502 
 
 .78061 
 
 .63854 
 
 .76959 
 
 19 
 
 42 
 
 .58354 
 
 .81208 
 
 .59763 
 
 .80178 
 
 .61153 
 
 .79122 
 
 .62524 
 
 .78043 
 
 .63877 
 
 .76940 
 
 18 
 
 43 
 
 .58378 
 
 .81191 
 
 .59786 
 
 .80160 
 
 .61176 
 
 .79105 
 
 .62547 
 
 .78025 
 
 .63899 
 
 .76921 
 
 17 
 
 44 
 
 .58401 
 
 .81174 
 
 .59809 
 
 .80143 
 
 .61199 
 
 .79087 
 
 .62570 
 
 .78007 
 
 .63922 
 
 .76903 
 
 16 
 
 45 
 
 .58425 
 
 .81157 
 
 .59832 
 
 .80125 
 
 .61222 
 
 .79069 
 
 .62592 
 
 .77988 
 
 .63944 
 
 .76884 
 
 15 
 
 46 
 
 .58449 
 
 .81140 
 
 .59856 
 
 .80108 
 
 .61245 
 
 .79051 
 
 .62615 
 
 .77970 
 
 .63966 
 
 .76866 
 
 14 
 
 47 
 
 .58472 
 
 .81123 
 
 .59879 
 
 .80091 
 
 .61268 
 
 .79033 
 
 .62638 
 
 .77952 
 
 .63989 
 
 .76847 
 
 13 
 
 48 
 
 .58496 
 
 .81106 
 
 .59902 
 
 .80073 
 
 .61291 
 
 .79016 
 
 .62660 
 
 .77934 
 
 .64011 
 
 .76828 
 
 12 
 
 49 
 
 .58519 
 
 .§1089 
 
 .59926 
 
 .80056 
 
 .61314 
 
 .78998 
 
 .62683 
 
 .77916 
 
 .64033 
 
 .76810 
 
 11 
 
 50 
 
 .58543 
 
 .81072 
 
 .59949 
 
 .80038 
 
 .61337 
 
 .78980 
 
 .62706 
 
 .77897 
 
 .64056 
 
 .76791 
 
 10 
 
 51 
 
 .58567 
 
 .81055 
 
 .59972 
 
 .80021 
 
 .61360 
 
 .78962 
 
 .62728 
 
 .77879 ■ 
 
 .64078 
 
 .76772 
 
 9 
 
 52 
 
 .58590 
 
 .81038 
 
 .59995 
 
 .80003 
 
 .61383 
 
 .78944 
 
 .62751 
 
 .77861 
 
 .64100 
 
 .76754 
 
 8 
 
 53 
 
 .58614 
 
 .81021 
 
 .60019 
 
 .79986 
 
 .61406 
 
 .78926 
 
 .62774 
 
 .77843 
 
 .64123 
 
 .76735 
 
 7 
 
 54 
 
 .58637 
 
 .81004 
 
 .60042 
 
 .79968 
 
 .61429 
 
 .78908 
 
 .62796 
 
 .77824 
 
 .64145 
 
 .76717 
 
 6 
 
 55 
 
 .58661 
 
 .80978 
 
 .60065 
 
 .79951 
 
 .61451 
 
 .78891 
 
 .62819 
 
 .77806 
 
 .64167 
 
 .76698 
 
 5 
 
 56 
 
 .58684 
 
 .80970 
 
 .60089 
 
 .79934 
 
 .61474 
 
 .78873 
 
 .62842 
 
 .77788 
 
 .64190 
 
 .76679 
 
 4 
 
 57 
 
 .58708 
 
 .80953 
 
 .60112 
 
 .79916 
 
 .61497 
 
 .78855 
 
 .62864 
 
 .77769 
 
 .64212 
 
 .76661 
 
 3 
 
 58 
 
 .58731 
 
 .80936 
 
 .60135 
 
 .79899 
 
 .61520 
 
 .78837 
 
 .62887 
 
 .77751 
 
 .64234 
 
 .76642 
 
 2 
 
 59 
 
 .58755 
 
 .80919 
 
 .60158 
 
 .79881 
 
 .61543 
 
 .78819 
 
 .62909 
 
 .77733 
 
 .64256 
 
 .76623 
 
 1 
 
 60 
 
 .58779 
 
 .80902 
 
 .60182 
 
 .79864 
 
 .61566 
 
 .78801 
 
 .62932 
 
 .77715 
 
 .64279 
 
 .76604 
 
 0 
 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 
 / 
 
 54 ° 
 
 ; 53 ° 
 
 52 ° 
 
 51 ° 
 
 50 ° 
 
 / 
 
NATURAL SINES AND COSINES. 
 
 463 
 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 44° 
 
 / 
 
 / - 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine 
 
 Cosine 
 
 Sine i 
 
 Cosine 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 . 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .64279 
 
 .64301 
 
 .64323 
 
 .64346 
 
 .64368 
 
 .64390 
 
 .64412 
 
 .64435 
 
 .64457 
 
 .64479 
 
 .64501 
 
 .64524 
 
 .64546 
 
 .64568 
 
 .64590 
 
 .64612 
 
 .64635 
 
 .64657 
 
 .64679 
 
 .64701 
 
 .64723 
 
 .64746 
 
 .64768 
 
 .64790 
 
 .64812 
 
 .64834 
 
 .64856 
 
 .64878 
 
 .64901 
 
 .64923 
 
 .64945 
 
 .64967 
 
 .64989 
 
 .65011 
 
 .65033 
 
 .65055 
 
 .65077 
 
 .65100 
 
 .65122 
 
 .65144 
 
 .65166 
 
 .65188 
 
 .65210 
 
 .65232 
 
 .65254 
 
 .65276 
 
 .65298 
 
 .65320 
 
 .65342 
 
 .65364 
 
 .65386 
 
 .65408 
 
 .65430 
 
 .65452 
 
 .65474 
 
 .65496 
 
 .65518 
 
 .65540 
 
 .65562 
 
 .65584 
 
 .65606 
 
 .76604 
 
 .76586 
 
 .76567 
 
 .76548 
 
 .76530 
 
 .76511 
 
 .76492 
 
 .76473 
 
 .76455 
 
 .76436 
 
 .76417 
 
 .76398 
 
 .76380 
 
 .76361 
 
 .76342 
 
 .76323 
 
 .76304 
 
 .76286 
 
 .76267 
 
 .76248 
 
 .76229 
 
 .76210 
 
 .76192 
 
 .76173 
 
 .76154 
 
 .76135 
 
 .76116 
 
 .76097 
 
 .76078 
 
 .76059 
 
 .76041 
 
 .76022 
 
 .76003 
 
 .75984 
 
 .75965 
 
 .75946 
 
 .75927 
 
 .75908 
 
 .75889 
 
 .75870 
 
 .75851 
 
 .75832 
 
 .75813 
 
 .75794 
 
 .75775 
 
 .75756 
 
 .75738 
 
 .75719 
 
 .75700 
 
 .75680 
 
 .75661 
 
 .75642 
 
 .75623 
 
 .75604 
 
 .75585 
 
 .75566 
 
 .75547 
 
 .75528 
 
 .75509 
 
 .75490 
 
 .75471 
 
 .65606 
 
 .65628 
 
 .65650 
 
 .65672 
 
 .65694 
 
 .65716 
 
 .65738 
 
 .65759 
 
 .65781 
 
 .65803 
 
 .65825 
 
 ' 
 
 .65847 
 
 .65869 
 
 .65891 
 
 .65913 
 
 .65935 
 
 .65956 
 
 .65978 
 
 .66000 
 
 .66022 
 
 .66044 
 
 .66066 
 
 .66088 
 
 .66109 
 
 .66131 
 
 .66153 
 
 .66175 
 
 .66197 
 
 .66218 
 
 .66240 
 
 .66262 
 
 .66284 
 
 .66306 
 
 .66327 
 
 .66349 
 
 .66371 
 
 .66393 
 
 .66414 
 
 .66436 
 
 .66458 
 
 .66480 
 
 .66501 
 .66523 
 .66545 
 .66566 
 .66588 
 .66610 
 .66632 
 .66653 
 .66675 
 - .66697 
 
 .66718 
 
 .66740 
 
 .66762 
 
 .66783 
 
 .66805 
 
 .66827 
 
 .66848 
 
 .66870 
 
 .66891 
 
 .66913 
 
 .75471 
 
 .75452 
 
 .75433 
 
 .75414 
 
 .75395 
 
 .75375 
 
 .75356 
 
 .75337 
 
 .75318 
 
 .75299 
 
 .75280 
 
 .75261 
 
 .75241 
 
 .75222 
 
 .75203 
 
 .75184 
 
 .75165 
 
 .75146 
 
 .75126 
 
 .75107 
 
 .75088 
 
 .75069 
 
 .75050 
 
 .75030 
 
 .75011 
 
 .74992 
 
 .74973 
 
 .74953 
 
 .74934 
 
 .74915 
 
 .74896 
 
 .74876 
 
 .74857 
 
 .74838 
 
 .74818 
 
 .74799 
 
 .74780 
 
 .74760 
 
 .74741 
 
 .74722 
 
 .74703 
 
 .74683 
 
 .74664 
 
 .74644 
 
 .74625 
 
 .74606 
 
 .74586 
 
 .74567 
 
 .74548 
 
 .74528 
 
 .74509 
 
 .74489 
 
 .74470 
 
 .74451 
 
 .74431 
 
 .74412 
 
 .74392 
 
 .74373 
 
 .74353 
 
 .74334 
 
 .74314 
 
 .66913 
 
 .66935 
 
 .66956 
 
 .66978 
 
 .66999 
 
 .67021 
 
 .67043 
 
 .67064 
 
 .67086 
 
 .67107 
 
 .67129 
 
 .67151 
 
 .67172 
 
 .67194 
 
 .67215 
 
 .67237 
 
 .67258 
 
 .67280 
 
 .67301 
 
 .67323 
 
 .67344 
 
 .67366 
 
 .67387 
 
 .67409 
 
 .67430 
 
 .67452 
 
 .67473 
 
 .67495 
 
 .67516 
 
 .67538 
 
 .67559 
 
 .67580 
 
 .67602 
 
 .67623 
 
 .67645 
 
 .67666 
 
 .67688 
 
 .67709 
 
 .67730 
 
 .67752 
 
 .67773 
 
 .67795 
 
 .67816 
 
 .67837 
 
 .67859 
 
 .67880 
 
 .67901 
 
 .67923 
 
 .67944 
 
 .67965 
 
 .67987 
 
 .68008 
 
 .68029 
 
 .68051 
 
 .68072 
 
 .68093 
 
 .68115 
 
 .68136 
 
 .68157 
 
 .68179 
 
 .68200 
 
 .74314 
 
 .74295 
 
 .74276 
 
 .74256 
 
 .74237 
 
 .74217 
 
 .74198 
 
 .74178 
 
 .74159 
 
 .74139 
 
 .74120 
 
 .74100 
 
 .74080 
 
 .74061 
 
 .74041 
 
 .74022 
 
 .74002 
 
 .73983 
 
 .73963 
 
 .73944 
 
 .73924 
 
 .73904 
 
 .73885 
 
 .73865 
 
 .73846 
 
 .73826 
 
 .73806 
 
 .73787 
 
 .73767 
 
 .73747 
 
 .73728 
 
 .73708 
 
 .73688 
 
 .73669 
 
 .73649 
 
 .73629 
 
 .73610 
 
 .73590 
 
 .73570 
 
 .73551 
 
 .73531 
 
 .73511 
 
 .73491 
 
 .73472 
 
 .73452 
 
 .73432 
 
 .73413 
 
 .73393 
 
 .73373 
 
 .73353 
 
 .73333 
 
 .73314 
 
 .73294 
 
 .73274 
 
 .73254 
 
 .73234 
 
 .73215 
 
 .73195 
 
 .73175 
 
 .73155 
 
 .73135 
 
 .68200 
 
 .68221 
 
 .68242 
 
 .68264 
 
 .68285 
 
 .68306 
 
 .68327 
 
 .68349 
 
 .68370 
 
 .68391 
 
 .68412 
 
 .68434 
 
 .68455 
 
 .68476 
 
 .68497 
 
 .68518 
 
 .68539 
 
 .68561 
 
 .68582 
 
 .68603 
 
 .68624 
 
 .68645 
 
 .68666 
 
 .68688 
 
 .68709 
 
 .68730 
 
 .68751 
 
 .68772 
 
 .68793 
 
 .68814 
 
 .68835 
 
 .68857 
 
 .68878 
 
 .68899 
 
 .68920 
 
 .68941 
 
 .68962 
 
 .68983 
 
 .69004 
 
 .69025 
 
 .69046 
 
 .69067 
 
 .69088 
 
 .69109 
 
 .69130 
 
 .69151 
 
 .69172 
 
 .69193 
 
 .69214 
 
 .69235 
 
 .69256 
 
 .69277 
 
 .69298 
 
 .69319 
 
 .69340 
 
 .69361 
 
 .69382 
 
 .69403 
 
 .69424 
 
 .69445 
 
 .69466 
 
 .73135 
 
 .73116 
 
 .73096 
 
 .73076 
 
 .73056 
 
 .73036 
 
 .73016 
 
 .72996 
 
 .72976 
 
 .72957 
 
 .72937 
 
 .72917 
 
 .72897 
 
 .72877 
 
 .72857 
 
 .72837 
 
 .72817 
 
 .72797 
 
 .72777 
 
 .72757 
 
 .72737 
 
 .72717 
 
 .72697 
 
 .72677 
 
 .72657 
 
 .72637 
 
 .72617 
 
 .72597 
 
 .72577 
 
 .72557 
 
 .72537 
 
 .72517 
 
 .72497 
 
 .72477 
 
 .72457 
 
 .72437 
 
 .72417 
 
 .72397 
 
 .72377 
 
 .72357 
 
 .72337 
 
 .72317 
 
 .72297 
 
 .72277 
 
 .72257 
 
 .72236 
 
 .72216 
 
 .72196 
 
 .72176 
 
 .72156 
 
 .72136 
 
 .72116 
 
 .72095 
 
 .72075 
 
 .72055 
 
 .72035 
 
 .72015 
 
 .71995 
 
 .71974 
 
 .71954 
 
 .71934 
 
 .69466 
 
 .69487 
 
 .69508 
 
 .69529 
 
 .69549 
 
 .69570 
 
 .69591 
 
 .69612 
 
 .69633 
 
 .69654 
 
 .69675 
 
 .69696 
 
 .69717 
 
 .69737 
 
 .69758 
 
 .69779 
 
 .69800 
 
 .69821 
 
 .69842 
 
 .69862 
 
 .69883 
 
 .69904 
 
 .69925 
 
 .69946 
 
 .69966 
 
 .69987 
 
 .70008 
 
 .70029 
 
 .70049 
 
 .70070 
 
 .70091 
 
 .70112 
 
 .70132 
 
 .70153 
 
 .70174 
 
 .70195 
 
 .70215 
 
 .70236 
 
 .70257 
 
 .70277 
 
 .70298 
 
 .70319 
 
 .70339 
 
 .70360 
 
 .70381 
 
 .70401 
 
 .70422 
 
 .70443 
 
 .70463 
 
 .70484 
 
 .70505 
 
 .70525 
 
 .70546 
 
 .70567 
 
 .70587 
 
 .70608 
 
 .70628 
 
 .70649 
 
 .70670 
 
 .70690 
 
 .70711 
 
 .71934 
 
 .71914 
 
 .71894 
 
 .71873 
 
 .71853 
 
 .71833 
 
 .71813 
 
 .71792 
 
 .71772 
 
 .71752 
 
 .71732 
 
 .71711 
 
 .71691 
 
 .71671 
 
 .71650 
 
 .71630 
 
 .71610 
 
 .71590 
 
 .71569 
 
 .71549 
 
 .71529 
 
 .71508 
 
 .71488 
 
 .71468 
 
 .71447 
 
 .71427 
 
 .71407 
 
 .71386 
 
 .71366 
 
 .71345 
 
 .71325 
 
 .71305 
 
 .71284 
 
 .71264 
 
 .71243 
 
 .71223 
 
 .71203 
 
 .71182 
 
 .71162 
 
 .71141 
 
 .71121 
 
 .71100 
 
 .71080 
 
 .71059 
 
 .71039 
 
 .71019 
 
 .70998 
 
 .70978 
 
 .70957 
 
 .70937 
 
 .70916 
 
 .70896 
 
 .70875 
 
 .70855 
 
 .70834 
 
 .70813 
 
 .70793 
 
 .70772 
 
 .70752 
 
 .70731 
 
 .70711 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 T 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 / 
 
 Cosine 
 
 s Sine 
 
 1 
 
 Cosine 
 
 } Sine 
 
 Cosine 
 
 : Sine 
 
 Cosine 
 
 ; Sine 
 
 Cosine 
 
 ! Sine 
 
 r 
 
 49° 
 
 48° 
 
 47° 
 
 46° 
 
 45° 
 
464 NATURAL TANGENTS AND COTANGENTS. 
 
 
 0 ° 
 
 1 ° 
 
 2 ° 
 
 3 ° 
 
 4 ° 
 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 0 
 
 .00000 
 
 Infin. 
 
 .01746 
 
 57.2900 
 
 .03492 
 
 28.6363 
 
 .05241 
 
 19.0811 
 
 .06993 
 
 14.3007 
 
 60 
 
 1 
 
 .00029 
 
 3437.75 
 
 .01775 
 
 56.3506 
 
 .03521 
 
 28.3994 
 
 .05270 
 
 18.9755 
 
 .07022 
 
 14.2411 
 
 59 
 
 2 
 
 .00058 
 
 1718.87 
 
 .01804 
 
 55.4415 
 
 .03550 
 
 28.1664 
 
 .05299 
 
 18.8711 
 
 .07051 
 
 14.1821 
 
 58 
 
 3 
 
 .00087 
 
 1145.92 
 
 .01833 
 
 54.5613 
 
 .03579 
 
 27.9372 
 
 .05328 
 
 18.7678 
 
 .07080 
 
 14.1235 
 
 57 
 
 4 
 
 .00116 
 
 859.436 
 
 .01862 
 
 53.7086 
 
 .03609 
 
 27.7117 
 
 .05357 
 
 18.6656 
 
 .07110 
 
 14.0655 
 
 56 
 
 5 
 
 .00145 
 
 687.549 
 
 .01891 
 
 52.8821 
 
 .03638 
 
 27.4899 
 
 .05387 
 
 18.5645 
 
 .07139 
 
 14.0079 
 
 55 
 
 6 
 
 .00175 
 
 572.957 
 
 .01920 
 
 52.0807 
 
 .03667 
 
 27.2715 
 
 .05416 
 
 18.4645 
 
 .07168 
 
 13.9507 
 
 54 
 
 7 
 
 .00204 
 
 491.106 
 
 .01949 
 
 51.3032 
 
 .03696 
 
 27.0566 
 
 .05445 
 
 18.3655 
 
 .07197 
 
 13.8940 
 
 53 
 
 8 
 
 .00233 
 
 429.718 
 
 .01978 
 
 50.5485 
 
 .03725 
 
 26.8450 
 
 .05474 
 
 18.2677 
 
 .07227 
 
 13.8378 
 
 52 
 
 9 
 
 .00262 
 
 381.971 
 
 .02007 
 
 49.8157 
 
 .03754 
 
 26.6367 
 
 .05503 
 
 18.1708 
 
 .07256 
 
 13.7821 
 
 51 
 
 10 
 
 .00291 
 
 343.774 
 
 .02036 
 
 49.1039 
 
 .03783 
 
 26.4316 
 
 .05533 
 
 18.0750 
 
 .07285 
 
 13.7267 
 
 50 
 
 11 
 
 .00320 
 
 312.521 
 
 .02066 
 
 48.4121 
 
 .03812 
 
 26.2296 
 
 .05562 
 
 17.9802 
 
 .07314 
 
 13.6719 
 
 49 
 
 12 
 
 .00349 
 
 286.478 
 
 .02095 
 
 47.7395 
 
 .03842 
 
 26.0307 
 
 .05591 
 
 17.8863 
 
 .07344 
 
 13.6174 
 
 48 
 
 13 
 
 .00378 
 
 264.441 
 
 .02124 
 
 47.0853 
 
 .03871 
 
 25.8348 
 
 .05620 
 
 17.7934 
 
 .07373 
 
 13.5634 
 
 47 
 
 14 
 
 .00407 
 
 245.552 
 
 .02153 
 
 46.4489 
 
 .03900 
 
 25.6418 
 
 .05649 
 
 17.7015 
 
 .07402 
 
 13.5098 
 
 46 
 
 15 
 
 .00436 
 
 229.182 
 
 .02182 
 
 45.8294 
 
 .03929 
 
 25.4517 
 
 .05678 
 
 17.6106 
 
 .07431 
 
 13.4566 
 
 45 
 
 16 
 
 .00465 
 
 214.858 
 
 .02211 
 
 45.2261 
 
 .03958 
 
 25.2644 
 
 .05708 
 
 17.5205 
 
 .07461 
 
 13.4039 
 
 44 
 
 17 
 
 .00495 
 
 202.219 
 
 .02240 
 
 44.6386 
 
 .03987 
 
 25.0798 
 
 .05737 
 
 17.4314 
 
 .07490 
 
 13.3515 
 
 43 
 
 18 
 
 .00524 
 
 190.984 
 
 .02269 
 
 44.0661 
 
 .04016 
 
 24.8978 
 
 .05766 
 
 17.3432 
 
 .07519 
 
 13.2996 
 
 42 
 
 19 
 
 .00553 
 
 180.932 
 
 .02298 
 
 43 5081 
 
 .04046 
 
 24.7185 
 
 .05795 
 
 17.2558 
 
 .07548 
 
 13.2480 
 
 41 
 
 20 
 
 .00582 
 
 171.885 
 
 .02328 
 
 42.9641 
 
 .04075 
 
 24.5418 
 
 .05824 
 
 17.1693 
 
 .07578 
 
 13.1969 
 
 40 
 
 21 
 
 .00611 
 
 163.700 
 
 .02357 
 
 42.4335 
 
 .04104 
 
 24.3675 
 
 .05854 
 
 17.0837 
 
 .07607 
 
 13.1461 
 
 39 
 
 22 
 
 .00640 
 
 156.259 
 
 .02386 
 
 41.9158 
 
 .04133 
 
 24.1957 
 
 .05883 
 
 16.9990 
 
 .07636 
 
 13.0958 
 
 38 
 
 23 
 
 .00669 
 
 149.465 
 
 .02415 
 
 41.4106 
 
 .04162 
 
 24.0263 
 
 .05912 
 
 16.9150 
 
 .07665 
 
 13.0458 
 
 37 
 
 24 
 
 .00698 
 
 143.237 
 
 .02444 
 
 40.9174 
 
 .04191 
 
 23.8593 
 
 .05941 
 
 16.8319 
 
 .07695 
 
 12.9962 
 
 36 
 
 25 
 
 .00727 
 
 137.507 
 
 .02473 
 
 40.4358 
 
 .04220 
 
 23.6945 
 
 .05970 
 
 16.7496 
 
 .07724 
 
 12.9469 
 
 35 
 
 26 
 
 .00756 
 
 132.219 
 
 .02502 
 
 39.9655 
 
 .04250 
 
 23.5321 
 
 .05999 
 
 16.6681 
 
 .07753 
 
 12.8981 
 
 34 
 
 27 
 
 .00785 
 
 127.321 
 
 .02531 
 
 39.5059 
 
 .04279 
 
 23.3718 
 
 .06029 
 
 16.5874 
 
 .07782 
 
 12.8496 
 
 33 
 
 28 
 
 .00815 
 
 122.774 
 
 .02560 
 
 39.0568 
 
 .04308 
 
 23.2137 
 
 .06058 
 
 16.5075 
 
 .07812 
 
 12.8014 
 
 32 
 
 29 
 
 .00844 
 
 118.540 
 
 .02589 
 
 38.6177 
 
 .04337 
 
 23.0577 
 
 .06087 
 
 16.4283 
 
 .07841 
 
 12.7536 
 
 31 
 
 30 
 
 .00873 
 
 114.589 
 
 .02619 
 
 38.1885 
 
 .04366 
 
 22.9038 
 
 .06116 
 
 16.3499 
 
 .07870 
 
 12.7062 
 
 30 
 
 31 
 
 .00902 
 
 110.892 
 
 .02648 
 
 37.7686 
 
 .04395 
 
 22.7519 
 
 .06145 
 
 16.2722 
 
 .07899 
 
 12.6591 
 
 29 
 
 32 
 
 .00931 
 
 107.426 
 
 .02677 
 
 37.3579 
 
 .04424 
 
 22.6020 
 
 .06175 
 
 16.1952 
 
 .07929 
 
 12.6124 
 
 28 
 
 33 
 
 .00960 
 
 104.171 
 
 .02706 
 
 36.9560 
 
 .04454 
 
 22.4541 
 
 .06204 
 
 16.1190 
 
 .07958 
 
 12.5660 
 
 27 
 
 34 
 
 .00989 
 
 101.107 
 
 .02735 
 
 36.5627 
 
 .04483 
 
 22.3081 
 
 .06233 
 
 16.0435 
 
 .07987 
 
 12.5199 
 
 26 
 
 35 
 
 .01018 
 
 98.2179 
 
 .02764 
 
 36.1776 
 
 .04512 
 
 22.1640 
 
 .06262 
 
 15.9687 
 
 .08017 
 
 12.4742 
 
 25 
 
 36 
 
 .01047 
 
 95.4895 
 
 .02793 
 
 35.8006 
 
 .04541 
 
 22.0217 
 
 .06291 
 
 15.8945 
 
 .08046 
 
 12.4288 
 
 24 
 
 37 
 
 .01076 
 
 92.9085 
 
 .02822 
 
 35.4313 
 
 .04570 
 
 21.8813 
 
 .06321 
 
 15.8211 
 
 .08075 
 
 12.3838 
 
 23 
 
 38 
 
 .01105 
 
 90.4633 
 
 .02851 
 
 35.0695 
 
 .04599 
 
 21.7426 
 
 .06350 
 
 15.7483 
 
 .08104 
 
 12.3390 
 
 22 
 
 39 
 
 .01135 
 
 88.1438 
 
 .02881 
 
 34.7151 
 
 .04628 
 
 21.6056 
 
 .06379 
 
 15.6762 
 
 .08134 
 
 12.2946 
 
 21 
 
 40 
 
 .01164 
 
 85.9398 
 
 .02910 
 
 34.3678 
 
 .04658 
 
 21.4704 
 
 .06408 
 
 15.6048 
 
 .08163 
 
 12.2505 
 
 20 
 
 41 
 
 .01193 
 
 83.8435 
 
 .02939 
 
 34.0273 
 
 .04687 
 
 21.3369 
 
 .06437 
 
 15.5340 
 
 .08192 
 
 12.2067 
 
 19 
 
 42 
 
 .01222 
 
 81.8470 
 
 .02968 
 
 33.6935 
 
 .04716 
 
 21,2049 
 
 .06467 
 
 15.4638 
 
 .08221 
 
 12.1632 
 
 18 
 
 43 
 
 .01251 
 
 79.9434 
 
 .02997 
 
 33.3662 
 
 .04745 
 
 21.0747 
 
 .06496 
 
 15.3943 
 
 .08251 
 
 12.1201 
 
 17 
 
 44 
 
 .01280 
 
 78.1263 
 
 .03026 
 
 33.0452 
 
 .04774 
 
 20.9460 
 
 .06525 
 
 15.3254 
 
 .08280 
 
 12.0772 
 
 16 
 
 45 
 
 .01309 
 
 76.3900 
 
 .03055 
 
 32.7303 
 
 .04803 
 
 20.8188 
 
 .06554 
 
 15.2571 
 
 .08309 
 
 12.0346 
 
 15 
 
 46 
 
 .01338 
 
 74.7292 
 
 .03084 
 
 32.4213 
 
 .04833 
 
 20.6932 
 
 .06584 
 
 15.1893 
 
 .08339 
 
 11.9923 
 
 14 
 
 47 
 
 .01367 
 
 73.1390 
 
 .03114 
 
 32.1181 
 
 .04862 
 
 20 5691 
 
 .06613 
 
 15.1222 
 
 .08368 
 
 11.9504 
 
 13 
 
 48 
 
 .01396 
 
 71.6151 
 
 .03143 
 
 31.8205 
 
 .04891 
 
 20.4465 
 
 .06642 
 
 15.0557 
 
 .08397 
 
 11.9087 
 
 12 
 
 49 
 
 .01425 
 
 70.1533 
 
 .03172 
 
 31.5284 
 
 .04920 
 
 20.3253 
 
 .06671 
 
 14.9898 
 
 .08427 
 
 11.8673 
 
 11 
 
 50 
 
 .01455 
 
 68.7501 
 
 .03201 
 
 31.2416 
 
 .04949 
 
 20.2056 
 
 .06700 
 
 14.9244 
 
 .08456 
 
 11.8262 
 
 10 
 
 51 
 
 .01484 
 
 67.4019 
 
 .03230 
 
 30.9599 
 
 .04978 
 
 20.0872 
 
 .06730 
 
 14.8596 
 
 .08485 
 
 11.7853 
 
 9 
 
 52 
 
 .01513 
 
 66.1055 
 
 .03259 
 
 30.6833 
 
 .05007 
 
 19.9702 
 
 .06759 
 
 14.7954 
 
 .08514 
 
 11.7448 
 
 8 
 
 53 
 
 .01542 
 
 64.8580 
 
 .03288 
 
 30.4116 
 
 .05037 
 
 19.8546 
 
 .06788 
 
 14.7317 
 
 .08544 
 
 11.7045 
 
 7 
 
 54 
 
 .01571 
 
 63.6567 
 
 .03317 
 
 30.1446 
 
 .05066 
 
 19.7403 
 
 .06817 
 
 14.6685 
 
 .08573 
 
 11.6645 
 
 6 
 
 55 
 
 .01600 
 
 62.4992 
 
 .03346 
 
 29.8823 
 
 .05095 
 
 19.6273 
 
 .06847 
 
 14.6059 
 
 .08602 
 
 11.6248 
 
 5 
 
 56 
 
 .01629 
 
 61.3829 
 
 .03376 
 
 29.6245 
 
 .05124 
 
 19.5156 
 
 .06876 
 
 14.5438 
 
 .08632 
 
 11.5853 
 
 4 
 
 57 
 
 .01658 
 
 60.3058 
 
 .03405 
 
 29.3711 
 
 .05153 
 
 19.4051 
 
 .06905 
 
 14.4823 
 
 .08661 
 
 11.5461 
 
 3 
 
 58 
 
 .01687 
 
 59.2659 
 
 .03434 
 
 29.1220 
 
 .05182 
 
 19.2959 
 
 .06934 
 
 14.4212 
 
 .08690 
 
 11.5072 
 
 2 
 
 59 
 
 .01716 
 
 58.2612 
 
 .03463 
 
 28.8771 
 
 .05212 
 
 19.1879 
 
 .06963 
 
 14.3607 
 
 .08720 
 
 11.4685 
 
 1 
 
 60 
 
 .01746 
 
 57.2900 
 
 .03492 
 
 28.6363 
 
 .05241 
 
 19.0811 
 
 .06993 
 
 14.3007 
 
 .08749 
 
 11.4301 
 
 0 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 1 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 
 89 ° 
 
 88 ° 
 
 87 ° 
 
 86 ° 
 
 85 ° 
 
 / 
 
NATURAL TANGENTS AND COTANGENTS. 
 
 465 
 
 
 5 ° 
 
 6 ° 
 
 7 ° 
 
 8 ° 
 
 9 ° 
 
 / 
 
 f 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .08749 
 
 .08778 
 
 .08807 
 
 .08837 
 
 .08866 
 
 .08895 
 
 .08925 
 
 .08954 
 
 .08983 
 
 .09013 
 
 .09042 
 
 .09071 
 
 .09101 
 
 .09130 
 
 .09159 
 
 .09189 
 
 .09218 
 
 .09247 
 
 .09277 
 
 .09306 
 
 .09335 
 
 .09365 
 
 .09394 
 
 .09423 
 
 .09453 
 
 .09482 
 
 .09511 
 
 .09541 
 
 .09570 
 
 .09600 
 
 .09629 
 
 .09658 
 
 .09688 
 
 .09717 
 
 .09746 
 
 .09776 
 
 .09805 
 
 .09834 
 
 .09864 
 
 .09893 
 
 .09923 
 
 .09952 
 
 .09981 
 
 .10011 
 
 .10040 
 
 .10069 
 
 .10099 
 
 .10128 
 
 .10158 
 
 .10187 
 
 .10216 
 
 .10246 
 
 .10275 
 
 .10305 
 
 .10334 
 
 .10363 
 
 .10393 
 
 .10422 
 
 .10452 
 
 .10481 
 
 .10510 
 
 11.4301 
 11.3919 
 11.3540 
 11.3163 
 11.2789 
 11.2417 
 11 .2048 
 11.1681 
 11.1316 
 11.0954 
 11.0594 
 
 11.0237 
 
 10.9882 
 
 10.9529 
 
 10.9178 
 
 10.8829 
 
 10.8483 
 
 10.8139 
 
 10.7797 
 
 10.7457 
 
 10.7119 
 
 10.6783 
 
 10.6450 
 
 10.6118 
 
 10.5789 
 
 10.5462 
 
 10.5136 
 
 10.4813 
 
 10.4491 
 
 10.4172 
 
 10.3854 
 
 10.3538 
 
 10.3224 
 
 10.2913 
 
 10.2602 
 
 10.2294 
 
 10.1988 
 
 10.1683 
 
 10.1381 
 
 10.1080 
 
 10.0780 
 
 10.0483 
 
 10.0187 
 
 9.98931 
 
 9.96007 
 
 9.93101 
 
 9.90211 
 
 9.87338 
 
 9.84482 
 
 9.81641 
 
 9.78817 
 
 9.76009 
 
 9.73217 
 
 9.70441 
 
 9.67680 
 
 9.64935 
 
 9.62205 
 
 9.59490 
 
 9.56791 
 
 9.54106 
 
 9.51436 
 
 .10510 
 
 .10540 
 
 .10569 
 
 .10599 
 
 .10628 
 
 .10657 
 
 .10687 
 
 .10716 
 
 .10746 
 
 .10775 
 
 .10805 
 
 .10834 
 
 .10863 
 
 .10893 
 
 .10922 
 
 .10952 
 
 .10981 
 
 .11011 
 
 .11040 
 
 .11070 
 
 .11099 
 
 .11128 
 
 .11158 
 
 .11187 
 
 .11217 
 
 .11246 
 
 .11276 
 
 .11305 
 
 .11335 
 
 .11364 
 
 .11394 
 
 .11423 
 
 .11452 
 
 .11482 
 
 .11511 
 
 .11541 
 
 .11570 
 
 .11600 
 
 .11629 
 
 .11659 
 
 .11688 
 
 .11718 
 
 .11747 
 
 .11777 
 
 .11806 
 
 .11836 
 
 .11865 
 
 .11895 
 
 .11924 
 
 .11954 
 
 .11983 
 
 .12013 
 
 .12042 
 
 .12072 
 
 .12101 
 
 .12131 
 
 .12160 
 
 .12190 
 
 .12219 
 
 .12249 
 
 .12278 
 
 9.51436 
 
 9.48781 
 
 9.46141 
 
 9.43515 
 
 9.40904 
 
 9.38307 
 
 9.35724 
 
 9.33155 
 
 9.30599 
 
 9.28058 
 
 9.25530 
 
 9.23016 
 
 9.20516 
 
 9.18028 
 
 9.15554 
 
 9.13093 
 
 9.10646 
 
 9.08211 
 
 9.05789 
 
 9.03379 
 
 9.00983 
 
 8.98598 
 
 8.96227 
 
 8.93867 
 
 8.91520 
 
 8.89185 
 
 8.86862 
 
 8.84551 
 
 8.82252 
 
 8.79964 
 
 8.77689 
 
 8.75425 
 
 8.73172 
 
 8.70931 
 
 8.68701 
 
 8.66482 
 
 8.64275 
 
 8.62078 
 
 8.59893 
 
 8.57718 
 
 8.55555 
 
 8.53402 
 
 8.51259 
 
 8.49128 
 
 8.47007 
 
 8.44896 
 
 8.42795 
 
 8.40705 
 
 8.38625 
 
 8.36555 
 
 8.34496 
 
 8.32446 
 
 8.30406 
 
 8.28376 
 
 8.26355 
 
 8.24345 
 
 8.22344 
 
 8.20352 
 
 8.1837C 
 
 8.16398 
 
 8.1443c 
 
 .12278 
 
 .12308 
 
 .12338 
 
 .12367 
 
 .12397 
 
 .12426 
 
 .12456 
 
 .12485 
 
 .12515 
 
 .12544 
 
 .12574 
 
 .12603 
 
 .12633 
 
 .12662 
 
 .12692 
 
 .12722 
 
 .12751 
 
 .12781 
 
 .12810 
 
 .12840 
 
 .12869 
 
 .12899 
 
 .12929 
 
 .12958 
 
 .12988 
 
 .13017 
 
 .13047 
 
 .13076 
 
 .13106 
 
 .13136 
 
 .13165 
 
 .13195 
 
 .13224 
 
 .13254 
 
 .13284 
 
 .13313 
 
 .13343 
 
 .13372 
 
 .13402 
 
 .13432 
 
 .13461 
 
 .13491 
 
 .13521 
 
 .13550 
 
 .13580 
 
 .13609 
 
 .13639 
 
 .13669 
 
 .13698 
 
 .13728 
 
 .13758 
 
 .13787 
 .13817 
 .13846 
 i .13876 
 i .13906 
 : .13935 
 
 ! .13965 
 
 1 .13995 
 
 ! .14024 
 
 i .ll054 
 
 8.14435 
 
 8.12481 
 
 8.10536 
 
 8.08600 
 
 8.06674 
 
 8.04756 
 
 8.02848 
 
 8.00948 
 
 7.99058 
 
 7.97176 
 
 7.95302 
 
 7.93438 
 
 7.91582 
 
 7.89734 
 
 7.87895 
 
 7.86064 
 
 7.84242 
 
 7.82428 
 
 7.80622 
 
 7.78825 
 
 7.77035 
 
 7.75254 
 
 7.73480 
 
 7.71715 
 
 7.69957 
 
 7.68208 
 
 7.66466 
 
 7.64732 
 
 7.63005 
 
 7.61287 
 
 7.59575 
 
 7.57872 
 
 7.56176 
 
 7.54487 
 
 7.52806 
 
 7.51132 
 
 7.49465 
 
 7.47806 
 
 7.46154 
 
 7.44509 
 
 7.42871 
 
 7.41240 
 
 7.39616 
 
 7.37999 
 
 7.36389 
 
 7.34786 
 
 7.33190 
 
 7.31600 
 
 7.30018 
 
 7.28442 
 
 7.26873 
 
 7.25310 
 
 7.23754 
 
 7.22204 
 
 7.20661 
 
 7.19125 
 
 7.17594 
 
 7.16071 
 
 7.14553 
 
 7.13042 
 
 7.11537 
 
 .14054 
 
 .14084 
 
 .14113 
 
 .14143 
 
 .14173 
 
 .14202 
 
 .14232 
 
 .14262 
 
 .14291 
 
 .14321 
 
 .14351 
 
 .14381 
 
 .14410 
 
 .14440 
 
 .14470 
 
 .14499 
 
 .14529 
 
 .14559 
 
 .14588 
 
 .14618 
 
 .14648 
 
 .14678 
 
 .14707 
 
 .14737 
 
 .14767 
 
 .14796 
 
 .14826 
 
 .14856 
 
 .14886 
 
 .14915 
 
 .14945 
 
 .14975 
 
 .15005 
 
 .15034 
 
 .15064 
 
 .15094 
 
 .15124 
 
 .15153 
 
 .15183 
 
 .15213 
 
 .15243 
 
 .15272 
 
 .15302 
 
 .15332 
 
 .15362 
 
 .15391 
 
 .15421 
 
 .15451 
 
 .15481 
 
 .15511 
 
 .15540 
 
 .15570 
 
 .15600 
 
 .15630 
 
 .15660 
 
 .15689 
 
 .15719 
 
 .15749 
 
 .15779 
 
 .15809 
 
 .15838 
 
 7.11537 
 
 7.10038 
 
 7.08546 
 
 7.07059 
 
 7.05579 
 
 7.04105 
 
 7.02637 
 
 7.01174 
 
 6.99718 
 
 6.98268 
 
 6.96823 
 
 6.95385 
 
 6.93952 
 
 6.92525 
 
 6.91104 
 
 6.89688 
 
 6.88278 
 
 6.86874 
 
 6.85475 
 
 6.84082 
 
 6.82694 
 
 6.81812 
 
 6.79936 
 
 6.78564 
 
 6.77199 
 
 6.75838 
 
 6.74483 
 
 6.73133 
 
 6.71789 
 
 6.70450 
 
 6.69116 
 
 6.67787 
 
 6.66463 
 
 6.65144 
 
 6.63831 
 
 6.62523 
 
 6.61219 
 
 6.59921 
 
 6.58627 
 
 6.57339 
 
 6.56055 
 
 6.54777 
 
 6.53503 
 
 6.52234 
 
 6.50970 
 
 6.49710 
 
 6.48456 
 
 6.47206 
 
 6.45961 
 
 6.44720 
 
 6.43484 
 
 6.42253 
 
 6.41026 
 
 6.39804 
 
 6.38587 
 
 6.37374 
 
 6.36165 
 
 6.34961 
 
 6.33761 
 
 6.32566 
 
 6.31375 
 
 .15838 
 
 .15868 
 
 .15898 
 
 .15928 
 
 .15958 
 
 .15988 
 
 .16017 
 
 .16047 
 
 .16077 
 
 .16107 
 
 .16137 
 
 .16167 
 
 .16196 
 
 .16226 
 
 .16256 
 
 .16286 
 
 .16316 
 
 .16346 
 
 .16376 
 
 .16405 
 
 .16435 
 
 .16465 
 
 .16495 
 
 .16525 
 
 .16555 
 
 .16585 
 
 .16615 
 
 .16645 
 
 .16674 
 
 .16704 
 
 .16734 
 
 .16764 
 
 .16794 
 
 .16824 
 
 .16854 
 
 .16884 
 
 .16914 
 
 .16944 
 
 .16974 
 
 .17004 
 
 .17033 
 
 .17063 
 
 .17093 
 
 .17123 
 
 .17153 
 
 .17183 
 
 .17213 
 
 .17243 
 
 .17273 
 
 .17303 
 
 .17333 
 
 .17363 
 
 .17393 
 
 .17423 
 
 .17453 
 
 .17483 
 
 .17513 
 
 .17543 
 
 .17573 
 
 .17603 
 
 .17633 
 
 6.31375 
 
 6.30189 
 
 6.29007 
 
 6.27829 
 
 6.26655 
 
 6.25486 
 
 6.24321 
 
 6.23160 
 
 6.22003 
 
 6.20851 
 
 6.19703 
 
 6.18559 
 
 6.17419 
 
 6.16283 
 
 6.15151 
 
 6.14023 
 
 6.12899 
 
 6.11779 
 
 6.10664 
 
 6.09552 
 
 6.08444 
 
 6.07340 
 
 6.06240 
 
 6.05143 
 
 6.04051 
 
 6.02962 
 
 6.01878 
 
 6.00797 
 
 5.99720 
 
 5.98646 
 
 5.97576 
 
 5.96510 
 
 5.95448 
 
 5.94390 
 
 5.93335 
 
 5.92283 
 
 5.91236 
 
 5.90191 
 
 5.89151 
 
 5.88114 
 
 5.87080 
 
 5.86051 
 
 5.85024 
 
 5.84001 
 
 5.82982 
 
 5.81966 
 
 5.80953 
 
 5.79944 
 
 5.78938 
 
 5.77936 
 
 5.76937 
 
 5.75941 
 
 5.74949 
 
 5.73960 
 
 5.72974 
 
 5.71992 
 
 5.71013 
 
 5.70037 
 
 5.69064 
 
 5.68094 
 
 5.67128 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 ' Tang 
 
 
 84 ° 
 
 83 ° 
 
 82 ° 
 
 81 ° 
 
 80 ° 
 
 
486 NATURAL TANGENTS AND COTANGENTS. 
 
 
 10° 
 
 11° 
 
 12° 
 
 18 ° 
 
 14 ° 
 
 f 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 0 
 
 .17633 
 
 5.67128 
 
 .19438 
 
 5.14455 
 
 .21256 
 
 4.70463 
 
 .23087 
 
 4.33148 
 
 .24933 
 
 4.01078 
 
 60 
 
 1 
 
 .17663 
 
 5.66165 
 
 .19468 
 
 5.13658 
 
 .21286 
 
 4.69791 
 
 .23117 
 
 4.32573 
 
 .24964 
 
 4.00582 
 
 59 
 
 2 
 
 .17693 
 
 5.65205 
 
 .19498 
 
 5.12862 
 
 .21316 
 
 4.69121 
 
 .23148 
 
 4.32001 
 
 .24995 
 
 4.00086 
 
 58 
 
 3 
 
 .17723 
 
 5.64248 
 
 .19529 
 
 5.12069 
 
 .21347 
 
 4.68452 
 
 .23179 
 
 4.31430 
 
 .25026 
 
 3.99592 
 
 57 
 
 4 
 
 .17753 
 
 5.63295 
 
 .19559 
 
 5.11279 
 
 .21377 
 
 4.67786 
 
 .23209 
 
 4.30860 
 
 .25056 
 
 3.99099 
 
 56 
 
 5 
 
 .17783 
 
 5.62344 
 
 .19589 
 
 5.10490 
 
 .21408 
 
 4.67121 
 
 .23240 
 
 4.30291 
 
 .25087 
 
 3.98607 
 
 55 
 
 6 
 
 .17813 
 
 5.61397 
 
 .19619 
 
 5.09704 
 
 .21438 
 
 4.66458 
 
 .23271 
 
 4.29724 
 
 .25118 
 
 3.98117 
 
 54 
 
 7 
 
 .17843 
 
 5.60452 
 
 .19649 
 
 5.08921 
 
 .21469 
 
 4.65797 
 
 .23301 
 
 4.29159 
 
 .25149 
 
 3.97627 
 
 53 
 
 8 
 
 .17873 
 
 5.59511 
 
 .19680 
 
 5.08139 
 
 .21499 
 
 4.65138 
 
 .23332 
 
 4.28595 
 
 .25180 
 
 3.97139 
 
 52 
 
 9 
 
 .17903 
 
 5.58573 
 
 .19710 
 
 5.07360 
 
 .21529 
 
 4.64480 
 
 .23363 
 
 4.28032 
 
 .25211 
 
 3.96651 
 
 51 
 
 10 
 
 .17933 
 
 5.57638 
 
 .19740 
 
 5.06584 
 
 .21560 
 
 4.63825 
 
 .23393 
 
 4.27471 
 
 .25242 
 
 3.96165 
 
 50 
 
 11 
 
 .17963 
 
 5.56706 
 
 .19770 
 
 5.05809 
 
 .21590 
 
 4.63171 
 
 .23424 
 
 4.26911 
 
 .25273 
 
 3.95680 
 
 49 
 
 12 
 
 .17993 
 
 5.55777 
 
 .19801 
 
 5.05037 
 
 .21621 
 
 4.62518 
 
 .23455 
 
 4.26352 
 
 .25304 
 
 3.95196 
 
 48 
 
 13 
 
 .18023 
 
 5.54851 
 
 .19831 
 
 5.04267 
 
 .21651 
 
 4.61868 
 
 .23485 
 
 4.25795 
 
 .25335 
 
 3.94713 
 
 47 
 
 14 
 
 .18053 
 
 5.53927 
 
 .19861 
 
 5.03499 
 
 .21682 
 
 4.61219 
 
 .23516 
 
 4.25239 
 
 .25366 
 
 3.94232 
 
 46 
 
 
 .18083 
 
 5.53007 
 
 .19891 
 
 5.02734 
 
 .21712 
 
 4.60572 
 
 .23547 
 
 4.24685 
 
 .25397 
 
 3.93751 
 
 45 
 
 16 
 
 .18113 
 
 5.52090 
 
 .19921 
 
 5.01971 
 
 .21743 
 
 4.59927 
 
 .23578 
 
 4.24132 
 
 .25428 
 
 3.93271 
 
 44 
 
 17 
 
 .18143 
 
 5.51176 
 
 .19952 
 
 5.01210 
 
 .21773 
 
 4.59283 
 
 .23608 
 
 4.23580 
 
 .25459 
 
 3.92793 
 
 43 
 
 18 
 
 .18173 
 
 5.50264 
 
 .19982 
 
 5.00451 
 
 .21804 
 
 4.58641 
 
 .23639 
 
 4.23030 
 
 .25490 
 
 3.92316 
 
 42 
 
 19 
 
 .18203 
 
 5.49356 
 
 .20012 
 
 4.99695 
 
 .21834 
 
 4.58001 
 
 .23670 
 
 4.22481 
 
 .25521 
 
 3.91839 
 
 41 
 
 20 
 
 .18233 
 
 5.48451 
 
 .20042 
 
 4.98940 
 
 .21864 
 
 4.57363 
 
 .23700 
 
 4.21933 
 
 .25552 
 
 3.91364 
 
 40 
 
 21 
 
 .18263 
 
 5.47548 
 
 .20073 
 
 4.98188 
 
 .21895 
 
 4.56726 
 
 .23731 
 
 4.21387 
 
 .25583 
 
 3.90890 
 
 39 
 
 22 
 
 .18293 
 
 5.46648 
 
 .20103 
 
 4.97438 
 
 .21925 
 
 4.56091 
 
 .23762 
 
 4.20842 
 
 .25614 
 
 3.90417 
 
 38 
 
 23 
 
 .18323 
 
 5.45751 
 
 .20133 
 
 4.96690 
 
 .21956 
 
 4.55458 
 
 .23793 
 
 4.20298 
 
 .25645 
 
 3.89945 
 
 37 
 
 24 
 
 .18353 
 
 5.44857 
 
 .20164 
 
 4.95945 
 
 .21986 
 
 4.54826 
 
 .23823 
 
 4.19756 
 
 .25676 
 
 3.89474 
 
 36 
 
 25 
 
 .18384 
 
 5.43966 
 
 .20194 
 
 4.95201 
 
 .22017 
 
 4.54196 
 
 .23854 
 
 4.19215 
 
 .25707 
 
 3.89004 
 
 35 
 
 26 
 
 .18414 
 
 5.43077 
 
 .20224 
 
 4.94460 
 
 .22047 
 
 4.53568 
 
 .23885 
 
 4.18675 
 
 .25738 
 
 3.88536 
 
 34 
 
 27 
 
 .18444 
 
 5.42192 
 
 .20254 
 
 4.93721 
 
 .22078 
 
 4.52941 
 
 .23916 
 
 4.18137 
 
 .25769 
 
 3.88068 
 
 33 
 
 28 
 
 .18474 
 
 5.41309 
 
 .20285 
 
 4.92984 
 
 .22108 
 
 4.52316 
 
 .23946 
 
 4.17600 
 
 .25800 
 
 3.87601 
 
 32 
 
 29 
 
 .18504 
 
 5.40429 
 
 .20315 
 
 4.92249 
 
 .22139 
 
 4.51693 
 
 .23977 
 
 4.17064 
 
 .25831 
 
 3.87136 
 
 31 
 
 30 
 
 .18534 
 
 5.39552 
 
 .20345 
 
 4.91516 
 
 .22169 
 
 4.51071 
 
 .24008 
 
 4.16530 
 
 .25862 
 
 3.86671 
 
 30 
 
 31 
 
 .18564 
 
 5.38677 
 
 .20376 
 
 4.90785 
 
 .22200 
 
 4.50451 
 
 .24039 
 
 4.15997 
 
 .25893 
 
 3.86208 
 
 29 
 
 32 
 
 .18594 
 
 5.37805 
 
 .20406 
 
 4.90056 
 
 .22231 
 
 4.49832 
 
 .24069 
 
 4.15465 
 
 .25924 
 
 3.85745 
 
 28 
 
 33 
 
 .18624 
 
 5.36936 
 
 .20436 
 
 4.89330 
 
 .22261 
 
 4.49215 
 
 .24100 
 
 4.1'4934 
 
 .25955 
 
 3.85284 
 
 27 
 
 34 
 
 .18654 
 
 5.36070 
 
 .20466 
 
 4.88605 
 
 .22292 
 
 4.48600 
 
 .24131 
 
 4.14405 
 
 .25986 
 
 3.84824 
 
 26 
 
 35 
 
 .18684 
 
 5.35206 
 
 .20497 
 
 4.87882 
 
 .22322 
 
 4.47986 
 
 .24162 
 
 4.13877 
 
 .26017 
 
 3.84364 
 
 25 
 
 36 
 
 .18714 
 
 5.34345 
 
 .20527 
 
 4.87162 
 
 .22353 
 
 4.47374 
 
 .24193 
 
 4.13350 
 
 .26048 
 
 3.83906 
 
 24 
 
 87 
 
 .18745 
 
 5.33487 
 
 .20557 
 
 4.86444 
 
 .22383 
 
 4.46764 
 
 .24223 
 
 4.12825 
 
 .26079 
 
 3.83449 
 
 23 
 
 38 
 
 .18775 
 
 5.32631 
 
 .20588 
 
 4.85727 
 
 .22414 
 
 4.46155 
 
 .24254 
 
 4.12301 
 
 .26110 
 
 3.82992 
 
 22 
 
 39 
 
 .18805 
 
 5.31778 
 
 .20618 
 
 4.85013 
 
 .22444 
 
 4.45548 
 
 .24285 
 
 4.11778 
 
 .26141 
 
 3.82537 
 
 21 
 
 40 
 
 .18835 
 
 5.30928 
 
 .20648 
 
 4.84300 
 
 .22475 
 
 4.44942 
 
 .24316 
 
 4.11256 
 
 .26172 
 
 3.82083 
 
 20 
 
 41 
 
 .18865 
 
 5.30080 
 
 .20679 
 
 4.83590 
 
 .22505 
 
 4.44338 
 
 .24347 
 
 4.10736 
 
 .26203 
 
 3.81630 
 
 19 
 
 42 
 
 .18895 
 
 5.29235 
 
 .20709 
 
 4.82882 
 
 .22536 
 
 4.43735 
 
 .24377 
 
 4.10216 
 
 .26235 
 
 3.81177 
 
 18 
 
 43 
 
 .18925 
 
 5.28393 
 
 .20739 
 
 4.82175 
 
 .22567 
 
 4.43134 
 
 .24408 
 
 4.09699 
 
 .26266 
 
 3.80726 
 
 17 
 
 44 
 
 .18955 
 
 5.27553 
 
 .20770 
 
 4.81471 
 
 .22597 
 
 4.42534 
 
 .24439 
 
 4.09182 
 
 .26297 
 
 3.80276 
 
 16 
 
 45 
 
 .18986 
 
 5.26715 
 
 .20800 
 
 4.80769 
 
 .22628 
 
 4.41936 
 
 .24470 
 
 4.08666 
 
 .26328 
 
 3.79827 
 
 15 
 
 46 
 
 .19016 
 
 5.25880 
 
 .20830 
 
 4.80068 
 
 .22658 
 
 4.41340 
 
 .24501 
 
 4.08152 
 
 .26359 
 
 3.79378 
 
 14 
 
 47 
 
 .19046 
 
 5.25048 
 
 .20861 
 
 4.79370 
 
 .22689 
 
 4.40745 
 
 .24532 
 
 4.07639 
 
 .26390 
 
 3.78931 
 
 13 
 
 48 
 
 .19076 
 
 5.24218 
 
 .20891 
 
 4.78673 
 
 .22719 
 
 4.40152 
 
 .24562 
 
 4.07127 
 
 .26421 
 
 3.78485 
 
 12 
 
 49 
 
 .19106 
 
 5.23391 
 
 .20921 
 
 4.77978 
 
 .22750 
 
 4.39560 
 
 .24593 
 
 4 06616 
 
 .26452 
 
 3.78040 
 
 11 
 
 50 
 
 .19136 
 
 5.22566 
 
 .20952 
 
 4.77286 
 
 .22781 
 
 4.38969 
 
 .24624 
 
 4.06107 
 
 .26483 
 
 3.77595 
 
 10 
 
 51 
 
 .19166 
 
 5.21744 
 
 .20982 
 
 4.76595 
 
 .22811 
 
 4.38381 
 
 .24655 
 
 4.05599 
 
 .26515 
 
 3.77152 
 
 9 
 
 52 
 
 .19197 
 
 5.20925 
 
 .21013 
 
 4.75906 
 
 .22842 
 
 4.37793 
 
 .24686 
 
 4.05092 
 
 .26546 
 
 3.76709 
 
 8 
 
 53 
 
 .19227 
 
 5.20107 
 
 .21043 
 
 4.75219 
 
 .22872 
 
 4.37207 
 
 .24717 
 
 4.04586 
 
 .26577 
 
 3.76268 
 
 7 
 
 54 
 
 .19257 
 
 5.19293 
 
 .21073 
 
 4.74534 
 
 .22903 
 
 4.36623 
 
 .24747 
 
 4.04081 
 
 .26608 
 
 3.75828 
 
 6 
 
 55 
 
 .19287 
 
 5.18480 
 
 .21104 
 
 4.73851 
 
 .22934 
 
 4.36040 
 
 .24778 
 
 4.03578 
 
 .26639 
 
 3.75388 
 
 5 
 
 56 
 
 .19317 
 
 5.17671 
 
 .21134 
 
 4.73170 
 
 .22964 
 
 4.35459 
 
 .24809 
 
 4.03076 
 
 .26670 
 
 3.74950 
 
 4 
 
 57 
 
 .19347 
 
 5.16863 
 
 .21164 
 
 4.72490 
 
 .22995 
 
 4.34879 
 
 .24840 
 
 4.02574 
 
 .26701 
 
 3,74512 
 
 3 
 
 58 
 
 .19378 
 
 5.16058 
 
 .21195 
 
 4.71813 
 
 .23026 
 
 4.34300 
 
 .24871 
 
 4.02074 
 
 .26733 
 
 3.74075 
 
 2 
 
 59 
 
 .19408 
 
 5.15256 
 
 .21225 
 
 4.71137 
 
 .23056 
 
 4.3J5723 
 
 .24902 
 
 4.01576 
 
 .26764 
 
 3.73640 
 
 1 
 
 60 
 
 .19438 
 
 5.14455 
 
 .21256 
 
 4.70463 
 
 .23087 
 
 4.33148 
 
 .24933 
 
 4.01078 
 
 .26795 
 
 3.73205 
 
 0 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 f 
 
 79 ° 
 
 78 ° 
 
 77 ° 
 
 76 ° 
 
 75 ° 
 
 
NATURAL TANGENTS AND COTANGENTS. 
 
 467 
 
 
 15° 
 
 16° 
 
 17° 
 
 18° 
 
 19° 
 
 
 / - 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 o 
 
 .26795 
 
 3.73205 
 
 .28675 
 
 3.48741 
 
 .30573 
 
 3.27085 
 
 .32492 
 
 3.07768 
 
 .34433 
 
 2.90421 
 
 60 
 
 
 26826 
 
 3.72771 
 
 .28706 
 
 3.48359 
 
 .30605 
 
 3.26745 
 
 .32524 
 
 3.07464 
 
 .34465 
 
 2.90147 
 
 59 
 
 
 26857 
 
 3.72338 
 
 .28738 
 
 3.47977 
 
 .30637 
 
 3.26406 
 
 .32556 
 
 3.07160 
 
 .34498 
 
 2.89873 
 
 58 
 
 3 
 
 .26888 
 
 3.71907 
 
 .28769 
 
 3.47596 
 
 .30669 
 
 3.26067 
 
 .32588 
 
 3.06857 
 
 .34530 
 
 2.89600 
 
 57 
 
 
 26920 
 
 3.71476 
 
 .28800 
 
 3.47216 
 
 .30700 
 
 3.25729 
 
 .32621 
 
 3.06554 
 
 .34563 
 
 2.89327 
 
 56 
 
 
 .26951 
 
 3.71046 
 
 .28832 
 
 3.46837 
 
 .30732 
 
 3.25392 
 
 .32653 
 
 3.06252 
 
 .34596 
 
 2.89055 
 
 55 
 
 
 .26982 
 
 3.70616 
 
 .28864 
 
 3.46458 
 
 .30764 
 
 3.25055 
 
 .32685 
 
 3.05950 
 
 .34628 
 
 2.88783 
 
 54 
 
 7 
 
 .27013 
 
 3.70188 
 
 .28895 
 
 3.46080 
 
 .30796 
 
 3.24719 
 
 .32717 
 
 3.05649 
 
 .34661 
 
 2.88511 
 
 53 
 
 
 .27044 
 
 3.69761 
 
 .28927 
 
 3.45703 
 
 .30828 
 
 3.24383 
 
 .32749 
 
 3.05349 
 
 .34693 
 
 2.88240 
 
 52 
 
 9 
 
 .27076 
 
 3.69335 
 
 .28958 
 
 3.45327 
 
 .30860 
 
 3.24049 
 
 .32782 
 
 3.05049 
 
 .34726 
 
 2.87970 
 
 51 
 
 10 
 
 .27107 
 
 3.68909 
 
 .28990 
 
 3.44951 
 
 .30891 
 
 3.23714 
 
 .32814 
 
 3.04749 
 
 .34758 
 
 2.87700 
 
 50 
 
 
 27138 
 
 3.68485 
 
 .29021 
 
 3.44576 
 
 .30923 
 
 3.23381 
 
 .32846 
 
 3.04450 
 
 .34791 
 
 2.87430 
 
 49 
 
 12 
 
 .27169 
 
 3.68061 
 
 .29053 
 
 3.44202 
 
 .30955 
 
 3.23048 
 
 .32878 
 
 3.04152 
 
 .34824 
 
 2.87161 
 
 48 
 
 13 
 
 .27201 
 
 3.67638 
 
 .29084 
 
 3.43829 
 
 .30987 
 
 3.22715 
 
 .32911 
 
 3.03854 
 
 .34856 
 
 2.86892 
 
 47 
 
 14 
 
 .27232 
 
 3.67217 
 
 .29116 
 
 3.43456 
 
 .31019 
 
 3.22384 
 
 .32943 
 
 3.03556 
 
 .34889 
 
 2.86624 
 
 46 
 
 15 
 
 27263 
 
 3.66796 
 
 .29147 
 
 3.43084 
 
 .31051 
 
 3.22053 
 
 .32975 
 
 3.03260 
 
 .34922 
 
 2 .86356 
 
 45 
 
 16 
 
 27294 
 
 3.66376 
 
 .29179 
 
 3.42713 
 
 .31083 
 
 3.21722 
 
 .33007 
 
 3.02963 
 
 .34954 
 
 2.86089 
 
 44 
 
 17 
 
 27326 
 
 3.65957 
 
 .29210 
 
 3.42343 
 
 .31115 
 
 3.21392 
 
 .33040 
 
 3.02667 
 
 .34987 
 
 2.85822 
 
 43 
 
 18 
 
 .27357 
 
 3.65538 
 
 .29242 
 
 3.41973 
 
 .31147 
 
 3.21063 
 
 .33072 
 
 3.02372 
 
 .35020 
 
 2.85555 
 
 42 
 
 19 
 
 .27388 
 
 3.65121 
 
 .29274 
 
 3.41604 
 
 .31178 
 
 3.20734 
 
 .33104 
 
 3.02077 
 
 .35052 
 
 2.85289 
 
 41 
 
 20 
 
 .27419 
 
 3.64705 
 
 .29305 
 
 3.41236 
 
 .31210 
 
 3.20406 
 
 .33136 
 
 3.01783 
 
 .35085 
 
 2.85023 
 
 40 
 
 21 
 
 27451 
 
 3.64289 
 
 .29337 
 
 3.40869 
 
 .31242 
 
 3.20079 
 
 .33169 
 
 3.01489 
 
 .35118 
 
 2.84758 
 
 39 
 
 
 27482 
 
 3.63874 
 
 .29368 
 
 3.40502 
 
 .31274 
 
 3.19752 
 
 .33201 
 
 3.01196 
 
 .35150 
 
 2.84494 
 
 38 
 
 23 
 
 27513 
 
 3.63461 
 
 .29400 
 
 3.40136 
 
 .31306 
 
 3.19426 
 
 .33233 
 
 3.00903 
 
 .35183 
 
 2.84229 
 
 37 
 
 
 27545 
 
 3.63048 
 
 .29432 
 
 3.39771 
 
 .31338 
 
 3.19100 
 
 .33266 
 
 3.00611 
 
 .35216 
 
 2.83965 
 
 36 
 
 25 
 
 27576 
 
 3.62636 
 
 .29463 
 
 3.39406 
 
 .31370 
 
 3.18775 
 
 .33298 
 
 3.00319 
 
 .35248 
 
 2.83702 
 
 35 
 
 
 27607 
 
 3.62224 
 
 .29495 
 
 3.39042 
 
 .31402 
 
 3 18451 
 
 .33330 
 
 3.00028 
 
 .35281 
 
 2.83439 
 
 34 
 
 
 27638 
 
 3.61814 
 
 .29526 
 
 3.38679 
 
 .31434 
 
 3.18127 
 
 .33363 
 
 2.99738 
 
 .35314 
 
 2.83176 
 
 33 
 
 28 
 
 .27670 
 
 3.61405 
 
 .29558 
 
 3.38317 
 
 .31466 
 
 3.17804 
 
 .33395 
 
 2.99447 
 
 .35346 
 
 2.82914 
 
 32 
 
 29 
 
 .27701 
 
 3.60996 
 
 .29590 
 
 3.37955 
 
 .31498 
 
 3.17481 
 
 .33427 
 
 2.99158 
 
 .35379 
 
 2.82653 
 
 31 
 
 30 
 
 .27732 
 
 3.60588 
 
 .29621 
 
 3.37594 
 
 .31530 
 
 3.17159 
 
 .33460 
 
 2.98868 
 
 .35412 
 
 2.82391 
 
 30 
 
 31 
 
 .27764 
 
 3.60181 
 
 .29653 
 
 3.37234 
 
 .31562 
 
 3.16838 
 
 .33492 
 
 2.98580 
 
 .35445 
 
 2.82130 
 
 29 
 
 32 
 
 .27795 
 
 3.59775 
 
 .29685 
 
 3.36875 
 
 .31594 
 
 3.16517 
 
 .33524 
 
 2.98292 
 
 .35477 
 
 2.81870 
 
 28 
 
 33 
 
 '.27826 
 
 3.59370 
 
 .29716 
 
 3.36516 
 
 .31626 
 
 3.16197 
 
 .33557 
 
 2.98004 
 
 .35510 
 
 2.81610 
 
 27 
 
 34 
 
 .27858 
 
 3.58966 
 
 .29748 
 
 3.36158 
 
 .31658 
 
 3.15877 
 
 .33589 
 
 2.97717 
 
 .35543* 
 
 2.81350 
 
 26 
 
 35 
 
 .27889 
 
 3.58562 
 
 .29780 
 
 3.35800 
 
 .31690 
 
 3.15558 
 
 .33621 
 
 2.97430 
 
 .35576 
 
 2.81091 
 
 25 
 
 36 
 
 .27921 
 
 3.58160 
 
 .29811 
 
 3.35443 
 
 .31722 
 
 3.15240 
 
 .33654 
 
 2.97144 
 
 .35608 
 
 2.80833 
 
 24 
 
 37 
 
 .27952 
 
 3.57758 
 
 .29843 
 
 3.35087 
 
 .31754 
 
 3.14922 
 
 .33686 
 
 2.96858 
 
 .35641 
 
 2.80574 
 
 23 
 
 38 
 
 .27983 
 
 3.57357 
 
 .29875 
 
 3.34732 
 
 .31786 
 
 3.14605 
 
 .33718 
 
 2.96573 
 
 <35674 
 
 2.80316 
 
 22 
 
 39 
 
 .28015 
 
 3.56957 
 
 .29906 
 
 3.34377 
 
 .31818 
 
 3.14288 
 
 .33751 
 
 2.96288 
 
 .35707 
 
 2.80059 
 
 21 
 
 40 
 
 !28046 
 
 3.56557 
 
 .29938 
 
 3.34023 
 
 .31850 
 
 3.13972 
 
 .33783 
 
 2.96004 
 
 .35740 
 
 2.79802 
 
 20 
 
 41 
 
 .28077 
 
 3.56159 
 
 .29970 
 
 3.33670 
 
 .31882 
 
 3.13656 
 
 .33816 
 
 2.95721 
 
 .35772 
 
 2.79545 
 
 19 
 
 42 
 
 .28109 
 
 3.55761 
 
 .30001 
 
 3.33317 
 
 .31914 
 
 3.13341 
 
 .33848 
 
 2.95437 
 
 .35805 
 
 2.79289 
 
 18 
 
 43 
 
 .28140 
 
 3.55364 
 
 .30033 
 
 3.32965 
 
 .31946 
 
 3.13027 
 
 .33881 
 
 2.95155 
 
 .35838 
 
 2.79033 
 
 17 
 
 44 
 
 .28172 
 
 3.54968 
 
 .30065 
 
 3.32614 
 
 .31978 
 
 3.12713 
 
 .33913 
 
 2.94872 
 
 .35871 
 
 2.78778 
 
 16 
 
 45 
 
 .28203 
 
 3.54573 
 
 .30097 
 
 3.32264 
 
 .32010 
 
 3.12400 
 
 .33945 
 
 2.94591 
 
 .35904 
 
 2 .78523 
 
 15 
 
 46 
 
 .28234 
 
 3.54179 
 
 .30128 
 
 3.31914 
 
 .32042 
 
 3.12087 
 
 .33978 
 
 2.94309 
 
 .35937 
 
 2.78269 
 
 14 
 
 47 
 
 .28266 
 
 3.53785 
 
 .30160 
 
 3.31565 
 
 .32074 
 
 3.11775 
 
 .34010 
 
 2.94028 
 
 .35969 
 
 2.78014 
 
 13 
 
 48 
 
 .28297 
 
 3.53393 
 
 .30192 
 
 3.31216 
 
 .32106 
 
 3.11464 
 
 .34043 
 
 2.93748 
 
 .36002 
 
 2.77761 
 
 12 
 
 49 
 
 .28329 
 
 3.53001 
 
 .30224 
 
 3.30868 
 
 .32139 
 
 3.11153 
 
 .34075 
 
 2.93468 
 
 .36035 
 
 2 .77507 
 
 11 
 
 50 
 
 .28360 
 
 3.52609 
 
 .30255 
 
 3.30521 
 
 .32171 
 
 3.10842 
 
 .34108 
 
 2.93189 
 
 .36068 
 
 2.77254 
 
 10 
 
 51 
 
 .28391 
 
 3.52219 
 
 .30287 
 
 3.30174 
 
 .32203 
 
 3.10532 
 
 .34140 
 
 2.92910 
 
 .36101 
 
 2.77002 
 
 9 
 
 52 
 
 .28423 
 
 3.51829 
 
 .30319 
 
 3.29829 
 
 .32235 
 
 3.10223 
 
 .34173 
 
 2.92632 
 
 .36134 
 
 2.76750 
 
 8 
 
 53 
 
 .28454 
 
 3.51441 
 
 .30351 
 
 3.29483 
 
 .32267 
 
 3.09914 
 
 .34205 
 
 2.92354 
 
 .36167 
 
 2.76498 
 
 7 
 
 54 
 
 .28486 
 
 3.51053 
 
 .30382 
 
 3.29139 
 
 .32299 
 
 3.09606 
 
 .34238 
 
 2.92076 
 
 .36199 
 
 2.76247 
 
 6 
 
 55 
 
 .28517 
 
 3.50666 
 
 .30414 
 
 3.28795 
 
 .32331 
 
 3.09298 
 
 .34270 
 
 2.91799 
 
 .36232 
 
 2.75996 
 
 5 
 
 56 
 
 .28549 
 
 3.50279 
 
 .30446 
 
 3.28452 
 
 .32363 
 
 3.08991 
 
 .34303 
 
 2.91523 
 
 .36265 
 
 2.75746 
 
 4 
 
 57 
 
 .28580 
 
 3.49894 
 
 .30478 
 
 3.28109 
 
 .32396 
 
 3.08685 
 
 .34335 
 
 2.91246 
 
 .36298 
 
 2.75496 
 
 3 
 
 58 
 
 .28612 
 
 3.49509 
 
 .30509 
 
 3.27767 
 
 .32428 
 
 3.08379 
 
 .34368 
 
 2.90971 
 
 .36331 
 
 2.75246 
 
 2 
 
 59 
 
 ’*28643 
 
 3.49125 
 
 .30541 
 
 3.27426 
 
 .32460 
 
 3.08073 
 
 .34400 
 
 2.90696 
 
 .36364 
 
 2.74997 
 
 1 
 
 60 
 
 .28675 
 
 3.48741 
 
 .30573 
 
 3.27085 
 
 .32492 
 
 3.07768 
 
 .34433 
 
 2.90421 
 
 .36397 
 
 2.74748 
 
 0 
 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 ; Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 j Tang 
 
 f 
 
 t 
 
 74° 
 
 73° 
 
 72° 
 
 71° 
 
 70° 
 
 
468 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 
 20° 
 
 21° 
 
 22° 
 
 23° 
 
 24° 
 
 / 
 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 0 
 
 .36397 
 
 2.74748 
 
 .38386 
 
 2.60509 
 
 .40403 
 
 2.47509 
 
 .42447 
 
 2.35585 
 
 .44523 
 
 2.24604 
 
 60 
 
 1 
 
 .36430 
 
 2.74499 
 
 .38420 
 
 2.60283 
 
 .40436 
 
 2.47302 
 
 .42482 
 
 2.35395 
 
 .44558 
 
 2.24428 
 
 59 
 
 2 
 
 .36463 
 
 2.74251 
 
 .38453 
 
 2.60057 
 
 .40470 
 
 2.47095 
 
 .42516 
 
 2.35205 
 
 .44593 
 
 2.24252 
 
 58 
 
 3 
 
 .36496 
 
 2.74004 
 
 .38487 
 
 2.59831 
 
 .40504 
 
 2.46888 
 
 .42551 
 
 2.35015 
 
 .44627 
 
 2.24077 
 
 57 
 
 4 
 
 .36529 
 
 2.73756 
 
 .38520 
 
 2.59606 
 
 .40538 
 
 2.46682 
 
 .42585 
 
 2.34825 
 
 .44662 
 
 2.23902 
 
 56 
 
 5 
 
 .36562 
 
 2.73509 
 
 .38553 
 
 2.59381 
 
 .40572 
 
 2.46476 
 
 .42619 
 
 2.34636 
 
 .44697 
 
 2.23727 
 
 55 
 
 6 
 
 .36595 
 
 2.73263 
 
 .38587 
 
 2.59156 
 
 .40606 
 
 2.46270 
 
 .42654 
 
 2.34447 
 
 .44732 
 
 2.23553 
 
 54 
 
 7 
 
 .36628 
 
 2.73017 
 
 .38620 
 
 2.58932 
 
 .40640 
 
 2.46065 
 
 .42688 
 
 2.34258 
 
 .44767 
 
 2.23378 
 
 53 
 
 8 
 
 .36661 
 
 2.72771 
 
 .38654 
 
 2.58708 
 
 .40674 
 
 2.45860 
 
 .42722 
 
 2.34069 
 
 .44802 
 
 2.23204 
 
 52 
 
 9 
 
 .36694 
 
 2.72526 
 
 .38687 
 
 2.58484 
 
 .40707 
 
 2.45655 
 
 .42757 
 
 2.33881 
 
 .44837 
 
 2.23030 
 
 51 
 
 10 
 
 .36727 
 
 2.72281 
 
 .38721 
 
 2.58261 
 
 .40741 
 
 2.45451 
 
 .42791 
 
 2.33693 
 
 .44872 
 
 2.22857 
 
 50 
 
 11 
 
 .36760 
 
 2.72036 
 
 .38754 
 
 2.58038 
 
 .40775 
 
 2.45246 
 
 .42826 
 
 2.33505 
 
 .44907 
 
 2.22683 
 
 49 
 
 12 
 
 .36793 
 
 2.71792 
 
 .38787 
 
 2.57815 
 
 .40809 
 
 2.45043 
 
 .42860 
 
 2.33317 
 
 .44942 
 
 2.22510 
 
 48 
 
 13 
 
 .36826 
 
 2.71548 
 
 .38821 
 
 2.57593 
 
 .40843 
 
 2.44839 
 
 .42894 
 
 2.33130 
 
 .44977 
 
 2.22337 
 
 47 
 
 14 
 
 .36859 
 
 2.71305 
 
 .38854 
 
 2.57371 
 
 .40877 
 
 2.44636 
 
 .42929 
 
 2.32943 
 
 .45012 
 
 2.22164 
 
 46 
 
 15 
 
 .36892 
 
 2.71062 
 
 .38888 
 
 2.57150 
 
 .40911 
 
 2.44433 
 
 .42963 
 
 2.32756 
 
 .45047 
 
 2.21992 
 
 45 
 
 16 
 
 .36925 
 
 2.70819 
 
 .38921 
 
 2.56928 
 
 .40945 
 
 2.44230 
 
 .42998 
 
 2.32570 
 
 .45082 
 
 2.21819 
 
 44 
 
 17 
 
 .36958 
 
 2.70577 
 
 .38955 
 
 2.56707 
 
 .40979 
 
 2.44027 
 
 .43032 
 
 2.32383 
 
 .45117 
 
 2.21647 
 
 43 
 
 18 
 
 .36991 
 
 2.70335 
 
 .38988 
 
 2.56487 
 
 .41013 
 
 2.43825 
 
 .43067 
 
 2.32197 
 
 .45152 
 
 2.21475 
 
 42 
 
 19 
 
 .37024 
 
 2.70094 
 
 .39022 
 
 2.56266 
 
 .41047 
 
 2.43623 
 
 .43101 
 
 2.32012 
 
 .45187 
 
 2.21304 
 
 41 
 
 20 
 
 .37057 
 
 2.69853 
 
 .39055 
 
 2.56046 
 
 .41081 
 
 2.43422 
 
 .43136 
 
 2.31826 
 
 .45222 
 
 2.21132 
 
 40 
 
 21 
 
 .37090 
 
 2.69612 
 
 .39089 
 
 2.55827 
 
 .41115 
 
 2.43220 
 
 .43170 
 
 2.31641 
 
 .45257 
 
 2.20961 
 
 39 
 
 22 
 
 .37123 
 
 2.69371 
 
 .39122 
 
 2.55608 
 
 .41149 
 
 2.43019 
 
 43205 
 
 2.31456 
 
 .45292 
 
 2.20790 
 
 38 
 
 23 
 
 .37157 
 
 2.69131 
 
 .39156 
 
 2.55389 
 
 .41183 
 
 2.42819 
 
 .43230 
 
 2.31271 
 
 .45327 
 
 2.20619 
 
 37 
 
 24 
 
 .37190 
 
 2.68892 
 
 .39190 
 
 2.55170 
 
 .41217 
 
 2.42618 
 
 .43274 
 
 2.31086 
 
 .45362 
 
 2.20449 
 
 36 
 
 25 
 
 .37223 
 
 2.68653 
 
 .39223 
 
 2.54952 
 
 .41251 
 
 2 42418 
 
 .43308 
 
 2.30902 
 
 .45397 
 
 2.20278 
 
 35 
 
 26 
 
 .37256 
 
 2.68414 
 
 .39257 
 
 2.54734 
 
 .41285 
 
 2.42218 
 
 .43343 
 
 2.30718 
 
 .45432 
 
 2.20108 
 
 34 
 
 27 
 
 .37289 
 
 2.68175 
 
 .39290 
 
 2.54516 
 
 .41319 
 
 2.42019 
 
 .43378 
 
 2.30534 
 
 .45467 
 
 2.19938 
 
 33 
 
 28 
 
 .37322 
 
 2.67937 
 
 .39324 
 
 2.54299 
 
 .41353 
 
 2.41819 
 
 .43412 
 
 2.30351 
 
 .45502 
 
 2.19769 
 
 32 
 
 29 
 
 .37355 
 
 2.67700 
 
 .39357 
 
 2.54082 
 
 .41387 
 
 2.41620 
 
 .43447 
 
 2.30167 
 
 .45538 
 
 2.19599 
 
 31 
 
 30 
 
 .37388 
 
 2.67462 
 
 .39391 
 
 2.53865 
 
 .41421 
 
 2.41421 
 
 .43481 
 
 2.29984 
 
 .45573 
 
 2.19430 
 
 30 
 
 31 
 
 .37422 
 
 2.67225 
 
 .39425 
 
 2.53648 
 
 .41455 
 
 2.41223 
 
 .43516 
 
 2.29801 
 
 .45608 
 
 2.19261 
 
 29 
 
 32 
 
 .37455 
 
 2.66989 
 
 .39458 
 
 2.53432 
 
 .41490 
 
 2.41025 
 
 .43550 
 
 2.29619 
 
 .45643 
 
 2.19092 
 
 28 
 
 33 
 
 .37488 
 
 2.66752 
 
 .39492 
 
 2.53217 
 
 .41524 
 
 2.40827 
 
 .43585 
 
 2.29437 
 
 .45678 
 
 2.18923 
 
 27 
 
 34 
 
 .37521 
 
 •2.66516 
 
 .39526 
 
 2.53001 
 
 .41558 
 
 2.40629 
 
 .43620 
 
 2.29254 
 
 .45713 
 
 2.18755 
 
 26 
 
 35 
 
 .37554 
 
 2.66281 
 
 .39559 
 
 2.52786 
 
 .41592 
 
 2.40432 
 
 .43654 
 
 2.29073 
 
 .45748 
 
 2.18587 
 
 25 
 
 36 
 
 .37588 
 
 2.66046 
 
 .39593 
 
 2.52571 
 
 .41626 
 
 2.40235 
 
 .43689 
 
 2.28891 
 
 .45784 
 
 2.18419 
 
 24 
 
 37 
 
 .37621 
 
 2.65811 
 
 .39626 
 
 2.52357 
 
 .41660 
 
 2.40038 
 
 .43724 
 
 2.28710 
 
 .45819 
 
 2.18251 
 
 23 
 
 38 
 
 .37654 
 
 2.65576 
 
 .39660 
 
 2.52142 
 
 .41694 
 
 2.39841 
 
 .43758 
 
 2.28528 
 
 .45854 
 
 2.18084 
 
 22 
 
 39 
 
 .37687 
 
 2.65342 
 
 .39694 
 
 2.51929 
 
 .41728 
 
 2.39645 
 
 .43793 
 
 2.28348 
 
 .45889 
 
 2.17916 
 
 21 
 
 40 
 
 .37720 
 
 2.65109 
 
 .39727 
 
 2.51715 
 
 .41763 
 
 2.39449 
 
 .43828 
 
 2.28167 
 
 .45924 
 
 2.17749 
 
 20 
 
 41 
 
 .37754 
 
 2.64875 
 
 .39761 
 
 2.51502 
 
 .41797 
 
 2.39253 
 
 .43862 
 
 2.27987 
 
 .45960 
 
 2.17582 
 
 19 
 
 42 
 
 .37787 
 
 2.64642 
 
 .39795 
 
 2.51289 
 
 .41831 
 
 2.39058 
 
 .43897 
 
 2.27806 
 
 .45995 
 
 2.17416 
 
 18 
 
 43 
 
 .37820 
 
 2.64410 
 
 .39829 
 
 2.51076 
 
 .41865 
 
 2.38863 
 
 .43932 
 
 2.27626 
 
 .46030 
 
 2.17249 
 
 17 
 
 44 
 
 .37853 
 
 2.64177 
 
 .39862 
 
 2.50864 
 
 .41899 
 
 2.38668 
 
 .43966 
 
 2.27447 
 
 .46065 
 
 2.17083 
 
 16 
 
 45 
 
 .37887 
 
 2.63945 
 
 .39896 
 
 2.50652 
 
 .41933 
 
 2.38473 
 
 .44001 
 
 2.27267 
 
 .46101 
 
 2.16917 
 
 15 
 
 46 
 
 .37920 
 
 2.63714 
 
 .39930 
 
 2.50440 
 
 .41968 
 
 2.38279 
 
 .44036 
 
 2.27088 
 
 .46136 
 
 2.16751 
 
 14 
 
 47 
 
 .37953 
 
 2.63483 
 
 .39963 
 
 2.50229 
 
 .42002 
 
 2.38084 
 
 .44071 
 
 2.26909 
 
 .46171 
 
 2.16585 
 
 13 
 
 48 
 
 .37986 
 
 2.63252 
 
 .39997 
 
 2.50018 
 
 .42036 
 
 2.37891 
 
 .44105 
 
 2.26730 
 
 .46206 
 
 2.16420 
 
 12 
 
 49 
 
 .38020 
 
 2.63021 
 
 .40031 
 
 2.49807 
 
 .42070 
 
 2.37697 
 
 .44140 
 
 2.26552 
 
 .46242 
 
 2.16255 
 
 11 
 
 50 
 
 .38053 
 
 2.62791 
 
 .40065 
 
 2.49597 
 
 .42105 
 
 2.37504 
 
 .44175 
 
 2.26374 
 
 .46277 
 
 2.16090 
 
 10 
 
 51 
 
 .38086 
 
 2.62561 
 
 .40098 
 
 2.49386 
 
 .42139 
 
 2.37311 
 
 .44210 
 
 2.26196 
 
 .46312 
 
 2.15925 
 
 9 
 
 52 
 
 .38120 
 
 2.62332 
 
 .40132 
 
 2.49177 
 
 .42173 
 
 2.37118 
 
 .44.244 
 
 2.26018 
 
 .46348 
 
 2.15760 
 
 8 
 
 53 
 
 .38153 
 
 2.62103 
 
 .40166 
 
 2.48967 
 
 .42207 
 
 2.36925 
 
 .44279 
 
 2.25840 
 
 .46383 
 
 2.15596. 
 
 7 
 
 54 
 
 .38186 
 
 2.61874 
 
 .40200 
 
 2.48758 
 
 .42242 
 
 2.36733 
 
 .44314 
 
 2.25663 
 
 .46418 
 
 2.15432 
 
 6 
 
 55 
 
 .38220 
 
 2.61646 
 
 .40234 
 
 2.48549 
 
 .42276 
 
 2.36541 
 
 .44349 
 
 2.25486 
 
 .46454 
 
 2.15268 
 
 5 
 
 56 
 
 .38253 
 
 2.61418 
 
 .40267 
 
 2.48340 
 
 .42310 
 
 2.36349 
 
 .44384 
 
 2.25309 
 
 .46489 
 
 2.15104 
 
 4 
 
 57 
 
 .38286 
 
 2.61190 
 
 .40301 
 
 2.48132 
 
 .42345 
 
 2.36158 
 
 .44418 
 
 2.25132 
 
 .46525 
 
 2.14940 
 
 3 
 
 58 
 
 .38320 
 
 2.60963 
 
 .40335 
 
 2.47924 
 
 .42379 
 
 2.35967 
 
 .44453 
 
 2.24956 
 
 .46560 
 
 2.14777 
 
 2 
 
 59 
 
 .38353 
 
 2.60736 
 
 .40369 
 
 2.47716 
 
 .42413 
 
 2.35776 
 
 .44488 
 
 2.24780 
 
 .46595 
 
 2.14614 
 
 1 
 
 60 
 
 .38386 
 
 2.60509 
 
 .40403 
 
 2.47509 
 
 .42447 
 
 2.35585 
 
 .44523 
 
 2.24604 
 
 .46631 
 
 2.14451 
 
 0 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotaug 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 f 
 
 69° 
 
 68° 
 
 67° 
 
 66° 
 
 65° 
 
 
NATURAL TANGENTS AND COTANGENTS. 
 
 469 
 
 
 25° 
 
 26° 
 
 27° 
 
 28° 
 
 29° 
 
 t 
 
 t 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 o 
 
 46631 
 
 2.14451 
 
 .48773 
 
 2.05030 
 
 .50953 
 
 1.96261 
 
 .53171 
 
 1.88073 
 
 .55431 
 
 1.80405 
 
 60 
 
 \ 
 
 46666 
 
 2.14288 
 
 .48809 
 
 2.04879 
 
 .50989 
 
 1.96120 
 
 .53208 
 
 1.87941 
 
 .55469 
 
 1.80281 
 
 59 
 
 
 46702 
 
 2.14125 
 
 .48845 
 
 2.04728 
 
 .51026 
 
 1.95979 
 
 .53246 
 
 1.87809 
 
 .55507 
 
 1.80158 
 
 58 
 
 3 
 
 46737 
 
 2.13963 
 
 .48881 
 
 2.04577 
 
 .51063 
 
 1.95838 
 
 .53283 
 
 1.87677 
 
 .55545 
 
 1.80034 
 
 57 
 
 4 
 
 46772 
 
 2.13801 
 
 .48917 
 
 2.04426 
 
 .51099 
 
 1.95698 
 
 .53320 
 
 1.87546 
 
 .55583 
 
 1.79911 
 
 56 
 
 
 46»n« 
 
 2 13639 
 
 .48953 
 
 2.04276 
 
 .51136 
 
 1.95557 
 
 .53358 
 
 1.87415 
 
 .55621 
 
 1.79788 
 
 55 
 
 
 46R43 
 
 2 13477 
 
 .48989 
 
 2.04125 
 
 .51173 
 
 1.95417 
 
 .53395 
 
 1.87283 
 
 .55659 
 
 1.79665 
 
 54 
 
 
 40R79 
 
 2 13316 
 
 .49026 
 
 2.03975 
 
 .51209 
 
 1.95277 
 
 .53432 
 
 1.87152 
 
 .55697 
 
 1.79542 
 
 53 
 
 
 46914 
 
 2 13154 
 
 .49062 
 
 2.03825 
 
 .51246 
 
 1.95137 
 
 .53470 
 
 1.87021 
 
 .55736 
 
 1.79419 
 
 52 
 
 g 
 
 46950 
 
 2.12993 
 
 .49098 
 
 2.03675 
 
 .51283 
 
 1.94997 
 
 .53507 
 
 1.86891 
 
 .55774 
 
 1.79296 
 
 51 
 
 10 
 
 .46985 
 
 2.12832 
 
 .49134 
 
 2.03526 
 
 .51319 
 
 1.94858 
 
 .53545 
 
 1.86760 
 
 .55812 
 
 1.79174 
 
 50 
 
 U 
 
 47021 
 
 2.12671 
 
 .49170 
 
 2.03376 
 
 .51356 
 
 1.94718 
 
 .53582 
 
 1.86630 
 
 .55850 
 
 1.79051 
 
 49 
 
 12 
 
 47056 
 
 2.12511 
 
 .49206 
 
 2.03227 
 
 .51393 
 
 1.94579 
 
 .53620 
 
 1.86499 
 
 .55888 
 
 1.78929 
 
 48 
 
 13 
 
 47092 
 
 2.12350 
 
 .49242 
 
 2.03078 
 
 .51430 
 
 1.94440 
 
 .53657 
 
 1.86369 
 
 .55926 
 
 1.78807 
 
 47 
 
 14 
 
 47128 
 
 2.12190 
 
 .49278 
 
 2.02929 
 
 .51467 
 
 1.94301 
 
 .53694 
 
 1.86239 
 
 .55964 
 
 1.78685 
 
 46 
 
 15 
 
 47163 
 
 2.12030 
 
 .49315 
 
 2.02780 
 
 .51503 
 
 1.94162 
 
 .53732 
 
 1.86109 
 
 .56003 
 
 1.78563 
 
 45 
 
 16 
 
 47199 
 
 2.11871 
 
 .49351 
 
 2.02631 
 
 .51540 
 
 1.94023 
 
 .53769 
 
 1.85979 
 
 .56041 
 
 1.78441 
 
 44 
 
 17 
 
 47234 
 
 2.11711 
 
 .49387 
 
 2.02483 
 
 .51577 
 
 1.93885 
 
 .53807 
 
 1.85850 
 
 .56079 
 
 1.78319 
 
 43 
 
 18 
 
 47270 
 
 2.11552 
 
 .49423 
 
 2.02335 
 
 .51614 
 
 1.93746 
 
 .53844 
 
 1.85720 
 
 .56117 
 
 1.78198 
 
 42 
 
 19 
 
 47305 
 
 2.11392 
 
 .49459 
 
 2.02187 
 
 .51651 
 
 1.93608 
 
 .53882 
 
 1.85591 
 
 .56156 
 
 1.78077 
 
 41 
 
 20 
 
 .47341 
 
 2.11233 
 
 .49495 
 
 2.02039 
 
 .51688 
 
 1.93470 
 
 .53920 
 
 1.85462 
 
 .56194 
 
 1.77955 
 
 40 
 
 21 
 
 47377 
 
 2 11075 
 
 .49532 
 
 2.01891 
 
 .51724 
 
 1.93332 
 
 .53957 
 
 1.85333 
 
 .56232 
 
 1.77834 
 
 39 
 
 22 
 
 .47412 
 
 2.10916 
 
 .49568 
 
 2.01743 
 
 .51761 
 
 1.93195 
 
 .53995 
 
 1.85204. 
 
 .56270 
 
 1.77713 
 
 38 
 
 23 
 
 .47448 
 
 2.10758 
 
 .49604 
 
 2.01596 
 
 .51798 
 
 1.93057 
 
 .54032 
 
 1.85075 
 
 .56309 
 
 1.77592 
 
 37 
 
 24 
 
 .47483 
 
 2.10600 
 
 .49640 
 
 2.01449 
 
 .51835 
 
 1.92920 
 
 .54070 
 
 1.84946 
 
 .56347 
 
 1.77471 
 
 36 
 
 25 
 
 47519 
 
 2.10442 
 
 .49677 
 
 2.01302 
 
 .51872 
 
 1.92782 
 
 .54107 
 
 1.84818 
 
 .56385 
 
 1.77351 
 
 35 
 
 26 
 
 
 2.10284 
 
 .49713 
 
 2.01155 
 
 .51909 
 
 1.92645 
 
 .54145 
 
 1.84689 
 
 .56424 
 
 1.77230 
 
 34 
 
 27 
 
 47590 
 
 2.10126 
 
 .49749 
 
 2.01008 
 
 .51946 
 
 1.92508 
 
 .54183 
 
 1.84561 
 
 .56462 
 
 1.77110 
 
 33 
 
 28 
 
 .47626 
 
 2.09969 
 
 .49786 
 
 2.00862 
 
 .51983 
 
 1.92371 
 
 .54220 
 
 1.84433 
 
 .56501 
 
 1.76990 
 
 32 
 
 29 
 
 .47662 
 
 2.09811 
 
 .49822 
 
 2.00715 
 
 .52020 
 
 1.92235 
 
 .54258 
 
 1.84305 
 
 .56539 
 
 1.76869 
 
 31 
 
 30 
 
 .47698 
 
 2.09654 
 
 .49858 
 
 2.00569 
 
 .52057 
 
 1.92098 
 
 .54296 
 
 1.84177 
 
 .56577 
 
 1.76749 
 
 30 
 
 31 
 
 .47733 
 
 2.09498 
 
 .49894 
 
 2.00423 
 
 .52094 
 
 1.91962 
 
 .54333 
 
 1.84049 
 
 .56616 
 
 1.76629 
 
 29 
 
 32 
 
 .47769 
 
 2.09341 
 
 .49931 
 
 2.00277 
 
 .52131 
 
 1.91826 
 
 .54371 
 
 1.83922 
 
 .56654 
 
 1.76510 
 
 28 
 
 33 
 
 .47805 
 
 2.09184 
 
 .49967 
 
 2.00131 
 
 .52168 
 
 1.91690 
 
 .54409 
 
 1.83794 
 
 .56693 
 
 1.76390 
 
 27 
 
 34 
 
 .47840 
 
 2.09028 
 
 .50004 
 
 1.99986 
 
 .52205 
 
 1.91554 
 
 .54446 
 
 1.83667 
 
 .56731 
 
 1.76271 
 
 26 
 
 35 
 
 .47876 
 
 2.08872 
 
 .50040 
 
 1.99841 
 
 .52242 
 
 1.91418 
 
 .54484 
 
 1.83540 
 
 .56769 
 
 1.76151 
 
 25 
 
 36 
 
 .47912 
 
 2.08716 
 
 .50076 
 
 1.99695 
 
 .52279 
 
 1.91282 
 
 .54522 
 
 1.83413 
 
 .56808 
 
 1.76032 
 
 24 
 
 37 
 
 .47948 
 
 2.08560 
 
 .50113 
 
 1.99550 
 
 .52316 
 
 1.91147 
 
 .54560 
 
 1.83286 
 
 .56846 
 
 1.75913 
 
 23 
 
 38 
 
 .47984 
 
 2.08405 
 
 .50149 
 
 1.99406 
 
 .52353 
 
 1.91012 
 
 .54597 
 
 1.83159 
 
 .56885 
 
 1.75794 
 
 22 
 
 39 
 
 • .48019 
 
 2.08250' 
 
 .50185 
 
 1.99261 
 
 .52390 
 
 1.90876 
 
 .54635 
 
 1.83033 
 
 .56923 
 
 1.75675 
 
 21 
 
 40 
 
 .48055 
 
 2.08094 
 
 .50222 
 
 1.99116 
 
 .52427 
 
 1.90741 
 
 .54673 
 
 1.82906 
 
 .56962 
 
 1.75556 
 
 20 
 
 41 
 
 .48091 
 
 2.07939 
 
 .50258 
 
 1.98972 
 
 .52464 
 
 1.90607 
 
 .54711 
 
 1.82780 
 
 .57000 
 
 1.75437 
 
 19 
 
 42 
 
 .48127 
 
 2.07785 
 
 .50295 
 
 1.98828 
 
 .52501 
 
 1.90472 
 
 .54748 
 
 1.82654 
 
 .57039 
 
 1.75319 
 
 18 
 
 43 
 
 .48163 
 
 2.07630 
 
 .50331 
 
 1.98684 
 
 .52538 
 
 1.90337 
 
 .54786 
 
 1.82528 
 
 .57078 
 
 1.75200 
 
 17 
 
 44 
 
 .48198 
 
 2.07476 
 
 .50368 
 
 1.98540 
 
 .52575 
 
 1.90203 
 
 .54824 
 
 1.82402 
 
 .57116 
 
 1.75082 
 
 16 
 
 *5 
 
 .48234 
 
 2.07321 
 
 .50404 
 
 1.98396 
 
 .52613 
 
 1.90069 
 
 .54862 
 
 1.82276 
 
 .57155 
 
 1.74964 
 
 15 
 
 46 
 
 .48270 
 
 2.07167 
 
 .50441 
 
 1.98253 
 
 .52650 
 
 1.89935 
 
 .54900 
 
 1.82150 
 
 .57193 
 
 1.74846 
 
 14 
 
 47 
 
 .48306 
 
 2.07014 
 
 .50477 
 
 1.98110 
 
 .52687 
 
 1.89801 
 
 .54938 
 
 1.82025 
 
 .57232 
 
 1.74728 
 
 13 
 
 48 
 
 .48342 
 
 2.06860 
 
 .50514 
 
 1.97966 
 
 .52724 
 
 1.89667 
 
 .54975 
 
 1.81899 
 
 .57271 
 
 1.74610 
 
 12 
 
 49 
 
 .48378 
 
 2.06706 
 
 .50550 
 
 1.97823 
 
 .52761 
 
 1.89533 
 
 .55013 
 
 1.81774 
 
 .57309 
 
 1.74492 
 
 11 
 
 50 
 
 .48414 
 
 2.06553 
 
 .50587 
 
 1.97681 
 
 .52798 
 
 1.89400 
 
 .55051 
 
 1.81649 
 
 .57348 
 
 1.74375 
 
 10 
 
 51 
 
 .48450 
 
 2.06400 
 
 .50623 
 
 1.97538 
 
 .52836 
 
 1.89266 
 
 .55089 
 
 1.81524 
 
 .57386 
 
 1.74257 
 
 9 
 
 52 
 
 .48486 
 
 2 06247 
 
 .50660 
 
 1.97395 
 
 .52873 
 
 1.89133 
 
 .55127 
 
 1.81399 
 
 .57425 
 
 1.74140 
 
 8 
 
 53 
 
 .48521 
 
 2.06094 
 
 .50696 
 
 1.97253 
 
 .52910 
 
 1.89000 
 
 .55165 
 
 1.81274 
 
 .57464 
 
 1.74022 
 
 7 
 
 54 
 
 .48557 
 
 2.05942 
 
 .50733 
 
 1.97111 
 
 .52947 
 
 1.88867 
 
 .55203 
 
 1.81150 
 
 .57503 
 
 1.73905 
 
 6 
 
 55 
 
 .48593 
 
 2.05790 
 
 .50769 
 
 1.96969 
 
 .52985 
 
 1.88734 
 
 .55241 
 
 1.81025 
 
 .57541 
 
 1.73788 
 
 5 
 
 56 
 
 .48629 
 
 2.05637 
 
 .50806 
 
 1.96827 
 
 .53022 
 
 1.88602 
 
 .55279 
 
 1.80901 
 
 .57580 
 
 1.73671 
 
 4 
 
 57 
 
 .48665 
 
 2.05485 
 
 .50843 
 
 1.96685 
 
 .53059 
 
 1.88469 
 
 .55317 
 
 1.80777 
 
 .57619 
 
 1.73555 
 
 3 
 
 58 
 
 .48701 
 
 2.05333 
 
 .50879 
 
 1.96544 
 
 .53096 
 
 1.88337 
 
 .55355 
 
 1.80653 
 
 .57657 
 
 1.73438 
 
 2 
 
 59 
 
 .48737 
 
 2.05182 
 
 .50916 
 
 1.96402 
 
 .53134 
 
 1.88205 
 
 .55393 
 
 1.80529 
 
 .57696 
 
 1.73321 
 
 1 
 
 60 
 
 .48773 
 
 2.05030 
 
 .50953 
 
 1.96261 
 
 .53171 
 
 1,88073 
 
 .55431 
 
 1.80405 
 
 .57735 
 
 1.73205 
 
 0 
 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 ; Tang 
 
 
 t 
 
 64° 
 
 I 
 
 63° 
 
 62° 
 
 61° 
 
 60° 
 
 
470 
 
 NATURAL TAX GENTS AND COTANGENTS. 
 
 
 30° 
 
 31° 
 
 32° 
 
 33° 
 
 2A° 
 
 t 
 
 
 Tang 
 
 Cotang 
 
 Tang Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 0 
 
 
 1.73205 
 
 .60086 1.66428 
 
 .62487 
 
 1.60033 
 
 .64941 
 
 1.53986 
 
 .67451 ; 
 
 1.48256 ' 
 
 60 
 
 1 ! 
 
 .57774 
 
 1.73089 
 
 .60126 1.66318 
 
 .62527 
 
 1.59930 
 
 .64982 
 
 1.53888 
 
 .67493 j 
 
 1.48163 
 
 59 
 
 2 
 
 .57813 ( 
 
 1.72973 
 
 .60165 1.66209 
 
 .62568 
 
 1.59826 
 
 .65024 1 
 
 1.53791 
 
 .67536 j 
 
 1.48070 ! 
 
 58 
 
 3 1 
 
 .57851 
 
 1.72857 
 
 .60205 1.66099 
 
 .62608 
 
 1.59723 
 
 .65065 i 
 
 1.53693 
 
 .67578 
 
 1.47977 j 
 
 57 
 
 4 | 
 
 .57890 
 
 1.72741 
 
 .60245 1.65990 
 
 .62649 
 
 1.59620 
 
 .65106 
 
 1.53595 
 
 .67620 i 
 
 1.47885 ; 
 
 56 
 
 
 .57929 
 
 1.72625 
 
 .60284 1.65881 
 
 .62689 
 
 1.59517 
 
 .65148 
 
 1.53497 
 
 .67663 1 
 
 1.47792 ! 
 
 55 
 
 6 
 
 .57968 
 
 1.72509 
 
 .60324 1.65772 
 
 .62730 
 
 1.59414 
 
 .65189 ! 
 
 1.53400 
 
 .67705 
 
 1.47699 : 
 
 54 
 
 
 
 1.72393 
 
 .60364 1.65663 
 
 .62770 : 
 
 1.59311 
 
 .65231 
 
 1.53302 
 
 .67748 I 
 
 1.47607 i 
 
 53 
 
 8 1 
 
 .55046 1 
 
 1.72278 
 
 .60403 1.65554 
 
 .62811 
 
 1.59208 
 
 .65272 
 
 1.53205 
 
 .67790 j 
 
 1.47514 | 
 
 52 
 
 9 
 
 .58085 
 
 1.72163 
 
 .60443 1.65445 
 
 .62852 
 
 1.59105 
 
 .65314 
 
 1.53107 
 
 .67832 | 
 
 1.47422 
 
 51 
 
 10 1 
 
 .58124 i 
 
 1.72047 
 
 .60483 1.65337 
 
 .62892 
 
 1.59002 
 
 .65355 i 
 
 1.53010 
 
 .67875 I 
 
 1.47330 ! 
 
 50 
 
 11 ; 
 
 .58162 ' 
 
 1.71932 
 
 .60522 1.65228 
 
 .62933 
 
 1.5S900 
 
 .65397 | 
 
 1.52913 
 
 .67917 
 
 1.47238 
 
 49 
 
 12 
 
 .58201 
 
 1.71817 
 
 .60562 1.65120 
 
 .62973 | 
 
 1.58797 
 
 .65438 i 
 
 1.52816 
 
 .67960 
 
 1.47146 ! 
 
 48 
 
 13 1 
 
 .58240 
 
 1.71702 
 
 .60602 1.65011 
 
 .63014 I 
 
 1.58695 
 
 .65480 
 
 1.52719 
 
 .68002 
 
 1.47053 | 
 
 47 
 
 14 | 
 
 .58279 
 
 1.71588 
 
 .60642 1.64903 
 
 .63055 
 
 1.58593 
 
 .65521 1 
 
 1.52622 
 
 .68045 
 
 1.46962 
 
 46 
 
 
 .58318 
 
 1.71473 
 
 .60681 1.64795 
 
 .63095 
 
 1.58490 
 
 .65563 ! 
 
 1.52525 
 
 .68088 ! 
 
 1.46870 
 
 45 
 
 16 j 
 
 .58357 
 
 1.71358 
 
 .60721 1.64687 
 
 .63136 
 
 1.58388 
 
 .65604 
 
 1.52429 
 
 .68130 1 
 
 1.4677S i 
 
 44 
 
 IT 1 
 
 .58396 
 
 1.71244 
 
 .60761 1.64579 
 
 .63177 
 
 1.5S286 
 
 .65646 
 
 1.52332 
 
 .68173 i 
 
 1.46686 
 
 43 
 
 18 j 
 
 .58435 
 
 1.71129 
 
 .60801 1.64471 
 
 .63217 
 
 1.58184 
 
 .65688 j 
 
 1.52235 
 
 .68215 
 
 1.46595 | 
 
 42 
 
 19 
 
 .58474 
 
 1.71015 
 
 .60841 1.64363 
 
 .63258 ! 
 
 1.58083 
 
 .657*29 ! 
 
 1.52139 
 
 .68258 
 
 1.46503 i 
 
 41 
 
 20 1 
 
 .58513 
 
 1.70901 
 
 .60881 ! 1.64256 
 
 .63299 
 
 1.57981 
 
 .65771 j 
 
 1.52043 
 
 .68301 
 
 1.46411 
 
 40 
 
 21 
 
 .58552 
 
 1.70787 
 
 .60921 1.64148 
 
 .63340 . 
 
 1.57879 
 
 .65813 j 
 
 1.51946 
 
 .68343 
 
 1.46320 
 
 39 
 
 22 
 
 .58591 
 
 1.70673 
 
 .60960 1.64041 
 
 .63380 
 
 1.57778 
 
 .65854 i 
 
 1.51850 
 
 .68386 
 
 1.46229 
 
 38 
 
 23 j 
 
 .58631 
 
 1.70560 
 
 .61000 1.63934 
 
 .63421 | 
 
 1 .57676 
 
 .65896 ! 
 
 1.51754 
 
 .68429 
 
 1.46137 
 
 37 
 
 24 1 
 
 .58670 
 
 1.70446 
 
 .61040 1.63826 
 
 .63462 ! 
 
 1.5i575 
 
 .65938 
 
 1.51658 
 
 .68471 ! 
 
 1.46046 I 
 
 36 
 
 25 1 
 
 .58709 
 
 1.70332 
 
 .610S0 1.63719 
 
 .63503 
 
 1.57474 
 
 .65980 
 
 1.51562 
 
 .68514 | 
 
 1.45955 
 
 
 26 
 
 .58748 
 
 1.70219 
 
 .61120 1.63612 
 
 .63544 
 
 1.57372 
 
 .66021 I 
 
 1.51466 
 
 .68557 
 
 1.45864 1 
 
 1 34 
 
 27 
 
 .58787 
 
 1.70106 
 
 .61160 1.63505 
 
 .63584 i 
 
 1.57271 
 
 .66063 1 
 
 1.51370 
 
 .68600 
 
 1.45773 
 
 1 33 
 
 28 
 
 .58826 
 
 1.69992 
 
 .61200 1.63398 
 
 .63625 
 
 1.57170 
 
 .66105 1 
 
 1.51275 
 
 .68642 
 
 1.45682 
 
 L 32 
 
 29 
 
 .58865 ! 
 
 1.69879 
 
 .61240 1.63292 
 
 .63666 ! 
 
 1.57069 
 
 .66147 
 
 J 1.51179 
 
 .68685 
 
 1.45592 
 
 31 
 
 30 1 
 
 .58905 
 
 1 1.69766 
 
 .61280 1.63185 
 
 .63707 
 
 1 1.56969 
 
 .66189 
 
 1.51084 
 
 .68728 1 
 
 1.45501 
 
 i 30 
 
 3! 
 
 .58944 
 
 i 1.69653 
 
 .61320 1 1.63079 
 
 .63748^1.56868 
 
 .66230 ! 
 
 1.50988 
 
 .68771 
 
 | 1.45410 
 
 : 29 
 
 32 
 
 .58983 
 
 1.69541 
 
 .61360 1.62972 
 
 .63789 ! 
 
 jl. 56767 
 
 .66272 ! 
 
 1.50893 
 
 .68814 
 
 ! 1.45320 
 
 -28 
 
 33 1 
 
 .59022 
 
 1.69428 
 
 .61400 1.62866 
 
 .63830 j 
 
 | 1.56667 
 
 .66314 
 
 1.50797 
 
 .68857 
 
 j 1.45229 
 
 . 27 
 
 34 | 
 
 .59061 
 
 1.69316 
 
 .61440 1.62760 
 
 .63871 
 
 i 1.56566 
 
 .66356 
 
 ! 1.50702 
 
 .68900 
 
 ! 1.45139 
 
 j 26 
 
 35 
 
 .59101 
 
 j 1.69203 
 
 .61480 1.62654 
 
 .63912 
 
 1 1.56466 
 
 .66398 
 
 : 1.50607 
 
 .68942 
 
 1 1.45049 
 
 25 
 
 36 i 
 
 .59149 
 
 ; 1.69091 
 
 .61520 1.62548 
 
 .63953 
 
 | 1.56366 
 
 .66440 
 
 1.50512 
 
 .68985 
 
 i 1.44958 
 
 24 
 
 37 
 
 1 .59179 
 
 1.68979 
 
 .61561 | 1.62442 
 
 .63994 
 
 1.56265 
 
 .66482 
 
 1.50417 
 
 .69028 
 
 j 1.44868 
 
 23 
 
 38 I 
 
 ' .59218 
 
 1.68S66 
 
 .61601 j 1.62336 
 
 .64035 
 
 1 1.56165 
 
 .66524 
 
 1.50322 
 
 .69071 
 
 ; 1.44778- 
 
 22 
 
 39 
 
 | .59253 
 
 1.68754 
 
 .61641 1.62230 
 
 .64076 
 
 I 1.56065 
 
 .66566 
 
 1.50228 
 
 .69114 
 
 ! 1.44688 
 
 i 21 
 
 40 
 
 | .59297 
 
 | 1.68643 
 
 .61681 : 1.62125 
 
 .64117 
 
 J 1.55966 
 
 .66608 
 
 1.50133 
 
 .69157 
 
 j 1.44598 
 
 20 
 
 41 
 
 .59336 
 
 * 1.68531 
 
 .61721 1.62019 
 
 .64158 
 
 ! 1.55866 
 
 .66650 
 
 1.50038 
 
 .69200 
 
 ! 1.4450S 
 
 J* 
 
 42 
 
 ! .59376 
 
 1.68419 
 
 .61761 1.61914 
 
 .64199 
 
 ! 1.55766 
 
 .66692 
 
 1.49944 
 
 .69243 
 
 1 1.44418 
 
 18 
 
 43 
 
 .59415 
 
 1.68308 
 
 .61801 1.61808 
 
 .64240 
 
 j 1.55666 
 
 .66734 
 
 1.49849 
 
 .69286 
 
 1.44329 
 
 17 
 
 44 
 
 .59454 
 
 1.68196 
 
 .61842 1.61703 
 
 .64281 
 
 1.55567 
 
 .66776 
 
 1.49755 
 
 .69329 
 
 1.44239 
 
 J® 
 
 45 
 
 .59494 
 
 1.68085 
 
 .61882 | 1.61598 
 
 .64322 
 
 1.55467 
 
 .66S18 
 
 1.49661 
 
 .69372 
 
 1.44149 
 
 J 3 
 
 46 
 
 j .59533 
 
 1.67974 
 
 .61922 1.61493 
 
 .64363 
 
 1.55368 
 
 .66860 
 
 1.49566 
 
 .69416 
 
 1.44060 
 
 14 
 
 47 
 
 I .59573 
 
 1.67863 
 
 .61962 | 1.61388 
 
 .64404 
 
 1.55269 
 
 .66902 
 
 1.49472 
 
 .69459 
 
 ! 1.43970 
 
 ! 13 
 
 48 
 
 .59612 
 
 1.67752 
 
 .62003 1.61283 
 
 .64446 
 
 1.55170 
 
 .66944 
 
 1.49378 
 
 .69502 
 
 1.43881 
 
 
 49 
 
 j .59651 
 
 1.67641 
 
 .62043 ! 1.61179 
 
 .64487 
 
 1.55071 
 
 .66986 
 
 1.49284 
 
 .69545 
 
 1.43792 
 
 1 11 
 
 50 
 
 ! .59691 
 
 1.67530 
 
 .62083 ! 1.61074 
 
 .64528 
 
 1.54972 
 
 .67028 
 
 1.49190 
 
 .69588 
 
 L. 43703 
 
 i 10 
 
 51 
 
 .59730 
 
 ' 1.67419 
 
 .62124 i 1.60970 
 
 .64569 
 
 1.54873 
 
 .67071 
 
 ' 1.49097 
 
 .69631 
 
 1.43614 
 
 9 
 
 52 
 
 i .59770 
 
 1 1.67309 
 
 .62164 1 1.60865 
 
 .64610 
 
 1.54774 
 
 .67113 
 
 I 1.49003 
 
 .69675 
 
 1.43525 
 
 8 
 
 53 
 
 ! .59809 
 
 i 1.67198 
 
 .62204 1.60761 
 
 .64652 
 
 1.54675 
 
 .67155 
 
 | 1.48909 
 
 .69718 
 
 1.43436 
 
 7 
 
 54 
 
 j .59849 
 
 1.67088 
 
 .62245 , 1.60657 
 
 .64693 
 
 I 1.54576 
 
 .67197 
 
 ' 1.48816 
 
 .69761 
 
 j 1.43347 
 
 6 
 
 55 
 
 .59888 
 
 1.66978 
 
 .62285 ! 1.60553 
 
 .64734 
 
 1.54478 
 
 .67239 
 
 , 1.48722 
 
 .69804 
 
 | 1.43258 
 
 5 
 
 56 
 
 ' .59928 
 
 1.66867 
 
 .62325 1.60449 
 
 .64775 
 
 1.54379 
 
 .67232 
 
 1.48629 
 
 .69847 
 
 1.43169 
 
 4 
 
 57 
 
 ■ .59967 
 
 1.66757 
 
 .62366 ! 1 160345 
 
 .64817 
 
 1.54281 
 
 .67324 
 
 1 1.48536 
 
 .69891 
 
 1.43080 
 
 3 
 
 58 
 
 .60007 
 
 1.66647 
 
 .62406 1.60241 
 
 .64858 
 
 1.54183 
 
 .67366 
 
 1 1.48442 
 
 .69934 
 
 1.42992 
 
 2 
 
 59 
 
 .60046 
 
 1.66538 
 
 .62446 1.60137 
 
 .64899 
 
 1.54085 
 
 .67409 
 
 1.48349 
 
 .69977 
 
 1.42903 
 
 1 
 
 60 
 
 .60086 
 
 1.66428 
 
 .62487 i 1.60033 
 
 .64941 
 
 1.53986 
 
 .67451 
 
 | 1.48256 
 
 .70021 
 
 1.42815 
 
 0 
 
 
 Cotang 
 
 Tang 
 
 Cotang ] Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 
 r 
 
 59 ° 
 
 58 ° 
 
 57 ° 
 
 56 ° 
 
 55 ° 
 
 
NATURAL TANGENTS AND COTANGENTS. 
 
 471 
 
 
 35° 
 
 36° 
 
 37° 
 
 38° 
 
 39° 
 
 / 
 
 f - 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang < 
 
 Cotang 
 
 Tang < 
 
 Cotang 
 
 Tang 1 
 
 Cotang 
 
 
 0 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 .70021 
 
 .70064 
 
 .70107 
 
 .70151 
 
 .70194 
 
 .70238 
 
 .70281 
 
 .70325 
 
 .70368 
 
 .70412 
 
 .70455 
 
 .70499 
 
 .70542 
 
 .70586 
 
 .70629 
 
 .70673 
 
 .70717 
 
 .70760 
 
 .70804 
 
 .70848 
 
 .70891 
 
 .70935 
 
 .70979 
 
 .71023 
 
 .71066 
 
 .71110 
 
 .71154 
 
 .71198 
 
 .71242 
 
 .71285 
 
 .71329 
 
 .71373 
 
 .71417 
 
 .71461 
 
 .71505 
 
 .71549 
 
 .71593 
 
 .71637 
 
 .71681 
 
 .71725 
 
 .71769 
 
 .71813 
 
 .71857 
 
 .71901 
 
 .71946 
 
 .71990 
 
 .72034 
 
 .72078 
 
 .72122 
 
 .72167 
 
 .72211 
 
 .72255 
 
 .72299 
 
 .72344 
 
 .72388 
 
 .72432 
 
 .72477 
 
 .72521 
 
 .72565 
 
 .72610 
 
 .72654 
 
 1.42815 
 L. 42726 
 L. 42638 
 L. 42550 
 1.42462 
 1.42374 
 1.42286 
 1.42198 
 1.42110 
 1.42022 
 1.41934 
 
 1.41847 
 
 1.41759 
 
 1.41672 
 
 1.41584 
 
 1.41497 
 
 1.41409 
 
 1.41322 
 
 1.41235 
 
 1.41148 
 
 1.41061 
 
 1.40974 
 
 1.40887 
 
 1.40800 
 
 1.40714 
 
 1.40627 
 
 1.40540 
 
 1.40454 
 
 1.40367 
 
 1.40281 
 
 1.40195 
 
 1.40109 
 1.40022 
 1.39936 
 1.39850 
 1.39764 
 1.39679 
 1.39593 
 1.39507 
 1 .39421 
 1.39336 
 
 1.39250 
 
 1.39165 
 
 1.39079 
 
 1.38994 
 
 1.38909 
 
 1.38824 
 
 1.38738 
 
 1.38653 
 
 1.38568 
 
 1.38484 
 
 1.38399 
 
 1.38314 
 
 1.38229 
 
 1.38145 
 
 1.38060 
 
 1.37976 
 
 1.37891 
 
 1.37807 
 
 1.37722 
 
 1.37638 
 
 .72654 
 
 .72699 
 
 .72743 
 
 .72788 
 
 .72832 
 
 .72877 
 
 .72921 
 
 .72966 
 
 .73010 
 
 .73055 
 
 .73100 
 
 .73144 
 
 .73189 
 
 .73234 
 
 .73278 
 
 .73323 
 
 .73368 
 
 .73413 
 
 .73457 
 
 .73502 
 
 .73547 
 
 .73592 
 
 .73637 
 
 .73681 
 
 .73726 
 
 .73771 
 
 .73816 
 
 .73861 
 
 .73906 
 
 .73951 
 
 .73996 
 
 .74041 
 
 .74086 
 
 .74131 
 
 .74176 
 
 .74221 
 
 .74267 
 
 .74312 
 
 .74357 
 
 .74402 
 
 .74447 
 
 .74492 
 
 .74538 
 
 .74583 
 
 .74628 
 
 .74674 
 
 .74719 
 
 .74764 
 
 .74810 
 
 .74855 
 
 .74900 
 
 .74946 
 .74991 
 .75037 
 .75082 
 .75128 
 .75173 
 .75219 
 .75264 
 ! .75310 
 
 1 .75355 
 
 L. 37638 
 .37554 
 1.37470 
 L. 37386 
 1.37302 
 1.37218 
 1.37134 
 1.37050 
 1.36967 
 1.36883 
 1.36800 
 
 1.36716 
 1.36633 
 1.36549 
 1.36466 
 1.36383 
 1.36300 
 1.36217 
 1.36134 
 1.36051 
 1 35968 
 
 1.35885 
 
 1.35802 
 
 1.35719 
 
 1.35637 
 
 1.35554 
 
 1.35472 
 
 1.35389 
 
 1.35307 
 
 1.35224 
 
 1.35142 
 
 1.35060 
 
 1.34978 
 
 1.34896 
 
 1.34814 
 
 1.34732 
 
 1.34650 
 
 1.34568 
 
 1.34487 
 
 1.34405 
 
 1.34323 
 
 1.34242 
 
 1.34160 
 
 1.34079 
 
 1.33998 
 
 1.33916 
 
 1.33835 
 
 1.33754 
 
 1.33673 
 
 1.33592 
 
 1.33511 
 
 1.33430 
 
 1.33349 
 
 1.33268 
 
 1.33187 
 
 1.33107 
 
 1.33026 
 
 1.32946 
 
 1.32865 
 
 1.32785 
 
 1.32704 
 
 .75355 ] 
 
 .75401 : 
 
 .75447 : 
 
 .75492 
 
 .75538 
 
 .75584 
 
 .75629 
 
 .75675 
 
 .75721 
 
 .75767 
 
 .75812 
 
 .75858 
 
 .75904 
 
 .75950 
 
 .75996 
 
 .76042 
 
 .76088 
 
 .76134 
 
 .76180 
 
 .76226 
 
 .76272 
 
 .76318 
 
 .76364 
 
 .76410 
 
 .76456 
 
 .76502 
 
 .76548 
 
 .76594 
 
 .76640 
 
 .76686 
 
 .76733 
 
 .76779 
 
 .76825 
 
 .76871 
 
 .76918 
 
 .76964 
 
 .77010 
 
 .77057 
 
 .77103 
 
 .77149 
 
 .77196 
 
 .77242 
 
 .77289 
 
 .77335 
 
 .77382 
 
 .77428 
 
 .77475 
 
 .77521 
 
 .77568 
 
 .77615 
 
 .77661 
 
 .77708 
 .77754 
 .77801 
 .77848 
 .77895 
 i .77941 
 i .77988 
 i .78035 
 ; .78082 
 
 L .78129 
 
 L. 32704 
 L. 32624 
 L. 32544 
 1.32464 
 1.32384 
 1.32304 
 1.32224 
 1.32144 
 1.32064 
 1.31984 
 1.31904 
 
 1.31825 
 
 1.31745 
 
 1.31666 
 
 1.31586 
 
 1.31507 
 
 1.31427 
 
 1.31348 
 
 1.31269 
 
 1.31190 
 
 1.31110 
 
 1.31031 
 1.30952 
 1.30873 
 1.30795 
 1.30716 
 1.30637 
 1.30558 
 1 .30480 
 1.30401 
 1.30323 
 
 1.30244 
 1.30166 
 1.30087 
 1.30009 
 1.29931 
 1.29853 
 1.29775 
 1 .29696 
 1 .29618 
 1.29541 
 
 1.29463 
 
 1.29385 
 
 1.29307 
 
 1.29229 
 
 1.29152 
 
 1.29074 
 
 1.28997 
 
 1.28919 
 
 1.28842 
 
 1.28764 
 
 1.28687 
 1 .28610 
 1 .28533 
 1.28456 
 1.28379 
 1.28302 
 1.28225 
 1.28148 
 1.28071 
 1.27994 
 
 .78129 ] 
 
 .78175 ] 
 
 .78222 : 
 .78269 : 
 
 .78316 : 
 
 .78363 
 .78410 
 .78457 
 .78504 
 .78551 
 .78598 
 
 .78645 
 
 .78692 
 
 .78739 
 
 .78786 
 
 .78834 
 
 .78881 
 
 .78928 
 
 .78975 
 
 .79022 
 
 .79070 
 
 .79117 
 
 .79164 
 
 .79212 
 
 .79259 
 
 .79306 
 
 .79354 
 
 .79401 
 
 .79449 
 
 .79496 
 
 .79544 
 
 .79591 
 
 .79639 
 
 .79686 
 
 .79734 
 
 .79781 
 
 .79829 
 
 .79877 
 
 .79924 
 
 .79972 
 
 .80020 
 
 .80067 
 
 .80115 
 
 .80163 
 
 .80211 
 
 .80258 
 
 .80306 
 
 .80354 
 
 .80402 
 
 .80450 
 
 .80498 
 
 .80546 
 .80594 
 .80642 
 .80690 
 .80738 
 .80786 
 i .80834 
 l .80882 
 . .80930 
 
 r .80978 
 
 L. 2 7994 , 
 
 L. 27917 , 
 
 L. 27841 
 
 1.27764 
 
 1.27688 
 
 1.27611 
 
 1.27535 
 
 1.27458 
 
 1.27382 
 
 1.27306 
 
 1.27230 
 
 1.27153 
 1.27077 
 1.27001 
 1 .26925 
 1.26849 
 1.26774 
 1.26698 
 1.26622 
 1.26546 
 1.26471 
 
 1.26395 
 1 .26319 
 1.26244 
 1 .26169 
 1 .26093 
 1 .26018 
 1 .25943 
 1.25867 
 1.25792 
 1.25717 
 
 1.25642 
 
 1.25567 
 
 1.25492 
 
 1.25417 
 
 1.25343 
 
 1.25268 
 
 1.25193 
 
 1.25118 
 
 1.25044 
 
 1.24969 
 
 1.24895 
 
 1.24820 
 
 1.24746 
 
 1.24672 
 
 1.24597 
 
 1.24523 
 
 1.24449 
 
 1.24375 
 
 1.24301 
 
 1.24227 
 
 1.24153 
 1 .24079 
 1.24005 
 1.23931 
 1.23858 
 1.23784 
 1.23710 
 1.23637 
 1.23563 
 1 .23490 
 
 .80978 : 
 
 .81027 
 .81075 : 
 
 .81123 : 
 
 .81171 
 .81220 
 .81268 
 .81316 
 .81364 
 .81413 
 .81461 
 
 .81510 
 
 .81558 
 
 .81606 
 
 .81655 
 
 .81703 
 
 .81752 
 
 .81800 
 
 .81849 
 
 .81898 
 
 .81946 
 
 .81995 
 
 .82044 
 
 .82092 
 
 .82141 
 
 .82190 
 
 .82238 
 
 .82287 
 
 .82336 
 
 .82385 
 
 .82434 
 
 .82483 
 
 .82531 
 
 .82580 
 
 .82629 
 
 .82678 
 
 .82727 
 
 .82776 
 
 .82825 
 
 .82874 
 
 .82923 
 
 .82972 
 
 .83022 
 
 .83071 
 
 .83120 
 
 .83169 
 
 .83218 
 
 .83268 
 
 .83317 
 
 .83366 
 
 .83415 
 
 .83465 
 .83514 
 .83564 
 .83613 
 .83662 
 .83712 
 i .83761 
 .83811 
 ; .83860 
 
 l .83910 
 
 L. 23490 
 1.23416 
 1.23343 
 1.23270 
 1.23196 
 1.23123 
 1.23050 
 1.22977 
 1.22904 
 1.22831 
 1.22758 
 
 1.22685 
 
 1.22612 
 
 1.22539 
 
 1.22467 
 
 1.22394 
 
 1.22321 
 
 1.22249 
 
 1.22176 
 
 1.22104 
 
 1.22031 
 
 1.21959 
 
 1.21886 
 
 1.21814 
 
 1.21742 
 
 1.21670 
 
 1.21598 
 
 1.21526 
 
 1.21454 
 
 1.21382 
 
 1.21310 
 
 1.21238 
 
 1.21166 
 
 1.21094 
 
 1.21023 
 
 1.20951 
 
 1.20879 
 
 1.20808 
 
 1.20736 
 
 1.20665 
 
 1.20593 
 
 1.20522 
 
 1.20451 
 
 1.20379 
 
 1.20308 
 
 1.20237 
 
 1.20166 
 
 1.20095 
 
 1.20024 
 
 1.19953 
 
 1.19882 
 
 1.19811 
 
 1.19740 
 
 1.19669 
 
 1.19599 
 
 1.19528 
 
 1.19457 
 
 1.19387 
 
 1.19316 
 
 1.19246 
 
 1.19175 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 1 2 
 
 ; l 
 
 , 0 
 
 
 Cotan j 
 
 ? Tang 
 
 Cotan £ 
 
 1 Tang 
 
 Cotang 
 
 l Tang 
 
 Cotang 
 
 ; Tang 
 
 Cotani 
 
 g Tang 
 
 t 
 
 f 
 
 54° 
 
 53° 
 
 i 
 
 52° 
 
 51° 
 
 50° 
 
 
472 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 
 40° 
 
 41° 
 
 42° 
 
 43° 
 
 440 
 
 / 
 
 ' 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 
 0 
 
 .83910 
 
 1.19175 
 
 .86929 
 
 1.15037 
 
 .90040 
 
 1.11061 
 
 .93252 
 
 1.07237 
 
 .96569 
 
 1.03553 
 
 60 
 
 1 
 
 .83960 
 
 1.19105 
 
 .86980 
 
 1.14969 
 
 .90093 
 
 1.10996 
 
 .93306 
 
 1.07174 
 
 .96625 
 
 1.03493 
 
 59 
 
 2 
 
 .84009 
 
 1.19035 
 
 .87031 
 
 1.14902 
 
 .90146 
 
 1.10931 
 
 .93360 
 
 1.07112 
 
 .96681 
 
 1.03433 
 
 58 
 
 3 
 
 .84059 
 
 1.18964 
 
 .87082 
 
 1.14834 
 
 .90199 
 
 1.10867 
 
 .93415 
 
 1.07049 
 
 .96738 
 
 1.03372 
 
 57 
 
 4 
 
 .84108 
 
 1.18894 
 
 .87133 
 
 1.14767 
 
 .90251 
 
 1.10802 
 
 .93469 
 
 1.06987 
 
 .96794 
 
 1.03312 
 
 56 
 
 5 
 
 .84158 
 
 1.18824 
 
 .87184 
 
 1.14699 
 
 .90304 
 
 1.10737 
 
 .93524 
 
 1.06925 
 
 .96850 
 
 1.03252 
 
 55 
 
 6 
 
 .84208 
 
 1.18754 
 
 .87236 
 
 1.14632 
 
 .90357 
 
 1.10672 
 
 .93578 
 
 1.06862 
 
 .96907 
 
 1.03192 
 
 54 
 
 7 
 
 .84258 
 
 1.18684 
 
 .87287 
 
 1.14565 
 
 .90410 
 
 1.10607 
 
 .93633 
 
 1.06800 
 
 .96963 
 
 1.03132 
 
 53 
 
 8 
 
 .84307 
 
 1.18614 
 
 .87338 
 
 1.14498 
 
 .90463 
 
 1.10543 
 
 .93688 
 
 1.06738 
 
 .97020 
 
 1.03072 
 
 52 
 
 9 
 
 .84357 
 
 1.18544 
 
 .87389 
 
 1.14430 
 
 .90516 
 
 1.10478 
 
 .93742 
 
 1.06676 
 
 .97076 
 
 1.03012 
 
 51 
 
 10 
 
 .84407 
 
 1.18474 
 
 .87441 
 
 1.14363 
 
 .90569 
 
 1.10414 
 
 .93797 
 
 1.06613 
 
 .97133 
 
 1.02952 
 
 50 
 
 n 
 
 .84457 
 
 1.18404 
 
 .87492 
 
 1.14296 
 
 .90621 
 
 1.10349 
 
 .93852 
 
 1.06551 
 
 .97189 
 
 1.02892 
 
 49 
 
 12 
 
 .84507 
 
 1.18334 
 
 .87543 
 
 1.14229 
 
 .90674 
 
 1.10285 
 
 .93906 
 
 1.06489 
 
 .97246 
 
 1.02832 
 
 48 
 
 13 
 
 .84556 
 
 1 .18264 
 
 .87595 
 
 1.14162 
 
 .90727 
 
 1.10220 
 
 .93961 
 
 1.06427 
 
 .97302 
 
 1.02772 
 
 47 
 
 14 
 
 .84606 
 
 1.18194 
 
 .87646 
 
 1.14095 
 
 .90781 
 
 1.10156 
 
 .94016 
 
 1.06365 
 
 .97359 
 
 1.02713 
 
 46 
 
 15 
 
 .84656 
 
 1.18125 
 
 .87698 
 
 1.14028 
 
 .90834 
 
 1.10091 
 
 .94071 
 
 1.06303 
 
 .97416 
 
 1.02653 
 
 45 
 
 16 
 
 .84706 
 
 1.18055 
 
 .87749 
 
 1.13961 
 
 .90887 
 
 1.10027 
 
 .94125 
 
 1.06241 
 
 .97472 
 
 1.02593 
 
 44 
 
 17 
 
 .84756 
 
 1.17986 
 
 .87801 
 
 1.13894 
 
 .90940 
 
 1.09963 
 
 .94180 
 
 1.06179 
 
 .07529 
 
 1.02533 
 
 43 
 
 18 
 
 .84806 
 
 1.17916 
 
 .87852 
 
 1.13828 
 
 .90993 
 
 1.09899 
 
 .94235 
 
 1.06117 
 
 .97586 
 
 1.02474 
 
 42 
 
 19 
 
 .84856 
 
 1.17846 
 
 .87904 
 
 1.13761 
 
 .91046 
 
 1.09834 
 
 .94290 
 
 1.06056 
 
 .97643 
 
 1.02414 
 
 41 
 
 20 
 
 .84906 
 
 1.17777 
 
 .87955 
 
 1.13694 
 
 .91099 
 
 1.09770 
 
 .94345 
 
 1.05994 
 
 .97700 
 
 1.02355 
 
 40 
 
 21 
 
 .84956 
 
 1.17708 
 
 .88007 
 
 1.13627 
 
 .91153 
 
 1.09706 
 
 .94400 
 
 1.05932 
 
 .97756 
 
 1.02295 
 
 39 
 
 22 
 
 .85006 
 
 1.17638 
 
 .88059 
 
 1.13561 
 
 .91206 
 
 1.09642 
 
 .94455 
 
 1.05870 
 
 .97813 
 
 1.02236 
 
 38 
 
 23 
 
 .85057 
 
 1.17569 
 
 .88110 
 
 1.13494 
 
 .91259 
 
 1.09578 
 
 .94510 
 
 1.05809 
 
 .97870 
 
 1.02176 
 
 37 
 
 24 
 
 .85107 
 
 1.17500 
 
 .88162 
 
 1.13428 
 
 .91313 
 
 1.09514 
 
 .94565 
 
 1.05747 
 
 .97927 
 
 1.02117 
 
 36 
 
 25 
 
 .85157 
 
 1.17430 
 
 .88214 
 
 1.13361 
 
 .91366 
 
 1.09450 
 
 .94620 
 
 1.05685 
 
 .97984 
 
 1.02057 
 
 35 
 
 26 
 
 .85207 
 
 1.17361 
 
 .88265 
 
 1.13295 
 
 .91419 
 
 1.09386 
 
 .94676 
 
 1.05624 
 
 .98041 
 
 1.01998 
 
 34 
 
 27 
 
 .85257 
 
 1.17292 
 
 .88317 
 
 1.13228 
 
 .91473 
 
 1.09322 
 
 .94731 
 
 1.05562 
 
 .98098 
 
 1.01939 
 
 33 
 
 28 
 
 .85308 
 
 1.17223 
 
 .88369 
 
 1.13162 
 
 .91526 
 
 1.09258 
 
 .94786 
 
 1.05501 
 
 .98155 
 
 1.01879 
 
 32 
 
 29 
 
 .85358 
 
 1.17154 
 
 .88421 
 
 1.13096 
 
 .91580 
 
 1.09195 
 
 .94841 
 
 1.05439 
 
 .98213 
 
 1.01820 
 
 31 
 
 30 
 
 .85408 
 
 1.17085 
 
 .88473 
 
 1.13029 
 
 .91633 
 
 1.09131 
 
 .94896 
 
 1.05378 
 
 .98270 
 
 1.01761 
 
 30 
 
 31 
 
 .85458 
 
 1.17016 
 
 .88524 
 
 1.12963 
 
 .91687 
 
 1.09067 
 
 .94952 
 
 1.05317 
 
 .98327 
 
 1.01702 
 
 29 
 
 32 
 
 .85509 
 
 1.16947 
 
 .88576 
 
 1.12897 
 
 .91740 
 
 1.09003 
 
 .95007 
 
 1.05255 
 
 .98384 
 
 1.01642 
 
 28 
 
 33 
 
 .85559 
 
 1.16878 
 
 .88628 
 
 1.12831 
 
 .91794 
 
 1.08940 
 
 .95062 
 
 1.05194 
 
 .98441 
 
 1.01583 
 
 27 
 
 34 
 
 .85609 
 
 1.16809 
 
 .88680 
 
 1.12765 
 
 .91847 
 
 1.08876 
 
 .95118 
 
 1.05133 
 
 .98499 
 
 1.01524 
 
 26 
 
 35 
 
 .85660 
 
 1.16741 
 
 .88732 
 
 1.12699 
 
 .91901 
 
 1.08813 
 
 .95173 
 
 1.05072 
 
 .98556 
 
 1.01465 
 
 25 
 
 36 
 
 .85710 
 
 1.16672 
 
 .88784 
 
 1.12633 
 
 .91955 
 
 1.08749 
 
 .95229 
 
 1.05010 
 
 .98613 
 
 1.01406 
 
 24 
 
 37 
 
 .85761 
 
 1.16603 
 
 .88836 
 
 1.12567 
 
 .92008 
 
 1.08686 
 
 .95284 
 
 1.04949 
 
 .98671 
 
 1.01347 
 
 23 
 
 38 
 
 .85811 
 
 1.16535 
 
 .88888 
 
 1.12501 
 
 .92062 
 
 1.08622 
 
 .95340 
 
 1.04888 
 
 .98728 
 
 1.01288 
 
 22 
 
 39 
 
 .85862 
 
 1.16466 
 
 .88940 
 
 1.12435 
 
 .92116 
 
 1.08559 
 
 .95395 
 
 1.04827 
 
 .98786 
 
 1.01229 
 
 21 
 
 40 
 
 .85912 
 
 1.16398 
 
 .88992 
 
 1.12369 
 
 .92170 
 
 1.08496 
 
 .95451 
 
 1.04766 
 
 .98843 
 
 1.01170 
 
 20 
 
 41 
 
 .85963 
 
 1.16329 
 
 .89045 
 
 1.12303 
 
 .92224 
 
 1.08432 
 
 .95506 
 
 1.04705 
 
 .98901 
 
 1.01112 
 
 19 
 
 42 
 
 .86014 
 
 1.16261 
 
 .89097 
 
 1.12238 
 
 .92277 
 
 1.08369 
 
 .95562 
 
 1.04644 
 
 .98958 
 
 1.01053 
 
 18 
 
 43 
 
 .86064 
 
 1.16192 
 
 .89149 
 
 1.12172 
 
 .92331 
 
 1.08306 
 
 .95618 
 
 1.04583 
 
 .99016 
 
 1.00994 
 
 17 
 
 44 
 
 .86115 
 
 1.16124 
 
 .89201 
 
 1.12106 
 
 .92385 
 
 1.08243 
 
 .95673 
 
 1.04522 
 
 .99073 
 
 1.00935 
 
 16 
 
 45 
 
 .86166 
 
 1.16056 
 
 .89253 
 
 1.12041 
 
 .92439 
 
 1.08179 
 
 .95729 
 
 1.04461 
 
 .99131 
 
 1.00876 
 
 15 
 
 46 
 
 .86216 
 
 1.15987 
 
 .89306 
 
 1.11975 
 
 .92493 
 
 1.08116 
 
 .95785 
 
 1.04401 
 
 .99189 
 
 1.00818 
 
 14 
 
 47 
 
 .86267 
 
 1.15919 
 
 .89358 
 
 1.11909 
 
 .92547 
 
 1.08053 
 
 .95841 
 
 1.04340 
 
 .99247 
 
 1.00759 
 
 13 
 
 48 
 
 .86318 
 
 1.15851 
 
 .89410 
 
 1.11844 
 
 .92601 
 
 1.07990 
 
 .95897 
 
 1.04279 
 
 .99304 
 
 1.00701 
 
 12 
 
 49 
 
 .86368 
 
 1.15783 
 
 .89463 
 
 1.11778 
 
 .92655 
 
 1.07927 
 
 .95952 
 
 1.04218 
 
 .99362 
 
 1.00642 
 
 11 
 
 50 
 
 .86419 
 
 1.15715 
 
 .89515 
 
 1.11713 
 
 .92709 
 
 1.07864 
 
 .96008 
 
 1.04158 
 
 .99420 
 
 1.00583 
 
 10 
 
 51 
 
 .86470 
 
 1.15647 
 
 .89567 
 
 1.11648 
 
 .92763 
 
 1.07801 
 
 .96064 
 
 1.04097 
 
 .99478 
 
 1.00525 
 
 9 
 
 52 
 
 .86521 
 
 1.15579 
 
 .89620 
 
 1.11582 
 
 .92817 
 
 1.07738 
 
 .96120 
 
 1.04036 
 
 .99536 
 
 1.00467 
 
 8 
 
 53 
 
 .86572 
 
 1.15511 
 
 .89672 
 
 1.11517 
 
 .92872 
 
 '1.07676 
 
 .96176 
 
 1.03976 
 
 .99594 
 
 1.00408 
 
 7 
 
 54 
 
 .86623 
 
 1.15443 
 
 .89725 
 
 1.11452 
 
 .92926 
 
 1.07613 
 
 .96232 
 
 1.03915 
 
 .99652 
 
 1.00350 
 
 6 
 
 55 
 
 .86674 
 
 1.15375 
 
 .89777 
 
 1.11387 
 
 .92980 
 
 1.07550 
 
 .96288 
 
 1.03855 
 
 .99710 
 
 1.00291 
 
 5 
 
 56 
 
 .86725 
 
 1.15308 
 
 .89830 
 
 1.11321 
 
 .93034 
 
 1.07487 
 
 .96344 
 
 1.03794 
 
 .99768 
 
 1.00233 
 
 4 
 
 57 
 
 .86776 
 
 1.15240 
 
 .89883 
 
 1.11256 
 
 .93088 
 
 1.07425 
 
 .96400 
 
 1.03734 
 
 .99826 
 
 1.00175 
 
 3 
 
 58 
 
 .86827 
 
 1.15172 
 
 .89935 
 
 1.11191 
 
 .93143 
 
 1.07362 
 
 .96457 
 
 1.03674 
 
 .99884 
 
 1.00116 
 
 2 
 
 59 
 
 .86878 
 
 1.15104 
 
 .89988 
 
 1.11126 
 
 .93197 
 
 1.07299 
 
 .96513 
 
 1.03613 
 
 .99942 
 
 1.00058 
 
 1 
 
 60 
 
 .86929 
 
 1.15037 
 
 .90040 
 
 1.11061 
 
 .93352 
 
 1.07237 
 
 .96569 
 
 1.03553 
 
 1.00000 
 
 1.00000 
 
 0 
 
 / 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 Cotang 
 
 Tang 
 
 / 
 
 
 49° 
 
 48° 
 
 47° 
 
 46° 
 
 45° 
 
 
logarithmic tables. 
 
 473 
 
 LOGARITHMIC TABLES. 
 
 rp m iPfth°and e folumn headS P (')i opposite 'the latter, and under the proper 
 2l® find the desired logarithmic sine, cosine, tangent, or cotangent. 
 
 in the sine, cosine, tangent, or cotangent 
 
 ^TnTincfthe L^MrUhmi^Rnictio^s^or an Angle Containing Degrees, Minutes, and 
 
 c j Tt'ind the logarithm for the degrees and minutes in the manner given 
 
 Seconds.— Find me logantnm , ded ,? d ta ke the number next below the 
 
 lo^Mithm^hus fo^d-^Mer toda^tglumn^ea e^ y 
 prope^vate/n^ he foun^^yJ^w^ating^etweeii^the^TOlues^gyra. 
 
 ina+°CphS S " ^4to%6° C tog T io7a C ^ log COtg a 
 
 log tang a + Cpl. T. _ — 
 
 COMMON LOGARITHMS OF NUMBERS. 
 
 To. 
 
 Log. 
 
 No.! 
 
 Log. 
 
 No. 
 
 Log. 
 
 0 
 
 — 00 
 
 20 
 
 30 103 
 
 40 
 
 60 206 
 
 1 
 
 00 000 
 
 21 
 
 32 222 
 
 41 
 
 61 278 
 
 2 
 
 30 103 
 
 22 
 
 34 242 
 
 42 
 
 62 325 
 
 3 
 
 47 712 
 
 23 
 
 36 173 
 
 43 
 
 63 347 
 
 4 
 
 60 206 
 
 24 
 
 38 021 
 
 44 
 
 64 345 
 
 5 
 
 69 897 
 
 25 
 
 39 794 
 
 45 
 
 65 321 
 
 6 
 
 77 815 
 
 26 
 
 41 497 
 
 46 
 
 66 276 
 
 7 
 
 84 510 
 
 27 
 
 43 136 
 
 47 
 
 67 210 
 
 8 
 
 90 309 
 
 28 
 
 44 716 
 
 48 
 
 68 124 
 
 9 
 
 95 424 
 
 29 
 
 46 240 
 
 49 
 
 69 020 
 
 10 
 
 00 000 
 
 30 
 
 47 712 
 
 50 
 
 69 897 
 
 11 
 
 04 139 
 
 31 
 
 49 136 
 
 51 
 
 70 757 
 
 12 
 
 07 918 
 
 32 
 
 50 515 
 
 52 
 
 71 600 
 
 13 
 
 11 394 
 
 33 
 
 51 851 
 
 53 
 
 72 428 
 
 14 
 
 14 613 
 
 34 
 
 53 148 
 
 54 
 
 73 239 
 
 15 
 
 17 609 
 
 35 
 
 54 407 
 
 55 
 
 74 036 
 
 16 
 
 20 412 
 
 36 
 
 55 630 
 
 56 
 
 74 819 
 
 17 
 
 23 045 
 
 37 
 
 56 820 
 
 57 
 
 75 587 
 
 18 
 
 25 527 
 
 38 
 
 57 978 
 
 58 
 
 76 343 
 
 19 
 
 27 875 
 
 39 
 
 59 106 
 
 59 
 
 77 085 
 
 20 
 
 30 103 
 
 40 
 
 60 206 
 
 60 
 
 77 815 
 
 No. 
 
 60 
 
 61 ' 
 
 62 
 
 68 
 
 64 
 
 65 
 
 66 
 
 67 
 
 68 
 
 69 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 78 
 
 79 
 
 80 
 
 Log. 
 
 77 815 
 
 78 533 
 
 79 239 
 
 79 934 
 
 80 618 
 81 291 
 
 81 954 
 
 82 607 
 
 83 251 
 83 885 
 
 84 510 
 
 85 126 
 
 85 733 
 
 86 332 
 
 86 923 
 
 87 506 
 
 88 081 
 
 88 649 
 
 89 209 
 89 763 
 
 90 309 
 
 No. 
 
 Log. 
 
 80 
 
 90 309 
 
 81 
 
 90 849 
 
 82 
 
 91 381 
 
 83 
 
 91 908 
 
 84 
 
 92 428 
 
 85 
 
 92 942 
 
 86 
 
 93 450 
 
 87 
 
 93 952 
 
 88 
 
 94 448 
 
 89 
 
 94 939 
 
 90 
 
 95 424 
 
 91 
 
 95 904 
 
 92 
 
 96 379 
 
 93 
 
 96 848 
 
 94 
 
 97 313 
 
 95 
 
 97 772 
 
 96 
 
 98 227 
 
 97 
 
 98 677 
 
 98 
 
 99 123 
 
 99 
 
 99 564 
 
 100 
 
 00 000 
 
474 
 
 LOGARITHMS. 
 
 N. 
 
 L. 0 
 
 , 
 
 T 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ' 7 
 
 8 
 
 9 
 
 P. P. 
 
 too 
 
 00 000 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 260 
 
 303 
 
 346 
 
 389 
 
 
 
 
 
 101 
 
 432 
 
 475 
 
 518 
 
 561 
 
 604 
 
 647 
 
 689 
 
 732 
 
 775 
 
 817 
 
 
 44 
 
 43 
 
 42 
 
 102 
 
 860 
 
 903 
 
 945 
 
 988 
 
 *030 
 
 *072 
 
 *115 
 
 *157 
 
 *199 
 
 *242 
 
 
 A A 
 
 A S 
 
 A 9 
 
 103 
 
 01 284 
 
 326 
 
 368 
 
 410 
 
 452 
 
 494 
 
 536 
 
 578 
 
 620 
 
 662 
 
 X 
 
 2 
 
 A. A 
 8.8 
 
 4.0 
 
 8.6 
 
 8.4 
 
 104 
 
 703 
 
 745 
 
 787 
 
 828 
 
 870 
 
 912 
 
 953 
 
 995 
 
 *036 
 
 *078 
 
 3 
 
 13.2 
 
 12.9 
 
 12.6 
 
 105 
 
 02 119 
 
 160 
 
 202 
 
 243 
 
 284 
 
 325 
 
 366 
 
 407 
 
 449 
 
 490 
 
 4 
 
 17.6 
 
 17.2 
 
 16.8 
 
 106 
 
 531 
 
 572 
 
 612 
 
 653 
 
 694 
 
 735 
 
 776 
 
 816 
 
 857 
 
 898 
 
 5 
 
 22.0 
 
 21.5 
 
 21.0 
 
 107 
 
 938 
 
 979 
 
 *019 
 
 *060 
 
 *100 
 
 *141 
 
 *181 
 
 *222 
 
 *262 
 
 *302 
 
 6 
 
 7 
 
 26.4 
 30 8 
 
 25.8 
 
 30.1 
 
 25.2 
 29 4 
 
 108 
 
 03 342 
 
 383 
 
 423 
 
 463 
 
 503 
 
 543 
 
 583 
 
 623 
 
 663 
 
 703 
 
 1 
 
 8 
 
 35^2 
 
 34.4 
 
 33^6 
 
 109 
 
 743 
 
 ■ 782 
 
 822 
 
 862 
 
 902 
 
 941 
 
 981 
 
 *021 
 
 *060 
 
 *100 
 
 9 
 
 39.6 
 
 38.7 
 
 37.8 
 
 110 
 
 04 139 
 
 179 
 
 218 
 
 258 
 
 297 
 
 336 
 
 376 
 
 415 
 
 454 
 
 493 
 
 
 
 
 
 111 
 
 532 
 
 571 
 
 610 
 
 650 
 
 689 
 
 727 
 
 766 
 
 805 
 
 844 
 
 883 
 
 
 41 
 
 40 
 
 39 
 
 112 
 
 922 
 
 961 
 
 999 
 
 *038 
 
 *077 
 
 *115 
 
 *154 
 
 *192 
 
 *231 
 
 *269 
 
 
 A 1 
 
 a n 
 
 *» Q 
 
 113 
 
 05 308 
 
 346 
 
 385 
 
 423 
 
 461 
 
 500 
 
 538 
 
 576 
 
 614 
 
 652 
 
 X 
 
 2 
 
 T. 1 
 8.2 
 
 8.0 
 
 O.i/ 
 
 7.8 
 
 114 
 
 690 
 
 729 
 
 767 
 
 805 
 
 843 
 
 881 
 
 918 
 
 956 
 
 994 
 
 *032 
 
 3 
 
 12.3 
 
 12.0 
 
 11.7 
 
 115 
 
 06 070 
 
 108 
 
 145 
 
 183 
 
 221 
 
 258 
 
 296 
 
 333 
 
 371 
 
 408 
 
 4 
 
 16.4 
 
 16.0 
 
 15.6 
 
 116 
 
 446 
 
 483 
 
 521 
 
 558 
 
 595 
 
 633 
 
 670 
 
 707 
 
 744 
 
 781 
 
 5 
 
 20.5 
 
 20.0 
 
 19.5 
 
 117 
 
 819 
 
 856 
 
 893 
 
 930 
 
 967 
 
 *004 
 
 *041 
 
 *078 
 
 *115 
 
 *151 
 
 6 
 
 7 
 
 24.6 
 
 28.7 
 
 24.0 
 
 28.0 
 
 23.4 
 
 27.3 
 
 118 
 
 07 188 
 
 225 
 
 262 
 
 298 
 
 335 
 
 372 
 
 408 
 
 445 
 
 482 
 
 518 
 
 1 
 
 8 
 
 32.8 
 
 32.0 
 
 31.2 
 
 119 
 
 555 
 
 591 
 
 628 
 
 664 
 
 700 
 
 737 
 
 773 
 
 809 
 
 846 
 
 882 
 
 9 
 
 36.9 
 
 36.0 
 
 35.1 
 
 120 
 
 918 
 
 954 
 
 990 
 
 *027 
 
 *063 
 
 *099 
 
 *135 
 
 *171 
 
 *207 
 
 *243 
 
 
 
 
 
 121 
 
 08 279 
 
 314 
 
 350 
 
 386 
 
 422 
 
 458 
 
 493 
 
 529 
 
 565 
 
 600 
 
 
 38 
 
 37 
 
 36 
 
 122 
 
 636 
 
 672 
 
 707 
 
 743 
 
 778 
 
 814 
 
 849 
 
 884 
 
 920 
 
 955 
 
 x 
 
 3.8 
 
 3.7 
 
 3.6 
 
 123 
 
 991 
 
 *026 
 
 *061 
 
 *096 
 
 *132 
 
 *167 
 
 *202 
 
 *237 
 
 *272 
 
 *307 
 
 2 
 
 7.6 
 
 7!4 
 
 7.2 
 
 124 
 
 09 342 
 
 377 
 
 412 
 
 447 
 
 482 
 
 517 
 
 552 
 
 587 
 
 621 
 
 656 
 
 3 
 
 11.4 
 
 11.1 
 
 10.8 
 
 125 
 
 691 
 
 726 
 
 760 
 
 795 
 
 830 
 
 864 
 
 899 
 
 934 
 
 968 
 
 *003 
 
 4 
 
 15.2 
 
 14.8 
 
 14.4 
 
 126 
 
 10 037 
 
 072 
 
 106 
 
 140 
 
 175 
 
 209 
 
 243 
 
 278 
 
 312 
 
 346 
 
 5 
 
 19.0 
 
 18.5 
 
 18.0 
 
 127 
 
 380 
 
 415 
 
 449 
 
 483 
 
 517 
 
 551 
 
 585 
 
 619 
 
 653 
 
 687 
 
 6 
 
 7 
 
 22.8 
 
 26.6 
 
 22.2 
 
 25.9 
 
 21. (> 
 25.2 
 
 128 
 
 721 
 
 755 
 
 789 
 
 823 
 
 857 
 
 890 
 
 924 
 
 958 
 
 992 
 
 *025 
 
 8 
 
 30^4 
 
 29 6 
 
 28^ 
 
 129 
 
 11 059 
 
 093 
 
 126 
 
 160 
 
 193 
 
 227 
 
 261 
 
 294 
 
 327 
 
 361 
 
 9 
 
 34.2 
 
 33.3 
 
 32.4 
 
 130 
 
 394 
 
 428 
 
 461 
 
 494 
 
 528 
 
 561 
 
 594 
 
 628 
 
 661 
 
 694 
 
 
 
 
 
 131 
 
 727 
 
 760 
 
 793 
 
 826 
 
 860 
 
 893 
 
 926 
 
 959 
 
 992 
 
 *024 
 
 
 35 
 
 34 
 
 33 
 
 132 
 
 12 057 
 
 090 
 
 123 
 
 156 
 
 189 
 
 222 
 
 254 
 
 287 
 
 320 
 
 352 
 
 1 
 
 3.5 
 
 3.4 
 
 3.3 
 
 133 
 
 385 
 
 418 
 
 450 
 
 483 
 
 516 
 
 548 
 
 581 
 
 613 
 
 646 
 
 678 
 
 2 
 
 7!o 
 
 6*8 
 
 6^6 
 
 134 
 
 710 
 
 743 
 
 775 
 
 808 
 
 840 
 
 872 
 
 905 
 
 937 
 
 969 
 
 *001 
 
 3 
 
 10.5 
 
 10.2 
 
 9.9 
 
 135 
 
 13 033 
 
 066 
 
 098 
 
 130 
 
 162 
 
 194 
 
 226 
 
 258 
 
 290 
 
 322 
 
 4 
 
 14.0 
 
 13.6 
 
 13.2 
 
 136 
 
 354 
 
 386 
 
 418 
 
 450 
 
 481 
 
 513 
 
 545 
 
 577 
 
 609 
 
 640 
 
 5 
 
 17.5 
 
 17.0 
 
 *)A A 
 
 16.5 
 
 TOO 
 
 137 
 
 672 
 
 704 
 
 735 
 
 767 
 
 799 
 
 830 
 
 862 
 
 893 
 
 925 
 
 956 
 
 6 
 
 7 
 
 21.0 
 
 24.5 
 
 zU.4 
 
 23.8 
 
 19.0 
 
 23.1 
 
 138 
 
 988 
 
 *019 
 
 *051 
 
 *082 
 
 *114 
 
 *145 
 
 *176 
 
 *208 
 
 *239 
 
 *270 
 
 8 
 
 28^0 
 
 27.2 
 
 26^4 
 
 139 
 
 14 301 
 
 333 
 
 364 
 
 395 
 
 426 
 
 457 
 
 489 
 
 520 
 
 551 
 
 582 
 
 9 
 
 31.5 
 
 30.6 
 
 29.7 
 
 140 
 
 613 
 
 644 
 
 675 
 
 706 
 
 737 
 
 768 
 
 799 
 
 829 
 
 860 
 
 891 
 
 
 
 
 
 141 
 
 922 
 
 953 
 
 983 
 
 *014 
 
 *045 
 
 *076 
 
 *106 
 
 *137 
 
 *168 
 
 *198 
 
 
 32 
 
 31 
 
 30 
 
 142 
 
 15 229 
 
 259 
 
 290 
 
 320 
 
 351 
 
 381 
 
 412 
 
 442 
 
 473 
 
 503 
 
 1 
 
 3.2 
 
 3.1 
 
 3.0 
 
 143 
 
 534 
 
 564 
 
 594 
 
 625 
 
 655 
 
 685 
 
 715 
 
 746 
 
 776 
 
 806 
 
 2 
 
 6^4 
 
 6^2 
 
 6^0 
 
 144 
 
 836 
 
 866 
 
 897 
 
 '927 
 
 957 
 
 987 
 
 *017 
 
 *047 
 
 *077 
 
 *107 
 
 3 
 
 9.6 
 
 9.3 
 
 9.0 
 
 145 
 
 16 137 
 
 167 
 
 197 
 
 227 
 
 256 
 
 286 
 
 316 
 
 346 
 
 376 
 
 406 
 
 4 
 
 12.8 
 
 12.4 
 
 12.0 
 
 146 
 
 435 
 
 465 
 
 495 
 
 524 
 
 554 
 
 584 
 
 613 
 
 643 
 
 673 
 
 702 
 
 5 
 
 16.0 
 
 15.5 
 
 15.0 
 
 147 
 
 732 
 
 761 
 
 791 
 
 820 
 
 850 
 
 879 
 
 909 
 
 938 
 
 967 
 
 997 
 
 6 
 
 7 
 
 19.2 
 
 22.4 
 
 18.6 
 
 21.7 
 
 18.0 
 
 21.0 
 
 148 
 
 17 026 
 
 056 
 
 085 
 
 114 
 
 143 
 
 173 
 
 202 
 
 231 
 
 260 
 
 289 
 
 8 
 
 25^6 
 
 24^8 
 
 24^0 
 
 149 
 
 319 
 
 348 
 
 377 
 
 406 
 
 435 
 
 464 
 
 493 
 
 522 
 
 551 
 
 580 
 
 9 
 
 28.8 
 
 27.9 
 
 27.0 
 
 150 
 
 609 
 
 638 
 
 667 
 
 696 
 
 725 
 
 754 
 
 782 
 
 811 
 
 840 
 
 869 
 
 
 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
LOGARITHMS. 
 
 475 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 P.P 
 
 • 
 
 150 1 
 
 7 609 
 
 638 
 
 667 
 
 696 
 
 725 
 
 754 
 
 782 
 
 811 
 
 840 
 
 869 
 
 
 
 
 
 151 
 
 898 
 
 926 
 
 955 
 
 984 * 
 
 <013 * 
 
 <041 * 
 
 *070 =< 
 
 <099 * 
 
 *127 * 
 
 *156 
 
 
 29 
 
 28 
 
 152 
 
 .8 184 
 
 213 
 
 241 
 
 270 
 
 298 
 
 327 
 
 355 
 
 384 
 
 412 
 
 441 
 
 i 
 
 2.9 
 
 2.8 
 
 153 
 
 469 
 
 498 
 
 526 
 
 554 
 
 583 
 
 611 
 
 639 
 
 667 
 
 696 
 
 724 
 
 2 
 
 5.8 
 
 5.6 
 
 154 
 
 752 
 
 780 
 
 808 
 
 837 
 
 865 
 
 893 
 
 921 
 
 949 
 
 977 * 
 
 *005 
 
 3 
 
 8.7 
 
 8.4 
 
 155 
 
 L 9 033 
 
 061 
 
 089 
 
 117 
 
 145 
 
 173 
 
 201 
 
 229 
 
 257 
 
 285 
 
 4 
 
 11.6 
 1 A 
 
 11.2 
 
 110 
 
 156 
 
 312 
 
 340 
 
 368 
 
 396 
 
 424 
 
 451 
 
 479 
 
 507 
 
 535 
 
 562 
 
 5 
 
 6 
 
 14.0 
 
 17.4 
 
 16.8 
 
 157 
 
 590 
 
 618 
 
 645 
 
 673 
 
 700 
 
 728 
 
 756 
 
 783 
 
 811 
 
 838 
 
 7 
 
 20.3 
 
 19.6 
 
 158 
 
 866 
 
 893 
 
 921 
 
 948 
 
 976 
 
 *003 5 
 
 *030 * 
 
 *058 5 
 
 *085 5 
 
 *112 
 
 8 
 
 23.2 
 
 22.4 
 
 159 
 
 20 140 
 
 167 
 
 194 
 
 222 
 
 249 
 
 276 
 
 303 
 
 330 
 
 358 
 
 385 
 
 9 
 
 26.1 
 
 25.2 
 
 160 
 
 412 
 
 439 
 
 466 
 
 493 
 
 520 
 
 548 
 
 575 
 
 602 
 
 629 
 
 656 
 
 
 
 
 
 161 
 
 683 
 
 710 
 
 737 
 
 763 
 
 790 
 
 817 
 
 844 
 
 871 
 
 898 
 
 925 
 
 
 27 
 
 26 
 
 162 
 
 952 
 
 978 
 
 *005 
 
 *032 
 
 *059 
 
 *085 : 
 
 *112 = 
 
 *139 : 
 
 *165 : 
 
 *192 
 
 1 
 
 2.7 
 
 2.6 
 
 163 
 
 21 219 
 
 245 
 
 272 
 
 299 
 
 325 
 
 352 
 
 378 
 
 405 
 
 431 
 
 458 
 
 2 
 
 5.4 
 
 5.2 
 
 164 
 
 484 
 
 511 
 
 537 
 
 564 
 
 590 
 
 617 
 
 643 
 
 669 
 
 696 
 
 722 
 
 3 
 
 8.1 
 
 7.8 
 
 165 
 
 748 
 
 775 
 
 801 
 
 827 
 
 854 
 
 880 
 
 906 
 
 932 
 
 958 
 
 985 
 
 4 
 
 10.8 
 IQ C 
 
 10.4 
 
 IQ ft 
 
 166 
 
 22 Oil 
 
 037 
 
 063 
 
 089 
 
 115 
 
 141 
 
 167 
 
 194 
 
 220 
 
 246 
 
 5 
 
 6 
 
 lo.D 
 
 16.2 
 
 lo.u 
 
 15.6 
 
 167 
 
 272 
 
 298 
 
 324 
 
 350 
 
 376 
 
 401 
 
 427 
 
 453 
 
 479 
 
 505 
 
 7 
 
 18.9 
 
 18.2 
 
 168 
 
 531 
 
 557 
 
 583 
 
 608 
 
 634 
 
 660 
 
 686 
 
 712 
 
 737 
 
 763 
 
 8 
 
 21.6 
 
 20.8 
 
 169 
 
 789 
 
 814 
 
 840 
 
 866 
 
 891 
 
 917 
 
 943 
 
 968 
 
 994 
 
 *019 
 
 9 
 
 24.3 
 
 23,4 
 
 170 
 
 23 045 
 
 070 
 
 096 
 
 121 
 
 147 
 
 172 
 
 198 
 
 223 
 
 249 
 
 274 
 
 
 
 
 
 171 
 
 300 
 
 325 
 
 350 
 
 376 
 
 401 
 
 426 
 
 452 
 
 477 
 
 502 
 
 528 
 
 
 
 25 
 
 172 
 
 553 
 
 578 
 
 603 
 
 629 
 
 654 
 
 679 
 
 704 
 
 729 
 
 754 
 
 779 
 
 
 1 
 
 2.5 
 
 173 
 
 805 
 
 830 
 
 855 
 
 880 
 
 905 
 
 930 
 
 955 
 
 980 
 
 *005 
 
 *030 
 
 
 2 
 
 5.0 
 
 174 
 
 24 055 
 
 080 
 
 105 
 
 130 
 
 155 
 
 180 
 
 204 
 
 229 
 
 254 
 
 279 
 
 
 3 
 
 7.5 
 
 175 
 
 304 
 
 329 
 
 353 
 
 378 
 
 403 
 
 428 
 
 452 
 
 477 
 
 502 
 
 527 
 
 
 4 
 
 10.0 
 1 o K 
 
 176 
 
 551 
 
 576 
 
 601 
 
 625 
 
 650 
 
 674 
 
 699 
 
 724 
 
 748 
 
 773 
 
 
 O 
 
 6 
 
 LA.O 
 
 15.0 
 
 .177 
 
 797 
 
 822 
 
 846 
 
 871 
 
 895 
 
 920 
 
 944 
 
 969 
 
 993 
 
 *018 
 
 
 7 
 
 17.5 
 
 178 
 
 25 042 
 
 066 
 
 091 
 
 115 
 
 139 
 
 164 
 
 188 
 
 212 
 
 237 
 
 261 
 
 
 8 
 
 20.0 
 
 179 
 
 285 
 
 310 
 
 334 
 
 358 
 
 382 
 
 406 
 
 431 
 
 455 
 
 479 
 
 503 
 
 
 9 
 
 22.5 
 
 180 
 
 527 
 
 551 
 
 575 
 
 600 
 
 624 
 
 648 
 
 672 
 
 696 
 
 720 
 
 744 
 
 
 
 
 
 181 
 
 768 
 
 792 
 
 816 
 
 840 
 
 864 
 
 888 
 
 912 
 
 935 
 
 959 
 
 983 
 
 
 24 
 
 23 
 
 182 
 
 26 007 
 
 031 
 
 055 
 
 079 
 
 102 
 
 126 
 
 150 
 
 174 
 
 198 
 
 221 
 
 1 
 
 2.4 
 
 2.3 ! 
 
 183 
 
 245 
 
 269 
 
 293 
 
 316 
 
 340 
 
 364 
 
 387 
 
 411 
 
 435 
 
 458 
 
 2 
 
 4.8 
 
 4.6 
 
 184 
 
 482 
 
 505 
 
 529 
 
 553 
 
 576 
 
 600 
 
 623 
 
 647 
 
 670 
 
 694 
 
 3 
 
 7.2 
 
 6.9 
 
 185 
 
 717 
 
 741 
 
 764 
 
 788 
 
 811 
 
 834 
 
 858 
 
 881 
 
 905 
 
 928 
 
 4 
 
 K 
 
 9.6 
 
 9.2 
 
 11.5 
 
 186 
 
 951 
 
 975 
 
 998 
 
 *021 
 
 *045 
 
 *068 
 
 *091 
 
 *114 
 
 *138 
 
 *161 
 
 o 
 
 6 
 
 LA .v 
 
 14.4 
 
 13^8 
 
 187 
 
 27 184 
 
 : 207 
 
 231 
 
 254 
 
 277 
 
 300 
 
 323 
 
 346 
 
 370 
 
 393 
 
 7 
 
 16.8 
 
 16.1 
 
 188 
 
 416 
 
 - 439 
 
 462 
 
 485 
 
 508 
 
 531 
 
 554 
 
 577 
 
 600 
 
 623 
 
 8 
 
 19.2 
 
 18.4 
 
 189 
 
 646 
 
 1 669 
 
 692 
 
 715 
 
 738 
 
 761 
 
 784 
 
 807 
 
 830 
 
 852 
 
 9 
 
 21.6 
 
 20.7 
 
 190 
 
 875 
 
 > 898 
 
 921 
 
 944 
 
 967 
 
 989 
 
 *012 
 
 *035 
 
 *058 
 
 *081 
 
 
 
 
 
 191 
 
 28 102 
 
 ; 126 
 
 149 
 
 171 
 
 194 
 
 217 
 
 240 
 
 262 
 
 285 
 
 307 
 
 
 22 
 
 2! 
 
 192 
 
 33 C 
 
 > 353 
 
 375 
 
 398 
 
 421 
 
 443 
 
 466 
 
 488 
 
 511 
 
 533 
 
 1 
 
 2.2 
 
 2.1 
 
 193 
 
 | 55 ( 
 
 > 578 
 
 601 
 
 623 
 
 646 
 
 668 
 
 691 
 
 713 
 
 735 
 
 758 
 
 2 
 
 4.4 
 
 4.2 
 
 194 
 
 78 C 
 
 ) 803 
 
 825 
 
 847 
 
 870 
 
 892 
 
 914 
 
 937 
 
 959 
 
 981 
 
 3 
 
 66 
 
 6.3 
 
 195 
 
 29 002 
 
 5 026 
 
 048 
 
 070 
 
 092 
 
 115 
 
 137 
 
 159 
 
 181 
 
 203 
 
 4 
 
 8.8 
 
 11.0 
 
 8.4 
 
 10.5 
 
 196 
 
 226 
 
 > 248 
 
 270 
 
 292 
 
 314 
 
 336 
 
 358 
 
 380 
 
 403 
 
 425 
 
 o 
 
 6 
 
 13.2 
 
 12.6 
 
 197 
 
 
 1 469 
 
 491 
 
 513 
 
 535 
 
 557 
 
 579 
 
 601 
 
 623 
 
 645 
 
 7 
 
 15.4 
 
 14.7 
 
 198 
 
 66 - 
 
 1 688 
 
 710 
 
 732 
 
 754 
 
 776 
 
 798 
 
 820 
 
 842 
 
 863 
 
 8 
 
 17.6 
 
 16.8 
 
 199 
 
 882 
 
 > 907 
 
 929 
 
 951 
 
 973 
 
 994 
 
 *016 
 
 *038 
 
 *060 
 
 *081 
 
 9 
 
 19.8 
 
 18.9 
 
 200 
 
 30 102 
 
 J 125 
 
 146 
 
 168 
 
 190 
 
 211 
 
 233 
 
 255 
 
 276 
 
 298 
 
 
 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
476 
 
 LOGARITHMS . 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 200 
 
 30 103 
 
 125 
 
 146 
 
 168 
 
 190 
 
 211 
 
 233 
 
 255 
 
 276 
 
 298 
 
 22 21 
 
 1 2.2 2.1 
 
 2 4.4 4.2 
 
 3 6.6 6.3 
 
 4 8.8 8.4 
 
 5 11.0 10.5 
 
 - 6 13.2 12.6 
 
 7 15.4 14.7 
 
 8 17.8 16.8 
 
 9 19.8 18.9 
 
 20 
 
 1 2.0 
 
 2 4.0 
 
 3 6.0 
 
 4 8.0 
 
 5 10.0 
 
 6 12.0 
 
 7 14.0 
 
 8 16.0 
 
 9 18.0 
 
 19 
 
 1 1.9 
 
 2 3.8 
 
 3 5.7 
 
 4 7.6 
 
 5 9.5 
 
 6 11.4 
 
 7 13.3 
 
 8 15.2 
 
 9 17.1 
 
 18 
 
 1 1.8 
 
 2 3.6 
 
 3 5.4 
 
 4 7.2 
 
 5 9.0 
 
 6 10.8 
 
 7 12.6 
 
 8 14.4 
 
 9 16.2 
 
 17 
 
 1 1.7 
 
 2 34 
 
 3 5.1 
 
 4 6.8 
 
 5 8.5 
 
 6 10.2 
 
 7 11.9 
 
 ! 8 13.6 
 
 9 15.3 
 
 201 
 
 202 
 
 203 
 
 204 
 
 205 
 
 206 
 
 207 
 
 208 
 
 209 
 
 210 
 
 320 
 
 535 
 
 750 
 
 963 
 
 31 175 
 387 
 597 
 806 
 
 32 015 
 
 341 
 
 557 
 
 771 
 
 984 
 
 197 
 
 408 
 
 618 
 
 827 
 
 035 
 
 363 
 
 578 
 
 792 
 
 *006 
 
 218 
 
 429 
 
 639 
 
 848 
 
 056 
 
 384 
 
 600 
 
 814 
 
 *027 
 
 239 
 
 450 
 
 660 
 
 869 
 
 077 
 
 406 
 
 621 
 
 835 
 
 *048 
 
 260 
 
 471 
 
 681 
 
 890 
 
 098 
 
 428 
 
 643 
 
 856 
 
 *069 
 
 281 
 
 492 
 
 702 
 
 911 
 
 118 
 
 449 
 
 664 
 
 878 
 
 *091 
 
 302 
 
 513 
 
 723 
 
 931 
 
 139 
 
 471 
 
 685 
 
 899 
 
 *112 
 
 323 
 
 534 
 
 744 
 
 952 
 
 160 
 
 492 
 
 707 
 
 920 
 
 *133 
 
 345 
 
 555 
 
 765 
 
 973 
 
 181 
 
 514 
 
 728 
 
 942 
 
 *154 
 
 366 
 
 576 
 
 785 
 
 994 
 
 201 
 
 222 
 
 243 
 
 263 
 
 284 
 
 305 
 
 325 
 
 346 
 
 366 
 
 387 
 
 408 
 
 211 
 
 212 
 
 213 
 
 214 
 
 215 
 
 216 
 
 217 
 
 218 
 219 
 
 220 
 
 221 
 
 222 
 
 223 
 
 224 
 
 225 
 
 226 
 
 227 
 
 228 
 229 
 
 230 
 
 231 
 
 232 
 
 233 
 
 234 
 
 235 
 
 236 
 
 237 
 
 238 
 
 239 
 
 240 
 
 241 
 
 242 
 
 243 
 
 244 
 
 245 
 
 246 
 
 247 
 
 248 
 
 249 
 
 250 
 
 428 
 
 634 
 
 838 
 
 33 041 
 244 
 445 
 646 
 846 
 
 34 044 
 
 449 
 
 654 
 
 858 
 
 062 
 
 264 
 
 465 
 
 666 
 
 866 
 
 064 
 
 469 
 
 675 
 
 879 
 
 082 
 
 284 
 
 486 
 
 686 
 
 885 
 
 084 
 
 490 
 
 695 
 
 899 
 
 102 
 
 304 
 
 506 
 
 706 
 
 905 - 
 
 104 
 
 510 
 
 715 
 
 919 
 
 122 
 
 325 
 
 526 
 
 726 
 
 925 
 
 124 
 
 531 
 
 736 
 
 940 
 
 143 
 
 345 
 
 546 
 
 746 
 
 945 
 
 143 
 
 552 
 
 756 
 
 960 
 
 163 
 
 365 
 
 566 
 
 766 
 
 965 
 
 163 
 
 572 
 
 777 
 
 980 
 
 183 
 
 385 
 
 586 
 
 786 
 
 985 
 
 183 
 
 593 
 
 797 
 
 *001 
 
 203 
 
 405 
 
 606 
 
 806 
 
 *005 
 
 203 
 
 613 
 
 818 
 
 *021 
 
 224 
 
 425 
 
 626 
 
 826 
 
 *025 
 
 223 
 
 242 
 
 262 
 
 282 
 
 301 
 
 321 
 
 341 
 
 361 
 
 380 
 
 400 
 
 420 
 
 439 
 635 
 830 
 35 025 
 218 
 411 
 603 
 793 
 984 
 
 459 
 
 655 
 
 850 
 
 044 
 
 238 
 
 430 
 
 622 
 
 813 
 
 *003 
 
 479 
 
 674 
 
 869 
 
 064 
 
 257 
 
 449 
 
 641 
 
 832 
 
 *021 
 
 498 
 
 694 
 
 889 
 
 083 
 
 276 
 
 468 
 
 660 
 
 851 
 
 *040 
 
 518 
 
 713 
 
 908 
 
 102 
 
 295 
 
 488 
 
 679 
 
 870 
 
 *059 
 
 537 
 
 733 
 
 928 
 
 122 
 
 315 
 
 507 
 
 698 
 
 889 
 
 *078 
 
 557 
 
 753 
 
 947 
 
 141 
 
 334 
 
 526 
 
 717 
 
 908 
 
 *097 
 
 577 
 
 772 
 
 967 
 
 160 
 
 353 
 
 545 
 
 736 
 
 927 
 
 *116 
 
 596 
 
 792 
 
 986 
 
 180 
 
 372 
 
 564 
 
 755 
 
 946 
 
 *135 
 
 616 
 
 811 
 
 *005 
 
 199 
 
 392 
 
 583 
 
 774 
 
 965 
 
 *154 
 
 36 173 
 
 192 
 
 211 
 
 229 
 
 248 
 
 267 
 
 286 
 
 305 
 
 324 
 
 342 
 
 361 
 549 
 736 
 922 
 37 107 
 291 
 475 
 658 
 840 
 
 380 
 
 568 
 
 754 
 
 940 
 
 125 
 
 310 
 
 493 
 
 676 
 
 858 
 
 399 
 
 586 
 
 773 
 
 959 
 
 144 
 
 328 
 
 511 
 
 694 
 
 876 
 
 418 
 
 605 
 
 791 
 
 977 
 
 162 
 
 346 
 
 530 
 
 712 
 
 894 
 
 436 
 
 624 
 
 810 
 
 996 
 
 181 
 
 365 
 
 548 
 
 731 
 
 912 
 
 455 
 
 642 
 
 829 
 
 *014 
 
 199 
 
 383 
 
 566 
 
 749 
 
 931 
 
 474 
 
 661 
 
 847 
 
 *033 
 
 218 
 
 401 
 
 585 
 
 767 
 
 949 
 
 493 
 
 680 
 
 866 
 
 *051 
 
 236 
 
 420 
 
 603 
 
 785 
 
 967 
 
 511 
 
 698 
 
 884 
 
 *070 
 
 254 
 
 438 
 
 621 
 
 803 
 
 985 
 
 530 
 
 717 
 
 903 
 
 *088 
 
 273 
 
 457 
 
 639 
 
 822 
 
 *003 
 
 38 021 
 
 039 
 
 057 
 
 075 
 
 093 
 
 112 
 
 130 
 
 148 
 
 166 
 
 184 
 
 202 
 382 
 561 
 739 
 917 
 39 094 
 270 
 445 
 ! 62 C 
 
 ! 220 
 ; 399 
 578 
 i 757 
 934 
 111 
 » 287 
 > 463 
 1 637 
 
 238 
 
 417 
 
 596 
 
 775 
 
 952 
 
 129 
 
 305 
 
 480 
 
 655 
 
 256 
 
 435 
 
 614 
 
 792 
 
 970 
 
 146 
 
 322 
 
 498 
 
 672 
 
 274 
 
 453 
 
 632 
 
 810 
 
 987 
 
 164 
 
 340 
 
 515 
 
 690 
 
 292 
 
 471 
 
 650 
 
 828 
 
 *005 
 
 182 
 
 358 
 
 533 
 
 707 
 
 310 
 489 
 ■ 668 
 846 
 *023 
 199 
 375 
 550 
 724 
 
 328 
 
 507 
 
 686 
 
 863 
 
 *041 
 
 217 
 
 393 
 
 568 
 
 742 
 
 346 
 
 525 
 
 703 
 
 881 
 
 *058 
 
 235 
 
 410 
 
 585 
 
 759 
 
 364 
 
 543 
 
 721 
 
 899 
 
 *076 
 
 252 
 
 428 
 
 602 
 
 777 
 
 794 
 
 t 811 
 
 829 
 
 846 
 
 863 
 
 881 
 
 898 
 
 915 
 
 933 
 
 950 
 
 K. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P.P. 
 
LOGARITHMS. 
 
 477 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 250 : 
 
 251 
 
 252 
 
 253 
 
 254 
 
 255 
 
 256 
 
 257 
 
 258 
 
 259 
 
 260 
 
 261 
 
 262 
 
 263 
 
 264 
 
 265 
 
 266 
 
 267 
 
 268 
 269 
 
 270 
 
 271 
 
 272 
 
 273 
 
 274 
 
 275 
 
 276 
 •‘277 
 
 278 
 
 279 
 
 280 
 
 281 
 
 282 
 
 283 
 
 284 
 
 285 
 
 286 
 
 287 
 
 288 
 289 
 
 290 
 
 291 
 
 292 
 
 293 
 
 294 
 
 295 
 
 296 
 
 297 
 
 298 
 
 299 
 
 300 
 
 59 794 
 
 811 
 
 829 
 
 846 
 
 863 
 
 881 
 
 898 
 
 915 
 
 933 
 
 950 
 
 18 
 
 1 1.8 
 
 2 3.6 
 
 3 5.4 
 
 4 7.2 
 
 5 9.0 
 
 6 10.8 
 7 12.6 
 
 8 14.4 
 
 9 16.2 
 
 17 
 
 1 1.7 
 
 2 3.4 
 
 3 5.1 
 
 4 6.8 
 
 5 8.5 
 
 6 10.2 
 
 7 11.9 
 
 8 13.6 
 
 9 15.3 
 
 16 
 
 1 1.6 
 
 2 3.2 
 
 3 4.8 
 
 4 6.4 
 
 5 8.0 
 
 6 9.6 
 
 7 11.2 
 
 8 12.8 
 
 9 14.4 
 
 15 
 
 1 1.5 
 
 2 3.0 
 
 3 4.5 
 
 4 6.0 
 
 5 7,5 
 
 6 9.0 
 
 7 10.5 
 
 8 12.0 
 
 9 13.5 
 
 14 
 
 1 1.4 
 
 2 2 8 
 
 3 4.2 
 
 4 5.6 
 
 5 7.0 
 
 6 8.4 
 
 7 9.8 
 
 8 11.2 
 
 9 12.6 
 
 967 
 10 140 
 312 
 483 
 654 
 824 
 993 
 41 162 
 330 
 
 985 
 
 157 
 
 329 
 
 500 
 
 671 
 
 841 
 
 *010 
 
 179 
 
 347 
 
 *002 
 
 175 
 
 346 
 
 518 
 
 688 
 
 858 
 
 *027 
 
 196 
 
 363 
 
 *019 
 
 192 
 
 364 
 
 535 
 
 705 
 
 875 
 
 *044 
 
 212 
 
 380 
 
 *037 
 
 209 
 
 381 
 
 552 
 
 722 
 
 892 
 
 *061 
 
 229 
 
 397 
 
 *054 5 
 226 
 398 
 569 
 739 
 909 
 *078 : 
 246 
 414 
 
 *071 5 
 243 
 415 
 586 
 756 
 926 
 *095 ; 
 263 
 430 
 
 *088 s 
 261 
 432 
 603 
 773 
 943 
 *111 : 
 280 
 447 
 
 *106 5 
 278 
 449 
 620 
 790 
 960 
 *128 ; 
 296 
 464 
 
 *123 
 
 295 
 
 466 
 
 637 
 
 807 
 
 976 
 
 *145 
 
 313 
 
 481 
 
 497 
 
 514 
 
 531 
 
 547 
 
 564 
 
 581 
 
 597 
 
 614 
 
 631 
 
 647 
 
 664 
 830 
 996 
 42 160 
 325 
 488 
 651 
 813 
 975 
 
 681 | 
 847 
 *012 
 177 
 341 
 504 
 667 
 830 
 991 
 
 697 
 
 863 
 
 *029 
 
 193 
 
 357 
 
 521 
 
 684 
 
 846 
 
 *008 
 
 714 
 
 880 
 
 *045 
 
 210 
 
 374 
 
 537 
 
 700 
 
 862 
 
 *024 
 
 731 
 
 896 
 
 *062 
 
 226 
 
 390 
 
 553 
 
 716 
 
 878 
 
 *040 
 
 747 
 
 913 
 
 *078 
 
 243 
 
 406 
 
 570 
 
 732 
 
 894 
 
 *056 
 
 764 
 
 929 
 
 *095 
 
 259 
 
 423 
 
 586 
 
 749 
 
 911 
 
 *072 
 
 780 
 
 946 
 
 *111 
 
 275 
 
 439 
 
 602 
 
 765 
 
 927 
 
 *088 
 
 797 
 
 963 
 
 *127 
 
 292 
 
 455 
 
 619 
 
 781 
 
 943 
 
 *104 
 
 814 
 
 979 
 
 *144 
 
 308 
 
 472 
 
 635 
 
 797 
 
 959 
 
 *120 
 
 43 136 
 
 152 
 
 169 
 
 185 
 
 201 
 
 217 
 
 233 
 
 249 
 
 265 
 
 281 
 
 297 
 457 
 616 
 775 
 933 
 44 091 
 248 
 404 
 560 
 
 313 
 
 473 
 
 632 
 
 791 
 
 949 
 
 107 
 
 264 
 
 420 
 
 576 
 
 329 
 
 489 
 
 648 
 
 807 
 
 965 
 
 122 
 
 279 
 
 436 
 
 592 
 
 345 
 
 505 
 
 664 
 
 823 
 
 981 
 
 138 
 
 295 • 
 
 451 
 
 607 
 
 361 
 
 521 
 
 680 
 
 838 
 
 996 
 
 154 
 
 311 
 
 467 
 
 623 
 
 377 
 
 537 
 
 696 
 
 854 
 
 *012 
 
 170 
 
 326 
 
 483 
 
 638 
 
 393 
 
 553 
 
 712 
 
 870 
 
 *028 
 
 185 
 
 342 
 
 498 
 
 654 
 
 409 
 
 569 * 
 
 727 
 
 886 
 
 *044 
 
 201 
 
 358 
 
 514 
 
 669 
 
 425 
 
 584 
 
 743 
 
 902 
 
 *059 
 
 217 
 
 373 
 
 529 
 
 685 
 
 441 
 
 600 
 
 759 
 
 917 
 
 *075 
 
 232 
 
 389 
 
 545 
 
 700 
 
 716 
 
 731 
 
 747 
 
 762 
 
 778 
 
 793 
 
 809 
 
 824 
 
 840 
 
 855 
 
 871 
 
 45 025 
 179 
 332 
 484 
 637 
 788 
 939 
 
 46 090 
 
 886 
 040 
 194 
 347 
 500 
 652 
 803 
 954 
 i 105 
 
 902 
 
 056 
 
 209 
 
 362 
 
 515 
 
 667 
 
 818 
 
 969 
 
 120 
 
 917 
 
 071 
 
 225 
 
 378 
 
 530 
 
 682 
 
 834 
 
 984 
 
 135 
 
 932 
 
 086 
 
 240 
 
 393 
 
 545 
 
 697 
 
 849 
 
 *000 
 
 150 
 
 948 
 
 102 
 
 255 
 
 408 
 
 561 
 
 712 
 
 864 
 
 *015 
 
 165 
 
 963 
 
 117 
 
 271 
 
 423 
 
 576 
 
 728 
 
 879 
 
 *030 
 
 180 
 
 979 
 
 133 
 
 286 
 
 439 
 
 591 
 
 743 
 
 894 
 
 *045 
 
 195 
 
 994 
 
 148 
 
 301 
 
 454 
 
 606 
 
 758 
 
 909 
 
 *060 
 
 210 
 
 *010 
 
 163 
 
 317 
 
 469 
 
 621 
 
 773 
 
 924 
 
 *075 
 
 225 
 
 240 
 
 i 255 
 
 270 
 
 285 
 
 300 
 
 315 
 
 330 
 
 345 
 
 359 
 
 374 
 
 380 
 
 538 
 
 m 
 
 835 
 982 
 47 120 
 27 1 
 422 
 567 
 
 > 404 
 i 553 
 ' 702 
 
 > 850 
 5 997 
 ) 144 
 5 290 
 
 > 436 
 I 582 
 
 419 
 
 568 
 
 716 
 
 864 
 
 *012 
 
 159 
 
 305 
 
 451 
 
 596 
 
 434 
 
 583 
 
 731 
 
 879 
 
 *026 
 
 173 
 
 319 
 
 465 
 
 611 
 
 449 
 
 598 
 
 746 
 
 894 
 
 *041 
 
 188 
 
 334 
 
 480 
 
 625 
 
 464 
 
 613 
 
 761 
 
 909 
 
 *056 
 
 202 
 
 349 
 
 494 
 
 640 
 
 479 
 
 627 
 
 776 
 
 923 
 
 *070 
 
 217 
 
 363 
 
 509 
 
 654 
 
 494 
 
 642 
 
 790 
 
 938 
 
 *085 
 
 232 
 
 378 
 
 524 
 
 669 
 
 509 
 
 657 
 
 805 
 
 953 . 
 
 *100 
 
 246 
 
 392 
 
 538 
 
 683 
 
 523 
 
 672 
 
 820 
 
 967 
 
 *114 
 
 261 
 
 407 
 
 553 
 
 698 
 
 712 
 
 2 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
478 
 
 LOGARITHMS. 
 
 N. 
 
 L.oj 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 p. 
 
 P. 
 
 300 
 
 47 712 
 
 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 
 
 301 
 
 857 
 
 871 
 
 885 
 
 900 
 
 914 
 
 929 
 
 943 
 
 958 
 
 972 
 
 986 
 
 
 
 302 
 
 48 001 
 
 015 
 
 029 
 
 044 
 
 058 
 
 073 
 
 087 
 
 101 
 
 116 
 
 130 
 
 
 
 303 
 
 144 
 
 159 
 
 173 
 
 187 
 
 202 
 
 216 
 
 230 
 
 244 
 
 259 
 
 273 
 
 
 
 304 
 
 287 
 
 302 
 
 316 
 
 330 
 
 344 
 
 359 
 
 373 
 
 387 
 
 401 
 
 416 
 
 
 15 
 
 305 
 
 430 
 
 444 
 
 458 
 
 473 
 
 487 
 
 501 
 
 515 
 
 530 
 
 544 
 
 558 
 
 
 306 
 
 572 
 
 586 
 
 601 
 
 615 
 
 629 
 
 643 
 
 657 
 
 671 
 
 686 
 
 700 
 
 1 
 
 1.5 
 
 307 
 
 714 
 
 728 
 
 742 
 
 756 
 
 770 
 
 785 
 
 799 
 
 813 
 
 827 
 
 841 
 
 2 
 
 3.0 
 
 308 
 
 855 
 
 869 
 
 883 
 
 897 
 
 911 
 
 926 
 
 940 
 
 954 
 
 968 
 
 982 
 
 3 
 
 A 
 
 4.5 
 a n 
 
 309 
 
 996 
 
 *010 
 
 *024 
 
 *038 
 
 *052 
 
 *066 
 
 *080 
 
 *094 
 
 *108 
 
 *122 
 
 4 
 
 5 
 
 o.u 
 
 7.5 
 
 310 
 
 49 136 
 
 150 
 
 104 
 
 178 
 
 192 
 
 206 
 
 220 
 
 234 
 
 248 
 
 262 
 
 7 | 
 
 9.0 
 
 10.5 
 
 311 
 
 276 
 
 290 
 
 304 
 
 318 
 
 332 
 
 346 
 
 360 
 
 374 
 
 388 
 
 402 
 
 8 
 
 9 j 
 
 12.0 
 
 13.5 
 
 312 
 
 415 
 
 429 
 
 443 
 
 457 
 
 471 
 
 485 
 
 499 
 
 513 
 
 527 
 
 541 
 
 
 
 313 
 
 554 
 
 568 
 
 582 
 
 596 
 
 610 
 
 624 
 
 638 
 
 651 
 
 665 
 
 679 
 
 
 
 314 
 
 693 
 
 707 
 
 721 
 
 734 
 
 748 
 
 762 
 
 776 
 
 790 
 
 803 
 
 817 
 
 
 
 315 
 
 831 
 
 845 
 
 859 
 
 872 
 
 886 
 
 900 
 
 914 
 
 927 
 
 941 
 
 955 
 
 
 14 
 
 316 
 
 969 
 
 982 
 
 996 
 
 *010 
 
 *024 
 
 *037 
 
 *051 
 
 *065 
 
 *079 
 
 *092 
 
 
 317 
 
 50 106 
 
 120 
 
 133 
 
 147 
 
 161 
 
 174 
 
 188 
 
 202 
 
 215 
 
 229 
 
 1 
 
 1.4 
 
 318 
 
 243 
 
 256 
 
 270 
 
 284 
 
 297 
 
 311 
 
 325 
 
 338 
 
 352 
 
 365 
 
 2 
 
 2.8 
 
 319 
 
 379 
 
 393 
 
 406 
 
 420 
 
 433 
 
 447 
 
 461 
 
 474 
 
 488 
 
 501 
 
 3 
 
 4 
 
 4.2 
 
 5.6 
 
 320 
 
 515 
 
 529 
 
 542 
 
 556 
 
 569 
 
 583 
 
 596 
 
 610 
 
 623 
 
 | 637 
 
 5 
 
 6 
 
 7.0 
 
 8.4 
 
 321 
 
 651 
 
 664 
 
 678 
 
 691 
 
 705 
 
 718 
 
 732 
 
 745 
 
 759 
 
 772 
 
 7 
 
 g 
 
 9.8 
 
 11.2 
 
 322 
 
 786 
 
 799 
 
 813 
 
 S 2 & 
 
 840 
 
 853 
 
 866 
 
 880 
 
 893 
 
 907 
 
 9 
 
 12^ 
 
 1 \ - 
 
 323 
 
 920 
 
 934 
 
 947 
 
 961 
 
 974 
 
 987 
 
 *001 
 
 *014 
 
 *028 
 
 *041 
 
 
 324 
 
 51 0551 
 
 068 
 
 081 
 
 095 
 
 108 
 
 121 
 
 135 
 
 148 
 
 162 
 
 175 
 
 
 
 325 
 
 188 
 
 202 
 
 215 
 
 228 
 
 242 
 
 255 
 
 268 
 
 282 
 
 295 
 
 308 
 
 
 
 326 
 
 322 
 
 335 
 
 348 
 
 362 
 
 375 
 
 388 
 
 402 
 
 415 
 
 428 
 
 441 
 
 
 13 
 
 327 
 
 455 
 
 468 
 
 481 
 
 495 
 
 508 
 
 521 
 
 534 
 
 548 
 
 561 
 
 574 
 
 
 328 
 
 587 
 
 601 
 
 614 
 
 627 
 
 640 
 
 654 
 
 667 
 
 680 
 
 693 
 
 706 
 
 1 
 
 1.3 
 
 329 
 
 720 
 
 733 
 
 746 
 
 759 
 
 772 
 
 786 
 
 799 
 
 812 
 
 825 
 
 838 
 
 2 
 
 3 
 
 2.6 
 
 3.9 
 
 330 
 
 851 
 
 865 
 
 878 
 
 891 
 
 904 
 
 917 
 
 930 
 
 943 
 
 957 
 
 970 
 
 4 
 
 5 
 
 5 2 
 6.5 
 
 331 
 
 983 
 
 996 
 
 *009 
 
 *022 
 
 *035 
 
 *048 
 
 *061 
 
 *075 
 
 *088 
 
 *101 
 
 6 
 
 7.8 
 
 Q 1 
 
 332 
 
 52 114 
 
 127 
 
 140 
 
 153 
 
 166 
 
 179 
 
 192 
 
 205 
 
 218 
 
 231 
 
 7 
 
 g 
 
 10.4 
 
 333 
 
 244 
 
 257 
 
 270 
 
 284 
 
 297 
 
 310 
 
 323 
 
 336 
 
 349 
 
 362 
 
 9 
 
 11.7 
 
 334 
 
 375 
 
 388 
 
 401 
 
 414 
 
 427 
 
 440 
 
 453 
 
 466 
 
 479 
 
 492 
 
 
 
 335 
 
 504 
 
 517 
 
 530 
 
 543 
 
 556 
 
 569 
 
 582 
 
 595 
 
 608 
 
 621 
 
 
 
 336 
 
 634 
 
 S 647 
 
 660 
 
 673 
 
 686 
 
 699 
 
 711 
 
 724 
 
 737 
 
 750 
 
 
 
 337 
 
 763 
 
 ; 776 
 
 789 
 
 802 
 
 815 
 
 827 
 
 840 
 
 853 
 
 866 
 
 879 
 
 
 12 
 
 338 
 
 892 : 905 
 
 917 
 
 930 
 
 943 
 
 956 
 
 969 
 
 982 
 
 994 
 
 *007 
 
 
 339 
 
 53 020 
 
 | 033 
 
 046 
 
 058 
 
 071 
 
 084 
 
 097 
 
 | 110 
 
 122 
 
 135 
 
 1 
 
 t) 
 
 1.2 
 
 O I 
 
 340 
 
 148 
 
 1 161 
 
 173 
 
 186 
 
 199 
 
 212 
 
 224 
 
 237 
 
 250 
 
 263 
 
 L 
 
 3 
 
 i 
 
 A.T 
 
 3.6 
 
 A W 
 
 311 
 
 275 
 
 | 288 
 
 301 
 
 314 
 
 326 
 
 339 
 
 352 
 
 364 
 
 377 
 
 390 
 
 4 
 
 5 
 
 4.0 
 
 6.0 
 
 342 
 
 403 
 
 ! 415 
 
 428 
 
 441 
 
 453 
 
 466 
 
 479 
 
 491 
 
 504 
 
 517 
 
 6 
 
 7.2 
 
 343 
 
 529 
 
 ! 542 
 
 555 
 
 567 
 
 580 
 
 593 
 
 605 . 
 
 618 
 
 631 
 
 m 
 
 7 
 
 g 
 
 8.4 
 
 9.6 
 
 344 
 
 656 
 
 ; 668 
 
 681 
 
 694 
 
 706 
 
 719 
 
 732 
 
 744 
 
 757 
 
 769 
 
 9 
 
 10.8 
 
 345 
 
 782 
 
 794 
 
 807 
 
 820 
 
 832 
 
 845 
 
 857 
 
 870 
 
 882 
 
 895 
 
 
 
 346 
 
 908 
 
 920 
 
 933 
 
 945 
 
 958 
 
 970 
 
 983 
 
 995 
 
 *008 
 
 *020 
 
 
 
 347 
 
 54 033 
 
 045 
 
 058 
 
 070 
 
 083 
 
 095 
 
 108 
 
 120 
 
 133 
 
 145 
 
 
 
 348 
 
 158 
 
 170 
 
 183 
 
 195 
 
 208 
 
 220 
 
 233 
 
 245 
 
 258 
 
 270 
 
 
 
 349 
 
 283 
 
 295 
 
 307 
 
 320 
 
 332 
 
 345 
 
 357 
 
 370 
 
 382 
 
 394 
 
 
 
 350 
 
 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 518 
 
 
 
 N. 
 
 L 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
LOGARITHMS. 
 
 479 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 350 
 
 351 
 
 352 
 
 353 
 
 354 
 
 355 
 
 356 
 
 357 
 
 358 
 
 359 
 
 360 
 
 361 
 
 362 
 
 363 
 
 364 
 
 365 
 
 366 
 
 367 
 
 368 
 
 369 
 
 370 
 
 371 
 
 372 
 
 373 
 
 374 
 
 375 
 
 376 
 
 377 
 
 378 
 
 379 
 
 380 
 
 381 
 
 382 
 
 383 
 
 384 
 
 385 
 
 386 
 
 387 
 
 388 
 
 389 
 
 390 
 
 391 
 
 392 
 
 393 
 
 394 
 
 395 
 
 396 
 
 397 
 
 398 
 
 399 
 
 400 
 
 54 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 518 
 
 13 
 
 1 1.3 
 
 2 2.6 
 
 3 3.9 
 
 4 5.2 
 
 5 6.5 
 
 6 7.8 ' 
 
 7 9.1 
 
 8 10.4 
 
 9 11.7 
 
 12 
 
 1 1.2 
 
 2 2.4 
 
 3 3.6 
 
 4 4.8 
 
 5 6.0 
 
 6 7,2 
 
 7 8.4 
 
 8 9.6 
 
 9 10.8 
 
 II 
 
 1 1.1 
 2 2.2 
 
 3 3.3 
 
 4 4.4 
 
 5 5.5 
 
 6 6.6 
 
 7 7.7 
 
 8 8.8 
 
 9 9.9 
 
 10 
 
 1 1.0 
 2 2,0 
 
 3 3.0 
 
 4 4.0 
 
 5 5.0 
 
 6 6.0 
 
 7 7.0 
 
 8 8.0 
 
 9 9.0 
 
 531 
 654 
 777 
 900 
 55 023 
 145 
 267 
 388 
 509 
 
 543 
 
 667 
 
 790 
 
 913 
 
 035 
 
 157 
 
 279 
 
 400 
 
 522 
 
 555 
 
 679 
 
 802 
 
 925 
 
 047 
 
 169 
 
 291 
 
 413 
 
 534 
 
 568 
 
 691 
 
 814 
 
 937 
 
 060 
 
 182 
 
 303 
 
 425 
 
 546 
 
 580 
 
 704 
 
 827 
 
 949 
 
 072 
 
 194 
 
 315 
 
 437 
 
 558 
 
 593 
 
 716 
 
 839 
 
 962 
 
 084 
 
 206 
 
 328 
 
 449 
 
 570 
 
 605 
 
 728 
 
 851 
 
 974 
 
 096 
 
 218 
 
 340 
 
 461 
 
 582 
 
 617 
 
 741 
 
 864 
 
 986 
 
 108 
 
 230 
 
 352 
 
 473 
 
 594 
 
 630 
 753 
 876 
 998 ! 
 121 
 242 
 364 
 485 
 606 
 
 642 
 
 765 
 
 888 
 
 *011 
 
 133 
 
 255 
 
 376 
 
 497 
 
 618 
 
 630 
 
 642 
 
 654 
 
 666 
 
 678 
 
 691 
 
 703 
 
 715 
 
 727 
 
 739 
 
 751 
 871 
 991 
 56 110 
 229 
 348 
 467 
 585 
 703 
 
 763 
 
 883 
 
 *003 
 
 122 
 
 241 
 
 360 
 
 478 
 
 597 
 
 714 
 
 775 
 
 895 
 
 *015 
 
 134 
 
 253 
 
 372 
 
 490 
 
 608 
 
 726 
 
 787 
 
 907 
 
 *027 
 
 146 
 
 265 
 
 384 
 
 502 
 
 620 
 
 738 
 
 799 
 
 919 
 
 *038 
 
 158 
 
 277 
 
 396 
 
 514 
 
 632 
 
 750 
 
 811 
 
 931 
 
 *050 
 
 170 
 
 289 
 
 407 
 
 526 
 
 644 
 
 761 
 
 823 
 
 943 
 
 *062 
 
 182 
 
 301 
 
 419 
 
 538 
 
 656 
 
 773 
 
 835 
 
 955 
 
 *074 
 
 194 
 
 312 
 
 431 
 
 549 
 
 667 
 
 785 
 
 847 
 
 967 
 
 *086 
 
 205 
 
 324 
 
 443 
 
 561 
 
 679 
 
 797 
 
 859 
 
 979 
 
 *098 
 
 217 
 
 336 
 
 455 
 
 573 
 
 691 
 
 808 
 
 820 
 
 832 
 
 844 
 
 855 
 
 867 
 
 879 
 
 891 
 
 902 
 
 914 
 
 926 
 
 937 
 57 054 
 171 
 287 
 * 403 
 
 519 
 634 
 749 
 864 
 
 949 
 
 066 
 
 183 
 
 299. 
 
 415 
 
 530 
 
 646 
 
 761 
 
 875 
 
 961 
 
 078 
 
 194 
 
 310 
 
 426 
 
 542 
 
 657 
 
 772 
 
 887 
 
 972 
 
 089 
 
 206 
 
 322 
 
 438 
 
 553 
 
 669 
 
 784 
 
 898 
 
 984 
 
 101 
 
 217 
 
 334 
 
 449 
 
 565 
 
 680 
 
 795 
 
 910 
 
 996 
 
 113 
 
 229 
 
 345 
 
 461 
 
 576 
 
 692 
 
 807 
 
 921 
 
 *008 
 
 124 
 
 241 
 
 357 
 
 473. 
 
 588 
 
 703 
 
 818 
 
 933 
 
 *019 
 
 136 
 
 252 
 
 368 
 
 484 
 
 600 
 
 715 
 
 830 
 
 944 
 
 *031 
 
 148 
 
 264 
 
 380 
 
 496 
 
 611 
 
 726 
 
 841 
 
 955 
 
 *043 
 
 159 
 
 276 
 
 392 
 
 507 
 
 623 
 
 738 
 
 852 
 
 967 
 
 978 
 
 990 
 
 *001 
 
 *013 
 
 *024 
 
 *035 
 
 *047 
 
 *058 
 
 *070 
 
 *081 
 
 58 092 
 206 
 320 
 433 
 546 
 659 
 771 
 883 
 995 
 
 104 
 
 218 
 
 331 
 
 444 
 
 557 
 
 670 
 
 782 
 
 894 
 
 *006 
 
 115 
 
 229 
 
 343 
 
 456 
 
 569 
 
 •681 
 
 794 
 
 906 
 
 *017 
 
 127 
 
 240 
 
 354 
 
 467 
 
 580 
 
 692 
 
 805 
 
 917 
 
 *028 
 
 138 
 
 252 
 
 365 
 
 478 
 
 591 
 
 704 
 
 816 
 
 928 
 
 *040 
 
 149 
 
 263 
 
 377 
 
 490 
 
 602 
 
 715 
 
 827 
 
 939 
 
 *051 
 
 161 
 
 274 
 
 388 
 
 501 
 
 614 
 
 726 
 
 838 
 
 950 
 
 *062 
 
 172 
 
 286 
 
 399 
 
 512 
 
 625 
 
 737 
 
 850 
 
 961 
 
 *073 
 
 184 
 
 297 
 
 410 
 
 524 
 
 636 
 
 749 
 
 861 
 
 973 
 
 *084 
 
 195 
 
 309 
 
 422 
 
 535 
 
 647 
 
 760 
 
 872 
 
 984 
 
 *095 
 
 59 106 
 
 118 
 
 129 
 
 140 
 
 151 
 
 162 
 
 173 
 
 184 
 
 195 
 
 207 
 
 218 
 
 329 
 
 439 
 
 55C 
 
 66C 
 
 77C 
 
 87? 
 
 m 
 
 60 
 
 229 
 340 
 450 
 l 561 
 1 671 
 1 780 
 ) 890 
 5 999 
 ’ 108 
 
 240 
 
 351 
 
 461 
 
 572 
 
 682 
 
 791 
 
 901 
 
 *010 
 
 119 
 
 251 
 
 362 
 
 472 
 
 583 
 
 693 
 
 802 
 
 912 
 
 *021 
 
 130 
 
 262 
 
 373 
 
 483 
 
 594 
 
 704 
 
 813 
 
 923 
 
 *032 
 
 141 
 
 273 
 
 384 
 
 494 
 
 605 
 
 715 
 
 824 
 
 934 
 
 *043 
 
 152 
 
 284 
 
 395 
 
 506 
 
 616 
 
 726 
 
 835 
 
 945 
 
 *054 
 
 163 
 
 295 
 
 406 
 
 517 
 
 627 
 
 737 
 
 846 
 
 956 
 
 *065 
 
 173 
 
 306 
 
 417 
 
 528 
 
 638 
 
 748 
 
 857 
 
 966 
 
 *076 
 
 184 
 
 318 
 
 428 
 
 539 
 
 649 
 
 759 
 
 868 
 
 977 
 
 *086 
 
 195 
 
 20( 
 
 3 217 
 
 228 
 
 239 
 
 249 
 
 260 
 
 271 
 
 282 
 
 293 
 
 304 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
480 
 
 LOGARITHMS. 
 
 N. 
 
 L 0 
 
 | 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 400 
 
 50 206 
 
 217 
 
 228. 
 
 239 
 
 249 
 
 260 
 
 271 
 
 282 
 
 293 
 
 304 
 
 
 
 401 
 
 314 
 
 325 
 
 336 
 
 347 
 
 358 
 
 369 
 
 379 
 
 390 
 
 401 
 
 412 
 
 
 
 402 
 
 423 
 
 433 
 
 444 
 
 455 
 
 466 
 
 477 
 
 487 
 
 498 
 
 509 
 
 520 
 
 
 
 403 
 
 531 
 
 541 
 
 552 
 
 563 
 
 574 
 
 584 
 
 595 
 
 606 
 
 617 
 
 627 
 
 
 
 404 
 
 638 
 
 649 
 
 660 
 
 670 
 
 681 
 
 692 
 
 703 
 
 713 
 
 724 
 
 735 
 
 
 
 405 
 
 746 
 
 756 
 
 767 
 
 778 
 
 788 
 
 799 
 
 810 
 
 821 
 
 831 
 
 842 
 
 
 
 406 
 
 853 
 
 863 
 
 874 
 
 885 
 
 895 
 
 906 
 
 917 
 
 927 
 
 938 
 
 949 
 
 
 H 
 
 407 
 
 959 
 
 970 
 
 981 
 
 991 
 
 *002 
 
 *013 
 
 *023 
 
 *034 
 
 *045 
 
 *055 
 
 
 
 408 
 
 61 066 
 
 077 
 
 087 
 
 098 
 
 109 
 
 119 
 
 130 
 
 140 
 
 151 
 
 162 
 
 l 
 
 i.i 
 
 409 
 
 172 
 
 183 
 
 194 
 
 204 
 
 215 
 
 225 
 
 236 
 
 247 
 
 257 
 
 268 
 
 2 
 
 3 
 
 2.2 
 
 3.3 
 
 410 
 
 278 
 
 289 
 
 300 
 
 310 
 
 321 
 
 331 
 
 342 
 
 352 
 
 363 
 
 374 
 
 4 
 
 5 
 
 4.4 
 
 5.5 
 
 411 
 
 384 
 
 395 
 
 405 
 
 416 
 
 426 
 
 437 
 
 448 
 
 458 
 
 469 
 
 479 
 
 6 
 
 6.6 
 
 412 
 
 490 
 
 500 
 
 511 
 
 521 
 
 532 
 
 542 
 
 553 
 
 563 
 
 574 
 
 584 
 
 7 
 
 g 
 
 7.7 
 
 8.8 
 
 413 
 
 595 
 
 606 
 
 616 
 
 627 
 
 637 
 
 648 
 
 658 
 
 669 
 
 679 
 
 690 
 
 9 
 
 9!9 
 
 414 
 
 700 
 
 711 
 
 721 
 
 731 
 
 742 
 
 752 
 
 763 
 
 773 
 
 784 
 
 794 
 
 
 
 415 
 
 805 
 
 815 
 
 826 
 
 836 
 
 847 
 
 857 
 
 868 
 
 878 
 
 888 
 
 899 
 
 
 
 416 
 
 909 
 
 920 
 
 930 
 
 941 
 
 951 
 
 962 
 
 972 
 
 982 
 
 993 
 
 *003 
 
 
 
 417 
 
 62 014 
 
 024 
 
 034 
 
 045 
 
 055 
 
 066 
 
 076 
 
 086 
 
 097 
 
 107 
 
 
 
 418 
 
 118 
 
 128 
 
 138 
 
 149 
 
 159 
 
 170 
 
 180 
 
 190 
 
 201 
 
 211 
 
 
 
 419 
 
 221 
 
 232 
 
 242 
 
 252 
 
 263 
 
 273 
 
 284 
 
 294 
 
 304 
 
 315 
 
 
 
 420 
 
 325 
 
 335 
 
 346 
 
 356 
 
 366 
 
 377 
 
 387 
 
 397 
 
 408 
 
 418 
 
 
 10 
 
 421 
 
 428 
 
 439 
 
 449 
 
 459 
 
 469 
 
 480 
 
 490 
 
 500 
 
 511 
 
 521 
 
 1 
 
 422 
 
 531 
 
 542 
 
 552 
 
 562 
 
 572 
 
 583 
 
 593 
 
 603 
 
 613 
 
 624 
 
 1 ! 
 
 o ! 
 
 1.0 
 
 423 
 
 634 
 
 644 
 
 655 
 
 665 
 
 675 
 
 685 
 
 696 
 
 706 
 
 716 
 
 726 
 
 3 
 
 z .u 
 3.0 
 
 424 
 
 737 
 
 747 
 
 757 
 
 767 
 
 778 
 
 788 | 
 
 798 
 
 808 
 
 818 
 
 829 
 
 
 4.0 
 
 425 
 
 839 
 
 849 
 
 859 
 
 87C 
 
 880 
 
 890 
 
 | 900 
 
 910 
 
 921 
 
 931 
 
 5 
 
 5.0 
 
 426 
 
 941 
 
 951 
 
 961 
 
 972 
 
 982 
 
 992 
 
 *002 
 
 *012 
 
 *022 
 
 *033 
 
 6 
 
 : 6.0 
 
 427 
 
 63 043 
 
 053 
 
 063 
 
 073 
 
 083 
 
 094 
 
 104 
 
 114 
 
 124 
 
 134 
 
 7 
 
 Q 
 
 i 7.0 
 
 Q A 
 
 428 
 
 144 
 
 155 
 
 165 
 
 175 
 
 185 
 
 195 
 
 205 
 
 215 
 
 225 
 
 236 
 
 O 
 
 9 
 
 j o.U 
 
 ! 9.0 
 
 429 
 
 246 
 
 256 
 
 266 
 
 276 
 
 286 
 
 296 
 
 306 
 
 317 
 
 327 
 
 337 
 
 
 1 
 
 430 
 
 347 
 
 357 
 
 367 
 
 377 
 
 387 
 
 397 
 
 407 
 
 417 
 
 428 
 
 438 
 
 
 
 431 
 
 448 
 
 458 
 
 468 
 
 478 
 
 488 
 
 498 
 
 508 
 
 518 
 
 528 
 
 538 
 
 
 
 432 
 
 548 
 
 558 
 
 568 
 
 579 
 
 589 
 
 599 
 
 609 
 
 619 
 
 629 
 
 639 
 
 
 
 433 
 
 649 
 
 659 
 
 669 
 
 679 
 
 689 
 
 699 
 
 709 
 
 719 
 
 729 
 
 739 
 
 
 
 434 
 
 749 
 
 759 
 
 769 
 
 779 
 
 789 
 
 799 
 
 809 
 
 819 
 
 829 
 
 839 
 
 
 
 435 
 
 849 
 
 859 
 
 869 
 
 879 
 
 889 
 
 899 
 
 909 
 
 919 
 
 929 
 
 939 
 
 
 Q 
 
 436 
 
 949 
 
 959 
 
 969 
 
 979 
 
 988 
 
 998 
 
 *008 
 
 *018 
 
 *028 
 
 *038 
 
 
 
 437 
 
 64 048 
 
 058 
 
 068 
 
 078 
 
 088 
 
 098 
 
 108 
 
 118 
 
 J.28 
 
 137 
 
 i 
 
 0.9 
 
 438 
 
 147 
 
 157 
 
 167 
 
 177 
 
 187 
 
 197 
 
 207 
 
 217 
 
 227 
 
 237 
 
 2 
 
 1.8 
 
 439 
 
 246 
 
 256 
 
 266 
 
 276 
 
 286 
 
 296 
 
 306 
 
 316 
 
 326 
 
 335 
 
 3 
 
 A 
 
 2.7 
 
 q 
 
 440 
 
 345 
 
 355 
 
 365 
 
 375 
 
 385 
 
 395 
 
 404 
 
 414 
 
 424 
 
 434 
 
 * 
 
 5 
 
 A 
 
 0.0 
 
 4.5 
 
 K A 
 
 441 
 
 444 
 
 454 
 
 464 
 
 473 
 
 483 
 
 493 
 
 503 
 
 513 
 
 523 
 
 532 
 
 D 
 
 7 
 
 O.T 
 
 6.3 
 
 442 
 
 542 
 
 ; 552 
 
 562 
 
 572 
 
 582 
 
 591 
 
 601 
 
 611 
 
 621 
 
 631 
 
 8 
 
 Q 
 
 7.2 
 
 Q 1 
 
 443 
 
 640 
 
 1 650 
 
 660 
 
 670 
 
 680 
 
 689 
 
 699 
 
 709 
 
 719 
 
 729 
 
 
 0.1 
 
 444 
 
 738 
 
 , 748 
 
 758 
 
 768 
 
 777 
 
 787 
 
 797 
 
 807 
 
 816 
 
 826 
 
 
 
 445 
 
 836 
 
 I 846 
 
 856 
 
 865 
 
 875 
 
 885 
 
 895 
 
 904 
 
 914 
 
 924 
 
 
 
 446 
 
 933 
 
 ; 943 
 
 953 
 
 963 
 
 972 
 
 982 
 
 992 
 
 *002 
 
 *011 
 
 *021 
 
 
 
 447 
 
 65 031 
 
 040 
 
 050 
 
 060 
 
 070 
 
 079 
 
 089 
 
 099 
 
 108 
 
 118 
 
 
 
 448 
 
 128 
 
 1 137 
 
 147 
 
 157 
 
 167 
 
 176 
 
 186 
 
 196 
 
 205 
 
 215 
 
 
 
 449 
 
 225 
 
 > 234 
 
 244 
 
 254 
 
 263 
 
 273 
 
 283 
 
 292 
 
 302 
 
 312 
 
 
 
 450 
 
 321 
 
 . 331 
 
 341 
 
 350 
 
 360 
 
 369 
 
 379 
 
 389 
 
 398 
 
 408 
 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 t-r 
 
 i 
 
 8 
 
 9 
 
 p. p. 
 
LOGARITHMS . 
 
 481 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 450 
 
 451 
 
 452 
 
 453 
 
 454 
 
 455 
 
 456 
 
 457 
 
 458 
 
 459 
 
 460 
 
 461 
 
 462 
 
 463 
 
 464 
 
 465 
 
 466 
 
 467 
 
 468 
 
 469 
 
 470 
 
 471 
 
 472 
 
 473 
 
 474 
 
 475 
 
 476 
 
 477 
 
 478 
 
 479 
 
 480 
 
 481 
 
 482 
 
 483 
 
 484 
 
 485 
 
 486 
 
 487 
 
 488 
 
 489 
 
 490 
 
 491 
 
 492 
 
 493 
 
 494 
 
 495 
 
 496 
 
 497 
 
 498 
 
 499 
 
 500 
 
 35 321 
 
 331 
 
 341 
 
 350 
 
 360 
 
 369 
 
 379 
 
 389 
 
 398 
 
 408 
 
 10 
 
 1 1.0 
 2 2.0 
 
 3 3.0 
 
 4 4.0 
 
 5 5.0 
 
 6 6.0 
 
 7 7.0 
 
 8 8.0 
 
 9 9.0 
 
 9 
 
 1 0.9 
 
 2 1.8 
 
 3 2.7 
 
 4 3.6 
 
 5 4.5 
 
 6 5.4 
 
 7 6.3 
 
 8 7.2 
 
 9 8.1 
 
 8 
 
 1 0.8 
 2 1.6 
 
 3 2.4 
 
 4 3.2 
 
 5 4.0 
 
 6 4.8 
 
 j 7 5.6 
 
 8 6.4 
 
 9 7.2 
 
 418 
 514 
 610 
 706 
 801 
 896 
 992 
 36 087 
 181 
 
 427 
 
 523 
 
 619 
 
 715 
 
 811 
 
 906 
 
 *001 
 
 096 
 
 191 
 
 437 
 
 533 
 
 629 
 
 725 
 
 820 
 
 916 
 
 *011 
 
 106 
 
 200 
 
 447 
 
 543 
 
 639 
 
 734 
 
 830 
 
 925 
 
 *020 
 
 115 
 
 210 
 
 456 
 
 552 
 
 648 
 
 744 
 
 839 
 
 935 
 
 *030 
 
 124 
 
 219 
 
 466 
 
 562 
 
 658 
 
 753 
 
 849 
 
 944 
 
 *039 
 
 134 
 
 229 
 
 475 
 571 
 667 
 763 
 858 
 954 
 *049 : 
 143 
 238 
 
 485 
 
 581 
 
 677 
 
 772 
 
 868 
 
 963 
 
 *058 
 
 153 
 
 247 
 
 495 
 
 591 
 
 686 
 
 782 
 
 877 
 
 973 
 
 *068 
 
 162 
 
 257 
 
 504 
 
 600 
 
 696 
 
 792 
 
 887 
 
 982 
 
 *077 
 
 172 
 
 266 
 
 276 
 
 285 
 
 295 
 
 304 
 
 314 
 
 323 
 
 332 
 
 342 
 
 351 
 
 361 
 
 370 
 464 
 558 
 652 
 745 
 839 
 932 
 67 025 
 117 
 
 380 
 
 474 
 
 567 
 
 661 
 
 755 
 
 848 
 
 941 
 
 034 
 
 127 
 
 389 
 
 483 
 
 577 
 
 671 
 
 764 
 
 857 
 
 950 
 
 043 
 
 136 
 
 398 
 
 492 
 
 586 
 
 680 
 
 773 
 
 867 
 
 960 
 
 052 
 
 145 
 
 408 
 
 502 
 
 596 
 
 689 
 
 783 
 
 876 
 
 969 
 
 062 
 
 154 
 
 417 
 
 511 
 
 605 
 
 699 
 
 792 
 
 885 
 
 978 
 
 071 
 
 164 
 
 427 
 
 521 
 
 614 
 
 708 
 
 801 
 
 894 
 
 987 
 
 080 
 
 173 
 
 436 
 
 530 
 
 624 
 
 717 
 
 811 
 
 904 
 
 997 
 
 089 
 
 182 
 
 445 
 
 539 
 
 633 
 
 727 
 
 820 
 
 913 
 
 *006 
 
 099 
 
 191 
 
 455 
 
 549 
 
 642 
 
 736 
 
 829 
 
 922 
 
 *015 
 
 108 
 
 201 
 
 210 
 
 219 
 
 228 
 
 237 
 
 247 
 
 256 
 
 265 
 
 274 
 
 284 
 
 293 
 
 302 
 394 
 486 
 578 
 669 
 761 
 852 
 943 
 68 034 
 
 311 
 
 403 
 
 495 
 
 587 
 
 679 
 
 770 
 
 861 
 
 952 
 
 043 
 
 321 
 
 413 
 
 504 
 
 596 
 
 688 
 
 779 
 
 870 
 
 961 
 
 052 
 
 330 
 
 422 
 
 514 
 
 605 
 
 697 
 
 788 
 
 879 
 
 970 
 
 061 
 
 339 
 
 431 
 
 523 
 
 614 
 
 706 
 
 797 
 
 888 
 
 979 
 
 070 
 
 348 
 
 440 
 
 532 
 
 624 
 
 715 
 
 806 
 
 897 
 
 988 
 
 079 
 
 357 
 
 449 
 
 541 
 
 633 
 
 724 
 
 815 
 
 906 
 
 997 
 
 088 
 
 367 
 
 459 
 
 550 
 
 642 
 
 733 
 
 825 
 
 916 
 
 *006 
 
 097 
 
 376 
 
 468 
 
 560 
 
 651 
 
 742 
 
 834 
 
 925 
 
 *015 
 
 106 
 
 385 
 
 477 
 
 569 
 
 660 
 
 752 
 
 843 
 
 934 
 
 *024 
 
 115 
 
 124 
 
 133 
 
 142 
 
 151 
 
 160 
 
 169 
 
 178 
 
 187 
 
 196 
 
 205 
 
 215 
 
 305 
 
 395 
 
 485 
 
 574 
 
 664 
 
 753 
 
 842 
 
 931 
 
 224 
 
 314 
 
 404 
 
 494 
 
 583 
 
 673 
 
 762 
 
 851 
 
 940 
 
 233 
 
 323 
 
 413 
 
 502 
 
 592 
 
 681 
 
 771 
 
 860 
 
 949 
 
 242 
 
 332 
 
 422 
 
 511 
 
 601 
 
 690 
 
 780 
 
 869 
 
 958 
 
 251 
 
 341 
 
 431 
 
 520 
 
 610 
 
 699 
 
 789 
 
 878 
 
 966 
 
 260 
 
 350 
 
 440 
 
 529 
 
 619 
 
 708 
 
 797 
 
 886 
 
 975 
 
 269 
 
 359 
 
 449 
 
 538 
 
 628 
 
 717 
 
 806 
 
 895 
 
 984 
 
 278 
 
 368 
 
 458 
 
 547 
 
 637 
 
 726 
 
 815 
 
 904 
 
 993 
 
 287 
 
 377 
 
 467 
 
 556 
 
 646 
 
 735 
 
 824 
 
 913 
 
 *002 
 
 296 
 
 386 
 
 476 
 
 565 
 
 655 
 
 744 
 
 833 
 
 922 
 
 *011 
 
 69 020 
 
 028 
 
 037 
 
 046 
 
 055 
 
 064 
 
 073 
 
 082 
 
 090 
 
 099 
 
 108 
 
 197 
 
 285 
 
 373 
 
 461 
 
 548 
 
 636 
 
 722 
 
 81 C 
 
 117 
 205 
 294 
 ; 381 
 469 
 l 557 
 i 644 
 ; 732 
 1 819 
 
 126 
 
 214 
 
 302 
 
 390 
 
 478 
 
 566 
 
 653 
 
 740 
 
 827 
 
 135 
 
 223 
 
 311 
 
 399 
 
 487 
 
 574 
 
 662 
 
 749 
 
 836 
 
 144 
 
 232 
 
 320 
 
 408 
 
 496 
 
 583 
 
 671 
 
 758 
 
 845 
 
 152 
 
 241 
 
 329 
 
 417 
 
 504 
 
 592 
 
 679 
 
 767 
 
 854 
 
 161 
 
 249 
 
 338 
 
 425 
 
 513 
 
 601 
 
 688 
 
 775 
 
 862 
 
 170 
 
 258 
 
 346 
 
 434 
 
 522 
 
 609 
 
 697 
 
 784 
 
 871 
 
 179 
 
 267 
 
 355 
 
 443 
 
 531 
 
 618 
 
 705 
 
 793 
 
 880 
 
 188 
 
 276 
 
 364 
 
 452 
 
 539 
 
 627 
 
 714 
 
 801 
 
 888 
 
 897 
 
 r 906 
 
 914 
 
 923 
 
 932 
 
 940 
 
 949 
 
 958 
 
 966 
 
 975 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
482 
 
 LOGARITHMS. 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 500 
 
 >9 897 
 
 906 
 
 914 
 
 923 1 
 
 932 
 
 940 
 
 949 
 
 958 1 
 
 966 
 
 975 
 
 9 
 
 1 0.9 
 
 2 i 1.8 
 
 3 2.7 
 
 4 3.6 
 
 5 | 4.5 
 
 6 1 5.4 
 
 7 I 6.3 
 
 8 7.2 
 9 | 8.1 
 
 8 
 
 1 0.8 
 2 1.6 
 
 3 2.4 
 
 4 3.2 
 
 5 4.0 
 
 6 4.8 
 
 7 5.6 
 
 8 6.4 
 
 9 7.2 
 
 7 
 
 | 
 
 1 0.7 
 
 2 1.4 
 
 3 2.1 
 
 4 2.8 
 
 501 
 
 502 
 
 503 
 
 504 
 
 505 
 
 506 
 
 507 
 
 508 
 
 509 
 
 984 
 '0 070 
 157 
 243 
 329 
 415 
 501 
 586 
 672 
 
 992 
 
 079 
 
 165 
 
 252 
 
 338 
 
 424 
 
 509 
 
 595 
 
 680 
 
 *001 
 
 088 
 
 174 
 
 260 
 
 346 
 
 432 
 
 518 
 
 603 
 
 689 
 
 *010 
 
 096 
 
 183 
 
 269 
 
 355 
 
 441 
 
 526 
 
 612 
 
 697 
 
 *018 
 
 105 
 
 191 
 
 278 
 
 364 
 
 449 
 
 535 
 
 621 
 
 706 
 
 *027 : 
 114 
 200 
 286 
 372 
 458 
 544 
 629 
 714 
 
 *036 : 
 122 
 209 
 295 
 381 
 467 
 552 
 638 
 723 
 
 *044 : 
 131 
 217 
 303 
 389 
 475 
 561 
 646 
 731 
 
 *053 : 
 140 
 226 
 312 
 398 
 484 
 569 
 655 
 740 
 
 *062 
 
 148 
 
 234 
 
 321 
 
 406 
 
 492 
 
 578 
 
 663 
 
 749 
 
 510 
 
 511 
 
 512 
 
 513 
 
 514 
 
 515 
 
 516 
 
 517 
 
 518 
 
 519 
 
 520 
 
 521 
 
 522 
 
 523 
 
 524 
 
 525 
 
 526 
 
 527 
 
 528 
 
 529 
 
 530 
 
 531 
 
 532 
 
 533 
 
 534 
 
 535 
 
 536 
 
 537 
 
 538 
 
 539 
 
 540 
 
 541 
 
 542 
 
 543 
 
 544 
 
 545 
 
 546 
 
 547 
 
 548 
 
 549 
 
 550 
 
 757 
 
 766 
 
 774 
 
 783 
 
 791 
 
 800 
 
 808 
 
 817 
 
 825 
 
 834 
 
 842 
 927 
 71 012 
 096 
 181 
 265 
 349 
 433 
 517 
 
 851 
 
 935 
 
 020 
 
 105 
 
 189 
 
 273 
 
 357 
 
 441 
 
 525 
 
 859 
 
 944 
 
 029 
 
 113 
 
 198 
 
 282 
 
 366 
 
 450 
 
 533 
 
 868 
 
 952 
 
 037 
 
 122 
 
 206 
 
 290 
 
 374 
 
 458 
 
 542 
 
 876 
 
 961 
 
 046 
 
 130 
 
 214 
 
 299 
 
 383 
 
 466 
 
 550 
 
 885 
 
 969 
 
 054 
 
 139 
 
 223 
 
 307 
 
 391 
 
 475 
 
 559 
 
 893 
 
 978 
 
 063 
 
 147 
 
 231 
 
 315 
 
 399 
 
 483 
 
 567 
 
 902 
 
 986 
 
 071 
 
 155 
 
 240 
 
 324 
 
 408 
 
 492 
 
 575 
 
 910 
 
 995 
 
 079 
 
 164 
 
 248 
 
 332 
 
 416 
 
 500 
 
 584 
 
 919 
 
 *003 
 
 088 
 
 172 
 
 257 
 
 341 
 
 425 
 
 508 
 
 592 
 
 600 
 
 609 
 
 617 
 
 625 
 
 634 
 
 642 
 
 650 
 
 659 
 
 667 
 
 675 
 
 684 
 767 
 850 
 933 
 72 016 
 099 
 181 
 263 
 346 
 
 692 
 
 775 
 
 858 
 
 941 
 
 024 
 
 107 
 
 189 
 
 272 
 
 354 
 
 700 
 
 784 
 
 867 
 
 950 
 
 032 
 
 115 
 
 198 
 
 280 
 
 362 
 
 709 
 
 792 
 
 875 
 
 958 
 
 041 
 
 123 
 
 206 
 
 288 
 
 370 
 
 717 
 
 800 
 
 883 
 
 966 
 
 049 
 
 132 
 
 214 
 
 296 
 
 378 
 
 725 
 
 809 
 
 892 
 
 975 
 
 057 
 
 140 
 
 222 
 
 304 
 
 387 
 
 734 
 
 817 
 
 900 
 
 983 
 
 066 
 
 148 
 
 230 
 
 313 
 
 395 
 
 742 
 825 
 908 
 991 ! 
 074 
 156 
 239 
 321 
 1 403 
 
 750 
 
 834 
 
 917 
 
 999 
 
 082 
 
 165 
 
 247 
 
 329 
 
 411 
 
 759 
 
 842 
 
 925 
 
 *008 
 
 090 
 
 173 
 
 255 
 
 337 
 
 419 
 
 428 
 
 436 
 
 444 
 
 452 
 
 460 
 
 469 
 
 477 
 
 485 
 
 493 
 
 501 
 
 509 
 591 
 673 
 754 
 835 
 916 
 997 
 73 078 
 159 
 
 518 
 599 
 681 
 762 
 843 
 925 
 *006 
 i 086 
 
 i 167 
 
 526 
 
 607 
 
 689 
 
 770 
 
 852 
 
 933 
 
 *014 
 
 094 
 
 175 
 
 534 
 
 616 
 
 697 
 
 779 
 
 860 
 
 941 
 
 *022 
 
 102 
 
 183 
 
 542 
 
 624 
 
 705 
 
 787 
 
 868 
 
 949 
 
 *030 
 
 111 
 
 191 
 
 550 
 
 632 
 
 713 
 
 795 
 
 876 
 
 957 
 
 *038 
 
 119 
 
 199 
 
 558 
 
 640 
 
 722 
 
 803 
 
 884 
 
 965 
 
 *046 
 
 127 
 
 207 
 
 567 
 
 648 
 
 730 
 
 811 
 
 892 
 
 973 
 
 *054 
 
 135 
 
 215 
 
 575 
 
 656 
 
 738 
 
 819 
 
 90 C 
 
 981 
 
 *062 
 
 143 
 
 223 
 
 583 
 | 665 
 j 746 
 I 827 
 908 
 989 
 1*070 
 151 
 231 
 
 239 
 
 i 247 
 
 255 
 
 263 
 
 272 
 
 280 
 
 288 
 
 296 
 
 304 
 
 312 
 
 5 3.5 
 
 6 4.2 
 
 32 C 
 
 40 C 
 
 48 C 
 
 56 C 
 
 64 C 
 
 711 
 
 791 
 
 87 * 
 
 951 
 
 1 328 
 ) 408 
 ) 488 
 ) 568 
 ) 648 
 ) 727 
 ) 807 
 5 886 
 J 965 
 
 336 
 
 416 
 
 496 
 
 576 
 
 656 
 
 735 
 
 815 
 
 894 
 
 973 
 
 344 
 
 424 
 
 504 
 
 584 
 
 664 
 
 743 
 
 823 
 
 902 
 
 981 
 
 352 
 
 432 
 
 512 
 
 592 
 
 672 
 
 751 
 
 830 
 
 910 
 
 989 
 
 360 
 
 440 
 
 520 
 
 600 
 
 679 
 
 759 
 
 838 
 
 918 
 
 997 
 
 368 
 
 448 
 
 528 
 
 608 
 
 687 
 
 767 
 
 846 
 
 926 
 
 *005 
 
 376 
 
 456 
 
 536 
 
 616 
 
 695 
 
 775 
 
 854 
 
 933 
 
 *013 
 
 384 
 
 464 
 
 544 
 
 624 
 
 703 
 
 783 
 
 862 
 
 941 
 
 *020 
 
 392 
 
 472 
 
 552 
 
 632 
 
 711 
 
 791 
 
 870 
 
 949 
 
 *028 
 
 7 4.9 
 
 8 56 
 
 9 6.3 
 
 74 03 ( 
 
 3 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 5 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 N 
 
 L.O 
 
 1 
 
 2 
 
 9 
 
 O 
 
 4 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P P. 
 
LOGARITHMS. 
 
 483 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. R 
 
 550 
 
 74 036 
 
 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 
 551 
 
 115 
 
 123 
 
 131 
 
 139 
 
 147 
 
 155 
 
 162 
 
 170 
 
 178 
 
 186 
 
 
 
 552 
 
 194 
 
 202 
 
 210 
 
 218 
 
 225 
 
 233 
 
 241 
 
 249 
 
 257 
 
 265 
 
 
 
 553 
 
 273 
 
 280 
 
 288 
 
 296 
 
 304 
 
 312 
 
 320 
 
 327 
 
 335 
 
 343 
 
 
 
 554 
 
 351 
 
 359 
 
 367 
 
 374 
 
 382 
 
 390 
 
 398 
 
 406 
 
 414 
 
 421 
 
 
 
 555 
 
 429 
 
 437 
 
 445 
 
 453 
 
 461 
 
 468 
 
 476 
 
 484 
 
 492 
 
 500 
 
 
 
 556 
 
 507 
 
 515 
 
 523 
 
 531 
 
 539 
 
 547 
 
 554 
 
 562 
 
 570 
 
 578 
 
 
 
 557 
 
 586 
 
 593 
 
 601 
 
 609 
 
 617 
 
 624 
 
 632 
 
 640 
 
 648 
 
 656 
 
 
 
 558 
 
 663 
 
 671 
 
 679 
 
 687 
 
 695 
 
 702 
 
 710 
 
 718 
 
 726 
 
 733 
 
 
 
 559 
 
 741 
 
 749 
 
 757 
 
 764 
 
 772 
 
 780 
 
 788 
 
 796 
 
 803 
 
 811 
 
 
 
 560 
 
 819 
 
 827 
 
 834 
 
 842 
 
 850 
 
 858 
 
 865 
 
 873 
 
 881 
 
 889 
 
 
 8 
 
 561 
 
 896 
 
 904 
 
 912 
 
 920 
 
 927 
 
 935 
 
 943 
 
 950 
 
 958 
 
 966 
 
 
 0.8 
 1 .6 
 
 562 
 
 974 
 
 981 
 
 989 
 
 997 
 
 *005 
 
 *012 
 
 *020 
 
 *028 
 
 *035 
 
 *043 
 
 1 
 
 2 
 
 563 
 
 75 051 
 
 059 
 
 066 
 
 074 
 
 082 
 
 089 
 
 097 
 
 105 
 
 113 
 
 120 
 
 3 
 
 2 A I 
 
 564 
 
 128 
 
 136 
 
 143 
 
 151 
 
 159 
 
 166 
 
 174 
 
 182 
 
 189 
 
 197 
 
 4 
 
 3.2 
 
 565 
 
 205 
 
 213 
 
 220 
 
 228 
 
 236 
 
 243 
 
 251 
 
 259 
 
 266 
 
 274 
 
 5 
 
 4.0 
 
 566 
 
 282 
 
 289 
 
 297 
 
 305 
 
 312 
 
 320 
 
 328 
 
 335 
 
 343. 
 
 351 
 
 6 
 
 4.8 
 
 5.6 
 
 6.4 
 
 567 
 
 358 
 
 366 
 
 374 
 
 381 
 
 389 
 
 397 
 
 404 
 
 412 
 
 420 
 
 427 
 
 7 
 
 8 
 
 568 
 
 435 
 
 442 
 
 450 
 
 458 
 
 465 
 
 473 
 
 481 
 
 488 
 
 496 
 
 504 
 
 9 
 
 7 >2 
 
 569 
 
 511 
 
 519 
 
 526 
 
 534 
 
 542 
 
 549 
 
 557 
 
 565 
 
 572 
 
 580 
 
 
 570 
 
 587 
 
 595 
 
 603 
 
 610 
 
 618 
 
 626 
 
 633 
 
 641 
 
 648 
 
 656 
 
 
 
 571 
 
 664 
 
 671 
 
 679 
 
 686 
 
 694 
 
 702 
 
 709 
 
 717 
 
 724 
 
 732 
 
 
 
 572 
 
 740 
 
 747 
 
 755 
 
 762 
 
 770 
 
 778 
 
 785 
 
 793 
 
 800 
 
 808 
 
 
 
 573 
 
 815 
 
 823 
 
 831 
 
 838 
 
 846 
 
 853 
 
 861 
 
 868 
 
 876 
 
 884 
 
 
 
 574 
 
 891 
 
 899 
 
 906 
 
 914 
 
 921 
 
 929 
 
 937 
 
 944 
 
 952 
 
 959 
 
 
 
 575 
 
 967 
 
 974 
 
 982 
 
 989 
 
 997 
 
 *005 
 
 *012 
 
 *020 
 
 *027 
 
 *035 
 
 
 
 576 
 
 76 042 
 
 050 
 
 057 
 
 065 
 
 072 
 
 080 
 
 087 
 
 095 
 
 103 
 
 110 
 
 
 
 577 
 
 118 
 
 125 
 
 133 
 
 140 
 
 148 
 
 155 
 
 163 
 
 170 
 
 178 
 
 185 
 
 
 
 578 
 
 193 
 
 200 
 
 208 
 
 215 
 
 223 
 
 230 
 
 238 
 
 245 
 
 253 
 
 260 
 
 
 
 579 
 
 268 
 
 275 
 
 283 
 
 290 
 
 298 
 
 305 
 
 313 
 
 320 
 
 328 
 
 335 
 
 
 
 580 
 
 343 
 
 350 
 
 358 
 
 365 
 
 373 
 
 380 
 
 388 
 
 395 
 
 403 
 
 410 
 
 
 7 
 
 581 
 
 418 
 
 425 
 
 433 
 
 440 
 
 448 
 
 455 
 
 462 
 
 470 
 
 477 
 
 485 
 
 1 
 
 0.7 
 
 582 
 
 492 
 
 500 
 
 507 
 
 515 
 
 522 
 
 530 
 
 537 
 
 545 
 
 552 
 
 559 
 
 2 
 
 1.4 
 
 583 
 
 567 
 
 574 
 
 582 
 
 589 
 
 597 
 
 604 
 
 612 
 
 619 
 
 '626 
 
 634 
 
 3 
 
 2.1 
 
 584 
 
 641 
 
 649 
 
 656 
 
 664 
 
 671 
 
 678 
 
 686 
 
 693 
 
 701 
 
 708 
 
 4 
 
 2.8 
 
 3.5 
 
 4.2 
 
 585 
 
 716 
 
 723 
 
 730 
 
 738 
 
 745 
 
 753 
 
 760 
 
 768 
 
 775 
 
 782 
 
 5 
 
 6 
 
 586 
 
 790 
 
 797 
 
 805 
 
 812 
 
 819 
 
 827 
 
 834 
 
 842 
 
 849 
 
 856 
 
 7 
 
 4 i 9 
 
 587 
 
 864 
 
 871 
 
 879 
 
 886 
 
 893 
 
 901 
 
 908 
 
 916 
 
 923 
 
 930 
 
 8 
 
 5.6 
 
 588 
 
 938 
 
 945 
 
 953 
 
 960 
 
 967 
 
 975 
 
 982 
 
 989 
 
 997 
 
 *004 
 
 9 
 
 6.3 
 
 589 
 
 77 012 
 
 019 
 
 026 
 
 034 
 
 041 
 
 048 
 
 056 
 
 063 
 
 070 
 
 078 
 
 
 
 590 
 
 085 
 
 093 
 
 100 
 
 107 
 
 115 
 
 122 
 
 129 
 
 137 
 
 144 
 
 151 
 
 
 
 591 
 
 159 
 
 166 
 
 173 
 
 181 
 
 188 
 
 195 
 
 203 
 
 210 
 
 217 
 
 225 
 
 
 
 592 
 
 232 
 
 240 
 
 247 
 
 254 
 
 262 
 
 269 
 
 276 
 
 283 
 
 291 
 
 298 
 
 
 
 593 
 
 305 
 
 313 
 
 320 
 
 327 
 
 335 
 
 342 
 
 349 
 
 357 
 
 364 
 
 371 
 
 
 
 594 
 
 i 379 
 
 1 386 
 
 393 
 
 401 
 
 408 
 
 415 
 
 422 
 
 430 
 
 437 
 
 444 
 
 
 
 595 
 
 452 
 
 : 459 
 
 466 
 
 474 
 
 481 
 
 488 
 
 495 
 
 503 
 
 510 
 
 517 
 
 
 
 596 
 
 525 
 
 . 532 
 
 539 
 
 546 
 
 554 
 
 561 
 
 568 
 
 576 
 
 583 
 
 590 
 
 
 
 597 
 
 597 
 
 605 
 
 612 
 
 619 
 
 627 
 
 634 
 
 641 
 
 648 
 
 656 
 
 663 
 
 
 
 598 
 
 670 
 
 i 677 
 
 685 
 
 692 
 
 699 
 
 706 
 
 714 
 
 721 
 
 728 
 
 735 
 
 
 
 599 
 
 743 
 
 i 750 
 
 757 
 
 764 
 
 772 
 
 779 
 
 786 
 
 793 
 
 801 
 
 808 
 
 
 
 600 
 
 81c 
 
 > 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 N. 
 
 L 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P P. 
 
484 
 
 LOGARITHMS. 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 600 
 
 77 815; 
 
 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 601 
 
 887 
 
 895 
 
 902 
 
 909 
 
 916 
 
 924 
 
 931 
 
 938 
 
 945 
 
 952 
 
 
 
 602 
 
 960 
 
 967 
 
 974 
 
 981 
 
 988 
 
 996 
 
 *003 : 
 
 *010 : 
 
 *017 1 
 
 *025 
 
 
 
 603 
 
 78 032 
 
 039 
 
 046 
 
 053 
 
 061 
 
 068 
 
 075 
 
 082 
 
 089 
 
 097 
 
 
 
 604 
 
 104 
 
 111 
 
 118 
 
 125 
 
 132 
 
 140 
 
 147 
 
 154 
 
 161 
 
 168 
 
 
 
 605 
 
 176 
 
 183 
 
 190 
 
 197 
 
 204 
 
 211 
 
 219 
 
 226 
 
 233 
 
 240 
 
 
 
 606 
 
 247 
 
 254 
 
 262 
 
 269 
 
 276 
 
 283 
 
 290 
 
 297 
 
 305 
 
 312 
 
 
 8 
 
 607 
 
 319! 
 
 326 
 
 333 
 
 340 
 
 347 
 
 355 
 
 362 
 
 369 
 
 376 
 
 383 
 
 | 
 
 
 608 
 
 390 ; 
 
 398 
 
 405 
 
 412 
 
 419 
 
 426 
 
 433 
 
 440 
 
 447 
 
 455 
 
 l 
 
 0.8 
 
 609 
 
 462| 
 
 469 
 
 476 
 
 483 
 
 490 
 
 497 
 
 504 
 
 512 
 
 519 
 
 526 
 
 2 : 
 3 
 
 1.6 
 
 2.4 
 
 610 
 
 533 
 
 540 
 
 547 
 
 554 
 
 561 
 
 569 
 
 576 
 
 583 
 
 590 
 
 597 
 
 4 
 
 5 
 
 3.2 
 
 4.0 
 
 611 
 
 604 
 
 611 
 
 618 
 
 625 
 
 633 
 
 640 
 
 647 
 
 654 
 
 661 
 
 668 
 
 6 
 
 ►7 
 
 4.8 
 
 c f; 
 
 612 
 
 675 
 
 682 
 
 689 
 
 696 
 
 704 
 
 711 
 
 718 
 
 725 
 
 732 
 
 739 
 
 7 
 
 g 
 
 0.0 
 
 6.4 
 
 613 
 
 746 
 
 753 
 
 760 
 
 767 
 
 774 
 
 781 
 
 789 
 
 796 
 
 803 
 
 810 
 
 9 
 
 7'.2 
 
 614 
 
 817 
 
 824 
 
 831 
 
 838 
 
 845 
 
 852 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 615 
 
 888 
 
 895 
 
 902 
 
 909 
 
 916 
 
 923 
 
 930 
 
 937 
 
 944 
 
 951 
 
 
 
 616 
 
 958 
 
 965 
 
 972 
 
 979 
 
 986 
 
 993 
 
 *000 
 
 *007 
 
 *014 
 
 *021 
 
 
 
 617 
 
 79 029 
 
 036 
 
 043 
 
 050 
 
 057 
 
 064 
 
 071 
 
 078 
 
 085 
 
 092 
 
 
 
 618 
 
 099 
 
 106 
 
 113 
 
 120 
 
 127 
 
 134 
 
 141 
 
 148 
 
 155 
 
 162 
 
 
 
 619 
 
 169 
 
 176 
 
 183 
 
 190 
 
 197 
 
 204 
 
 211 
 
 218 
 
 225 
 
 232 
 
 
 
 620 
 
 239 
 
 246 
 
 253 
 
 260 
 
 267 
 
 274 
 
 281 
 
 288 
 
 295 
 
 302 
 
 
 7 
 
 621 
 
 309 
 
 316 
 
 323 
 
 330 
 
 337 
 
 344 
 
 351 
 
 358 
 
 365 
 
 372 
 
 
 
 622 
 
 379 
 
 386 
 
 393 
 
 400 
 
 407 
 
 414 
 
 421 
 
 428 
 
 435 
 
 442 
 
 1 
 
 2 
 
 0.7 
 
 1.4 
 
 623 
 
 449 
 
 456 
 
 463 
 
 470 
 
 477 
 
 484 
 
 491 
 
 498 
 
 505 
 
 511 
 
 3 
 
 2*1 
 
 624 
 
 518 
 
 525 
 
 532 
 
 539 
 
 546 
 
 553 
 
 560 
 
 567 
 
 574 
 
 581 
 
 4 
 
 2.8 
 
 625 
 
 588 
 
 595 
 
 602 
 
 609 
 
 616 
 
 623 
 
 630 
 
 637 
 
 644 
 
 650 
 
 5 
 
 3.5 
 
 626 
 
 657 
 
 664 
 
 671 
 
 678 
 
 685 
 
 692 
 
 699 
 
 706 
 
 713 
 
 720 
 
 6 
 
 4.2 
 
 627 
 
 727 
 
 734 
 
 741 
 
 748 
 
 754 
 
 761 
 
 768 
 
 775 
 
 782 
 
 789 
 
 7 
 
 g 
 
 4.9 
 
 5.6 
 
 628 
 
 796 
 
 803 
 
 810 
 
 817 
 
 824 
 
 831 
 
 837 
 
 844 
 
 851 
 
 858 
 
 9 
 
 | 6.3 
 
 629 
 
 865 
 
 872 
 
 879 
 
 886 
 
 893 
 
 900 
 
 906 
 
 913 
 
 920 
 
 927 
 
 
 
 630 
 
 934 
 
 941 
 
 948 
 
 955 
 
 962 
 
 969 
 
 975 
 
 982 
 
 989 
 
 996 
 
 
 
 631 
 
 80 003 
 
 010 
 
 017 
 
 024 
 
 030 
 
 037 
 
 044 
 
 051 
 
 058 
 
 065 
 
 
 
 632 
 
 072 
 
 079 
 
 085 
 
 092 
 
 099 
 
 106 
 
 113 
 
 120 
 
 127 
 
 1 34 
 
 
 
 633 
 
 140 
 
 147 
 
 154 
 
 161 
 
 168 
 
 175 
 
 182 
 
 188 
 
 195 
 
 202 
 
 
 
 634 
 
 209 
 
 216 
 
 223 
 
 229 
 
 236 
 
 243 
 
 250 
 
 257 
 
 264 
 
 271 
 
 
 
 635 
 
 277 
 
 284 
 
 291 
 
 298 
 
 305 
 
 312 
 
 318 
 
 325 
 
 332 
 
 339 
 
 
 6 
 
 636 
 
 346 
 
 353 
 
 359 
 
 366 
 
 373 
 
 380 
 
 387 
 
 393 
 
 400 
 
 407 
 
 
 
 637 
 
 414 
 
 421 
 
 428 
 
 434 
 
 441 
 
 448 
 
 455 
 
 462 
 
 468 
 
 475 
 
 1 
 
 0.6 
 
 638 
 
 482 
 
 489 
 
 496 
 
 502 
 
 509 
 
 516 
 
 523 
 
 530 
 
 536 
 
 543 
 
 2 
 
 1.2 
 
 639 
 
 550 
 
 557 
 
 504 
 
 570 
 
 577 
 
 584 
 
 591 
 
 598 
 
 604 
 
 611 
 
 3 
 
 4 
 
 1.8 
 i 2.4 
 
 640 
 
 618 
 
 625 
 
 632 
 
 638 
 
 645 
 
 652 
 
 659 
 
 665 
 
 672 
 
 679 
 
 5 
 
 6 
 
 3.0 
 3 6 
 
 641 
 
 686 
 
 693 
 
 699 
 
 706 
 
 713 
 
 720 
 
 726 
 
 733 
 
 740 
 
 747 
 
 7 
 
 4.2 
 
 4 Q 
 
 642 
 
 754 
 
 760 
 
 767 
 
 774 
 
 781 
 
 787 
 
 794 
 
 801 
 
 808 
 
 814 
 
 8 
 
 9 
 
 ! 4.0 
 
 1 5.4 
 
 643 
 
 821 
 
 828 
 
 835 
 
 841 
 
 848 
 
 855 
 
 862 
 
 868 
 
 875 
 
 882 
 
 
 
 644 
 
 889 
 
 i 895 
 
 902 
 
 909 
 
 916 
 
 922 
 
 929 
 
 936 
 
 943 
 
 949 
 
 
 
 645 
 
 956 
 
 i 963 
 
 969 
 
 976 
 
 983 
 
 990 
 
 996 
 
 *003 
 
 *010 
 
 *017 
 
 
 
 646 
 
 81 023 
 
 1 030 
 
 037 
 
 043 
 
 050 
 
 057 
 
 064 
 
 070 
 
 077 
 
 084 
 
 
 
 647 
 
 090 
 
 i 097 
 
 104 
 
 111 
 
 117 
 
 124 
 
 131 
 
 137 
 
 144 
 
 151 
 
 
 
 648 
 
 158 
 
 ! 164 
 
 171 
 
 178 
 
 184 
 
 191 
 
 198 
 
 204 
 
 211 
 
 218 
 
 
 
 649 
 
 224 
 
 : 231 
 
 238 
 
 245 
 
 251 
 
 258 
 
 265 
 
 271 
 
 278 
 
 285 
 
 
 
 650 
 
 291 
 
 . 298 
 
 305 
 
 311 
 
 318 
 
 325 
 
 331 
 
 338 
 
 345 
 
 351 
 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 <J 
 
 F 
 
 \ P. 
 
LOGARITHMS. 
 
 485 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 650 
 
 81 291 
 
 298 
 
 305 
 
 311 
 
 318 
 
 325 
 
 331 
 
 338 
 
 345 
 
 351 
 
 
 
 651 
 
 358 
 
 365 
 
 371 
 
 378 
 
 385 
 
 391 
 
 398 
 
 405 
 
 411 
 
 418 
 
 
 
 652 
 
 425 
 
 431 
 
 438 
 
 445 
 
 451 
 
 458 
 
 465 
 
 471 
 
 478 
 
 485 
 
 
 
 653 
 
 491 
 
 498 
 
 505 
 
 511 
 
 518 
 
 525 
 
 531 
 
 538 
 
 544 
 
 551 
 
 
 
 654 
 
 558 
 
 564 
 
 571 
 
 578 
 
 584 
 
 591 
 
 598 
 
 604 
 
 611 
 
 617 
 
 
 
 655 
 
 624 
 
 631 
 
 637 
 
 644 
 
 651 
 
 657 
 
 664 
 
 671 
 
 677 
 
 684 
 
 
 
 656 
 
 690 
 
 697 
 
 704 
 
 710 
 
 717 
 
 723 
 
 730 
 
 737 
 
 743 
 
 750 
 
 
 
 657 
 
 757 
 
 763 
 
 770 
 
 776 
 
 783 
 
 790 
 
 796 
 
 803 
 
 809 
 
 816 
 
 
 
 658 
 
 823 
 
 829 
 
 836 
 
 842 
 
 849 
 
 856 
 
 862 
 
 869 
 
 875 
 
 882 
 
 
 
 659 
 
 889 
 
 895 
 
 902 
 
 908 
 
 915 
 
 921 
 
 928 
 
 935 
 
 941 
 
 948 
 
 
 
 660 
 
 954 
 
 961 
 
 968 
 
 974 
 
 981 
 
 987 
 
 994 
 
 *000 
 
 *007 
 
 *014 
 
 
 7 
 
 661 
 
 82 020 
 
 027 
 
 033 
 
 040 
 
 046 
 
 053 
 
 060 
 
 066 
 
 073 
 
 079 
 
 1 
 
 
 662 
 
 086 
 
 092 
 
 099 
 
 105 
 
 112 
 
 119 
 
 125 
 
 132 
 
 138 
 
 145 
 
 1 
 
 2 
 
 U.7 
 
 1,4 
 
 663 
 
 151 
 
 158 
 
 164 
 
 171 
 
 178 
 
 184 
 
 191 
 
 197 
 
 204 
 
 210 
 
 3 
 
 2.1 
 
 664 
 
 217 
 
 223 
 
 230 
 
 236 
 
 243 
 
 249 
 
 256 
 
 263 
 
 269 
 
 276 
 
 4 
 
 2.8 
 
 665 
 
 282 
 
 289 
 
 295 
 
 302 
 
 308 
 
 315 
 
 321 
 
 328 
 
 334 
 
 341 
 
 5 
 
 3.5 
 
 666 
 
 347 
 
 354 
 
 360 
 
 367 
 
 373 
 
 380 
 
 387 
 
 393 
 
 400 
 
 406 
 
 6 
 
 h 
 
 4.2 
 
 A Q 
 
 667 
 
 413 
 
 419 
 
 426 
 
 432 
 
 439 
 
 445 
 
 452 
 
 458 
 
 465 
 
 471 
 
 7 
 
 8 
 
 5 6 
 
 668 
 
 478 
 
 484 
 
 491 
 
 497 
 
 504 
 
 510 
 
 517 
 
 523 
 
 530 
 
 536 
 
 9 
 
 6.3 
 
 669 
 
 543 
 
 549 
 
 556 
 
 562 
 
 569 
 
 575 
 
 582 
 
 588 
 
 595 
 
 601 
 
 
 
 670 
 
 607 
 
 614 
 
 620 
 
 627 
 
 633 
 
 S40 
 
 646 
 
 653 
 
 659 
 
 666 
 
 
 
 671 
 
 672 
 
 679 
 
 685 
 
 692 
 
 698 
 
 705 
 
 711 
 
 718 
 
 724 
 
 730 
 
 
 
 672 
 
 737 
 
 743 
 
 750 
 
 756 
 
 763 
 
 769 
 
 776 
 
 782 
 
 789 
 
 795 
 
 
 
 673 
 
 802 
 
 808 
 
 814 
 
 821 
 
 827 
 
 834 
 
 840 
 
 847 
 
 853 
 
 860 
 
 
 
 674 
 
 866 
 
 872 
 
 879 
 
 885 
 
 892 
 
 898 
 
 905 
 
 911 
 
 918 
 
 924 
 
 
 
 675 
 
 930 
 
 937 
 
 943 
 
 950 
 
 956 
 
 963 
 
 969 
 
 975 
 
 982 
 
 988 
 
 
 
 676 
 
 995 
 
 *001 
 
 *008 
 
 *014 
 
 *020 
 
 *027 
 
 *033 
 
 *040 
 
 *046 
 
 *052 
 
 
 
 677 
 
 83 059 
 
 065 
 
 072 
 
 078 
 
 085 
 
 091 
 
 097 
 
 104 
 
 110 
 
 117 
 
 
 
 678 
 
 123 
 
 129 
 
 136 
 
 142 
 
 149 
 
 155 
 
 161 
 
 168 
 
 174 
 
 181 
 
 
 
 679 
 
 187 
 
 193 
 
 200 
 
 206 
 
 213 
 
 219 
 
 225 
 
 232 
 
 238 
 
 245 
 
 
 
 680 
 
 251 
 
 257 
 
 264 
 
 270 
 
 276 
 
 283 
 
 289 
 
 296 
 
 302 
 
 308 
 
 
 6 ! 
 
 681 
 
 315 
 
 321 
 
 327 
 
 334 
 
 340 
 
 347 
 
 353 
 
 359 
 
 366 
 
 372 
 
 1 
 
 0.6 
 
 682 
 
 378 
 
 385 
 
 391 
 
 398 
 
 404 
 
 410 
 
 417 
 
 423 
 
 429 
 
 436 
 
 2 
 
 1.2 
 
 683 
 
 442 
 
 448 
 
 455 
 
 461 
 
 467 
 
 474 
 
 480 
 
 487 
 
 493 
 
 499 
 
 3 
 
 1.8 
 
 684 
 
 506 
 
 512 
 
 518 
 
 525 
 
 531 
 
 537 
 
 544 
 
 550 
 
 556 
 
 563 
 
 4 
 
 5 
 
 2.4 
 
 3.0 
 
 685 
 
 569 
 
 575 
 
 582 
 
 588 
 
 594 
 
 601 
 
 607 
 
 613 
 
 620 
 
 626 
 
 6 
 
 3.6 
 
 686 
 
 632 
 
 639 
 
 645 
 
 651 
 
 658 
 
 664 
 
 670 
 
 677 
 
 683 
 
 689 
 
 7 
 
 4.2 i 
 
 687 
 
 696 
 
 702 
 
 708 
 
 715 
 
 721 
 
 727 
 
 734 
 
 740 
 
 746 
 
 753 
 
 8 
 
 4.8 
 
 688 
 
 759 
 
 765 
 
 771 
 
 778 
 
 784 
 
 790 
 
 797 
 
 803 
 
 809 
 
 816 
 
 9 
 
 5.4 ; 
 
 689 
 
 822 
 
 828 
 
 835 
 
 841 
 
 847 
 
 853 
 
 860 
 
 866 
 
 872 
 
 879 
 
 
 
 690 
 
 885 
 
 891 
 
 897 
 
 904 
 
 910 
 
 916 
 
 923 
 
 929 
 
 935 
 
 942 
 
 
 
 691 
 
 948 
 
 954 
 
 960 
 
 967 
 
 973 
 
 979 
 
 985 
 
 992 
 
 998 
 
 *004 
 
 
 
 692 
 
 84 Oil 
 
 017 
 
 023 
 
 029 
 
 036 
 
 042 
 
 048 
 
 055 
 
 061 
 
 067 
 
 
 
 693 
 
 073 
 
 080 
 
 086 
 
 092 
 
 098 
 
 105 
 
 111 
 
 117 
 
 123 
 
 130 
 
 
 
 694 
 
 136 
 
 142 
 
 148 
 
 155 
 
 161 
 
 167 
 
 173 
 
 180 
 
 186 
 
 192 
 
 
 
 695 
 
 198 
 
 205 
 
 211 
 
 217 
 
 223 
 
 230 
 
 236 
 
 242 
 
 248 
 
 255 
 
 
 
 696 
 
 261 
 
 267 
 
 273 
 
 280 
 
 286 
 
 292 
 
 298 
 
 305 
 
 311 
 
 317 
 
 
 
 697 
 
 323 
 
 330 
 
 336 
 
 342 
 
 348 
 
 354 
 
 361 
 
 367 
 
 373 
 
 379 
 
 
 
 698 
 
 ! 386 
 
 392 
 
 398 
 
 404 
 
 410 
 
 417 
 
 423 
 
 429 
 
 435 
 
 442 
 
 
 
 699 
 
 448 
 
 454 
 
 460 
 
 466 
 
 473 
 
 479 
 
 485 
 
 491 
 
 497 
 
 504 
 
 
 
 700 
 
 510 
 
 516 
 
 522 
 
 528 
 
 535 
 
 541 
 
 547 
 
 553 
 
 559 
 
 566 
 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
486 
 
 LOGARITHMS . 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 535 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 700 
 
 701 
 
 702 
 
 703 
 
 704 
 
 705 
 
 706 
 
 707 
 
 708 
 
 709 
 
 710 
 
 711 
 
 712 
 
 713 
 
 714 
 
 715 
 
 716 
 
 717 
 
 718 
 
 719 
 
 720 
 
 721 
 
 722 
 
 723 
 
 724 
 
 725 
 
 726 
 
 727 
 
 728 
 
 729 
 
 730 
 
 731 
 
 732 
 
 733 
 
 734 
 
 735 
 
 736 
 '737 
 
 738 
 
 739 
 
 740 
 
 741 
 
 742 
 
 743 
 
 744 
 
 745 
 
 746 
 
 747 
 
 748 
 
 749 
 
 750 
 
 H 510 
 
 516 
 
 522 
 
 528 
 
 541 
 
 547 
 
 553 
 
 559 
 
 566 
 
 7 
 
 1 0.7 
 
 2 1.4 
 
 3 2.1 
 
 4 2.8 
 
 5 3.5 
 
 6 4.2 
 
 7 4.9 
 
 8 5.6 
 
 9 6.3 
 
 6 
 
 1 1 0.6 
 
 2 I 1.2 
 
 3 1.8 
 
 4 2.4 
 
 5 1 3.0 
 
 6 3.6 
 
 7 4.2 
 
 8 4.8 
 
 9 | 5.4 
 
 5 
 
 1 0.5 
 
 2 1.0 
 
 3 1.5 
 
 4 2.0 
 
 5 2.5 
 
 6 3.0 
 
 7 3.5 
 
 8 4.0 
 
 9 45 
 
 572 
 634 
 696 
 757 
 819 
 880 
 942 
 35 003 
 065 
 
 578 
 
 640 
 
 702 
 
 763 
 
 825 
 
 887 
 
 948 
 
 009 
 
 071 
 
 584 
 
 646 
 
 708 
 
 770 
 
 831 
 
 893 
 
 954 
 
 016 
 
 077 
 
 590 
 
 652 
 
 714 
 
 776 
 
 837 
 
 899 
 
 960 
 
 022 
 
 083 
 
 597 
 
 658 
 
 720 
 
 782 
 
 844 
 
 905 
 
 967 
 
 028 
 
 089 
 
 603 
 
 665 
 
 726 
 
 788 
 
 850 
 
 911 
 
 973 
 
 034 
 
 095 
 
 609 
 
 671 
 
 733 
 
 794 
 
 856 
 
 917 
 
 979 
 
 040 
 
 101 
 
 615 
 
 677 
 
 739 
 
 800 
 
 862 
 
 924 
 
 985 
 
 046 
 
 107 
 
 621 
 
 683 
 
 745 
 
 807 
 
 868 
 
 930 
 
 991 
 
 052 
 
 114 
 
 628 
 
 689 
 
 751 
 
 813 
 
 874 
 
 936 
 
 997 
 
 058 
 
 120 
 
 126 
 
 132 
 
 138 
 
 144 
 
 150 
 
 156 
 
 163 
 
 169 
 
 175 
 
 181 
 
 187 
 
 248 
 
 309 
 
 370 
 
 431 
 
 491 
 
 552 
 
 612 
 
 673 
 
 193 
 
 254 
 
 315 
 
 376 
 
 437 
 
 497 
 
 558 
 
 618 
 
 679 
 
 199 
 
 260 
 
 321 
 
 382 
 
 443 
 
 503 
 
 564 
 
 625 
 
 685 
 
 205 
 
 266 
 
 327 
 
 388 
 
 449 
 
 509 
 
 570 
 
 631 
 
 691 
 
 211 
 
 272 
 
 333 
 
 394 
 
 455 
 
 516 
 
 576 
 
 637 
 
 697 
 
 217 
 
 278 
 
 339 
 
 400 
 
 461 
 
 522 
 
 582 
 
 643 
 
 703 
 
 224 
 
 285 
 
 345 
 
 406 
 
 467 
 
 528 
 
 588 
 
 649 
 
 709 
 
 230 
 
 291 
 
 352 
 
 412 
 
 473 
 
 534 
 
 594 
 
 655 
 
 715 
 
 236 
 
 297 
 
 358 
 
 418 
 
 479 
 
 540 
 
 600 
 
 661 
 
 721 
 
 242 
 
 303 
 
 364 
 
 425 
 
 485 
 
 546 
 
 606 
 
 667 
 
 727 
 
 733 
 
 739 
 
 745 
 
 751 
 
 757 
 
 818 
 
 878 
 
 938 
 
 998 
 
 058 
 
 118 
 
 177 
 
 237 
 
 297 
 
 763 
 
 769 
 
 775 
 
 781 
 
 788 
 
 794 
 854 
 914 
 974 
 86 034 
 094 
 153 
 213 
 273 
 
 800 
 
 860 
 
 920 
 
 980 
 
 040 
 
 100 
 
 159 
 
 219 
 
 279 
 
 806 
 
 866 
 
 926 
 
 986 
 
 016 
 
 106 
 
 165 
 
 225 
 
 285 
 
 812 
 
 872 
 
 932 
 
 992 
 
 052 
 
 112 
 
 171 
 
 231 
 
 291 
 
 824 
 
 884 
 
 944 
 
 *004 
 
 064 
 
 124 
 
 183 
 
 243 
 
 303 
 
 830 
 
 890 
 
 950 
 
 *010 
 
 070 
 
 130 
 
 189 
 
 249 
 
 308 
 
 836 
 
 896 
 
 956 
 
 *016 
 
 076 
 
 136 
 
 195 
 
 255 
 
 314 
 
 842 
 
 902 
 
 962 
 
 *022 
 
 082 
 
 141 
 
 201 
 
 261 
 
 320 
 
 848 
 
 908 
 
 968 
 
 *028 
 
 088 
 
 147 
 
 207 
 
 267 
 
 326 
 
 332 
 
 338 
 
 344 
 
 350 
 
 356 
 
 362 
 
 368 
 
 374 
 
 380 
 
 386_ 
 
 392 
 
 451 
 
 510 
 
 570 
 
 629 
 
 688 
 
 747 
 
 806 
 
 864 
 
 398 
 
 457 
 
 516 
 
 576 
 
 635 
 
 694 
 
 753 
 
 812 
 
 870 
 
 404 
 
 463 
 
 522 
 
 581 
 
 641 
 
 700 
 
 759 
 
 817 
 
 876 
 
 410 
 
 469 
 
 528 
 
 587 
 
 646 
 
 705 
 
 764 
 
 823 
 
 882 
 
 415 
 
 475 
 
 534 
 
 593 
 
 652 
 
 711 
 
 770 
 
 829 
 
 888 
 
 421 
 
 481 
 
 540 
 
 599 
 
 658 
 
 717 
 
 776 
 
 835 
 
 894 
 
 427 
 
 487 
 
 546 
 
 605 
 
 664 
 
 723 
 
 782 
 
 841 
 
 900 
 
 433 
 
 493 
 
 552 
 
 611 
 
 670 
 
 729 
 
 788 
 
 847 
 
 906 
 
 439 
 
 499 
 
 558 
 
 617 
 
 676 
 
 735 
 
 794 
 
 853 
 
 911 
 
 445 
 
 504 
 
 564 
 
 623 
 
 682 
 
 741 
 
 800 
 
 859 
 
 917 
 
 923 
 
 ; 929 
 
 935 
 
 941 
 
 947 
 
 953 
 
 958 
 
 964 
 
 970 
 
 976^ 
 
 982 
 87 04C 
 099 
 157 
 2ie 
 27 4 
 332 
 39C 
 44* 
 
 ! 988 
 l 046 
 l 105 
 r 163 
 i 221 
 L 280 
 ! 338 
 ) 396 
 l 454 
 
 994 
 
 052 
 
 111 
 
 169 
 
 227 
 
 286 
 
 344 
 
 402 
 
 460 
 
 999 
 
 058 
 
 116 
 
 175 
 
 233 
 
 291 
 
 349 
 
 408 
 
 466 
 
 *005 
 
 064 
 
 122 
 
 181 
 
 239 
 
 297 
 
 355 
 
 413 
 
 471 
 
 *011 
 
 070 
 
 128 
 
 186 
 
 245 
 
 303 
 
 361 
 
 419 
 
 477 
 
 *017 
 
 075 
 
 134 
 
 192 
 
 251 
 
 309 
 
 367 
 
 425 
 
 483 
 
 *023 
 
 081 
 
 140 
 
 198 
 
 256 
 
 315 
 
 373 
 
 431 
 
 489 
 
 *029 
 
 087 
 
 146 
 
 204 
 
 262 
 
 320 
 
 379 
 
 437 
 
 495 
 
 *035 
 
 093 
 
 151 
 
 210 
 
 268 
 
 326 
 
 384 
 
 442 
 
 500 
 
 50( 
 
 5 512 
 
 518 
 
 523 
 
 529 
 
 535 
 
 541 
 
 547 
 
 552 
 
 558 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P.P. 
 
LOGARITHMS. 
 
 48 7 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P.P. 
 
 750 
 
 B 7 506 
 
 512 
 
 518 
 
 523 
 
 529 
 
 535 
 
 541 
 
 547 
 
 552 
 
 558 
 
 
 
 751 
 
 564 
 
 570 
 
 576 
 
 581 
 
 587 
 
 593 
 
 599 
 
 604 
 
 610 
 
 616 
 
 
 
 752 
 
 622 
 
 628 
 
 633 
 
 639 
 
 645 
 
 651 
 
 656 
 
 662 
 
 668 
 
 674 
 
 
 
 753 
 
 679 
 
 685 
 
 691 
 
 697 
 
 703 
 
 708 
 
 714 
 
 720 
 
 726 
 
 731 
 
 
 
 754 
 
 737 
 
 743 
 
 749 
 
 754 
 
 760 
 
 766 
 
 772 
 
 777 
 
 783 
 
 789 
 
 
 
 755 
 
 795 
 
 800 
 
 806 
 
 812 
 
 818 
 
 823 
 
 829 
 
 835 
 
 841 
 
 846 
 
 
 
 756 
 
 852 
 
 858 
 
 864 
 
 869 
 
 875 
 
 881 
 
 887 
 
 892 
 
 898 
 
 904 
 
 
 
 757 
 
 910 
 
 915 
 
 921 
 
 927 
 
 933 
 
 938 
 
 944 
 
 950 
 
 955 
 
 961 
 
 
 
 758 
 
 967 
 
 973 
 
 978 
 
 984 
 
 990 
 
 996 
 
 *001 
 
 *007 
 
 *013 
 
 *018 
 
 
 
 759 
 
 88 024 
 
 030 
 
 036 
 
 041 
 
 047 
 
 053 
 
 058 
 
 064 
 
 070 
 
 076 
 
 
 
 760 
 
 081 
 
 087 
 
 093 
 
 098 
 
 104 
 
 110 
 
 116 
 
 121 
 
 127 
 
 133 
 
 
 6 
 
 761 
 
 138 
 
 144 
 
 150 
 
 156 
 
 161 
 
 167 
 
 173 
 
 178 
 
 184 
 
 190 
 
 
 762 
 
 195 
 
 201 
 
 207 - 
 
 213 
 
 218 
 
 224 
 
 230 
 
 235 
 
 241 
 
 247 
 
 1 
 
 2 
 
 0.6 
 1 2 
 
 763 
 
 252 
 
 258 
 
 264 
 
 270 
 
 275 
 
 281 
 
 287 
 
 292 
 
 298 
 
 304 
 
 3 
 
 1.8 
 
 764 
 
 309 
 
 315 
 
 321 
 
 326 
 
 332 
 
 338 
 
 343 
 
 349 
 
 355 
 
 360 
 
 4 
 
 2.4 
 
 765 
 
 366 
 
 372 
 
 377 
 
 383 
 
 389 
 
 395 
 
 400 
 
 406 
 
 412 
 
 417 
 
 5 
 
 3.0 
 
 766 
 
 423 
 
 429 
 
 434 
 
 440 
 
 446 
 
 451 
 
 457 
 
 463 
 
 468 
 
 474 
 
 6 
 
 3.6 
 
 767 
 
 480 
 
 485 
 
 491 
 
 497 
 
 502 
 
 508 
 
 513 
 
 519 
 
 525 
 
 530 
 
 7 
 
 g 
 
 4.2 
 
 4.8 
 
 768 
 
 536 
 
 542 
 
 547 
 
 553 
 
 559 
 
 564 
 
 570 
 
 576 
 
 581 
 
 587 
 
 9 
 
 5^4 
 
 769 
 
 593 
 
 598 
 
 604 
 
 610 
 
 615 
 
 621 
 
 627 
 
 632 
 
 638 
 
 643 
 
 
 
 770 
 
 649 
 
 655 
 
 660 
 
 666 
 
 672 
 
 677 
 
 683 
 
 689 
 
 694 
 
 700 
 
 
 
 771 
 
 705 
 
 711 
 
 717 
 
 722 
 
 728 
 
 734 
 
 739 
 
 745 
 
 750 
 
 756 
 
 
 
 772 
 
 762 
 
 767 
 
 773 
 
 779 
 
 784 
 
 790 
 
 795 
 
 801 
 
 807 
 
 812 
 
 
 
 773 
 
 818 
 
 824 
 
 829 
 
 835 
 
 840 
 
 846 
 
 852 
 
 857 
 
 863 
 
 868 
 
 
 
 774 
 
 874 
 
 880 
 
 885 
 
 891 
 
 897 
 
 902 
 
 908 
 
 913 
 
 919 
 
 925 
 
 
 
 775 
 
 930 
 
 936 
 
 941 
 
 947 
 
 953 
 
 958 
 
 964 
 
 969 
 
 975 
 
 981 
 
 
 
 776 
 
 986 
 
 992 
 
 997 
 
 *003 
 
 *009 
 
 *014 
 
 *020 
 
 *025 
 
 *031 
 
 *037 
 
 
 
 777 
 
 89 042 
 
 048 
 
 053 
 
 059 
 
 064 
 
 070 
 
 076 
 
 081 
 
 087 
 
 092 
 
 
 
 778 
 
 098 
 
 104 
 
 109 
 
 115 
 
 120 
 
 126 
 
 131 
 
 137 
 
 143 
 
 148 
 
 
 
 779 
 
 154 
 
 159 
 
 165 
 
 170 
 
 176 
 
 182 
 
 187 
 
 193 
 
 198 
 
 204 
 
 
 
 780 
 
 209 
 
 215 
 
 221 
 
 226 
 
 232 
 
 237 
 
 243 
 
 248 
 
 254 
 
 260 
 
 
 5 
 
 781 
 
 265 
 
 271 
 
 276 
 
 282 
 
 287 
 
 293 
 
 298 
 
 304 
 
 310 
 
 315 
 
 1 
 
 0.5 
 
 782 
 
 321 
 
 326 
 
 332 
 
 337 
 
 343 
 
 348 
 
 354 
 
 360 
 
 365 
 
 37 L 
 
 2 
 
 1.0 
 
 783 
 
 376 
 
 382 
 
 387 
 
 393 
 
 398 
 
 404 
 
 409 
 
 415 
 
 421 
 
 426 
 
 3 
 
 1.5 
 
 784 
 
 432 
 
 437 
 
 443 
 
 448 
 
 454 
 
 459 
 
 465 
 
 470 
 
 476 
 
 481 
 
 4 
 
 2.0 
 
 785 
 
 487 
 
 492 
 
 498 
 
 504 
 
 509 
 
 515 
 
 520 
 
 526 
 
 531 
 
 537 
 
 5 
 
 « 
 
 2.5 
 
 3.0 
 
 786 
 
 542 
 
 548 
 
 553 
 
 559 
 
 564 
 
 570 
 
 575 
 
 581 
 
 586 
 
 592 
 
 | D 
 
 7 
 
 3.5 
 
 787 
 
 597 
 
 603 
 
 609 
 
 614 
 
 620 
 
 625 
 
 631 
 
 636 
 
 642 
 
 647 
 
 8 
 
 4.0 
 
 788 
 
 653 
 
 658 
 
 664 
 
 669 
 
 675 
 
 680 
 
 686 
 
 691 
 
 697 
 
 702 
 
 9 
 
 4.5 
 
 789 
 
 708 
 
 713 
 
 719 
 
 724 
 
 730 
 
 735 
 
 741 
 
 746 
 
 752 
 
 757 
 
 
 
 790 
 
 763 
 
 768 
 
 774 
 
 779 
 
 785 
 
 790 
 
 796 
 
 801 
 
 807 
 
 812 
 
 
 
 791 
 
 818 
 
 . 823 
 
 829 
 
 834 
 
 840 
 
 845 
 
 851 
 
 856 
 
 862 
 
 867 
 
 
 
 792 
 
 873 
 
 878 
 
 883 
 
 889 
 
 894 
 
 900 
 
 905 
 
 911 
 
 916 
 
 922 
 
 
 
 793 
 
 927 
 
 933 
 
 938 
 
 944 
 
 949 
 
 955 
 
 960 
 
 966 
 
 971 
 
 977 
 
 
 
 794 
 
 982 
 
 : 988 
 
 993 
 
 998 
 
 *004 
 
 *009 
 
 *015 
 
 *020 
 
 *026 
 
 *031 
 
 
 
 795 
 
 90 037 
 
 042 
 
 048 
 
 053 
 
 059 
 
 064 
 
 069 
 
 075 
 
 080 
 
 086 
 
 
 
 796 
 
 091 
 
 097 
 
 102 
 
 108 
 
 113 
 
 119 
 
 124 
 
 129 
 
 135 
 
 140 
 
 
 
 797 
 
 146 
 
 l 151 
 
 157 
 
 162 
 
 168 
 
 173 
 
 179 
 
 184 
 
 189 
 
 195 
 
 
 
 798 
 
 ; 200 
 
 i 206 
 
 211 
 
 217 
 
 222 
 
 227 
 
 233 
 
 238 
 
 244 
 
 249 
 
 
 
 799 
 
 255 
 
 > 260 
 
 266 
 
 271 
 
 276 
 
 282 
 
 287 
 
 293 
 
 298 
 
 304 
 
 
 
 800 
 
 30 S 
 
 1 314 
 
 320 
 
 325 
 
 331 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
488 
 
 LOGARITHMS. 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. 
 
 P. 
 
 800 
 
 90 309 
 
 314 
 
 320 
 
 325 
 
 331 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 
 
 801 
 
 363 
 
 369 
 
 374 
 
 380 
 
 385 
 
 390 
 
 396 
 
 401 
 
 407 
 
 412 
 
 
 
 802 
 
 417 
 
 423 
 
 428 
 
 434 
 
 439 
 
 445 
 
 450 
 
 455 
 
 461 
 
 466 
 
 
 
 803 
 
 472 
 
 477 
 
 482 
 
 488 
 
 493 
 
 499 
 
 504 
 
 509 
 
 515 
 
 520 
 
 
 
 804 
 
 526 
 
 531 
 
 536 
 
 542 
 
 547 
 
 553 
 
 558 
 
 563 
 
 569 
 
 574 
 
 
 
 805 
 
 580 
 
 585 
 
 590 
 
 596 
 
 601 
 
 607 
 
 612 
 
 617 
 
 623 
 
 628 
 
 
 
 806 
 
 634 
 
 639 
 
 644 
 
 650 
 
 655 
 
 660 
 
 666 
 
 671 
 
 677 
 
 682 
 
 
 
 807 
 
 687 
 
 693 
 
 698 
 
 703 
 
 709 
 
 714 
 
 720 
 
 725 
 
 730 
 
 736 
 
 
 
 808 
 
 741 
 
 747 
 
 752 
 
 757 
 
 763 
 
 768 
 
 773 
 
 779 
 
 784 
 
 789 
 
 
 
 809 
 
 795 
 
 800 
 
 806 
 
 811 
 
 816 
 
 822 
 
 827 
 
 832 
 
 838 
 
 843 
 
 
 
 810 
 
 849 
 
 854 
 
 859 
 
 865 
 
 870 
 
 875 
 
 881 
 
 886 
 
 891 
 
 897 
 
 
 6 
 
 811 
 
 902 
 
 907 
 
 913 
 
 918 
 
 924 
 
 929 
 
 934 
 
 940 
 
 945 
 
 950 
 
 
 812 
 
 956 
 
 961 
 
 966 
 
 972 
 
 977 
 
 982 
 
 988 
 
 993 
 
 998 
 
 *004 
 
 i 
 
 o 
 
 0.6 
 1 0 
 
 813 
 
 91 009 
 
 014 
 
 020 
 
 025 
 
 030 
 
 036 
 
 041 
 
 046 
 
 052 
 
 057 
 
 L 
 
 3 
 
 Y.L 
 
 1.8 
 
 814 
 
 062 
 
 068 
 
 073 
 
 078 
 
 084 
 
 089 
 
 094 
 
 100 
 
 105 
 
 110 
 
 4 
 
 2.4 
 
 815 
 
 116 
 
 121 
 
 126 
 
 132 
 
 137 
 
 142 
 
 148 
 
 153 
 
 158 
 
 164 
 
 5 
 
 3.0 
 
 816 
 
 169 
 
 174 
 
 180 
 
 185 
 
 190 
 
 196 
 
 201 
 
 206 
 
 212 
 
 217 
 
 6 
 
 3.6 
 
 817 
 
 222 
 
 228 
 
 233 
 
 238 
 
 243 
 
 249 
 
 254 
 
 259 
 
 265 
 
 270 
 
 7 
 
 Q 
 
 4.2 
 ± a 
 
 818 
 
 275 
 
 281 
 
 286 
 
 291 
 
 297 
 
 302 
 
 307 
 
 312 
 
 318 
 
 323 
 
 O 
 
 9 
 
 5.4 
 
 819 
 
 328 
 
 334 
 
 339 
 
 344 
 
 350 
 
 355 
 
 360 
 
 365 
 
 371 
 
 376 
 
 
 
 820 
 
 381 
 
 387 
 
 392 
 
 397 
 
 403 
 
 408 
 
 413 
 
 418 
 
 424 
 
 429 
 
 
 
 821 
 
 434 
 
 440 
 
 445 
 
 450 
 
 455 
 
 461 
 
 466 
 
 471 
 
 477 
 
 482 
 
 
 
 822 
 
 487 
 
 492 
 
 498 
 
 503 
 
 508 
 
 514 
 
 519 
 
 524 
 
 529 
 
 535 
 
 
 
 823 
 
 540 
 
 545 
 
 551 
 
 556 
 
 561 
 
 566 
 
 572 
 
 577 
 
 582 
 
 587 
 
 
 
 824 
 
 593 
 
 598 
 
 603 
 
 609 
 
 614 
 
 619 
 
 624 
 
 630 
 
 635 
 
 640 
 
 
 
 825 
 
 645 
 
 651 
 
 656 
 
 661 
 
 666 
 
 672 
 
 677 
 
 682 
 
 687 
 
 693 
 
 
 
 826 
 
 698 
 
 703 
 
 709 
 
 714 
 
 719 
 
 724 
 
 730 
 
 735 
 
 740 
 
 745 
 
 
 
 827 
 
 751 
 
 756 
 
 761 
 
 766 
 
 772 
 
 777 
 
 782 
 
 787 
 
 793 
 
 798 
 
 
 
 828 
 
 803 
 
 808 
 
 814 
 
 819 
 
 824 
 
 829 
 
 834 
 
 840 
 
 845 
 
 850 
 
 
 
 829 
 
 855 
 
 861 
 
 866 
 
 871 
 
 876 
 
 882 
 
 887 
 
 892 
 
 897 
 
 903 
 
 
 
 830 
 
 908 
 
 913 
 
 918 
 
 924 
 
 929 
 
 934 
 
 939 
 
 944 
 
 950 
 
 955 
 
 
 5 
 
 831 
 
 960 
 
 965 
 
 971 
 
 976 
 
 981 
 
 986 
 
 991 
 
 997 
 
 *002 
 
 *007 
 
 1 
 
 0.5 
 
 832 
 
 92 012 
 
 018 
 
 023 
 
 028 
 
 033 
 
 038 
 
 044 
 
 049 
 
 054 
 
 059 
 
 2 
 
 1.0 
 
 833 
 
 065 
 
 070 
 
 075 
 
 080 
 
 085 
 
 091 
 
 096 
 
 101 
 
 106 
 
 111 
 
 3 
 
 1.5 
 
 834 
 
 117 
 
 122 
 
 127 
 
 132 
 
 137 
 
 143 
 
 148 
 
 153 
 
 158 
 
 163 
 
 4 
 
 2.0 
 
 835 
 
 169 
 
 174 
 
 179 
 
 184 
 
 189 
 
 195 
 
 200 
 
 205 
 
 210 
 
 215 
 
 5 
 
 c 
 
 2.5 
 9 n 
 
 836 
 
 221 
 
 226 
 
 231 
 
 236 
 
 241 
 
 247 
 
 252 
 
 257 
 
 262 
 
 267 
 
 D 
 
 7 
 
 o.u 
 
 3.5 
 
 837 
 
 273 
 
 278 
 
 283 
 
 288 
 
 293 
 
 298 
 
 304 
 
 309 
 
 314 
 
 319 
 
 8 
 
 4.0 
 
 838 
 
 324 
 
 330 
 
 335 
 
 340 
 
 345 
 
 350 
 
 355 
 
 361 
 
 366 
 
 371 
 
 9 
 
 4.5 
 
 839 
 
 376 
 
 381 
 
 387 
 
 392 
 
 397 
 
 402 
 
 407 
 
 412 
 
 418 
 
 423 
 
 
 
 840 
 
 428 
 
 433 
 
 438 
 
 443 
 
 449 
 
 454 
 
 459 
 
 464 
 
 469 
 
 474 
 
 
 
 841 
 
 480 
 
 485 
 
 490 
 
 495 
 
 500 
 
 505 
 
 511 
 
 516 
 
 521 
 
 526 
 
 
 
 842 
 
 531 
 
 536 
 
 542 
 
 547 
 
 552 
 
 557 . 
 
 562 
 
 567 
 
 572 
 
 578 
 
 
 
 843 
 
 583 
 
 588 
 
 593 
 
 598 
 
 603 
 
 609 
 
 614 
 
 619 
 
 624 
 
 629 
 
 
 
 844 
 
 634 
 
 639 
 
 645 
 
 650 
 
 655 
 
 660 
 
 665 
 
 670 
 
 675 
 
 681 
 
 
 
 845 
 
 586 
 
 ; 691 
 
 696 
 
 701 
 
 706 
 
 711 
 
 716 
 
 722 
 
 727 
 
 732 
 
 
 
 846 
 
 737 
 
 742 
 
 747 
 
 752 
 
 758 
 
 763 
 
 768 
 
 773 
 
 778 
 
 783 
 
 
 
 847 
 
 78 £ 
 
 ; 793 
 
 799 
 
 804 
 
 809 
 
 814 
 
 819 
 
 824 
 
 829 
 
 834 
 
 
 
 848 
 
 84 C 
 
 1 845 
 
 850 
 
 855 
 
 860 
 
 865 
 
 870 
 
 875 
 
 881 
 
 886 
 
 
 
 849 
 
 891 
 
 . 896 
 
 901 
 
 906 
 
 911 
 
 916 
 
 921 
 
 927 
 
 932 
 
 937 
 
 
 
 850 
 
 942 
 
 1 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 988 
 
 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 I 
 
 \ P. 
 
LOGARITHMS. 
 
 483 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 850 1 
 
 851 
 
 852 
 
 853 
 
 854 
 
 855 
 
 856 
 
 857 
 
 858 
 
 859 
 
 860 
 
 861 
 
 862 
 
 863 
 
 864 
 
 865 
 
 866 
 
 867 
 
 868 
 869 
 
 870 
 
 871 
 
 872 
 
 873 
 
 874 
 
 875 
 
 876 
 
 877 
 
 878 
 
 879 
 
 880 
 
 881 
 
 882 
 
 883 
 
 884 
 
 885 
 
 886 
 
 887 
 
 888 
 889 
 
 890 
 
 891 ‘ 
 
 892 
 
 893 
 
 894 
 
 895 
 
 896 
 
 897 
 
 898 
 
 899 
 
 900 
 
 12 942 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 988 
 
 6 
 
 1 0.6 
 2 1.2 
 
 3 1.8 
 
 4 2.4 
 
 5 3.0 
 
 6 3.6 
 
 7 4.2 
 
 8 4.8 
 
 9 5.4 
 
 5 
 
 1 0.5 
 
 2 1.0 
 
 3 1.5 
 
 4 2.0 
 
 5 2,5 
 
 6 3.0 
 
 7 3.5 
 
 8 4.0 
 
 9 4.5 
 
 4 
 
 1 0.4 
 
 2 0.8 
 
 3 1.2 
 
 4 1.6 
 
 5 2.0 
 
 6 2.4 
 
 7 2.8 
 
 8 3.2 
 
 9 3.6 
 
 993 
 33 044 
 095 
 146 
 197 
 247 
 298 
 349 
 399 
 
 998 
 
 049 
 
 100 
 
 151 
 
 202 
 
 252 
 
 303 
 
 354 
 
 404 
 
 *003 
 
 054 
 
 105 
 
 156 
 
 207 
 
 258 
 
 308 
 
 359 
 
 409 
 
 *008 
 
 059 
 
 110 
 
 161 
 
 212 
 
 263 
 
 313 
 
 364 
 
 414 
 
 *013 
 
 064 
 
 115 
 
 166 
 
 217 
 
 268 
 
 318 
 
 369 
 
 420 
 
 *018 : 
 069 
 120 
 171 
 222 
 273 
 323 
 374 
 425 
 
 *024 : 
 075 
 125 
 176 
 227 
 278 
 328 
 379 
 430 
 
 *029 : 
 080 
 131 
 181 
 232 
 283 
 334 
 384 
 435 
 
 *034 : 
 085 
 136 
 186 
 237 
 288 
 339 
 389 
 440 
 
 *039 
 
 090 
 
 141 
 
 192 
 
 242 
 
 293 
 
 344 
 
 394 
 
 445 
 
 450 
 
 455 
 
 460 
 
 465 
 
 470 
 
 475 
 
 480 
 
 485 
 
 490 
 
 495 
 
 500 
 
 551 
 
 601 
 
 651 
 
 702 
 
 752 
 
 802 
 
 852 
 
 902 
 
 505 
 
 556 
 
 606 
 
 656 
 
 707 
 
 757 
 
 807 
 
 857 
 
 907 
 
 510 
 
 561 
 
 611 
 
 661 
 
 712 
 
 762 
 
 812 
 
 862 
 
 912 
 
 515 
 
 566 
 
 616 
 
 666 
 
 717 
 
 767 
 
 817 
 
 867 
 
 917 
 
 520 
 
 571 
 
 621 
 
 671 
 
 722 
 
 772 
 
 822 
 
 872 
 
 922 
 
 526 
 
 576 
 
 626 
 
 676 
 
 727 
 
 777 
 
 827 
 
 877 
 
 927 
 
 531 
 
 581 
 
 631 
 
 682 
 
 732 
 
 782 
 
 832 
 
 882 
 
 932 
 
 536 
 
 586 
 
 636 
 
 687 
 
 737 
 
 787 
 
 837 
 
 887 
 
 937 
 
 541 
 
 591 
 
 641 
 
 692 
 
 742 
 
 792 
 
 842 
 
 892 
 
 942 
 
 546 
 
 596 
 
 646 
 
 697 
 
 747 
 
 797 
 
 847 
 
 897 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 972 
 
 977 
 
 982 
 
 987 
 
 992 
 
 997 
 
 94 002 
 052 
 101 
 151 
 201 
 250 
 300 
 349 
 399 
 
 007 
 
 057 
 
 106 
 
 156 
 
 206 
 
 255 
 
 305 
 
 354 
 
 404 
 
 012 
 
 062 
 
 111 
 
 161 
 
 211 
 
 260 
 
 310 
 
 359 
 
 409 
 
 017 
 
 067 
 
 116 
 
 166 
 
 216 
 
 265 
 
 315 
 
 364 
 
 414 
 
 022 
 
 072 
 
 121 
 
 171 
 
 221 
 
 270 
 
 320 
 
 369 
 
 419 
 
 027 
 
 077 
 
 126 
 
 176 
 
 226 
 
 275 
 
 325 
 
 374 
 
 424 
 
 032 
 
 082 
 
 131 
 
 181 
 
 231 
 
 280 
 
 330 
 
 379 
 
 429 
 
 037 
 
 086 
 
 136 
 
 186 
 
 236 
 
 285 
 
 335 
 
 384 
 
 433 
 
 042 
 
 091 
 
 141 
 
 191 
 
 240 
 
 290 
 
 340 
 
 389 
 
 438 
 
 047 
 
 096 
 
 146 
 
 196 
 
 245 
 
 295 
 
 345 
 
 394 
 
 443 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 473 
 
 478 
 
 483 
 
 488 
 
 493 
 
 498 
 
 547 
 
 596 
 
 645 
 
 694 
 
 743 
 
 792 
 
 841 
 
 890 
 
 503 
 
 552 
 
 601 
 
 650 
 
 699 
 
 748 
 
 797 
 
 846 
 
 895 
 
 507 
 
 557 
 
 606 
 
 655 
 
 704 
 
 753 
 
 802 
 
 851 
 
 900 
 
 512 
 
 562 
 
 611 
 
 660 
 
 709 
 
 758 
 
 807 
 
 856 
 
 905 
 
 517 
 
 567 
 
 616 
 
 665 
 
 714 
 
 763 
 
 812 
 
 861 
 
 910 
 
 522 
 
 571 
 
 621 
 
 670 
 
 719 
 
 768 
 
 817 
 
 866 
 
 915 
 
 527 
 
 576 
 
 626 
 
 675 
 
 724 
 
 773 
 
 822 
 
 871 
 
 919 
 
 532 
 
 581 
 
 630 
 
 680 
 
 729 
 
 778 
 
 827 
 
 876 
 
 924 
 
 537 
 
 586 
 
 635 
 
 685 
 
 734 
 
 783 
 
 832 
 
 880 
 
 929 
 
 542 
 
 591 
 
 640 
 
 689 
 
 738 
 
 787 
 
 836 
 
 885 
 
 934 
 
 939 
 
 944 
 
 949 
 
 954 
 
 959 
 
 963 
 
 968 
 
 973 
 
 978 
 
 983 
 
 988 
 95 036 
 085 
 134 
 182 
 231 
 279 
 328 
 37 € 
 
 993 
 041 
 090 
 139 
 i 187 
 236 
 ' 284 
 1 332 
 i 381 
 
 998 
 
 046 
 
 095 
 
 143 
 
 192 
 
 240 
 
 289 
 
 337 
 
 386 
 
 *002 
 
 051 
 
 100 
 
 148 
 
 197 
 
 245 
 
 294 
 
 342 
 
 390 
 
 *007 
 
 056 
 
 105 
 
 153 
 
 202 
 
 -250 
 
 299 
 
 347 
 
 395 
 
 *012 
 
 061 
 
 109 
 
 158 
 
 207 
 
 255 
 
 303 
 
 352 
 
 400 
 
 *017 
 
 066 
 
 114 
 
 163 
 
 211 
 
 260 
 
 308 
 
 357 
 
 405 
 
 *022 
 
 071 
 
 119 
 
 168 
 
 216 
 
 265 
 
 313 
 
 361 
 
 410 
 
 *027 
 
 075 
 
 124 
 
 173 
 
 221 
 
 270 
 
 318 
 
 366 
 
 415 
 
 *032 
 
 080 
 
 129 
 
 177 
 
 226 
 
 274 
 
 323 
 
 371 
 
 419 
 
 424 
 
 t 429 
 
 434 
 
 439 
 
 444 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 N. * 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
490 
 
 LOGARITHMS . 
 
 N. 
 
 L. 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 soo 
 
 901 
 
 902 
 
 903 
 
 904 
 
 905 
 
 906 
 
 907 
 
 908 
 
 909 
 
 910 
 
 911 
 
 912 
 
 913 
 
 914 
 
 915 
 
 916 
 
 917 
 
 918 
 
 919 
 
 920 
 
 921 
 
 922 
 
 923 
 
 924 
 
 925 
 
 926 
 
 927 
 
 928 
 
 929 
 
 930 
 
 931 
 
 932 
 
 933 
 
 934 
 
 935 
 
 936 
 
 937 
 
 938 
 
 939 
 
 940 
 
 941 
 
 942 
 
 943 
 
 944 
 
 945 
 
 946 
 
 947 
 
 948 
 
 949 
 
 950 
 
 & 424 
 
 429 
 
 434 
 
 439 
 
 444 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 5 
 
 1 0.5 
 
 2 1.0 
 
 3 1.5 
 
 4 2.0 
 
 5 2.5 
 
 6 3.0 
 
 7 3.5 
 
 8 4.0 
 
 9 4.5 
 
 4 
 
 1 0.4 
 
 2 0.8 
 
 3 1.2 
 
 4 1.6 
 
 5 2.0 
 
 6 2.4 
 
 7 2.8 
 
 8 3.2 | 
 
 9 3.6 
 
 472 
 
 521 
 
 569 
 
 617 
 
 665 
 
 713 
 
 761 
 
 809 
 
 856 
 
 477 
 
 525 
 
 574 
 
 622 
 
 670 
 
 718 
 
 766 
 
 813 
 
 861 
 
 482 
 
 530 
 
 578 
 
 626 
 
 674 
 
 722 
 
 770 
 
 818 
 
 866 
 
 487 
 
 535 
 
 583 
 
 631 
 
 679 
 
 727 
 
 775 
 
 823 
 
 871 
 
 492 
 
 540 
 
 588 
 
 636 
 
 684 
 
 732 
 
 780 
 
 828 
 
 875 
 
 497 
 
 545 
 
 593 
 
 641 
 
 689 
 
 737 
 
 785 
 
 832 
 
 880 
 
 501 
 
 550 
 
 598 
 
 646 
 
 694 
 
 742 
 
 789 
 
 837 
 
 885 
 
 506 
 
 554 
 
 602 
 
 650 
 
 698 
 
 746 
 
 794 
 
 842 
 
 890 
 
 511 
 
 559 
 
 607 
 
 655 
 
 703 
 
 751 
 
 799 
 
 847 
 
 895 
 
 516 
 
 564 
 
 612 
 
 660 
 
 708 
 
 756 
 
 804 
 
 852 
 
 899 
 
 904 
 
 909 
 
 914 
 
 918 
 
 923 
 
 928 
 
 933 
 
 938 
 
 942 
 
 947 
 
 952 
 999 
 96 047 
 095 
 142 
 190 
 237 
 284 
 332 
 
 957 
 
 *004 
 
 052 
 
 099 
 
 147 
 
 194 
 
 242 
 
 289 
 
 336 
 
 961 
 
 *009 
 
 057 
 
 104 
 
 152 
 
 199 
 
 246 
 
 294 
 
 341 
 
 966 
 
 *014 
 
 061 
 
 109 
 
 156 
 
 204 
 
 251 
 
 298 
 
 346 
 
 971 
 
 *019 
 
 066 
 
 114 
 
 161 
 
 209 
 
 256 
 
 303 
 
 350 
 
 976 
 
 *023 
 
 071 
 
 118 
 
 166 
 
 213 
 
 261 
 
 308 
 
 355 
 
 980 
 
 *028 
 
 076 
 
 123 
 
 171 
 
 218 
 
 265 
 
 313 
 
 360 
 
 985 
 
 *033 
 
 080 
 
 128 
 
 175 
 
 223 
 
 270 
 
 317 
 
 365 
 
 990 
 
 *038 
 
 085 
 
 133 
 
 180 
 
 227 
 
 275 
 
 322 
 
 369 
 
 995 
 
 *042 
 
 090 
 
 137 
 
 185 
 
 232 
 
 280 
 
 327 
 
 374 
 
 379 
 
 384 
 
 388 
 
 393 
 
 398 
 
 402 
 
 407 
 
 412 
 
 417 
 
 421 
 
 426 
 
 473 
 
 520 
 
 567 
 
 614 
 
 661 
 
 708 
 
 755 
 
 802 
 
 431 
 
 478 
 
 525 
 
 572 
 
 619 
 
 666 
 
 713 
 
 759 
 
 806 
 
 435 
 
 483 
 
 530 
 
 577 
 
 624 
 
 670 
 
 717 
 
 764 
 
 811 
 
 440 
 
 487 
 
 534 
 
 581 
 
 628 
 
 675 
 
 722 
 
 769 
 
 .816 
 
 445 
 
 492 
 
 539 
 
 586 
 
 633 
 
 680 
 
 727 
 
 774 
 
 820 
 
 450 
 
 497 
 
 544 
 
 591 
 
 638 
 
 685 
 
 731 
 
 778 
 
 825 
 
 454 
 
 501 
 
 548 
 
 595 
 
 642 
 
 689 
 
 736 
 
 783 
 
 830 
 
 459 
 
 506 
 
 553 
 
 600 
 
 647 
 
 694 
 
 741 
 
 788 
 
 834 
 
 464 
 
 511 
 
 558 
 
 605 
 
 652 
 
 699 
 
 745 
 
 792 
 
 839 
 
 468 
 
 515 
 
 562 
 
 609 
 
 656 
 
 703 
 
 750 
 
 797 
 
 844 
 
 848 
 
 853 
 
 858 
 
 862 
 
 867 
 
 872 
 
 876 
 
 881 
 
 886 
 
 890 
 
 895 
 942 
 988 
 97 035 
 081 
 128 
 174 
 220 
 267 
 
 900 
 
 946 
 
 993 
 
 039 
 
 086 
 
 132 
 
 179 
 
 225 
 
 271 
 
 904 
 
 951 
 
 997 
 
 044 
 
 090 
 
 137 
 
 183 
 
 230 
 
 276 
 
 909 
 
 956 
 
 *002 
 
 049 
 
 095 
 
 142 
 
 188 
 
 234 
 
 280 
 
 914 
 
 960 
 
 *007 
 
 053 
 
 100 
 
 146 
 
 192 
 
 239 
 
 285 
 
 918 
 
 965 
 
 *011 
 
 058 
 
 104 
 
 151 
 
 197 
 
 243 
 
 290 
 
 923 
 
 970 
 
 *016 
 
 063 
 
 109 
 
 155 
 
 202 
 
 248 
 
 294 
 
 928 
 
 974 
 
 *021 
 
 067 
 
 114 
 
 160 
 
 206 
 
 253 
 
 299 
 
 932 
 
 979 
 
 *025 
 
 072 
 
 118 
 
 165 
 
 211 
 
 257 
 
 304 
 
 937 
 
 984 
 
 *030 
 
 077 
 
 123 
 
 169 
 
 216 
 
 262 
 
 308 
 
 313 
 
 317 
 
 322 
 
 327 
 
 331 
 
 336 
 
 340 
 
 345 
 
 350 
 
 354 
 
 
 359 
 405 
 451 
 I 497 
 543 
 589 
 635 
 681 
 727 
 
 364 
 410 
 456 
 502 
 , 548 
 i 594 
 > 640 
 685 
 ' 731 
 
 368 
 
 414 
 
 460 
 
 506 
 
 552 
 
 598 
 
 644 
 
 690 
 
 736 
 
 373 
 
 419 
 
 465 
 
 511 
 
 557 
 
 603 
 
 649 
 
 695 
 
 740 
 
 377 
 
 424 
 
 470 
 
 516 
 
 562 
 
 607 
 
 653 
 
 699 
 
 745 
 
 382 
 
 428 
 
 474 
 
 520 
 
 566 
 
 612 
 
 658 
 
 704 
 
 749 
 
 387 
 
 433 
 
 479 
 
 525 
 
 571 
 
 617 
 
 663 
 
 708 
 
 754 
 
 391 
 
 437 
 
 483 
 
 529 
 
 575 
 
 621 
 
 667 
 
 713 
 
 759 
 
 396 
 
 442 
 
 488 
 
 534 
 
 580 
 
 626 
 
 672 
 
 717 
 
 763 
 
 400 
 
 447 
 
 493 
 
 539 
 
 585 
 
 630 
 
 676 
 
 722 
 
 768 
 
 
 772 
 
 \ 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 813 
 
 
 N. 
 
 L.O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
LOGARITHMS 
 
 491 
 
 1 
 
 L 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
 950 
 
 Y7 772 ' 
 
 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 813 
 
 5 
 
 1 0.5 
 
 2 1.0 
 
 3 1.5 
 
 4 2.0 
 
 5 2.5 
 
 6 3.0 
 
 7 3.5 
 
 8 4.0 
 
 9 4.5 
 
 4 
 
 1 0.4 
 
 2 0.8 
 
 3 1.2 
 
 4 1.6 
 
 5 2.0 
 
 6 2.4 
 
 7 2.8 
 
 8 3.2 
 
 9 3.6 
 
 951 
 
 952 
 
 953 
 
 954 
 
 955 
 
 956 
 
 957 
 
 958 
 
 959 
 
 960 
 
 961 
 
 962 
 
 963 
 
 964 
 
 965 
 
 966 
 
 967 
 
 968 
 
 969 
 
 970 
 
 971 
 
 972 
 
 973 
 
 974 
 
 975 
 
 976 
 
 977 
 978. 
 979 
 
 980 
 
 981 
 
 982 
 
 983 
 
 984 
 
 985 
 
 986 
 
 987 
 
 988 
 
 989 
 
 990 
 
 991 
 
 992 
 
 993 
 
 994 
 
 995 
 
 996 
 
 997 
 
 998 
 
 999 
 
 I00C 
 
 818 
 864 
 909 
 955 
 )8 000 
 046 
 091 
 137 
 182 
 
 823 
 
 868 
 
 914 
 
 959 
 
 005 
 
 050 
 
 096 
 
 141 
 
 186 
 
 827 
 
 873 
 
 918 
 
 964 
 
 009 
 
 055 
 
 100 
 
 146 
 
 191 
 
 832 
 
 877 
 
 923 
 
 968 
 
 014 
 
 059 
 
 105 
 
 150 
 
 195 
 
 836 
 
 882 
 
 928 
 
 973 
 
 019 
 
 064 
 
 109 
 
 155 
 
 200 
 
 841 
 
 886 
 
 932 
 
 978 
 
 023 
 
 068 
 
 114 
 
 159 
 
 204 
 
 845 
 
 891 
 
 937 
 
 982 
 
 028 
 
 073 
 
 118 
 
 164' 
 
 209 
 
 850 
 
 896 
 
 941 
 
 987 
 
 032 
 
 078 
 
 123 
 
 168 
 
 214 
 
 855 
 
 900 
 
 946 
 
 991 
 
 037 
 
 082 
 
 127 
 
 173 
 
 218 
 
 859 
 
 905 
 
 950 
 
 996 
 
 041 
 
 087 
 
 132 
 
 177 
 
 223 
 
 227 
 
 232 
 
 236 
 
 241 
 
 245 
 
 250 
 
 254 
 
 259 
 
 263 
 
 268 
 
 272 
 
 318 
 
 363 
 
 408 
 
 453 
 
 498 
 
 543 
 
 588 
 
 632 
 
 277 
 
 322 
 
 367 
 
 412 
 
 457 
 
 502 
 
 547 
 
 592 
 
 637 
 
 281 
 
 327 
 
 372 
 
 417 
 
 462 
 
 507 
 
 552 
 
 597 
 
 641 
 
 286 
 
 331 
 
 376 
 
 421 
 
 466 
 
 511 
 
 556 
 
 601 
 
 646 
 
 290 
 
 336 
 
 381 
 
 426 
 
 471 
 
 516 
 
 561 
 
 605 
 
 650 
 
 295 
 
 340 
 
 385 
 
 430 
 
 475 
 
 520 
 
 565 
 
 610 
 
 655 
 
 299 
 
 345 
 
 390 
 
 435 
 
 480 
 
 525 
 
 570 
 
 614 
 
 659 
 
 304 
 
 349 
 
 394 
 
 439 
 
 484 
 
 529 
 
 574 
 
 619 
 
 664 
 
 308 
 
 354 
 
 399 
 
 444 
 
 489 
 
 534 
 
 579 
 
 623 
 
 668 
 
 313 
 
 358 
 
 403 
 
 448 
 
 493 
 
 538 
 
 583 
 
 628 
 
 673 
 
 677 
 
 682 
 
 686 
 
 691 
 
 695 
 
 700 
 
 704 
 
 709 
 
 713 
 
 717 
 
 722 
 767 
 811 
 856 
 900 
 945 
 989 
 99 034 
 078 
 
 726 
 
 771 
 
 816 
 
 860 
 
 905 
 
 949 
 
 994 
 
 038 
 
 083 
 
 731 
 
 776 
 
 820 
 
 865 
 
 909 
 
 954 
 
 998 
 
 043 
 
 087 
 
 735 
 
 780 
 
 825 
 
 869 
 
 914 
 
 958 
 
 *003 
 
 047 
 
 092 
 
 740 
 
 784 
 
 829 
 
 874 
 
 918 
 
 963 
 
 *007 
 
 052 
 
 096 
 
 744 
 
 789 
 
 834 
 
 878 
 
 923 
 
 967 
 
 *012 
 
 056 
 
 100 
 
 749 
 
 793 
 
 838 
 
 883 
 
 927 
 
 972 
 
 *016 
 
 061 
 
 105 
 
 753 
 
 798 
 
 843 
 
 887 
 
 932 
 
 976 
 
 *021 
 
 065 
 
 109 
 
 758 
 
 802 
 
 847 
 
 892 
 
 936 
 
 981 
 
 *025 
 
 069 
 
 114 
 
 762 
 
 807 
 
 851 
 
 896 
 
 941 
 
 985 
 
 *029 
 
 074 
 
 118 
 
 123 
 
 127 
 
 131 
 
 136 
 
 140 
 
 145 
 
 149 
 
 154 
 
 158 
 
 162 
 
 167 
 
 211 
 
 255 
 
 300 
 
 344 
 
 388 
 
 432 
 
 476 
 
 52C 
 
 171 
 216 
 260 
 304 
 348 
 i 392 
 ! 436 
 i 480 
 1 524 
 
 176 
 
 220 
 
 264 
 
 308 
 
 352 
 
 396 
 
 441 
 
 484 
 
 528 
 
 180 
 
 224 
 
 269 
 
 313 
 
 357 
 
 401 
 
 445 
 
 489 
 
 533 
 
 185 
 
 229 
 
 273 
 
 317 
 
 361 
 
 405 
 
 449 
 
 493 
 
 537 
 
 189 
 
 233 
 
 277 
 
 322 
 
 366 
 
 410 
 
 454 
 
 498 
 
 542 
 
 193 
 
 238 
 
 282 
 
 326 
 
 370 
 
 414 
 
 458 
 
 502 
 
 546 
 
 198 
 
 242 
 
 286 
 
 330 
 
 374 
 
 4ia 
 
 463 
 
 506 
 
 550 
 
 202 
 
 247 
 
 291 
 
 335 
 
 379 
 
 423 
 
 467 
 
 511 
 
 555 
 
 207 
 
 251 
 
 295 
 
 339 
 
 383 
 
 427 
 
 471 
 
 515 
 
 559 
 
 56 4 
 
 t 568 
 
 572 
 
 577 
 
 581 
 
 585 
 
 590 
 
 594 
 
 599 
 
 603 
 
 607 
 651 
 69 1 
 73< 
 782 
 '82( 
 87( 
 91; 
 95' 
 
 ? 612 
 [ 656 
 > 699 
 ) 743 
 1 787 
 3 830 
 ) 874 
 3 917 
 7 961 
 
 616 
 
 660 
 
 704 
 
 747 
 
 791 
 
 835 
 
 878 
 
 922 
 
 965 
 
 621 
 
 664 
 
 708 
 
 752 
 
 795 
 
 839 
 
 883 
 
 926 
 
 970 
 
 625 
 
 669 
 
 712 
 
 756 
 
 800 
 
 843 
 
 887 
 
 930 
 
 974 
 
 629 
 
 673 
 
 717 
 
 760 
 
 804 
 
 848 
 
 891 
 
 935 
 
 978 
 
 634 
 
 677 
 
 721 
 
 765 
 
 808 
 
 852 
 
 896 
 
 939 
 
 983 
 
 638 
 
 682 
 
 726 
 
 769 
 
 813 
 
 856 
 
 900 
 
 944 
 
 987 
 
 642 
 
 686 
 
 730 
 
 774 
 
 817 
 
 861 
 
 904 
 
 948 
 
 991 
 
 647 
 
 691 
 
 734 
 
 778 
 
 822 
 
 865 
 
 909 
 
 952 
 
 996 
 
 | 00 001 
 
 1 004 
 
 009 
 
 013 
 
 017 
 
 022 
 
 026 
 
 030 
 
 035 
 
 039 
 
 
 N. 
 
 L.O 
 
 1 1 
 
 2 
 
 3 
 
 I 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P. P. 
 
492 
 
 LOGAixs m ^ nv TRIGONOMETRIC FUNCTIONS. 
 
 0 ° 
 
 // 
 
 ~ o 
 60 
 120 
 180 
 240 
 
 / 
 
 L. Sin. 1 d. 
 
 Cpl. S. 
 
 Cpl. T. 
 
 L. Tang.j d. c. 
 
 . L. Cotg. 
 
 L. Cos. 
 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6.46373 
 
 6.76476 
 
 6.94085 
 
 7.06579 
 
 30103 
 
 17609 
 
 12494 
 
 9691 
 
 7918 
 
 6694 
 
 5800 
 
 5115 
 
 4576 
 
 4139 
 
 3779 
 
 3476 
 
 3218 
 
 2997 
 
 2802 
 
 2633 
 
 2483 
 
 2348 
 
 2227 
 
 2119 
 
 2021 
 
 1930 
 
 1848 
 
 1773 
 
 1704 
 
 1639 
 
 1579 
 
 1524 
 
 1472 
 
 1424 
 
 1379 
 
 1336 
 
 1297 
 
 1259 
 
 1223 
 
 1190 
 
 1158 
 
 1128 
 
 1100 
 
 1072 
 
 1046 
 
 1022 
 
 999 
 
 976 
 
 954 
 
 934 
 
 914 
 
 896 
 
 877 
 
 860 
 
 843 
 
 827 
 
 812 
 
 797 
 
 782 
 
 769 
 
 755 
 
 743 
 
 730 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 6.46373 
 
 6.76476 
 
 6.94085 
 
 7.06579 
 
 30103 
 
 17609 
 
 12494 
 
 9691 
 
 7918 
 
 6694 
 
 5800 
 
 5115 
 
 4576 
 
 4139 
 
 3779 
 
 3476 
 
 3219 
 
 2996 
 
 2803 
 
 2633 
 
 2482 
 
 2348 
 
 2228 
 
 2119 
 
 2020 
 
 1931 
 
 1848 
 
 1773 
 
 1704 
 
 1639 
 
 1579 
 
 1524 
 
 1473 
 
 1424 
 
 1379 
 
 1336 
 
 1297 
 
 1259 
 
 1223 
 
 1190 
 
 1159 
 
 1128 
 
 1100 
 
 1072 
 
 1047 
 
 1022 
 
 998 
 
 976 
 
 955 
 
 934 
 
 915 
 
 895 
 
 878 
 
 860 
 
 843 
 
 828 
 
 812 
 
 797 
 
 782 
 
 769 
 
 756 
 
 742 
 
 730 
 
 3.53627 
 
 3.23524 
 
 3.05915 
 
 2.93421 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 300 
 
 360 
 
 420 
 
 480 
 
 540 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 7.16270 
 
 7.24188 
 
 7.30882 
 
 7.36682 
 
 7.41797 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 7.16270 
 
 7.24188 
 
 7.30882 
 
 7.36682 
 
 7.41797 
 
 2.83730 
 
 2.75812 
 
 2.69118 
 
 2.63318 
 
 2.58203 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 600 
 
 660 
 
 720 
 
 780 
 
 840 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 7.46373 
 
 7.50512 
 
 7.54291 
 
 7.57767 
 
 7.60985 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 7.46373 
 
 7.50512 
 
 7.54291 
 
 7.57767 
 
 7.60986 
 
 2.53627 
 
 2.49488 
 
 2.45709 
 
 2.42233 
 
 2.39014 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 0.00000 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 900 
 
 960 
 
 1020 
 
 1080 
 
 1140 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 7.63982 
 
 7.66784 
 
 7.69417 
 
 7.71900 
 
 7.74248 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 7.63982 
 
 7.66785 
 
 7.69418 
 
 7.71900 
 
 7.74248 
 
 2.36018 
 
 2.33215 
 
 2.30582 
 
 2.28100 
 
 2.25752 
 
 0.00000 
 
 0.00000 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 1200 
 
 1260 
 
 1320 
 
 1380 
 
 1440 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 7.76475 
 
 7.78594 
 
 7.80615 
 
 7.82545 
 
 7.84393 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 7.76476 
 
 7.78595 
 
 7.80615 
 
 7.82546 
 
 7.84394 
 
 2.23524 
 
 2.21405 
 
 2.19385 
 
 2.17454 
 
 2.15606 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 1500 
 
 1560 
 
 1620 
 
 1680 
 
 1740 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 7.86166 
 
 7.87870 
 
 7.89509 
 
 7.91088 
 
 7.92612 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31442 
 
 5.31441 
 
 7.86167 
 
 7.87871 
 
 7.89510 
 
 7.91089 
 
 7.92613 
 
 2.13833 
 
 2.12129 
 
 2.10490 
 
 2.08911 
 
 2.07387 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 9.99999 
 
 9.99998 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 1800 
 
 1860 
 
 1920 
 
 1980 
 
 2040 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 7.94084 
 
 7.95508 
 
 7.96887 
 
 7.98223 
 
 7.99520 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 7.94086 
 
 7.95510 
 
 7.96889 
 
 7.98225 
 
 7.99522 
 
 2.05914 
 
 2.04490 
 
 2.03111 
 
 2.01775 
 
 2.00478 
 
 9.99998 
 
 9.99998 
 
 9.99998 
 
 9.99998 
 
 9.99998 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 2100 
 
 2160 
 
 2220 
 
 2280 
 
 2340 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 8.00779 
 
 8.02002 
 
 8.03192 
 
 8.04350 
 
 8.05478 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31443 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 5.31441 
 
 8.00781 
 
 8.02004 
 
 8.03194 
 
 8.04353 
 
 8.05481 
 
 1.99219 
 
 1.97996 
 
 1.96806 
 
 1.95647 
 
 1.94519 
 
 9.99998 
 
 9.99998 
 
 9.99997 
 
 9.99997 
 
 9.99997 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 2400 
 
 2460 
 
 2520 
 
 2580 
 
 2640 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 8.06578 
 
 8.07650 
 
 8.08696 
 
 8.09718 
 
 8.10717 
 
 5.31443 
 
 5.31444 
 5.31444 
 5.31444 
 5.31444 
 
 5.31441 
 
 5.31440 
 
 5.31440 
 
 5.31440 
 
 5.31440 
 
 8.06581 
 
 8.07653 
 
 8.08700 
 
 8.09722 
 
 8.10720 
 
 1.93419 
 
 1.92347 
 
 1.91300 
 
 1.90278 
 
 1.89280 
 
 9.99997 
 
 9.99997 
 
 9.99997 
 
 9.99997 
 
 9.99996 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 2700 
 
 2760 
 
 2820 
 
 2880 
 
 2940 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 8.11693 
 
 8.12647 
 
 8.13581 
 
 8.14495 
 
 8.15391 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31440 
 
 5.31440 
 
 5.31440 
 
 5.31440 
 
 5.31440 
 
 8.11696 
 
 8.12651 
 
 8.13585 
 
 8.14500 
 
 8.15395 
 
 1.88304 
 
 1.87349 
 
 1.86415 
 
 1.85500 
 
 1.84605 
 
 9.99996 
 
 9.99996 
 
 9.99996 
 
 9.99996 
 
 9.99996 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 3000 
 
 3060 
 
 3120 
 
 3180 
 
 3240 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 8.16268 
 
 8.17128 
 
 8.17971 
 
 8.18798 
 
 8.19610 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31444 
 
 5.31439 
 
 5.31439 
 
 5.31439 
 
 5.31439 
 
 5.31439 
 
 8.16273 
 
 8.17133 
 
 8.17976 
 
 8.18804 
 
 8.19616 
 
 1.83727 
 
 1.82867 
 
 1.82024 
 
 1.81196 
 
 1.80384 
 
 9.99995 
 
 9.99995 
 
 9.99995 
 
 9.99995 
 
 9.99995 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 3300 
 
 3360 
 
 3420 
 
 3480 
 
 3540 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 8.20407 
 
 8.21189 
 
 8.21958 
 
 8.22713 
 
 8.23456 
 
 5.31444 
 
 5.31444 
 
 5.31445 
 5.31445 
 5.31445 
 
 5.31439 
 
 5.31439 
 
 5.31439 
 
 5.31438 
 
 5.31438 
 
 8.20413 
 
 8.21195 
 
 8.21964 
 
 8.22720 
 
 8.23462 
 
 1.79587 
 
 1.78805 
 
 1.78036 
 
 1.77280 
 
 1.76538 
 
 9.99994 
 
 9.99994 
 
 9.99994 
 
 9.99994 
 
 9.99994 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 3600 
 
 60 
 
 8.24186 
 
 5.31445 
 
 5.31438 
 
 8.24192 
 
 1.75808 
 
 9.99993 
 
 0 
 
 
 
 L. Cos. 1 d. 
 
 
 
 L. Cotg. 
 
 d. c. 
 
 L. Tang. 
 
 L. Sin. 
 
 / 
 
 89 ° 
 
493 
 
 LOGARITHMS OF TRIGONOMETRIC FTT 
 
 1 v 
 
 n 
 
 / 
 
 L. Sin. 
 
 d. 
 
 Cpl. S. 
 
 Cpl. T. I 
 
 j. Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 3600 
 
 3660 
 
 3720 
 
 3780 
 
 3840 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 8.24186 
 
 8.24903 
 
 8.25609 
 
 8.26304 
 
 8.26988 
 
 717 
 
 706 
 
 695 
 
 684 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31438 
 
 5.31438 
 
 5.31438 
 
 5.31438 
 
 5.31437 
 
 8.24192 
 
 8.24910 
 
 8.25616 
 
 8.26312 
 
 8.26996 
 
 718 
 706 
 696 
 684 
 673 - 
 
 1.75808 ' 
 
 1.75090 
 
 1.74384 
 
 1.73688 
 
 1.73004 
 
 9.99993 
 
 9.99993 
 
 9.99993 
 
 9.99993 
 
 9.99992 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 3900 
 
 3960 
 
 4020 
 
 4080 
 
 4140 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 8.27661 
 
 8.28324 
 
 8.28977 
 
 8.29621 
 
 8.30255 
 
 673 
 
 663 
 
 653 
 
 644 
 
 634 
 
 624 
 
 616 
 
 608 
 
 599 
 
 590 
 
 583 
 
 575 
 
 568 
 
 560 
 
 553 
 
 547 
 
 539 
 
 533 
 
 526 
 
 520 
 
 514 
 
 508 
 
 502 
 
 496 
 
 491 
 
 485 
 
 480 
 
 474 
 
 470 
 
 464 
 
 459 
 
 455 
 
 450 
 
 445 
 
 441 
 
 436 
 
 433 
 
 427 
 
 424 
 
 419 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31445 
 
 5.31437 
 
 5.31437 
 
 5.31437 
 
 5.31437 
 
 5.31437 
 
 8.27669 
 
 8.28332 
 
 8.28986 
 
 8.29629 
 
 8.30263 
 
 663 
 654 
 643 
 634 
 625 - 
 
 1.72331 
 
 1.71668 
 
 1.71014 
 
 1.70371 
 
 1.69737 
 
 9.99992 
 
 9.99992 
 
 9.99992 
 
 9.99992 
 
 9.99991 
 
 4200 
 
 4260 
 
 4320 
 
 4380 
 
 4440 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 8.30879 
 
 8,31495 
 
 8.32103 
 
 8.32702 
 
 8.33292 
 
 5.31446 
 
 5.31446 
 
 5.31446 
 
 5.31446 
 
 5.31446 
 
 5.31437 
 
 5.31436 
 
 5.31436 
 
 5.31436 
 
 5.31436 
 
 8.30888 
 
 8.31505 
 
 8.32112 
 
 8.32711 
 
 8.33302 
 
 617 
 607 
 599 
 591 
 584 - 
 
 1.69112 
 
 1.68495 
 
 1.67888 
 
 1.67289 
 
 1.66698 
 
 9.99991 
 
 9.99991 
 
 9.99990 
 
 9.99990 
 
 9.99990 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 4500 
 
 4560 
 
 4620 
 
 4680 
 
 4740 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 8.33875 
 
 8.34450 
 
 8.35018 
 
 8.35578 
 
 8.36131 
 
 5.31446 
 5.31446 
 5 31446 
 5.31446 
 5.31446 
 
 5.31436 
 
 5.31435 
 
 5.31435 
 
 5.31435 
 
 5.31435 
 
 8.33886 
 
 8.34461 
 
 8.35029 
 
 8.35590 
 
 8.36143 
 
 575 
 
 568 
 
 561 
 
 553 
 
 546 
 
 1.66114 
 
 1.65539 
 
 1.64971 
 
 1.64410 
 
 1.63857 
 
 9.99990 
 
 9.99989 
 
 9.99989 
 
 9.99989 
 
 9.99989 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 4800 
 
 4860 
 
 4920 
 
 4980 
 
 5040 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 8.36678 
 
 8.37217 
 
 8.37750 
 
 8.38276 
 
 8.38796 
 
 5.31446 
 
 5.31447 
 5.31447 
 5.31447 
 5.31447 
 
 5.31435 
 
 5.31434 
 
 5.31434 
 
 5.31434 
 
 5.31434 
 
 8.36689 
 
 8.37229 
 
 8.37762 
 
 8.38289 
 
 8.38809 
 
 540 
 
 533 
 
 527 
 
 520 
 
 514 
 
 1.63311 
 
 1.62771 
 
 1.62238 
 
 1.61711 
 
 1.61191 
 
 9.99988 
 
 9.99988 
 
 9.99988 
 
 9.99987 
 
 9.99987 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 5100 
 
 5160 
 
 5220 
 
 5280 
 
 5340 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 8.39310 
 
 8.39818 
 
 8.40320 
 
 8.40816 
 
 8.41307 
 
 5.31447 
 
 5.31447 
 
 5.31447 
 
 5.31447 
 
 5.31447 
 
 5.31434 
 
 5.31433 
 
 5.31433 
 
 5.31433 
 
 5.31433 
 
 8.39323 
 
 8.39832 
 
 8.40334 
 
 8.40830 
 
 8.41321 
 
 509 
 
 502 
 
 496 
 
 491 
 
 486 
 
 1.60677 
 
 1.60168 
 
 1.59666 
 
 1.59170 
 
 1.58679 
 
 9.99987 
 
 9.99986 
 
 9.99986 
 
 9.99986 
 
 9.99985 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 5400 
 
 5460 
 
 5520 
 
 5580 
 
 5640 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 8.41792 
 
 8.42272 
 
 8.42746 
 
 8.43216 
 
 8.43680 
 
 5.31447 
 
 5.31448 
 5.31448 
 5.31448 
 5.31448 
 
 5.31433 
 
 5.31432 
 
 5.31432 
 
 5.31432 
 
 5.31432 
 
 8.41807 
 
 8.42287 
 
 8.42762 
 
 8.43232 
 
 8.43696 
 
 480 
 
 475 
 
 470 
 
 464 
 
 460 
 
 1.58193 
 
 1.57713 
 
 1.57238 
 
 1.56768 
 
 1.56304 
 
 9.99985 
 
 9.99985 
 
 9.99984 
 
 9.99984 
 
 9.99984 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 5700 
 
 5760 
 
 5820 
 
 5880 
 
 5940 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 8.44139 
 
 8.44594 
 
 8.45044 
 
 8.45489 
 
 8.45930 
 
 5.31448 
 
 5.31448 
 
 5.31448 
 
 5.31448 
 
 5.31449 
 
 5.31431 
 
 5.31431 
 
 5.31431 
 
 5.31431 
 
 5.31431 
 
 8.44156 
 
 8.44611 
 
 8.45061 
 
 8.45507 
 
 8.45948 
 
 455 
 
 450 
 
 446 
 
 441 
 
 437 
 
 1.55844 
 
 1.55389 
 
 1.54939 
 
 1.54493 
 
 1.54052 
 
 9.99983 
 
 9.99983 
 
 9.99983 
 
 9.99982 
 
 9.99982 
 
 25 
 
 24 
 
 23 ‘ 
 22 
 21 
 
 6000 
 
 6060 
 
 6120 
 
 6180 
 
 6240 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 8.46366 
 
 8.46799 
 
 8.47226 
 
 8.47650 
 
 8.48069 
 
 5.31449 
 
 5.31449 
 
 5.31449 
 
 5.31449 
 
 5.31449 
 
 5.31430 
 
 5.31430 
 
 5.31430 
 
 5.31430 
 
 5.31429 
 
 8.46385 
 
 8.46817 
 
 8.47245 
 
 8.47669 
 
 8.48089 
 
 432 
 428 
 424 
 420 
 • 416 
 412 
 408 
 404 
 401 
 - 397 
 393 
 390 
 386 
 383 
 380 
 
 1.53615 
 
 1.53183 
 
 1.52755 
 
 1.52331 
 
 1.51911 
 
 9.99982 
 
 9.99981 
 
 9.99981 
 
 9.99981 
 
 9.99980 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 6300 
 
 6360 
 
 6420 
 
 6480 
 
 6540 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 8.48485 
 
 8.48896 
 
 8.49304 
 
 8.49708 
 
 8.50108 
 
 416 
 
 411 
 
 408 
 
 404 
 
 400 
 
 5.31449 
 
 5.31449 
 
 5.31450 
 5.31450 
 5.31450 
 
 5.31429 
 
 5.31429 
 
 5.31428 
 
 5.31428 
 
 5.31428 
 
 8.48505 
 
 8.48917 
 
 8.49325 
 
 8.49729 
 
 8.50130 
 
 1.51495 
 
 1.51083 
 
 1.50675 
 
 1.50271 
 
 1.49870 
 
 9.99980 
 
 9.99979 
 
 9.99979 
 
 9.99979 
 
 9.99978 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 6600 
 
 6660 
 
 6720 
 
 6780 
 
 6840 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 8.50504 
 
 8.50897 
 
 8.51287 
 
 8.51673 
 
 8.52055 
 
 - 396 
 393 
 390 
 386 
 382 
 
 5.31450 
 
 5.31450 
 
 5.31450 
 
 5.31450 
 
 5.31450 
 
 5.31428 
 
 5.31427 
 
 5.31427 
 
 5.31427 
 
 5.31427 
 
 8.50527 
 8.50920 
 8.51310 
 8.51696 
 : 8.52079 
 
 1.49473 
 
 1.49080 
 
 1.48690 
 
 1.48304 
 
 1.47921 
 
 9.99978 
 
 9.99977 
 
 9.99977 
 
 9.99977 
 
 9.99976 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 6900 
 
 6960 
 
 7020 
 
 7080 
 
 7140 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 8.52434 
 
 8.52810 
 
 8.53183 
 
 8.53552 
 
 8.53919 
 
 - 379 
 376 
 373 
 i 369 
 i 367 
 
 5.31451 
 
 5.31451 
 
 5.31451 
 
 5.31451 
 
 5.31451 
 
 5.31426 
 
 5.31426 
 
 5.31426 
 
 5.31425 
 
 5.31425 
 
 8.52459 
 8.52835 
 8.53208 
 ■ 8.53578 
 i 8.53945 
 
 376 
 
 373 
 
 370 
 
 367 
 
 363 
 
 1.47541 
 
 1.47165 
 
 1.46792 
 
 1.46422 
 
 1.46055 
 
 9.99976 
 
 9.99975 
 
 9.99975 
 
 9.99974 
 
 9.99974 
 
 5 
 
 4 
 
 3 
 
 2' 
 
 1 
 
 7200 
 
 60 
 
 8.54282 
 
 - 363 
 
 5.31451 
 
 5.31425 
 
 > 8.54308 
 
 
 1.45692 
 
 9.99974 
 
 0 
 
 
 
 L. Cos 
 
 . d. 
 
 
 
 L. Cots 
 
 % d.c.lL. Tans 
 
 L. Sin. 
 
 
 88 ° 
 
4»4 
 
 mn A T! ITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 e" 
 
 u 
 
 t 
 
 L. Sin. 
 
 d. 
 
 Cpl. S. 
 
 Cpl. T. 
 
 L. Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 7200 
 
 7260 
 
 7320 
 
 7380 
 
 7440 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 8.54282 
 
 8.54642 
 
 8.54999 
 
 8.55354 
 
 8.55705 
 
 360 
 
 357 
 
 355 
 
 351 
 
 349 
 
 346 
 
 343 
 
 341 
 
 337 
 
 336 
 
 332 
 
 330 
 
 328 
 
 325 
 
 323 
 
 320 
 
 318 
 
 316 
 
 313 
 
 311 
 
 309 
 
 307 
 
 305 
 
 302 
 
 301 
 
 298 
 
 296 
 
 294 
 
 293 
 
 290 
 
 288 
 
 287 
 
 284 
 
 283 
 
 281 
 
 279 
 
 277 
 
 276 
 
 274 
 
 272 
 
 270 
 
 269 
 
 267 
 
 266 
 
 263 
 
 263 
 
 260 
 
 259 
 
 258 
 
 256 
 
 254 
 
 253 
 
 252 
 
 250 
 
 249 
 
 247 
 
 246 
 
 244 
 
 243 
 
 242 
 
 5.31451 
 
 5.31451 
 
 5.31452 
 5.31452 
 5.31452 
 
 5.31425 
 
 5.31425 
 
 5.31424 
 
 5.31424 
 
 5.31424 
 
 8.54308 
 
 8.54669 
 
 8.55027 
 
 8.55382 
 
 8.55734 
 
 361 
 
 358 
 
 355 
 
 352 
 
 349 
 
 346 
 
 344 
 
 341 
 
 338 
 
 336 
 
 333 
 
 330 
 
 328 
 
 326 
 
 323 
 
 321 
 
 319 
 
 316 
 
 314 
 
 311 
 
 310 
 
 307 
 
 305 
 
 303 
 
 301 
 
 299 
 
 297 
 
 295 
 
 292 
 
 291 
 
 289 
 
 287 
 
 285 
 
 284 
 
 281 
 
 280 
 
 278 
 
 276 
 
 274 
 
 273 
 
 271 
 
 269 
 
 268 
 
 266 
 
 264 
 
 263 
 
 261 
 
 260 
 
 258 
 
 257 
 
 255 
 
 254 
 
 252 
 
 251 
 
 249 
 
 248 
 
 246 
 
 245 
 
 244 
 
 243 
 
 1.45692 
 
 1.45331 
 
 1.44973 
 
 1.44618 
 
 1.44266 
 
 9.99974 
 
 9.99973 
 
 9.99973 
 
 9.99972 
 
 9.99972 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 7500 
 
 7560 
 
 7620 
 
 7680 
 
 7740 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 8.56054 
 
 8.56400 
 
 8.56743 
 
 8.57084 
 
 8.57421 
 
 5.31452 
 
 5.31452 
 
 5.31452 
 
 5.31453 
 5.31453 
 
 5.31423 
 
 5.31423 
 
 5.31423 
 
 5.31422 
 
 5.31422 
 
 8.56083 
 
 8.56429 
 
 8.56773 
 
 8.57114 
 
 8.57452 
 
 1.43917 
 
 1.43571 
 
 1.43227 
 
 1.42886 
 
 1.42548 
 
 9.99971 
 
 9.99971 
 
 9.99970 
 
 9.99970 
 
 9.99969 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 7800 
 
 7860 
 
 7920 
 
 7980 
 
 8040 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 8.57757 
 
 8.58089 
 
 8.58419 
 
 8.58747 
 
 8.59072 
 
 5.31453 
 
 5.31453 
 
 5.31453 
 
 5.31453 
 
 5.31454 
 
 5.31422 
 
 5.31421 
 
 5.31421 
 
 5.31421 
 
 5.31421 
 
 8.57788 
 
 8.58121 
 
 8.58451 
 
 8.58779 
 
 8.59105 
 
 1.42212 
 
 1.41879 
 
 1.41549 
 
 1.41221 
 
 1.40895 
 
 9.99969 
 
 9.99968 
 
 9.99968 
 
 9.99967 
 
 9.99967 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 8100 
 
 8160 
 
 8220 
 
 8280 
 
 8340 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 8.59395 
 
 8.59715 
 
 8.60033 
 
 8.60349 
 
 8.60662 
 
 5.31454 
 
 5.31454 
 
 5.31454 
 
 5.31454 
 
 5.31454 
 
 5.31420 
 
 5.31420 
 
 5.31420 
 
 5.31419 
 
 5.31419 
 
 8.59428 
 
 8.59749 
 
 8.60068 
 
 8.60384 
 
 8.60698 
 
 1.40572 
 
 1.40251 
 
 1.39932 
 
 1.39616 
 
 1.39302 
 
 9.99967 
 
 9.99966 
 
 9.99966 
 
 9.99965 
 
 9.99964 
 
 8400 
 
 8460 
 
 8520 
 
 8580 
 
 8640 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 8.60973 
 
 8.61282 
 
 8.61589 
 
 8.61894 
 
 8.62196 
 
 5.31455 
 
 5.31455 
 
 5.31455 
 
 5.31455 
 
 5.31455 
 
 5.31418 
 
 5.31418 
 
 5.31418 
 
 5.31417 
 
 5.31417 
 
 8.61009 
 
 8.61319 
 
 8.61626 
 
 8.61931 
 
 8.62234 
 
 1.38991 
 
 1.38681 
 
 1.38374 
 
 1.38069 
 
 1.37766 
 
 9.99964 
 
 9.99963 
 
 9.99963 
 
 9.99962 
 
 9.99962 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 8700 
 
 8760 
 
 8820 
 
 8880 
 
 8940 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 8.62497 
 
 8.62795 
 
 8.63091 
 
 8.63385 
 
 8.63678 
 
 5.31455 
 
 5.31456 
 5.31456 
 5.31456 
 5.31456 
 
 5.31417 
 
 5.31416 
 
 5.31416 
 
 5.31416 
 
 5.31415 
 
 8.62535 
 
 8.62834 
 
 8.63131 
 
 8.63426 
 
 8.63718 
 
 1.37465 
 
 1.37166 
 
 1.36869 
 
 1.36574 
 
 1.36282 
 
 9.99961 
 
 9.99961 
 
 9.99960 
 
 9.99960 
 
 9.99959 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 9000 
 
 9060 
 
 9120 
 
 9180 
 
 9240 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 8.63968 
 
 8.64256 
 
 8.64543 
 
 8.64827 
 
 8.65110 
 
 5.31456 
 
 5.31456 
 
 5.31457 
 5.31457 
 5.31457 
 
 5.31415 
 
 5.31415 
 
 5.31414 
 
 5.31414 
 
 5.31413 
 
 8.64009 
 
 8.64298 
 
 8.64585 
 
 8.64870 
 
 8.65154 
 
 1.35991 
 
 1.35702 
 
 1.35415 
 
 1.35130 
 
 1.34846 
 
 9.99959 
 
 9.99958 
 
 9.99958 
 
 9.99957 
 
 9.99956 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 9300 
 
 9360 
 
 9420 
 
 9480 
 
 9540 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 8.65391 
 
 8.65670 
 
 8.65947 
 
 8.66223 
 
 8.66497 
 
 5.31457 
 
 5.31457 
 
 5.31458 
 5.31458 
 5.31458 
 
 5.31413 
 
 5.31413 
 
 5.31412 
 
 5.31412 
 
 5.31412 
 
 8.65435 
 
 8.65715 
 
 8.65993 
 
 8.66269 
 
 8.66543 
 
 1.34565 
 
 1.34285 
 
 1.34007 
 
 1.33731 
 
 1.33457 
 
 9.99956 
 
 9.99955 
 
 9.99955 
 
 9.99954 
 
 9.99954 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 9600 
 
 9660 
 
 9720 
 
 9780 
 
 9840 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 8.66769 
 
 8.67039 
 
 8.67308 
 
 8.67575 
 
 8.67841 
 
 5.31458 
 
 5.31458 
 
 5.31459 
 5.31459 
 5.31459 
 
 5.31411 
 
 5.31411 
 
 5.31410 
 
 5.31410 
 
 5.31410 
 
 8.66816 
 
 8.67087 
 
 8.67356 
 
 8.67624 
 
 8.67890 
 
 1.33184 
 
 1.32913 
 
 1.32644 
 
 1.32376 
 
 1.32110 
 
 9.99953 
 
 9.99952 
 
 9.99952 
 
 9.99951 
 
 9.99951 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 9900 
 
 9960 
 
 10020 
 
 10080 
 
 10140 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 8.68104 
 
 8.68367 
 
 8.68627 
 
 8.68886 
 
 8.69144 
 
 5.31459 
 
 5.31459 
 
 5.31460 
 5.31460 
 5.31460 
 
 5.31409 
 
 5.31409 
 
 5.31408 
 
 5.31408 
 
 5.31408 
 
 8.68154 
 
 8.68417 
 
 8.68678 
 
 8.68938 
 
 8.69196 
 
 1.31846 
 
 1.31583 
 
 1.31322 
 
 1.31062 
 
 1.30804 
 
 9.99950 
 
 9.99949 
 
 9.99949 
 
 9.99948 
 
 9.99948 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10200 
 
 10260 
 
 10320 
 
 10380 
 
 10440 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 8.69400 
 
 8.69654 
 
 8.69907 
 
 8.70159 
 
 8.70409 
 
 5.31460 
 
 5.31460 
 
 5.31461 
 5.31461 
 5.31461 
 
 5.31407 
 
 5.31407 
 
 5.31406 
 
 5.31406 
 
 5.31405 
 
 8.69453 
 
 8.69708 
 
 8.69962 
 
 8.70214 
 
 8.70465 
 
 1.30547 
 
 1.30292 
 
 1.30038 
 
 1.29786 
 
 1.29535 
 
 9.99947 
 
 9.99946 
 
 9.99946 
 
 9.99945 
 
 9.99944 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 10500 
 10560 . 
 10620 
 10680 
 10740 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 8.70658 
 
 8.70905 
 
 8.71151 
 
 8.71395 
 
 8.71638 
 
 5.31461 
 
 5.31461 
 
 5.31462 
 5.31462 
 5.31462 
 
 5.31405 
 
 5.31405 
 
 5.31404 
 
 5.31404 
 
 5.31403 
 
 8.70714 
 
 8.70962 
 
 8.71208 
 
 8.71453 
 
 8.71697 
 
 1.29286 
 
 1.29038 
 
 1.28792 
 
 1.28547 
 
 1.28303 
 
 9.99944 
 
 9.99943 
 
 9.99942 
 
 9.99942 
 
 9.99941 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 10800 
 
 60 
 
 8.71880 
 
 5.31462 
 
 5.31403 
 
 8.71940 
 
 1.28060 
 
 9.99940 
 
 0 
 
 
 
 L. Cos. 
 
 d. 
 
 
 
 L. Cotg. 
 
 d. c. 
 
 L. Tang. 
 
 L. Sin. 
 
 / 
 
 87 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 495 
 
 ' ] 
 
 L. Sin. ( 
 
 0 1 
 
 3.71880 0 
 
 1 * 
 
 3.72120 l 
 
 2 1 
 
 3.72359 i 
 
 3 1 
 
 3.72597 i 
 
 4 
 
 3.72834 % 
 
 5 ! 
 
 8.73069 0 
 
 6 : 
 
 8.73303 i 
 
 7 
 
 8.73535 i 
 
 8 
 
 8.73767 i 
 
 9 
 
 8.73997 % 
 
 10 
 
 8.74226 
 
 11 
 
 8.74454 i 
 
 12 
 
 8.74680 i 
 
 13 
 
 8.74906 f 
 
 14 
 
 8.75130 i 
 
 15 
 
 8.75353 J 
 
 16 
 
 8.75575 i 
 
 17 
 
 8.75795 i 
 
 18 
 
 8.76015 i 
 
 19 
 
 8.76234 ; 
 
 20 
 
 8.76451 ; 
 
 21 
 
 8.76667 ; 
 
 22 
 
 8.76883 ; 
 
 23 
 
 8.77097 ; 
 
 24 
 
 8.77310 ; 
 
 25 
 
 8.77522 ; 
 
 26 
 
 8.77733 
 
 27 
 
 8.77943 
 
 28 
 
 8.78152 
 
 29 
 
 8.78360 
 
 30 
 
 8.78568 
 
 31 
 
 8.78774 
 
 32 
 
 8.78979 
 
 33 
 
 8.79183 
 
 34 
 
 8.79386 
 
 35 
 
 8.79588 
 
 36 
 
 8.79789 
 
 37 
 
 8.79990 
 
 38 
 
 8.80189 
 
 39 
 
 8.80388 
 
 40 
 
 8.80585 
 
 41 
 
 8.80782 
 
 42 
 
 8.80978 
 
 43 
 
 8.81173 
 
 44 
 
 8.81367 
 
 45 
 
 ' 8.81560 
 
 46 
 
 8.81752 
 
 47 
 
 8.81944 
 
 48 
 
 8.82134 
 
 49 
 
 8.82324 
 
 50 
 
 8.82513 
 
 51 
 
 8.82701 
 
 52 
 
 8.82888 
 
 53 
 
 8.83075 
 
 54 
 
 8.83261 
 
 55 
 
 8.83446 
 
 56 
 
 8.83630 
 
 57 
 
 8.83813 
 
 58 
 
 8.83996 
 
 59 
 
 8.84177 
 
 60 
 
 8.84358 
 
 
 L. Cos. 
 
 229 
 
 228 
 
 226 
 
 226 
 
 224 
 
 223 
 
 222 
 
 220 
 
 220 
 
 219 
 
 217 
 
 216 
 
 216 
 
 214 
 
 213 
 
 212 
 
 211 
 
 210 
 
 209 
 
 208 
 
 L.Tang. d. 
 
 8.71940 
 8.72181 £ 
 8.72420 £ 
 
 8.72659 % 
 
 8.72896 £ 
 
 8.73132 % 
 8.73366 % 
 
 8.73600 % 
 
 8.73832 2- 
 
 8.74063 % 
 
 8.74292 o ; 
 8.74521 2 
 
 8.74748 2 
 
 8.74974 * 
 
 8.75199 2 
 
 8.75423 0 
 
 8.75645 2 
 8.75867 2 
 
 8.76087 2 
 
 8.76306 2 
 
 8.76525 . 
 
 8.76742 i 
 8.76958 i 
 8.77173 i 
 8.77387 i 
 
 8.77600 ' 
 
 8.77811 ; 
 8.78022 * 
 ' 8.78232 ; 
 
 ; 8.78441 ; 
 
 ! 8.78649 ; 
 
 » 8.78855 ; 
 
 | 8.79061 ; 
 
 8.79266 ; 
 
 5 8.79470 
 
 ' 8.79673 
 
 \ 8.79875 
 
 { 8.80076 
 
 l 8.80277 
 ^ 8.80476 
 
 i 8.80674 
 l 8.80872 
 ° 8.81068 
 5 8.81264 
 
 i 8.81459 
 
 t 8.81653 
 2 8.81846 
 
 2 8.82038 
 
 5 8.82230 
 
 5> 8.82420 
 
 7 8.82610 
 
 » 8.82799 
 
 4 8.82987 
 
 7 8.83175 
 
 8.83361 
 
 , 8.83547 
 
 $4 8.83732 
 
 3 8.83916 
 
 g 8.84100 
 *1 8.84282 
 
 31 8.84464 
 
 d. 
 
 L. Cotg. 
 
 201 
 
 201 
 
 199 
 
 198 
 
 198 
 
 196 
 
 196 
 
 195 
 
 194 
 
 193 
 
 192 
 
 192 
 
 190 
 
 190 
 
 189 
 
 188 
 
 188 
 
 186 
 
 186 
 
 185 
 
 184 
 
 184 
 
 182 
 
 182 
 
 Ic 
 
 j. Cotg. ] 
 
 L. Cos. 
 
 
 1.28060 1 
 
 3.99940 ( 
 
 50 
 
 1.27819 1 
 
 3.99940 
 
 59 
 
 1.27580 ! 
 
 3.99939 
 
 58 
 
 1.27341 1 
 
 3.99938 
 
 57 
 
 1.27104 ' 
 
 3.99938 
 
 56 
 
 1.26868 1 
 
 9.99937 
 
 55 
 
 1.26634 
 
 9.99936 
 
 54 
 
 1.26400 
 
 9.99936 
 
 53 
 
 1.26168 
 
 9.99935 
 
 52 
 
 1.25937 
 
 9.99934 
 
 51 
 
 1.25708 
 
 9.99934 
 
 50 
 
 1.25479 
 
 9.99933 
 
 49 
 
 1.25252 
 
 9.99932 
 
 48 
 
 1.25026 
 
 9.99932 
 
 47 
 
 1.24801 
 
 9.99931 
 
 46 
 
 1.24577 
 
 9.99930 
 
 45 
 
 1.24355 
 
 9.99929 
 
 44 
 
 1.24133 
 
 9.99929 
 
 43 
 
 1.23913 
 
 9.99928 
 
 42 
 
 1.23694 
 
 9.99927 
 
 41 
 
 1.23475 
 
 9.99926 
 
 40 
 
 1.23258 
 
 9.99926 
 
 39 
 
 1.23042 
 
 9.99925 
 
 38 
 
 1.22827 
 
 9.99924 
 
 37 
 
 1.22613 
 
 9.99923 
 
 36 
 
 1.22400 
 
 9.99923 
 
 35 
 
 1.22189 
 
 9.99922 
 
 34 
 
 1.21978 
 
 9.99921 
 
 33 
 
 1.21768 
 
 9.99920 
 
 32 
 
 1.21559 
 
 9.99920 
 
 31 
 
 1.21351 
 
 9.99919 
 
 30 
 
 1.21145 
 
 9.99918 
 
 29 
 
 1.20939 
 
 9.99917 
 
 28 
 
 1.20734 
 
 9.99917 
 
 27 
 
 1.20530 
 
 9.99916 
 
 26 
 
 1.20327 
 
 9.99915 
 
 25 
 
 1.20125 
 
 9.99914 
 
 24 
 
 1.19924 
 
 9.99913 
 
 23 
 
 1.19723 
 
 9.99913 
 
 22 
 
 1.19524 
 
 9.99912 
 
 21 
 
 1.19326 
 
 9.99911 
 
 20 
 
 1.19128 
 
 9.99910 
 
 19 
 
 1.18932 
 
 9.99909 
 
 18 
 
 1.18736 
 
 9.99909 
 
 17 
 
 1.18541 
 
 9.99908 
 
 16 
 
 1.18347 
 
 9.99907 
 
 15 
 
 1.18154 
 
 9.99906 
 
 14 
 
 1.17962 
 
 9.99905 
 
 13 
 
 1.17770 
 
 9.99904 
 
 12 
 
 1.17580 
 
 9.99904 
 
 11 
 
 1.17390 
 
 9.99903 
 
 10 
 
 1.17201 
 
 9.99902 
 
 : 9 
 
 1.17013 
 
 , 9.99901 
 
 8 
 
 1.16825 
 
 i 9.99900 
 
 i 7 
 
 1.16639 
 
 i 9.99899 
 
 i 6 
 
 1.16453 
 
 i 9.99898 
 
 i 5 
 
 1.1626? 
 
 ; 9.99898 
 
 1 4 
 
 1.16084 
 
 [ 9.99897 
 
 r 3 
 
 1.1590C 
 
 > 9.99896 
 
 i 2 
 
 ; 1.1571? 
 
 * 9.9989£ 
 
 ► 1 
 
 1 1.1553* 
 
 J 9.99894 
 
 L 0 
 
 5. L.Tang 
 
 r. L. Sin 
 
 / 
 
 P. P. 
 
 238 
 
 23.8 
 
 27.8 
 
 31.7 
 
 35.7 
 
 39.7 
 79.3 
 
 119.0 
 
 158.7 
 
 198.3 
 
 234 
 
 23.4 
 
 27.3 
 
 31.2 
 
 35.1 
 
 39.0 
 
 78.0 
 
 117.0 
 
 156.0 
 
 195.0 
 
 229 
 
 22.9 
 
 26.7 
 30.5 
 34.4 
 
 38.2 
 
 76.3 
 114.5 
 
 152.7 
 
 190.8 
 
 225 
 
 220 
 
 216 
 
 22.5 
 
 22.0 
 
 21.6 
 
 26.3 
 
 25.7 
 
 25.2 
 
 30.0 
 
 29.3 
 
 28.8 
 
 33.8 
 
 33.0 
 
 32.4 
 
 37.5 
 
 36.7 
 
 36.0 
 
 75.0 
 
 73.3 
 
 72.0 
 
 112.5 
 
 110.0 
 
 108.0 
 
 150.0 
 
 146.7 
 
 144.0 
 
 187.5 
 
 183.3 
 
 180.0 
 
 212 
 
 208 
 
 204 
 
 21.2 
 
 20.8 
 
 20.4 
 
 24.7 
 
 24.3 
 
 23.8 
 
 28.3 
 
 27.7 
 
 27.2 
 
 31.8 
 
 31.2 
 
 30.6 
 
 35.3 
 
 34.7 
 
 34.0 
 
 70.7 
 
 69.3 
 
 68.0 
 
 106.0 
 
 104.0 
 
 102.0 
 
 141.3 
 
 138.7 
 
 136.0 
 
 176.7 
 
 173.3 
 
 170.0 
 
 201 
 
 197 
 
 193 
 
 20.1 
 
 23.5 
 26.8 
 
 30.2 
 
 33.5 
 
 67.0 
 
 100.5 
 
 134.0 
 
 167.5 
 
 189 
 
 18.9 
 
 22.1 
 
 25.2 
 
 28.4 
 
 31.5 
 63.0 
 
 94.5 
 
 126.0 
 l 157.5 
 
 19.7 
 23.0 
 
 26.3 
 
 29.6 
 
 32.8 
 
 65.7 
 
 98.5 
 
 131.3 
 
 164.2 
 
 185 
 
 18.5 
 
 21.6 
 
 24.7 
 
 27.8 
 
 30.8 
 61.7 
 92.5 
 
 123.3 
 154.2 
 
 
 4 
 
 3 
 
 2 
 
 6 
 
 0.4 
 
 0.3 
 
 0.2 
 
 7 
 
 0.5 
 
 0.4 
 
 0.2 
 
 8 
 
 0.5 
 
 0.4 
 
 0.3 
 
 9 
 
 0.6 
 
 0.5 
 
 0.3 
 
 10 
 
 0.7 
 
 0.5 
 
 0.3 
 
 20 
 
 1.3 
 
 1.0 
 
 0.7 
 
 30 
 
 2.0 
 
 1.5 
 
 1.0 
 
 40 
 
 2.7 
 
 2.0 
 
 1.3 
 
 50 
 
 3.3 
 
 2.5 
 
 1.7 
 
 19.3 
 
 22.5 
 
 25.7 
 
 29.0 
 
 32.2 
 
 64.3 
 
 96.5 
 
 128.7 
 
 160.8 
 
 181 
 
 18.1 
 
 21.1 
 
 24.1 
 
 27.2 
 
 30.2 
 
 60.3 
 
 90.5 
 
 120.7 
 
 150.8 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.8 
 
 P. P. 
 
 86 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 4 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 8.84358 
 
 8.84539 
 
 8.84718 
 
 8.84897 
 
 8.85075 
 
 181 
 
 179 
 
 179 
 
 178 
 
 177 
 
 177 
 
 176 
 
 175 
 
 175 
 
 173 
 
 173 
 
 173 
 
 171 
 
 171 
 
 171 
 
 169 
 
 169 
 
 169 
 
 167 
 
 168 
 166 
 166 
 165 
 164 
 164 
 163 
 163 
 162 
 162 
 160 
 161 
 159 
 159 
 159 
 158 
 157 
 157 
 156 
 155 
 155 
 155 
 154 
 153 
 153 
 152 
 152 
 151 
 151 
 150 
 150 
 149 
 149 
 148 
 147 
 147 
 147 
 146 
 146 
 145 
 145 
 
 8.84464 
 
 8.84646 
 
 8.84826 
 
 8.85006 
 
 8.85185 
 
 182 
 
 180 
 
 180 
 
 179 
 
 178 
 
 177 
 
 177 
 
 176 
 
 176 
 
 174 
 
 174 
 
 174 
 
 172 
 
 172 
 
 171 
 
 171 
 
 170 
 
 169 
 
 169 
 
 168 
 
 167 
 
 167 
 
 166 
 
 165 
 
 165 
 
 165 
 
 163 
 
 163 
 
 163 
 
 161 
 
 162 
 
 160 
 
 160 
 
 160 
 
 159 
 
 158 
 
 158 
 
 157 
 
 157 
 
 156 
 
 155 
 
 155 
 
 155 
 
 153 
 
 154 
 153 
 152 
 152 
 151 
 151 
 150 
 150 
 149 
 
 148 
 
 149 
 147 
 147 
 147 
 146 
 146 
 
 1.15536 
 
 1.15354 
 
 1.15174 
 
 1.14994 
 
 1.14815 
 
 9.99894 
 
 9.99893 
 
 9.99892 
 
 9.99891 
 
 9.99891 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 181 
 
 6 18.1 
 
 7 21.1 
 
 8 24.1 
 
 9 27.2 
 
 10 30.2 
 
 20 60.3 
 
 30 90.5 
 
 40 120.7 
 50 150.8 
 
 175 
 
 6 17.5 
 
 7 20.4 
 
 8 23.3 
 
 9 26.3 
 
 10 29.2 
 
 20 58.3 
 
 30 87.5 
 
 40 116.7 
 50 145.8 
 
 168 
 
 6 16.8 
 
 7 19.6 
 
 8 22.4 
 
 9 25.2 
 
 10 28.0 
 20 56.0 
 
 30 84.0 
 
 40 112.0 
 50 140.0 
 
 162 
 
 6 16.2 
 
 7 18.9 
 
 8 21.6 
 
 9 24.3 
 
 10 27.0 
 
 20 54.0 
 
 30 81.0 
 
 40 108.0 
 50 135.0 
 
 155 
 
 6 15.5 
 
 7 18.1 
 
 8 20.7 
 
 9 23.3 
 
 10 25.8 
 
 20 51.7 
 
 30 77.5 
 
 40 103.3 
 50 129.2 
 
 149 
 
 6 14.9 
 
 7 17.4 
 
 8 19.9 
 
 9 22.4 
 
 10 24.8 
 
 20 49.7 
 
 30 74.5 
 
 40 99.3 
 
 50 124.2 
 
 179 
 
 17.9 
 
 20.9 
 
 23.9 
 
 26.9 
 29.8 
 
 59.7 
 
 89.5 
 119.3 
 
 149.2 
 
 173 
 
 17.3 
 20.2 
 
 23.1 
 26.0 
 
 28.8 
 
 57.7 
 
 86.5 
 
 115.3 
 
 144.2 
 
 166 
 
 16.6 
 
 19.4 
 
 22.1 
 
 24.9 
 
 27.7 
 
 55.3 
 
 83.0 
 110.7 
 
 138.3 
 
 159 
 
 15.9 
 18.6 
 21.2 
 
 23.9 
 
 26.5 
 
 53.0 
 
 79.5 
 106.0 
 
 132.5 
 
 153 
 
 15.3 
 
 17.9 
 
 20.4 
 
 23.0 
 
 25.5 
 
 51.0 
 
 76.5 
 102.0 
 
 127.5 
 
 147 
 
 14.7 
 
 17.2 
 
 19.6 
 
 22.1 
 
 24.5 
 
 49.0 
 
 73.5 
 
 98.0 
 
 122.5 
 
 177 
 
 17.7 
 
 20.7 
 
 23.6 
 
 26.6 
 
 29.5 
 
 59.0 
 
 88.5 
 118.0 
 
 147.5 
 
 171 
 
 17.1 
 20.0 
 
 22.8 
 25.7 
 
 28.5 
 
 57.0 
 
 85.5 
 114.0 
 
 142.5 
 
 164 
 
 16.4 
 
 19.1 
 
 21.9 
 
 24.6 
 
 27.3 
 
 54.7 
 82.0 
 
 109.3 
 
 136.7 
 
 157 
 
 15.7 
 
 18.3 
 
 20.9 
 23.6 
 
 26.2 
 
 52.3 
 
 78.5 
 
 104.7 
 
 130.8 
 
 151 
 
 15.1 
 
 17.6 
 
 20.1 
 
 22.7 
 
 25.2 
 
 50.3 
 75.5 
 
 100.7 
 
 125.8 
 
 1 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.8 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 8.85252 
 
 8.85429 
 
 8.85605 
 
 8.85780 
 
 8.85955 
 
 8.85363 
 
 8.85540 
 
 8.85717 
 
 8.85893 
 
 8.86069 
 
 1.14637 
 
 1.14460 
 
 1.14283 
 
 1.14107 
 
 1.13931 
 
 9.99890 
 
 9.99889 
 
 9.99888 
 
 9.99887 
 
 9.99886 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 8.86128 
 
 8.86301 
 
 8.86474 
 
 8.86645 
 
 8.86816 
 
 8.86243 
 
 8.86417 
 
 8.86591 
 
 8.86763 
 
 8.86935 
 
 1.13757 
 
 1.13583 
 
 1.13409 
 
 1.13237 
 
 1.13065 
 
 9.99885 
 
 9.99884 
 
 9.99883 
 
 9.99882 
 
 9.99881 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 8.86987 
 
 8.87156 
 
 8.87325 
 
 8.87494 
 
 8.87661 
 
 8.87106 
 
 8.87277 
 
 8.87447 
 
 8.87616 
 
 8.87785 
 
 1.12894 
 
 1.12723 
 
 1.12553 
 
 1.12384 
 
 1.12215 
 
 9.99880 
 9.99879 
 9 99879 
 9.99878 
 9.99877 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 8.87829 
 
 8.87995 
 
 8.88161 
 
 8.88326 
 
 8.88490 
 
 8.87953 
 
 8.88120 
 
 8.88287 
 
 8.88453 
 
 8.88618 
 
 1.12047 
 
 1.11880 
 
 1.11713 
 
 1.11547 
 
 1.11382 
 
 9.99876 
 
 9.99875 
 
 9.99874 
 
 9.99873 
 
 9.99872 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 8.88654 
 
 8.88817 
 
 8.88980 
 
 8.89142 
 
 8.89304 
 
 8.88783 
 
 8.88948 
 
 8.89111 
 
 8.89274 
 
 8.89437 
 
 1.11217 
 
 1.11052 
 
 1.10889 
 
 1.10726 
 
 1.10563 
 
 9.99871 
 
 9.99870 
 
 9.99869 
 
 9.99868 
 
 9.99867 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 8.89464 
 
 8.89625 
 
 8.89784 
 
 8.89943 
 
 8.90102 
 
 8.89598 
 
 8.89760 
 
 8.89920 
 
 8.90080 
 
 8.90240 
 
 1.10402 
 
 1.10240 
 
 1.10080 
 
 1.09920 
 
 1.09760 
 
 9.99866 
 
 9.99865 
 
 9.99864 
 
 9.99863 
 
 9.99862 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 8.90260 
 
 8.90417 
 
 8.90574 
 
 8.90730 
 
 8.90885 
 
 8.90399 
 
 8.90557 
 
 8.90715 
 
 8.90872 
 
 8.91029 
 
 1.09601 
 
 1.09443 
 
 1.09285 
 
 1.09128 
 
 1.08971 
 
 9.99861 
 
 9.99860 
 
 9.99859 
 
 9.99858 
 
 9.99857 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 8.91040 
 
 8.91195 
 
 8.91349 
 
 8.91502 
 
 8.91655 
 
 8.91185 
 
 8.91340 
 
 8.91495 
 
 8.91650 
 
 8.91803 
 
 1.08815 
 
 1.08660 
 
 1.08505 
 
 1.08350 
 
 1.08197 
 
 9.99856 
 
 9.99855 
 
 9.99854 
 
 9.99853 
 
 9.99852 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 8.91807 
 
 8.91959 
 
 8.92110 
 
 8.92261 
 
 8.92411 
 
 8.91957 
 
 8.92110 
 
 8.92262 
 
 8.92414 
 
 8.92565 
 
 1.08043 
 
 1.07890 
 
 1.07738 
 
 1.07586 
 
 1.07435 
 
 9.99851 
 
 9.99850 
 
 9.99848 
 
 9.99847 
 
 9.99846 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 8.92561 
 
 8.92710 
 
 8.92859 
 
 8.93007 
 
 8.93154 
 
 8.92716 
 
 8.92866 
 
 8.93016 
 
 8.93165 
 
 8.93313 
 
 1.07284 
 
 1.07134 
 
 1.06984 
 
 1.06835 
 
 1.06687 
 
 9.99845 
 
 9.99844 
 
 9.99843 
 
 9.99842 
 
 9.99841 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 8.93301 
 
 8.93448 
 
 8.93594 
 
 8.93740 
 
 8.93885 
 
 8.93462 
 
 3.93609 
 
 8.93756 
 
 8.93903 
 
 8.94049 
 
 1.06538 
 
 1.06391 
 
 1.06244 
 
 1.06097 
 
 1.05951 
 
 9.99840 
 
 9.99839 
 
 9.99838 
 
 9.99837 
 
 9.99836 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 8.94030 
 
 8.94195 
 
 1.05805 
 
 9.99834 
 
 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 / 
 
 P. P. 
 
 85 ° 
 
497 
 
 Logarithms of trigonometric functions. 
 
 5 ° . 
 
 ) 
 
 L. Sin. 
 
 d. I 
 
 j.Tang. 
 
 d. c. 1 
 
 Cotg. 
 
 L. Cos. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 8794030 “ 
 
 8.94174 
 
 8.94317 
 
 8.94461 
 
 8.94603 
 
 144 
 
 143 
 
 144 
 142 
 
 8.94195 
 
 8.94340 
 
 8.94485 
 
 8.94630 
 
 8.94773 
 
 145 
 
 145 
 
 145 
 
 143 
 
 I'M 
 
 1.05805 
 
 1.05660 
 
 1.05515 
 
 1.05370 
 
 1.05227 
 
 9.99834 
 
 9.99833 
 
 9.99832 
 
 9.99831 
 
 9.99830 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 6 
 
 7 
 
 8 
 9 
 
 145 
 
 14.5 
 
 16.9 
 
 19.3 
 
 21.8 
 
 143 
 
 14.3 
 
 16.7 
 
 19.1 
 
 21.5 
 
 141 
 
 14.1 
 
 16.5 
 18.8 
 
 21.2 
 
 23.5 
 
 47.0 
 
 70.5 
 
 94.0 
 117.5 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 8.94746 
 
 8.94887 
 
 8.95029 
 
 8.95170 
 
 8.95310 
 
 143 
 
 141 
 
 142 
 141 
 140 
 140 
 139 
 139 
 139 
 138 
 138 
 137 
 137 
 136 
 136 
 136 
 135 
 135 
 134 
 134 
 133 
 133 
 133 
 132 
 132 
 131 
 131 
 131 
 130 
 130 
 129 
 129 
 129 
 128 
 128 
 128 
 127 
 127 
 126 
 126 
 
 8.94917 
 
 8.95060 
 
 8.95202 
 
 8.95344 
 
 8.95486 
 
 143 
 
 142 
 
 142 
 
 142 
 
 i/ii 
 
 1.05083 
 
 1.04940 
 
 1.04798 
 
 1.04656 
 
 1.04514 
 
 9.99829 
 
 9.99828 
 
 9.99827 
 
 9.99825 
 
 9.99824 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 24.2 
 
 48.3- 
 
 72.5 
 
 96.7 
 
 120.8 
 
 23.8 
 
 47.7 
 
 71.5 
 
 95.3 
 
 119.2 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 8.95450 
 
 8.95589 
 
 8.95728 
 
 8.95867 
 
 8.96005 
 
 8.95627 
 
 8.95767 
 
 8.95908 
 
 8.96047 
 
 8.96187 
 
 140 
 
 141 
 
 139 
 
 140 
 
 138 
 
 139 
 138 
 
 137 
 
 138 
 
 1.04373 
 
 1.04233 
 
 1.04092 
 
 1.03953 
 
 1.03813 
 
 9.99823 
 
 9.99822 
 
 9.99821 
 
 9.99820 
 
 9.99819 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 6 
 
 7 
 
 8 
 9 
 
 139 
 
 13.9 
 16.2 
 18.5 
 
 20.9 
 
 138 
 
 13.8 
 
 16.1 
 
 18.4 
 
 20.7 
 
 136 
 
 13.6 
 
 15.9 
 
 18.1 
 
 20.4 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 8.96143 
 
 8.96280 
 
 8.96417 
 
 8.96553 
 
 8.96689 
 
 8.96325 
 
 8.96464 
 
 8.96602 
 
 8.96739 
 
 8.96877 
 
 1.03675 
 
 1.03536 
 
 1.03398 
 
 1.03261 
 
 1.03123 
 
 9.99817 
 
 9.99816 
 
 9.99815 
 
 9.99814 
 
 9.99813 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 23.2 
 
 46.3 
 69.5 
 92.7 
 
 115.8 
 
 23.0 
 
 46.0 
 
 69.0 
 
 92.0 
 115.0 
 
 22.7 
 45.3 
 68.0 
 
 90.7 
 113.3 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 8.96825 
 
 8.96960 
 
 8.97095 
 
 8.97229 
 
 8.97363 
 
 8.97013 
 
 8.97150 
 
 8.97285 
 
 8.97421 
 
 8.97556 
 
 137 
 
 135 
 
 136 
 135 
 
 1.02987 
 
 1.02850 
 
 1.02715 
 
 1.02579 
 
 1.02444 
 
 9.99812 
 
 9.99810 
 
 9.99809 
 
 9.99808 
 
 9.99807 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 6 
 
 7 
 
 8 
 9 
 
 135 
 
 13.5 
 
 15.8 
 
 18.0 
 
 20.3 
 
 133 
 
 13.3 
 
 15.5 
 
 17.7 
 
 20.0 
 
 131 
 
 13.1 
 
 15.3 
 
 17.5 
 
 19.7 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 8.97496 
 
 8.97629 
 
 8.97762 
 
 8.97894 
 
 8.98026 
 
 8.97691 
 
 8.97825 
 
 8.97959 
 
 8.98092 
 
 8.98225 
 
 134 
 
 134 
 
 133 
 
 133 
 
 133 
 
 1.02309 
 
 1.02175 
 
 1.02041 
 
 1.01908 
 
 1.01775 
 
 9.99806 
 
 9.99804 
 
 9.99803 
 
 9.99802 
 
 9.99801 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 22.5 
 
 45.0 
 
 67.5 
 
 90.0 
 112.5 
 
 22.2 
 
 44.3 
 
 66.5 
 
 88.7 
 
 110.8 
 
 21.8 
 
 43.7 
 
 65.5 
 
 87.3 
 
 109.2 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 8.98157 
 
 8.98288 
 
 8.98419 
 
 8.98549 
 
 8.98679 
 
 8.98358 
 
 8.98490 
 
 8.98622 
 
 8.98753 
 
 8.98884 
 
 132 
 
 132 
 
 131 
 
 131 
 
 131 
 
 130 
 
 130 
 
 130 
 
 129 
 
 198 
 
 1.01642 
 
 1.01510 
 
 1.01378 
 
 1.01247 
 
 1.01116 
 
 9.99800 
 
 9.99798 
 
 9.99797 
 
 9.99796 
 
 9.99795 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 6 
 
 7 
 
 8 
 9 
 
 129 
 
 12.9 
 
 15.1 
 
 17.2 
 19.4 
 
 128 
 
 12.8 
 
 14.9 
 
 17.1 
 
 19.2 
 
 126 
 
 12.6 
 
 14.7 
 
 16.8 
 18.9 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 8.98808 
 
 8.98937 
 
 8.99066 
 
 8.99194 
 
 8.99322 
 
 8.99015 
 
 8.99145 
 
 8.99275 
 
 8.99405 
 
 8.99534 
 
 1.00985 
 
 1.00855 
 
 1.00725 
 
 1.00595 
 
 1.00466 
 
 9.99793 
 
 9.99792 
 
 9.99791 
 
 9.99790 
 
 9.99788 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 21.5 
 43;0 
 
 64.5 
 86.0 
 
 107,5 
 
 21.3 
 42.7 
 64.0 
 
 85.3 
 106.7 
 
 21.0 
 
 42.0 
 
 63.0 
 
 84.0 
 105.0 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 8.99450 
 
 8.99577 
 
 8.99704 
 
 8.99830 
 
 8.99956 
 
 8.99662 
 
 8.99791 
 
 8.99919 
 
 9.00046 
 
 9.00174 
 
 129 
 
 128 
 
 127 
 
 128 
 127 
 126 
 126 
 126 
 126 
 196 
 
 1.00338 
 
 1.00209 
 
 1.00081 
 
 0.99954 
 
 0.99826 
 
 9.99787 
 
 9.99786 
 
 9.99785 
 
 9.99783 
 
 9.99782 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 6 
 
 7 
 
 8 
 q 
 
 
 125 
 
 12.5 
 
 14.6 
 
 16.7 
 
 1ft 8 
 
 
 123 
 
 12.3 
 
 14.4 
 
 16.4 
 
 18.5 
 
 20.5 
 
 41.0 
 
 61.5 
 
 82.0 
 102.5 
 
 
 122 
 
 12.2 
 
 14.2 
 
 16.3 
 
 18.3 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.00082 
 
 9.00207 
 
 9.00332 
 
 9.00456 
 
 9.00581 
 
 126 
 
 125 
 
 125 
 
 124 
 
 125 
 
 9.00301 
 
 9.00427 
 
 9.00553 
 
 9.00679 
 
 9.00805 
 
 0.99699 
 
 0.99573 
 
 0.99447 
 
 0.99321 
 
 0.99195 
 
 9.99781 
 
 9.99780 
 
 9.99778 
 
 9.99777 
 
 9.99776 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 20.8 
 
 41.7 
 
 62.5 
 
 83.3 
 
 104.2 
 
 
 20.3 
 40.7 
 61.0 
 
 81.3 
 101.7 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 9.00704 
 
 9.00828 
 
 9.00951 
 
 9.01074 
 
 9.01196 
 
 - 123 
 124 
 123 
 123 
 | 122 
 
 9.00930 
 
 9.01055 
 
 9.01179 
 
 9.01303 
 
 9.01427 
 
 125 
 124 
 124 
 124 
 1 92 
 
 0.99070 
 
 0.98945 
 
 0.98821 
 
 0.98697 
 
 0.98573 
 
 9.99775 
 
 9.99773 
 
 9.99772 
 
 9.99771 
 
 9.99769 
 
 to 
 
 9 
 8 
 - 7 
 6 
 
 
 6 
 
 7 
 
 8 
 
 121 
 
 12.1 
 
 14.1 
 
 16.1 
 18.2 
 20.2 
 40.3 
 60.5 
 80.7 
 
 100.8 
 
 120 
 
 12.0 
 
 14.0 
 
 16.0 
 18 0 
 
 1 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.01318 
 
 9.01440 
 
 9.01561 
 
 9.01682 
 
 9.01802 
 
 - 122 
 
 1 m 
 ► 121 
 ; 121 
 
 9.01550 
 
 9.01673 
 
 9.01796 
 
 9.01918 
 
 9.0204C 
 
 ' 123 
 
 ; 123 
 122 
 122 
 122 
 
 0.98450 
 
 0.98327 
 
 0.98204 
 
 0.98082 
 
 0.97960 
 
 9.99768 
 9.99767 
 9.99765 
 9.99764 
 i 9.99763 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 V 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 iO.v 
 
 20.0 
 
 40.0 
 
 60.0 
 80.0 
 
 100.0 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.8 
 
 60 
 
 9.01922 
 
 - 120 
 
 9.0216S 
 
 0.97838 
 
 1 9.99761 
 
 0 
 
 
 
 
 
 
 
 
 
 L. Cos 
 
 . d. 
 
 L. Cots 
 
 r. d. c 
 
 *. L.Tang 
 
 ; L. Sin. 
 
 / 
 
 P. P. 
 
 84 ° 
 
498 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 6 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.01923 
 
 9.02043 
 
 9.02163 
 
 9.02283 
 
 9.02402 
 
 120 
 
 120 
 
 120 
 
 119 
 
 118 
 
 119 
 
 118 
 
 117 
 
 118 
 117 
 117 
 116 
 116 
 116 
 116 
 115 
 115 
 
 114 
 
 115 
 
 113 
 
 114 
 114 
 113 
 112 
 113 
 112 
 112 
 112 
 111 
 111 
 111 
 110 
 110 
 110 
 110 
 109 
 109 
 109 
 108 
 109 
 108 
 
 107 
 
 108 
 107 
 107 
 106 
 107 
 106 
 
 105 
 
 106 
 105 
 105 
 105 
 105 
 104 
 104 
 104 
 103 
 103 
 103 
 
 9.02162 
 
 9.02283 
 
 9.02404 
 
 9.02525 
 
 9.02645 
 
 121 
 
 121 
 
 121 
 
 120 
 
 121 
 
 119 
 
 120 
 119 
 118 
 119 
 118 
 118 
 
 117 
 
 118 
 116 
 117 
 116 
 116 
 116 
 115 
 115 
 115 
 115 
 114 
 114 
 
 113 
 
 114 
 113 
 112 
 113 
 112 
 112 
 112 
 111 
 111 
 111 
 110 
 111 
 110 
 
 109 
 
 110 
 109 
 109 
 108 
 109 
 108 
 108 
 
 107 
 
 108 
 107 
 106 
 107 
 106 
 106 
 106 
 106 
 105 
 105 
 105 
 104 
 
 0.97838 
 
 0.97717 
 
 0.97596 
 
 0.97475 
 
 0.97355 
 
 9.99761 
 
 9.99760 
 
 9.99759 
 
 9.99757 
 
 9.99756 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 12! 
 
 6 12.1 
 
 7 14.1 
 
 8 16.1 
 9 ' 18.2 
 
 10 20.2 
 20 40.3 
 
 3C 60.5 
 40 80.7 
 
 50 100.8 
 
 120 
 
 12.0 
 
 14.0 
 
 16.0 
 18.0 
 20.0 
 
 40.0 
 
 60.0 
 80.0 
 
 100.0 
 
 117 
 
 11.7 
 
 13.7 
 
 15.6 
 
 17.6 
 
 19.5 
 
 39.0 
 
 58.5 
 
 78.0 
 
 97.5 
 
 114 
 
 11.4 
 
 13.3 
 
 15.2 
 
 17.1 
 
 19.0 
 
 38.0 
 
 57.0 
 
 76.0 
 
 95.0 
 
 II! 
 
 11.1 
 
 13.0 
 
 14.8 
 
 16.7 
 
 18.5 
 
 37.0 
 
 55.5 
 
 74.0 
 
 92.5 
 
 108 
 
 10.8 
 
 12.6 
 
 14.4 
 
 16.2 
 
 18.0 
 
 36.0 
 
 54.0 
 
 72.0 
 
 90.0 
 
 105 
 
 10.5 
 12.3 
 
 14.0 
 15.8 
 
 17.5 
 
 35.0 
 
 52.5 
 
 70.0 
 
 87.5 
 
 119 
 
 11.9 
 
 13.9 
 
 15.9 
 
 17.9 
 19.8 
 39.7 
 59.5 
 79.3 
 99.2 
 
 116 
 
 11.6 
 
 13.5 
 
 15.5 
 
 17.4 
 
 19.3 
 
 38.7 
 
 58.0 
 
 77.3 
 
 96.7 
 
 113 
 
 11.3 
 
 13.2 
 
 15.1 
 
 17.0 
 
 18.8 
 
 37.7 
 
 56.5 
 
 75.3 
 
 94.2 
 
 110 
 
 11.0 
 
 12.8 
 
 14.7 
 
 16.5 
 
 18.3 
 
 36.7 
 
 55.0 
 
 73.3 
 
 91.7 
 
 107 
 
 10.7 
 
 12.5 
 
 14.3 
 
 16.1 
 
 17.8 
 35.7 
 
 53.5 
 
 71.3 
 
 89.2 
 
 IQ4 
 
 10.4 
 12.1 
 
 13.9 
 
 15.6 
 
 17.3 
 
 34.7 
 52.0 
 
 69.3 
 
 86.7 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.02520 
 
 9.02639 
 
 9.02757 
 
 9.02874 
 
 9.02992 
 
 9.02766 
 
 9.02885 
 
 9.03005 
 
 9.03124 
 
 9.03242 
 
 0.97234 
 
 0.97115 
 
 0.96995 
 
 0.96876 
 
 0.96758 
 
 9.99755 
 
 9.99753 
 
 9.99752 
 
 9.99751 
 
 9.99749 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.03109 
 
 9.03226 
 
 9.03342 
 
 9.03458 
 
 9.03574 
 
 9.03361 
 
 9.03479 
 
 9.03597 
 
 9.03714 
 
 9.03832 
 
 0.96639 
 
 0.96521 
 
 0.96403 
 
 0.96286 
 
 0.96168 
 
 9.99748 
 
 9.99747 
 
 9.99745 
 
 9.99744 
 
 9.99742 
 
 60 
 
 49 
 
 48 
 
 47 
 
 46 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 20 
 30 
 40 
 1 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 Ii8 
 
 11.8 
 
 13.8 
 
 15.7 
 
 17.7 
 
 19.7 
 
 39.3 
 
 59.0 
 
 78.7 
 
 98.3 
 
 115 
 
 11.5 
 
 13.4 
 
 15.3 
 
 17.3 
 
 19.2 
 
 38.3 
 
 57.5 
 
 76.7 
 
 95.8 
 
 112 
 
 11.2 
 
 13.1 
 
 14.9 
 16.8 
 
 18.7 
 
 37.3 
 
 56.0 
 
 74.7 
 
 93.3 
 
 <09 
 
 10.9 
 
 12.7 
 
 14.5 
 
 16.4 
 
 18.2 
 
 36.3 
 
 54.5 
 
 72.7 
 
 90.8 
 
 106 
 
 10.6 
 
 12.4 
 
 14.1 
 
 15.9 
 
 17.7 
 
 35.3 
 53.0 
 
 70.7 
 
 88.3 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.03690 
 
 9.03805 
 
 9.03920 
 
 9.04034 
 
 9.04149 
 
 9.03948 
 
 9.04065 
 
 9.04181 
 
 9.04297 
 
 9.04413 
 
 0.96052 
 
 0.95935 
 
 0.95819 
 
 0.95703 
 
 0.95587 
 
 9.99741 
 
 9.99740 
 
 9.99738 
 
 9.99737 
 
 9.99736 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.04262 
 
 9.04376 
 
 9.04490 
 
 9.04603 
 
 9.04715 
 
 9.04528 
 
 9.04643 
 
 9.04758 
 
 9.04873 
 
 9.04987 
 
 0.95472 
 
 0.95357 
 
 0.95242 
 
 0.95127 
 
 0.95013 
 
 9.99734 
 
 9.99733 
 
 9.99731 
 
 9.99730 
 
 9.99728 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.04828 
 
 9.04940 
 
 9.05052 
 
 9.05164 
 
 9.05275 
 
 9.05101 
 
 9.05214 
 
 9.05328 
 
 9.05441 
 
 9.05553 
 
 0.94899 
 
 0.94786 
 
 0.94672 
 
 0.94559 
 
 0.94447 
 
 9.99727 
 
 9.99726 
 
 9.99724 
 
 9.99723 
 
 9.99721 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.05386 
 
 9.05497 
 
 9.05607 
 
 9.05717 
 
 9.05827 
 
 9.05666 
 
 9.05778 
 
 9.05890 
 
 9.06002 
 
 9.06113 
 
 0.94334 
 
 0.94222 
 
 0.94110 
 
 0.93998 
 
 0.93887 
 
 9.99720 
 
 9.99718 
 
 9.99717 
 
 9.99716 
 
 9.99714 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.05937 
 
 9.06046 
 
 9.06155 
 
 9.06264 
 
 9.06372 
 
 9.06224 
 
 9.06335 
 
 9.06445 
 
 9.06556 
 
 9.06666 
 
 0.93776 
 
 0.93665 
 
 0.93555 
 
 0.93444 
 
 0.93334 
 
 9.99713 
 
 9.99711 
 
 9.99710 
 
 9.99708 
 
 9.99707 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.06481 
 
 9.06589 
 
 9.06696 
 
 9.06804 
 
 9.06911 
 
 9.06775 
 
 9.06885 
 
 9.06994 
 
 9.07103 
 
 9.07211 
 
 0.93225 
 
 0.93115 
 
 0.93006 
 
 0.92897 
 
 0.92789 
 
 9.99705 
 
 9.99704 
 
 9.99702 
 
 9.99701 
 
 9.99699 
 
 20 
 
 15 
 18 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.07018 
 
 9.07124 
 
 9.07231 
 
 9.07337 
 
 9.07442 
 
 9.07320 
 
 9.07428 
 
 9.07536 
 
 9.07643 
 
 9.07751 
 
 0.92680 
 
 0.92572 
 
 0.92464 
 
 0,92357 
 
 0.92249 
 
 9.99698 
 
 9.99696 
 
 9.99695 
 
 9.99693 
 
 9.99692 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 9.07548 
 
 9.07653 
 
 9.07758 
 
 9.07863 
 
 9.07968 
 
 9.07858 
 
 9.07964 
 
 9.08071 
 
 9.08177 
 
 9.08283 
 
 0.92142 
 
 0.92036 
 
 0.91929 
 
 0.91823 
 
 0.91717 
 
 9.99690 
 
 9.99689 
 
 9.99687 
 
 9.99686 
 
 9.99684 
 
 to 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.08072 
 
 9.08176 
 
 9.08280 
 
 9.08383 
 
 9.08486 
 
 9.08389 
 
 9.08495 
 
 9.08600 
 
 9.08705 
 
 9.08810 
 
 0.91611 
 
 0.91505 
 
 0.91400 
 
 0.91295 
 
 0.91190 
 
 9.99683 
 
 9.99681 
 
 9.99680 
 
 9.99678 
 
 9.99677 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 9.08589 
 
 9.08914 
 
 0.91086 
 
 9.99675 
 
 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 / 
 
 P. P. 
 
 83 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 7 ° 
 
 499 
 
 ' ] 
 
 L Sin. c 
 
 0 * 
 
 >.08589 
 
 1 * 
 
 >.08692 { 
 
 2 * 
 
 >.08795 | 
 
 3 t 
 
 >.08897 ]) 
 
 4 < 
 
 >.08999 l 
 
 5 ! 
 
 >.09101 
 
 6 ! 
 
 >.09202 }) 
 
 7 1 
 
 >.09304 1 
 
 8 ! 
 
 >.09405 j 
 
 9 1 
 
 9.09506 * 
 
 10 1 
 
 9.09606 , 
 
 11 1 
 
 9.09707 } 
 
 12 
 
 9.09807 } 
 
 13 
 
 9.09907 1 
 
 14 
 
 9.10006 
 
 15 
 
 9.10106 
 
 16 
 
 9.10205 
 
 17 
 
 9.10304 
 
 18 
 
 9.10402 
 
 19 
 
 9.10501 
 
 20 
 
 9.10599 
 
 21 
 
 9.10697 
 
 22 
 
 9.10795 
 
 23 
 
 9.10893 
 
 24 
 
 9.10990 
 
 25 
 
 9.11087 
 
 26 
 
 9.11184 
 
 27 
 
 9.11281 
 
 28 
 
 9.11377 
 
 29 
 
 9.11474 
 
 30 
 
 9.11570 
 
 31 
 
 9.11666 
 
 32 
 
 9.11761 
 
 33 
 
 9.11857 
 
 34 
 
 9.11952 
 
 35 
 
 9.12047 
 
 36 
 
 9.12142 
 
 37 
 
 9.12236 
 
 38 
 
 9.12331 
 
 39 
 
 9.12425 
 
 40 
 
 9.12519 
 
 41 
 
 9.12612 
 
 42 
 
 9.12706 
 
 43 
 
 9.12799 
 
 44 
 
 9.12892 
 
 45 
 
 9.12985 
 
 46 
 
 9.13078 
 
 47 
 
 9.13171 
 
 48 
 
 9.13263 
 
 49 
 
 9.13355 
 
 50 
 
 9.13447 
 
 51 
 
 9.13539 
 
 52 
 
 9.13630 
 
 53 
 
 9.13722 
 
 54 
 
 9.13813 
 
 55 
 
 9.13904 
 
 56 
 
 9.13994 
 
 57 
 
 9.14085 
 
 58 
 
 9.14175 
 
 59 
 
 9.14266 
 
 60 
 
 9.14356 
 
 
 L. Cos. 
 
 d. L.Tang. 
 9.08914 
 9.09019 
 9.09123 
 9.09227 
 9.09330 
 
 9.09434 
 
 9.09537 
 
 9.09640 
 
 9.09742 
 
 9.09845 
 
 99 
 00 
 99 
 99 
 
 98 
 
 99 
 98 
 98 
 98 
 98 
 97 
 97 
 97 
 97 
 
 96 
 
 97 
 96 
 96 
 
 95 
 
 96 
 95 
 95 
 95 
 
 94 
 
 95 
 94 
 94 
 
 93 
 
 94 
 93 
 93 
 93 
 93 
 93 
 92 
 92 
 92 
 92 
 
 91 
 
 92 
 91 
 91 
 
 90 
 
 91 
 
 90 
 
 91 
 90 
 
 9.09947 
 
 9.10049 
 
 9.10150 
 
 9.10252 
 
 9.10353 
 
 9.10454 
 
 9.10555 
 
 9.10656 
 
 9.10756 
 
 9.10856 
 
 9.10956 
 
 9.11056 
 
 9.11155 
 
 9.11254 
 
 9.11353 
 
 9.11452 
 
 9.11551 
 
 9.11649 
 
 9.11747 
 
 9.11845 
 
 9.11943 
 
 9.12040 
 
 9.12138 
 
 9.12235 
 
 9.12332 
 
 105 
 
 104 
 
 104 
 
 103 
 
 104 
 103 
 103 
 102 
 103 
 102 
 102 
 101 
 102 
 101 
 101 
 101 
 101 
 100 
 100 
 100 
 100 
 
 99 
 
 99 
 
 9.12428 
 
 9.12525 
 
 9.12621 
 
 9.12717 
 
 9.12813 
 
 9.12909 
 
 9.13004 
 
 9.13099 
 
 9.13194 
 
 9.13289 
 
 9.13384 
 
 9.13478 
 
 9.13573 
 
 9.13667 
 
 9.13761 
 
 9.13854 
 
 9.13948 
 
 9.14041 
 
 9.14134 
 
 9.14227 
 
 9.14320 
 
 9.14412 
 
 9.14504 
 
 9.14597 
 
 9.14688 
 
 99 
 
 99 
 
 99 
 
 98 
 
 98 
 
 98 
 
 98 
 
 97 
 
 98 
 97 
 97 
 
 96 
 
 97 
 96 
 96 
 96 
 96 
 95 
 95 
 95 
 95 
 95 
 
 94 
 
 95 
 94 
 94 
 
 9.14780 
 
 d. L. Cotg 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 0.91086 
 
 9.99675 f 
 
 >0 
 
 0.90981 
 
 9.99674 J 
 
 59 
 
 0.90877 
 
 9.99672 J 
 
 58 
 
 0.90773 
 
 9.99670 ; 
 
 57 
 
 0.90670 
 
 9.99669 
 
 56 
 
 0.90566 
 
 9.99667 
 
 55 
 
 0.90463 
 
 9.99666 
 
 54 
 
 0.90360 
 
 9.99664 
 
 53 
 
 0.90258 
 
 9.99663 
 
 52 
 
 0.90155 
 
 9.99661 
 
 51 
 
 0.90053 
 
 9.99659 ! 
 
 50 
 
 0.89951 
 
 9.99658 
 
 49 
 
 0.89850 
 
 9.99656 
 
 48 
 
 0.89748 
 
 9.99655 
 
 47 
 
 0.89647 
 
 9.99653 
 
 46 
 
 0.89546 
 
 9.99651 
 
 45 
 
 0.89445 
 
 9.99650 
 
 44 
 
 0.89344 
 
 9.99648 
 
 43 
 
 0.89244 
 
 9.99647 
 
 42 
 
 0.89144 
 
 9.99645 
 
 41 
 
 0.89044 
 
 9.99643 
 
 40 
 
 0.88944 
 
 9.99642 
 
 39 
 
 0.88845 
 
 9.99640 
 
 38 
 
 0.88746 
 
 9.99638 
 
 37 
 
 0.88647 
 
 9.99637 
 
 36 
 
 0.88548 
 
 9.99635 
 
 35 
 
 0.88449 
 
 9.99633 
 
 34 
 
 0.88351 
 
 9.99632 
 
 33 
 
 0.88253 
 
 9.99630 
 
 32 
 
 0.88155 
 
 9.99629 
 
 31 
 
 0.88057 
 
 9.99627 
 
 30 
 
 0.87960 
 
 9.99625 
 
 29 
 
 0.87862 
 
 9.99624 
 
 28 
 
 0.87765 
 
 9.99622 
 
 27 
 
 0.87668 
 
 9.99620 
 
 26 
 
 0.87572 
 
 9.99618 
 
 25 
 
 0.87475 
 
 9.99617 
 
 24 
 
 0.87379 
 
 9.99615 
 
 23 
 
 0.87283 
 
 9.99613 
 
 22 
 
 0.87187 
 
 9.99612 
 
 21 
 
 0.87091 
 
 9.99610 
 
 20 
 
 0.86996 
 
 9.99608 
 
 19 
 
 0.86901 
 
 9.99607 
 
 18 
 
 0.86806 
 
 9.99605 
 
 17 
 
 0.86711 
 
 9.99603 
 
 16 
 
 0.86616 
 
 9.99601 
 
 15 
 
 0.86522 
 
 9.99600 
 
 14 
 
 0.86427 
 
 9.99598 
 
 13 
 
 0.86333 
 
 , 9.99596 
 
 12 
 
 j 0.86239 
 
 i 9.99595 
 
 11 
 
 ’ 0.86146 
 
 1 9.99593 
 
 10 
 
 : 0.86052 
 
 5 9.99591 
 
 9 
 
 I 0.85952 
 
 ) 9.99589 
 
 8 
 
 » 0.8586( 
 
 } 9.99588 
 
 7 
 
 5 0.8577; 
 
 5 9.99586 
 
 6 
 
 ' 0.8568( 
 
 ) 9.99584 
 
 5 
 
 \ 0.85585 
 
 3 9.99582 
 
 1 4 
 
 2 0.85491 
 
 5 9.99581 
 
 3 
 
 } 0.8540: 
 
 3 9.99572 
 
 ) 2 
 
 \ 0.8531! 
 
 2 9.99577 
 
 * 1 
 
 2 0.8522 
 
 C 9.9957c 
 
 ) 0 
 
 c. L.Tanj 
 
 g. L. Sin 
 
 ! 
 
 P. P. 
 
 105 
 
 10.5 
 12.3 
 
 14.0 
 15.8 
 
 17.5 
 
 35.0 
 
 52.5 
 
 70.0 
 
 87.5 
 
 102 
 
 6 ! 10.2 
 11.9 
 13.6 
 15.3 
 
 17.0 
 
 34.0 
 
 51.0 
 
 68.0 
 
 85.0 
 
 104 
 
 10.4 
 12.1 
 13.9 
 
 15.6 
 
 17.3 
 
 34.7 
 
 52.0 
 
 69.3 
 
 86.7 
 
 101 
 
 10.1 
 
 11.8 
 
 13.5 
 
 15.2 
 16.8 
 33.7 
 
 50.5 
 
 67.3 
 84.2 
 
 103 
 
 10.3 
 
 12.0 
 
 13.7 
 
 15.5 
 
 17.2 
 
 34.3 
 
 51.5 
 
 68.7 
 
 85.8 
 
 100 
 
 10.0 
 
 11.7 
 
 13.3 
 -15.0 
 
 16.7 
 
 33.3 
 50.0 
 
 66.7 
 
 83.3 
 
 99 
 
 9.9 
 
 11.6 
 
 13.2 
 
 14.9 
 
 16.5 
 
 33.0 
 
 49.5 
 
 66.0 
 
 82.5 
 
 98 
 
 9.8 
 
 11.4 
 
 13.1 
 
 14.7 
 
 16.3 
 
 32.7 
 49.0 
 
 65.3 
 
 81.7 
 
 97 
 
 9.7 
 
 11.3 
 12.9 
 
 14.6 
 16.2 
 
 32.3 
 
 48.5 
 
 64.7 
 
 80.8 
 
 94 
 
 9.4 
 
 11.0 
 
 12.5 
 14.1 
 
 15.7 
 
 31.3 
 47.0 
 
 62.7 
 
 78.3 
 
 96 
 
 9.6 
 
 11.2 
 
 12.8 
 
 14.4 
 16.0 
 
 32.0 
 
 48.0 
 
 64.0 
 
 80.0 
 
 93 
 
 9.3 
 
 10.9 
 
 12.4 
 
 14.0 
 
 15.5 
 
 31.0 
 
 46.5 
 
 62.0 
 
 77.5 
 
 91 
 
 9.1 
 
 10.6 
 
 12.1 
 
 13.7 
 
 15.2 
 
 30.3 
 45.5 
 
 60.7 
 
 75.8 
 
 90 I 
 
 9.0 1 
 
 10.5 
 
 12.0 
 
 13.5 
 
 15.0 
 
 30.0 
 
 45.0 
 
 60.0 
 75.0 
 
 95 
 
 9.5 
 
 11.1 
 
 12.7 
 
 14.3 
 
 15.8 
 
 31.7 
 
 47.5 
 
 63.3 
 
 79.2 
 
 92 
 
 9.2 
 
 10.7 
 
 12.3 
 
 13.8 
 
 15.3 
 
 30.7 
 46.0 
 
 61.3 
 
 76.7 
 
 I 2 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.3 
 
 0.7 
 
 1.0 
 
 1.3 
 1.7 
 
 P. P. 
 
 82 ° 
 
500 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 
 
 8 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 
 
 P. 
 
 p. 
 
 
 
 0 
 
 9.14356 
 
 
 9.14780 
 
 
 0.85220 
 
 9.99575 
 
 60 
 
 
 
 
 
 
 
 1 
 
 9.14445 
 
 89 
 
 9.14872 
 
 92 
 
 0.85128 
 
 9.99574 
 
 59 
 
 
 
 ! SI 
 
 90 
 
 2 
 
 9.14535 
 
 90 
 
 9.14963 
 
 91 
 
 0.85037 
 
 9.99572 
 
 58 
 
 6 
 
 9.2 
 
 | 9.1 
 
 9.0 
 
 B 
 
 9.14624 
 
 89 
 
 9.15054 
 
 91 
 
 0.84946 
 
 9.99570 
 
 57 
 
 7 
 
 10.7 
 
 10.6 
 
 10.5 
 
 4 
 
 9.14714 
 
 90 
 
 89 
 
 9.15145 
 
 91 
 
 91 
 
 0.84855 
 
 9.99568 
 
 56 
 
 8 
 
 9 
 
 12.3 
 
 J3.8 
 
 1 12.1 
 13.7 
 
 12.0 
 
 13.5 
 
 5 
 
 9.14803 
 
 9.15236 
 
 0.84764 
 
 9.99566 
 
 55 
 
 10 
 
 15^3 
 
 15^2 
 
 15.0 
 
 6 
 
 9.14891 
 
 88 
 
 9.15327 
 
 91 
 
 0.84673 
 
 9.99565 
 
 54 
 
 20 
 
 30.7 
 
 30.3 
 
 30.0 
 
 7 
 
 9.14980 
 
 sy 
 
 9.15417 
 
 90 
 
 0.84583 
 
 9.99563 
 
 53 
 
 30 
 
 46.0 
 
 ! 45.5 
 
 45.0 
 
 8 
 
 9.15069 
 
 »y 
 
 9.15508 
 
 9L 
 
 0.84492 
 
 9.99561 
 
 52 
 
 40 
 
 61.3 
 
 1 60.7 
 
 60.0 
 
 9 
 
 9.15157 
 
 8 o 
 
 88 
 
 9.15598 
 
 90 
 
 90 
 
 0.84402 
 
 9.99559 
 
 51 
 
 50 
 
 76.7 
 
 i 75.8 
 
 75.0 
 
 10 
 
 9.15245 
 
 9.15688 
 
 0.84312 
 
 9.99557 
 
 50 
 
 
 
 
 
 
 
 11 
 
 9.15333 
 
 88 
 
 9.15777 
 
 89 
 
 0.84223 
 
 9.99556 
 
 49 
 
 
 
 89 
 
 88 
 
 12 
 
 9.15421 
 
 88 
 
 9.15867 
 
 90 
 
 0.84133 
 
 9.99554 
 
 48 
 
 
 6 
 
 1 8.9 
 
 8.8 
 
 13 
 
 9.15508 
 
 87 
 
 9.15956 
 
 89 
 
 0.84044 
 
 9.99552 
 
 47 
 
 
 7 
 
 i 10.4 
 
 10.3 
 
 14 
 
 9.15596 
 
 88 
 
 87 
 
 9.16046 
 
 90 
 
 89 
 
 0.83954 
 
 9.99550 
 
 46 
 
 
 8 
 
 A 
 
 i 11.9 
 
 ! 1 Q A 
 
 11.7 
 
 IQ O 
 
 15 
 
 9.15683 
 
 
 9.16135 
 
 0.83865 
 
 9.99548 
 
 45 
 
 V 
 
 10 
 
 lo.4 
 
 14.8 
 
 lo. Z 
 
 14.7 
 
 16 
 
 9.15770 
 
 o7 
 
 9.16224 
 
 89 
 
 0.83776 
 
 9.99546 
 
 44 
 
 20 
 
 1 20.7 
 
 90 
 
 17 
 
 9.15857 
 
 8 / 
 
 9.16312 
 
 88 
 
 0.83688 
 
 9.99545 
 
 43 
 
 30 
 
 44.5 
 
 L. J.O 
 
 44.0 
 
 18 
 
 9.15944 
 
 g7 
 
 9.16401 
 
 89 
 
 0.83599 
 
 9.99543 
 
 42 
 
 40 
 
 1 593 
 
 58.7 
 
 19 
 
 9.16030 
 
 ob 
 
 86 
 
 9.16489 
 
 88 
 
 88 
 
 0.83511 
 
 9.99541 
 
 41 
 
 50 
 
 ! 74.2 
 
 73^3 
 
 20 
 
 9.16116 
 
 9.16577 
 
 0.83423 
 
 9.99539 
 
 40 
 
 
 
 
 
 
 
 21 
 
 9.16203 
 
 8 / 
 
 9.16665 
 
 88 
 
 0.83335 
 
 9.99537 
 
 39 
 
 
 
 87 
 
 86 
 
 22 
 
 9.16289 
 
 86 
 
 9.16753 
 
 88 
 
 0.83247 
 
 9.99535 
 
 38 
 
 
 6 
 
 8.7 
 
 8.6 
 
 23 
 
 9.16374 
 
 85 
 
 9.16841 
 
 88 
 
 0.83159 
 
 9.99533 
 
 37 
 
 
 7 
 
 10.2 
 
 10.0 
 
 24 
 
 9.16460 
 
 86 
 
 85 
 
 9.16928 
 
 87 
 
 qq 
 
 0.83072 
 
 9.99532 
 
 36 
 
 
 8 
 
 11.6 
 
 11.5 
 
 25 
 
 9.16545 
 
 9.17016 
 
 
 0.82984 
 
 9.99530 
 
 35 
 
 y 
 
 1 A 
 
 13.1 
 
 12.9 
 
 26 
 
 9.16631 
 
 86 
 
 9.17103 
 
 87 
 
 0.82897 
 
 9.99528 
 
 34 
 
 10 
 
 14.5 
 
 14.3 
 
 27 
 
 9.16716 
 
 85 
 
 9.17190 
 
 87 
 
 0.82810 
 
 9.99526 
 
 33 
 
 20 
 
 on 
 
 29.0 
 
 AO K 
 
 28.7 
 
 28 
 
 9.16801 
 
 85 
 
 9.17277 
 
 87 
 
 0.82723 
 
 9.99524 
 
 32 
 
 30 
 
 A A 
 
 CQ A 
 
 43.0 
 
 cr? o 
 
 29 
 
 9.16886 
 
 85 
 
 84 
 
 or; 
 
 9.17363 
 
 86 
 
 87 
 
 0.82637 
 
 9.99522 
 
 31 
 
 40 
 
 RO 
 
 Oo.U 
 79 * 
 
 5/.o 
 71 7 
 
 30 
 
 9.16970 
 
 9.17450 
 
 0.82550 
 
 9.99520 
 
 30 
 
 
 
 
 
 
 
 31 
 
 9.17055 
 
 85 
 
 O/l 
 
 9.17536 
 
 86 
 
 0.82464 
 
 9.99518 
 
 29 
 
 
 
 85 
 
 84 
 
 32 
 
 9.17139 
 
 84 
 
 9.17622 
 
 86 
 
 0.82378 
 
 9.99517 
 
 28 
 
 
 6 
 
 8.5 
 
 8.4 
 
 33 
 
 9.17223 
 
 84 
 
 0/4 
 
 9.17708 
 
 86 
 
 0.82292 
 
 9.99515 
 
 27 
 
 
 7 
 
 9.9 
 
 9.8 
 
 34 
 
 9.17307 
 
 84 
 
 84 
 
 9.17794 
 
 86 
 
 86 
 
 0.82206 
 
 9.99513 
 
 26 
 
 
 8 
 
 11.3 
 
 11.2 
 
 35 
 
 9.17391 
 
 9.17880 
 
 0.82120 
 
 9.99511 
 
 25 
 
 
 9 
 
 12.8 
 
 12.6 
 
 36 
 
 9.17474 
 
 83 
 
 9.17965 
 
 85 
 
 0.82035 
 
 9.99509 
 
 24 
 
 10 
 
 14.2 
 
 14.0 
 
 37 
 
 9.17558 
 
 84 
 
 9.18051 
 
 86 
 
 0.81949 
 
 9.99507 
 
 23 
 
 20 
 
 28.3 
 
 28.0 
 
 38 
 
 9.17641 
 
 83 
 
 9.18136 
 
 85 
 
 0.81864 
 
 9.99505 
 
 22 
 
 30 
 
 42.5 
 
 42.0 
 
 39 
 
 9.17724 
 
 83 
 
 83 
 
 QO 
 
 9.18221 
 
 85 
 
 85 
 
 0.81779 
 
 9.99503 
 
 21 
 
 40 
 
 ! £;a 
 
 56.7 
 
 '7A Q 
 
 56.0 
 
 >7 n n 
 
 40 
 
 9.17807 
 
 9.18306 
 
 0.81694 
 
 9.99501 
 
 20 
 
 
 
 
 
 
 • V / ;j 
 
 41 
 
 9.17890 
 
 oS 
 
 QO 
 
 9.18391 
 
 85 
 
 0.81609 
 
 9.99409 
 
 19 
 
 
 
 83 
 
 82 
 
 42 
 
 9.17973 
 
 GO 
 
 QO 
 
 9.18475 
 
 84 
 
 0.81525 
 
 9.99497 
 
 18 
 
 
 6 
 
 8.3 
 
 8.2 
 
 43 
 
 9.18055 
 
 oZ 
 
 QO 
 
 9.18560 
 
 85 
 
 O A 
 
 0.81440 
 
 9.99495 
 
 17 
 
 
 7 
 
 9.7 
 
 9.6 
 
 44 
 
 9.18137 
 
 oZ 
 
 83 
 
 9.18644 
 
 g4 
 
 84 
 
 0.81356 
 
 9.99494 
 
 16 
 
 
 8 
 
 11.1 
 
 10.9 
 
 45 
 
 9.18220 
 
 9.18728“ 
 
 0.81272 
 
 9.99492 
 
 15 
 
 
 9 
 
 12.5 
 
 12.3 
 
 46 
 
 9.18302 
 
 82 
 
 Q 1 
 
 9.18812 
 
 84 
 
 0.81188 
 
 9.99490 
 
 14 
 
 JO 
 
 13.8 
 
 13.7 
 
 47 
 
 9.18383 
 
 ol 
 
 QO 
 
 9.18896 
 
 84 
 
 0.81104 
 
 9.99488 
 
 13 
 
 20 
 
 27.7 
 
 27.3 ! 
 
 48 
 
 9.18465 
 
 oZ 
 
 9.18979 
 
 83 
 
 0.81021 
 
 9.99486 
 
 12 
 
 30 
 
 41.5 
 
 41.0 
 
 49 
 
 9.18547 
 
 82 
 
 81 
 
 9.19063 
 
 84 
 
 83 
 
 0.80937 
 
 9.99484 
 
 11 
 
 40 
 
 55.3 
 
 £Q O 
 
 54.7 
 
 £Q O 
 
 50 
 
 9.18628 
 
 Q t 
 
 9.19146 
 
 0.80854 
 
 9.99482 
 
 10 
 
 OU 
 
 oy . z 
 
 DO.O | 
 
 51 
 
 9.18709 
 
 ol 
 
 Q1 
 
 9.19229 
 
 83 
 
 0.80771 
 
 9.99480 
 
 9 
 
 
 81 
 
 80 
 
 9 
 
 52 
 
 9.18790 
 
 ol 
 
 9.19312 
 
 83 
 
 0.80688 
 
 9.99478 
 
 8 
 
 6 
 
 8.1 
 
 8.0 
 
 0.2 
 
 53 
 
 9.18871 
 
 G1 
 
 Q1 
 
 9.19395 
 
 83 
 
 QO 
 
 0.80605 
 
 9.99476 
 
 fT 
 
 ! 
 
 7 
 
 9.5 
 
 9.3 
 
 0 2 
 
 54 
 
 9.18952 
 
 G± 
 
 81 
 
 9.19478 
 
 GO 
 
 83 
 
 0.80522 
 
 9.99474 
 
 6 
 
 8 
 
 10.8 
 
 10.7 
 
 0.3 
 
 55 
 
 9.19033 
 
 qh 
 
 9.19561 
 
 0.80439 
 
 9.99472 
 
 5 
 
 9 
 
 12.2 
 
 12.0 
 
 0.3 
 
 56 
 
 9.19113 
 
 oU 
 
 QO 
 
 9.19643 
 
 82 
 
 0.80357 
 
 9.99470 
 
 4 
 
 10 
 
 13.5 
 
 13.3 
 
 0.3 
 
 57 
 
 9.19193 
 
 oU 
 
 QO 
 
 9.19725 
 
 82 
 
 0.80275 
 
 9.99468 
 
 3 
 
 20 
 
 27.0 
 
 26.7 
 
 0.7 
 
 58 
 
 9.19273 
 
 GU 
 QO * 
 
 9.19807 
 
 82 
 
 0.80193 
 
 9.99466 
 
 2 
 
 30 
 
 40.5 
 
 40.0 
 
 1.0 
 
 59 
 
 9.19353 
 
 oU 
 
 80 
 
 9.19889 
 
 82 
 
 82 
 
 0.80111 
 
 9.99464 
 
 1 
 
 40 
 
 54.0 ; 
 
 53.3 
 
 1.3 
 
 60 
 
 9.19433 
 
 
 9.19971 
 
 0.80029 
 
 9.99462 
 
 0 
 
 50 
 
 67.5 i 
 
 66.7 
 
 1.7 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 t 
 
 
 
 p. 
 
 P. 
 
 
 
 81 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 501 
 
 t 
 
 L. Sin. < 
 
 0 
 
 9.19433 , 
 
 1 
 
 9.19513 ; 
 
 2 
 
 9.19592 
 
 3 
 
 9.19672 ; 
 
 4 
 
 9.19751 
 
 5 
 
 9.19830 
 
 6 
 
 9.19909 
 
 7 
 
 9.19988 
 
 8 
 
 9.20067 
 
 9 
 
 9.20145 
 
 10 
 
 9.20223 
 
 11 
 
 9.20302 
 
 12 
 
 9.20380 
 
 13 
 
 9.20458 
 
 14 
 
 9.20535 
 
 15 
 
 9.20613 
 
 16 
 
 9.20691 
 
 17 
 
 9.20768 
 
 18 
 
 9.20845 
 
 19 
 
 9.20922 
 
 20 
 
 0.20999 
 
 21 
 
 9.21076 
 
 22 
 
 9.21153 
 
 23 
 
 9.21229 
 
 24 
 
 9.21306 
 
 25 
 
 9.21382 
 
 26 
 
 9.21458 
 
 27 
 
 9.21534 
 
 28 
 
 9.21610 
 
 29 
 
 9.21685 
 
 30 
 
 9.21761 
 
 31 
 
 9.21836 
 
 32 
 
 9.21912 
 
 33 
 
 9.21987 
 
 34 
 
 9.22062 
 
 35 
 
 9.22137 
 
 36 
 
 9.22211 
 
 37 
 
 9.22286 
 
 38 
 
 9.22361 
 
 39 
 
 9.22435 
 
 40 
 
 9.22509 
 
 41 
 
 9.22583 
 
 42 
 
 9.22657 
 
 43 
 
 9.22731 
 
 44 
 
 9.22805 
 
 45 
 
 9.22878 
 
 46 
 
 9.22952 
 
 47 
 
 9.23025 
 
 48 
 
 9.23098 
 
 49 
 
 9.23171 
 
 50 
 
 ' 9.23244 
 
 51 
 
 9.23317 
 
 52 
 
 9.23390 
 
 53 
 
 9.23462 
 
 54 
 
 9.23535 
 
 55 
 
 9.23607 
 
 56 
 
 9.23679 
 
 57 
 
 9.23752 
 
 58 
 
 9.23823 
 
 59 
 
 9.23895 
 
 60 
 
 9.23967 
 
 
 L. Cos. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 0.80029 
 
 9.99462 l 
 
 BO 
 
 0.79947 
 
 9.99460 
 
 59 
 
 0.79866 
 
 9.99458 
 
 58 
 
 0.79784 
 
 9.99456 
 
 57 
 
 0.79703 
 
 9.99454 
 
 56 
 
 0.79622 
 
 9.99452 
 
 55 ] 
 
 0.79541 
 
 9.99450 
 
 54 $ 
 
 0.79460 
 
 9.99448 
 
 53 £ 
 
 0.79379 
 
 9.99446 
 
 52 t 
 
 0.79299 
 
 9 99444 
 
 51 l 
 
 0.79218 
 
 9.99442 
 
 50 
 
 0.79138 
 
 9.99440 
 
 49 
 
 0.79058 
 
 9.99438 
 
 48 
 
 0.78978 
 
 9.99436 
 
 47 
 
 0.78898 
 
 9.99434 
 
 46 
 
 0.78818 
 
 9.99432 
 
 45 
 
 0.78739 
 
 9.99429 
 
 44 
 
 0.78659 
 
 9.99427 
 
 43 
 
 0.78580 
 
 9.99425 
 
 42 
 
 0.78501 
 
 9.99423 
 
 41 
 
 0.78422 
 
 9.99421 
 
 40 
 
 0.78343 
 
 9.99419 
 
 39 
 
 0.78264 
 
 9.99417 
 
 38 
 
 0.78186 
 
 9.99415 
 
 37 
 
 0.78107 
 
 9.99413 
 
 36 
 
 0.78029 
 
 9.99411 
 
 35 
 
 0.77951 
 
 9.99409 
 
 34 
 
 0.77873 
 
 9.99407 
 
 33 
 
 0.77795 
 
 9.99404 
 
 32 
 
 0.77717 
 
 9.99402 
 
 31 
 
 0.77639 
 
 9.99400 
 
 30 
 
 0.77562 
 
 9.99398 
 
 29 
 
 0.77484 
 
 9.99396 
 
 28 
 
 0.77407 
 
 9.99394 
 
 27 
 
 0.77330 
 
 9.99392 
 
 26 
 
 0.77253 
 
 9.99390 
 
 25 
 
 0.77176 
 
 9.99388 
 
 24 
 
 0.77099 
 
 9.99385 
 
 23 
 
 0.77023 
 
 9.99383 
 
 22 
 
 0.76946 
 
 9.99381 
 
 21 
 
 0.76870 
 
 9.99379 
 
 20 
 
 0.76794 
 
 9.99377 
 
 19 
 
 0.76717 
 
 9.99375 
 
 18 
 
 0.76641 
 
 9.99372 
 
 17 
 
 0.76565 
 
 9.99370 
 
 16 
 
 0.76490 
 
 9.99368 
 
 15 
 
 0.76414 
 
 9.99366 
 
 14 
 
 0.76339 
 
 9.99364 
 
 13 
 
 0.76263 
 
 9.99362 
 
 12 
 
 0.76188 
 
 9.99359 
 
 ' 11 
 
 0.76113 
 
 " 9.99357 
 
 10 
 
 0.76038 
 
 9.99355 
 
 9 
 
 0.75963 
 
 9.99353 
 
 8 
 
 0.75888 
 
 9.99351 
 
 7 
 
 0.75814 
 
 9.99348 
 
 6 
 
 0.75739 
 
 ' 9.99346 
 
 5 
 
 0.75665 
 
 . 9.99344 
 
 4 
 
 1 0.75590 
 
 i 9.99342 
 
 ; 3 
 
 0.75516 
 
 i 9.99340 
 
 i 2 
 
 ■ 0.75442 
 
 \ 9.99337 
 
 ’ 1 
 
 0.7536* 
 
 5 9.9933E 
 
 » 0 
 
 c. L.Tang 
 
 f. L. Sin. 
 
 f 
 
 d. 
 
 79 
 
 80 
 79 
 79 
 79 
 79 
 79 
 78 
 
 78 
 
 79 
 78 
 78 
 
 77 
 
 78 
 78 
 77 
 77 
 77 
 77 
 77 
 77 
 
 76 
 
 77 
 76 
 76 
 76 
 76 
 
 75 
 
 76 
 
 75 
 
 76 
 75 
 75 
 75 
 
 74 
 
 75 
 75 
 74 
 74 
 74 
 74 
 74 
 74 
 
 73 
 
 74 
 73 
 73 
 73 
 73 
 73 
 73 
 
 72 
 
 73 
 72 
 
 72 
 
 73 
 
 71 
 
 72 
 72 
 
 d. 
 
 Tang. 
 
 .19971 
 
 .20053 
 
 .20134 
 
 .20216 
 
 .20297 
 
 20378 
 
 20459 
 
 .20540 
 
 .20621 
 
 .20701 
 
 20782 
 
 9.20862 
 
 9.20942 
 
 9.21022 
 
 9.21102 
 
 d. c. 
 
 9.21182 
 
 9.21261 
 
 9.21341 
 
 9.21420 
 
 9.21499 
 
 9.21578 
 
 9.21657 
 
 9.21736 
 
 9.21814 
 
 9.21893 
 
 9.21971 
 
 9.22049 
 
 9.22127 
 
 9.22205 
 
 9.22283 
 
 9.22361 
 
 9.22438 
 
 9.22516 
 
 9.22593 
 
 9.22670 
 
 9.22747 
 
 9.22824 
 
 9.22901 
 
 9.22977 
 
 9.23054 
 
 9.23130 
 
 9.23206 
 
 9.23283 
 
 9.23359 
 
 9.23435 
 
 9.23510 
 
 9.23586 
 
 9.23661 
 
 9.23737 
 
 9.23812 
 
 9.23887 
 
 9.23962 
 
 9.24037 
 
 9.24112 
 
 9.24186 
 
 9.24261 
 
 9.24335 
 
 9.24410 
 
 9.24484 
 
 9.24558 
 
 82 
 
 81 
 
 82 
 
 81 
 
 81 
 
 81 
 
 81 
 
 81 
 
 80 
 
 81 
 
 80 
 
 80 
 
 80 
 
 80 
 
 80 
 
 79 
 
 80 
 79 
 79 
 79 
 79 
 79 
 
 78 
 
 79 
 78 
 78 
 78 
 78 
 78 
 78 
 
 77 
 
 78 
 77 
 77 
 77 
 77 
 77 
 
 76 
 
 77 
 76 
 
 76 
 
 77 
 76 
 76 
 
 75 
 
 76 
 
 75 
 
 76 
 75 
 75 
 75 
 75 
 75 
 74 
 
 9.24632 
 
 L. Cotg. 
 
 P. P. 
 
 82 
 
 81 
 
 80 
 
 8.2 
 
 8.1 
 
 8.0 
 
 9.6 
 
 9.5 
 
 9.3 
 
 10.9 
 
 10.8 
 
 10.7 
 
 12.3 
 
 12.2 
 
 12.0 
 
 13.7 
 
 13.5 
 
 13.3 
 
 27.3 
 
 27.0 
 
 26.7 
 
 41.0 
 
 40.5 
 
 40.0 
 
 54.7 
 
 54.0 
 
 53.3 
 
 68.3 
 
 67.5 
 
 66.7 
 
 6 i 
 
 ?! 
 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 20 
 30 
 40 
 50 
 
 79 
 
 7.9 
 
 9.2 
 
 10.5 
 11.9 
 
 13.2 
 
 26.3 
 
 39.5 
 
 52.7 
 
 65.8 
 
 77 
 
 7.7 
 
 9.0 
 
 10.3 
 
 11.6 
 
 12.8 
 25.7 
 
 38.5 
 
 51.3 
 
 64.2 
 
 75 
 
 7.5 
 
 8.8 
 
 10.0 
 
 11.3 
 
 12.5 
 
 25.0 
 
 37.5 
 
 50.0 
 
 62.5 
 
 73 
 
 7.3 
 
 8.5 
 9.7 
 
 11.0 
 12.2 
 
 24.3 
 
 36.5 
 
 48.7 
 
 60.8 
 
 71 
 
 78 
 
 7.8 
 9.1 
 
 10.4 
 
 11.7 
 
 13.0 
 
 26.0 
 
 39.0 
 
 52.0 
 
 65.0 
 
 76 
 
 7.6 
 
 8.9 
 10 . 1 ' 
 
 11.4 
 
 12.7 
 
 25.3 
 
 38.0 
 
 50.7 
 
 63.3 
 
 74 
 
 7.4 
 
 8.6 
 
 9.9 
 
 11.1 
 
 12.3 
 
 24.7 
 
 37.0 
 
 49.3 
 
 61.7 
 
 72 
 
 7.2 
 
 8.4 
 9.6 
 
 10.8 
 
 12.0 
 
 24.0 
 
 36.0 
 
 48.0 
 
 60.0 
 
 3 
 
 80 
 
 7.1 
 
 0.3 
 
 0.2 
 
 8.3 
 
 0.4 
 
 0.2 
 
 9.5 
 
 0.4 
 
 0.3 
 
 10.7 
 
 0.5 
 
 0.3 
 
 11.8 
 
 0.5 
 
 0.3 
 
 23.7 
 
 1.0 
 
 0.7 
 
 35.5 
 
 1.5 
 
 1.0 
 
 47.3 
 
 2.0 
 
 1.3 
 
 59.2 
 
 2.5 
 
 1.7 
 
 P. 
 
 P. 
 
 
502 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 10 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.23967 
 
 9.24039 
 
 9.24110 
 
 9.24181 
 
 9.24253 
 
 72 
 
 71 
 
 71 
 
 72 
 71 
 71 
 71 
 
 70 
 
 71 
 
 70 
 
 71 
 70 
 70 
 70 
 70 
 70 
 70 
 
 69 
 
 70 
 69 
 69 
 69 
 69 
 69 
 69 
 69 
 68 
 69 
 68 
 68 
 68 
 68 
 68 
 68 
 68 
 
 67 
 
 68 
 67 
 67 
 67 
 67 
 67 
 67 
 67 
 66 
 67 
 66 
 67 
 66 
 66 
 66 
 66 
 
 65 
 
 66 
 66 
 65 
 
 65 
 
 66 
 65 
 65 
 
 9.24632 
 
 9.24706 
 
 9.24779 
 
 9.24853 
 
 9.24926 
 
 74 
 
 73 
 
 74 
 
 73 
 
 74 
 73 
 73 
 73 
 73 
 73 
 
 72 
 
 73 
 
 72 
 
 73 
 72 
 72 
 72 
 72 
 72 
 
 71 
 
 72 
 
 71 
 
 72 
 71 
 71 
 71 
 71 
 
 70 
 
 71 
 71 
 70 
 
 70 
 
 71 
 70 
 70 
 70 
 70 
 
 69 
 
 70 
 
 69 
 
 70 
 69 
 69 
 69 
 69 
 69 
 69 
 69 
 68 
 69 
 68 
 69 
 68 
 68 
 68 
 68 
 
 67 
 
 68 
 68 
 67 
 
 0.75368 
 
 0.75294 
 
 0.75221 
 
 0.75147 
 
 0.75074 
 
 9.99335 
 
 9.99333 
 
 9.99331 
 
 9.99328 
 
 9.99326 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 20 
 30 
 40 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 20 
 30 
 40 
 50 
 
 6 
 
 7 
 
 1 8 
 9 
 10 
 20 
 30 
 40 
 50 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 20 
 30 
 40 
 50 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 10 
 30 
 40 
 50 
 
 74 
 
 7.4 
 8.6 
 
 9.9 
 11.1 
 
 12.3 
 
 24.7 
 
 37.0 
 
 49.3 
 
 61.7 
 
 72 
 
 7.2 
 
 8.4 
 
 9.6 
 
 10.8 
 
 12.0 
 
 24.0 
 
 36.0 
 
 48.0 
 
 60.0 
 
 70 
 
 7.0 
 
 8.2 
 9.3 
 
 10.5 
 
 11.7 
 
 23.3 
 
 35.0 
 
 46.7 
 
 58.3 
 
 68 
 
 6.8 
 
 7.9 
 
 9.1 
 10.2 
 
 11.3 
 
 22.7 
 
 34.0 
 
 45.3 
 
 56.7 
 
 66 
 
 6.6 
 
 7.7 
 
 8.8 
 
 9.9 
 
 11.0 
 22.0 
 
 33.0 
 
 44.0 
 
 55.0 
 
 3 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 1.0 
 
 1.5 
 2.0 
 
 2.5 
 
 73 
 
 7.3 
 
 8.5 
 
 9.7 
 11.0 
 12.2 
 
 24.3 
 
 36.5 
 
 48.7 
 60.8. 
 
 71 
 
 7.1 
 
 8.3 
 
 9.5 
 
 10.7 
 
 11.8 
 
 23.7 
 
 35.5 
 
 47.3 
 
 59.2 
 
 69 
 
 6.9 
 
 8.1 
 9.2 
 
 10.4 
 
 11.5 
 
 23.0 
 
 34.5 
 
 46.0 
 
 57.5 
 
 67 
 
 6.7 
 
 7.8 
 
 8.9 
 
 10.1 
 
 11.2 
 
 22.3 
 
 33.5 
 
 44.7 
 
 55.8 
 
 65 
 
 6.5 , 
 
 7.6 
 
 8.7 
 
 9.8 
 
 10.8 
 
 21.7 
 
 32.5 
 
 43.3 
 54.2 
 
 2 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.3 
 
 0.7 ! 
 
 1.0 
 
 1.3 
 
 1.7 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.24324 
 
 9.24395 
 
 9.24466 
 
 9.24536 
 
 9.24607 
 
 9.25000 
 
 9.25073 
 
 9.25146 
 
 9.25219 
 
 9.25292 
 
 0.75000 
 
 0.74927 
 
 0.74854 
 
 0.74781 
 
 0.74708 
 
 9.99324 
 
 9.99322 
 
 9.99319 
 
 9.99317 
 
 9.99315 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.24677 
 
 9.24748 
 
 9.24818 
 
 9.24888 
 
 9.24958 
 
 9,25365 
 
 9.25437 
 
 9.25510 
 
 9.25582 
 
 9.25655 
 
 0.74635 
 
 0.74563 
 
 0.74490 
 
 0.74418 
 
 0.74345 
 
 9.99313 
 
 9.99310 
 
 9.99308 
 
 9.99306 
 
 9.99304 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.25028 
 
 9.25098 
 
 9.25168 
 
 9.25237 
 
 9.25307 
 
 9.25727 
 
 9.25799 
 
 9.25871 
 
 9.25943 
 
 9.26015 
 
 0.74273 
 
 0.74201 
 
 0.74129 
 
 0.74057 
 
 0.73985 
 
 9.99301 
 
 9.99299 
 
 9.99297 
 
 9.99294 
 
 9.99292 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.25376 
 
 9.25445 
 
 9.25514 
 
 9.25583 
 
 9.25652 
 
 9.26086 
 
 9.26158 
 
 9.26229 
 
 9.26301 
 
 9.26372 
 
 0.73914 
 
 0.73842 
 
 0.73771 
 
 0.73699 
 
 0.73628 
 
 9.99290 
 
 9.99288 
 
 9.99285 
 
 9.99283 
 
 9.99281 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.25721 
 
 9.25790 
 
 9.25858 
 
 9.25927 
 
 9.25995 
 
 9.26443 
 
 9.26514 
 
 9.26585 
 
 9.26655 
 
 9.26726 
 
 0,73557 
 
 0.73486 
 
 0.73415 
 
 0.73345 
 
 0.73274 
 
 9.99278 
 
 9.99276 
 
 9.99274 
 
 9.99271 
 
 9.99269 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.26063 
 
 9.26131 
 
 9.26199 
 
 9.26267 
 
 9.26335 
 
 9.26797 
 
 9.26867 
 
 9.26937 
 
 9.27008 
 
 9.27078 
 
 0.73203 
 
 0.73133 
 
 0.73063 
 
 0.72992 
 
 0.72922 
 
 9.99267 
 
 9.99264 
 
 9.99262 
 
 9.99260 
 
 9.99257 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.26403 
 
 9.26470 
 
 9.26538 
 
 9.26605 
 
 9.26672 
 
 9.27148 
 
 9.27218 
 
 9.27288 
 
 9.27357 
 
 9.27427 
 
 0.72852 
 
 0.72782 
 
 0.72712 
 
 0.72643 
 
 0.72573 
 
 9.99255 
 
 9.99252 
 
 9.99250 
 
 9.99248 
 
 9.99245 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.26739 
 
 9.26806 
 
 9.26873 
 
 9.26940 
 
 9.27007 
 
 9.27496 
 
 9.27566 
 
 9.27635 
 
 9.27704 
 
 9.27773 
 
 0.72504 
 
 0.72434 
 
 0.72365 
 
 0.72296 
 
 0.72227 
 
 9.99243 
 
 9.99241 
 
 9.99238 
 
 9.99236 
 
 9.99233 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.27073 
 
 9.27140 
 
 9.27206 
 
 9.27273 
 
 9.27339 
 
 9.27842 
 
 9.27911 
 
 9.27980 
 
 9.28049 
 
 9.28117 
 
 0.72158 
 
 0.72089 
 
 0.72020 
 
 0.71951 
 
 0.71883 
 
 9.99231 
 
 9.99229 
 
 9.99226 
 
 9.99224 
 
 9.99221 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 9.27405 
 
 9.27471 
 
 9.27537 
 
 9.27602 
 
 9.27668 
 
 9.28186 
 
 9.28254 
 
 9.28323 
 
 9.28391 
 
 9.28459 
 
 0.71814 
 
 0.71746 
 
 0.71677 
 
 0.71609 
 
 0.71541 
 
 9.99219 
 
 9.99217 
 
 9.99214 
 
 9.99212 
 
 9.99209 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 t>5 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.27734 
 
 9.27799 
 
 9.27864 
 
 9.27930 
 
 9.27995 
 
 9.28527 
 
 9.28595 
 
 9.28662 
 
 9.28730 
 
 9.28798 
 
 0.71473 
 
 0.71405 
 
 0.71338 
 
 0.71270 
 
 0.71202 
 
 9.99207 
 
 9.99204 
 
 9.99202 
 
 9.99200 
 
 9.99197 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 9.28060 
 
 9.28865 
 
 0.71135 
 
 9.99195 
 
 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 f 
 
 P. P. 
 
 79 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 11 ° 
 
 503 
 
 / 
 
 L. Sin. 
 
 d. I 
 
 j. Tang. 
 
 d. c. L. Cotg. 
 
 L. Cos. 
 
 
 
 P. 
 
 P. 
 
 o 
 
 9.28060 
 
 
 9.28865 
 
 
 0.71135 
 
 9.99195 
 
 60 
 
 
 
 fif 
 
 t 
 
 6 
 
 1 
 
 9.28125 
 
 65 
 
 9.28933 
 
 68 
 
 0.71007 
 
 9.99192 
 
 59 
 
 6 
 
 6.8 
 
 6 
 
 2 
 
 9.28190 
 
 65 
 
 9 29000 
 
 67 
 
 0.71000 
 
 9.99190 
 
 58 
 
 7 
 
 7.9 
 
 7 
 
 3 
 
 9.28254 
 
 64 
 
 9.29067 
 
 67 
 
 0.70933 
 
 9.99187 
 
 57 
 
 8 
 
 9.1 
 
 8 
 
 4 
 
 9.28319 
 
 65 
 
 9.29134 
 
 67 
 
 0.70866 
 
 9.99185 
 
 56 
 
 9 
 
 10.2 
 
 10 
 
 
 9.28384 
 
 65 
 
 9.29201 
 
 
 0.70799 
 
 9.99182 
 
 55 
 
 10 
 
 11.3 
 
 11 
 
 6 
 
 9.28448 
 
 64 
 
 9.29268 
 
 67 
 
 0.70732 
 
 9.99180 
 
 54 
 
 20 
 
 22.7 
 
 22 
 
 7 
 
 9.28512 
 
 64 
 
 9.29335 
 
 67 
 
 0.70665 
 
 9 99177 
 
 53 
 
 30 
 
 34.0 
 
 33 
 
 8 
 
 9.28577 
 
 65 
 
 9.29402 
 
 67 
 
 0.70598 
 
 9.99175 
 
 52 
 
 40 
 
 45 3 
 
 44 
 
 9 
 
 9.28641 
 
 64 
 
 9.29468 
 
 & , 
 
 0.70532 
 
 9.99172 
 
 51 
 
 50 
 
 56.7 
 
 55 
 
 10 
 
 9.28705 
 
 64 
 
 9.29535 
 
 
 0.70465 
 
 9.99170 
 
 50 
 
 
 
 cc 
 
 
 11 
 
 9.28769 
 
 64 
 
 9.29601 
 
 66 
 
 0.70399 
 
 9.99167 
 
 49 
 
 A 
 
 uu 
 
 A A 
 
 12 
 
 9.28833 
 
 64 
 
 9.29668 
 
 67 
 
 0.70332 
 
 9.99165 
 
 48 
 
 O 
 
 7 
 
 0.0 
 7 7 
 
 13 
 
 9.28896 
 
 63 
 
 9.29734 
 
 66 
 
 0.70266 
 
 9.99162 
 
 47 
 
 i 
 
 Q 
 
 ft ft 
 
 14 
 
 9.28960 
 
 64 
 
 9.29800 
 
 66 
 
 66 - 
 
 0.70200 
 
 9.99160 
 
 46 
 
 O 
 
 9 
 
 0.0 
 
 9.9 
 
 15 
 
 9.29024 
 
 64 
 
 9.29866 
 
 0.70134 
 
 9.99157 
 
 45 
 
 10 
 
 li.o : 
 
 16 
 
 9.29087 
 
 63 
 
 9.29932 
 
 66 
 
 0.70068 
 
 9.99155 
 
 44 
 
 20 
 
 22.0 : 
 
 17 
 
 9.29150 
 
 63 
 
 9.29998 
 
 66 
 
 0.70002 
 
 9.99152 
 
 43 
 
 30 
 
 33.0 ! 
 
 18 
 
 9.29214 
 
 64 
 
 9.30064 
 
 66 
 
 0.69936 
 
 9.99150 
 
 42 
 
 40 
 
 44.0 - 
 
 19 
 
 9.29277 
 
 63 
 
 9.30130 
 
 66 
 
 65 
 
 0.69870 
 
 9.99147 
 
 41 
 
 50 
 
 55.0 1 
 
 20 
 
 9.29340 
 
 bo 
 
 9.30195 
 
 
 0.69805 
 
 9.99145 
 
 40 
 
 
 
 
 
 
 21 
 
 9.29403 
 
 63 
 
 9.30261 
 
 66 
 
 0.69739 
 
 9.99142 
 
 39 
 
 
 
 64 
 
 22 
 
 9.29466 
 
 63 
 
 9.30326 
 
 65 
 
 0.69674 
 
 9.99140 
 
 38 
 
 6 
 
 
 0 .3. : 
 
 M C 
 
 23 
 
 9.29529 
 
 63 
 
 9.30391 
 
 65 
 
 0.69609 
 
 9.99137 
 
 37 
 
 
 
 
 7.5 
 
 24 
 
 9.29591 
 
 62 
 
 9.30457 
 
 66 
 
 65 
 
 0.69543 
 
 9.99135 
 
 36 
 
 9 
 
 
 8.5 j 
 
 Q fi 
 
 25 
 
 9.29654 
 
 60 
 
 9.30522 
 
 
 0.69478 
 
 9.99132 
 
 35 
 
 10 
 
 10.7 j 
 
 26 
 
 9.29716 
 
 62 
 
 9.30587 
 
 65 
 
 0.69413 
 
 9.99130 
 
 34 
 
 20 
 
 21.3 
 
 27 
 
 9.29779 
 
 63 
 
 9.30652 
 
 65 
 
 0.69348 
 
 9.99127 
 
 33 
 
 30 
 
 32.0 
 
 28 
 
 9.29841 
 
 62 
 
 9.30717 
 
 65 
 
 0.69283 
 
 9.99124 
 
 32 
 
 40 
 
 42.7 
 
 29 
 
 9.29903 
 
 62 
 
 no 
 
 9.30782 
 
 65 
 
 64 
 
 0.69218 
 
 9.99122 
 
 31 
 
 50 
 
 53.3 
 
 30 
 
 9.29966 
 
 DO 
 
 9.30846 
 
 vr± 
 
 0.69154 
 
 9.99119 
 
 30 
 
 
 
 
 
 
 31 
 
 9.30028 
 
 62 
 
 9.30911 
 
 65 
 
 0.69089 
 
 9.99117 
 
 29 
 
 
 
 
 62 
 
 32 
 
 9.30090 
 
 62 
 
 9.30975 
 
 64 
 
 0.69025 
 
 9.99114 
 
 28 
 
 
 6 
 
 
 6.2 
 
 33 
 
 9.30151 
 
 61 
 
 9.31040 
 
 65 
 
 0.68960 
 
 9.99112 
 
 27 
 
 
 7 
 
 
 7.2 
 
 34 
 
 9.30213 
 
 62 
 
 9.31104 
 
 64 
 
 64 
 
 0.68896 
 
 9.99109 
 
 26 
 
 
 8 
 
 
 8.3 
 
 35 
 
 9.30275 
 
 62 
 
 9.31168 
 
 Cr± 
 
 /»r 
 
 0.68832 
 
 9.99106 
 
 25 
 
 y 
 
 10 
 
 y.o 
 10 8 
 
 36 
 
 9.30336 
 
 61 
 
 9.31233 
 
 65 
 
 0.68767 
 
 9.99104 
 
 24 
 
 J.U 
 
 90 
 
 1U.O 
 
 90 7 
 
 37 
 
 9.30398 
 
 62 
 
 9.31297 
 
 64 
 
 0.68703 
 
 9.99101 
 
 23 
 
 ZU 
 
 Qfk 
 
 ZU. 1 
 
 31.0 
 
 38 
 
 9.30459 
 
 61 
 
 9.31361 
 
 64 
 
 0.68639 
 
 9.99099 
 
 22 
 
 OU 
 
 40 
 
 41.3 
 
 39 
 
 9.30521 
 
 62 
 
 61 
 
 9.31425 
 
 64 
 
 64 
 
 0.68575 
 
 9.99096 
 
 21 
 
 1U 
 
 50 
 
 51.7 
 
 40 
 
 9.30582 
 
 9.31489 
 
 
 0.68511 
 
 9.99093 
 
 20 
 
 
 
 
 
 
 41 
 
 9.30643 
 
 61 
 
 9.31552 
 
 63 
 
 0.68448 
 
 9.99091 
 
 19 
 
 
 
 
 60 
 
 42 
 
 9.30704 
 
 61 
 
 9.31616 
 
 64 
 
 0.68384 
 
 9.99088 
 
 18 
 
 
 6 
 
 
 6.0 
 
 43 
 
 9.30765 
 
 61 
 
 9.31679 
 
 63 
 
 0.68321 
 
 9.99086 
 
 17 
 
 
 7 
 
 
 7.0 
 
 44 
 
 9.30826 
 
 61 
 
 9.31743 
 
 64 
 
 63 
 
 0.68257 
 
 9.99083 
 
 16 
 
 
 8 
 
 
 8.0 
 
 45 
 
 9.30887 
 
 61 
 
 9.31806 
 
 0.68194 
 
 9.99080 
 
 15 
 
 
 9 
 
 
 9.0 
 
 46 
 
 9.30947 
 
 60 
 
 9.31870 
 
 64 
 
 0.68130 
 
 9.99078 
 
 14 
 
 1U 
 
 1U.U 
 
 47 
 
 9.31008 
 
 61 
 
 9.31933 
 
 63 
 
 0.68067 
 
 9.99075 
 
 13 
 
 20 
 
 20.0 
 
 48 
 
 9.31068 
 
 60 
 
 9.31996 
 
 63 
 
 0.68004 
 
 9.99072 
 
 12 
 
 30 
 
 30.0 
 
 49 
 
 9.31129 
 
 61 
 
 V»A 
 
 9.32059 
 
 63 
 
 63 
 
 0.67941 
 
 9.99070 
 
 11 
 
 40 
 
 *0 
 
 40.0 
 
 fiO 0 
 
 50 
 
 9.31189 
 
 oU 
 
 9.32122 
 
 0.67878 
 
 9.99067 
 
 10 
 
 
 
 
 
 
 51 
 
 9.31250 
 
 61 
 
 9.32185 
 
 63 
 
 0.67815 
 
 9.99064 
 
 9 
 
 
 
 
 
 3 
 
 52 
 
 9.31310 
 
 60 
 
 9.32248 
 
 63 
 
 0.67752 
 
 9.99062 
 
 8 
 
 
 < 
 
 5 
 
 0.3 
 
 53 
 
 9.31370 
 
 60 
 
 9.32311 
 
 63 
 
 0.67689 
 
 9.99059 
 
 7 
 
 
 1 
 
 7 
 
 0.4 
 
 54 
 
 9.31430 
 
 60 
 
 AH 
 
 9.32373 
 
 62 
 
 63 
 
 0.67627 
 
 9.99056 
 
 6 
 
 
 8 
 
 0.4 
 
 55 
 
 9.31490 
 
 bU 
 
 9.32436 
 
 
 0.67564 
 
 9.99054 
 
 5 
 
 
 . ( 
 
 3 
 
 0.5 
 
 56 
 
 9.31549 
 
 59 
 
 9.32498 
 
 62 
 
 0.67502 
 
 : 9.99051 
 
 4 
 
 
 10 
 
 0.5 
 
 57 
 
 9.31609 
 
 60 
 
 9.32561 
 
 63 
 
 0.67439 
 
 9.99048 
 
 3 
 
 
 20 
 
 1.0 
 
 58 
 
 9.31669 
 
 60 
 
 9.32623 
 
 62 
 
 0.67377 
 
 9.99046 
 
 i 2 
 
 
 30 
 
 1.5 
 
 1 59 
 
 9.31728 
 
 59 
 
 ' AH 
 
 9.32685 
 
 62 
 
 62 
 
 0.67315 
 
 i 9.99043 
 
 1 
 
 
 40 
 
 2.0 
 
 9 
 
 60 
 
 9.31788 
 
 r 60 
 
 9.32747 
 
 
 0.67253 
 
 ; 9.99040 
 
 i 0 
 
 
 OU 
 
 Z.D 
 
 I 
 
 L. Cos 
 
 d. 
 
 L. Cotg 
 
 d. c. 
 
 L.Tang 
 
 L. Sin. 
 
 t 
 
 
 
 
 P 
 
 . P 
 
 65 
 
 6.5 
 
 7.6 
 
 8.7 
 
 9.8 
 
 63 
 
 6.3 
 
 7.4 
 
 8.4 
 
 9.5 
 
 10.5 
 
 21.0 
 
 31.5 
 42.0 
 
 52.5 
 
 6i 
 
 6.1 
 
 7.1 
 
 8.1 
 
 9.2 
 
 10.2 
 
 20.3 
 
 30.5 
 
 40.7 
 
 50.8 
 
 59 
 
 5.9 
 
 6.9 
 
 7.9 
 
 8.9 
 9.8 
 
 19.7 
 
 29.5 
 
 39.3 
 
 49.2 
 
 2 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.3 
 
 0.7 
 
 1.0 
 
 1.3 
 
 1.7 
 
 78 ° 
 
504 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 12 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 P. P. 
 
 0 
 
 9.31788 
 
 
 9.32747 
 
 
 0.67253 
 
 9.99040 
 
 60 
 
 
 
 
 
 
 1 
 
 9.31847 
 
 59 
 
 9.32810 
 
 63 
 
 0.67190 
 
 9.99038 
 
 59 
 
 
 
 DO 
 
 62 
 
 2 
 
 9.31907 
 
 60 
 
 9.32872 
 
 62 
 
 0.67128 
 
 9.99035 
 
 58 
 
 6 
 
 
 6.3 
 
 ff A 
 
 6.2 
 
 3 
 
 9.31966 
 
 59 
 
 9.32933 
 
 61 
 
 0.67067 
 
 9.99032 
 
 57 
 
 7 
 
 
 7.4 
 
 7.2 j 
 
 4 
 
 9.32025 
 
 59 
 
 59 
 
 9.32995 
 
 62 
 
 62 
 
 0.67005 
 
 9.99030 
 
 56 
 
 8 
 
 9 
 
 
 8.4 
 
 9.5 
 
 8.3 
 
 9.3 ' 
 
 5 
 
 9.32084 
 
 9.33057 
 
 0.66943 
 
 9.99027 
 
 55 
 
 10 
 
 
 10*5 
 
 io!3 
 
 6 
 
 9.32143 
 
 59 
 
 9.33119 
 
 62 
 
 0.66881 
 
 9.99024 
 
 54 
 
 20 
 
 21.0 
 
 20.7 
 
 7 
 
 9.32202 
 
 59 
 
 9.33180 
 
 61 
 
 0.66820 
 
 9.99022 
 
 53 
 
 30 
 
 31.5 
 
 31.0 
 
 8 
 
 9.32261 
 
 59 
 
 9.33242 
 
 62 
 
 0.66758 
 
 9.99019 
 
 52 
 
 40 
 
 42.0 
 
 41.3 
 
 9 
 
 9.32319 
 
 58 
 
 9.33303 
 
 61 
 
 62 
 
 0.66697 
 
 9.99016 
 
 51 
 
 50 
 
 52.5 
 
 51.7 
 
 10 
 
 9.32378 
 
 
 9.33365 
 
 0.66635 
 
 9.99013 
 
 50 
 
 
 
 
 
 
 11 
 
 9.32437 
 
 59 
 
 9.33426 
 
 61 
 
 0.66574 
 
 9.99011 
 
 49 
 
 
 
 61 
 
 60 
 
 12 
 
 9.32495 
 
 58 
 
 9.33487 
 
 61 
 
 0.66513 
 
 9.99008 
 
 48 
 
 6 
 
 
 6.1 
 
 6.0 
 
 13 
 
 9.32553 
 
 58 
 
 9.33548 
 
 61 
 
 0.66452 
 
 9.99005 
 
 47 
 
 7 
 
 
 7.1 
 
 7.0 
 
 14 
 
 9.32612 
 
 59 
 
 58 
 
 9.33609 
 
 61 
 
 61 
 
 0.66391 
 
 9.99002 
 
 46 
 
 8 
 
 Q 
 
 
 8.1 
 Q O 
 
 8.0 
 . Q A 
 
 15 
 
 9.32670 
 
 9.33670 
 
 0.66330 
 
 9.99000 
 
 45 
 
 y 
 
 10 
 
 
 y.z 
 
 10.2 
 
 y.u 
 
 10.0 
 
 16 
 
 9.32728 
 
 58 
 
 9.33731 
 
 61 
 
 0.66269 
 
 9.98997 
 
 44 
 
 20 
 
 
 on q 
 
 20.0 
 
 17 
 
 9.32786 
 
 58 
 
 9.33792 
 
 61 
 
 0.66208 
 
 9.98994 
 
 43 
 
 30 
 
 so 5 
 
 30.0 
 
 18 
 
 9.32844 
 
 58 
 
 9.33853 
 
 61 
 
 0.66147 
 
 9.98991 
 
 42 
 
 40 
 
 
 407 
 
 4o!o 
 
 19 
 
 9.32902 
 
 58 
 
 9.33913 
 
 60 
 
 61 
 
 0.66087 
 
 9.98989 
 
 41 
 
 50 
 
 50^8 
 
 5o!o 
 
 20 
 
 9.32960 
 
 
 9.33974 
 
 0.66026 
 
 9.98986 
 
 40 
 
 
 
 
 
 
 21 
 
 9.33018 
 
 58 
 
 9.34034 
 
 60 
 
 0.65966 
 
 9.98983 
 
 39 
 
 
 
 
 59 
 
 22 
 
 9.33075 
 
 57 
 
 9.34095 
 
 61 
 
 0.65905 
 
 9.98980 
 
 38 
 
 
 
 6 
 
 5.9 
 
 23 
 
 9.33133 
 
 58 
 
 9.34155 
 
 60 
 
 0.65845 
 
 9.98978 
 
 37 
 
 
 
 7 
 
 6.9 
 
 24 
 
 9.33190 
 
 57 
 
 9.34215 
 
 60 
 
 61 
 
 0.65785 
 
 9.98975 
 
 36 
 
 
 8 
 
 7.9 
 
 25 
 
 9.33248 
 
 Oo 
 
 9.34276 
 
 0.65724 
 
 9.98972 
 
 35 
 
 
 9 
 
 1 A 
 
 8.9 
 
 Q Q 
 
 26 
 
 9.33305 
 
 57 
 
 9.34336 
 
 60 
 
 0.65664 
 
 9.98969 
 
 34 
 
 
 10 
 
 OA 
 
 y.s 
 
 in 7 
 
 27 
 
 9.33362 
 
 57 
 
 9.34396 
 
 60 
 
 0.65604 
 
 9.98967 
 
 33 
 
 
 20 
 
 QA 
 
 iy.7 
 
 on 
 
 28 
 
 9.33420 
 
 58 
 
 9.34456 
 
 60 
 
 0.65544 
 
 9.98964 
 
 32 
 
 
 30 
 
 zy.o 
 
 on o 
 
 29 
 
 9.33477 
 
 57 
 
 9.34516 
 
 60 
 
 60 
 
 0.65484 
 
 9.98961 
 
 31 
 
 
 40 
 
 Aft 
 
 i oy .6 
 1 4Q 9 
 
 30 
 
 9.33534 
 
 O/ 
 
 9.34576 
 
 0.65424 
 
 9.98958 
 
 30 
 
 
 
 
 
 
 31 
 
 9.33591 
 
 57 
 
 9.34635 
 
 59 
 
 0.65365 
 
 9.98955 
 
 29 
 
 
 
 58 
 
 57 
 
 32 
 
 9.33647 
 
 56 
 
 9.34695 
 
 60 
 
 0.65305 
 
 9.98953 
 
 28 
 
 6 
 
 
 5.8 
 
 5.7 
 
 33 
 
 9.33704 
 
 57 
 
 9.34755 
 
 60 
 
 0.65245 
 
 9.98950 
 
 27 
 
 7 
 
 
 6.8 
 
 6.7 
 
 34 
 
 9.33761 
 
 57 
 
 57 
 
 9.34814 
 
 59 
 
 60 
 
 0.65186 
 
 9.98947 
 
 26 
 
 8 
 
 
 7.7 
 
 7.6 
 
 35 
 
 9.33818 
 
 9.34874 
 
 0.65126 
 
 9.98944 
 
 25 
 
 9 
 
 
 8.7 
 
 8.6 
 
 36 
 
 9.33874 
 
 56 
 
 9.34933 
 
 59 
 
 0.65067 
 
 9.98941 
 
 24 
 
 10 
 
 
 9.7 
 
 9.5 
 
 37 
 
 9.33931 
 
 57 
 
 9.34992 
 
 59 
 
 0.65008 
 
 9.98938 
 
 23 
 
 20 
 
 
 19.3 
 
 19.0 
 
 38 
 
 9.33987 
 
 56 
 
 9.35051 
 
 59 
 
 0.64949 
 
 9.98936 
 
 22 
 
 30 
 
 
 29.0 
 
 28.5 
 
 39 
 
 9.34043 
 
 56 
 
 9.35111 
 
 60 
 
 fSQ 
 
 0.64889 
 
 9.98933 
 
 21 
 
 40 
 
 50 
 
 
 38.7 
 
 AQ Q 
 
 38.0 
 
 47.5 
 
 40 
 
 9.34100 
 
 0/ 
 
 9.35170 
 
 V** 
 
 0.64830 
 
 9.98930 
 
 20 
 
 
 
 
 41 
 
 9.34156 
 
 56 
 
 9.35229 
 
 59 
 
 0.64771 
 
 9.98927 
 
 19 
 
 
 
 56 
 
 55 
 
 42 
 
 9.34212 
 
 56 
 
 9.35288 
 
 59 
 
 0.64712 
 
 9.98924 
 
 18 
 
 6 
 
 
 5.6 
 
 5.5 
 
 43 
 
 9.34268 
 
 56 
 
 9.35347 
 
 59 
 
 0.64653 
 
 9.98921 
 
 17 
 
 7 
 
 
 6.5 
 
 6.4 
 
 44 
 
 9.34324 
 
 56 
 
 56 
 
 9.35405 
 
 58 
 
 59 
 
 0.64595 
 
 9.98919 
 
 16 
 
 8 
 
 
 7.5 
 
 7.3 
 
 45 
 
 9.34380 
 
 9.35464 
 
 0.64536 
 
 9.98916 
 
 15 
 
 9 
 
 
 8.4 
 
 8.3 
 
 46 
 
 9.34436 
 
 56 
 
 9.35523 
 
 59 
 
 0.64477 
 
 9.98913 
 
 14 
 
 10 
 
 
 9.3 
 
 9.2 
 
 47 
 
 9.34491 
 
 55 
 
 9.35581 
 
 58 
 
 0.64419 
 
 9.98910 
 
 13 
 
 20 
 
 
 18.7 
 
 .18.3 
 
 48 
 
 9.34547 
 
 56 
 
 9.35640 
 
 59 
 
 0.64360 
 
 9.98907 
 
 12 
 
 30 
 
 
 28.0 
 
 27.5 
 
 49 
 
 9.34602 
 
 55 
 
 9.35698 
 
 58 
 
 0.64302 
 
 9.98904 
 
 11 
 
 40 
 
 
 37.3 
 
 36.7 
 
 
 
 
 
 59 
 
 
 
 
 50 
 
 
 
 
 45.8 
 
 50 
 
 9.34658 
 
 oo 
 
 9.35757 
 
 0.64243 
 
 9.98901 
 
 10 
 
 
 4:0. t 
 
 51 
 
 9.34713 
 
 55 
 
 9.35815 
 
 58 
 
 0.64185 
 
 9.98898 
 
 9 
 
 
 
 
 3 
 
 2 
 
 52 
 
 9.34769 
 
 56 
 
 9.35873 
 
 58 
 
 0.64127 
 
 9.98896 
 
 8 
 
 a 
 
 0.3 
 
 0.2 
 
 53 
 
 9.34824 
 
 55 
 
 9.35931 
 
 58 
 
 0.64069 
 
 9.98893 
 
 7 
 
 7 
 
 
 0.4 
 
 0.2 
 
 54 
 
 9.34879 
 
 55 
 
 55 
 
 9.35989 
 
 58 
 
 58 
 
 0.64011 
 
 9.98890 
 
 6 
 
 8 
 
 0.4 
 
 0.3 
 
 55 
 
 9.34934 
 
 9.36047 
 
 0.63953 
 
 9.98887 
 
 5 
 
 9 
 
 0.5 
 
 0.3 
 
 56 
 
 9.34989 
 
 55 
 
 9.36105 
 
 58 
 
 0.63895 
 
 9.98884 
 
 4 
 
 10 
 
 0.5 
 
 0.3 
 
 57 
 
 9.35044 
 
 55 
 
 9.36163 
 
 58 
 
 0.63837 
 
 9.98881 
 
 3 
 
 20 
 
 1.0 
 
 0.7 
 
 58 
 
 9.35099 
 
 55 
 
 9.36221 
 
 58 ■ 
 
 0.63779 
 
 9.98878 
 
 2 
 
 30 
 
 1.5 
 
 1.0 
 
 59 
 
 9.35154 
 
 55 
 
 9.36279 
 
 58 
 
 0.63721 
 
 9.98875 
 
 1 
 
 40 
 
 2.0 
 
 1.3 
 
 
 
 55 
 
 
 57 
 
 
 
 
 
 
 
 
 1.7 
 
 60 
 
 9.35209 
 
 9.36336 
 
 0.63664 
 
 9.98872 
 
 0 
 
 i ou 
 
 A. 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 t 
 
 P. P. 
 
 77 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 13 ° 
 
 505 
 
 r 
 
 L. Sin. 
 
 0 
 
 9.35209 
 
 1 
 
 9.35263 
 
 2 
 
 9.35318 
 
 3 
 
 9.35373 
 
 4 
 
 9.35427 
 
 5 
 
 9.35481 
 
 6 
 
 9.35536 
 
 7 
 
 9.35590 
 
 8 
 
 9.35644 
 
 9 
 
 9.35698 
 
 10 
 
 9.35752 
 
 11 
 
 9.35806 
 
 12 
 
 9.35860 
 
 13 
 
 9.35914 
 
 14 
 
 9.35968 
 
 15 
 
 9.36022 
 
 16 
 
 9.36075 
 
 17 
 
 9.36129 
 
 18 
 
 9.36182 
 
 19 
 
 9.36236 
 
 20 
 
 9.36289 
 
 21 
 
 9.36342 
 
 22 
 
 9.36395 
 
 23 
 
 9.36449 
 
 24 
 
 9.36502 
 
 25 
 
 9.36555 
 
 26 
 
 9.36608 
 
 27 
 
 9.36660 
 
 28 
 
 9.36713 
 
 29 
 
 9.36766 
 
 30 
 
 9.36819 
 
 31 
 
 9.36871 
 
 32 
 
 9.36924 
 
 33 
 
 9.36976 
 
 34 
 
 9.37028 
 
 35 
 
 9.37081 
 
 36 
 
 9.37133 
 
 37 
 
 9.37185 
 
 38 
 
 9.37237 
 
 39 
 
 9.37289 
 
 40 
 
 9.37341 
 
 41 
 
 9.37393 
 
 42 
 
 9.37445 
 
 43 
 
 9.37497 
 
 44 
 
 9.37549 
 
 45 
 
 9.37600 
 
 46 
 
 9.37652 
 
 47 
 
 9.37703 
 
 48 
 
 9.37755 
 
 49 
 
 9.37806 
 
 50 
 
 9.37858 
 
 51 
 
 9.37909 
 
 52 
 
 9.37960 
 
 53 
 
 9.38011 
 
 54 
 
 9.38062 
 
 55 
 
 9.38113 
 
 56 
 
 9.38164 
 
 57 
 
 9.38215 
 
 58 
 
 9.38266 
 
 59 
 
 9.38317 
 
 60 
 
 9.38368 
 
 
 L. Cos. 
 
 d. 
 
 54 
 
 55 
 55 
 54 
 
 54 
 
 55 
 54 
 54 
 54 
 54 
 54 
 54 
 54 
 54 
 54 
 
 53 
 
 54 
 
 53 
 
 54 
 53 
 53 
 
 53 
 
 54 
 53 
 53 
 53 
 
 52 
 
 53 
 53 
 53 
 
 52 
 
 53 
 52 
 
 52 
 
 53 
 52 
 52 
 52 
 52 
 52 
 52 
 52 
 52 
 52 
 
 51 
 
 52 
 
 51 
 
 52 
 
 51 
 
 52 
 51 
 51 
 51 
 51 
 51 
 51 
 51 
 51 
 51 
 
 d. 
 
 L.Tang. 6 
 
 9.36336 
 
 9.36394 
 
 9.36452 
 
 9.36509 
 
 9.36566 
 
 9.36624 
 
 9.36681 
 
 9.36738 
 
 9.36795 
 
 9.36852 
 
 9.36909 
 
 9.36966 
 
 9.37023 
 
 9.37080 
 
 9.37137 
 
 9.37193 
 
 9.37250 
 
 9.37306 
 
 9.37363 
 
 9.37419 
 
 9.37476 
 
 9.37532 
 
 9.37588 
 
 9.37644 
 
 9.37700 
 
 9.37756 
 
 9.37812 
 
 9.37868 
 
 9.37924 
 
 9.37980 
 
 9.38035 
 
 9.38091 
 
 9.38147 
 
 9.38202 
 
 9.38257 
 
 9.38313 
 
 9.38368 
 
 9.38423 
 
 9.38479 
 
 9.38534 
 
 9.38589 
 
 9.38644 
 
 9.38699 
 
 9.38754 
 
 9.38808 
 
 9.38863 
 
 9.38918 
 
 9.38972 
 
 9.39027 
 
 9.39082 
 
 9.39136 
 
 9.39190 
 
 9.39245 
 
 9.39299 
 
 9.3935a 
 
 9.39407 
 
 9.39461 
 
 9.39515 
 
 9.39569 
 
 9.39623 
 
 9.39677 
 
 L. Cotg. 
 
 d .c. 
 
 58 
 
 58 
 
 57 
 
 57 
 
 58 
 57 
 57 
 57 
 57 
 57 
 57 
 57 
 57 
 57 
 
 56 
 
 57 
 
 56 
 
 57 
 
 56 
 
 57 
 56 
 56 
 56 
 56 
 56 
 56 
 56 
 56 
 56 
 
 55 
 
 56 
 56 
 55 
 
 55 
 
 56 
 55 
 
 55 
 
 56 
 55 
 55 
 55 
 55 
 55 
 
 54 
 
 55 
 55 
 
 54 
 
 55 
 55 
 54 
 
 54 
 
 55 
 54 
 54 
 54 
 54 
 54 
 54 
 54 
 
 L. Cotg. 
 
 L. Cos. 
 
 
 0.63004 
 
 9.98872 
 
 60 
 
 0.63606 
 
 9.98869 
 
 59 
 
 0.63548 
 
 9.98867 
 
 58 
 
 0.63491 
 
 9.98864 
 
 57 
 
 0.63434 
 
 9.98861 
 
 56 
 
 0.63376 
 
 9.98858 
 
 55 
 
 0.63319 
 
 9.98855 
 
 54 
 
 0.63262 
 
 9.98852 
 
 53 
 
 0.63205 
 
 9.98849 
 
 52 
 
 0.63148 
 
 9.98846 
 
 51 
 
 0.63091 
 
 9.98843 
 
 50 
 
 0.63034 
 
 9.98840 
 
 49 
 
 0.62977 
 
 9.98837 
 
 48 
 
 0.62920 
 
 9.98834 
 
 47 
 
 0.62863 
 
 9.98831 
 
 46 
 
 0.62807 
 
 9.98828 
 
 45 
 
 0.62750 
 
 9.98825 
 
 44 
 
 0.62694 
 
 9.98822 
 
 43 
 
 0.62637 
 
 9.98819 
 
 42 
 
 0.62581 
 
 9.98816 
 
 41 
 
 0.62524 
 
 9.98813 
 
 40 
 
 0.62468 
 
 9.98810 
 
 39 
 
 0.62412 
 
 9.98807 
 
 38 
 
 0.62356 
 
 9.98804 
 
 37 
 
 0.62300 
 
 9.98801 
 
 36 
 
 0.62244 
 
 9.98798 
 
 35 
 
 0.62188 
 
 9.98795 
 
 34 
 
 0.62132 
 
 9.98792 
 
 33 
 
 0.62076 
 
 9.98789 
 
 32 
 
 0.62020 
 
 9.98786 
 
 31 
 
 0.61965 
 
 9.98783 
 
 30 
 
 0.61909 
 
 9.98780 
 
 29 
 
 0.61853 
 
 9.98777 
 
 28 
 
 0.61798 
 
 9.98774 
 
 27 
 
 0.61743 
 
 9.98771 
 
 26 
 
 0.61687 
 
 9.98768 
 
 25 
 
 0.61632 
 
 9.98765 
 
 24 
 
 0.61577 
 
 9.98762 
 
 23 
 
 0.61521 
 
 9.98759 
 
 22 
 
 0.61466 
 
 9.98756 
 
 21 
 
 0.61411 
 
 9.98753 
 
 20 
 
 0.61356 
 
 9.98750 
 
 19 
 
 0.61301 
 
 9.98746 
 
 18 
 
 0.61246 
 
 9.98743 
 
 17 
 
 0.61192 
 
 9.98740 
 
 16 
 
 0.61137 
 
 9.98737 
 
 15 
 
 0.61082 
 
 9.98734 
 
 14 
 
 0.61028 
 
 9.98731 
 
 13 
 
 0.60973 
 
 9.98728 
 
 12 
 
 0.60918 
 
 9.98725 
 
 11 
 
 0.60864 
 
 9.98722 
 
 10 
 
 0.60810 
 
 9.98719 
 
 9 
 
 0.60755 
 
 i 9.98715 
 
 8 
 
 0.60701 
 
 9.98712 
 
 7 
 
 0.60647 
 
 ’ 9.98709 
 
 6 
 
 0.60593 
 
 1 9.98706 
 
 5 
 
 0.6053S 
 
 1 9.98703 
 
 : 4 
 
 0.60485 
 
 1 9.98700 
 
 i 3 
 
 0.60431 
 
 L 9.98697 
 
 ' 2 
 
 0.60377 
 
 t 9.98694 
 
 l 1 
 
 0.6032; 
 
 1 9.9869C 
 
 ) 0 
 
 3. L.Tang 
 
 r. L. Sin, 
 
 / 
 
 P. P. 
 
 58 
 
 5.8 
 
 6.8 
 
 7.7 
 
 8.7 
 
 9.7 
 
 19.3 
 
 29.0 
 
 38.7 
 
 48.3 
 
 56 
 
 5.6 
 
 6.5 
 
 7.5 
 8.4 
 9.3 
 
 18.7 
 
 28.0 
 
 37.3 
 
 46.7 
 
 57 
 
 5.7 
 
 6.7 
 
 7.6 
 
 8.6 
 
 9.5 
 
 19.0 
 
 28.5 
 
 38.0 
 
 47.5 
 
 55 
 
 5.5 
 6.4 
 
 7.3 
 
 8.3 
 9.2 
 
 18.3 
 
 27.5 
 
 36.7 
 
 45.8 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 54 
 
 5.4 
 
 6.3 
 
 7.2 
 
 8.1 
 
 9.0 
 
 18.0 
 
 27.0 
 
 36.0 
 
 45.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 53 
 
 5.3 
 
 6.2 
 
 7.1 
 8.0 
 8.8 
 
 17.7 
 
 26.5 
 35.3 
 44.2 
 
 51 
 
 5.1 
 6.0 
 6.8 
 7.7 
 
 8.5 
 
 17.0 
 
 25.5 
 
 34.0 
 
 42.5 
 
 3 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 1.0 
 
 1.5 
 
 2.0 
 
 2.5 
 
 52 
 
 5.2 
 
 6.1 
 
 6.9 
 
 7.8 
 
 8.7 
 
 17.3 
 26.0 
 34.7 
 
 43.3 
 
 4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.3 
 
 2.0 
 
 2.7 
 
 3.3 
 
 2 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.3 
 
 0.7 
 
 1.0 
 
 1.3 
 
 1.7 
 
 P. P. 
 
 76 
 
506 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 14 ° 
 
 t 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 
 P. P 
 
 
 0 
 
 9.38368 
 
 
 9.39677 
 
 
 0.60323 
 
 9.98690 
 
 
 60 
 
 
 
 
 1 
 
 9.38418 
 
 50 
 
 9.39731 
 
 54 
 
 0.60269 
 
 9.98687 
 
 3 
 
 59 
 
 
 
 
 2 
 
 9.38469 
 
 51 
 
 9.39785 
 
 54 
 
 0.60215 
 
 9.98684 
 
 3 
 
 58 
 
 
 
 
 3 
 
 9.38519 
 
 50 
 
 9.39838 
 
 53 
 
 0.60162 
 
 9.98681 
 
 3 
 
 57 
 
 
 54 
 
 53 
 
 4 
 
 9.38570 
 
 51 
 
 50 
 
 9.39892 
 
 54 
 
 53 
 
 0.60108 
 
 9.98678 
 
 3 
 
 q 
 
 56 
 
 6 
 
 7 
 
 5.4 
 
 6.3 
 
 5.3 
 
 fi 9 
 
 5 
 
 9.38620 
 
 9.39945 
 
 0.60055 
 
 9.98675 
 
 6 
 
 55 
 
 8 
 
 in 
 
 7.1 
 
 6 
 
 9.38670 
 
 50 
 
 9.39999 
 
 54 
 
 0.60001 
 
 9.98671 
 
 4 
 
 54 
 
 9 
 
 8.1 
 
 8*0 
 
 7 
 
 9.38721 
 
 51 
 
 9.40052 
 
 53 
 
 0.59948 
 
 9.98668 
 
 3 
 
 53 
 
 10 
 
 9.0 
 
 8.8 
 
 8 
 
 9.38771 
 
 50 
 
 9.40106 
 
 54 
 
 0.59894 
 
 9.98665 
 
 3 
 
 52 
 
 20 
 
 18.0 
 
 ii .7 
 
 9 
 
 9.38821 
 
 50 
 
 50 
 
 9.40159 
 
 53 
 
 53 
 
 0.59841 
 
 9.98662 
 
 3 
 
 q 
 
 51 
 
 30 
 
 27.0 
 
 26.5 
 
 10 
 
 9.38871 
 
 9.40212 
 
 0.59788 
 
 9.98659 
 
 o 
 
 50 
 
 40 
 
 36.0 
 
 35.3 
 
 11 
 
 9.38921 
 
 50 
 
 9.40266 
 
 54 
 
 0.59734 
 
 9.98656 
 
 3 
 
 49 
 
 50 
 
 45.0 
 
 44.2 
 
 12 
 
 9.38971 
 
 50 
 
 9.40319 
 
 53 
 
 0.59681 
 
 9.98652 
 
 4 
 
 48 
 
 
 
 
 13 
 
 9.39021 
 
 50 
 
 9.40372 
 
 53 
 
 0.59628 
 
 9.98649 
 
 3 
 
 47 
 
 
 
 
 14 
 
 9.39071 
 
 50 
 
 50 
 
 9.40425 
 
 53 
 
 53 
 
 0.59575 
 
 9.98646 
 
 3 
 
 q 
 
 46 
 
 
 52 
 
 51 
 
 15 
 
 9.39121 
 
 9.40478 
 
 0.59522 
 
 9.98643 
 
 6 
 
 45 
 
 6 
 
 5.2 
 
 5.1 
 
 16 
 
 9.39170 
 
 49 
 
 9.40531 
 
 53 
 
 0.59469 
 
 9.98640 
 
 3 
 
 44 
 
 7 
 
 6.1 
 
 6.0 
 
 17 
 
 9.39220 
 
 50 
 
 9.40584 
 
 53 
 
 0.59416 
 
 9.98636 
 
 4 
 
 43 
 
 8 
 
 6.9 
 
 6.8 
 
 18 
 
 9.39270 
 
 50 
 
 9.40636 
 
 52 
 
 0.59364 
 
 9.98633 
 
 3 
 
 42 
 
 9 
 
 7.8 
 
 7.7 
 
 19 
 
 9.39319 
 
 49 
 
 50 
 
 9.40689 
 
 53 
 
 53 
 
 0.59311 
 
 9.98630 
 
 3 
 
 q 
 
 41 
 
 10 
 
 8.7 
 
 8.5 
 
 20 
 
 9.39369 
 
 9.40742 
 
 0.59258 
 
 9.98627 
 
 o 
 
 40 
 
 20 
 
 17.3 
 
 17.0 
 
 21 
 
 9.39418 
 
 49 
 
 9.40795 
 
 53 
 
 0.59205 
 
 9.98623 
 
 4 
 
 39 
 
 30 
 
 26.0 
 
 25.5 
 
 22 
 
 9.39467 
 
 49 
 
 9.40847 
 
 52 
 
 0.59153 
 
 9.98620 
 
 3 
 
 38 
 
 40 
 
 34.7 
 
 34.0 
 
 23 
 
 9.39517 
 
 50 
 
 9.40900 
 
 53 
 
 0.59100 
 
 9.98617 
 
 3 
 
 37 
 
 50 
 
 43.3 
 
 42.5 
 
 24 
 
 9.39566 
 
 49 
 
 9.40952 
 
 52 
 
 0.59048 
 
 9.98614 
 
 3 
 
 36 
 
 
 
 
 
 
 49 
 
 
 53 
 
 
 
 A 
 
 
 
 
 
 25 
 
 9.39615 
 
 9.41005 
 
 0.58995 
 
 9.98610 
 
 
 35 
 
 
 
 
 26 
 
 9.39664 
 
 49 
 
 9.41057 
 
 52 
 
 0.58943 
 
 9.98607 
 
 3 
 
 34 
 
 
 50 
 
 49 
 
 27 
 
 9.39713 
 
 49 
 
 9.41109 
 
 52 
 
 0.58891 
 
 9.98604 
 
 3 
 
 33 
 
 6 
 
 5.0 
 
 4.9 
 
 28 
 
 9.39762 
 
 49 
 
 9.41161 
 
 52 
 
 0.58839 
 
 9.98601 
 
 3 
 
 32 
 
 7 
 
 5.8 
 
 5.7 
 
 29 
 
 9.39811 
 
 49 
 
 49 
 
 9.41214 
 
 53 
 
 52 
 
 0.58786 
 
 9.98597 
 
 4 
 
 q 
 
 31 
 
 8 
 
 6.7 
 
 6.5 
 
 30 
 
 9.39860 
 
 9.41266 
 
 0.58734 
 
 9.98594 
 
 o 
 
 30 
 
 9 
 
 7 .5 
 
 7.4 
 
 31 
 
 9.39909 
 
 49 
 
 9.41318 
 
 52 
 
 0.58682 
 
 9.98591 
 
 3 
 
 29 
 
 10 
 
 8.3 
 
 8.2 
 
 32 
 
 9.39958 
 
 49 
 
 9.41370 
 
 52 
 
 0.58630 
 
 9.98588 
 
 3 
 
 28 
 
 20 
 
 16.7 
 
 16.3 
 
 33 
 
 9.40006 
 
 48 
 
 9.41422 
 
 52 
 
 0.58578 
 
 9.98584 
 
 4 
 
 27 
 
 30 
 
 25.0 
 
 24.5 
 
 34 
 
 9.40055 
 
 49 
 
 48 
 
 9.41474 
 
 52 
 
 52 
 
 0.58526 
 
 9.98581 
 
 3 
 
 q 
 
 26 
 
 40 
 
 50 
 
 
 66.6 
 
 41.7 
 
 32.7 
 
 40.8 
 
 35 
 
 9.40103 
 
 9.41526 
 
 0.58474 
 
 9.98578 
 
 o 
 
 25 
 
 
 36 
 
 9.40152 
 
 49 
 
 9.41578 
 
 52 
 
 0.58422 
 
 9.98574 
 
 4 
 
 24 
 
 
 
 
 37 
 
 9.40200 
 
 48 
 
 9.41629 
 
 51 
 
 0.58371 
 
 9.98571 
 
 3 
 
 23 
 
 
 
 
 38 
 
 9.40249 
 
 49 
 
 9.41681 
 
 52 
 
 0.58319 
 
 9.98568 
 
 3 
 
 22 
 
 
 48 
 
 47 
 
 39 
 
 9.40297 
 
 48 
 
 9.41733 
 
 52 
 
 0.58267 
 
 9.98565 
 
 3 
 
 21 
 
 6 
 
 4.8 
 
 4.7 
 
 
 
 49 
 
 
 51 
 
 
 
 A 
 
 
 I *7 
 
 
 E K 
 
 40 
 
 9.40346 
 
 9.41784 
 
 0.58216 
 
 9.98561 
 
 
 20 
 
 / 
 
 Q 
 
 0.0 
 ft d 
 
 0.0 
 
 41 
 
 9.40394 
 
 48 
 
 9.41836 
 
 52 
 
 0.58164 
 
 9.98558 
 
 3 
 
 19 
 
 O 
 
 Q 
 
 u.i 
 7 9 
 
 0.0 
 7 i 
 
 42 
 
 9.40442 
 
 48 
 
 9.41887 
 
 51 
 
 0.58113 
 
 9.98555 
 
 3 
 
 18 
 
 10 
 
 / .z 
 q ft 
 
 /.i 
 7 S 
 
 43 
 
 9.40490 
 
 48 
 
 9.41939 
 
 52 
 
 0.58061 
 
 9.98551 
 
 4 
 
 17 
 
 iU 
 
 90 
 
 o.U 
 1 ft ft 
 
 / .o 
 i f; 7 
 
 44 
 
 9.40538 
 
 48 
 
 48 
 
 9.41990 
 
 51 
 
 51 
 
 0.58010 
 
 9.98548 
 
 3 
 
 q 
 
 16 
 
 ZA) 
 
 30 
 
 10.0 
 
 24.0 
 
 19 . / 
 
 23.5 
 
 45 
 
 9.40586 
 
 9.42041 
 
 0.57959 
 
 9.98545 
 
 6 
 
 15 
 
 40 
 
 32.0 
 
 31.3 
 
 46 
 
 9.40634 
 
 48 
 
 9.42093 
 
 52 
 
 0.57907 
 
 9.98541 
 
 4 
 
 14 
 
 50 
 
 40.0 
 
 39.2 
 
 47 
 
 9.40682 
 
 48 
 
 9.42144 
 
 51 
 
 0.57856 
 
 9.98538 
 
 3 
 
 13 
 
 
 
 
 48 
 
 9.40730 
 
 48 
 
 9.42195 
 
 51 
 
 0.57805 
 
 9.98535 
 
 3 
 
 12 
 
 
 
 
 49 
 
 9.40778 
 
 48 
 
 47 
 
 9.42246 
 
 51 
 
 51 
 
 0.57754 
 
 9.98531 
 
 4 
 
 q 
 
 11 
 
 
 A 
 
 9 
 
 50 
 
 9.40825 
 
 9.42297 
 
 0.57703 
 
 9.98528 
 
 o 
 
 10 
 
 6 
 
 0.4 
 
 V 
 
 0.3 
 
 51 
 
 9.40873 
 
 48 
 
 9.42348 
 
 51 
 
 0.57652 
 
 9.98525 
 
 3 
 
 9 
 
 7 
 
 0.5 
 
 0.4 
 
 52 
 
 9.40921 
 
 48 
 
 9.42399 
 
 51 
 
 0.5760L 
 
 9.98521 
 
 4 
 
 8 
 
 8 
 
 0.5 
 
 0.4 
 
 53 
 
 9.40968 
 
 47 
 
 9.42450 
 
 51 
 
 0.57550 
 
 9.98518 
 
 3 
 
 7 
 
 9 
 
 0.6 
 
 0.5 
 
 54 
 
 9.41016 
 
 48 
 
 47 
 
 9.42501 
 
 51 
 
 51 
 
 0.57499 
 
 9.98515 
 
 3 
 
 A 
 
 6 
 
 10 
 
 0.7 
 
 0.5 
 
 55 
 
 9.41063 
 
 9.42552 
 
 0.57448 
 
 9.98511 
 
 
 ' 5 
 
 20 
 
 1.3 
 
 1.0 
 
 56 
 
 9.41111 
 
 48 
 
 9.42603 
 
 51 
 
 0.57397 
 
 9.98508 
 
 3 
 
 4 
 
 30 
 
 2.0 
 
 1.5 
 
 57 
 
 9.41158 
 
 47 
 
 9.42653 
 
 50 
 
 0.57347 
 
 9.98505 
 
 3 
 
 3 
 
 40 
 
 2.7 
 
 2.0 
 
 58 
 
 9.41205 
 
 47 
 
 9.42704 
 
 51 
 
 0.57296 
 
 9.98501 
 
 4 
 
 2 
 
 50 
 
 3.3 
 
 2.5 
 
 59 
 
 9.41252 
 
 47 
 
 48 
 
 9.42755 
 
 51 
 
 50 
 
 0.57245 
 
 9.98498 
 
 3 
 
 A 
 
 1 
 
 
 
 
 60 
 
 9.41300 
 
 9.42805 
 
 0.57195 
 
 9.98494 
 
 
 0 
 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 t 
 
 P. P. 
 
 75 ° 
 
logarithms of trigonometric functions. 
 
 507 
 
 ' ] 
 
 Li. Sin. 6 
 
 o ( . 
 
 141300 . 
 
 1 < 
 
 141347 \ 
 
 2 1 
 
 141394 4 
 
 3 1 
 
 141441 4 
 
 4 ' 
 
 141488 4 
 
 5 1 
 
 141535 . 
 
 1 6 1 
 
 141582 4 
 
 1 7 1 
 
 9.41628 ; 
 
 8 1 
 
 9.41675 
 
 9 
 
 9.41722 ; 
 
 10 
 
 9.41768 
 
 11 
 
 9.41815 
 
 12 
 
 9.41861 
 
 13 
 
 9.41908 ; 
 
 14 
 
 9.41954 ; 
 
 15 
 
 9.42001 
 
 16 
 
 9.42047 
 
 17 
 
 9.42093 
 
 18 
 
 9.42140 
 
 19 
 
 9.42186 
 
 20 
 
 9.42232 
 
 21 
 
 9.42278 
 
 22 
 
 9.42324 
 
 23 
 
 9.42370 
 
 24 
 
 9.42416 
 
 25 
 
 9.42461 
 
 26 
 
 9.42507 
 
 27 
 
 9.42553 
 
 28 
 
 9.42599 
 
 29 
 
 9.42644 
 
 30 
 
 9.42690 
 
 31 
 
 9.42735 
 
 32 
 
 9.42781 
 
 33 
 
 9.42826 
 
 34 
 
 9.42872 
 
 35 
 
 9.42917 
 
 36 
 
 9.42962 
 
 37 
 
 9.43008 
 
 38 
 
 9.43053 
 
 39 
 
 9.43098 
 
 40 
 
 0.43143 
 
 41 
 
 9.43188 
 
 42 
 
 9.43233 
 
 43 
 
 9.43278 
 
 44 
 
 9.43323 
 
 45 
 
 ' 9.43367 
 
 46 
 
 9.43412 
 
 47 
 
 9.43457 
 
 48 
 
 9.43502 
 
 49 
 
 9.43546 
 
 50 
 
 “ 9.43591 
 
 51 
 
 9.43635 
 
 52 
 
 9.43680 
 
 53 
 
 9.43724 
 
 54 
 
 9.43769 
 
 55 
 
 ~ 9.43813 
 
 56 
 
 9.43857 
 
 57 
 
 9.43901 
 
 58 
 
 9.43946 
 
 59 
 
 i 9.43990 
 
 60 
 
 i 9.44034 
 
 | 
 
 L. Cos. 
 
 L.Tang. 
 
 9.42805 
 
 9.42856 
 
 9.42906 
 
 9.42957 
 
 9.43007 
 
 9.43057 
 
 9.43108 
 
 9.43158 
 
 9.43208 
 
 9.43258 
 
 9.43308 
 
 9.43358 
 
 9.43408 
 
 9.43458 
 
 9.43508 
 
 46 
 
 46 
 
 46 
 
 46 
 
 46 
 
 45 
 
 46 
 46 
 46 
 
 45 
 
 46 
 
 45 
 
 46 
 
 45 
 
 46 
 45 
 
 45 
 
 46 
 45 
 45 
 45 
 45 
 45 
 45 
 45 
 
 44 
 
 45 
 45 
 45 
 
 44 
 
 45 
 
 44 
 
 45 
 
 44 
 
 45 
 44 
 44 
 
 44 
 
 45 
 44 
 44 
 
 9.43558 
 
 9.43607 
 
 9.43657 
 
 9.43707 
 
 9.43756 
 
 9.43806 
 
 9.43855 
 
 9.43905 
 
 9.43954 
 
 9.44004 
 
 d. c. 
 
 9.44053 
 
 9.44102 
 
 9.44151 
 
 9.44201 
 
 9.44250 
 
 9.44299 
 
 9.44348 
 
 9.44397 
 
 9.44446 
 
 9.44495 
 
 9.44544 
 
 9.44592 
 
 9.44641 
 
 9.44690 
 
 9.44738 
 
 9.44787 
 
 9.44836 
 
 9.44884 
 
 9.44933 
 
 9.44981 
 
 9.45029 
 
 9.45078 
 
 9.45126 
 
 9.45174 
 
 9.45222 
 
 9.45271 
 
 9.45319 
 
 9.45367 
 
 9.45415 
 
 9.45463 
 
 51 
 
 50 
 
 51 
 50 
 
 50 
 
 51 
 50 
 50 
 50 
 50 
 50 
 50 
 50 
 50 
 50 
 
 49 
 
 50 
 50 
 
 49 
 
 50 
 
 49 
 
 50 
 
 49 
 
 50 
 49 
 49 
 
 49 
 
 50 
 49 
 49 
 49 
 49 
 49 
 49 
 49 
 
 48 
 
 49 
 49 
 
 48 
 
 49 
 49 
 
 48 
 
 49 
 48 
 
 48 
 
 49 
 
 9.45511 
 
 9.45559 
 
 9.45606 
 
 9.45654 
 
 9.45702 
 
 9.45750 
 
 d. L. Cotg 
 
 L. Cotg. 1 
 
 Li. Cos. d 
 
 1157195 
 
 9.98494 Q 
 
 0.57144 
 
 9.98491 * 
 
 0.57094 
 
 9.98488 6 . 
 
 0.57043 
 
 9.98484 % 
 
 0.56993 
 
 9.98481 « 
 
 0.56943 
 
 9.98477 Q 
 
 0.56892 
 
 9.98474 % 
 
 0.56842 
 
 9.98471 « 
 
 0.56792 
 
 9.98467 4 
 
 0.56742 
 
 9.98464 ‘ 
 
 0.56692 
 
 9.98460 f 
 
 0.56642 
 
 9.98457 * 
 
 0.56592 
 
 9.98453 ; 
 
 0.56542 
 
 9.98450 i 
 
 0.56492 
 
 9.98447 ; 
 
 0.56442 
 
 9.98443 , 
 
 0.56393 
 
 9.98440 1 
 
 0.56343 
 
 9.98436 : 
 
 0.56293 
 
 9.98433 1 
 
 0.56244 
 
 9.98429 ; 
 
 0.56194 
 
 9.98426 
 
 0.56145 
 
 9.98422 
 
 0.56095 
 
 9.98419 
 
 0.56046 
 
 9.98415 
 
 0.55996 
 
 9.98412 
 
 0.55947 
 
 9.98409 
 
 0.55898 
 
 9.98405 
 
 0.55849 
 
 9.98402 
 
 0.55799 
 
 9.98398 
 
 0.55750 
 
 9.98395 
 
 0.55701 
 
 9.98391 
 
 0.55652 
 
 9.98388 
 
 0.55603 
 
 9.98384 
 
 0.55554 
 
 9.98381 
 
 0.55505 
 
 9.98377 
 
 0.55456 
 
 9.98373 
 
 0.55408 
 
 9.98370 
 
 0.55359 
 
 9.98366 
 
 0.55310 
 
 9.98363 
 
 0.55262 
 
 9.98359 
 
 0.55213 
 
 9.98356 
 
 0.55164 
 
 9.98352 
 
 0.55116 
 
 9.98349 
 
 0.55067 
 
 9.98345 
 
 0.55019 
 
 9.98342 
 
 0.54971 
 
 9.98338 
 
 1 0.54922 
 
 9.98334 
 
 ’ 0.54874 
 
 9.98331 
 
 ’ 0.54826 
 
 9.98327 
 
 1 0.54778 
 
 9.98324 
 
 1 0.54729 
 
 9.98320 
 
 1 0.54681 
 
 9.98317 
 
 * 0.54633 
 
 9.98313 
 
 * 0.54585 
 
 , 9.98309 
 
 \ 0.54537 
 
 9.98306 
 
 5 0.54488 
 
 > 9.98302 
 
 3 0.54441 
 
 . 9.98299 
 
 7 0.5439-1 
 
 t 9.98295 
 
 l 0.5434( 
 
 > 9.98291 
 
 5 0.5429* 
 
 5 9.98288 
 
 8 0.5425( 
 
 ) 9.98284 
 
 c. L.Tang 
 
 r. L. Sin. 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56_ 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 _31 
 
 30 
 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 
 10 
 
 9 
 
 d. 
 
 P. P. 
 
 
 51 
 
 50 
 
 6 
 
 5.1 
 
 5.0 
 
 7 
 
 6.0 
 
 5.8 
 
 8 
 
 6.8 
 
 6.7 
 
 9 
 
 7.7 
 
 7.5 
 
 10 
 
 8.5 
 
 8.3 
 
 20 
 
 17.0 
 
 16.7 
 
 30 
 
 25.5 
 
 25.0 
 
 40 
 
 34.0 
 
 33.3 
 
 50 
 
 42.5 
 
 41.7 
 
 
 49 
 
 48 
 
 6 
 
 4.9 
 
 4.8 
 
 7 
 
 5.7 
 
 5.6 
 
 8 
 
 6.5 
 
 6.4 
 
 9 
 
 7.4 
 
 7.2 
 
 10 
 
 8.2 
 
 8.0 
 
 20 
 
 16.3 
 
 16.0 
 
 30 
 
 24.5 
 
 24.0 
 
 40 
 
 32.7 
 
 32.0 
 
 50 
 
 40.8 
 
 40.0 
 
 
 47 
 
 46 
 
 6 
 
 4.7 
 
 4.a 
 
 7 
 
 5.5 
 
 5.4 
 
 8 
 
 6.3 
 
 6.1 
 
 9 
 
 7.1 
 
 6.9 
 
 10 
 
 7.8 
 
 7.7 
 
 20 
 
 15.7 
 
 15.3 
 
 30 
 
 23.5 
 
 23.0 
 
 40 
 
 31.3 
 
 30.7 
 
 50 
 
 39.2 
 
 38.3 
 
 45 
 
 4.5 
 5.3 
 6.0 
 6.8 
 
 7.5 
 15.0 
 22.5 
 
 44 
 
 4.4 
 
 5.1 
 
 5.9 
 
 6.6 
 
 7.8 
 
 14.7 
 
 22.0 
 
 30.0 29.3 
 37.5 36.7 
 
 4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.3 
 2.0 
 2.7 
 
 3.3 
 
 3 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 1.0 
 
 1.5 
 
 2.0 
 
 2.5 
 
 P. P. 
 
 74 
 
508 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 16 ° 
 
 f 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 
 P. P 
 
 
 0 
 
 9.44034 
 
 
 9.45750 
 
 
 0.54250 
 
 9.98284 
 
 
 60 
 
 
 
 
 1 
 
 9.44078 
 
 44 
 
 9.45797 
 
 47 
 
 0.54203 
 
 9.98281 
 
 3 
 
 59 
 
 
 
 
 2 
 
 9.44122 
 
 44 
 
 9.45845 
 
 48 
 
 0.54155 
 
 9.98277 
 
 4 
 
 58 
 
 
 48 
 
 A O 
 
 
 3 
 
 9.44166 
 
 44 
 
 9.45892 
 
 47 
 
 0.54108 
 
 9.98273 
 
 4 
 
 57 
 
 
 47 
 
 A *7 
 
 4 
 
 9.44210 
 
 44 
 
 43 
 
 9.45940 
 
 48 
 
 47 
 
 0.54060 
 
 9.98270 
 
 3 
 
 A 
 
 56 
 
 6 
 
 7 
 
 4.8 
 
 5.6 
 
 4.7 
 
 5.5 
 
 5 
 
 9.44253 
 
 9.45987 
 
 0.54013 
 
 9.98266 
 
 4 
 
 55 
 
 8 
 
 6!4 
 
 6^3 
 
 6 
 
 9.44297 
 
 44 
 
 9.46035 
 
 48 
 
 0.53965 
 
 9.98262 
 
 4 
 
 54 
 
 9 
 
 7.2 
 
 7.1 
 
 7 
 
 9.44341 
 
 44 
 
 9.46082 
 
 47 
 
 0.53918 
 
 9.98259 
 
 3 
 
 53 
 
 10 
 
 8.0 
 
 7.8 
 
 8 
 
 9.44385 
 
 44 
 
 9.46130 
 
 48 
 
 0.53870 
 
 9.98255 
 
 4 
 
 52 
 
 20 
 
 16.0 
 
 15.7 
 
 9 
 
 9.44428 
 
 43 
 
 44 
 
 9.46177 
 
 47 
 
 0.53823 
 
 9.98251 
 
 4 
 
 q 
 
 51 
 
 30 
 
 24.0 
 
 23.5 
 
 10 
 
 9.44472 
 
 9.46224 
 
 4/ 
 
 0.53776 
 
 9.98248 
 
 o 
 
 50 
 
 40 
 
 32.0 
 
 31.3 
 
 11 
 
 9.44516 
 
 44 
 
 9.46271 
 
 47 
 
 0.53729 
 
 9.98244 
 
 4 
 
 49 
 
 50 
 
 40.0 
 
 39.2 
 
 12 
 
 9.44559 
 
 43 
 
 9.46319 
 
 48 
 
 0.53681 
 
 9.98240 
 
 4 
 
 48 
 
 
 
 
 13 
 
 9.44602 
 
 43 
 
 9.46366 
 
 47 
 
 0.53634 
 
 9.98237 
 
 3 
 
 47 
 
 
 
 
 14 
 
 9.44646 
 
 44 
 
 AO 
 
 9.46413 
 
 47 
 
 0.53587 
 
 9.98233 
 
 4 
 
 A 
 
 46 
 
 
 46 
 
 45 
 
 15 
 
 9.44689 
 
 
 9.46460 
 
 4/ 
 
 0.53540 
 
 9.98229 
 
 4 
 
 45 
 
 6 
 
 4.6 
 
 4.5 
 
 16 
 
 9.44733 
 
 44 
 
 9.46507 
 
 47 
 
 0.53493 
 
 9.98226 
 
 3 
 
 44 
 
 7 
 
 5.4 
 
 5.3 
 
 17 
 
 9.44776 
 
 43 
 
 9.46554 
 
 47 
 
 0.53446 
 
 9.98222 
 
 4 
 
 43 
 
 8 
 
 6.1 
 
 6.0 
 
 18 
 
 9.44819 
 
 43 
 
 9.46601 
 
 47 
 
 0.53399 
 
 9.98218 
 
 4 
 
 42 
 
 9 
 
 6.9 
 
 6.8 
 
 19 
 
 9.44862 
 
 43 
 
 43 
 
 9.46648 
 
 47 
 
 0.53352 
 
 9.98215 
 
 3 
 
 A 
 
 41 
 
 10 
 
 7.7 
 
 7.5 
 
 20 
 
 9.44905 
 
 9.46694 
 
 40 
 
 0.53306 
 
 9.98211 
 
 4 
 
 40 
 
 20 
 
 15.3 
 
 15.0 
 
 21 
 
 9.44948 
 
 43 
 
 9.46741 
 
 47 
 
 0.53259 
 
 9.98207 
 
 4 
 
 39 
 
 30 
 
 23.0 
 
 22.5 
 
 22 
 
 9.44992 
 
 44 
 
 9.46788 
 
 47 
 
 0.53212 
 
 9.98204 
 
 3 
 
 38 
 
 40 
 
 30.7 
 
 30.0 
 
 23 
 
 9.45035 
 
 43 
 
 9.46835 
 
 47 
 
 0.53165 
 
 9.98200 
 
 4 
 
 37 
 
 50 
 
 38.3 
 
 37.5 
 
 24 
 
 9.45077 
 
 42 
 
 A Q 
 
 9.46881 
 
 46 
 
 0.53119 
 
 9.98196 
 
 4 
 
 A 
 
 36 
 
 
 
 
 25 
 
 9.45120 
 
 
 9.46928 
 
 4/ 
 
 0.53072 
 
 9.98192 
 
 4 
 
 35 
 
 
 
 
 26 
 
 9.45163 
 
 43 
 
 9.46975 
 
 47 
 
 0.53025 
 
 9.98189 
 
 3 
 
 34 
 
 
 44 
 
 43 
 
 27 
 
 9.45206 
 
 43 
 
 9.47021 
 
 46 
 
 0.52979 
 
 9.98185 
 
 4 
 
 33 
 
 6 
 
 4.4 
 
 4.3 
 
 28 
 
 9.45249 
 
 43 
 
 9.47068 
 
 47 
 
 0.52932 
 
 9.98181 
 
 4 
 
 32 
 
 7 
 
 5.1 
 
 5.0 
 
 29 
 
 9.45292 
 
 43 
 
 AO 
 
 9.47114 
 
 46 
 
 An 
 
 0.52886 
 
 9.98177 
 
 4 
 
 q 
 
 31 
 
 8 
 
 5.9 
 
 5.7 
 
 30 
 
 9.45334 
 
 
 9.47160 
 
 4o 
 
 0.52840 
 
 9.98174 
 
 o 
 
 30 
 
 9 
 
 6.6 
 *1 O 
 
 6.5 
 
 31 
 
 9.45377 
 
 43 
 
 9.47207 
 
 47 
 
 0.52793 
 
 9.98170 
 
 4 
 
 29 
 
 10 
 
 OA 
 
 7.0 
 
 -f A hr 
 
 7.2 
 
 1/IO 
 
 32 
 
 9.45419 
 
 42 
 
 9.47253 
 
 46 
 
 0.52747 
 
 9.98166 
 
 4 
 
 28 
 
 20 
 
 14.7 
 
 14.3 
 
 33 
 
 9.45462 
 
 43 
 
 9.47299 
 
 46 
 
 0.52701 
 
 9.98162 
 
 4 
 
 27 
 
 30 
 
 22.0 
 
 21.5 
 
 no 17 
 
 34 
 
 9.45504 
 
 42 
 
 9.47346 
 
 47 
 
 0.52654 
 
 9.98159 
 
 3 
 
 A 
 
 26 
 
 40 
 
 50 
 
 29.3 
 
 36.7 
 
 28.7 
 
 35.8 
 
 35 
 
 9.45547 
 
 40 
 
 9.47392 
 
 40 
 
 0.52608 
 
 9.98155 
 
 4 
 
 25 
 
 36 
 
 9.45589 
 
 42 
 
 9.47438 
 
 46 
 
 0.52562 
 
 9.98151 
 
 4 
 
 24 
 
 
 
 
 37 
 
 9.45632 
 
 43 
 
 9.47484 
 
 46 
 
 0.52516 
 
 9.98147 
 
 4 
 
 23 
 
 
 
 
 38 
 
 9.45674 
 
 42 
 
 9.47530 
 
 46 
 
 0.52470 
 
 9.98144 
 
 3 
 
 22 
 
 
 42 
 
 41 
 
 39 
 
 9.45716 
 
 42 
 
 AO 
 
 9.47576 
 
 46 
 
 A(K 
 
 0.52424 
 
 9.98140 
 
 4 
 
 A 
 
 21 
 
 6 
 
 7 
 
 4.2 
 
 A Q 
 
 4.1 
 
 A 8 
 
 40 
 
 9.45758 
 
 4Z 
 
 9.47622 
 
 40 
 
 0.52378 
 
 9.98136 
 
 4 
 
 20 
 
 g 
 
 4.y 
 
 5.6 
 
 4.0 
 
 5.5 
 
 41 
 
 9.45801 
 
 43 
 
 9.47668 
 
 46 
 
 0.52332 
 
 9.98132 
 
 4 
 
 19 
 
 9 
 
 6.3 
 
 6^2 
 
 42 
 
 9.45843 
 
 42 
 
 9.47714 
 
 46 
 
 0.52286 
 
 9.98129 
 
 3 
 
 18 
 
 10 
 
 7.0 
 
 6.8 
 
 43 
 
 9.45885 
 
 42 
 
 9.47760 
 
 46 
 
 0.52240 
 
 9.98125 
 
 4 
 
 17 
 
 20 
 
 140 
 
 13.7 
 
 44 
 
 9.45927 
 
 42 
 
 AO 
 
 9.47806 
 
 46 
 
 Ad 
 
 0.52194 
 
 9.98121 
 
 4 
 
 A 
 
 16 
 
 30 
 
 2L0 
 
 20i5 
 
 45 
 
 9.45969 
 
 4Z 
 
 9.47852 
 
 40 
 
 0.52148 
 
 9.98117 
 
 
 15 
 
 40 
 
 28.0 
 
 27.3 
 
 46 
 
 9.46011 
 
 42 
 
 9.47897 
 
 45 
 
 0.52103 
 
 9.98113 
 
 4 
 
 14 
 
 50 
 
 35.0 
 
 34.2 
 
 47 
 
 9.46053 
 
 42 
 
 9.47943 
 
 46 
 
 0.52057 
 
 9.98110 
 
 3 
 
 13 
 
 
 
 
 48 
 
 9.46095 
 
 42 
 
 9.47989 
 
 46 
 
 0.52011 
 
 9.98106 
 
 4 
 
 12 
 
 
 
 
 49 
 
 9.46136 
 
 41 
 
 AO 
 
 9.48035 
 
 46 
 
 AK 
 
 0.51965 
 
 9.98102 
 
 4 
 
 A 
 
 11 
 
 
 4 
 
 3 
 
 50 
 
 9.46178 
 
 4Z 
 
 9.48080 
 
 40 
 
 0.51920 
 
 9.98098 
 
 ‘x 
 
 10 
 
 6 
 
 0.4 
 
 0.3 
 
 51 
 
 9.46220 
 
 42 
 
 9.48126 
 
 46 
 
 0.51874 
 
 9.98094 
 
 4 
 
 9 
 
 7 
 
 0.5 
 
 0.4 
 
 52 
 
 9.46262 
 
 42 
 
 9.48171 
 
 45 
 
 0,51829 
 
 9.98090 
 
 4 
 
 8 
 
 8 
 
 0.5 
 
 0.4 
 
 53 
 
 9.46303 
 
 41 
 
 9.48217 
 
 46 
 
 0.51783 
 
 9.98087 
 
 3 
 
 7 
 
 9 
 
 0.6 
 
 0.5 
 
 54 
 
 9.46345 
 
 42 
 
 41 
 
 9.48262 
 
 45 
 
 45 
 
 0.51738 
 
 9.98083 
 
 4 
 
 4 
 
 6 
 
 10 
 
 0.7 
 
 0.5 
 
 55 
 
 9.46386 
 
 9.48307 
 
 0.51693 
 
 9.98079 
 
 
 5 
 
 20 
 
 1.3 
 
 1.0 
 
 56 
 
 9.46428 
 
 42 
 
 8.48353 
 
 46 
 
 0.51647 
 
 9.98075 
 
 4 
 
 4 
 
 30 
 
 2.0 
 
 1.5 
 
 57 
 
 9.46469 
 
 41 
 
 9.48398 
 
 45 
 
 0.51602 
 
 9.98071 
 
 4 
 
 3 
 
 40 
 
 2.7 
 
 2.0 
 
 58 
 
 9.46511 
 
 42 
 
 9.48443 
 
 45 
 
 0.51557 
 
 9.98067 
 
 4 
 
 2 
 
 50 
 
 | 3.3 
 
 2.5 
 
 59 
 
 9.46552 
 
 41 
 
 42 
 
 9.48489 
 
 46 
 
 A*\ 
 
 0.51511 
 
 9.98063 
 
 4 
 
 q 
 
 1 
 
 
 
 
 60 
 
 9.46594 
 
 9.48534 
 
 40 
 
 0.51466 
 
 9.98060 
 
 o 
 
 0 
 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 f 
 
 P. P. 
 
 73 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 17 ° 
 
 509 
 
 ' 
 
 [j. Sin. 
 
 d. I 
 
 ,.Tang. c 
 
 0 
 
 1 
 
 ).46594 
 ). 46635 
 
 41 
 
 {148534 
 
 9.48579 
 
 2 
 
 D.46676 
 
 41 
 
 9.48624 
 
 3 
 
 3.46717 
 
 41 
 
 9.48669 
 
 4 
 
 3.46758 
 
 41 
 
 42 
 
 9.48714 
 
 5 
 
 6 
 
 3.46800 
 
 3.46841 
 
 41 
 
 9.48759 
 
 9.48804 
 
 7 
 
 9.46882 
 
 41 
 
 9.48849 
 
 8 
 
 9.46923 
 
 41 
 
 9.48894 
 
 9 
 
 9.46964 
 
 41 
 
 41 
 
 9.48939 
 
 10 
 
 11 
 
 9.47005 
 
 9.47045 
 
 'll 
 
 40 
 
 9.48984 
 
 9.49029 
 
 12 
 
 9.47086 
 
 41 
 
 9.49073 
 
 13 
 
 9.47127 
 
 41 
 
 9.49118 
 
 14 
 
 9.47168 
 
 41 
 
 41 
 
 9.49163 
 
 15 
 
 16 
 
 9.47209 
 
 9.47249 
 
 ‘i-L 
 
 40 
 
 9.49207 
 
 9.49252 
 
 17 
 
 9.47290 
 
 41 
 
 9.49296 
 
 18 
 
 9.47330 
 
 40 
 
 9.49341 
 
 19 
 
 9.47371 
 
 41 
 
 40 
 
 9.49385 
 
 20 
 
 21 
 
 9.47411 
 
 9.47452 
 
 41 
 
 9.49430 
 
 9.49474 
 
 22 
 
 9.47492 
 
 40 
 
 9.49519 
 
 23 
 
 9.47533 
 
 41 
 
 9.49563 
 
 24 
 
 9.47573 
 
 40 
 
 40 
 
 41 
 
 9.49607 
 
 25 
 
 26 
 
 9.47613 
 
 9.47654 
 
 9.49652 
 
 9.49696 
 
 27 
 
 9.47694 
 
 40 
 
 9.49740 
 
 28 
 
 9.47734 
 
 40 
 
 9.49784 
 
 29 
 
 9.47774 
 
 ‘ 40 
 40 
 40 
 
 9.49828 
 
 30 
 
 31 
 
 9.47814 
 
 9.47854 
 
 9.49872 
 
 9.49916 
 
 32 
 
 9.47894 
 
 40 
 
 9.49960 
 
 33 
 
 34 
 
 9.47934 
 
 9.47974 
 
 40 
 
 40 
 
 40 
 
 40 
 
 9.50004 
 
 9.50048 
 
 35 
 
 36 
 
 9.48014 
 
 9.48054 
 
 9.50092 
 
 9.50136 
 
 37 
 
 9.48094 
 
 40 
 
 9.50180 
 
 38 
 
 9.48133 
 
 39 
 
 9.50223 
 
 39 
 
 9.48173 
 
 40 
 
 40 
 
 39 
 
 9.50267 
 
 40 
 
 41 
 
 9.48213 
 
 9.48252 
 
 9.50311 
 
 9.50355 
 
 42 
 
 9.48292 
 
 40 
 
 9.50398 
 
 43 
 
 9.48332 
 
 40 
 
 9.50442 
 
 44 
 
 9.48371 
 
 39 
 
 40 
 39 
 
 9.50485 
 
 45 
 
 46 
 
 9.48411 
 
 9.48450 
 
 9.50529 
 
 9.50572 
 
 47 
 
 9.48490 
 
 40 
 
 9.50616 
 
 48 
 
 9.48529 
 
 39 
 
 9.50659 
 
 49 
 
 9.48568 
 
 39 
 
 39 
 
 40 
 
 9.50703 
 
 50 
 
 51 
 
 ' 9.48607 
 9.48647 
 
 9.50746 
 
 9.50789 
 
 52 
 
 9.48686 
 
 39 
 
 9.50833 
 
 53 
 
 9.48725 
 
 39 
 
 9.50876 
 
 54 
 
 9.48764 
 
 39 
 
 39 
 
 9.50919 
 
 55 
 
 56 
 
 9.48803 
 
 9.48842 
 
 QQ 
 
 l OU 
 
 1 OA 
 
 9.50962” 
 
 9.51005 
 
 57 
 
 9.48881 
 
 39 
 
 9.51048 
 
 58 
 
 9.4892C 
 
 1 
 
 9.51092 
 
 59 
 
 9.4895c 
 
 ’ 39 
 
 9.51135 
 
 60 
 
 9.4899* 
 
 » 
 
 9.51178 
 
 
 L. Cos 
 
 . 1 d. 
 
 L. Cotg 
 
 d.c. 
 
 L. Cotg. 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 45 
 
 44 
 
 45 
 45 
 
 44 
 
 45 
 
 44 
 
 45 
 
 44 
 
 45 
 
 44 
 
 45 
 44 
 
 44 
 
 45 
 44 
 44 
 44 
 44 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 44 
 
 43 
 
 44 
 44 
 44 
 
 43 
 
 44 
 
 43 
 
 44 
 
 43 
 
 44 
 
 43 
 
 44 
 43 
 
 43 
 
 44 
 43 
 43 
 43 
 43 
 
 43 
 
 44 
 43 
 43 
 
 0.51466 
 
 0.51421 
 
 0.51376 
 
 0.51331 
 
 0.51286 
 
 0.51241 
 
 0.51196 
 
 0.51151 
 
 0.51106 
 
 0.51061 
 
 0.51016 
 
 0.50971 
 
 0.50927 
 
 0.50882 
 
 0.50837 
 
 0.50793 
 
 0.50748 
 
 0.50704 
 
 0.50659 
 
 0.50615 
 
 0.50570 
 
 0.50526 
 
 0.50481 
 
 0.50437 
 
 0.50393 
 
 0.50348 
 
 0.50304 
 
 0.50260 
 
 0.50216 
 
 0.50172 
 
 0.50128 
 
 0.50084 
 
 0.50040 
 
 0.49996 
 
 0.49952 
 
 0.49908 
 
 0.49864 
 
 0.49820 
 
 0.49777 
 
 0.49733 
 
 0.49689 
 
 0.49645 
 
 0.49602 
 
 0.49558 
 
 0.49515 
 
 0.49471 
 
 0.49428 
 
 0.49384 
 
 0.49341 
 
 0.49297 
 
 0.49254 
 
 0.49211 
 
 0.49167 
 
 0.49124 
 
 0.49081 
 
 0.49038 
 
 0.48995 
 
 0.48952 
 
 0.48908 
 
 0.48865 
 
 0.48822 
 
 d. c. IL.Tang 
 
 Cos. d. 
 
 .98060 
 
 .98056 
 
 98052 
 
 98048 
 
 .98044 
 
 98040 
 
 98036 
 
 .98032 
 
 .98029 
 
 98025 
 
 .98021 
 
 98017 
 
 9.98013 
 
 9.98009 
 
 9.98005 
 
 9.98001 
 
 9.97997 
 
 9.97993 
 
 9.97989 
 
 9.97986 
 
 9.97982 
 
 9.97978 
 
 9.97974 
 
 9.97970 
 
 9.97966 
 
 9.97962 
 
 9.97958 
 
 9.97954 
 
 9.97950 
 
 9.97946 
 
 9.97942 
 
 9.97938 
 
 9.97934 
 
 9.97930 
 
 9.97926 
 
 9.97922 
 
 9.97918 
 
 9.97914 
 
 9.97910 
 
 9.97906 
 
 9.97902 
 
 9.97898 
 
 9.97894 
 
 9.97890 
 
 9.97886 
 
 9.97882 
 
 9.97878 
 
 9.97874 
 
 9.97870 
 
 9.97866 
 
 9.97861 
 
 9.97857 
 
 9.97853 
 
 9.97849 
 
 9.97845 
 
 9.97841 
 
 9.97837 
 
 9.97833 
 
 9.97829 
 
 9.97825 
 
 9.97821 
 
 . L. Sin. d 
 
 4 
 
 4 
 
 3 
 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 
 3 
 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 
 4 
 
 5 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 ^ 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 P. P. 
 
 45 
 
 4.5 
 5.3 
 6.0 
 6.8 
 
 7.5 
 
 15.0 
 
 22.5 
 
 30.0 
 
 37.5 
 
 44 
 
 4.4 
 
 5.1 
 
 5.9 
 
 6.6 
 
 7.3 
 
 14.7 
 22.0 
 29.3 
 
 36.7 
 
 43 
 
 4.3 
 
 5.0 
 
 5.7 
 
 6.5 
 
 7.2 
 
 14.3 
 
 21.5 
 
 28.7 
 
 35.8 
 
 42 
 
 4.2 
 4.9 
 5.6 
 
 6.3 
 7.0 
 
 14.0 
 
 21.0 
 28.0 
 35.0 
 
 40 
 
 4.0 
 
 4.7 
 5.3 
 
 6.0 
 
 6.7 
 
 13.3 
 20.0 
 26.7 
 
 33.3 
 
 41 
 
 4.1 
 
 4.8 
 
 5.5 
 
 6.2 
 
 6.8 
 
 13.7 
 
 20.5 
 27.3 
 34.2 
 
 39 
 
 3.9 
 4.6 
 5.2 
 
 5.9 
 6.5 
 
 13.0 
 
 19.5 
 
 26.0 
 
 32.5 
 
 11 
 
 
 5 
 
 4 
 
 3 
 
 JO 
 
 6 
 
 0.5 
 
 0.4 
 
 0.3 
 
 9 
 
 7 
 
 0.6 
 
 0.5 
 
 0.4 
 
 8 
 
 8 
 
 0.7 
 
 0.5 
 
 0.4 
 
 7 
 
 9 
 
 0.8 
 
 0.6 
 
 0.5 
 
 6 
 
 10 
 
 0.8 
 
 0.7 
 
 0.5 
 
 5 
 
 ' 20 
 
 1.7 
 
 1.3 
 
 1.0 
 
 4 
 
 30 
 
 2.5 
 
 2.0 
 
 1.5 
 
 
 40 
 
 3.3 
 
 2.7 
 
 2.0 
 
 0 
 2 
 
 1 
 
 50 
 
 4.2 
 
 3.3 
 
 2.5 
 
 0 
 
 
 
 
 
 t 
 
 P. P. 
 
 72 ° 
 
510 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 18 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 0 
 
 9.48998 
 
 
 9.51178 
 
 
 0.48822 
 
 9.97821 
 
 
 60 
 
 1 
 
 9.49037 
 
 39 
 
 9.51221 
 
 43 
 
 0.48779 
 
 9.97817 
 
 4 
 
 59 
 
 2 
 
 9.49076 
 
 39 
 
 9.51264 
 
 43 
 
 0.48736 
 
 9.97812 
 
 5 
 
 58 
 
 3 
 
 9.49115 
 
 39 
 
 9.51306 
 
 42 
 
 0.48694 
 
 9.97808 
 
 4 
 
 57 
 
 4 
 
 9.49153 
 
 38 
 
 39 
 
 9.51349 
 
 43 
 
 43 
 
 0.48651 
 
 9.97804 
 
 4 
 
 A 
 
 56 
 
 5 
 
 9.49192 
 
 9.51392 
 
 0.48608 
 
 9.97800 
 
 4 
 
 55 
 
 6 
 
 9.49231 
 
 39 
 
 9.51435 
 
 43 
 
 0.48565 
 
 9.97796 
 
 4 
 
 54 
 
 7 
 
 9.49269 
 
 38 
 
 9.51478 
 
 43 
 
 0.48522 
 
 9.97792 
 
 4 
 
 53 
 
 8 
 
 9.49308 
 
 39 
 
 9.515:4) 
 
 42 
 
 0.48480 
 
 9.97788 
 
 4 
 
 52 
 
 9 
 
 9.49347 
 
 39 
 
 38 
 
 9.51563 
 
 43 
 
 43 
 
 0.48437 
 
 9.97784 
 
 4 
 
 51 
 
 10 
 
 9.49385 
 
 9.51606 
 
 0.48394 
 
 9.97779 
 
 0 
 
 50 
 
 11 
 
 9.49424 
 
 39 
 
 9.51648 
 
 42 
 
 0.48352 
 
 9.977“5 
 
 4 
 
 49 
 
 12 
 
 9.49462 
 
 38 
 
 9.51691 
 
 43 
 
 0.48309 
 
 9.97771 
 
 4 
 
 48 
 
 13 
 
 9.49500 
 
 38 
 
 9.51734 
 
 43 
 
 0.48266 
 
 9.97767 
 
 4 
 
 47 
 
 14 
 
 9.49539 
 
 39 
 
 38 
 
 9.51776 
 
 42 
 
 43 
 
 0.48224 
 
 9.97763 
 
 4 
 
 A 
 
 46 
 
 15 
 
 9.49577 
 
 9.51819 
 
 0.48181 
 
 9.97759 
 
 *± 
 
 45 
 
 16 
 
 9.49615 
 
 38 
 
 9.51861 
 
 42 
 
 0.48139 
 
 9.97754 
 
 5 
 
 44 
 
 17 
 
 9.49654 
 
 39 
 
 9.51903 
 
 42 
 
 0.48097 
 
 9.97750 
 
 4 
 
 43 
 
 18 
 
 9.49692 
 
 38 
 
 9.51946 
 
 43 
 
 0.48054 
 
 9.97746 
 
 4 
 
 42 
 
 19 
 
 9.49730 
 
 38 
 
 9.51988 
 
 42 
 
 0.48012 
 
 9.97742 
 
 4 
 
 41 
 
 
 
 38 
 
 
 
 
 
 A 
 
 
 20 
 
 9.49768 
 
 9.52031 
 
 *±0 
 
 0.47969 
 
 9.97738 
 
 'i 
 
 40 
 
 21 
 
 9.49806 
 
 oS 
 
 9.52073 
 
 42 
 
 0.47927 
 
 9.97734 
 
 4 
 
 39 
 
 22 
 
 9.49844 
 
 38 
 
 9.52115 
 
 42 
 
 0.47885 
 
 9.97729 
 
 5 
 
 38 
 
 23 
 
 9.49882 
 
 38 
 
 9.52157 
 
 42 
 
 0.47843 
 
 9.97725 
 
 4 
 
 37 
 
 24 
 
 9.49920 
 
 38 
 
 38 
 
 9.52200 
 
 43 
 
 42 
 
 0.47800 
 
 9.97721 
 
 4 
 
 A 
 
 36 
 
 25 
 
 9.49958 
 
 9.52242 
 
 0.47758 
 
 9.97717 
 
 Tt 
 
 35 
 
 26 
 
 9.49996 
 
 38 
 
 9.52284 
 
 42 
 
 0.47716 
 
 9.97713 
 
 4 
 
 34 
 
 27 
 
 9.50034 
 
 38 
 
 9.52326 
 
 42 
 
 0.47674 
 
 9.97708 
 
 5 
 
 33 
 
 28 
 
 9.50072 
 
 38 
 
 9.52368 
 
 42 
 
 0.47632 
 
 9.97704 
 
 4 
 
 32 
 
 29 
 
 9.50110 
 
 38 
 
 38 
 
 9.52410 
 
 42 
 
 42 
 
 0 47590 
 
 9.97700 
 
 4 
 
 A 
 
 31 
 
 30 
 
 9.50148 
 
 9.52452 
 
 0.47548 
 
 9.97696 
 
 rk 
 
 30 
 
 31 
 
 9.50185 
 
 37 
 
 oo 
 
 9.52494 
 
 42 
 
 0.47506 
 
 9.97691 
 
 5 
 
 29 
 
 32 
 
 9.50223 
 
 38 
 
 oo 
 
 9.52536 
 
 42 
 
 0.47464 
 
 9.97687 
 
 4 
 
 28 
 
 33 
 
 9.50261 
 
 38 
 
 0*7 
 
 9.52578 
 
 42 
 
 0.47422 
 
 9.97683 
 
 4 
 
 27 
 
 34 
 
 9.50298 
 
 37 
 
 38 
 
 oo 
 
 9.52620 
 
 42 
 
 41 
 
 0.47380 
 
 9.97679 
 
 4 
 
 5 
 
 26 
 
 35 
 
 9.50336 
 
 9.52661 
 
 0.47339 
 
 9.97674 
 
 25 
 
 36 
 
 9.50374 
 
 38 
 
 9.52703 
 
 42 
 
 0.47297 
 
 9.97670 
 
 4 
 
 24 
 
 37 
 
 9.50411 
 
 37 
 
 oo 
 
 9.52745 
 
 42 
 
 0.47255 
 
 9.97666 
 
 4 
 
 23 
 
 38 
 
 9.50449 
 
 38 
 
 0*7 
 
 9.52787 
 
 42 
 
 0.47213 
 
 9.97662 
 
 4 
 
 22 
 
 39 
 
 9.50486 
 
 37 
 
 37 
 
 OQ 
 
 9.52829 
 
 42 
 
 0.47171 
 
 9.97657 
 
 5 
 
 A 
 
 21 
 
 40 
 
 9.50523 
 
 9.52870 
 
 “±1 
 
 0.47130 
 
 9.97653 
 
 
 20 
 
 41 
 
 9.50561 
 
 38 
 
 0*7 
 
 9.52912 
 
 42 
 
 0.47088 
 
 9.97649 
 
 4 
 
 19 
 
 42 
 
 9-50598 
 
 37 
 
 9.52953 
 
 41 
 
 0.47047 
 
 9.97645 
 
 4 
 
 18 
 
 43 
 
 9.50635 
 
 37 
 
 oo 
 
 9.52995 
 
 42 
 
 0.47005 
 
 9.97640 
 
 5 
 
 17 
 
 44 
 
 9.50673 
 
 38 
 
 37 
 
 9.53037 
 
 42 
 
 41 
 
 0.46963 
 
 9 97636 
 
 4 
 
 4 
 
 16 
 
 45 
 
 9.50710 
 
 0*7 
 
 9.53078 
 
 0.46922 
 
 9.97632 
 
 
 15 
 
 46 
 
 9.50747 
 
 3 / 
 
 0*7 
 
 9.53120 
 
 42 
 
 0.46880 
 
 9.97628 
 
 4 
 
 14 
 
 47 
 
 9.50784 
 
 37 
 
 9.53161 
 
 41 
 
 0.46839 
 
 9.97623 
 
 5 
 
 13 
 
 48 
 
 9.50821 
 
 37 
 
 0*7 
 
 9.53202 
 
 41 
 
 0.46798 
 
 9.97619 
 
 4 
 
 12 
 
 49 
 
 9.50858 
 
 37 
 
 38 
 
 9.53244 
 
 42 
 
 41 
 
 0.46756 
 
 9.97615 
 
 4 
 
 11 
 
 50 
 
 9.50896 
 
 0*7 
 
 9.53285 
 
 0.46715 
 
 9.97610 
 
 u 
 
 A 
 
 10 
 
 51 
 
 9.50933 
 
 37 
 
 0*7 
 
 9.53327 
 
 42 
 
 0.46673 
 
 9.97606 
 
 4 
 
 9 
 
 52 
 
 9.50970 
 
 37 
 
 9.53368 
 
 41 
 
 0.46632 
 
 9.97602 
 
 4 
 
 8 
 
 53 
 
 9.51007 
 
 37 
 
 oo 
 
 9.53409 
 
 41 
 
 0.46591 
 
 9.97597 
 
 5 
 
 7 
 
 54 
 
 9.51043 
 
 3b 
 
 37 
 
 9.53450 
 
 41 
 
 42 
 
 0.46550 
 
 9.97593 
 
 4 
 
 4 
 
 6 
 
 55 
 
 9.51080 
 
 9.53492 
 
 0.46508 
 
 9.97589 
 
 
 5 
 
 56 
 
 9.51117 
 
 37 
 
 9.53533 
 
 41 
 
 0.46467 
 
 9.97584 
 
 5 
 
 4 
 
 57 
 
 9.51154 
 
 37 
 
 9.53574 
 
 41 
 
 0.46426 
 
 9.97580 
 
 4 
 
 3 
 
 58 
 
 9.51191 
 
 37 
 
 9.53615 
 
 41 
 
 0.46385 
 
 9.97576 
 
 4 
 
 2 
 
 59 
 
 9.51227 
 
 36 
 
 37 
 
 9.53656 
 
 41 
 
 0.46344 
 
 9.97571 
 
 5 
 
 A 
 
 1 
 
 60 
 
 9.51264 
 
 9.53697 
 
 
 0.46303 
 
 9.97567 
 
 
 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 43 
 
 4.3 
 
 5.0 
 
 5.7 
 
 6.5 
 
 7.2 
 
 14.3 
 
 21.5 
 
 28.7 
 
 35.8 
 
 42 
 
 4.2 
 4.9 
 5.6 
 
 6.3 
 7.0 
 
 14.0 
 
 21.0 
 28.0 
 35.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 41 
 
 4.1 
 
 4.8 
 5.5 
 
 6.2 
 
 6.8 
 13.7 
 20.5 
 27.3 
 34.2 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 I 
 50 ! 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 39 
 
 3.9 
 4.6 
 5.2 
 
 5.9 
 6.5 
 
 13.0 
 
 19.5 
 
 26.0 
 
 32.5 
 
 37 
 
 3.7 
 
 4.3 
 
 4.9 
 
 5.6 
 
 6.2 
 
 12.3 
 
 18.5 
 
 24.7 
 
 30.8 
 
 5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4.2 
 
 P. P. 
 
 38 
 
 3.8 
 
 4.4 
 
 5.1 
 
 5.7 
 
 6.3 
 
 12.7 
 19.0 
 25.3 
 
 31.7 
 
 36 
 
 3.6 
 
 4.2 
 
 4.8 
 
 5.4 
 
 6.0 
 
 12.0 
 
 18.0 
 
 24.0 
 
 30.0 
 
 4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.3 
 2.0 
 2.7 
 
 3.3 
 
 71 ° 
 
/ 1 ] 
 
 L. Sin. i 
 
 0 < 
 
 1 h 
 
 2 1 
 
 3 ! 
 
 4 1 
 
 2.51264 , 
 
 2.51301 : 
 
 2.51338 ; 
 
 2.51374 ; 
 
 2.51411 
 
 5 ' 
 
 2.51447 
 
 6 1 
 
 9.51484 
 
 7 
 
 9.51520 
 
 8 
 
 9.51557 
 
 9 
 
 9.51593 
 
 10 
 
 9.51629 
 
 11 
 
 9.51666 
 
 12 
 
 9.51702 
 
 13 
 
 9.51738 
 
 14 L 
 
 9.51774 
 
 15 
 
 9.51811 
 
 16 
 
 9.51847 
 
 17 
 
 9.51883 
 
 18 
 
 9.51919 
 
 19 
 
 9.51955 
 
 20 
 
 9.51991 
 
 21 
 
 9.52027 
 
 22 
 
 9.52063 
 
 23 
 
 9.52099 
 
 24 
 
 9.52135 
 
 25 
 
 9.52171 
 
 26 
 
 9.52207 
 
 27 
 
 9.52242 
 
 28 
 
 9.52278 
 
 29 
 
 9.52314 
 
 30 
 
 | 9.52350 
 
 31 
 
 9.52385 
 
 32 
 
 9.52421 
 
 33 
 
 9.52456 
 
 34 1 
 
 9.52492 
 
 35 
 
 9.52527 
 
 36 
 
 9.52563 
 
 37 
 
 9.52598 
 
 38 
 
 9.52634 
 
 39 
 
 9.52669 
 
 40 
 
 9.52705 
 
 41 
 
 9.52740 
 
 42 
 
 9.52775 
 
 43 
 
 9.52811 
 
 44 
 
 9.52846 
 
 45 
 
 9.52881 
 
 46 
 
 9.52916 
 
 47 
 
 9.52951 
 
 48 
 
 9.52986 
 
 49 
 
 9.53021 
 
 50 
 
 9.53056 
 
 51 
 
 9.53092 
 
 52 
 
 9.53126 
 
 53 
 
 9.53161 
 
 54 
 
 9.53196 
 
 55 
 
 9.53231 
 
 56 
 
 9.53266 
 
 57 
 
 9.53301 
 
 58 
 
 9.53336 
 
 59 
 
 i 9.53370 
 
 JO 
 
 9.53405 
 1 L. Cos. 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 19 ° 
 
 511 
 
 d. L.Tang. 
 9.53697 
 9.53738 
 9.53779 
 9.53820 
 9.53861 
 
 37 
 
 36 
 
 37 
 36 
 
 36 
 
 37 
 36 
 36 
 
 36 
 
 37 
 36 
 36 
 36 
 36 
 36 
 36 
 36 
 36 
 36 
 
 9.53902 
 
 9.53943 
 
 9.53984 
 
 9.54025 
 
 9.54065 
 
 9.54106 
 
 9.54147 
 
 9.54187 
 
 9.54228 
 
 9.54269 
 
 36 
 
 35 
 
 36 
 36 
 36 
 
 35 
 
 36 
 
 35 
 
 36 
 
 35 
 
 36 
 
 35 
 
 36 
 
 35 
 
 36 
 35 
 
 35 
 
 36 
 35 
 35 
 35 
 35 
 35 
 35 
 
 35 
 
 36 
 
 34 
 
 35 
 35 
 35 
 35 
 35 
 35 
 
 34 
 
 35 
 
 d. c. 
 
 9.54309 
 
 9.54350 
 
 9.54390 
 
 9.54431 
 
 9.5 4471 
 
 9.54512 
 
 9.54552 
 
 9.54593 
 
 9.54633 
 
 9.54673 
 
 9.54714 
 
 9.54754 
 
 9.54794 
 
 9.54835 
 
 9.54875 
 
 9.54915 
 
 9.54955 
 
 9.54995 
 
 9.55035 
 
 9.55075 
 
 9.55115 
 9.55155 
 9.55195 
 9 55235 
 9.55275 
 
 9.55315 
 
 9.55355 
 
 9.55395 
 
 9.55434 
 
 9.55474 
 
 9.55514 
 
 9.55554 
 
 9.55593 
 
 9.55633 
 
 9.55673 
 
 9.55712 
 
 9.55752 
 
 9.55791 
 
 9.55831 
 
 9.55870 
 
 9.55910 
 
 9.55949 
 
 9.55989 
 
 9.56028 
 
 9.56067 
 
 9.56107 
 
 41 
 41 
 41 
 41 
 41 
 41 
 41 
 41 
 
 40 
 
 41 
 41 
 
 40 
 
 41 
 41 
 
 40 
 
 41 
 
 40 
 
 41 
 
 40 
 
 41 
 
 40 
 
 41 
 40 
 
 40 
 
 41 
 
 40 
 
 40 
 
 41 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 40 
 
 39 
 
 40 
 40 
 40 
 
 39 
 
 40 
 40 
 
 39 
 
 40 
 
 39 
 
 40 
 
 39 
 
 40 
 
 39 
 
 40 
 39 
 
 39 
 
 40 
 
 d. L. Cotg.l d. c. 
 
 j. Cotg. : 
 
 L. Cos. 
 
 d. 
 
 
 I 
 
 0.46303 
 
 9.97567 
 
 
 50 
 
 
 
 0.46262 
 
 9.97563 
 
 4 
 
 E 
 
 59 
 
 
 
 0.46221 
 
 9.97558 
 
 5 
 
 58 
 
 
 
 0.46180 
 
 9.97554 
 
 4 
 
 57 
 
 5 
 
 
 0.46139 
 
 9.97550 
 
 4 
 
 5 
 
 56 
 
 7 
 
 
 0.46098 
 
 9.97545 
 
 A 
 
 55 
 
 8 
 
 
 0.46057 
 
 9.97541 
 
 4 
 
 54 
 
 9 
 
 
 0.46016 
 
 9.97536 
 
 5 
 
 A 
 
 53 
 
 10 
 
 
 0.45975 
 
 9.97532 
 
 4 
 
 52 
 
 20 
 
 •1 
 
 0.45935 
 
 9.97528 
 
 4 
 
 5 
 
 51 
 
 30 
 
 ( 
 
 0.45894 
 
 9.97523 
 
 
 50 
 
 40 
 
 
 0.45853 
 
 9.97519 
 
 4 
 
 49 
 
 50 
 
 
 0.45813 
 
 9.97515 
 
 4 
 
 E 
 
 48 
 
 
 
 0.45772 
 
 9 97510 
 
 0 
 
 A 
 
 47 
 
 
 
 0.45731 
 
 9.97506 
 
 4 
 
 5 - 
 
 46 
 
 
 
 0.45691 
 
 9.97501 
 
 V 
 
 A 
 
 '45 
 
 
 ( 
 
 0.45650 
 
 9.97497 
 
 4 
 
 44 
 
 
 
 0.45610 
 
 9.97492 
 
 5 
 
 43 
 
 
 i 
 
 0.45569 
 
 9.97488 
 
 4 
 
 42 
 
 
 < 
 
 0.45529 
 
 9.97484 
 
 4 
 
 41 
 
 
 li 
 
 "0.45488 
 
 9.97479 
 
 o 
 
 A 
 
 40 
 
 
 2 < 
 
 0.45448 
 
 9.97475 
 
 4 
 
 39 
 
 
 o' 
 
 At 
 
 0.45407 
 
 9.97470 
 
 5 
 
 38 
 
 
 4 1 
 
 E, 
 
 0.45367 
 
 9.97466 
 
 4 
 
 E 
 
 37 
 
 
 O 1 
 
 0.45327 
 
 9.97461 
 
 0 
 
 4 
 
 36 
 
 
 
 0.45286 
 
 9.97457 
 
 
 35 
 
 
 
 0.45246 
 
 9.97453 
 
 4 
 
 34 
 
 
 
 0.45206 
 
 9.97448 
 
 5 
 
 33 
 
 6 
 
 
 0.45165 
 
 9.97444 
 
 4 
 
 E 
 
 32 
 
 7 
 
 
 0.45125 
 
 9.97439 
 
 0 
 
 4 
 
 31 
 
 8 
 
 Q 
 
 1 
 
 0.45085 
 
 9.97435 
 
 E 
 
 30 
 
 10 
 
 0.45045 
 
 9.97430 
 
 O 
 
 A 
 
 29 
 
 20 
 
 0.45005 
 
 9.97426 
 
 4 
 
 E 
 
 28 
 
 30 
 
 0.44965 
 
 9.97421 
 
 0 
 
 A 
 
 27 
 
 40 
 
 0.44925 
 
 9.97417 
 
 4 
 
 5 
 
 26 
 
 50 
 
 044885 
 
 9.97412 
 
 A 
 
 25 
 
 
 
 0.44845 
 
 9.97408 
 
 4 
 
 E 
 
 24 
 
 
 
 0.44805 
 
 9.97403 
 
 0 
 
 A 
 
 23 
 
 
 
 0.44765 
 
 9.97399 
 
 4 
 
 c 
 
 22 
 
 a 
 
 0.44725 
 
 9.97394 
 
 O 
 
 4 
 
 21 
 
 u 
 
 7 
 
 0.44685 
 
 9.97390 
 
 
 20 
 
 8 
 
 0.44645 
 
 9.97385 
 
 0 
 
 A 
 
 19 
 
 9 
 
 0.44605 
 
 9.97381 
 
 4 
 
 E 
 
 18 
 
 10 
 
 0.44566 
 
 9.97376 
 
 o 
 
 A 
 
 17 
 
 20 
 
 0.44526 
 
 9.97372 
 
 4 
 
 5 
 
 16 
 
 30 
 
 0.44486 
 
 9.97367" 
 
 A 
 
 15" 
 
 40 
 
 0.44446 
 
 9.97363 
 
 4 
 
 E 
 
 14 
 
 50 
 
 0.44407 
 
 9.97358 
 
 0 
 
 E 
 
 13 
 
 
 
 0.44367 
 
 9.97353 
 
 0 
 
 12 
 
 
 
 0.44327 
 
 9.97349 
 
 4 
 
 5 
 
 11 
 
 
 
 0.44288 
 
 9.97344 
 
 
 '0 
 
 
 6 
 
 0.44248 
 
 9.97340 
 
 ; i 
 
 9 
 
 
 *3 
 
 0.44209 
 
 i 9.97335 
 
 • l 
 
 8 
 
 
 * 
 
 0.44169 
 
 i 9.97331 
 
 4 
 
 7 
 
 
 c 
 
 0.44130 
 
 l 9.97326 
 
 L t 
 
 6 
 
 1C 
 
 0.4409C 
 
 1 9.97322 
 
 
 5 
 
 2 ( 
 
 0.44051 
 
 . 9.9731*3 
 
 r 0 
 
 4 
 
 
 OK 
 
 At 
 
 0.44011 
 
 l 9.97312 
 
 > 5 
 
 3 
 
 
 41 
 
 0.43971 
 
 > 9.9730* 
 
 S \ 
 
 2 
 
 
 Ol 
 
 0.4393* 
 
 5 9.9730* 
 
 \ l 
 
 1 
 
 
 
 0.4389* 
 
 5 9.97295 
 
 
 0 
 
 
 
 i. L.Tani 
 
 I L. Sin 
 
 . d. 
 
 / 
 
 
 P. P. 
 
 41 
 
 4.1 
 
 4.8 
 5.5 
 
 6.2 
 
 6.8 
 
 40 
 
 4.0 
 
 4.7 
 5.3 
 
 6.0 
 
 6.7 
 
 13.3 
 20.0 
 26.7 
 
 33.3 
 
 39 
 
 3.9 
 4.6 
 5.2 
 
 5.9 
 6.5 
 
 13.0 
 
 19.5 
 
 26.0 
 
 32.5 
 
 37 
 
 3.7 
 4.3 
 4.9 
 5.6 
 6.2 
 
 12.3 
 
 18.5 
 
 24.7 
 
 30.8 
 
 35 
 
 3.5 
 4.1 
 
 4.7 
 5.3 
 
 5.8 
 11.7 
 
 17.5 
 23.3 
 29.2 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4.2 
 
 36 
 
 3.6 
 
 4.2 
 
 4.8 
 
 5.4 
 
 6.0 
 
 12.0 
 
 18.0 
 
 24.0 
 
 30.0 
 
 34 
 
 3.4 
 
 4.0 
 
 4.5 
 
 5.1 
 5.7 
 
 11.3 
 17.0 
 22.7 
 
 28.3 
 
 4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.3 
 2.0 
 2.7 
 
 3.3 
 
 P. P. 
 
 70 ° 
 
512 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 20 ° 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.53405 
 
 1 
 
 9.53440 
 
 2 
 
 9.53475 
 
 3 
 
 9.53509 
 
 4 
 
 9.53544 
 
 5 
 
 9.53578 
 
 6 
 
 9.53613 
 
 7 
 
 9.53647 
 
 8 
 
 9.53682 
 
 9 
 
 9.53716 
 
 10 
 
 9.53751 
 
 11 
 
 9.53785 
 
 12 
 
 9.53819 
 
 13 
 
 9.53854 
 
 14 
 
 9.53888 
 
 15 
 
 9.53922 
 
 16 
 
 9.53957 
 
 17 
 
 9.53991 
 
 18 
 
 9.54025 
 
 19 
 
 9.54059 
 
 20 
 
 9.54093 
 
 21 
 
 9 54127 
 
 22 
 
 9.54161 
 
 23 
 
 9.54195 
 
 24 
 
 9.54229 
 
 25 
 
 9.54263 
 
 26 
 
 9.54297 
 
 27 
 
 9.54331 
 
 28 
 
 9.54365 
 
 29 
 
 9.54399 
 
 30 
 
 9.54433 
 
 31 
 
 9.54466 
 
 32 
 
 9.54500 
 
 33 
 
 9.54534 
 
 34 
 
 9.54567 
 
 35 
 
 9.54601 
 
 36 
 
 9.54635 
 
 37 
 
 9.54668 
 
 38 
 
 9.54702 
 
 39 
 
 9.54735 
 
 40 
 
 9.54769 
 
 41 
 
 9.54802 
 
 42 
 
 9.54836 
 
 43 
 
 9.54869 
 
 44 
 
 9.54903 
 
 45 
 
 9.54936 
 
 46 
 
 9.54969 
 
 47 
 
 9.55003 
 
 48 
 
 9.55036 
 
 49 
 
 9.55069 
 
 50 
 
 9.55102 
 
 51 
 
 9.55136 
 
 52 
 
 9.55169 
 
 53 
 
 9.55202 
 
 54 
 
 9.55235 
 
 55 
 
 9.55268 
 
 56 
 
 9.55301 
 
 57 
 
 9.55334 
 
 58 
 
 9.55367 
 
 59 
 
 9.55400 
 
 60 
 
 9.55433 
 
 
 L. Cos. 
 
 d. 
 
 35 
 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 34 
 
 34 
 
 35 
 34 
 
 34 
 
 35 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 
 33 
 
 34 
 34 
 
 33 
 
 34 
 34 
 
 33 
 
 34 
 
 33 
 
 34 
 
 33 
 
 34 
 
 33 
 
 34 
 33 
 
 33 
 
 34 
 33 
 33 
 
 33 
 
 34 
 33 
 33 
 33 
 33 
 33 
 33 
 33 
 33 
 33 
 
 L.Tang. 
 
 9.56107 
 
 9.56146 
 
 9.56185 
 
 9.56224 
 
 9.56264 
 
 9.56303 
 
 9.56342 
 
 9.56381 
 
 9.56420 
 
 9.56459 
 
 9.56498 
 
 9.56537 
 
 9.56576 
 
 9.56615 
 
 9.56654 
 
 9.56693 
 
 9.56732 
 
 9.56771 
 
 9.56810 
 
 9.56849 
 
 9.56887 
 
 9.56926 
 
 9.56965 
 
 9.57004 
 
 9.57042 
 
 9.57081 
 
 9.57120 
 
 9.57158 
 
 9.57197 
 
 9.57235 
 
 9.57274 
 
 9.57312 
 
 9.57351 
 
 9.57389 
 
 9.57428 
 
 9.57466 
 
 9.57504 
 
 9.57543 
 
 9.57581 
 
 9.57619 
 
 9.57658 
 
 9.57696 
 
 9.57734 
 
 9.57772 
 
 9.57810 
 
 9.57849 
 
 9.57887 
 
 9.57925 
 
 9.57963 
 
 9.58001 
 
 9.58039 
 
 9.58077 
 
 9.58115 
 
 9.58153 
 
 9.58191 
 
 9.58229 
 
 9.58267 
 
 9.58304 
 
 9.58342 
 
 9.58380 
 
 9.58418 
 
 d. L. Cotg, 
 
 d. c. 
 
 39 
 
 39 
 
 39 
 
 40 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 39 
 
 38 
 
 39 
 39 
 39 
 
 38 
 
 39 
 39 
 
 38 
 
 39 
 
 38 
 
 39 
 
 38 
 
 39 
 
 38 
 
 39 
 38 
 
 38 
 
 39 
 38 
 
 38 
 
 39 
 38 
 38 
 38 
 
 38 
 
 39 
 38 
 38 
 38 
 38 
 38 
 38 
 38 
 38 
 38 
 38 
 38 
 
 37 
 
 38 
 38 
 38 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0.43893 
 
 9.97299 
 
 
 60 
 
 
 
 
 
 0.43854 
 
 9.97294 
 
 5 
 
 59 
 
 
 
 
 
 0.43815 
 
 9.97289 
 
 5 
 
 58 
 
 
 
 
 
 0.43776 
 
 9.97285 
 
 4 
 
 57 
 
 
 4U 
 
 39 
 
 0.43736 
 
 9.97280 
 
 5 
 
 A 
 
 56 
 
 6 
 
 7 
 
 4.0 
 
 4.7 
 
 3.9 
 
 4.6 
 
 0.43697 
 
 9.97276 
 
 ‘i. 
 
 55 
 
 8 
 
 5.3 
 
 5.2, 
 
 0.43658 
 
 9.97271 
 
 5 
 
 54 
 
 9 
 
 6.0 
 
 5.9 
 
 0.43619 
 
 9.97266 
 
 5 
 
 53 
 
 10 
 
 6.7 
 
 6.5 
 
 0.43580 
 
 9.97262 
 
 4 
 
 52 
 
 20 
 
 13.3 
 
 13.0 
 
 0.43541 
 
 9.97257 
 
 5 
 
 51 
 
 30 ! 
 
 20.0 
 
 19.5 
 
 0.43502 
 
 9.97252 
 
 0 
 
 50 
 
 40 
 
 26.7 
 
 26.0 
 
 0.43463 
 
 9.97248 
 
 4 
 
 49 
 
 50 1 
 
 33.3 
 
 32.5 
 
 0.43424 
 
 9.97243 
 
 5 
 
 48 
 
 
 
 
 
 0.43385 
 
 9.97238 
 
 5 
 
 47 
 
 
 
 
 
 0.43346 
 
 9.97234 
 
 4 
 
 46 
 
 
 38 
 
 37 
 
 0.43307 
 
 9.97229 
 
 o 
 
 45 
 
 6 
 
 3.8 
 
 3.7 
 
 0.43268 
 
 9.97224 
 
 5 
 
 44 
 
 7 
 
 4.4 
 
 4.3 
 
 0.43229 
 
 9.97220 
 
 4 
 
 43 
 
 8 
 
 5.1 
 
 4.9 
 
 0.43190 
 
 9.97215 
 
 c 
 
 42 
 
 9 
 
 5.7 
 
 5.6 
 
 0.43151 
 
 9.97210 
 
 5 
 
 A 
 
 41 
 
 10 
 
 6.3 
 
 6.2 
 
 0.43113 
 
 9.97206 
 
 ‘i 
 
 40 
 
 20 
 
 12.7 
 
 12.3 
 
 0.43074 
 
 9.97201 
 
 5 
 
 39 
 
 30 
 
 19.0 
 
 18.5 
 
 0.43035 
 
 9.97196 
 
 5 
 
 38 
 
 40 
 
 25.3 
 
 24.7 
 
 0.42996 
 
 9.97192 
 
 4 
 
 37 
 
 50 
 
 31.7 
 
 30.8 
 
 0.42958 
 
 9.97187 
 
 5 
 
 36 
 
 
 
 
 
 0.42919 
 
 9.97182 
 
 O 
 
 35 
 
 
 
 
 
 0.42880 
 
 9.97178 
 
 4 
 
 34 
 
 
 
 35 
 
 0.42842 
 
 9.97173 
 
 5 
 
 33 
 
 
 6 
 
 3.5 
 
 0.42803 
 
 9.97168 
 
 5 
 
 32 
 
 
 7 
 
 4.1 
 
 0.42765 
 
 9.97163 
 
 5 
 
 A 
 
 31 
 
 
 8 
 
 4.7 
 
 0.42726 
 
 9.97159 
 
 
 30 
 
 y 
 
 in 
 
 5.3 
 
 p; Q 
 
 0.42688 
 
 9.97154 
 
 5 
 
 29 
 
 on 
 
 O.o 
 
 ii 7 
 
 0.42649 
 
 9.97149 
 
 5 
 
 28 
 
 ZU 
 
 on 
 
 11. / 
 ir c 
 
 0.42611 
 
 9.97145 
 
 4 
 
 27 
 
 30 
 
 An 
 
 1/.0 
 OQ Q 
 
 0.42572 
 
 9.97140 
 
 5 
 
 F» 
 
 26 
 
 
 Z6.6 
 9Q 9 
 
 0.42534 
 
 9.97135 
 
 o 
 
 25 
 
 
 
 
 
 0.42496 
 
 9.97130 
 
 5 
 
 24 
 
 
 
 
 
 0.42457 
 
 9.97126 
 
 4 
 
 23 
 
 
 
 
 
 0.42419 
 
 9.9712] 
 
 5 
 
 22 
 
 
 34 
 
 33 
 
 0.42381 
 
 9.97116 
 
 5 
 
 r 
 
 21 
 
 6 
 
 7 
 
 3.4 
 4 0 
 
 3.3 
 q Q 
 
 0.42342 
 
 9.97111 
 
 o 
 
 20 
 
 i 
 
 8 
 
 4.5 
 
 4.4 
 
 0.42304 
 
 9.97107 
 
 4 
 
 19 
 
 9 
 
 5.1 
 
 5.0 
 
 0.42266 
 
 9.97102 
 
 5 
 
 18 
 
 10 
 
 5.7 
 
 5.5 
 
 0.42228 
 
 9.97097 
 
 5 
 
 17 
 
 20 
 
 11.3 
 
 11.0 
 
 0.42190 
 
 9.97092 
 
 5 
 
 K 
 
 16 
 
 30 
 
 17.0 
 
 16.5 
 
 0.42151 
 
 9.97087 
 
 0 
 
 15 
 
 40 
 
 22.7 
 
 22.0 
 
 0.42113 
 
 9.97083 
 
 4 
 
 14 
 
 50 
 
 28.3 
 
 27.5 
 
 0.42075 
 
 9.97078 
 
 5 
 
 13 
 
 
 
 
 
 0.42037 
 
 9.97073 
 
 5 
 
 12 
 
 
 
 
 
 0.41999 
 
 9.97068 
 
 5 
 
 k 
 
 11 
 
 
 
 5 
 
 4 
 
 0.41961 
 
 9.97063 
 
 o 
 
 A 
 
 10 
 
 6 
 
 0.5 
 
 0.4 
 
 0.41923 
 
 9.97059 
 
 4 
 
 9 
 
 7 
 
 0.6 
 
 0.5 
 
 0.41885 
 
 9.97054 
 
 5 
 
 8 
 
 8 
 
 0.7 
 
 0.5 
 
 0.41847 
 
 9.97049 
 
 5 
 
 7 
 
 9 
 
 0.8 
 
 0.6 
 
 0.41809 
 
 9.97044 
 
 5 
 
 f; 
 
 6 
 
 10 
 
 0.8 
 
 0.7 
 
 0.41771 
 
 9.97039 
 
 o 
 
 5 
 
 20 
 
 1.7 
 
 1.3 
 
 0.41733 
 
 9.97035 
 
 4 
 
 4 
 
 30 
 
 2.5 
 
 2.0 
 
 0.41696 
 
 9.97030 
 
 5 
 
 3 
 
 40 
 
 3.3 
 
 2.7 
 
 0.41658 
 
 9.97025 
 
 5 
 
 2 
 
 50 
 
 4.2 
 
 3.3 
 
 0.41620 
 
 9.97020 
 
 5 
 
 1 
 
 
 
 
 
 0.41582 
 
 9.97015 
 
 o 
 
 0 
 
 
 
 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 21 ° 
 
 513 
 
 / 
 
 L. Sin. 
 
 0 1 
 
 3.55433 
 
 1 1 
 
 3.55466 
 
 2 ! 
 
 9.55499 
 
 3 ( 
 
 9.55532 
 
 4 ! 
 
 9.55564 
 
 5 1 
 
 9.55597 
 
 6 
 
 9.55630 
 
 7 
 
 9.55663 
 
 8 
 
 9.55695 
 
 9 
 
 9.55728 
 
 10 
 
 9.55761 
 
 11 
 
 9.55793 
 
 12 
 
 9.55826 
 
 13 
 
 9.55858 
 
 14 
 
 9.55891 
 
 15 
 
 9.55923 
 
 16 
 
 9.55956 
 
 17 
 
 9.55988 
 
 18 
 
 9.56021 
 
 19 
 
 9.56053 
 
 20 
 
 9.56085 
 
 21 
 
 9.56118 
 
 22 
 
 9.56150 
 
 23 
 
 9.56182 
 
 24 
 
 9.56215 
 
 25 
 
 9.56247 
 
 26 
 
 9.56279 
 
 27 
 
 9.56311 
 
 28 
 
 9.56343 
 
 29 
 
 9.56375 
 
 30 
 
 9.56408 
 
 31 
 
 9.56440 
 
 32 
 
 9.56472 
 
 33 
 
 9.56504 
 
 34 
 
 9.56536 
 
 35 
 
 9.56568 
 
 36 
 
 9.56599 
 
 37 
 
 9.56631 
 
 38 
 
 9.56663 
 
 39 
 
 9.56695 
 
 40 
 
 9.56727 
 
 41 
 
 9.56759 
 
 42 
 
 9.56790 
 
 43 
 
 9.56822 
 
 44 
 
 9.56854 
 
 45 
 
 9.56886 
 
 46 
 
 9.56917 
 
 47 
 
 9.56949 
 
 48 
 
 9.56980 
 
 49 
 
 9.57012 
 
 50 
 
 9.57044 
 
 51 
 
 9.57075 
 
 52 
 
 9.57107 
 
 53 
 
 9.57138 
 
 54 
 
 9.57169 
 
 55 
 
 ‘ 9.57201 
 
 56 
 
 9.57232 
 
 57 
 
 9.57264 
 
 58 
 
 9.57295 
 
 59 
 
 9.57326 
 
 60 
 
 9.57358 
 
 
 L. Cos. 
 
 33 
 
 33 
 
 33 
 
 32 
 
 33 
 33 
 33 
 
 32 
 
 33 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 32 
 
 32 
 
 33 
 32 
 
 32 
 
 33 
 32 
 32 
 32 
 32 
 
 32 
 
 33 
 32 
 32 
 32 
 32 
 32 
 
 31 
 
 32 
 32 
 32 
 32 
 32 
 
 31 
 
 32 
 32 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 32 
 
 31 
 
 32 
 31 
 
 31 
 
 32 
 
 31 
 
 32 
 31 
 
 31 
 
 32 
 
 If. 
 
 .Tang. 
 
 d.c. I 
 
 j. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 158418 
 
 158455 
 
 >.58493 
 
 158531 
 
 >.58569 
 
 37 
 
 38 
 38 
 38 
 
 Q7 
 
 0.41582 
 
 0.41545 
 
 0.41507 
 
 0.41469 
 
 0.41431 
 
 9.97015 
 
 9.97010 
 
 9.97005 
 
 9.97001 
 
 9.96996 
 
 5 
 
 5 
 
 4 
 
 5 
 5 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 >.58606 
 
 61 
 
 0.41394 
 
 9.96991 
 
 
 55 
 
 >.58644 
 
 38 
 
 0.41356 
 
 9.96986 
 
 5 
 
 54 
 
 >.58681 
 
 37 
 
 0.41319 
 
 9.96981 
 
 5 
 
 53 
 
 >.58719 
 
 38 
 
 0.41281 
 
 9.96976 
 
 5 
 
 52 
 
 >.58757 
 
 38 
 
 Q7 
 
 0.41243 
 
 9.96971 
 
 5 
 
 5 
 
 51 
 
 >.58794 
 
 61 
 
 0.41206 
 
 9.96966 
 
 A 1 
 
 50 
 
 >.58832 
 
 38 
 
 0.41168 
 
 9.96962 
 
 4 
 
 49 
 
 9.58869 
 
 37 
 
 0.41131 
 
 9.96957 
 
 5 
 
 E 
 
 48 
 
 9.58907 
 
 38 
 
 0.41093 
 
 9.96952 
 
 0 
 
 r 
 
 47 
 
 3.58944 
 
 37 
 
 07 
 
 0.41056 
 
 9.96947 
 
 5 
 
 5 
 
 46 
 
 9.58981 
 
 6/ 
 
 0.41019 
 
 9.96942 
 
 e 
 
 45 
 
 9.59019 
 
 38 
 
 0.40981 
 
 9.96937 
 
 0 
 
 e 
 
 44 
 
 9.59056 
 
 37 
 
 0.40944 
 
 9.96932 
 
 0 
 
 E 
 
 43 
 
 9.59094 
 
 38 
 
 0.40906 
 
 9.96927 
 
 O 
 
 e 
 
 42 
 
 9.59131 
 
 37 
 
 Q7 
 
 0.40869 
 
 9.96922 . 
 
 5 
 1 5 
 
 41 
 
 9.59168 
 
 61 
 
 0.40832 
 
 9.96917 
 
 e 
 
 40 
 
 9.59205 
 
 37 
 
 0.40795 
 
 9.96912 
 
 0 
 
 e 
 
 39 
 
 9.59243 
 
 38 
 
 0.40757 
 
 9.96907 
 
 0 
 
 A 
 
 38 
 
 9.59280 
 
 37 
 
 0.40720 
 
 9.96903 
 
 4 
 
 E 
 
 37 
 
 9.59317 
 
 37 
 
 07 
 
 0.40683 
 
 9.96898 
 
 5 
 
 5 
 
 36 
 
 9.59354 
 
 Oi 
 
 0.40646 
 
 9.96893 
 
 e 
 
 35 
 
 9.59391 
 
 37 
 
 0.40609 
 
 9.96888 
 
 O 
 
 E 
 
 34 
 
 9.59429 
 
 38 
 
 0.40571 
 
 9.96883 
 
 5 
 
 e 
 
 33 
 
 9.59466 
 
 37 
 
 0.40534 
 
 9.96878 
 
 O 
 
 E 
 
 32 
 
 9.59503 
 
 37 
 
 07 
 
 0.40497 
 
 9.96873 
 
 0 
 
 5 
 
 31 
 
 9.59540 
 
 61 
 
 0.40460 
 
 9.96868 
 
 c 
 
 30 
 
 9.59577 
 
 37 
 
 0.40423 
 
 9.96863 
 
 0 
 
 E 
 
 29 
 
 9.59614 
 
 37 
 
 0.40386 
 
 9.96858 
 
 0 
 
 E 
 
 28 
 
 9.59651 
 
 37 
 
 0.40349 
 
 9.96853 
 
 D 
 
 E 
 
 27 
 
 9.59688 
 
 37 
 
 Q7 
 
 0.40312 
 
 9.96848 
 
 0 
 
 5 
 
 26 
 
 9.59725 
 
 04. 
 
 0.40275 
 
 9.96843 
 
 E 
 
 25 
 
 9.59762 
 
 37 
 
 0.40238 
 
 9.96838 
 
 0 
 
 E 
 
 24 
 
 9.59799 
 
 37 
 
 0.40201 
 
 9.96833 
 
 0 
 
 E 
 
 23 
 
 9.59835 
 
 36 
 
 0.40165 
 
 9.96828 
 
 0 
 
 E 
 
 22 
 
 9.59872 
 
 37 
 
 Q7 
 
 0.40128 
 
 9.96823 
 
 0 
 
 5 
 
 21 
 
 9.59909 
 
 04 
 
 0.40091 
 
 9.96818 
 
 E 
 
 20 
 
 9.59946 
 
 37 
 
 0.40054 
 
 9.96813 
 
 0 
 
 E 
 
 19 
 
 9.59983 
 
 37 
 
 0.40017 
 
 9.96808 
 
 0 
 
 E 
 
 18 
 
 9.60019 
 
 36 
 
 0.39981 
 
 9.96803 
 
 0 
 
 E 
 
 17 
 
 9.60056 
 
 37 
 
 37 
 
 0.39944 
 
 9.96798 
 
 0 
 
 5 
 
 16 
 
 9.60093 
 
 0.39907 
 
 9.96793 
 
 E 
 
 15 
 
 9.60130 
 
 37 
 
 0.39870 
 
 9.96788 
 
 0 
 
 E 
 
 14 
 
 9.60166 
 
 36 
 
 0.39834 
 
 9.96783 
 
 0 
 
 E 
 
 13 
 
 9.60203 
 
 37 
 
 0.39797 
 
 9.96778 
 
 0 
 
 a 
 
 12 
 
 9.60240 
 
 37 
 
 oa 
 
 0.39760 
 
 9.96772 
 
 D 
 
 - 5 
 
 11 
 
 9.60276 
 
 DO 
 
 0.39724 
 
 9.96767 
 
 E 
 
 10 
 
 9.60313 
 
 37 
 
 0.39687 
 
 9.96762 
 
 0 
 
 E 
 
 9 
 
 9.60349 
 
 36 
 
 0.39651 
 
 9.96757 
 
 0 
 
 E 
 
 8 
 
 9.60386 
 
 37 
 
 0.39614 
 
 9.96752 
 
 D 
 
 E 
 
 7 
 
 9.60422 
 
 36 
 
 *37 
 
 0.39578 
 
 9.96747 
 
 0 
 
 5 
 
 6 
 
 9.60459 
 
 oi 
 
 0.39541 
 
 9.96742 
 
 
 5 
 
 9.60495 
 
 gg 
 
 0.39505 
 
 9.96737 
 
 5 
 
 4 
 
 9.60532 
 
 i 37 
 
 0.39468 
 
 ; 9.96732 
 
 > 5 
 
 3 
 
 9.60568 
 
 l jj® 
 
 0.39432 
 
 : 9.96727 
 
 ’ 5 
 
 2 
 
 9.60605 
 
 ' 36 
 
 0.39395 
 
 > 9.96722 
 
 ! ^ 
 r & 
 
 1 
 
 9.60641 
 
 
 0.3935? 
 
 > 9.96717 
 
 
 0 
 
 L. Cotg 
 
 d. c. 
 
 , L.Tang 
 
 ;. L. Sin, 
 
 . d. 
 
 / 
 
 P. P. 
 
 38 
 
 3.8 
 
 4.4 
 
 5.1 
 
 5.7 
 
 6.3 
 
 12.7 
 19.0 
 25.3 
 
 31.7 
 
 38 
 
 3.6 
 
 4.2 
 
 4.8 
 
 5.4 
 
 6.0 
 
 12.0 
 
 18.0 
 
 24.0 
 
 30.0 
 
 37 
 
 3.7 
 4.3 
 4.9 
 5.6 
 6.2 
 
 12.3 
 
 18.5 
 
 24.7 
 
 30.8 
 
 33 
 
 3.3 
 3.9 
 
 4.4 
 5.0 
 
 5.5 
 11.0 
 
 16.5 
 
 22.0 
 
 27.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 32 
 
 3.2 
 
 3.7 
 
 4.3 
 
 4.8 
 
 5.3 
 
 10.7 
 16.0 
 21.3 
 
 26.7 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 31 
 
 3.1 
 
 3.6 
 
 4.1 
 
 4.7 
 
 5.2 
 10.3 
 15.5 
 
 20.7 
 
 25.8 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4.2 
 
 6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.3 
 2.0 
 2.7 
 
 3.3 
 
 P. P. 
 
 lO 
 
514 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 22 ° 
 
 r 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 
 P 
 
 . P. 
 
 0 
 
 9.57358 
 
 
 9.60641 
 
 
 0.39359 
 
 9.96717 
 
 
 60 
 
 
 
 
 
 1 
 
 9.57389 
 
 31 
 
 9.60677 
 
 36 
 
 0.39323 
 
 9.96711 
 
 6 
 
 59 
 
 
 
 
 
 2 
 
 9.57420 
 
 31 
 
 9.60714 
 
 37 
 
 0.39286 
 
 9.96706 
 
 5 
 
 58 
 
 
 
 
 
 3 
 
 9.57451 
 
 31 
 
 9.60750 
 
 36 
 
 0.39250 
 
 9.96701 
 
 5 
 
 57 
 
 ct 
 
 a/ 
 
 O *7 
 
 36 
 
 4 
 
 9.57482 
 
 31 
 
 QO 
 
 9.60786 
 
 36 
 
 37 
 
 0.39214 
 
 9.96696 
 
 5 
 
 56 
 
 D 
 
 7 
 
 3.7 
 
 4.3 
 
 3.6 
 
 4.2 
 
 5 
 
 9.57514 
 
 
 9.60823 
 
 0.39177 
 
 9.96691 
 
 o 
 
 '55 
 
 8 
 
 4!9 
 
 4^8 
 
 6 
 
 9.57545 
 
 31 
 
 9.60859 
 
 36 
 
 0.39141 
 
 9.96686 
 
 5 
 
 54 
 
 9 
 
 5.6 
 
 5.4 
 
 7 
 
 9.57576 
 
 31 
 
 9.60895 
 
 36 
 
 0.39105 
 
 9.96681 
 
 5 
 
 53 
 
 10 
 
 6.2 
 
 6.0 
 
 . 8 
 
 9.57607 
 
 31 
 
 9.60931 
 
 36 
 
 0.39069 
 
 9.96676 
 
 5 
 
 52 
 
 20 
 
 12.3 
 
 12.0 
 
 9 
 
 9.57638 
 
 31 
 
 9.60967 
 
 36 
 
 37 
 
 0.39033 
 
 9.96670 
 
 6 
 
 51 
 
 30 
 
 18.5 
 
 18.0 
 
 10 
 
 9.57669 
 
 
 9.61004 
 
 0.38996 
 
 9.96665 
 
 O 
 
 50 
 
 40 
 
 24.7 
 
 24.0 
 
 11 
 
 9.57700 
 
 31 
 
 9.61040 
 
 36 
 
 0.38960 
 
 9.96660 
 
 5 
 
 49 
 
 50 
 
 30.8 
 
 30.0 
 
 12 
 
 9.57731 
 
 31 
 
 9.61076 
 
 36 
 
 0.38924 
 
 9.96655 
 
 5 
 
 48 
 
 
 
 
 
 13 
 
 9.57762 
 
 31 
 
 9.61112 
 
 36 
 
 0.38888 
 
 9.96650 
 
 5 
 
 47 
 
 
 
 
 
 14 
 
 9.57793 
 
 31 
 
 q-i 
 
 9.61148 
 
 36 
 
 36 
 
 0.38852 
 
 9.96645 
 
 5 
 
 46 
 
 
 
 35 
 
 15 
 
 9.57824 
 
 
 9.61184 
 
 0.38816 
 
 9.96640 
 
 0 
 
 45 
 
 
 6 
 
 3.5 
 
 16 
 
 9.57855 
 
 31 
 
 9.61220 
 
 36 
 
 0.38780 
 
 9.96634 
 
 6 
 
 44 
 
 
 7 
 
 4.1 
 
 17 
 
 9.57885 
 
 30 
 
 9.61256 
 
 36 
 
 0.38744 
 
 9.96629 
 
 5 
 
 43 
 
 
 8 
 
 4.7 
 
 18 
 
 9.57916 
 
 31 
 
 9.61292 
 
 36 
 
 0.38708 
 
 9.96624 
 
 5 
 
 42 
 
 
 9 
 
 5.3 
 
 19 
 
 9.57947 
 
 31 
 
 9.61328 
 
 36 
 
 36 
 
 0.38672 
 
 9.96619 
 
 5 
 
 41 
 
 10 
 
 5.8 
 
 20 
 
 9.57978 
 
 o± 
 
 9.61364 
 
 0.38636 
 
 9.96614 
 
 0 
 
 40 
 
 20 
 
 11.7 
 
 21 
 
 9.58008 
 
 30 
 
 9.61400 
 
 36 
 
 0.38600 
 
 9.96608 
 
 6 
 
 39 
 
 30 
 
 17 .5 
 
 22 
 
 9.58039 
 
 31 
 
 9.61436 
 
 36 
 
 0.38564 
 
 9.96603 
 
 5 
 
 38 
 
 40 
 
 23.3 
 
 23 
 
 9.58070 
 
 31 
 
 9.61472 
 
 36 
 
 0.38528 
 
 9.96598 
 
 5 
 
 37 
 
 50 
 
 29.2 
 
 24 
 
 9.58101 
 
 31 
 
 9.61508 
 
 36 
 
 36 
 
 0.38492 
 
 9.96593 
 
 5 
 
 36 
 
 
 
 
 
 25 
 
 9.58131 
 
 
 9.61544 
 
 0.38456 
 
 9.96588 
 
 0 
 
 35 
 
 
 
 
 
 26 
 
 9.58162 
 
 31 
 
 9.61579 
 
 35 
 
 0.38421 
 
 9.96582 
 
 6 
 
 34 
 
 
 32 
 
 31 
 
 27 
 
 9.58192 
 
 30 
 
 9.61615 
 
 36 
 
 0.38385 
 
 9.96577 
 
 5 
 
 33 
 
 6 
 
 3.2 
 
 3.1 
 
 28 
 
 9.58223 
 
 31 
 
 9.61651 
 
 36 
 
 0.38349 
 
 9.96572 
 
 5 
 
 32 
 
 7 
 
 3.7 
 
 3.6 
 
 29 
 
 9.58253 
 
 30 
 
 31 
 
 9.61687 
 
 36 
 
 35 
 
 0.38313 
 
 9.96567 
 
 5 
 
 31 
 
 8 
 
 4.3 
 
 4.1 
 
 30 
 
 9.58284 
 
 9.61722 
 
 0.38278 
 
 9.96562 
 
 o 
 
 30 
 
 9 
 
 1 A 
 
 4.8 
 
 K Q 
 
 4.7 
 
 K O 
 
 31 
 
 9.58314 
 
 30 
 
 9.61758 
 
 36 
 
 0.38242 
 
 9.96556 
 
 6 
 
 29 
 
 1U 
 
 on 
 
 O.o 
 in *7 
 
 O.Z 
 in q 
 
 32 
 
 9.58345 
 
 31 
 
 9.61794 
 
 36 
 
 0.38206 
 
 9.96551 
 
 5 
 
 28 
 
 ZU 
 
 on 
 
 1U. / 
 i c* n 
 
 1 KJ.O 
 
 ICC 
 
 33 
 
 9.58375 
 
 30 
 
 9.61830 
 
 36 
 
 0.38170 
 
 9.96546 
 
 5 
 
 27 
 
 oU 
 
 An 
 
 Lo.U 
 
 Ol Q 
 
 10. 0 
 on h 
 
 34 
 
 9.58406 
 
 31 
 
 30 
 
 9.61865 
 
 35 
 
 36 
 
 0.38135 
 
 9.96541 
 
 5 
 
 a 
 
 26 
 
 50 
 
 Zl.o 
 
 9fi.7 
 
 Zkj. / 
 
 25.8 
 
 35 
 
 9.58436 
 
 9.61901 
 
 0.38099 
 
 9.96535 
 
 0 
 
 25 
 
 
 
 36 
 
 9.58467 
 
 31 
 
 9.61936 
 
 . 35 
 
 0.38064 
 
 9.96530 
 
 5 
 
 24 
 
 
 
 
 
 37 
 
 9.58497 
 
 30 
 
 9.61972 
 
 36 
 
 0.38028 
 
 9.96525 
 
 5 
 
 23 
 
 
 
 
 
 38 
 
 9.58527 
 
 30 
 
 9.62008 
 
 36 
 
 0.37992 
 
 9.96520 
 
 5 
 
 22 
 
 
 ou 
 
 29 
 
 39 
 
 9.58557 
 
 30 
 
 31 
 
 9.62043 
 
 35 
 
 36 
 
 0.37957 
 
 9.96514 
 
 6 
 
 21 
 
 6 
 
 7 
 
 3.0 
 
 q f; 
 
 2.9 
 
 Q 1 
 
 40 
 
 9.58588 
 
 9.62079 
 
 0.37921 
 
 9.96509 
 
 o 
 
 20 
 
 i 
 
 3 
 
 0.0 
 
 4.0 
 
 O.i 
 
 3.9 
 
 41 
 
 9.58618 
 
 30 
 
 9.62114 
 
 35 
 
 0.37886 
 
 9.96504 
 
 5 
 
 19 
 
 9 
 
 4/5 
 
 4/4 
 
 42 
 
 9.58648 
 
 30 
 
 9.62150 
 
 36 
 
 0.37850 
 
 9.96498 
 
 6 
 
 18 
 
 10 
 
 5^0 
 
 4*8 
 
 43 
 
 9.58678 
 
 30 
 
 9.62185 
 
 35 
 
 0.37815 
 
 9.96493 
 
 5 
 
 17 
 
 20 
 
 io!o 
 
 9^7 
 
 44 
 
 9.58709 
 
 31 
 
 30 
 
 9.62221 
 
 36 
 
 35 
 
 0.37779 
 
 9.96488 
 
 5 
 
 K 
 
 16 
 
 30 
 
 15!o 
 
 14.5 
 
 45 
 
 9.58739 
 
 9.62256 
 
 0.37744 
 
 9.96483 
 
 o 
 
 15 
 
 40 
 
 20.0 
 
 19.3 
 
 46 
 
 9.58769 
 
 30 
 
 9.62292 
 
 36 
 
 0.37708 
 
 9.96477 
 
 6 
 
 14 
 
 50 
 
 25.0 
 
 24.2 
 
 47 
 
 9.58799 
 
 30 
 
 9.62327 
 
 35 
 
 0.37673 
 
 9.96472 
 
 5 
 
 13 
 
 
 
 
 
 48 
 
 9.58829 
 
 30 
 
 9.62362 
 
 35 
 
 0.37638 
 
 9.96467 
 
 5 
 
 12 
 
 
 
 
 
 49 
 
 9.58859 
 
 30 
 
 30 
 
 9.62398 
 
 36 
 
 35 
 
 0.37602 
 
 9.96461 
 
 6 
 
 11 
 
 
 
 fi 
 
 5 
 
 50 
 
 9.58889 
 
 9.62433 
 
 0.37567 
 
 9.96456 
 
 o 
 
 10 
 
 6 
 
 0.6 
 
 0.5 
 
 51 
 
 9.58919 
 
 30 
 
 9.62468 
 
 35 
 
 0.37532 
 
 9.96451 
 
 5 
 
 9 
 
 7 
 
 0.7 
 
 0.6 
 
 52 
 
 9.58949 
 
 30 
 
 9.62504 
 
 3 6 l 
 
 0.37496 
 
 9.96445 
 
 6 
 
 8 
 
 8 
 
 0.8 
 
 0.7 
 
 53 
 
 9.58979 
 
 30 
 
 9.62539 
 
 35 
 
 0.37461 
 
 9.96440 
 
 5 
 
 7 
 
 9 
 
 0.9 
 
 0.8 
 
 54 
 
 9.59009 
 
 30 
 
 30 
 
 9.62574 
 
 35 
 
 35 
 
 0.37426 
 
 9.96435 
 
 5 
 
 6 
 
 6 
 
 10 
 
 1.0 
 
 0.8 
 
 55 
 
 9.59039 
 
 9.62609 
 
 0.37391 
 
 9.96429 
 
 5 
 
 20 
 
 2.0 
 
 1.7 
 
 56 
 
 9.59069 
 
 30 
 
 9.62645 
 
 36 
 
 0.37355 
 
 9.96424 
 
 5 
 
 4 
 
 30 
 
 3.0 
 
 2.5 
 
 57 
 
 9.59098 
 
 29 
 
 9.62680 
 
 35 
 
 0.37320 
 
 9.96419 
 
 5 
 
 3 
 
 40 
 
 4.0 
 
 3.3 
 
 58 
 
 9.59128 
 
 30 
 
 9.62715 
 
 35 
 
 0.37285 
 
 9.96413 
 
 6 
 
 2 
 
 50 
 
 5.0 
 
 4.2 
 
 59 
 
 9.59158 
 
 30 
 
 30 
 
 9.62750 
 
 35 
 
 35 
 
 0.37250 
 
 9.96408 
 
 5 
 
 * 
 
 1 
 
 
 
 
 
 60 
 
 9.59188 
 
 9.62785 
 
 0.37215 
 
 9.96403 
 
 O 
 
 0 
 
 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 t 
 
 P. P. 
 
 67 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 23 ° 
 
 515 
 
 t 
 
 L. Sin.^ 
 
 0 
 
 9.59188 
 
 1 
 
 9.59218 
 
 2 
 
 9.59247 
 
 3 
 
 9.59277 
 
 4 
 
 9.59307 
 
 5 
 
 9.59336 
 
 6 
 
 9.59366 
 
 7 
 
 9.59396 
 
 8 
 
 9.59425 
 
 9 
 
 9.59455 
 
 10 
 
 9.59484 
 
 11 
 
 9.59514 
 
 12 
 
 9.59543 
 
 13 
 
 9.59573 
 
 14 
 
 9.59602 
 
 15 
 
 9.59632 
 
 16 
 
 9.59661 
 
 17 
 
 9.59690 
 
 18 
 
 9.59720 
 
 19 
 
 9.59749 
 
 20 
 
 9.59778 
 
 21 
 
 9.59808 
 
 22 
 
 9.59837 
 
 23 
 
 9.59866 
 
 24 
 
 9.59895 
 
 25 
 
 9.59924 
 
 26 
 
 9.59954 
 
 27 
 
 9.59983 
 
 28 
 
 9.60012 
 
 29 
 
 9.60041 
 
 30 
 
 9.60070 
 
 31 
 
 9.60099 
 
 32 
 
 9.60128 
 
 33 
 
 9.60157 
 
 34 
 
 9.60186 
 
 35 
 
 9.60215 
 
 36 
 
 9.60244 
 
 37 
 
 9.60273 
 
 38 
 
 9.60302 
 
 39 
 
 9.60331 
 
 40 
 
 9.60359 
 
 41 
 
 9.60388 
 
 42 
 
 9.60417 
 
 43 
 
 9.60446 
 
 44 
 
 9.60474 
 
 45 
 
 *9.60503 
 
 46 
 
 9.60532 
 
 47 
 
 9.60561 
 
 48 
 
 9.60589 
 
 49 
 
 9.60618 
 
 50 
 
 9.60646 
 
 51 
 
 9.60675 
 
 52 
 
 9.60704 
 
 53 
 
 9.60732 
 
 54 
 
 9.60761 
 
 55 
 
 9.60789 
 
 56 
 
 9.60818 
 
 57 
 
 9.60846 
 
 58 
 
 9.60875 
 
 59 
 
 9.60903 
 
 60 
 
 9.60931 
 
 
 L. Cos. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 0.37215 
 
 9.96403 
 
 
 60 
 
 0.37180 
 
 9.96397 
 
 6 
 
 59 
 
 0.37145 
 
 9.96392 
 
 5 
 
 58 
 
 0.37110 
 
 9.96387 
 
 5 
 
 57 
 
 0.37074 
 
 9.96381 
 
 6 
 
 f; 
 
 56 
 
 0.37039 
 
 9.96376 
 
 u 
 
 55 
 
 0.37004 
 
 9.96370 
 
 6 
 
 54 
 
 0.36969 
 
 9.96365 
 
 5 
 
 53 
 
 0.36934 
 
 9.96360 
 
 5 
 
 52 
 
 0.36899 
 
 9.96354 
 
 6 
 
 5 
 
 51 
 
 0.36865 
 
 9.96349 
 
 50 
 
 0.36830 
 
 9.96343 
 
 6 
 
 49 
 
 0.36795 
 
 9.96338 
 
 5 
 
 48 
 
 0.36760 
 
 9.96333 
 
 5 
 
 47 
 
 0.36725 
 
 9.96327 
 
 6 
 
 46 
 
 0.36690 
 
 9.96322 
 
 o 
 
 6 
 
 45 
 
 0.36655 
 
 9.96316 
 
 44 
 
 0.36621 
 
 9.96311 
 
 5 
 
 43 
 
 0.36586 
 
 9.96305 
 
 6 
 
 42 
 
 0.36551 
 
 9.96300 
 
 5 
 
 6 
 
 41 
 
 0.36516 
 
 9.96294 
 
 40 
 
 0.36481 
 
 9.96289 
 
 5 
 
 39 
 
 0.36447 
 
 9.96284 
 
 5 
 
 38 
 
 0.36412 
 
 9.96278 
 
 6 
 
 37 
 
 0.36377 
 
 9.96273 
 
 5 
 
 6 
 
 36 
 
 0.36343 
 
 9.96267 
 
 35 
 
 0.36308 
 
 9.96262 
 
 5 
 
 34 
 
 0.36274 
 
 9.96256 
 
 6 
 
 33 
 
 0.36239 
 
 9.96251 
 
 5 
 
 32 
 
 0.36204 
 
 9.96245 
 
 6 
 
 5 
 
 31 
 
 0.36170 
 
 9.9624C 
 
 30 
 
 0.36135 
 
 9.96234 
 
 6 
 
 29 
 
 0.36101 
 
 9.96229 
 
 5 
 
 28 
 
 0.36066 
 
 9.96223 
 
 6 
 
 27 
 
 0.36032 
 
 9.96218 
 
 5 
 
 6 
 
 26 
 
 0.35997 
 
 9.96212 
 
 25 
 
 0.35963 
 
 9.96207 
 
 5 
 
 24 
 
 0.35928 
 
 9.96201 
 
 6 
 
 23 
 
 0.35894 
 
 9.96196 
 
 5 
 
 22 
 
 0.35860 
 
 9.96190 
 
 6 
 
 5 
 
 21 
 
 0.35825 
 
 9.96185 
 
 20 
 
 0.35791 
 
 9.96179 
 
 6 
 
 19 
 
 0.35757 
 
 9.96174 
 
 5 
 
 18 
 
 0.35722 
 
 9.96168 
 
 6 
 
 17 
 
 0.35688 
 
 9.96162 
 
 6 
 
 5 
 
 16 
 
 0.35654 
 
 9.96157 
 
 15 
 
 0.35619 
 
 9.96151 
 
 6 
 
 14 
 
 0.35585 
 
 9.96146 
 
 5 
 
 6 
 
 13 
 
 0.35551 
 
 9.96140 
 
 12 
 
 0.35517 
 
 9.96135 
 
 5 
 
 6 
 
 11 
 
 0.35483 
 
 9.96129 
 
 10 
 
 0.35448 
 
 9.96123 
 
 6 
 
 9 
 
 0.35414 
 
 9.96118 
 
 5 
 
 8 
 
 0.35380 
 
 9.96112 
 
 6 
 
 7 
 
 0.35346 
 
 9.96107 
 
 5 
 
 6 
 
 6 
 
 0.35312 
 
 9.96101 
 
 5 
 
 0.35278 
 
 9.96095 
 
 6 
 
 4 
 
 0.35244 
 
 9.96090 
 
 5 
 
 3 
 
 0.35210 
 
 9.96084 
 
 6 
 
 2 
 
 0.35176 
 
 9.96079 
 
 5 
 
 6 
 
 1 
 
 0.35142 
 
 9.96073 
 
 0 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 t 
 
 d. L.Tang. 
 
 30 
 
 29 
 
 30 
 30 
 
 29 
 
 30 
 30 
 
 29 
 
 30 
 
 29 
 
 30 
 
 29 
 
 30 
 
 29 
 
 30 
 29 
 
 29 
 
 30 
 29 
 
 29 
 
 30 
 29 
 29 
 29 
 
 29 
 
 30 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 28 
 29 
 29 
 29 
 28 
 29 
 29 
 29 
 28 
 29 
 28 
 29 
 29 
 28 
 29 
 28 
 29 
 28 
 29 
 28 
 28 
 
 9.62785 
 
 9.62820 
 
 9.62855 
 
 9.62890 
 
 9.62926 
 
 9.62961 
 
 9.62996 
 
 9.63031 
 
 9.63066 
 
 9.63101 
 
 9.63135 
 
 9.63170 
 
 9.63205 
 
 9.63240 
 
 9.63275 
 
 9.63310 
 
 9.63345 
 
 9.63379 
 
 9.63414 
 
 9.63449 
 
 9.63484 
 
 9.63519 
 
 9.63553 
 
 9.63588 
 
 9.63623 
 
 9.63657 
 
 9.63692 
 
 9.63726 
 
 9.63761 
 
 9.63796 
 
 9.63830 
 
 9.63865 
 
 9.63899 
 
 9.63934 
 
 9.63968 
 
 d. c. 
 
 9.64003 
 
 9.64037 
 
 9.64072 
 
 9.64106 
 
 9.64140 
 
 9.64175 
 
 9.64209 
 
 9.64243 
 
 9.64278 
 
 9.64312 
 
 9.64346 
 
 9.64381 
 
 9.64415 
 
 9.64449 
 
 9.64483 
 
 9.64517 
 
 9.64552 
 
 9.64586 
 
 9.64620 
 
 9.64654 
 
 1164688 
 
 9.64722 
 
 9.64756 
 
 9.64790 
 
 9.64824 
 
 9164858 
 
 d. L. Cotg 
 
 35 
 
 35 
 
 35 
 
 36 
 35 
 35 
 35 
 35 
 35 
 
 34 
 
 35 
 35 
 35 
 35 
 35 
 35 
 
 34 
 
 35 
 35 
 35 
 
 35 
 
 34 
 
 35 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 
 34 
 
 35 
 34 
 
 34 
 
 35 
 34 
 
 34 
 
 35 
 34 
 
 34 
 
 35 
 34 
 34 
 34 
 
 34 
 
 35 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 34 
 
 d. c. 
 
 P. P. 
 
 36 
 
 3.6 
 
 4.2 
 
 4.8 
 
 5.4 
 
 6.0 
 
 12.0 
 
 18.0 
 
 24.0 
 
 30.0 
 
 35 
 
 3.5 
 
 4.1 
 
 4.7 
 
 5.3 
 
 5.8- 
 
 11.7* 
 
 17.5 
 
 23.3 
 
 29.2 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 34 
 
 3.4 
 
 4.0 
 
 4.5 
 
 5.1 
 5.7 
 
 11.3 
 17.0 
 22.7 
 
 28.3 
 
 30 
 
 3.0 
 
 3.5 
 
 4.0 
 
 4.5 
 
 5.0 
 10.0 
 
 15.0 
 
 20.0 
 
 25.0 
 
 29 
 
 2.9 
 3.4 
 
 3.9 
 •4.4 
 4.8 
 9.7 
 
 14.5 
 
 19.3 
 
 24.2 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 28 
 
 2.8 
 
 3.3 
 
 3.7 
 
 4.2 
 
 4.7 
 
 9.3 
 14.0 
 18.7 
 23.3 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 .20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4.2 
 
 P. P. 
 
 66 
 
516 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 24 ° 
 
 f 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. ] 
 
 L. Cotg. 
 
 L. Cos. 
 
 0 
 
 1 
 
 9.60931 
 
 9.60960 
 
 29 
 
 9.64858 
 
 9.64892 
 
 34 
 
 0.35142 
 
 0.35108 
 
 9.96073 
 
 9.96067 
 
 2 
 
 9.60988 
 
 28 
 
 9.64926 
 
 34 
 
 0.35074 
 
 9.96062 
 
 3 
 
 9.61016 
 
 28 
 
 9.64960 
 
 34 
 
 0.35040 
 
 9.96056 
 
 4 
 
 9.61045 
 
 29 
 
 28 
 
 28 
 
 9.64994 
 
 34 
 
 34 - 
 34 
 
 0.35006 
 
 9.96050 
 
 5 
 
 6 
 
 9.61073 
 
 9.61101 
 
 9.65028 
 
 9.65062 
 
 0.34972 
 
 0.34938 
 
 9.96045 
 
 9.96039 
 
 7 
 
 9.61129 
 
 28 
 
 9.65096 
 
 34 
 
 0.34904 
 
 9.96034 
 
 8 
 
 9.61158 
 
 29 
 
 9.65130 
 
 34 
 
 0.34870 
 
 9.96028 
 
 9 
 
 9.61186 
 
 28 
 
 28 
 
 28 
 
 9.65164 
 
 34 
 
 33 
 
 34 
 
 0.34836 
 
 9.96022 
 
 10 
 
 11 
 
 9.61214 
 
 9.61242 
 
 9.65197 
 
 9.65231 
 
 0.34803 
 
 0.34769 
 
 9.96017 
 
 9.96011 
 
 12 
 
 9.61270 
 
 28 
 
 9.65265 
 
 34 
 
 0.34735 
 
 9.96005 
 
 13 
 
 9.61298 
 
 28 
 
 9.65299 
 
 34 
 
 0.34701 
 
 9.96000 
 
 14 
 
 9.61326 
 
 28 
 
 28 
 
 28 
 
 9.65333 
 
 34 
 
 qq 
 
 0.34667 
 
 9.95994 
 
 15 
 
 16 
 
 9.61354 
 
 9.61382 
 
 9.65366 
 
 9.65400 
 
 OO 
 
 34 
 
 0.34634 
 
 0.34600 
 
 9.95988 
 
 9.95982 
 
 17 
 
 9.61411 
 
 29 
 
 9.65434 
 
 34 
 
 0.34566 
 
 9.95977 
 
 18 
 
 9.61438 
 
 27 
 
 9.65467 
 
 33 
 
 0.34533 
 
 9.95971 
 
 19 
 
 9.61466 
 
 28 
 
 28 
 
 28 
 
 9.65501 
 
 34 
 
 04 . 
 
 0.34499 
 
 9.95965 
 
 20 
 
 21 
 
 9.61494 
 
 9.61522 
 
 9.65535 
 
 9.65568 
 
 cyi 
 
 33 
 
 0.34465 
 
 0.34432 
 
 9.95960 
 
 9.95954 
 
 22 
 
 9.61550 
 
 28 
 
 9.65602 
 
 34 
 
 0.34398 
 
 9.95948 
 
 23 
 
 9.61578 
 
 28. 
 
 9.65636 
 
 34 
 
 0.34364 
 
 9.95942 
 
 24 
 
 9.61606 
 
 28 
 
 28 
 
 28 
 
 9.65669 
 
 33 
 
 °A 
 
 0.34331 
 
 9.95937 
 
 25 
 
 26 
 
 9.61634 
 
 9.61662 
 
 9.65703 
 
 9.65736 
 
 irt 
 
 33 
 
 0.34297 
 
 0.34264 
 
 9.95931 
 
 9.95925 
 
 27 
 
 9.61689 
 
 27 
 
 9.65770 
 
 34 
 
 0.34230 
 
 9.95920 
 
 28 
 
 9.61717 
 
 28 
 
 9.65803 
 
 33 
 
 0.34197 
 
 9.95914 
 
 29, 
 
 9.61745 
 
 28 
 
 28 
 
 27 
 
 9.65837 
 
 34 
 
 QQ 
 
 0.34163 
 
 9.95908 
 
 30 
 
 31 
 
 9.61773 
 
 9.61800 
 
 9.65870 
 
 9.65904 
 
 Ot) 
 
 34 
 
 0.34130 
 
 0.34096 
 
 9.95902 
 
 9.95897 
 
 32 
 
 9.61828 
 
 28 
 
 9.65937 
 
 33 
 
 0.34063 
 
 9.95891 
 
 33 
 
 9.61856 
 
 28 
 
 9.65971 
 
 34 
 
 0.34029 
 
 9.95885 
 
 34 
 
 9.61883 
 
 27 
 
 28 
 28 
 
 9.66004 
 
 33 
 
 0.33996 
 
 9.95879 
 
 35 
 
 36 
 
 9.61911 
 
 9.61939 
 
 9.66038 
 
 9.66071 
 
 33 
 
 0.33962 
 
 0.33929 
 
 9.95873 
 
 9.95868 
 
 37 
 
 9.61966 
 
 27 
 
 9.66104 
 
 33 
 
 0.33896 
 
 9.95862 
 
 38 
 
 9.61994 
 
 28 
 
 9.66138 
 
 34 
 
 0.33862 
 
 9.95856 
 
 39 
 
 9.62021 
 
 27 
 
 28 
 27 
 
 9.66171 
 
 33 
 
 0.33829 
 
 9.95850 
 
 40 
 
 41 
 
 1162049 
 
 9.62076 
 
 9.66204 
 
 9.66238 
 
 OO 
 
 34 
 
 0.33796 
 
 0.33762 
 
 9.95844 
 
 9.95839 
 
 42 
 
 9.62104 
 
 28 
 
 9.66271 
 
 33 
 
 0.33729 
 
 9.95833 
 
 43 
 
 9.62131 
 
 27 
 
 9.66304 
 
 33 
 
 0.33696 
 
 9.95827 
 
 44 
 
 9.62159 
 
 28 
 
 27 
 
 28 
 
 9.66337 
 
 33 
 
 34 
 33 
 
 0.33663 
 
 9.95821 
 
 45 
 
 46 
 
 9.62186 
 
 9.62214 
 
 9.66371 
 
 9.66404 
 
 0.33629 
 
 0.33596 
 
 9.95815 
 
 9.95810 
 
 47 
 
 9.62241 
 
 27 
 
 9.66437 
 
 33 
 
 0.33563 
 
 9.95804 
 
 48 
 
 9.62268 
 
 27 
 
 9.66470 
 
 33 
 
 0.33530 
 
 9.95798 
 
 49 
 
 9.62296 
 
 28 
 
 27 
 
 27 
 
 9.66503 
 
 33 
 
 34 
 33 
 
 0.33497 
 
 9.95792 
 
 50 
 
 51 
 
 9.62323 
 
 9.62350 
 
 9.66537 
 
 9.66570 
 
 0.33463 
 
 0.33430 
 
 9.95786 
 
 9.95780 
 
 52 
 
 9.62377 
 
 27 
 
 9.66603 
 
 33 
 
 0.33397 
 
 9.95775 
 
 53 
 
 9.62405 
 
 28 
 
 9.66636 
 
 33 
 
 0.33364 
 
 9.95769 
 
 54 
 
 9.62432 
 
 27 
 
 27 
 
 9.66669 
 
 33 
 
 33 
 
 0.33331 
 
 9.95763 
 
 55 
 
 9.62459 
 
 9.66702 
 
 0.33298 
 
 9.95757 
 
 56 
 
 9.62486 
 
 27 
 
 9.66735 
 
 33 
 
 0.33265 
 
 9.95751 
 
 57 
 
 9.62513 
 
 27 
 
 9.66768 
 
 33 
 
 0.33232 
 
 9.95745 
 
 58 
 
 9.62541 
 
 28 
 
 9.66801 
 
 33 
 
 0.33199 
 
 9.95739 
 
 59 
 
 9.62568 
 
 27 
 
 07 
 
 9.66834 
 
 33 
 
 QQ 
 
 0.33166 
 
 9.95733 
 
 60 
 
 ' 9.62595 
 
 
 9.66867 
 
 OO 
 
 0.33133 
 
 9.95728 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg 
 
 . d. c. 
 
 L.Tang 
 
 . L. Sin. 
 
 d. 
 
 6 
 
 5 
 
 6 
 6 
 
 5 
 
 6 
 
 5 
 
 6 
 6 
 
 5 
 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 5 
 
 ~dT 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 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 
 
 14 
 
 13 
 
 12 
 
 11 
 
 to 
 
 9 
 
 8 
 
 7 
 
 _6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 P. P. 
 
 34 
 
 3.4 
 
 4.0 
 
 4.5 
 
 5.1 
 5.7 
 
 11.3 
 17.0 
 22.7 
 
 28.3 
 
 33 
 
 3.3 
 3.9 
 
 4.4 
 5.0 
 
 5.5 
 11.0 
 
 16.5 
 22.0 
 
 27.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 29 
 
 2.9 
 
 3.4 
 
 3.9 
 
 4.4 
 4.8 
 9.7 
 
 14.5 
 
 19.3 
 
 24.2 
 
 28 
 
 2.8 
 
 3.3 
 
 3.7 
 
 4.2 
 
 4.7 
 
 9.3 
 14.0 
 18.7 
 23.3 
 
 27 
 
 2.7 
 
 3.2 
 
 3.6 
 
 4.1 
 
 4.5 
 
 9.0 
 
 13.5 
 18.0 
 
 22.5 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4-2 
 
 P. P. 
 
 65 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 25 ° 
 
 517 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.62595 
 
 1 
 
 9.62622 
 
 2 
 
 9.62649 
 
 3 
 
 9.62676 
 
 4 
 
 9.62703 
 
 5 
 
 9.62730 
 
 6 
 
 9.62757 
 
 7 
 
 9.62784 
 
 8 
 
 9.62811 
 
 9 
 
 9.62838 
 
 10 
 
 '9.62865 
 
 11 
 
 9.62892 
 
 12 
 
 9.62918 
 
 13 
 
 9.62945 
 
 14 
 
 9.62972 
 
 15 
 
 9.62999 
 
 16 
 
 9.63026 
 
 17 
 
 9.63052 
 
 18 
 
 9.63079 
 
 19 
 
 9.63106 
 
 20 
 
 9.63133 
 
 21 
 
 9.63159 
 
 22 
 
 9.63186 
 
 23 
 
 9.63213 
 
 24 
 
 9.63239 
 
 25 
 
 9.63266 
 
 26 
 
 9.63292 
 
 27 
 
 9.63319 
 
 28 
 
 9.63345 
 
 29 
 
 9.63372 
 
 30 
 
 9.63398 
 
 31 
 
 9.63425 
 
 32 
 
 9.63451 
 
 33 
 
 9.63478 
 
 34 
 
 9.63504 
 
 35 
 
 9.63531 
 
 36 
 
 9.63557 
 
 37 
 
 9.63583 
 
 38 
 
 9.63610 
 
 39 
 
 9.63636 
 
 40 
 
 9.63662 
 
 41 
 
 9.63689 
 
 42 
 
 9.63715 
 
 43 
 
 9.63741 
 
 44 
 
 9.63767 
 
 45 
 
 9.63794 
 
 46 
 
 9.63820 
 
 47 
 
 9.63846 
 
 48 
 
 9.63872 
 
 49 
 
 9.63898 
 
 50 
 
 9.63924 
 
 51 
 
 9.63950 
 
 52 
 
 9.63976 
 
 53 
 
 9.64002 
 
 54 
 
 9.64028 
 
 55 
 
 9.64054 
 
 56 
 
 9.64080 
 
 57 
 
 9.64106 
 
 58 
 
 9.64132 
 
 59 
 
 9.64158 
 
 60 
 
 9.64184 
 
 
 L. Cos. 
 
 d. 
 
 27 
 27 . 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 26 
 27 
 27 
 27 
 27 
 26 
 27 
 27 
 27 
 26 
 27 
 27 
 26 
 27 
 26 
 27 
 26 
 27 
 26 
 27 
 26 
 27 
 26 
 27 
 26 
 26 
 27 
 26 
 26 
 27 
 26 
 26 
 26 
 27 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 26 
 
 L.Tang. d. c. |L. Cotg. 
 
 9.66867 
 
 9.66900 
 
 9.66933 
 
 9.66966 
 
 9.66999 
 
 9.67032 
 
 9.67065 
 
 9.67098 
 
 9.67131 
 
 9.67163 
 
 9.67196 
 
 9.67229 
 
 9.67262 
 
 9.67295 
 
 9.67327 
 
 9.67360 
 
 9.67393 
 
 9.67426 
 
 9.67458 
 
 9.67491 
 
 9.67524 
 
 9.67556 
 
 9.67589 
 
 9.67622 
 
 9.67654 
 
 d. 
 
 9.67687 
 
 9.67719 
 
 9.67752 
 
 9.67785 
 
 9.67817 
 
 9.67850 
 
 9.67882 
 
 9.67915 
 
 9.67947 
 
 9.67980 
 
 9.68012 
 
 9.68044 
 
 9.68077 
 
 9.68109 
 
 9.68142 
 
 9.68174 
 
 9.68206 
 
 9.68239 
 
 9.68271 
 
 9.68303 
 
 9.68336 
 
 9.68368 
 
 9.68400 
 
 9.68432 
 
 9.68465 
 
 9.68497 
 
 9.68529 
 
 9.68561 
 
 9.68593 
 
 9.68626 
 
 9.68658 
 
 9.68690 
 
 9.68722 
 
 9.68754 
 
 9.68786 
 
 9.68818 
 
 L. Cotg. 
 
 33 
 
 33 
 
 33 
 
 33 
 
 33 
 
 33 
 
 33 
 
 33 
 
 32 
 
 33 
 33 
 33 
 33 
 
 32 
 
 33 
 33 
 33 
 
 32 
 
 33 
 33 
 
 32 
 
 33 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 
 32 
 
 33 
 32 
 
 32 
 
 33 
 
 32 
 
 33 
 32 
 
 32 
 
 33 
 32 
 
 32 
 
 33 
 32 
 32 
 
 32 
 
 33 
 32 
 32 
 32 
 
 32 
 
 33 
 32 
 32 
 32 
 32 
 32 
 32 
 
 0.33133 
 
 0.33100 
 
 0.33067 
 
 0.33034 
 
 0.33001 
 
 0.32968 
 
 0.32935 
 
 0.32902 
 
 0.32869 
 
 0.32837 
 
 0.32804 
 
 0.32771 
 
 0.32738 
 
 0.32705 
 
 0.32673 
 
 0.32640 
 
 0.32607 
 
 0.32574 
 
 0.32542 
 
 0.32509 
 
 d. c. 
 
 0.32476 
 
 0.32444 
 
 0.32411 
 
 0.32378 
 
 0.32346 
 
 0.32313 
 
 0.32281 
 
 0.32248 
 
 0.32215 
 
 0.32183 
 
 L. Cos^ 
 9.95728 
 9.95722 
 9.95716 
 9.95710 
 9.95704 
 
 9.95698 
 
 9.95692 
 
 9.95686 
 
 9.95680 
 
 9.95674 
 
 9.95668 
 
 9.95663 
 
 9.95657 
 
 9.95651 
 
 9.95645 
 
 9.95639 
 
 9.95633 
 
 9.95627 
 
 9.95621 
 
 9.95615 
 
 9.95609 
 
 9.95603 
 
 9.95597 
 
 9.95591 
 
 9.95585 
 
 0.32150 
 
 0.32118 
 
 0.32085 
 
 0.32053 
 
 0.32020 
 
 0.31988 
 
 0.31956 
 
 0.31923 
 
 0.31891 
 
 0.31858 
 
 0.31826 
 
 0.31794 
 
 0.31761 
 
 0.31729 
 
 0.31697 
 
 0.31664 
 
 0.31632 
 
 0.31600 
 
 0.31568 
 
 0.31535 
 
 0.31503 
 
 0.31471 
 
 0.31439 
 
 0.31407 
 
 0.31374 
 
 0.31342 
 
 0.31310 
 
 0.31278 
 
 0.34246 
 
 0.31214 
 
 0.31182 
 
 9.95579 
 
 9.95573 
 
 9.95567 
 
 9.95561 
 
 9.95555 
 
 9.95549 
 
 9.95543 
 
 9.95537 
 
 9.95531 
 
 9.95525 
 
 9.95519 
 
 9.95513 
 
 9.95507 
 
 9.95500 
 
 9.95494 
 
 9.95488 
 
 9.95482 
 
 9.95476 
 
 9.95470 
 
 9.95464 
 
 9.95458 
 
 9.95452 
 
 9.95446 
 
 9.95440 
 
 9.95434 
 
 9.95427 
 
 9.95421 
 
 9.95415 
 
 9.95409 
 
 9.95403 
 
 9.95397 
 
 9.95391 
 
 9.95384 
 
 9.95378 
 
 9.95372 
 
 9.95366 
 
 _d._ 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 6 
 
 5 
 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 7 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 7 
 6 
 6 
 6 
 6 
 6 
 6 
 7 
 6 
 6 
 6 
 
 L.Tang. L. Sin. 
 
 64 ° 
 
 d. 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 _36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 P. P. 
 
 33 
 
 3.3 
 3.9 
 
 4.4 
 5.0 
 
 5.5 
 11.0 
 
 16.5 
 
 22.0 
 
 27.5 
 
 32 
 
 3.2 
 
 3.7 
 
 4.3 
 
 4.8 
 
 5.3 
 
 10.7 
 16.0 
 21.3 
 
 26.7 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 27 
 
 2.7 
 
 3.2 
 
 3.6 
 
 4.1 
 
 4.5 
 
 9.0 
 
 13.5 
 18.0 
 
 22.5 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 7 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.1 
 
 1.2 
 
 2.3 
 
 3.5 
 
 4.7 
 
 5.8 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3.3 
 
 4.2 
 
 P. P. 
 
518 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 ' 26 ° 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.64 L84 
 
 1 
 
 9.64210 
 
 2 
 
 9.64236 
 
 3 
 
 9.64262 
 
 4 
 
 9.64288 
 
 5 
 
 9.64313 
 
 6 
 
 9.64339 
 
 7 
 
 9.64365 
 
 8 
 
 9.64391 
 
 9 
 
 9.64417 
 
 10 
 
 9.64442 
 
 11 
 
 9.64468 
 
 12 
 
 9.64494 
 
 13 
 
 9.64519 
 
 14 
 
 9.64545 
 
 15 
 
 9.64571 
 
 16 
 
 9.64596 
 
 17 
 
 9.64622 
 
 18 
 
 9.64647 
 
 19 
 
 9.64673 
 
 20 
 
 9 64698 
 
 21 
 
 9.34724 
 
 22 
 
 9.64749 
 
 23 
 
 9.64775 
 
 24 
 
 9.64800 
 
 25 
 
 9.64826 
 
 26 
 
 9.64851 
 
 27 
 
 9.64877 
 
 28 
 
 9.64902 
 
 29 
 
 9.64927 
 
 30 
 
 9.64953 
 
 31 
 
 9.64978 
 
 32 
 
 9.65003 
 
 33 
 
 9.65029 
 
 34 
 
 9.65054 
 
 35 
 
 9.65079 
 
 36 
 
 9.65104 
 
 37 
 
 9.65130 
 
 38 
 
 9.65155 
 
 39 
 
 9.65180 
 
 40 
 
 9.65205 
 
 41 
 
 9.65230 
 
 42 
 
 9.65255 
 
 43 
 
 9.65281 
 
 44 
 
 9.65306 
 
 45 
 
 9.65331 
 
 46 
 
 9.65356 
 
 47 
 
 9.65381 
 
 48 
 
 9.65406 
 
 49 
 
 9.65431 
 
 50 
 
 9.65456 
 
 51 
 
 9.65481 
 
 52 
 
 9.65506 
 
 53 
 
 9.65531 
 
 54 
 
 9.65556 
 
 55 
 
 9.65580 
 
 56 
 
 9.65605 
 
 57 
 
 9.65630 
 
 58 
 
 9.65655 
 
 59 
 
 9.65680 
 
 60 
 
 9.65705 
 
 L.Tang. 
 
 ¥.'68818 
 
 9.68850 
 
 9.68882 
 
 9.68914 
 
 9.68946 
 
 9.68978 
 
 9.69010 
 
 9.69042 
 
 9.69074 
 
 9.69106 
 
 9.69138 
 
 9.69170 
 
 9.69202 
 
 9.69234 
 
 9.69266 
 
 9.69298 
 
 9.69329 
 
 9.69361 
 
 9.69393 
 
 9.69425 
 
 9.69457 
 
 9.69488 
 
 9.69520 
 
 9.69552 
 
 9.69584 
 
 9.69615 
 
 9.69647 
 
 9.69679 
 
 9.69710 
 
 9.69742 
 
 9.69774 
 
 9.69805 
 
 9.69837 
 
 9.69868 
 
 9.69900 
 
 9.69932 
 . 9.69963 
 9.69995 
 9.70026 
 9.70058 
 
 9.70089 
 
 9.70121 
 
 9.70152 
 
 9.70184 
 
 9.70215 
 
 9.70247 
 
 9.70278 
 
 9.70309 
 
 9.70341 
 
 9.70372 
 
 9.70404 
 
 9.70435 
 
 9.70466 
 
 9.70498 
 
 9.70529 
 
 9.70560 
 
 9.70592 
 
 9.70623 
 
 9.70654 
 
 9.70685 
 
 9.70717 
 
 L. Cotg. 
 
 L. Cotg. 
 
 L. Cos. 
 
 0.31182 
 
 9.95366 
 
 0.31150 
 
 9.95360 
 
 0.31118 
 
 9.95354 
 
 0.31086 
 
 9.95348 
 
 0.31054 
 
 9.95341 
 
 0.31022 
 
 9.95335 
 
 0.30990 
 
 9.95329 
 
 0.30958 
 
 9.95323 
 
 0.30926 
 
 9.95317 
 
 0.30894 
 
 9.95310 
 
 0.30862 
 
 9.95304 
 
 0.30830 
 
 9.95298 
 
 0.30798 
 
 9.95292 
 
 0.30766 
 
 9.95286 
 
 0.30734 
 
 9.95279 
 
 0.30702 
 
 9.95273 
 
 0.30671 
 
 9.95267 
 
 0.30639 
 
 9.95261 
 
 0.30607 
 
 9.95254 
 
 0.30575 
 
 9.95248 
 
 0.30543 
 
 9.95242 
 
 0.30512 
 
 9.95236 
 
 0.30480 
 
 9.95229 
 
 0.30448 
 
 9.95223 
 
 0.30416 
 
 9.95217 
 
 0.30385 
 
 9.95211 
 
 0.30353 
 
 9.95204 
 
 0.30321 
 
 9.95198 
 
 0.30290 
 
 9.95192 
 
 0.30258 
 
 9.95185 
 
 0.30226 
 
 9.95179 
 
 0.30195 
 
 9.95173 
 
 0.30163 
 
 9.95167 
 
 0.30132 
 
 9.95160 
 
 0.30100 
 
 9.95154 
 
 0.30068 
 
 9.95148 
 
 0.30037 
 
 9.95141 
 
 0.30005 
 
 9.95135 
 
 0.29974 
 
 9.95129 
 
 0.29942 
 
 9.95122 
 
 0.29911 
 
 9.95116 
 
 0.29879 
 
 9.95110 
 
 0.29848 
 
 9.95103 
 
 0.29816 
 
 9.95097 
 
 0.29785 
 
 9.95090 
 
 0.29753 
 
 9.95084 
 
 0.29722 
 
 9.95078 
 
 0.29691 
 
 9.95071 
 
 0.29659 
 
 9.95065 
 
 0.29628 
 
 9.95059 
 
 0.29596 
 
 9.95052 
 
 0.29565 
 
 9.95046 
 
 0.29534 
 
 9.95039 
 
 0.29502 
 
 9.95033 
 
 0.29471 
 
 9.95027 
 
 0.29440 
 
 9.95020 
 
 ■ 0.29408 
 
 9.95014 
 
 0.29?77 
 
 9.95007 
 
 0.29346 
 
 9.95001 
 
 0.29315 
 
 9.94995 
 
 0.29283 
 
 9.94988 
 
 L.Tang. 
 
 L. Sin. 
 
 L. Cos. 
 
 d. 
 
 26 
 
 26 
 
 26 
 
 26 
 
 25 
 
 26 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 25 
 25 
 25 
 25 
 25 
 25 
 25 
 25 
 25 
 
 24 
 
 25 
 25 
 25 
 25 
 25 
 
 d. c. 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 32 
 
 31 
 
 32 
 32 
 32 
 32 
 
 31 
 
 32 
 32 
 32 
 
 31 
 
 32 
 32 
 
 31 
 
 32 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 
 31 
 
 32 
 31 
 
 31 
 
 32 
 
 31 
 
 32 
 31 
 
 31 
 
 32 
 31 
 
 31 
 
 32 
 31 
 31 
 
 31 
 
 32 
 
 d. 
 
 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 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 P. P. 
 
 32 
 
 3.2 
 
 3.7 
 
 4.3 
 
 4.8 
 
 5.3 
 
 10.7 
 16.0 
 21.3 
 
 26.7 
 
 31 
 
 3.1 
 
 3.6 
 
 4.1 
 
 4.7 
 
 5.2 
 10.3 
 15.5 
 
 20.7 
 
 25.8 
 
 6 
 
 n 
 
 3 
 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 25 
 
 2.5 
 
 2.9 
 
 3.3 
 
 3.8 
 
 4.2 
 
 8.3 
 12.5 
 
 16.7 
 
 20.8 
 
 24 
 
 2.4 
 
 2.8 
 
 3.2 
 
 3.6 
 
 4.0 
 
 8.0 
 12.0 
 16.0 
 20.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 7 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.1 
 
 1.2 
 
 2.3 
 
 3.5 
 
 4.7 
 
 5.8 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 P. P. 
 
 63 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 27 ° 
 
 519 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.65705" 
 
 1 
 
 9.65729 
 
 2 
 
 9.65754 
 
 3 
 
 9.65779 
 
 4 
 
 9.65804 
 
 5 
 
 9.65828 
 
 6 
 
 9.65853 
 
 7 
 
 9.65878 
 
 8 
 
 9.65902 
 
 9 
 
 9.65927 
 
 10 
 
 9.65952 
 
 11 
 
 9.65976 
 
 12 
 
 9.66001 
 
 13 
 
 9.66025 
 
 14 
 
 9.66050 
 
 15 
 
 9.66075 
 
 16 
 
 9.66099 
 
 17 
 
 9.66124 
 
 18 
 
 9.66148 
 
 19 
 
 9.66173 
 
 20 
 
 9.66197 
 
 21 
 
 9.66221 
 
 22 
 
 9.66246 
 
 23 
 
 9.66270 
 
 24 
 
 9.66295 
 
 25 
 
 9.66319 
 
 26 
 
 9.66343 
 
 27 
 
 9.66368 
 
 28 
 
 9.66392 
 
 29 
 
 9.66416 
 
 30 
 
 9.66441 
 
 31 
 
 9.66465 
 
 32 
 
 9.66489 
 
 33 
 
 9.66513 
 
 34 
 
 9.66537 
 
 35 
 
 9.66562 
 
 36 
 
 9.66586 
 
 37 
 
 9.66610 
 
 38 
 
 9.66634 
 
 39 
 
 9.66658 
 
 40 
 
 9.66682 
 
 41 
 
 9.66706 
 
 42 
 
 9.66731 
 
 43 
 
 9.66755 
 
 44 
 
 9.66779 
 
 45 
 
 9.66803 
 
 46 
 
 9.66827 
 
 47 
 
 9.66851 
 
 48 
 
 9.66875 
 
 49 
 
 9.66899 
 
 50 
 
 9.66922 
 
 51 
 
 9.66946 
 
 52 
 
 9.66970 
 
 53 
 
 9.66994 
 
 54 
 
 9.67018 
 
 55 
 
 9.67042 
 
 56 
 
 9.67066 
 
 57 
 
 9.67090 
 
 58 
 
 9.67113 
 
 59 
 
 9.67137 
 
 60 
 
 9.67161 
 
 
 L. Cos. 
 
 L.Tang. 
 
 9.70717 
 
 9.70748 
 
 9.70779 
 
 9.70810 
 
 9.70841 
 
 9.70873 
 
 9.70904 
 
 9.70935 
 
 9.70966 
 
 9.70997 
 
 9.71028 
 
 9.71059 
 
 9.71090 
 
 9.71121 
 
 9.71153 
 
 9.71184 
 
 9.71215 
 
 9.71246 
 
 9.71277 
 
 9.71308 
 
 9.71339 
 
 9.71370 
 
 9.71401 
 
 9.71431 
 
 9.71462 
 
 9.71493 
 
 9.71524 
 
 9.71555 
 
 9.71586 
 
 9.71617 
 
 9.71648 
 
 9.71679 
 
 9.71709 
 
 9.71740 
 
 9.71771 
 
 9.71802 
 
 9.71833 
 
 9.71863 
 
 9.71894 
 
 9.71925 
 
 9.71955 
 
 9.71986 
 
 9.72017 
 
 9.72048 
 
 9.72078 
 
 9.72109 
 
 9.72140 
 
 9.72170 
 
 9.72201 
 
 9.72231 
 
 9.72262 
 
 9.72293 
 
 9.72323 
 
 9.72354 
 
 9.72384 
 
 9.72415 
 
 9.72445 
 
 9.72476 
 
 9.72506 
 
 9.72537 
 
 9.72567 
 
 L. Cotg. 
 
 d. 
 
 24 
 
 25 
 25 
 25 
 
 24 
 
 25 
 25 
 
 24 
 
 25 
 25 
 
 24 
 
 25 
 
 24 
 
 25 
 25 
 
 24 
 
 25 
 
 24 
 
 25 
 24 
 
 24 
 
 25 
 
 24 
 
 25 
 24 
 
 24 
 
 25 
 24 
 
 24 
 
 25 
 24 
 24 
 24 
 
 24 
 
 25 
 24 
 24 
 24 
 24 
 24 
 
 24 
 
 25 
 24 
 24 
 24 
 24 
 24 
 24 
 24 
 
 23 
 
 24 
 24 
 24 
 24 
 24 
 24 
 24 
 
 23 
 
 24 
 24 
 
 d. 
 
 d. c. 
 
 31 
 
 31 
 
 31 
 
 31 
 
 32 
 31 
 31 
 31 
 31 
 31 
 31 
 31 
 
 31 
 
 32 
 31 
 31 
 31 
 31 
 31 
 31 
 31 
 31 
 
 30 
 
 31 
 31 
 31 
 31 
 31 
 31 
 31 
 31 
 
 30 
 
 31 
 31 
 31 
 31 
 
 30 
 
 31 
 31 
 
 30 
 
 31 
 31 
 31 
 
 30 
 
 31 
 31 
 
 30 
 
 31 
 
 30 
 
 31 
 31 
 
 30 
 
 31 
 
 30 
 
 31 
 
 30 
 
 31 
 
 30 
 
 31 
 30 
 
 |L. Cot g. 
 0.29283 
 0.29252 
 0.29221 
 0.29190 
 0.29159 
 
 0.29127 
 
 0.29096 
 
 0.29065 
 
 0.29034 
 
 0.29003 
 
 0.28972 
 
 0.28941 
 
 0.28910 
 
 0.28879 
 
 0.28847 
 
 0.28816 
 
 0.28785 
 
 0.28754 
 
 0.28723 
 
 0.28692 
 
 0.28661 
 
 0.28630 
 
 0.28599 
 
 0.28569 
 
 0.28538 
 
 0.28507 
 
 0.28476 
 
 0.28445 
 
 0.28414 
 
 0.28383 
 
 0.28352 
 
 0.28321 
 
 0.28291 
 
 0.28260 
 
 0.28229 
 
 0.28198 
 
 0.28167 
 
 0.28137 
 
 0.28106 
 
 0.28075 
 
 0.28045 
 
 0.28014 
 
 0.27983 
 
 0.27952 
 
 0.27922 
 
 0.27891 
 
 0.27860 
 
 0.27830 
 
 0.27799 
 
 0.27769 
 
 0.27738 
 
 0.27707 
 
 0.27677 
 
 0.27646 
 
 0.27616 
 
 0.27585 
 
 0.27555 
 
 0.27524 
 
 0.27494 
 
 0.27463 
 
 0.27433 
 
 d. c. L.Tang 
 
 L. Cos. d. 
 9.94988" 
 9.94982 
 9.94975 
 9.94969 
 9.94962 
 
 9.94956 
 
 9.94949 
 
 9.94943 
 
 9.91936 
 
 9.94930 
 
 9.94923 
 
 9.94917 
 
 9.94911 
 
 9.94904 
 
 9.94898 
 
 9.94891 
 
 9.94885 
 
 9.94878 
 
 9.94871 
 
 9.94865 
 
 9.94858 
 
 9.94852 
 
 9.94845 
 
 9.94839 
 
 9.94832 
 
 9.94826 
 
 9.94819 
 
 9.94813 
 
 9.94806 
 
 9.94799 
 
 9.94793 
 
 9.94786 
 
 9.94780 
 
 9.94773 
 
 9.94767 
 
 9.94760 
 
 9.94753 
 
 9.94747 
 
 9.94740 
 
 9.94734 
 
 9.94727 
 
 9.94720 
 
 9.94714 
 
 9.94707 
 
 9.94700 
 
 9.94694 
 
 9.94687 
 
 9.94680 
 
 9.94674 
 
 9.94667 
 
 9.94660 
 
 9.94654 
 
 9.94647 
 
 9.94640 
 
 9.94634 
 
 9.94627 
 
 9.94620 
 
 9.94614 
 
 9.94607 
 
 9.94600 
 
 9.94593 
 
 L. Sin. d 
 
 P. P. 
 
 32 
 
 3.2 
 
 3.7 
 
 4.3 
 
 4.8 
 
 5.3 
 
 10.7 
 16.0 
 21.3 
 
 26.7 
 
 31 
 
 3.1 
 
 3.6 
 
 4.1 
 
 4.7 
 
 5.2 
 10.3 
 15.5 
 
 20.7 
 
 25.8 
 
 6 
 
 7 
 
 •8 
 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 30 
 
 3.0 
 
 3.5 
 
 4.0 
 
 4.5 
 
 5.0 
 10.0 
 
 15.0 
 
 20.0 
 25.0 
 
 25 
 
 2.5 
 
 2.9 
 
 3.3 
 
 3.8 
 
 4.2 
 
 8.3 
 12.5 
 
 16.7 
 
 20.8 
 
 24 
 
 2.4 
 
 2.8 
 
 3.2 
 
 3.6 
 
 4.0 
 
 8.0 
 12.0 
 16.0 
 20.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 23 
 
 2.3 
 
 2.7 
 3.1 
 3.5 
 
 3.8 
 7.7 
 
 11.5 
 
 15.3 
 
 19.2 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 7 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.1 
 
 1.2 
 
 2.3 
 
 3.5 
 
 4.7 
 
 5.8 
 
 6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 2.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 P. P. 
 
 62 ° 
 
520 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 28 ° 
 
 ' 1 
 
 L. Sin. 
 
 0 
 
 9.67161 
 
 1 
 
 9.67185 
 
 2 
 
 9.67208 
 
 3 
 
 9.67232 
 
 4 
 
 9.67256 
 
 5 
 
 9.67280 
 
 6 
 
 9.67303 
 
 7 
 
 9.67327 
 
 8 
 
 9.67350 
 
 9 
 
 9.67374 
 
 10 
 
 9.67398 
 
 11 
 
 9.67421 
 
 12 
 
 9.67445 
 
 13 
 
 9.67468 
 
 14 
 
 9.67492 
 
 15 
 
 9.67515 
 
 16 
 
 9.67539 
 
 17 
 
 9.67562 
 
 18 
 
 9.67586 
 
 19 
 
 9.67609 
 
 20 
 
 9„67633 
 
 21 
 
 9.67656 
 
 22 
 
 9.67680 
 
 23 
 
 9.67703 
 
 24 
 
 9.67726 
 
 25 
 
 9.67750 
 
 26 
 
 9.67773 
 
 27 
 
 9.67796 
 
 28 
 
 9.67820 
 
 29 
 
 9.67843 
 
 30 
 
 9.67866 
 
 31 
 
 9.67890 
 
 32 
 
 9.67913 
 
 33 
 
 9.67936 
 
 34 
 
 9.67959 
 
 35 
 
 9.67982 
 
 36 
 
 9.68006 
 
 37 
 
 9.68029 
 
 38 
 
 9.68052 
 
 39 
 
 9.68075 
 
 40 
 
 9.68098 
 
 41 
 
 9.68121 
 
 42 
 
 9.68144 
 
 43 
 
 9.68167 
 
 44 
 
 9.68190 
 
 45 
 
 9.68213 
 
 46 
 
 9.68237 
 
 47 
 
 9.68260 
 
 48 
 
 9.68283 
 
 49 
 
 9.68305 
 
 50 
 
 9.68328 
 
 51 
 
 9.68351 
 
 52 
 
 9.68374 
 
 53 
 
 9.68397 
 
 54 
 
 9.68420 
 
 55 
 
 9.68443 
 
 56 
 
 9.68466 
 
 57 
 
 9.68489 
 
 58 
 
 9.68512 
 
 59 
 
 9.68534 
 
 60 
 
 9.68557 
 
 
 L. Cos. 
 
 
 P. 
 
 P. 
 
 
 
 31 
 
 30 
 
 1 6 
 
 3.1 
 
 3.0 
 
 7 
 
 3.6 
 
 3.5 
 
 8 
 
 4.1 
 
 4.0 
 
 9 
 
 4.7 
 
 4.5 
 
 10 
 
 5.2 
 
 5.0 
 
 20 
 
 10.3 
 
 10.0 
 
 30 
 
 15.5 
 
 15.0 
 
 40 
 
 20.7 
 
 20.0 
 
 50 
 
 25.8 
 
 25.0 
 
 
 
 29 
 
 
 6 
 
 2.9 
 
 
 7 
 
 3.4 
 
 
 8 
 
 3.9 
 
 
 9 
 
 4.4 
 
 10 
 
 4.8 
 
 20 
 
 9.7 1 
 
 30 
 
 14.5 
 
 40 
 
 19.3 
 
 50 
 
 24.2 
 
 
 24 
 
 23 
 
 6 
 
 2.4 
 
 2.3 
 
 7 
 
 2.8 
 
 2.7 
 
 8 
 
 3.2 
 
 3.1 
 
 - . 9 
 
 3.6 
 
 3.5 
 
 10 
 
 4.0 
 
 3.8 
 
 20 
 
 8.0 
 
 7.7 
 
 30 
 
 12.0 
 
 11.5 > 
 
 40 
 
 16.0 
 
 15.3 
 
 - 50 
 
 20.0 
 
 19.2 
 
 
 
 22 
 
 
 6 
 
 2.2 
 
 
 7 
 
 2.6 
 
 
 8 
 
 2.9 
 
 
 9 
 
 3.3 
 
 
 10 
 
 3.7 
 
 
 20 
 
 7.3 
 
 
 30 
 
 11.0 
 
 
 40 
 
 14.7 
 
 
 50 
 
 18.3 
 
 
 
 7 
 
 6 
 
 i 
 
 6 
 
 0.7 
 
 0.6 
 
 i 
 
 7 
 
 0.8 
 
 0.7 
 
 i 
 
 8 
 
 0.9 
 
 0.8 
 
 
 9 
 
 1.1 
 
 0.9 
 
 * 10 
 
 1.2 
 
 1.0 
 
 , 20 
 
 2.3 
 
 2.0 
 
 t 30 
 
 3.5 
 
 3.0 
 
 ; 40 
 
 4.7 
 
 4.0 
 
 » 50 
 
 5.8 
 
 5.0 
 
 1 
 
 
 
 
 d. 
 
 L.Tang. 
 
 24 
 
 23 
 
 24 
 24 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 
 23 
 
 24 
 23 
 
 23 
 
 24 
 23 
 
 23 
 
 24 
 23 
 
 23 
 
 24 
 23 
 23 
 23 
 
 23 
 
 24 
 23 
 23 
 23 
 23 
 23 
 23 
 23 
 23 
 
 23 
 
 24 
 23 
 23 
 22 
 23 
 23 
 23 
 23 
 23 
 23 
 23 
 23 
 23 
 22 
 23 
 
 d. 
 
 9.72567 
 
 9.72598 
 
 9.72628 
 
 9.72659 
 
 9.72689 
 
 9.72720 
 
 9o72750 
 
 9.72780 
 
 9.72811 
 
 9.72841 
 
 9.72872 
 
 9.72902 
 
 9.72932 
 
 9.72963 
 
 9.72993 
 
 9.73023 
 
 9.73054 
 
 9.73084 
 
 9.73114 
 
 9.73144 
 
 9.73175 
 
 9.73205 
 
 9.73235 
 
 9.73265 
 
 9.73295 
 
 d. c. 
 
 9.73326 
 
 9.73356 
 
 9.73386 
 
 9.73416 
 
 9.73446 
 
 9.73476 
 
 9.73507 
 
 9.73537 
 
 9.73567 
 
 9.73597 
 
 9.73627 
 
 9.73657 
 
 9.73687 
 
 9.73717 
 
 9.73747 
 
 9.73777 
 
 9.73807 
 
 9.73837 
 
 9.73867 
 
 9.73897 
 
 9.73927 
 
 9.73957 
 
 9.73987 
 
 9.74017 
 
 9.74047 
 
 9.74077 
 
 9.74107 
 
 9.74137 
 
 9.74166 
 
 9.74196 
 
 9.74226 
 
 9.74256 
 
 9.74286 
 
 9.74316 
 
 9.74345 
 
 31 
 
 30 
 
 31 
 
 30 
 
 31 
 30 
 
 30 
 
 31 
 
 30 
 
 31 
 30 
 
 30 
 
 31 
 30 
 
 30 
 
 31 
 30 
 30 
 
 30 
 
 31 
 30 
 30 
 30 
 
 30 
 
 31 
 30 
 30 
 30 
 30 
 
 30 
 
 31 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 30 
 
 9.74375 
 
 L. Cotg. 
 
 L. Cotg. 
 
 30 
 
 30 
 
 30 
 
 30 
 
 29 
 
 30 
 30 
 30 
 30 
 30 
 
 29 
 
 30 
 
 0.27433 
 
 0.27402 
 
 0.27372 
 
 0.27341 
 
 0.27311 
 
 9.94593 
 
 9.94587 
 
 9.94580 
 
 9.94573 
 
 9.94567 
 
 0.27280 
 
 0.27250 
 
 0.27220 
 
 0.27189 
 
 0.27159 
 
 0.27128 
 
 0.27098 
 
 0.27068 
 
 0.27037 
 
 0.27007 
 
 0.26977 
 
 0.26946 
 
 0.26916 
 
 0.26886 
 
 0.26856 
 
 0.26825 
 
 0.26795 
 
 0.26765 
 
 0.26735 
 
 0.26705 
 
 0.26674 
 
 0.26644 
 
 0.26614 
 
 0.26584 
 
 0.26554 
 
 0.26524 
 
 0.26493 
 
 0.26463 
 
 0.26433 
 
 0.26403 
 
 0.26373 
 
 0.26343 
 
 0.26313 
 
 0.26283 
 
 0.26253 
 
 0.26223 
 
 0.26193 
 
 0.26163 
 
 0.26133 
 
 0.26103 
 
 0.26073 
 
 0.26043 
 
 0.26013 
 
 0.25983 
 
 0.25953 
 
 0.25923 
 
 0.25893 
 
 0.25863 
 
 0.25834 
 
 0.25804 
 
 0.25774 
 
 0.25744 
 
 0.25714 
 
 0.25684 
 
 0.25655 
 
 0.25625 
 
 d. c. L.Tang 
 
 61 ° 
 
 L. Cos. d. 
 
 9.94560 
 
 9.94553 
 
 9.94546 
 
 9.94540 
 
 9.94533 
 
 9.94526 
 
 9.94519 
 
 9.94513 
 
 9.94506 
 
 9.94499 
 
 9.94492 
 
 9.94485 
 
 9.94479 
 
 9.94472 
 
 9.94465 
 
 9.94458 
 
 9.94451 
 
 9.94445 
 
 9.94438 
 
 9.94431 
 
 9.94424 
 
 9.94417 
 
 9.94410 
 
 9.94404 
 
 9.94397 
 
 9.94390 
 
 9.94383 
 
 9.94376 
 
 9.94369 
 
 9.94362 
 
 9.94355 
 
 9.94349 
 
 9.94342 
 
 9.94335 
 
 9.94328 
 
 9.94321 
 
 9.94314 
 
 9.94307 
 
 9.94300 
 
 9.94293 
 
 9.94286 
 
 9.94279 
 
 9.94273 
 
 9.94266 
 
 9.94259 
 
 9.94252 
 
 9.94245 
 
 9.94238 
 
 9.94231 
 
 9.94224 
 
 9.94217 
 
 9.94210 
 
 9.94203 
 
 9.94196 
 
 9.94189 
 
 9.94182 
 
 L. Sin. d 
 
 6 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 6 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 7 
 
 10 
 
 P. P. 
 
52.1 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 29 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 9.68557 
 
 
 9.74375 
 
 
 0.25625 
 
 9.94182 
 
 
 60 
 
 
 
 
 
 1 
 
 9.68580 
 
 23 
 
 9.74405 
 
 30 
 
 0.25595 
 
 9.94175 
 
 7 
 
 59 
 
 
 
 
 
 2 
 
 9.68603 
 
 23 
 
 9.74435 
 
 30 
 
 0.25565 
 
 9.94168 
 
 7 
 
 58 
 
 
 
 
 
 3 
 
 9.68625 
 
 22 
 
 9.74465 
 
 30 
 
 0.25535 
 
 9.94161 
 
 7 
 
 57 
 
 a 
 
 ou 
 
 O A 
 
 4 
 
 9.68648 
 
 23 
 
 Oq 
 
 9.74494 
 
 29 
 
 QO 
 
 0.25506 
 
 9.94154 
 
 7 
 
 7 
 
 56 
 
 D 
 
 7 
 
 3.0 
 
 3.5 
 
 5 
 
 9.68671 
 
 
 9.74524 
 
 
 0.25476 
 
 9.94147 
 
 by 
 
 55 
 
 8 
 
 4.0 
 
 6 
 
 9.68694 
 
 23 
 
 9.74554 
 
 30 
 
 0.25446 
 
 9.94140 
 
 7 
 
 54 
 
 9 
 
 4.5 
 
 7 
 
 9.68716 
 
 22 
 
 9.74583 
 
 29 
 
 0.25417 
 
 9.94133 
 
 7 
 
 53 
 
 10 
 
 5.0 
 
 8 
 
 9.68739 
 
 23 
 
 9.74613 
 
 30 
 
 0.25387 
 
 9.94126 
 
 7 
 
 52 
 
 20 
 
 10.0 
 
 9 
 
 9.68762 
 
 23 
 
 22 
 
 9.74643 
 
 30 
 
 30 
 
 0.25357 
 
 9.94119 
 
 7 
 
 7 
 
 51 
 
 30 
 
 15.0 
 
 10 
 
 9.68784 
 
 9.74673 
 
 0.25327 
 
 9.94112 
 
 
 50 
 
 40 
 
 20.0 
 
 11 
 
 9.68807 
 
 23 
 
 9.74702 
 
 29 
 
 0.25298 
 
 9.94105 
 
 7 
 
 49 
 
 50 
 
 25.0 
 
 12 
 
 9.68829 
 
 22 
 
 9.74732 
 
 30 
 
 0.25268 
 
 9.94098 
 
 7 
 
 48 
 
 
 
 
 
 13 
 
 9.68852 
 
 23 
 
 9.74762 
 
 30 
 
 0.25238 
 
 9.94090 
 
 8 
 
 47 
 
 
 
 
 
 14 
 
 9.68875 
 
 23 
 
 99 
 
 9.74791 
 
 29 
 
 30 
 
 0.25209 
 
 9.94083 
 
 7 
 
 7 
 
 46 
 
 
 
 29 
 
 15 
 
 9.68897 
 
 
 9.74821 
 
 0.25179 
 
 9.94076 
 
 « 
 
 by 
 
 45 
 
 6 
 
 2.9 
 
 16 
 
 9.68920 
 
 23 
 
 9.74851 
 
 30 
 
 0.25149 
 
 9.94069 
 
 7 
 
 44 
 
 
 1 
 
 3.4 
 
 17 
 
 9.68942 
 
 22 
 
 9.74880 
 
 29 
 
 0.25120 
 
 9.94062 
 
 7 
 
 43 
 
 8 
 
 3.9 
 
 18 
 
 9.68965 
 
 23 
 
 9.74910 
 
 30 
 
 0.25090 
 
 9.94055 
 
 7 
 
 42 
 
 9 
 
 4.4 
 
 19 
 
 9.68987 
 
 22 ) 
 23 
 
 9.74939 
 
 29 
 
 30 
 
 0.25061 
 
 9.94048 
 
 7 
 
 7 
 
 41 
 
 10 
 
 4.8 
 
 20 
 
 9.69010 
 
 9.749&9 
 
 0.25031 
 
 9.94041 
 
 
 40 
 
 20 
 
 9.7 
 
 21 
 
 9.69032 
 
 22 
 
 9.74998 
 
 29 
 
 0.25002 
 
 9.94034 
 
 7 
 
 39 
 
 30 
 
 14.5 
 
 22 
 
 9.69055 
 
 23 
 
 9.75028 
 
 30 
 
 0.24972 
 
 9.94027 
 
 7 
 
 38 
 
 40 
 
 19.3 
 
 23 
 
 9.69077 
 
 22 
 
 9.75058 
 
 30 
 
 0.24942 
 
 9.94020 
 
 7 
 
 37 
 
 50 
 
 24.2 
 
 24 
 
 9.69100 
 
 23 
 
 99 
 
 9.75087 
 
 29 
 
 30 
 
 0.24913 
 
 9.94012 
 
 8 
 
 7 
 
 36 
 
 
 
 
 
 25 
 
 9.69122 
 
 
 9.75117 
 
 0.24883 
 
 9.94005 
 
 fy 
 
 35 
 
 
 
 
 
 26 
 
 9.69144 
 
 22 
 
 9.75146 
 
 29 
 
 0.24854 
 
 9.93998 
 
 7 
 
 34 
 
 
 
 23 
 
 27 
 
 9.69167 
 
 23 
 
 9.75176 
 
 30 
 
 0.24824 
 
 9.93991 
 
 7 
 
 33 
 
 6 
 
 2.3 
 
 28 
 
 9.69189 
 
 22 
 
 9.75205 
 
 29 
 
 0.24795 
 
 9.93984 
 
 7 
 
 32 
 
 
 7 
 
 2.7 
 
 29 
 
 9.69212 
 
 23 
 
 9.75235 
 
 30 
 
 0.24765 
 
 9.93977 
 
 7 
 
 31 
 
 8 
 
 3.1 
 
 
 
 22 
 
 
 29 
 
 
 
 *1 
 
 
 
 
 
 
 30 
 
 9.69234 
 
 9.75264 
 
 0.24736 
 
 9.93970 
 
 4 
 
 by 
 
 30 
 
 1 A 
 
 6.0 
 o o 
 
 31 
 
 9.69256 
 
 22 
 
 9.75294 
 
 30 
 
 0.24706 
 
 9.93963 
 
 7 
 
 29 
 
 10 
 
 3.8 
 
 <7 >1 
 
 32 
 
 9.69279 
 
 23 
 
 9.75323 
 
 29 
 
 0.24677 
 
 9.93955 
 
 8 
 
 28 
 
 20 
 
 OA 
 
 7.7 
 
 33 
 
 9.69301 
 
 22 
 
 9.75353 
 
 30 
 
 0.24647 
 
 9.93948 
 
 7 
 
 27 
 
 30 
 
 A(\ 
 
 11.0 
 1C Q 
 
 34 
 
 9.69323 
 
 22 
 
 22 
 
 9.75382 
 
 29 
 
 9Q 
 
 0.24618 
 
 9.93941 
 
 7 
 
 7 
 
 26 
 
 R0 
 
 10.3 
 IQ 9 
 
 35 
 
 9.69345 
 
 9.75411 
 
 
 0.24589 
 
 9.93934 
 
 f 
 
 by 
 
 25 
 
 
 
 
 
 36 
 
 9.69368 
 
 23 
 
 9.75441 
 
 30 
 
 0.24559 
 
 9.93927 
 
 7 
 
 24 
 
 
 
 
 
 37 
 
 9.69390 
 
 22 
 
 9.75470 
 
 29 
 
 0.24530 
 
 9.93920 
 
 7 
 
 23 
 
 
 
 
 
 38 
 
 9.69412 
 
 22 
 
 9.75500 
 
 30 
 
 0.24500 
 
 9.93912 
 
 8 
 
 22 
 
 
 
 ZZ 
 
 39 
 
 9.69434 
 
 22 
 
 22 
 
 9.75529 
 
 29 
 
 29 
 
 0.24471 
 
 9.93905 
 
 7 
 
 7 
 
 21 
 
 6 
 
 7 
 
 2.2 
 9 A 
 
 40 
 
 9.69456 
 
 9.75558 
 
 0.24442 
 
 9.93898 
 
 f 
 
 ty 
 
 20 
 
 « 
 
 Z.O 
 
 2.9 
 
 41 
 
 9.69479 
 
 23 
 
 9.75588 
 
 30 
 
 0.24412 
 
 9.93891 
 
 7 
 
 19 
 
 l 
 
 3 
 
 3*3 
 
 42 
 
 9.69501 
 
 22 
 
 9.75617 
 
 29 
 
 0.24383 
 
 9.93884 
 
 7 
 
 18 
 
 10 
 
 8.7 
 
 43 
 
 9.69523 
 
 22 
 
 9.75647 
 
 30 
 
 0.24353 
 
 9.93876 
 
 8 
 
 17 
 
 20 
 
 7.3 
 
 44 
 
 9.69545 
 
 22 
 
 22 
 
 9.75676 
 
 29 
 
 29 
 
 0.24324 
 
 9.93869 
 
 7 
 
 7 
 
 16 
 
 30 
 
 11.0 
 
 45 
 
 9.69567 
 
 9.75705 
 
 0.24295 
 
 9.93862 
 
 i 
 
 by 
 
 15 
 
 40 
 
 14.7 
 
 46 
 
 9.69589 
 
 22 
 
 9.75735 
 
 30 
 
 0.24265 
 
 9.93855 
 
 7 
 
 14 
 
 50 
 
 18.3 
 
 47 
 
 9.69611 
 
 22 
 
 9.75764 
 
 29 
 
 0.24236 
 
 9.93847 
 
 8 
 
 13 
 
 
 
 
 
 48 
 
 9.69633 
 
 22 
 
 9.75793 
 
 29 
 
 0.24207 
 
 9 93840 
 
 7 
 
 12 
 
 
 
 
 
 49 
 
 9.69655 
 
 22 
 
 22 
 
 9.75822 
 
 29 
 
 qn 
 
 0.24178 
 
 9.93833 
 
 7 
 
 7 
 
 11 
 
 
 
 ft 
 
 a 
 
 50 
 
 9.69677 
 
 9.75852 
 
 ou 
 
 0.24148 
 
 9.93826 
 
 i 
 
 10 
 
 6 
 
 0.8 
 
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 51 
 
 9.69699 
 
 22 
 
 9.75881 
 
 29 
 
 0.24119 
 
 9.93819 
 
 7 
 
 9 
 
 7 
 
 0.9 
 
 0.8 
 
 52 
 
 9.69721 
 
 22 
 
 9.75910 
 
 29 
 
 0.24090 
 
 9.93811 
 
 8 
 
 8 
 
 8 
 
 1.1 
 
 0.9 
 
 53 
 
 9.69743 
 
 22 
 
 9.75939 
 
 29 
 
 0.24061 
 
 9.93804 
 
 7 
 
 7 
 
 9 
 
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 54 
 
 9.69765 
 
 22 
 
 22 
 
 9.75969 
 
 30 
 
 29 
 
 0.24031 
 
 9.93797 
 
 7 
 
 8 
 
 6 
 
 10 
 
 1.3 
 
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 55 
 
 9.69787 
 
 9.75998 
 
 0.24002 
 
 9.93789 
 
 5 
 
 20 
 
 2.7 
 
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 56 
 
 9.69809 
 
 22 
 
 9.76027 
 
 29 
 
 0.23973 
 
 9.93782 
 
 7 
 
 4 
 
 30 
 
 4.0 
 
 3.5 
 
 57 
 
 9.69831 
 
 22 
 
 9.76056 
 
 29 
 
 0.23944 
 
 9.93775 
 
 7 
 
 3 
 
 40 
 
 5.3 
 
 4.7 
 
 58 
 
 9.69853 
 
 22 
 
 9.76086 
 
 30 
 
 0.23914 
 
 9.93768 
 
 7 
 
 2 
 
 50 
 
 6.7 
 
 5.8 
 
 59 
 
 9.69875 
 
 22 
 
 22 
 
 9.76115 
 
 29 
 
 29 
 
 0.23885 
 
 9.93760 
 
 8 
 
 7 
 
 1 
 
 
 
 
 
 60 
 
 9.69897 
 
 9.76144 
 
 0.23856 
 
 9.93753 
 
 4 
 
 0 
 
 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 60 ° 
 
522 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 30 ° 
 
 r 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.69897 
 
 9.69919 
 
 9.69941 
 
 9.69963 
 
 9.69984 
 
 22 
 
 22 
 
 22 
 
 21 
 
 22 
 
 22 
 
 22 
 
 22 
 
 21 
 
 22 
 
 22 
 
 22 
 
 21 
 
 22 
 
 22 
 
 21 
 
 22 
 
 21 
 
 22 
 
 22 
 
 21 
 
 22 
 
 21 
 
 22 
 
 21 
 
 22 
 
 21 
 
 22 
 
 21 
 
 22 
 
 A 
 
 22 
 
 21 
 
 22 
 
 21 
 
 21 
 
 22 
 
 21 
 
 21 
 
 22 
 
 21 
 
 21 
 
 21 
 
 22 
 
 21 
 
 21 
 
 21 
 
 22 
 
 21 
 
 - 21 
 21 
 21 
 21 
 22 
 
 - 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 91 
 
 9.76144 
 
 9.76173 
 
 9.76202 
 
 9.76231 
 
 9.76261 
 
 29 
 
 29 
 
 29 
 
 30 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 
 29 
 
 30 
 29 
 29 
 28 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 29 
 28 
 29 
 29 
 
 29 
 
 29 
 
 29 
 
 28 
 
 29 
 
 29 
 
 29 
 
 29 
 
 28 
 
 29 
 
 29 
 
 29 
 
 ( 28 
 29 
 29 
 28 
 29 
 29 
 29 
 
 - 28 
 29 
 28 
 29 
 29 
 
 Oft 
 
 0.23856 
 
 0.23827 
 
 0.23798 
 
 0.23769 
 
 0.23739 
 
 9.93753 
 
 9.93746 
 
 9.93738 
 
 9.93731 
 
 9.93724 
 
 7 
 
 8 
 7 
 7 
 
 7 
 
 8 
 7 
 
 7 
 
 8 
 7 
 
 7 
 
 8 
 
 7 
 
 8 
 7 
 
 7 
 
 8 
 
 7 
 7. 
 
 8 
 
 7 
 
 8 
 7 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 7 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 7 
 
 - 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 - 8 
 
 7 
 
 8 
 
 7 
 
 8 
 
 - 7 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 30 29 
 
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 7 3.5 3.4 
 
 8 4.0 3.9 
 
 9 4.5 4.4 
 
 10 5.0 4.8 
 
 20 10.0 9.7 
 
 30 15.0 14.5 
 40 20.0 19.3 
 50 25.0 24.2 
 
 28 
 
 6 2.8 
 
 7 3.3 
 
 8 3.7 
 
 9 4.2 
 
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 20 9.3 
 
 30 14.0 
 40 18.7 
 50 23.3 
 
 22 
 
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 8 2.9 
 
 9 3.3 
 
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 40 14.7 
 50 18.3 
 
 2^ 
 
 6 2.1 
 
 7 2.5 
 
 8 2.8 
 
 9 3.2 
 
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 20 7.0 
 
 30 10.5 
 40 14.0 
 50 17.5 
 
 8 7 
 
 6 0.8 0.7 
 
 7 0.9 0.8 
 
 8 1.1 0.9 
 
 9 1.2 1.1 
 10 1.3 1.2 
 20 2.7 2.3 
 30 4.0 3.5 
 40 5.3 4.7 
 50 6.7 5.8 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.70006 
 
 9.70028 
 
 9.70050 
 
 9.70072 
 
 9.70093 
 
 9.76290 
 
 9.76319 
 
 9.76348 
 
 9.76377 
 
 9.76406 
 
 0.23710 
 
 0.23681 
 
 0.23652 
 
 0.23623 
 
 0.23594 
 
 9.93717 
 
 9.93709 
 
 9.93702 
 
 9.93695 
 
 9.93687 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.70115 
 
 9.70137 
 
 9.70159 
 
 9.70180 
 
 9.70202 
 
 9.76435 
 
 9.76464 
 
 9.76493 
 
 9.76522 
 
 9.76551 
 
 0.23565 
 
 0.23536 
 
 0.23507 
 
 0.23478 
 
 0.23449 
 
 9.93680 
 
 9.93673 
 
 9.93665 
 
 9.93658 
 
 9.93650 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.70224 
 
 9.70245 
 
 9.70267 
 
 9.70288 
 
 9.70310 
 
 9.76580 
 
 9.76609 
 
 9.76639 
 
 9.76668 
 
 9.76697 
 
 0.23420 
 
 0.23391 
 
 0.23361 
 
 0.23332 
 
 0.23303 
 
 9.93643 
 
 9.93636 
 
 9.93628 
 
 9.93621 
 
 9.93614 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.70332 
 
 9.70353 
 
 9.70375 
 
 9.70396 
 
 9.70418 
 
 9.76725 
 
 9.76754 
 
 9.76783 
 
 9.76812 
 
 9.76841 
 
 0.23275 
 
 0.23246 
 
 0.23217 
 
 0.23188 
 
 0.23159 
 
 9.93606 
 
 9.93599 
 
 9.93591 
 
 9.93584 
 
 9.93577 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.70439 
 
 9.70461 
 
 9.70482 
 
 9.70504 
 
 9.70525 
 
 9.76870 
 
 9.76899 
 
 9.76928 
 
 9.76957 
 
 9.76986 
 
 0.23130 
 
 0.23101 
 
 0.23072 
 
 0.23043 
 
 0.23014 
 
 9.93569 
 
 9.93562 
 
 9.93554 
 
 9.93547 
 
 9.93539 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.70547 
 
 9.70568 
 
 9.70590 
 
 9.70611 
 
 9.70633 
 
 9.77015 
 
 9.77044 
 
 9.77073 
 
 9.77101 
 
 9.77130 
 
 0.22985 
 
 0.22956 
 
 0.22927 
 
 0.22899 
 
 0.22870 
 
 9.93532 
 
 9.93525 
 
 9.93517 
 
 9.93510 
 
 9.93502 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.70654 
 
 9.70675 
 
 9.70697 
 
 9.70718 
 
 9.70739 
 
 9.77159 
 
 9.77188 
 
 9.77217 
 
 9.77246 
 
 9.77274 
 
 0.22841 
 
 0.22812 
 
 0.22783 
 
 0.22754 
 
 0.22726 
 
 9.93495 
 
 9.93487 
 
 9.93480 
 
 9.93472 
 
 9.93465 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.70761 
 
 9.70782 
 
 9.70803 
 
 9.70824 
 
 9.70846 
 
 9.77303 
 
 9.77332 
 
 9.77361 
 
 9.77390 
 
 9.77418 
 
 0.22697 
 
 0.22668 
 
 0.22639 
 
 0.22610 
 
 0.22582 
 
 9.93457 
 
 9.93450 
 
 9.93442 
 
 9.93435 
 
 9.93427 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.70867 
 
 9.70888 
 
 9.70909 
 
 9.70931 
 
 9.70952 
 
 • 9.77447 
 9.77476 
 9.77505 
 9.77533 
 9.77562 
 
 0.22553 
 
 0.22524 
 
 0.22495 
 
 0.22467 
 
 0.22438 
 
 9.93420 
 
 9.93412 
 
 9.93405 
 
 9.93397 
 
 9.93390 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 ’ 9.70973 
 9.70994 
 9.71015 
 9.71036 
 9.71058 
 
 9.77591 
 
 9.77619 
 
 9.77648 
 
 9.77677 
 
 9.77706 
 
 0.22409 
 
 0.22381 
 
 0.22352 
 
 0.22323 
 
 0.22294 
 
 9.93382 
 
 9.93375 
 
 9.93367 
 
 9.93360 
 
 9.93352 
 
 lo 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.71079 
 
 9.71100 
 
 9.71121 
 
 9.71142 
 
 9.71163 
 
 9.77734 
 
 9.77763 
 
 9.77791 
 
 9.77820 
 
 9.77849 
 
 0.22266 
 
 0.22237 
 
 0.22209 
 
 0.22180 
 
 0.22151 
 
 9.93344 
 
 9.93337 
 
 9.93329 
 
 9.93322 
 
 9.93314 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 " 9771184 
 
 
 9.77877 
 
 AdCj 
 
 0.22123 
 
 9.93307 
 
 
 0 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg 
 
 . d. c. 
 
 L.Tang 
 
 . L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 59 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 523 
 
 31 ° 
 
 f 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.71184 
 
 9.71205 
 
 9.71226 
 
 9.71247 
 
 9.71268 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 20 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 20 
 
 21 
 
 21 
 
 21 
 
 20 
 
 21 
 
 21 
 
 21 
 
 20 
 
 21 
 
 21 
 
 20 
 
 21 
 
 21 
 
 20 
 
 21 
 
 20 
 
 21 
 
 20 
 
 21 
 
 20 
 
 21 
 
 21 
 
 20 
 
 20 
 
 21 
 
 20 
 
 21 
 
 20 
 
 21 
 
 20 
 
 20 
 
 21 
 
 ,20 
 
 20 
 
 21 
 
 20 
 
 20 
 
 21 
 
 20 
 
 20 
 
 21 
 
 20 
 
 20 
 
 9.77877 
 
 9.77906 
 
 9.77935 
 
 9.77963 
 
 .9.77992 
 
 29 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 28 
 
 29 
 
 28 
 
 29 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 - 28 
 
 0.22123 
 
 0.22094 
 
 0.22065 
 
 0.22037 
 
 0.22008 
 
 9.93307 
 
 9.93299 
 
 9.93291 
 
 9.93284 
 
 9.93276 
 
 8 
 
 8 
 
 7 
 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 8 
 8 
 8 
 
 7 
 
 8 
 8 
 8 
 8 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 29 
 
 6 2.9 
 
 7 3.4 
 
 8 3.9 
 
 9 4.4 
 
 10 4.8 
 
 20 9.7 
 
 30 14.5 
 40 19.3. 
 
 50 24.2 
 
 28 
 
 6 1 2.8 
 
 7 3.3 
 
 8 3.7 
 
 9 4.2 
 
 10 4.7 
 
 20 9.3 
 
 30 14.0 
 40 18.7 
 50 23.3 
 
 21 
 
 6 2.1 
 
 7 2.5 
 
 8 2.8 
 
 9 3.2 
 
 10 3.5 
 
 20 7.0 
 
 30 10.5 
 40 14.0 
 50 17.5 
 
 20 
 
 6 2.0 
 
 7 2.3 
 
 8 2.7 
 
 9 3.0 
 
 10 3.3 
 
 20 6.7 
 
 30 10.0 
 40 13.3 
 50 16.7 
 
 8 7 
 
 6 0.8 0.7 
 
 7 0.9 0.8 
 
 8 1.1 0.9 
 
 9 1.2 1.1 
 10 1.3 1.2 
 20 2.7 2.3 
 30 4.0 3.5 
 40 5.3 4.7 
 50 6.7 5.8 
 
 9.71289 
 
 9.71310 
 
 9.71331 
 
 9.71352 
 
 .9.71373 
 
 9.78020 
 
 9.78049 
 
 9.78077 
 
 9.78106 
 
 9.78135 
 
 0.21980 
 
 0.21951 
 
 0.21923 
 
 0.21894 
 
 0.21865 
 
 9.93269 
 
 9.93261 
 
 9.93253 
 
 9.93246 
 
 9.93238 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.71393 
 
 9.71414 
 
 9.71435 
 
 9.71456 
 
 9.71477 
 
 9.78163 
 
 9.78192 
 
 9.78220 
 
 9.78249 
 
 9.78277 
 
 0.21837 
 
 0.21808 
 
 0.21780 
 
 0.21751 
 
 0.21723 
 
 9.93230 
 
 9.93223 
 
 9.93215 
 
 9.93207 
 
 9.93200 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.71498 
 
 9.71519 
 
 9.71539 
 
 9.71560 
 
 9.71581 
 
 9.78306 
 
 9.78334 
 
 9.78363 
 
 9.78391 
 
 9.78419 
 
 0.21694 
 
 0.21666 
 
 0.21637 
 
 0.21609 
 
 0.21581 
 
 9.93192 
 
 9.93184 
 
 9.93177 
 
 9.93169 
 
 9.93161 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.71602 
 
 9.71622 
 
 9.71643 
 
 9.71664 
 
 9.71685 
 
 9.78448 
 
 9.78476 
 
 9.78505 
 
 9.78533 
 
 9.78562 
 
 0.21552 
 
 0.21524 
 
 0.21495 
 
 0.21467 
 
 0.21438 
 
 9.93154 
 
 9.93146 
 
 9.93138 
 
 9.93131 
 
 9.93123 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.71705 
 
 9.71726 
 
 9.71747 
 
 9.71767 
 
 9.71788 
 
 9.78590 
 
 9.78618 
 
 9.78647 
 
 9.78675 
 
 9.78704 
 
 0.21410 
 
 0.21382 
 
 0.21353 
 
 0.21325 
 
 0.21296 
 
 9.93115 
 
 9.93108 
 
 9.93100 
 
 9.93092 
 
 9.93084 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.71809 
 
 9.71829 
 
 9.71850 
 
 9.71870 
 
 9.71891 
 
 9.78732 
 
 9.78760 
 
 9.78789 
 
 9.78817 
 
 9.78845 
 
 0.21268 
 
 0.21240 
 
 0.21211 
 
 0.21183 
 
 0.21155 
 
 9.93077 
 
 9.93069 
 
 9.93061 
 
 9.93053 
 
 9.93046 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.71911 
 
 9.71932 
 
 9.71952 
 
 9.71973 
 
 9.71994 
 
 9.78874 
 
 9.78902 
 
 9.78930 
 
 9.78959 
 
 9.78987 
 
 0.21126 
 
 0.21098 
 
 0.21070 
 
 0.21041 
 
 0.21013 
 
 9.93038 
 
 9.93030 
 
 9.93022 
 
 9.93014 
 
 9.93007 
 
 25 
 
 24 
 
 ^23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.72014 
 
 9.72034 
 
 9.72055 
 
 9.72075 
 
 9.72096 
 
 9.79015 
 
 9.79043 
 
 9.79072 
 
 9.79100 
 
 9.79128 
 
 0.20985 
 
 0.20957 
 
 0.20928 
 
 0.20900 
 
 0.20872 
 
 9.92999 
 
 9.92991 
 
 9.92983 
 
 9.92976 
 
 9.92968 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.72116 
 
 9.72137 
 
 9.72157 
 
 9.72177 
 
 9.72198 
 
 9.79156 
 
 9.79185 
 
 9.79213 
 
 9.79241 
 
 9.79269 
 
 0.20844 
 
 0.20815 
 
 0.20787 
 
 0.20759 
 
 0.20731 
 
 9.92960 
 
 9.92952 
 
 9.92944 
 
 9.92936 
 
 9.92929 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 9.72218' 
 
 9.72238 
 
 9.72259 
 
 9.72279 
 
 9.72299 
 
 9.79297 
 
 9.79326 
 
 9.79354 
 
 9.79382 
 
 9.79410 
 
 0.20703 
 
 0.20674 
 
 0.20646 
 
 0.20618 
 
 0.20590 
 
 9.92921 
 
 9.92913 
 
 9.92905 
 
 9.92897 
 
 9.92889 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.72320 
 
 9.72340 
 
 9.72360 
 
 9.72381 
 
 9.72401 
 
 9.79438 
 
 9.79466 
 
 9.79495 
 
 9.79523 
 
 9.79551 
 
 0.20562 
 
 0.20534 
 
 0.20505 
 
 0.20477 
 
 0.20449 
 
 9.92881 
 
 9.92874 
 
 9.92866 
 
 9.92858 
 
 9.92850 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 9.72421 
 L. Cos. 
 
 9.79579 
 
 0.20421 
 
 9.92842 
 
 0 
 
 
 d. 
 
 L. Cotg 
 
 . d. c. 
 
 L.Tang. 
 
 . L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 58 ° 
 
524 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 32 ° 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.72421 
 
 1 
 
 9.72441 
 
 2 
 
 9.72461 
 
 3 
 
 9.72482 
 
 4 
 
 9.72502 
 
 5 
 
 9.72522 
 
 6 
 
 9.72542 
 
 7 
 
 9.72562 
 
 8 
 
 9.72582 
 
 9 
 
 9.72602 
 
 10 
 
 9.72622 
 
 11 
 
 9.72643 
 
 12 
 
 9.72663 
 
 13 
 
 9.72683 
 
 14 
 
 9.72703 
 
 15 
 
 9.72723 
 
 16 
 
 9.72743 
 
 17 
 
 9.72763 
 
 18 
 
 9.72783 
 
 19 
 
 9.72803 
 
 20 
 
 9.72823 
 
 21 
 
 9.72843 
 
 22 
 
 9.72863 
 
 23 
 
 9.72883 
 
 24 
 
 9.72902 
 
 25 
 
 9.72922 
 
 26 
 
 9.72942 
 
 27 
 
 9.72962 
 
 28 
 
 9.72982 
 
 29 
 
 9.73002 
 
 30 
 
 9.73022 
 
 31 
 
 9.73041 
 
 32 
 
 9.73061 
 
 33 
 
 9.73081 
 
 34 
 
 9.73101 
 
 35 
 
 9.73121 
 
 36 
 
 9.73140 
 
 37 
 
 9.73160 
 
 38 
 
 9.73180 
 
 39 
 
 9.73200 
 
 40 
 
 9.73219 
 
 41 
 
 9.73239 
 
 42 
 
 9.73259 
 
 43 
 
 9.73278 
 
 44 
 
 9.73298 
 
 45 
 
 9.73318 
 
 46 
 
 9.73337 
 
 47 
 
 9.73357 
 
 48 
 
 9.73377 
 
 49 
 
 9.73396 
 
 50 
 
 9.73416 
 
 51 
 
 9.73435 
 
 52 
 
 9.73455 
 
 53 
 
 9.73474 
 
 54 
 
 9.73494 
 
 55 
 
 9.73513 
 
 56 
 
 9.73533 
 
 57 
 
 9.73552 
 
 58 
 
 9.73572 
 
 59 
 
 9.73591 
 
 60 
 
 9.73611 
 
 
 L.Cos. 
 
 d. 
 
 L.Tang. 
 
 20 
 
 20 
 
 21 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 21 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 20 
 
 19 
 
 20 
 20 
 20 
 20 
 20 
 20 
 
 19 
 
 20 
 20 
 20 
 20 
 
 19 
 
 20 
 20 
 20 
 
 19 
 
 20 
 20 
 
 19 
 
 20 
 20 
 
 19 
 
 20 
 20 
 
 19 
 
 20 
 
 19 
 
 20 
 
 19 
 
 20 
 
 19 
 
 20 
 
 19 
 
 20 
 
 19 
 
 20 
 
 9.79579 
 
 9.79607 
 
 9.79635 
 
 9.79663 
 
 9.79691 
 
 9.79719 
 
 9.79747 
 
 9.79776 
 
 9.79804 
 
 9.79832 
 
 9.79860 
 
 9.79888 
 
 9.79916 
 
 9.79944 
 
 9.79972 
 
 9.80000 
 
 9.80028 
 
 9.80056 
 
 9.80084 
 
 9.80112 
 
 9.80140 
 
 9.80168 
 
 9.80195 
 
 9.80223 
 
 9.80251 
 
 9.80279 
 
 9.80307 
 
 9.80335 
 
 9.80363 
 
 9.80391 
 
 9.80419 
 
 9.80447 
 
 9.80474 
 
 9.80502 
 
 9.80530 
 
 9.80558 
 
 9.80586 
 
 9.80614 
 
 9.80642 
 
 9.80669 
 
 9.80697 
 
 9.80725 
 
 9.80753 
 
 9.80781 
 
 9.80808 
 
 9.80836 
 
 9.80864 
 
 9.80892 
 
 9.80919 
 
 9.80947 
 
 9.80975 
 
 9.81003 
 
 9.81030 
 
 9.81058 
 
 9.810 86 
 
 9.81113 
 
 9.81141 
 
 9.81169 
 
 9.81196 
 
 9.81224 
 
 9.81252 
 
 d. c. 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 29 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 27 
 
 28 
 28 
 28 
 28 
 28 
 28 
 28 
 28 
 28 
 
 27 
 
 28 
 28 
 28 
 28 
 28 
 28 
 
 27 
 
 28 
 28 
 28 
 28 
 
 27 
 
 28 
 28 
 28 
 
 27 
 
 28 
 28 
 28 
 
 27 
 
 28 
 28 
 
 27 
 
 28 
 28 
 
 27 
 
 28 
 28 
 
 d. L. Cotg. d. c 
 
 L. Cotg. 
 
 L. Cos. 
 
 0.20421 
 
 9.92842 
 
 0.20393 
 
 9.92834 
 
 0.20365 
 
 9.92826 
 
 0.20337 
 
 9.92818 
 
 0.20309 
 
 9.92810 
 
 0.20281 
 
 9.92803 
 
 0.20253 
 
 9.92795 
 
 0.20224 
 
 9.92787 
 
 0.20196 
 
 9.92779 
 
 0.20168 
 
 9.92771 
 
 0.20140 
 
 9.92763 
 
 0.20112 
 
 9.92755 
 
 0.20084 
 
 9.92747 
 
 0.20056 
 
 9.92739 
 
 0.20028 
 
 9.92731 
 
 0.20000 
 
 9.92723 
 
 0.19972 
 
 9.92715 
 
 0.19944 
 
 9.92707 
 
 0.19916 
 
 9.92699 
 
 0.19888 
 
 9.92691 
 
 0.19860 
 
 9.92683 
 
 0.19832 
 
 9.92675 
 
 0.19805 
 
 9.92667 
 
 0.19777 
 
 9.92659 
 
 0.19749 
 
 9.92651 
 
 0.19721 
 
 9.92643 
 
 0.19693 
 
 9.92635 
 
 0.19665 
 
 9.92627 
 
 0.19637 
 
 9.92619 
 
 0.19609 
 
 9.92611 
 
 0.19581 
 
 9.92603 
 
 0.19553 
 
 9.92595 
 
 0.19526 
 
 9.92587 
 
 0.19498 
 
 9.92579 
 
 0.19470 
 
 9.92571 
 
 0.19442 
 
 9.92563 
 
 0.19414 
 
 9.92555 
 
 0.19386 
 
 9.92546 
 
 0.19358 
 
 9.92538 
 
 0.19331 
 
 9.92530 
 
 0.19303 
 
 9.92522 
 
 0.19275 
 
 9.92514 
 
 0.19247 
 
 9.92506 
 
 0.19219 
 
 9.92498 
 
 0.19192 
 
 9.92490 
 
 0.19164 
 
 9.92482 
 
 0.19136 
 
 9.92473 
 
 0.19108 
 
 9.92465 
 
 0.19081 
 
 9.92457 
 
 0.19053 
 
 9.92449 
 
 0.19025 
 
 9.92441 
 
 0.18997 
 
 9.92433 
 
 0.18970 
 
 9.92425 
 
 0.18942 
 
 9.92416 
 
 0.18914 
 
 9.92408 
 
 0.18887 
 
 9.92400 
 
 0.18859 
 
 9.92392 
 
 0.18831 
 
 9.92384 
 
 0.18804 
 
 9.92376 
 
 0.18776 
 
 9.92367 
 
 0.18748 
 
 9.92359 
 
 L.Tang, 
 
 . L. Sin. 
 
 8 
 
 8 
 
 8 
 
 8 
 
 7 
 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 
 P. 
 
 P. 
 
 
 
 
 29 
 
 28 
 
 6 
 
 2.9 
 
 2.8 
 
 7 
 
 3.4 
 
 3.3 
 
 8 
 
 3.9 
 
 3.7 
 
 9 
 
 4.4 
 
 4.2 
 
 10 
 
 4.8 
 
 4.7 
 
 20 
 
 9.7 
 
 9.3 
 
 30 
 
 14.5 
 
 14.0 
 
 40 
 
 19.3 
 
 18.7 
 
 50 
 
 24.2 
 
 23.3 
 
 
 
 27 
 
 
 
 6 
 
 2.7 
 
 
 
 7 
 
 3.2 
 
 
 
 8 
 
 3.6 
 
 
 
 9 
 
 4.1 
 
 
 
 10 
 
 4.5 
 
 
 
 20 
 
 9.0 
 
 
 
 30 
 
 13.5 
 
 
 
 40 
 
 18.0 
 
 
 
 50 
 
 22.5 
 
 
 
 21 
 
 20 
 
 6 
 
 ! 2.1 
 
 2.0 
 
 7 
 
 2.5 
 
 2.3 
 
 8 
 
 2.8 
 
 2.7 
 
 - 9 
 
 3.2 
 
 3.0 
 
 10 
 
 3.5 
 
 3.3 
 
 20 
 
 7.0 
 
 6.7 
 
 30 
 
 10.5 
 
 10.0 
 
 40 
 
 14.0 
 
 13.3 
 
 - 50 
 
 17.5 
 
 16.7 
 
 
 
 19 
 
 
 
 6 
 
 1.9 
 
 
 
 7 
 
 2.2 
 
 
 
 8 
 
 2.5 
 
 
 
 9 
 
 2.9 
 
 
 
 10 
 
 3.2 
 
 
 
 20 
 
 6.3 
 
 
 
 30 
 
 9.5 
 
 
 
 40 
 
 12.7 
 
 
 
 50 
 
 15.8 
 
 
 
 9 
 
 
 8 
 
 7 
 
 6 
 
 0.9 
 
 0.8 
 
 0.7 
 
 7 
 
 1.1 
 
 0.9 
 
 0.8 
 
 8 
 
 1.2 
 
 1.1 
 
 0.9 
 
 9 
 
 1.4 
 
 1.2 
 
 1.1 
 
 10 
 
 1.5 
 
 1.3 
 
 1.2 
 
 20 
 
 3.0 
 
 2.7 
 
 2.3 
 
 30 
 
 4.5 
 
 4.0 
 
 3.5 
 
 40 
 
 6.0 
 
 5.3 
 
 4.7 
 
 50 
 
 7.5 
 
 6.7 
 
 5.8 
 
 
 P. 
 
 , I 
 
 > 
 
 
 57 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 525 
 
 f 
 
 L. Sin. 
 
 0 
 
 9.73611 
 
 1 
 
 9,73630 
 
 2 
 
 9.73650 
 
 3 
 
 9.73669 
 
 4 
 
 9.73689 
 
 5 
 
 9.73708 
 
 6 
 
 9.73727 
 
 7 
 
 9.73747 
 
 8 
 
 9.73766 
 
 9 
 
 9.73785 
 
 10 
 
 9.73805 
 
 11 
 
 9.73824 
 
 12 
 
 9.73843 
 
 13 
 
 9.73863 
 
 14 
 
 9.73882 
 
 15 
 
 9.73901 
 
 16 
 
 9.73921 
 
 17 
 
 9.73940 
 
 18 
 
 9.73959 
 
 19 
 
 9.73978 
 
 20 
 
 9.73997 
 
 21 
 
 9.74017 
 
 22 
 
 9.74036 
 
 23 
 
 9.74055 
 
 24 
 
 9.74074 
 
 25 
 
 9.74093 
 
 26 
 
 9.74113 
 
 27 
 
 9.74132 
 
 28 
 
 9.74151 
 
 29 
 
 9.74170 
 
 30 
 
 9.74189 
 
 31 
 
 9.74208 
 
 32 
 
 9.74227 
 
 33 
 
 9.74246 
 
 34 
 
 9.74265 
 
 35 
 
 9.74284 
 
 36 
 
 9.74303 
 
 37 
 
 9.74322 
 
 38 
 
 9.74341 
 
 39 
 
 9.74360 
 
 40 
 
 9.74379 
 
 41 
 
 9.74398 
 
 42 
 
 9.74417 
 
 43 
 
 9.74436 
 
 44 
 
 9.74455 
 
 45 
 
 9.74474 
 
 46 
 
 9.74493 
 
 47 
 
 9.74512 
 
 48 
 
 9.74531 
 
 49 
 
 9.74549 
 
 50 
 
 9.74568 
 
 51 
 
 9.74587 
 
 52 
 
 9.74606 
 
 53 
 
 9.74625 
 
 54 
 
 9.74644 
 
 55 
 
 9.74662 
 
 56 
 
 9.74681 
 
 57 
 
 9.74700 
 
 58 
 
 9.74719 
 
 59 
 
 9.74737 
 
 60 
 
 9.74756 
 
 
 L. Cos. 
 
 d. 
 
 19 
 
 20 
 
 19 
 
 20 
 19 
 
 19 
 
 20 
 19 
 
 19 
 
 20 
 19 
 
 19 
 
 20 
 19 
 
 19 
 
 20 
 19 
 19 
 19 
 
 19 
 
 20 
 19 
 19 
 19 
 
 19 
 
 20 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 19 
 18 
 19 
 19 
 19 
 19 
 19 
 18 
 19 
 19 
 19 
 18 
 19 
 
 L.Tang. 
 
 d. c. : 
 
 L. Cotg. 
 
 L. Cos. 
 
 9.81252 
 
 
 0.18748 
 
 9.92359 
 
 9.81279 
 
 27 
 
 0.18721 
 
 9.92351 
 
 9.81307 
 
 28 
 
 0.18693 
 
 9.92343 
 
 9.81335 
 
 28 
 
 0.18665 
 
 9.92335 
 
 9.81362 
 
 27 
 
 OG 
 
 0.18638 
 
 9.92326 
 
 9.81390 
 
 Zo 
 
 0.18610 
 
 9.92318 
 
 9.81418 
 
 28 
 
 0.18582 
 
 9.92310 
 
 9.81445 
 
 27 
 
 0.18555 
 
 9.92302 
 
 9.81473 
 
 28 
 
 0.18527 
 
 9.92293 
 
 9.81500 
 
 27 
 
 OG 
 
 0.18500 
 
 9.92285 
 
 9.81528 
 
 Zo 
 
 0.18472 
 
 9.92277 
 
 9.81556 
 
 28 
 
 0.18444 
 
 9.92269 
 
 9.81583 
 
 27 
 
 0.18417 
 
 9.92260 
 
 9.81611 
 
 28 
 
 0.18389 
 
 9.92252 
 
 9.81638 
 
 27 
 
 OQ 
 
 0.18362 
 
 9.92244 
 
 9.81666 
 
 Zo 
 
 0.18334 
 
 9.92235 
 
 9.81693 
 
 27 
 
 0.18307 
 
 9.92227 
 
 9.81721 
 
 28 
 
 0.18279 
 
 9.92219 
 
 9.81748 
 
 27 
 
 0.18252 
 
 9.92211 
 
 9.81776 
 
 28 
 
 07 
 
 0.18224 
 
 9.92202 
 
 9.81803 
 
 Zl 
 
 0.18197 
 
 9.92194 
 
 9.81831 
 
 28 
 
 0.18169 
 
 9.92186 
 
 9.81858 
 
 27 
 
 0.18142 
 
 9.92177 
 
 9.81886 
 
 28 
 
 0.18114 
 
 9.92169 
 
 9.81913 
 
 27 
 
 OG 
 
 0.18087 
 
 9.92161 
 
 9.81941 
 
 Zo 
 
 0.18059 
 
 9.92152 
 
 9.81968 
 
 27 
 
 0.18032 
 
 9.92144 
 
 9.81996 
 
 28 
 
 0.18004 
 
 9.92136 
 
 9.82023 
 
 27 
 
 0.17977 
 
 9.92127 
 
 9.82051 
 
 28 
 
 07 
 
 0.17949 
 
 9.92119 
 
 9.82078 
 
 Zl 
 
 0.17922 
 
 9.92111 
 
 9.82106 
 
 28 
 
 0.17894 
 
 9.92102 
 
 9.82133 
 
 27 
 
 0.17867 
 
 9.92094 
 
 9.82161 
 
 28 
 
 0.17839 
 
 9.92086 
 
 9.82188 
 
 27 
 
 07 
 
 0.17812 
 
 9.92077 
 
 9.82215 
 
 Zl 
 
 0.17785 
 
 9.92069 
 
 9.82243 
 
 28 
 
 0.17757 
 
 9.92060 
 
 9.82270 
 
 27 
 
 0.17730 
 
 9.92052 
 
 9.82298 
 
 28 
 
 0.17702 
 
 9.92044 
 
 9.82325 
 
 27 
 
 07 
 
 0.17675 
 
 9.92035 
 
 9.82352 
 
 Zl 
 
 0.17648 
 
 9.92027 
 
 9.82380 
 
 28 
 
 0.17620 
 
 9.92018 
 
 9.82407 
 
 27 
 
 0.17593 
 
 9.92010 
 
 9.82435 
 
 28 
 
 0.17565 
 
 9.92002 
 
 9.82462 
 
 27 
 
 07 
 
 0.17538 
 
 9.91993 
 
 9.82489 
 
 Zl 
 
 0.17511 
 
 9.91985 
 
 9.82517 
 
 28 
 
 0.17483 
 
 9.91976 
 
 9.82544 
 
 27 
 
 0.17456 
 
 9.91968 
 
 9.82571 
 
 27 
 
 0.17429 
 
 9.91959 
 
 9.82599 
 
 28 
 
 27 
 
 0.17401 
 
 9.91951 
 
 9.82626 
 
 0.17374 
 
 9.91942 
 
 9.82653 
 
 27 
 
 0.17347 
 
 9.91934 
 
 9.82681 
 
 28 
 
 0.17319 
 
 9.91925 
 
 9.82708 
 
 27 
 
 0.17292 
 
 9.91917 
 
 9.82735 
 
 27 
 
 07 
 
 0.17265 
 
 9.91908 
 
 9.82762 
 
 Zl 
 
 0.17238 
 
 9.91900 
 
 9.82790 
 
 28 
 
 0.17210 
 
 9.91891 
 
 9.82817 
 
 27 
 
 0.17183 
 
 9.91883 
 
 9.82844 
 
 27 
 
 0.17156 
 
 9.91874 
 
 9.82871 
 
 27 
 
 - 28 
 
 0.17129 
 
 9.91866 
 
 9.82899 
 
 0.17101 
 
 9.91857 
 
 L. Cotg 
 
 . d. c. 
 
 L.Tang 
 
 . L. Sin. 
 
 d. 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 ~ d7 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56_ 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 35 
 34 
 33 
 32 
 31 
 
 36 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 a 
 
 ii 
 
 to 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 P. P. 
 
 28 
 
 2.8 
 
 3.3 
 
 3.7 
 
 4.2 
 
 4.7 
 
 9.3 
 14.0 
 18.7 
 23.3 
 
 27 
 
 2.7 
 
 3.2 
 
 3.6 
 
 4.1 
 
 4.5 
 
 9.0 
 
 13.5 
 18.0 
 
 22.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 20 
 
 2.0 
 
 2.3 
 
 2.7 
 3.0 
 
 3.3 
 
 6.7 
 10.0 
 13.3 
 16.7 
 
 19 
 
 1.9 
 
 2.2 
 
 2.5 
 
 2.9 
 
 3.2 
 
 6.3 
 
 9.5 
 
 12.7 
 
 15.8 
 
 18 
 
 1.8 
 
 2.1 
 
 2.4 
 
 2.7 
 
 3.0 
 
 6.0 
 
 9.0 
 
 12.0 
 
 15.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 9 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.4 
 
 1.5 
 
 3.0 
 
 4.5 
 
 6.0 
 
 7.5 
 
 0.8 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.3 
 
 2.7 
 4.0 
 
 5.3 
 
 6.7 
 
 P. P. 
 
 5S C 
 
526 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 34 ° 
 
 t 
 
 L. Sin. 
 
 0 
 
 9.74756 
 
 1 
 
 9.74775 
 
 2 
 
 9.74794 
 
 3 
 
 9.74812 
 
 4 
 
 9.74831 
 
 5 
 
 9.74850 
 
 6 
 
 9.74868 
 
 7 
 
 9.74887 
 
 8 
 
 9.74906 
 
 9 
 
 9.74924 
 
 10 
 
 9.74943 
 
 11 
 
 9.74961 
 
 12 
 
 9.74980 
 
 13 
 
 9.74999 
 
 14 
 
 9.75017 
 
 15 
 
 9.75036 
 
 16 
 
 9.75054 
 
 17 
 
 9.75073 
 
 18 
 
 9.75091 
 
 19 
 
 9.75110 
 
 20 
 
 9.75128 
 
 21 
 
 9.75147 
 
 22 
 
 9.75165 
 
 23 
 
 9.75184 
 
 24 
 
 9.75202 
 
 25 
 
 9.75221 
 
 26 
 
 9.75239 
 
 27 
 
 9.75258 
 
 28 
 
 9.75276 
 
 29 
 
 9.75294 
 
 30 
 
 9.75313 
 
 31 
 
 9.75331 
 
 32 
 
 9.75350 
 
 33 
 
 9.75368 
 
 34 
 
 9.75386 
 
 35 
 
 9.75405 
 
 36 
 
 9.75423 
 
 37 
 
 9.75441 
 
 38 
 
 9.75459 
 
 39 
 
 9.75478 
 
 40 
 
 9.75496 
 
 41 
 
 9.75514 
 
 42 
 
 9.75533 
 
 43 
 
 9.75551 
 
 44 
 
 9.75569 
 
 45 
 
 9.75587 
 
 46 
 
 9.75605 
 
 47 
 
 9.75624 
 
 48 
 
 9.75642 
 
 49 
 
 9.75660 
 
 50 
 
 9.75678 
 
 51 
 
 9.75696 
 
 52 
 
 9.75714 
 
 53 
 
 9.75733 
 
 54 
 
 9.75751 
 
 55 
 
 9.75769 
 
 56 
 
 9.75787 
 
 57 
 
 9.75805 
 
 58 
 
 9.75823 
 
 59 
 
 9.75841 
 
 60 
 
 9.75859 
 
 d. L.Tang. 
 
 L. Cos. 
 
 19 
 
 19 
 
 18 
 
 19 
 
 19 
 
 18 
 
 19 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 19 
 
 18 
 
 18 
 
 19 
 
 18' 
 19 
 18 
 18 
 19 
 18 
 18 
 18 
 19 
 18 
 18 
 19 
 18 
 18 
 18 
 18 
 19 
 18 • 
 18 
 18 
 18 
 18 
 19 
 18 
 18 
 18 
 18 
 18 
 18 
 18 
 
 9.82899 
 
 9.82926 
 
 9.82953 
 
 9.82980 
 
 9.83008 
 
 9.83035 
 
 9.83062 
 
 9.83089 
 
 9.83117 
 
 9.83144 
 
 9.83171 
 
 9.83198 
 
 9.83225 
 
 9.83252 
 
 9.83280 
 
 9.83307 
 
 9.83334 
 
 9.83361 
 
 9.83388 
 
 9.83415 
 
 9.83442 
 
 9.83470 
 
 9.83497 
 
 9.83524 
 
 9.83551 
 
 9.83578 
 
 9.83605 
 
 9.83632 
 
 9.83659 
 
 9.83686 
 
 9.83713 
 
 9.83740 
 
 9.83768 
 
 9.83795 
 
 9.83822 
 
 9.83849 
 
 9.83876 
 
 9.83903 
 
 9.83930 
 
 9.83957 
 
 9.83984 
 
 9.84011 
 
 9.84038 
 
 9.84065 
 
 9.84092 
 
 9.84119 
 
 9.84146 
 
 9.84173 
 
 9.84200 
 
 9.84227 
 
 9.84254 
 
 9.84280 
 
 9.84307 
 
 9.84334 
 
 9.84361 
 
 9.84388 
 
 9.84415 
 
 9.84442 
 
 9.84469 
 
 9.84496 
 
 9.84523 
 
 L. Cotgf. 
 
 27 
 
 27, 
 
 27 
 
 28 
 27 
 27 
 
 27 
 
 28 
 27 
 27 
 27 
 27 
 
 27 
 
 28 
 27 
 27 
 27 
 27 
 27 
 
 27 
 
 28 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 
 27 
 
 28 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 26 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 27 
 
 L. Cotg. 
 
 L. Cos. 
 
 0.17101 
 
 0.17074 
 
 0.17047 
 
 0.17020 
 
 0.16992 
 
 9.91857 
 
 9.91849 
 
 9.91840 
 
 9.91832 
 
 9.91823 
 
 0.16965 
 
 0.16938 
 
 0.16911 
 
 0.16883 
 
 0.16856 
 
 9.91815 
 
 9.91806 
 
 9.91798 
 
 9.91789 
 
 9.91781 
 
 0.16829 
 
 0.16802 
 
 0.16775 
 
 0.16748 
 
 0.16720 
 
 9.91772 
 
 9.91763 
 
 9.91755 
 
 9.91746 
 
 9.91738 
 
 0.16693 
 
 0.16666 
 
 0.16639 
 
 0.16612 
 
 0.16585 
 
 9.91729 
 
 9.91720 
 
 9.91712 
 
 9.91703 
 
 9.91695 
 
 0.16558 
 
 0.16530 
 
 0.16503 
 
 0.16476 
 
 0.16449 
 
 9.91686 
 
 9.91677 
 
 9.91669 
 
 9.91660 
 
 9.91651 
 
 0.16422 
 
 0.16395 
 
 0.16368 
 
 0.16341 
 
 0.16314 
 
 9.91643 
 
 9.91634 
 
 9.91625 
 
 9.91617 
 
 9.91608 
 
 0.16287 
 
 0.16260 
 
 0.16232 
 
 0.16205 
 
 0.16178 
 
 9.91599 
 
 9.91591 
 
 9.91582 
 
 9.91573 
 
 9.91565 
 
 0.16151 
 
 0.16124 
 
 0.16097 
 
 0.16070 
 
 0.16043 
 
 9.91556 
 
 9.9154? 
 
 9.91538 
 
 9.91530 
 
 9.91521 
 
 0.16016 
 
 0.15989 
 
 0.15962 
 
 0.15935 
 
 0.15908 
 
 9.91512 
 
 9.91504 
 
 9.91495 
 
 9.91486 
 
 9.91477 
 
 0.15881 
 
 0.15854 
 
 0.15827 
 
 0.15800 
 
 0.15773 
 
 9.91469 
 
 9.91460 
 
 9.91451 
 
 9.91442 
 
 9.91433 
 
 0.15746 
 
 0.15720 
 
 0.15693 
 
 0.15666 
 
 0.15639 
 
 9.91425 
 
 9.91416 
 
 9.91407 
 
 9.91398 
 
 9.91389 
 
 0.15612 
 
 0.15585 
 
 0.15558 
 
 0.15531 
 
 0.15504 
 
 9.91381 
 
 9.91372 
 
 9.91363 
 
 9.91354 
 
 9.91345 
 
 0.15477 
 
 9.91336 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 d. 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 0 
 
 P. P. 
 
 28 
 
 2.8 
 
 3.3 
 
 3.7 
 
 4.2 
 
 4.7 
 
 9.3 
 14.0 
 18.7 
 23.3 
 
 27 
 
 2.7 
 
 3.2 
 
 3.6 
 
 4.1 
 
 4.5 
 
 9.0 
 
 13.5 
 18.0 
 
 22.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 13 
 
 1.9 
 
 2.2 
 
 2.5 
 
 2.9 
 
 3.2 
 
 6.3 
 
 9.5 
 
 12.7 
 
 15.8 
 
 18 
 
 1.8 
 
 2.1 
 
 2.4 
 
 2.7 
 
 3.0 
 
 6.0 
 
 9.0 
 
 12.0 
 
 15.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 9 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.4 
 
 1.5 
 
 3.0 
 
 4.5 
 
 6.0 
 
 7.5 
 
 0.8 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.3 
 
 2.7 
 4.0 
 
 5.3 
 
 6.7 
 
 P. P. 
 
 55 ° 
 
f 
 
 L. Sin. 
 
 0 
 
 9.75859 
 
 1 
 
 9.75877 
 
 2 
 
 9.75895 
 
 3 
 
 9.75913 
 
 4 
 
 9.75931 
 
 5 
 
 9.75949 
 
 6 
 
 9.75967 
 
 7 
 
 9.75985 
 
 8 
 
 9.76003 
 
 9 
 
 9.76021 
 
 10 
 
 9.76039 
 
 11 
 
 9.76057 
 
 12 
 
 9.76075 
 
 13 
 
 9.76093 
 
 14 
 
 9.76111 
 
 15 
 
 9.76129 
 
 16 
 
 9.76146 
 
 17 
 
 9.76164 
 
 18 
 
 9.76182 
 
 19 
 
 9.76200 
 
 20 
 
 9.76218 
 
 21 
 
 9.76236 
 
 22 
 
 9.76253 
 
 23 
 
 9.76271 
 
 24 
 
 9.76289 
 
 25 
 
 9.76307 
 
 26 
 
 9.76324 
 
 27 
 
 9.76342 
 
 28 
 
 9.76360 
 
 29 
 
 9.76378 
 
 30 
 
 9.76395 
 
 31 
 
 9.76413 
 
 32 
 
 9.76431 
 
 33 
 
 9.76448 
 
 34 
 
 9.76466 
 
 35 
 
 9.76484 
 
 36 
 
 9.76501 
 
 37 
 
 9.76519 
 
 38 
 
 9.76537 
 
 39 
 
 9.76554 
 
 40 
 
 9.76572 
 
 41 
 
 9.76590 
 
 42 
 
 9.76607 
 
 43 
 
 9.76625 
 
 44 
 
 9.76642 
 
 45 
 
 9.76660 
 
 46 
 
 9.76677 
 
 47 
 
 9.76695 
 
 48 
 
 9.76712 
 
 49 
 
 9.76730 
 
 50 
 
 9.76747 
 
 51 
 
 9.76765 
 
 52 
 
 9.76782 
 
 53 
 
 9.76800 
 
 54 
 
 9.76817 
 
 55 
 
 9.76835 
 
 56 
 
 9.76852 
 
 57 
 
 9.76870 
 
 58 
 
 9.76887 
 
 59 
 
 9.76904 
 
 60 
 
 9.76922 
 
 
 L. Cos. 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 35 ° 
 
 527 
 
 d. 
 
 L.Tang. d. c. L. Cotg. 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 18 
 
 17 
 
 18 
 18 
 18 
 18 
 18 
 
 17 
 
 18 
 18 
 18 
 
 17 
 
 18 
 18 
 18 
 
 17 
 
 18 
 18 
 
 17 
 
 18 
 18 
 
 17 
 
 18 
 18 
 
 17 
 
 18 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 
 17 
 
 18 
 17 
 
 17 
 
 18 
 
 9.84528 
 
 9.84550 
 
 9.84576 
 
 9.84603 
 
 9.84630 
 
 9.84657 
 
 9.84684 
 
 9.84711 
 
 9.84738 
 
 9.84764 
 
 9.84791 
 
 9.84818 
 
 9.84845 
 
 9.84872 
 
 9.84899 
 
 9.84925 
 
 9.84952 
 
 9.84979 
 
 9.85006 
 
 9.85033 
 
 9.85059 
 
 9.85086 
 
 9.85113 
 
 9.85140 
 
 9.85166 
 
 9.85193 
 
 9.85220 
 
 9.85247 
 
 9.85273 
 
 9.85300 
 
 9.85327 
 
 9.85354 
 
 9.85380 
 
 9.85407 
 
 9.85434 
 
 9.85460 
 
 9.85487 
 
 9.85514 
 
 9.85540 
 
 9.85567 
 
 9.85594 
 
 9.85620 
 
 9.85647 
 
 9.85674 
 
 9.85700 
 
 9.85727 
 
 9.85754 
 
 9.85780 
 
 9.85807 
 
 9.85834 
 
 9.85860 
 
 9.85887 
 
 9.85913 
 
 9.85940 
 
 9.85967 
 
 9.85993 
 
 9.86020 
 
 9.86046 
 
 9.86073 
 
 9.86100 
 
 9.86126 
 
 d. L. Cotg. 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 0.15477 
 
 0.15450 
 
 0.15424 
 
 0.15397 
 
 0.15370 
 
 0.15343 
 
 0.15316 
 
 0.15289 
 
 0.15262 
 
 0.15236 
 
 0.15209 
 
 0.15182 
 
 0.15155 
 
 0.15128 
 
 0.15101 
 
 0.15075 
 
 0.15048 
 
 0.15021 
 
 0.14994 
 
 0.14967 
 
 0.14941 
 
 0.14914 
 
 0.14887 
 
 0.14860 
 
 0.14834 
 
 0.14807 
 
 0.14780 
 
 0.14753 
 
 0.14727 
 
 0.14700 
 
 L. Cos. 
 
 9.91336 
 
 9.91328 
 
 9.91319 
 
 9.91310 
 
 9.91301 
 
 9.91292 
 
 9.91283 
 
 9.91274 
 
 9.91266 
 
 9.91257 
 
 9.91248 
 
 9.91239 
 
 9.91230 
 
 9.91221 
 
 9.91212 
 
 d. 
 
 9.91203 
 
 9.91194 
 
 9.91185 
 
 9.91176 
 
 9.91167 
 
 9.91158 
 
 .9.91149 
 
 9.91141 
 
 9.91132 
 
 9.91123 
 
 0.14673 
 
 0.14646 
 
 0.14620 
 
 0.14593 
 
 0.14566 
 
 0.14540 
 
 0.14513 
 
 0.14486 
 
 0.14460 
 
 0.14433 
 
 0.14406 
 
 0.14380 
 
 0.14353 
 
 0.14326 
 
 0.14300 
 
 9.91114 
 
 9.91105 
 
 9.91096 
 
 9.91087 
 
 9.91078 
 
 9.91069 
 
 9.91060 
 
 9.91051 
 
 9.91042 
 
 9.91033 
 
 9.91023 
 
 9.91014 
 
 9.91005 
 
 9.90996 
 
 9.90987 
 
 0.14273 
 
 0.14246 
 
 0.14220 
 
 0.14193 
 
 0.14166 
 
 0.14140 
 
 0.14113 
 
 0.14087 
 
 0.14060 
 
 0.14033 
 
 0.14007 
 
 0.13980 
 
 0.13954 
 
 0.13927 
 
 0.13900 
 
 d. c. 
 
 0.13874 
 I L.Tang 
 
 9.90978 
 
 9.90969 
 
 9.90960 
 
 9.90951 
 
 9.90942 
 
 9.90933 
 
 9.90924 
 
 9.90915 
 
 9.90906 
 
 9.90896 
 
 9.90887 
 
 9.90878 
 
 9.90869 
 
 9.90860 
 
 9.90851 
 
 9.90842 
 
 9.90832 
 
 9.90823 
 
 9.90814 
 
 9.90805 
 
 9.90796 
 
 L. Sin. 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 9 
 9 
 9 
 
 10 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 9 
 
 9 
 9 
 9 
 9 
 9 
 
 10 
 9 
 9 
 
 9 
 9 
 9 
 9 
 
 10 
 9 
 9 
 9 
 9 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 ^ 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 P. P. 
 
 
 27 
 
 26 
 
 6 
 
 2.7 
 
 2.6 
 
 7 
 
 3.2 
 
 3.0 
 
 8 
 
 3.6 
 
 3.5 
 
 9 
 
 4.1 
 
 3.9 
 
 10 
 
 4.5 
 
 4.3 
 
 20 
 
 9.0 
 
 8.7 
 
 30 
 
 13.5 
 
 13.0 
 
 40 
 
 18.0 
 
 17.3 
 
 50 
 
 22.5 
 
 21.7 
 
 
 18 
 
 6 
 
 1.8 
 
 7 
 
 2.1 
 
 8 
 
 2.4 
 
 9 
 
 2.7 
 
 10 
 
 3.0 
 
 20 
 
 6.0 
 
 30 
 
 9.0 
 
 40 
 
 12.0 
 
 50 
 
 15.0 
 
 17 
 
 1.7 
 2.0 
 2.3 
 2.6 
 
 2.8 
 5.7 
 8.5 
 
 11.3 
 
 14.2 
 
 10 
 
 1.0 
 
 1.2 
 
 1.3 
 1.5 
 
 1.7 
 
 3.3 
 5.0 
 
 6.7 
 
 8.3 
 
 
 9 
 
 8 
 
 6 
 
 0.9 
 
 0.8 
 
 7 
 
 1.1 
 
 0.9 
 
 8 
 
 1.2 
 
 1.1 
 
 9 
 
 1.4 
 
 1.2 
 
 10 
 
 1.5 
 
 1.3 
 
 20 
 
 3.0 
 
 2.7 
 
 30 
 
 4.5 
 
 4.0 
 
 40 
 
 6.0 
 
 5.3 
 
 50 
 
 7.5 
 
 6.7 
 
 P. P. 
 
 54 ° 
 
528 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 36 ° 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 0.13874 
 
 9.90796 
 
 
 60 
 
 0.13847 
 
 9.90787 
 
 9 
 
 59 
 
 0.13821 
 
 9.90777 
 
 10 
 
 58 
 
 0.13794 
 
 9.90768 
 
 9 
 
 57 
 
 0.13768 
 
 9.90759 
 
 9 
 
 q 
 
 56 
 
 0.13741 
 
 9.90750 
 
 «7 
 
 A 
 
 55 
 
 0.13715 
 
 9.90741 
 
 9 
 
 54 
 
 0.13688 
 
 9.90731 
 
 10 
 
 53 
 
 0.13662 
 
 9.90722 
 
 9 
 
 52 
 
 0.13635 
 
 9.90713 
 
 9 
 
 q 
 
 51 
 
 0.13608 
 
 9.90704 
 
 V 
 
 -? rv 
 
 50 
 
 0.13582 
 
 9.90694 
 
 iO 
 
 49 
 
 0.13555 
 
 9.90685 
 
 9 
 
 48 
 
 0.13529 
 
 9.90676 
 
 9 
 
 47 
 
 0.13502 
 
 9.90667 
 
 9 
 
 10 
 
 46 
 
 0.13476 
 
 9.90657 
 
 -LV/ 
 
 A 
 
 45 
 
 0.13449 
 
 9.90648 
 
 9 
 
 a 
 
 44 
 
 0.13423 
 
 9.90639 
 
 9 
 
 43 
 
 0.13397 
 
 9.90630 
 
 9 
 
 42 
 
 0.13370 
 
 9.90620 
 
 10 
 
 q 
 
 41 
 
 0.13344 
 
 9.90611 
 
 A 
 
 40 
 
 0.13317 
 
 9.90602 
 
 9 
 
 1 A 
 
 39 
 
 0.13291 
 
 9.90592 
 
 10 
 
 38 
 
 0.13264 
 
 9.90583 
 
 9 
 
 37 
 
 0.13238 
 
 9.90574 
 
 9 
 
 9 
 
 36 
 
 0.13211 
 
 9.90565 
 
 
 35 
 
 0.13185 
 
 9.90555 
 
 10 
 
 34 
 
 0.13158 
 
 9.90546 
 
 9 
 
 A 
 
 33 
 
 0.13132 
 
 9.90537 
 
 9 
 
 32 
 
 0.13106 
 
 9.90527 
 
 10 
 
 9 
 
 31 
 
 0.13079 
 
 9.90518 
 
 
 30 
 
 0.13053 
 
 9.90509 
 
 9 
 
 29 
 
 0.13026 
 
 9.90499 
 
 10 
 
 A 
 
 28 
 
 0.13000 
 
 9.90490 
 
 9 
 
 27 
 
 0.12973 
 
 9.90480 
 
 10 
 
 9 
 
 26 
 
 0.12947 
 
 9.90471 
 
 
 25 
 
 0.12921 
 
 9.90462 
 
 9 
 
 24 
 
 0.12894 
 
 9.90452 
 
 10 
 
 23 
 
 0.12868 
 
 9.90443 
 
 9 
 
 22 
 
 0.12842 
 
 9.90434 
 
 9 
 
 10 
 
 21 
 
 0.12815 
 
 9.90424 
 
 20 
 
 0.12789 
 
 9.90415 
 
 9 
 
 19 
 
 0.12762 
 
 9.90405 
 
 10 
 
 18 
 
 0.12736 
 
 9.90396 
 
 9 
 
 17 
 
 0.12710 
 
 9.90386 
 
 10 
 
 9 
 
 16 
 
 0.12683 
 
 9.90377 
 
 
 15 
 
 0.12657 
 
 9.90368 
 
 9 
 
 14 
 
 0.12631 
 
 9.90358 
 
 10 
 
 13 
 
 0.12604 
 
 9.90349 
 
 9 
 
 12 
 
 0.12578 
 
 9.90339 
 
 10 
 
 9 
 
 11 
 
 0.12552 
 
 9.90330 
 
 
 10 
 
 0.12525 
 
 9.90320 
 
 10 
 
 9 
 
 0.12499 
 
 9.90311 
 
 9 
 
 *1 A 
 
 8 
 
 0.12473 
 
 9.90301 
 
 10 
 
 7 
 
 0.12446 
 
 9.90292 
 
 9 
 
 10 
 
 6 
 
 0.12420 
 
 9.90282 
 
 
 5 
 
 0.12394 
 
 9.90273 
 
 9 
 
 4 
 
 0.12367 
 
 9.90263 
 
 10 
 
 3 
 
 0.12341 
 
 9.90254 
 
 9 
 
 2 
 
 0.12315 
 
 9.90244 
 
 10 
 
 9 
 
 1 
 
 0.12289 
 
 9.90235 
 
 
 0 
 
 L.Tang. 
 
 L. Sin 
 
 | d.. 
 
 / 
 
 P. P. 
 
 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 
 
 45 
 
 46 
 
 47 
 48' 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.76922 
 
 9.76939 
 
 9.76957 
 
 9.76974 
 
 9.76991 
 
 9.77009 
 
 9.77026 
 
 9.77043 
 
 9.77061 
 
 9.77078 
 
 9.77095 
 
 9.77112 
 
 9.77130 
 
 9.77147 
 
 9.77164 
 
 9.77181 
 
 9.77199 
 
 9.77216 
 
 9.77233 
 
 9.77250 
 
 9.77268 
 
 9.77285 
 
 9.77302 
 
 9.77319 
 
 9.77336 
 
 9.77353 
 
 9.77370 
 
 9.77387 
 
 9.77405 
 
 9.7/422 
 
 9.77439 
 
 9.77456 
 
 9.77473 
 
 9.77490 
 
 977507 
 
 9.77524 
 
 9.77541 
 
 9.77558 
 
 9.77575 
 
 9.77592 
 
 9.77609 
 
 9.77626 
 
 9.77643 
 
 9.77660 
 
 9.77677 
 
 9.77694 
 
 9.77711 
 
 9.77728 
 
 9.77744 
 
 9.77761 
 
 9.77778 
 
 9.77795 
 
 9.77812 
 
 9.77829 
 
 9.77846 
 
 9.77862 
 
 9.77879 
 
 9.77896 
 
 9.77913 
 
 9.77930 
 
 60 
 
 9.77946 
 L. Cos. 
 
 17 
 
 18 
 17 
 
 17 
 
 18 
 17 
 
 17 
 
 18 
 17 
 17 
 
 17 
 
 18 
 17 
 17 
 
 17 
 
 18 
 17 
 17 
 
 17 
 
 18 
 17 
 17 
 17 
 17 
 17 
 17 
 
 17 
 
 18 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 17 
 16 
 17 
 17 
 17 
 17 
 17 
 17 
 16 
 
 17 
 
 17 
 
 17 
 
 17 
 
 16 
 
 9.86126 
 
 9.86153 
 
 9.86179 
 
 9.86206 
 
 9.86232 
 
 9.86259 
 
 9.86285 
 
 9.86312 
 
 9.86338 
 
 9.86365 
 
 9.86392 
 
 9.86418 
 
 9.86445 
 
 9.86471 
 
 9.86498 
 
 9.86524 
 
 9.86551 
 
 9.86577 
 
 9.86603 
 
 9.86630 
 
 9.86656 
 
 9.86683 
 
 9.86709 
 
 9.86736 
 
 9. 86762 
 
 9.86789 
 
 9.86815 
 
 9.86842 
 
 9.86868 
 
 9.86894 
 
 9.86921 
 
 9=86947 
 
 9.86974 
 
 9.87000 
 
 9.87027 
 
 9.87053 
 
 9.87079 
 
 9.87106 
 
 9.87132 
 
 9.87158 
 
 9.87185 
 
 9.87211 
 
 9.87238 
 
 9.87264 
 
 9.87290 
 
 9.87317 
 
 9.87343 
 
 9.87369 
 
 9.87396 
 
 9.87422 
 
 9.87448 
 
 9.87475 
 
 9.87501 
 
 9.87527 
 
 9.87554 
 
 9.87580 
 
 9.87606 
 
 9.87633 
 
 9.87659 
 
 9.87685 
 
 9.87711 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 27 
 
 26 
 
 26 
 
 26 
 
 27 
 
 2.7 
 
 3.2 
 
 3.6 
 
 4.1 
 
 4.5 
 
 9.0 
 
 13.5 
 18.0 
 
 22.5 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 18 
 
 1.8 
 
 2.1 
 
 2.4 
 
 2.7 
 
 3.0 
 
 6.0 
 
 9.0 
 
 40 12.0 
 50 i 15.0 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 17 
 
 1.7 
 2.0 
 2.3 
 2.6 
 
 2.8 
 5.7 
 8.5 
 
 11.3 
 
 14.2 
 
 16 
 
 1.6 
 
 1.9 
 
 2.1 
 
 2.4 
 
 2.7 
 
 5.3 
 
 8.0 
 
 10.7 
 
 13.3 
 
 iO 
 
 1.0 
 
 1.2 
 
 1.3 
 1.5 
 
 1.7 
 
 3.3 
 5.0 
 
 6.7 
 
 8.3 
 
 9 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.4 
 
 1.5 
 
 3.0 
 
 4.5 
 
 6.0 
 
 7.5 
 
 d. L. Cotg, 
 
 d. c. 
 
 P. P. 
 
 53 c 
 
logarithms of trigonometric functions. 
 
 529 
 
 / I 
 
 L. Sin. c 
 
 0 
 
 9.77946 n 
 
 1 
 
 9.77963 
 
 2 
 
 9.77980 } 
 
 3 
 
 9.77997 
 
 4 
 
 9.78013 
 
 5 
 
 9.78030 . 
 
 6 
 
 9.78047 ] 
 
 7 
 
 9.78063 ; 
 
 8 
 
 9.78080 : 
 
 9 
 
 9.78097 : 
 
 10 
 
 9.78113 ; 
 
 11 
 
 9.78130 
 
 12 
 
 9.78147 
 
 13 
 
 9.78163 
 
 14 
 
 9.78180 
 
 15 
 
 9.78197 
 
 16 1 
 
 9.78213 
 
 17 
 
 9.78230 
 
 18 
 
 9.78246 
 
 19 
 
 9.78263 
 
 20 
 
 9.78280 
 
 21 
 
 9.78296 
 
 22 
 
 9.78313 
 
 23 
 
 9.78329 
 
 24 
 
 9.78346 
 
 25 
 
 9.78362 
 
 26 
 
 9.78379 
 
 27 
 
 9.78395 
 
 28 
 
 9.78412 
 
 29 
 
 ; 9.78428 
 
 30 
 
 9.78445 
 
 31 
 
 9.78461 
 
 32 
 
 9.78478 
 
 33 
 
 9.78494 
 
 34 1 
 
 9.78510 
 
 35 
 
 9.78527 
 
 36 
 
 9.78543 
 
 37 
 
 9.78560 
 
 38 
 
 9.78576 
 
 39 
 
 9.78592 
 
 40 
 
 9.78609 
 
 41 
 
 9.78625 
 
 42 
 
 9.78642 
 
 43 
 
 9.78658 
 
 44 
 
 9.78674 1 
 
 45 
 
 9.78691" 
 
 46 
 
 9.78707 
 
 47 
 
 9.78723 
 
 48 
 
 9.78739 
 
 49 
 
 9.78756 
 
 50 
 
 | 9.78772 
 
 51 
 
 9.78788 
 
 52 
 
 9.78805 
 
 53 
 
 9.78821 
 
 54 
 
 9.78837 
 
 55 
 
 9.78853 
 
 56 
 
 9.78869 
 
 57 
 
 9.78886 
 
 58 
 
 . 9.78902 
 
 59 
 
 i 9.78918 
 
 60 
 
 i 9.78934 
 
 16 
 
 17 
 
 16 
 
 17 
 
 17 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 16 
 
 17 
 
 16 
 
 17 
 
 16 
 
 16 
 
 17 
 
 16 
 
 16 
 
 16 
 
 17 
 
 16 
 
 d. 
 
 L.Tang. d. 
 
 9.87711 r 
 9.87738 i 
 9.87764 i 
 9.87790 ; 
 
 9.87817 t 
 
 9.87843 J 
 9.87869 ; 
 
 9.87895 ; 
 
 9.87922 ; 
 
 9.87948 ; 
 
 9.87974 i 
 9.88000 ; 
 
 9.88027 ; 
 
 9.88053 ; 
 
 9.88079 ; 
 
 9.88105 
 
 9.88131 
 
 9.88158 
 
 9.88184 
 
 9.88210 
 
 9.88236 
 
 9.88262 
 
 9.88289 
 
 9.88315 
 
 9.88341 
 
 9.88367 
 
 9.88393 
 
 9.88420 
 
 9.88446 
 
 9.88472 
 
 9.88498 
 
 9.88524 
 
 9.88550 
 
 9.88577 
 
 9.88603 
 
 9.88629 
 
 9.88655 
 
 9.88681 
 
 9.88707 
 
 9.88733 
 
 9.88759 
 
 9.88786 
 
 9.88812 
 
 9.88838 
 
 9.88864 
 
 9.88890 
 
 9.88916 
 
 9.88942 
 
 9.88968 
 
 9.88994 
 
 9.89020 
 9.89046 
 9.89073 
 ' 9.89099 
 
 1 9.89125 
 
 ’ 9.89151 
 
 » 9.89177 
 
 ' 9.89203 
 
 > 9.89229 
 
 > 9.89255 
 
 3 9.89281 
 
 L. Cotg 
 
 .c. I 
 
 j. Cotg. ] 
 
 L. Cos. 
 
 d. 
 
 
 P. 
 
 P. 
 
 
 
 0.12289 
 
 9.90235 
 
 i 
 
 60 
 
 
 
 
 27- 
 
 0.12262 
 
 9.90225 
 
 10 
 
 59 
 
 
 
 
 26 
 
 0.12236 
 
 9.90216 
 
 9 
 
 58 
 
 
 27 
 
 
 26 
 
 0.12210 
 
 9.90206 
 
 10 
 
 57 
 
 6 
 
 2.7 
 
 
 27 
 
 0.12183 
 
 9.90197 
 
 9 
 
 10 
 
 56 
 
 7 
 
 3.2 
 
 
 26 
 
 0.12157 
 
 9.90187 
 
 55 
 
 8 
 
 3.6 
 
 
 26 
 
 0.12131 
 
 9.90178 
 
 9 
 
 54 
 
 9 
 
 4.1 
 
 
 26 
 
 0.12105 
 
 9.90168 
 
 10 
 
 53 
 
 10 
 
 4.5 
 
 
 27 
 
 0.12078 
 
 9.90159 
 
 9 
 
 52 
 
 20 
 
 9.0 
 
 
 26 
 
 0.12052 
 
 9.90149 
 
 10 
 
 10 - 
 
 51 
 
 30 
 
 13.5 
 
 
 2b 
 
 0.12026 
 
 9.90139 
 
 50 
 
 40 
 
 18.0 
 
 
 26 
 
 0.12000 
 
 9.90130 
 
 9 
 
 49 
 
 50 
 
 22.5 
 
 
 27 
 
 0.11973 
 
 9.90120 
 
 10 
 
 48 
 
 
 
 
 26 
 
 0.11947 
 
 9.90111 
 
 9 
 
 47 
 
 
 
 
 26 
 
 0.11921 
 
 9.90101 
 
 10 
 
 10 
 
 46 
 
 
 26 
 
 
 Zb 
 
 0.11895 
 
 9.90091 
 
 45 
 
 6 
 
 2.6 
 
 26 
 
 0.11869 
 
 9.90082 
 
 9 
 
 44 
 
 7 
 
 3.0 
 
 27 
 
 0.11842 
 
 9.90072 
 
 10 
 
 43 
 
 8 
 
 3.5 
 
 26 
 
 0.11816 
 
 9.90063 
 
 9 
 
 42 
 
 9 
 
 3.9 
 
 26 
 
 0.11790 
 
 9.90053 
 
 10 
 
 41 
 
 10 
 
 4.3 
 
 26 
 
 
 
 10 
 
 
 20 
 
 
 
 0.11764 
 
 9.90043 
 
 40 
 
 o ./ 
 
 1 o A 
 
 26 
 
 0.11738 
 
 9.90034 
 
 9 
 
 39 
 
 30 
 
 13.0 
 
 27 
 
 0.11711 
 
 9.90024 
 
 10 
 
 38 
 
 40 
 
 17.3 
 
 HI *7 j 
 
 26 
 
 0.11685 
 
 9.90014 
 
 10 
 
 37 
 
 50 
 
 21.7 
 
 26 
 
 26 
 
 0.11659 
 
 9.90005 
 
 9 
 
 10 
 
 36 
 
 
 N 
 
 
 0.11633 
 
 9.89995 
 
 35 
 
 
 
 
 26 
 
 0.11607 
 
 9.89985 
 
 10 
 
 34 
 
 
 17 
 
 
 27 
 
 0.11580 
 
 9.89976 
 
 9 
 
 33 
 
 6 
 
 1.7 
 
 26 
 
 0.11554 
 
 9.89966 
 
 10 
 
 32 
 
 7 
 
 2.0 
 
 26 
 
 0.11528 
 
 9.89956 
 
 10 
 
 q 
 
 31 
 
 8 
 
 Q 
 
 2.3 
 9 fi 
 
 Zo 
 
 0.11502 
 
 9.89947 
 
 U 
 -1 A 
 
 30 
 
 10 
 
 2.8 
 
 26 
 
 0.11476 
 
 9.89937 
 
 10 
 
 29 
 
 20 
 
 5.7 
 
 26 
 
 0.11450 
 
 9.89927 
 
 10 
 
 28 
 
 30 
 
 8.5 
 
 27 
 
 0.11423 
 
 9.89918 
 
 9 
 
 27 
 
 40 
 
 11.3 
 
 26 
 O a 
 
 0.11397 
 
 9.89908 
 
 10 
 
 10 
 
 26 
 
 50 
 
 14.2 
 
 ZO 
 
 0.11371 
 
 9.89898 
 
 25 
 
 
 
 
 26 
 
 0.11345 
 
 9.89888 
 
 10 
 
 24 
 
 
 
 
 .26 
 
 0.11319 
 
 9.89879 
 
 9 
 
 23 
 
 
 ic 
 
 26 
 
 0.11293 
 
 9.89869 
 
 10 
 
 22 
 
 o 
 
 lb 
 
 1 A 
 
 26 
 
 OA 
 
 0.11267 
 
 9.89859 
 
 10 
 
 10 
 
 21 
 
 b 
 
 7 
 
 i.b 
 
 1.9 
 
 Zb 
 
 0.11241 
 
 9.89849 
 
 20 
 
 8 
 
 2.1 
 
 27 
 
 0.11214 
 
 9.89840 
 
 9 
 
 19 
 
 9 
 
 2.4 
 
 26 
 
 0.11188 
 
 9.89830 
 
 10 
 
 18 
 
 10 
 
 2.7 
 
 26 
 
 0.11162 
 
 9.89820 
 
 10 
 
 17 
 
 20 
 
 5.3 
 
 26 
 
 OA 
 
 0.11136 
 
 9.89810 
 
 10 
 
 9 
 
 16 
 
 30 
 
 8.0 
 
 Zb 
 
 0.11110 
 
 9.89801 
 
 15 
 
 40 
 
 10.7 
 
 26 
 
 0.11084 
 
 9.89791 
 
 10 
 
 14 
 
 50 
 
 i 13.3 
 
 26 
 
 0.11058 
 
 9.89781 
 
 10 
 
 13 
 
 
 
 
 26 
 
 0.11032 
 
 9.89771 
 
 10 
 
 12 
 
 
 
 
 26 
 
 OA 
 
 0.11006 
 
 9.89761 
 
 10 
 
 Q 
 
 11 
 
 
 10 
 
 9 
 
 Zb 
 
 0.10980 
 
 9.89752" 
 
 -f A 
 
 10 
 
 6 
 
 1.0 
 
 0.9 
 
 26 
 
 0.10954 
 
 9.89742 
 
 10 
 
 9 
 
 7 
 
 1.2 
 
 1.1 
 
 27 
 
 0.10927 
 
 9.89732 
 
 10 
 
 8 
 
 8 
 
 1.3 
 
 1.2 
 
 26 
 
 0.10901 
 
 9.89722 
 
 10 
 
 7 
 
 9 
 
 1.5 
 
 1.4 
 
 26 
 
 OA 
 
 0.10875 
 
 9.89712 
 
 10 
 
 ' 10 
 
 6 
 
 10 
 
 1.7 
 
 1.5 
 
 ZO 
 
 0.10849 
 
 9.89702 
 
 
 5 
 
 20 
 
 3.3 
 
 3.0 
 
 26 
 
 0.10823 
 
 9.89693 
 
 i 9 
 
 4 
 
 30 
 
 5.0 
 
 4.5 
 
 26 
 
 0.10797 
 
 9.89683 
 
 1 K 
 
 3 
 
 40 
 
 6.7 
 
 6.0 
 
 26 
 
 0.10771 
 
 9.89673 
 
 ; In 
 
 2 
 
 50 
 
 8.3 
 
 7.5 
 
 26 
 o Ct 
 
 0.10745 
 
 > 9.89663 
 
 5 10 
 
 1 
 
 
 
 
 ZO 
 
 0.10719 
 
 1 9.8965* 
 
 i 10 
 
 0 
 
 
 
 
 d. c 
 
 . L.Tang 
 
 f. L. Sin 
 
 . d. 
 
 / 
 
 P. P. 
 
 52 ° 
 
530 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS . 
 
 38 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.78934 
 
 9.78950 
 
 9.78967 
 
 9.78983 
 
 9.78999 
 
 16 
 
 17 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 16 
 
 9.89281 
 
 9.89307 
 
 9.89333 
 
 9.89359 
 
 9.89385 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 26 
 
 0.10719 
 
 0.10693 
 
 0.10667 
 
 0.10641 
 
 0.10615 
 
 9.89653 
 
 9.89643 
 
 9.89633 
 
 9.89624 
 
 9.89614 
 
 10 
 
 10 
 
 9 
 
 ! 10 
 j 10 
 
 10 
 : io 
 ! io 
 ! 10 
 10 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 26 
 
 6 2.6 
 
 7 3.0 
 
 8 I 3.5 
 
 9 j 3.9 
 
 10 4.3 
 
 20 8.7 
 
 30 13.0 
 
 25 
 
 2.5 
 2.9 
 I 3.3 
 1 3.8 
 ! 4.2 
 8.3 
 12.5 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.79015 
 
 9.79031 
 
 9.79047 
 
 9.79063 
 
 9.79079 
 
 9.80411 
 
 9.89437 
 
 9.89463 
 
 9.89489 
 
 9.89515 
 
 0.10589 
 
 0.10563 
 
 0.10537 
 
 0.10511 
 
 0.10485 
 
 9.89604 
 
 9.89594 
 
 9.89584 
 
 9.89574 
 
 9.89564 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 9.79095 
 
 9.89541 
 
 0.10459 
 
 9.89554 
 
 50 
 
 40 17.3 
 
 16.7 
 
 11 
 
 9.79111 
 
 16 
 
 9.89567 
 
 26 
 
 0.10433 
 
 9.89544 
 
 10 
 
 49 
 
 50 | 21.7 
 
 20.S 
 
 12 
 
 9.79128 
 
 17 
 
 9.89593 
 
 26 
 
 0.10407 
 
 9.89534 
 
 ! 10 
 
 48 
 
 
 
 13 
 
 9.79144 
 
 16 
 
 9.89619 
 
 26 
 
 0.10381 
 
 9.89524 
 
 1 10 
 
 47 
 
 
 
 14 
 
 9.79160 
 
 16 
 
 16 
 
 9.89645 
 
 26 
 
 OA 
 
 0.10355 
 
 9.89514 
 
 10 
 
 10 
 
 46 
 
 17 
 
 15 
 
 9.79176 
 
 9.89671 
 
 
 0.10329 
 
 9.89504 
 
 45 
 
 6 17 
 
 16 
 
 9.79192 
 
 16 
 
 9.89697 
 
 26 
 
 0.10303 
 
 9.89495 
 
 9 
 
 44 
 
 7 ? 5 
 
 > 0 
 
 17 
 
 9.79208 
 
 16 
 
 9.89723 
 
 26 
 
 0.10277 
 
 9.89485 
 
 ! 10 
 
 43 
 
 8 2.3 
 
 18 
 
 9.79224 
 
 16 
 
 9.89749 
 
 26 
 
 0.10251 
 
 9.S9475 
 
 i 10 
 
 42 
 
 9 2.6 
 
 19 
 
 9.79240 
 
 16 
 
 16 
 
 9.89775 
 
 26 
 
 26 
 
 0.10225 
 
 9.89465 
 
 1 10 
 ! 10 
 
 41 
 
 io *: 
 
 >.8 
 
 20 
 
 9.79256 
 
 9.89801 
 
 0.10199 
 
 9.89455 
 
 40 
 
 20 ! 5.7 
 
 21 
 
 9.79272 
 
 16 
 
 9.89827 
 
 26 
 
 0.10173 
 
 9.S9445 
 
 1 10 
 
 39 
 
 30 | 8.5 
 
 22 
 
 9.79288 
 
 16 
 
 9.89S53 
 
 26 
 
 0.10147 
 
 9.89435 
 
 j 10 
 
 38 
 
 40 j 11.3 
 
 23 
 
 9.79304 
 
 16 
 
 9.89879 
 
 26 
 
 0.10121 
 
 9.89425 
 
 i 10 
 
 37 
 
 50 j 14.2 
 
 24 
 
 9.79319 
 
 15 
 
 16 
 
 9.89905 
 
 26 
 
 26 
 
 0.10095 
 
 9.89415 
 
 I 10 
 10 
 
 36 
 
 
 
 25 
 
 9.79335 
 
 9.89931 
 
 0.10069 
 
 9.89405 
 
 35 
 
 
 
 26 
 
 9.79351 
 
 16 
 
 9.89957 
 
 26 
 
 0.10043 
 
 9.89395 
 
 10 
 
 34 
 
 16 
 
 15 
 
 27 
 
 9.79367 
 
 16 
 
 9.89983 
 
 26 
 
 0.10017 
 
 9.89385 
 
 10 
 
 33 
 
 6 1.6 
 
 1.5 
 
 28 
 
 9.79383 
 
 16 
 
 9.90009 
 
 26 
 
 0.09991 
 
 9.89375 
 
 10 
 
 32 
 
 7 1.9 
 
 1.8 
 
 29 
 
 9.79399 
 
 16 
 
 9.90035 I 
 
 26 
 
 j 0.09965 
 
 9.89364 
 
 ! 11 
 
 31 
 
 8 2.1 
 
 2.0 
 
 
 
 16 
 
 
 1 Ofx 
 
 
 
 ! 10 
 
 
 
 
 30 
 
 9.79415 
 
 9.90061 
 
 zo 
 
 ! 0.09939 
 
 9.89354 
 
 30 
 
 9 2.4 
 
 | 2.3 
 
 31 
 
 9.79431 ! 
 
 16 
 
 9.90086 
 
 25 
 
 0.09914 
 
 9.S9344 
 
 ! 10 
 
 29 
 
 10 2.7 
 
 : 2.5 
 
 32 
 
 9.79447 ! 
 
 16 
 
 9.90112 
 
 26 
 
 i 0.09888 
 
 9.89334 
 
 j 10 
 
 28 
 
 20 i o.3 
 
 ; 5.0 
 
 33 
 
 9.79463 
 
 16 
 
 9.90138 
 
 j 26 
 
 0.09862 
 
 9.89324 
 
 ! 10 
 
 27 
 
 30 8.0 
 
 * 7.5 
 
 34 
 
 9.79478 
 
 15 
 
 16 
 
 9.90164 ! 
 
 26 
 
 26 
 
 i 0.09836 
 
 9.89314 
 
 j 10 
 10 
 
 26 
 
 40 10.7 
 50 , 13.3 | 
 
 10.0 
 
 12.5 
 
 35 
 
 9.79494 1 
 
 9.90190 | 
 
 i 0.09810 
 
 9.89304 
 
 25 
 
 
 36 
 
 9.79510 1 
 
 16 
 
 9.90216 ! 
 
 26 
 
 ! 0.09784 
 
 9.89294 
 
 10 
 
 24 
 
 
 
 37 
 
 9.79526 
 
 16 
 
 9.90242 1 
 
 
 1 0.09758 
 
 9.89284 
 
 10 
 
 23 
 
 
 
 38 
 
 9.79542 
 
 16 
 
 9.90268 | 
 
 26 
 
 | 0.09732 
 
 9.89274 
 
 10 
 
 22 
 
 II 
 
 39 
 
 9.79558 
 
 16 
 
 9.90294 | 
 
 26 
 
 0.09706 
 
 9.89264 
 
 10 
 
 21 
 
 6 1.1 
 
 
 
 15 
 
 
 
 
 
 10 
 
 
 T 1 Q 
 
 40 
 
 9.79573 | 
 
 9.90320 j 
 
 zo 
 
 0.09680 
 
 9.89254 
 
 20 
 
 i j 1.0 
 <2 1 ^ 
 
 41 
 
 9.79589 | 
 
 16 
 
 9.90346 
 
 26 
 
 0.09654 
 
 9.89244 ! 
 
 10 
 
 19 
 
 O 1.0 
 
 Q 1 7 
 
 42 
 
 9.79605 
 
 16 
 
 9.90371 I 
 
 25 
 
 ! 0.09629 
 
 9.89233 
 
 11 
 
 18 
 
 St j A. / 
 10 1 ^ 
 
 43 
 
 9.79621 
 
 16 
 
 9.90397 | 
 
 26 
 
 0.09603 
 
 9.89223 
 
 10 
 
 17 
 
 1U ; J..O 
 
 20 ^ 7 
 
 44 
 
 9.79636 
 
 15 
 
 16 
 
 9.90423 | 
 
 26 
 
 •>A 
 
 0.09577 
 
 9.89213 ! 
 
 10 
 
 10 
 
 16 
 
 _u o. / 
 
 30 ' 5.5 
 
 45 
 
 9.79652 ! 
 
 9.90449 j 
 
 ZD 
 
 0.09551 
 
 9.89203 | 
 
 15 
 
 40 7. 
 
 3 
 
 46 
 
 9.79668 
 
 16 
 
 9.90475 
 
 26 
 
 0.09525 
 
 9.89193 
 
 10 
 
 14 
 
 50 j 9. 
 
 o 
 
 47 
 
 9.79684 
 
 16 
 
 9.90501 
 
 26 
 
 0.09499 
 
 9.89183 
 
 10 
 
 13 
 
 
 
 48 
 
 9.79699 | 
 
 15 
 
 9.90527 
 
 26 
 
 0.09473 
 
 9.89173 i 
 
 10 
 
 12 
 
 
 
 49 
 
 9.79715 
 
 16 
 
 16 
 
 9.90553 
 
 26 j 
 25 | 
 
 0.09447 
 
 9.89162 I 
 
 11 
 
 10 
 
 11 
 
 10 
 
 9 
 
 50 
 
 9.79731 i 
 
 9.90578 
 
 0.09422 
 
 9.89152 
 
 10 
 
 6 1 1.0 
 
 0.9 
 
 51 
 
 9.79746 1 
 
 15 
 
 9.90604 
 
 26 i 
 
 0.09396 
 
 9.89142 i 
 
 10 
 
 9 
 
 7 1.2 | 
 
 1.1 
 
 52 
 
 9.79762 ! 
 
 16 
 
 9.90630 
 
 26 
 
 0.09370 
 
 9.89132 i 
 
 10 
 
 8 
 
 8 1 1.3 
 
 1.2 
 
 53 
 
 9.79778 
 
 16 
 
 9.90656 
 
 26 i 
 
 0.09344 
 
 9.89122 i 
 
 10 
 
 7 
 
 9 1.5 
 
 1.4 
 
 54 
 
 9.79793 | 
 
 15 
 
 16 
 
 9.90682 | 
 
 26 
 
 26 
 
 0.09318 
 
 9.89112 
 
 10 
 
 11 
 
 6 
 
 10 j 1.7 
 
 1.5 
 
 55 
 
 9.79809 i 
 
 9.90708 
 
 0.09292 
 
 9.89101 
 
 5 
 
 20 ! 3.3 
 
 3.0 
 
 56 
 
 9.79825 1 
 
 16 
 
 9.90734 
 
 26 
 
 0.09266 
 
 9.89091 i 
 
 10 
 
 4 
 
 30 1 5.0 
 
 4.5 
 
 57 
 
 9.79840 | 
 
 15 
 
 9.90759 ! 
 
 25 
 
 0.09241 
 
 9.S9081 
 
 10 
 
 3 
 
 40 6.7 
 
 6.0 
 
 58 
 
 9.79856 i 
 
 16 
 
 9.90785 j 
 
 26 
 
 0.09215 
 
 9.89071 
 
 10 
 
 2 
 
 50 | 8.3 
 
 7.6 
 
 59 
 
 9.79872 ! 
 
 16 
 
 15 
 
 9.90811 1 
 
 26 . 
 
 0.09189 
 
 9.89060 ! 
 
 11 
 
 10 
 
 1 
 
 
 
 60 
 
 9.79887 
 
 9.90837 
 
 — U , 
 
 0.09163 
 
 9.89050 
 
 0 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 51 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 
 
 531 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.79887 
 
 1 
 
 9.79903 
 
 2 
 
 9.79918 
 
 3 
 
 9.79934 
 
 4 
 
 9.79950 
 
 5 
 
 9.79965 
 
 6 
 
 9.79981 
 
 7 
 
 9.79996 
 
 8 
 
 9.80012 
 
 9 
 
 9.80027 
 
 10 
 
 9.80043 
 
 11 
 
 9.80058 
 
 12 
 
 9.80074 
 
 13 
 
 9.80089 
 
 14 
 
 9.80105 
 
 15 
 
 9.80120 
 
 16 
 
 9.80136 
 
 17 
 
 9.80151 
 
 18 
 
 9 V 80166 
 
 19 
 
 9.80182 
 
 20 
 
 9.80197 
 
 21 
 
 9.80213 
 
 22 
 
 9.80228 
 
 23 
 
 9.80244 
 
 24 
 
 9.80259 
 
 25 
 
 9.80274 
 
 26 
 
 9.80290 
 
 27 
 
 9.8G305 
 
 28 
 
 9.80320 
 
 29 
 
 9.80336 
 
 30 
 
 9.80351 
 
 31 
 
 9.80366 
 
 32 
 
 9.80382 
 
 33 
 
 9.80397 
 
 34 
 
 9.80412 
 
 35 
 
 9.80428 
 
 36 
 
 9.80443 
 
 37 
 
 9.80458 
 
 38 
 
 9.80473 
 
 39 
 
 9.80489 
 
 40 
 
 9.80504 
 
 41 
 
 9.80519 
 
 42 
 
 9.80534 
 
 43 
 
 9.80550 
 
 44 
 
 9.80565 
 
 45 
 
 9.80580 
 
 46 
 
 9.80595 
 
 47 
 
 9.80610 
 
 48 
 
 9.80625 
 
 49 
 
 9.80641 
 
 50 
 
 9.80656 
 
 51 
 
 9.80671 
 
 52 
 
 9.80686 
 
 53 
 
 9.80701 
 
 54 
 
 9.80716 
 
 55 
 
 9.80731 
 
 56 
 
 9.80746 
 
 57 
 
 9.80762 
 
 58 
 
 9.80777 
 
 59 
 
 9.80792 
 
 60 
 
 9.80807 
 
 
 L. Cos. 
 
 d. 
 
 16 
 
 15 
 
 16 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 15 
 
 15 
 
 16 
 
 15 
 
 16 
 
 15 
 
 16 
 15 
 
 15 
 
 16 
 15 
 
 15 
 
 16 
 15 
 
 15 
 
 16 
 15 
 
 15 
 
 16 
 15 
 15 
 
 15 
 
 16 
 15 
 15 
 
 15 
 
 16 
 15 
 15 
 15 
 15 
 
 15 
 
 16 
 15 
 15 
 15 
 15 
 15 
 15 
 
 15 
 
 16 
 15 
 15 
 
 L.Tang. d 
 
 “9790837 
 
 9.90863 
 
 9.90889 
 
 9.90914 
 
 9.90940 
 
 9.90966 
 
 9.90992 
 
 9.91018 
 
 9.91043 
 
 9.91069 
 
 9.91095 
 
 9.91121 
 
 9.91147 
 
 9.91172 
 
 9.91198 
 
 9.91224 
 
 9.91250 
 
 9.91276 
 
 9.91301 
 
 9.91327 
 
 9.91353 
 
 9.91379 
 
 9.91404 
 
 9.91430 
 
 9.91456 
 
 9.91482 
 
 9.91507 
 
 9.91533 
 
 9.91559 
 
 9.91585 
 
 9.91610 
 
 9.91636 
 
 9.91662 
 
 9.91688 
 
 9.91713 
 
 9.91739 
 
 9.91765 
 
 9.91791 
 
 9.91816 
 
 9.91842 
 
 9.91868 
 
 9.91893 
 
 9.91919 
 
 9.91945 
 
 9.91971 
 
 9.91996 
 
 9.92022 
 
 9.92048 
 
 9.92073 
 
 9.92099 
 
 9.92125 
 
 9.92150 
 
 9.92176 
 
 9.92202 
 
 9.92227 
 
 9.92253 
 
 9.92279 
 
 9.92304 
 
 9.92330 
 
 9.92356 
 
 9.92381 
 
 L. Cotg. 
 
 26 
 
 26 
 
 25 
 
 26 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 
 25 
 . 26 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 25 
 
 L. Cotg. 
 
 L. Cos. c 
 
 0.09163 " 
 
 9.89050 . 
 
 0.09137 
 
 9.89040 ] 
 
 0.09111 
 
 9.89030 } 
 
 0.09086 
 
 9.89020 } 
 
 0.09060 
 
 9.89009 1 
 
 0.09034 
 
 9.88999 , 
 
 0.09008 
 
 9.88989 
 
 0.08982 
 
 9.88978 
 
 0.08957 
 
 9.88968 
 
 0.08931 
 
 9.88958 
 
 0.08905 
 
 9.88948 . 
 
 0.08879 
 
 9.88937 3 
 
 0.08853 
 
 9.88927 : 
 
 0.08828 
 
 9.88917 : 
 
 0.08802 
 
 9.8^906 : 
 
 0.08776 
 
 9.88896 ! 
 
 0.08750 
 
 9.88886 : 
 
 0.08724 
 
 9.88875 : 
 
 0.08699 
 
 9.88865 : 
 
 0.08673 
 
 9.88855 
 
 0.08647 
 
 9.88844 
 
 0.08621 
 
 9.88834 
 
 0.08596 
 
 9.88824 
 
 0.08570 
 
 9.88813 
 
 0.08544 
 
 9.88803 
 
 0.08518 
 
 9.88793 
 
 0.08493 
 
 9.88782 
 
 0.08467 
 
 9.88772 
 
 0.08441 
 
 9.88761 
 
 0.08415 
 
 9.88751 
 
 0.08390 
 
 9.88741 
 
 0.08364 
 
 9.88730 
 
 0.08338 
 
 9.88720 
 
 0.08312 
 
 9.88709 
 
 0.08287 
 
 9.88699 
 
 0.08261 
 
 9.88688 
 
 0.08235 
 
 9.88678 
 
 0.08209 
 
 9.88668 
 
 0.08184 
 
 9.88657 
 
 0.08158 
 
 9.88647 
 
 0.08132 
 
 9.88636 
 
 0.08107 
 
 9.88626 
 
 0.08081 
 
 9.88615 
 
 0.08055 
 
 9.88605 
 
 0.08029 
 
 9.88594 
 
 0.08004 
 
 9.88584 
 
 0.07978 
 
 9.88573 
 
 0.07952 
 
 9.88563 
 
 0.07927 
 
 9.88552 
 
 0.07901 
 
 9.88542 
 
 0.07875 
 
 9.88531 
 
 0.07850 
 
 9.88521 
 
 0.07824 
 
 9.88510 
 
 0.07798 
 
 9.88499 
 
 0.07773 
 
 9.88489 
 
 0.07747 
 
 9.88478 
 
 0.07721 
 
 9.88468 
 
 0.07696 
 
 9.88457 
 
 0.07670 
 
 9.88447 
 
 0.07644 
 
 9.88436 
 
 0.07619 
 
 9.88425 
 
 3. L.Tang 
 
 . L. Sin. 
 
 60 
 
 59 
 
 58 
 
 57 
 
 _56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 P. P. 
 
 10 
 
 10 
 
 11 
 
 10 
 
 10 
 
 11 
 
 10 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 10 
 
 11 
 
 11 
 
 d. 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 25 
 
 2.5 
 
 2.9 
 
 3.3 
 
 3.8 
 
 4.2 
 
 8.3 
 12.5 
 
 16.7 
 
 20.8 
 
 16 
 
 1.6 
 
 1.9 
 
 2.1 
 
 2.4 
 
 2.7 
 
 5.3 
 
 8.0 
 
 10.7 
 
 13.3 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 15 
 
 1.5 
 1.8 
 2.0 
 2.3 
 
 2.5 
 5.0 
 
 7.5 
 10.0 
 12.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 II 
 
 1.1 
 
 1.3 
 
 1.5 
 
 1.7 
 
 1.8 
 
 3.7 
 
 5.5 
 
 7.3 
 9.2 
 
 10 
 
 1.0 
 
 1.2 
 
 1.3 
 1.5 
 
 1.7 
 
 3.3 
 5.0 
 
 6.7 
 
 8.3 
 
 P. P. 
 
 50 ° 
 
532 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS, 
 
 40 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.80807 
 
 9.80822 
 
 9.80837 
 
 9.80852 
 
 9.80867 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 15 
 
 14 . 
 
 15 
 15 
 15 
 15 
 15 
 15 
 15 
 
 14 
 
 15 
 15 
 15 
 15 
 
 14 
 
 15 
 15 
 15 
 15 
 
 14 
 
 15 
 15 
 
 14 
 
 15 
 15 
 15 
 
 14 
 
 15 
 15 
 
 14 
 
 15 
 15 
 
 14 
 
 15 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 14 
 
 9.92381 
 
 9.92407 
 
 9.92433 
 
 9.92458 
 
 9.92484 
 
 26 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 
 25 
 
 26 
 25 
 
 0.07619 
 
 0.07593 
 
 0.07567 
 
 0.07542 
 
 0.07516 
 
 9.88425 
 
 9.88415 
 
 9.88404 
 
 9.88394 
 
 9.88383 
 
 10 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 26 
 
 6 2.6 
 
 7 3.0 
 
 8 3.5 
 
 9 3.9 
 
 10 4.3 
 
 20 8.7 
 
 30 13.0 
 40 17.3 
 50 21.7 
 
 25 
 
 6 2.5 
 
 7 2.9 
 
 8 3.3 
 
 9 3.8 
 
 10 4.2 
 
 20 8.3 ! 
 
 30 12.5 
 40 16.7 
 50 20.8 
 
 15 
 
 6 1.5 | 
 
 7 1.8 
 
 8 2.0 
 
 9 2.3 
 
 10 2.5 
 
 20 5.0 
 
 30 7.5 
 
 40 10.0 
 50 12.5 
 
 14 ! 
 
 6 1.4 
 
 7 1.6 
 
 8 1.9 
 
 9 2.1 
 
 10 2.3 
 
 20’ 4.7 
 
 30 7.0 
 
 40 9.3 
 
 50 11.7 
 
 II 10 
 
 6 1.1 1.0 
 
 7 1.3 1.2 
 
 8 1.5 1.3 
 
 9 1.7 1.5 
 10 1.8 1.7 
 
 20 3.7 3.3 j 
 30 5.5 5.0 
 40 7.3 6.7 
 50 9.2 8.3 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.80882 
 
 9.80897 
 
 9.80912 
 
 9.80927 
 
 9.80942 
 
 9.92510 
 
 9.92535 
 
 9.92561 
 
 9.92587 
 
 9.92612 
 
 0.07490 
 
 0.07465 
 
 0.07439 
 
 0.(57413 
 
 0.07388 
 
 9.88372 
 
 9.88362 
 
 9.88351 
 
 9.88340 
 
 9.88330 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.80957 
 
 9.80972 
 
 9.80987 
 
 9.81002 
 
 9.81017 
 
 9.92638 
 
 9.92663 
 
 9.92689 
 
 9.92715 
 
 9.92740 
 
 0.07362 
 
 0.07337 
 
 0.07311 
 
 0.07285 
 
 0.07260 
 
 9.88319 
 
 9.88308 
 
 9.88298 
 
 9.88287 
 
 9.88276 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.81032 
 
 9.81047 
 
 9.81061 
 
 9.81076 
 
 9.81091 
 
 9.92766 
 
 9.92792 
 
 9.92817 
 
 9.92843 
 
 9.92868 
 
 0.07234 
 
 0.07208 
 
 0.07183 
 
 0.07157 
 
 0.07132 
 
 9.88266 
 
 9.88255 
 
 9.88244 
 
 9.88234 
 
 9.88223 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.81106 
 
 9.81121 
 
 9.81136 
 
 9.81151 
 
 9.81166 
 
 9.92894 
 
 9.92920 
 
 9.92945 
 
 9.92971 
 
 9.92996 
 
 0.07106 
 
 0.07080 
 
 0.07055 
 
 0.07029 
 
 0.07004 
 
 9.88212 
 
 9.88201 
 
 9.88191 
 
 9.88180 
 
 9.88169 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.81180 
 
 9.81195 
 
 9.81210 
 
 9.81225 
 
 9.81240 
 
 9.93022 
 
 9.93048 
 
 9.93073 
 
 9.93099 
 
 9.93124 
 
 0.06978 
 
 0.06952 
 
 0.06927 
 
 0.06901 
 
 0.06876 
 
 9.88158 
 
 9.88148 
 
 9.88137 
 
 9.88126 
 
 9.88115 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.81254 
 
 9.81269 
 
 9.81284 
 
 9.81299 
 
 9.81314 
 
 9.93150 
 
 9.93175 
 
 9.93201 
 
 9.93227 
 
 9.93252 
 
 0.06850 
 
 0.06825 
 
 0.06799 
 
 0.06773 
 
 0.06748 
 
 9.88105 
 
 9.88094 
 
 9.88083 
 
 9.88072 
 
 9.88061 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.81328 
 
 9.81343 
 
 9.81358 
 
 9.81372 
 
 9.81387 
 
 9.93278 
 
 9.93303 
 
 9.93329 
 
 9.93354 
 
 9.93380 
 
 0.06722 
 
 0.06697 
 
 0.06671 
 
 0.06646 
 
 0.06620 
 
 9.88051 
 
 9.88040 
 
 9.88029 
 
 9.88018 
 
 9.88007 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.81402 
 
 9.81417 
 
 9.81431 
 
 9.81446 
 
 9.81461 
 
 9.93406 
 
 9.93431 
 
 9.93457 
 
 9.93482 
 
 9.93508 
 
 0.06594 
 
 0.06569 
 
 0.06543 
 
 0.06518 
 
 0.06492 
 
 9.87996 
 
 9.87985 
 
 9.87975 
 
 9.87964 
 
 9.87953 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.81475 
 
 9.81490 
 
 9.81505 
 
 9.81519 
 
 9.81534 
 
 9.93533 
 
 9.93559 
 
 9.93584 
 
 9.93610 
 
 9.93636 
 
 0.06467 
 
 0.06441 
 
 0.06416 
 
 0.06390 
 
 0.06364 
 
 9.87942 
 
 9.87931 
 
 9.87920 
 
 9.87909 
 
 9.87898 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 1 52 
 
 53 
 
 54 
 
 9.81549 
 
 9.81563 
 
 9.81578 
 
 9.81592 
 
 9.81607 
 
 9.93661 
 
 9.93687 
 
 9.93712 
 
 9.93738 
 
 9.93763 
 
 0.06339 
 
 0.06313 
 
 0.06288 
 
 0.06262 
 
 0.06237 
 
 9.87887 
 
 9.87877 
 
 9.87866 
 
 9.87855 
 
 9.87844 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 60 
 
 9.81622 
 
 9.81636 
 
 9.81651 
 
 9.81665 
 
 9.81680 
 
 9.93789 
 
 9.93814 
 
 9.93840 
 
 9.93865 
 
 9.93891 
 
 0.06211 
 
 0.06186 
 
 0.06160 
 
 0.06135 
 
 0.06109 
 
 9.87833 
 
 9.87822 
 
 9.87811 
 
 9.87800 
 
 9.87789 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 9.81694 
 
 9.93916 
 
 0.06084 
 
 9.87778 
 
 0 
 
 
 L. Cos. 
 
 ~~A. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 49 ° 
 
1 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 533 
 
 / 
 
 L. Sin. 
 
 0 
 
 9.81694 
 
 1 
 
 9.81709 
 
 2 
 
 9.81723 
 
 3 
 
 9.81738 
 
 4 
 
 9.81752 
 
 5 
 
 9.81767 
 
 6 
 
 9.81781 
 
 7 
 
 9.81796 
 
 8 
 
 9.81810 
 
 9 
 
 9.81825 
 
 10 
 
 9.81839 
 
 11 
 
 9.81854 
 
 12 
 
 9.81868 
 
 13 
 
 9.81882 
 
 14 
 
 9.81897 
 
 15 
 
 9.81911 
 
 16 
 
 9.81926 
 
 17 
 
 9.81940 
 
 18 
 
 9.81955 
 
 19 
 
 9.81969 
 
 20 
 
 9.81983 
 
 21 
 
 9.81998 
 
 22 
 
 9.82012 
 
 23 
 
 9.82026 
 
 24 
 
 9.82041 
 
 25 
 
 9.82055 
 
 26 
 
 9.82069 
 
 27 
 
 9.82084 
 
 28 
 
 9.82098 
 
 29 
 
 9.82112 
 
 30 
 
 9.82126 
 
 31 
 
 9.82141 
 
 32 
 
 9.82155 
 
 33 
 
 9.82169 
 
 34 
 
 9.82184 
 
 35 
 
 9.82198 
 
 36 
 
 9.82212 
 
 37 
 
 9.82226 
 
 38 
 
 9.82240 
 
 39 
 
 9.82255 
 
 40 
 
 9.82269 
 
 41 
 
 9.82283 
 
 42 
 
 9.82297 
 
 43 
 
 9.82311 
 
 44 
 
 9.82326 
 
 45 
 
 9.82340 
 
 46 
 
 9.82354 
 
 47 
 
 9.82368 
 
 48 
 
 9.82382 
 
 49 
 
 9.82396 
 
 50 
 
 9.82410 
 
 51 
 
 9.82424 
 
 52 
 
 9.82439 
 
 53 
 
 9.82453 
 
 54 
 
 9.82467 
 
 55 
 
 9.82481 
 
 56 
 
 9.82495 
 
 57 
 
 9.82509 
 
 58 
 
 9.82523 
 
 59 
 
 9.82537 
 
 60 
 
 9.82551 
 
 
 L. Cos. 
 
 d. L.Tang. 
 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 14 
 
 14 
 
 15 
 
 14 
 
 15 
 
 14 
 
 15 
 14 
 
 14 
 
 15 
 14 
 
 14 
 
 15 
 14 
 
 14 
 
 15 
 14 
 14 
 
 14 
 
 15 
 14 
 
 14 
 
 15 
 14 
 14 
 14 
 
 14 
 
 15 
 14 
 14 
 14 
 
 14 
 
 15 
 14 
 14 
 14 
 14 
 14 
 14 
 
 14 
 
 15 
 14 
 14 
 14 
 14 
 14 
 14 
 14 
 14 
 
 9.93916 
 
 9.93942 
 
 9.93967 
 
 9.93993 
 
 9.94018 
 
 9.94044 
 
 9.94069 
 
 9.94095 
 
 9.94120 
 
 9.94146 
 
 9.94171 
 
 9.94197 
 
 9.94222 
 
 9.94248 
 
 9.94273 
 
 9.94299 
 
 9.94324 
 
 9.94350 
 
 9.94375 
 
 9.94401 
 
 9.94426 
 
 9.94452 
 
 9.94477 
 
 9.94503 
 
 9.94528 
 
 9.94554 
 
 9.94579 
 
 9.94604 
 
 9.94630 
 
 9.94655 
 
 9.94681 
 
 9.94706 
 
 9.94732 
 
 5.94757 
 
 9.94783 
 
 d.c. J 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 
 0.06084 
 
 9.87778 
 
 
 60 
 
 26 
 
 0.06058 
 
 9.87767 
 
 11 
 
 59 
 
 25 
 
 0.06033 
 
 9.87756 
 
 11 
 
 58 
 
 26 
 
 0.06007 
 
 9.87745 
 
 11 
 
 57 
 
 25 
 
 0.05982 
 
 9.87734 
 
 11 
 
 11 
 
 56 
 
 2b 
 
 0.05956 
 
 9.87723 
 
 55 
 
 25 
 
 0.05931 
 
 9.87712 
 
 11 
 
 54 
 
 26 
 
 0.05905 
 
 9.87701 
 
 11 
 
 53 
 
 25 
 
 0.05880 
 
 9.87690 
 
 11 
 
 52 
 
 26 
 
 0.05854 
 
 9.87679 
 
 11 
 
 11 
 
 51 
 
 Zo 
 
 0.05829 
 
 9.87668 
 
 50 
 
 26 
 
 0.05803 
 
 9.87657 
 
 11 
 
 49 
 
 25 
 
 0.05778 
 
 9.87646 
 
 11 
 
 48 
 
 26 
 
 0.05752 
 
 9.87635 
 
 11 
 
 47 
 
 25 
 
 26 
 
 0.05727 
 
 9.87624 
 
 11 
 
 11 
 
 46 
 
 0.05701 
 
 9.87613 
 
 45 
 
 25 
 
 0.05676 
 
 9.87601 
 
 12 
 
 44 
 
 26 
 
 0.05650 
 
 9.87590 
 
 11 
 
 43 
 
 25 
 
 0.05625 
 
 9.87579 
 
 11 
 
 42 
 
 26 
 
 25 
 
 0.65599 
 
 9.87568 
 
 11 
 
 41 
 
 0.05574 
 
 9.87557 
 
 -LX 
 1 1 
 
 40 
 
 26 
 
 0.05548 
 
 9.87546 
 
 11 
 
 39 
 
 25 
 
 0.05523 
 
 9.87535 
 
 11 
 
 38 
 
 26 
 
 0.05497 
 
 9.87524 
 
 11 
 
 37 
 
 25 
 
 0.05472 
 
 9.87513 
 
 11 
 
 12 
 
 36 
 
 Zo 
 
 0.05446 
 
 9.87501 
 
 35 
 
 25 
 
 0.05421 
 
 9.87490 
 
 11 
 
 34 
 
 25 
 
 0.05396 
 
 9.87479 
 
 11 
 
 33 
 
 26 
 
 0.05370 
 
 9.87468 
 
 11 
 
 32 
 
 25 
 
 0.05345 
 
 9.87457 
 
 11 
 11 
 1 o 
 
 31 
 
 ZO 
 
 0.05319 
 
 9.87446 
 
 30 
 
 25 
 
 0.05294 
 
 9.87434 
 
 12 
 
 29 
 
 26 
 
 0.05268 
 
 9.87423 
 
 11 
 
 28 
 
 25 
 
 0.05243 
 
 9.87412 
 
 11 
 
 27 
 
 26 
 - 25 
 
 0.05217 
 
 9.87401 
 
 11 
 11 
 1 o 
 
 26 
 
 0.05192 
 
 9.87390 
 
 25 
 
 26 
 
 0.05166 
 
 9.87378 
 
 12 
 
 24 
 
 i 25 
 
 0.05141 
 
 9.87367 
 
 11 
 1 1 
 
 23 
 
 25 
 
 0.05116 
 
 9.87356 
 
 11 
 
 22 
 
 i 1 
 
 0.05090 
 
 9.87345 
 
 11 
 11 
 1 o 
 
 21 
 
 > ^ 
 
 0.05065 
 
 9.87334 
 
 20 
 
 26 
 
 0.05039 
 
 9.87322 
 
 12 
 
 19 
 
 i 25 
 
 0.05014 
 
 9.87311 
 
 11 
 
 18 
 
 » 26 
 
 0.04988 
 
 9.87300 
 
 1] 
 
 17 
 
 r 25 
 
 0.04963 
 
 9.87288 
 
 12 
 
 11 
 
 16 
 
 > 25 
 
 0.04938 
 
 9.87277 
 
 15 
 
 i 26 
 
 0.04912 
 
 9.87266 
 
 11 
 
 14 
 
 > 25 
 
 0.04887 
 
 9.87255 
 
 11 
 1 o 
 
 13 
 
 > It 
 
 0.04861 
 
 9.87243 
 
 12 
 
 12 
 
 1 26 
 
 0.04836 
 
 9.87232 
 
 11 
 
 11 
 
 11 
 
 ) It 
 
 0.04810 
 
 9.87221 
 
 1 o 
 
 10 
 
 
 0.04785 
 
 9.87209 
 
 12 
 i -| 
 
 9 
 
 ) 25 
 
 0.04760 
 
 9.87198 
 
 11 
 
 8 
 
 5 It 
 
 0.04734 
 
 9.87187 
 
 11 
 
 7 
 
 1 26 
 
 0.04709 
 
 9.87175 
 
 12, 
 
 11 
 
 6 
 
 7 it 
 
 0.04683 
 
 9.87164 
 
 5 
 
 2 25 
 
 0.04658 
 
 9.87153 
 
 11 
 
 4 
 
 3 ?6 
 
 0.04632 
 
 9.87141 
 
 12 
 
 3 
 
 i 25 
 
 0.04607 
 
 9.87130 
 
 11 
 
 2 
 
 8 25 
 
 0.04582 
 
 9.87119 
 
 11 
 
 - 12 
 
 , 1 
 
 4 26 
 
 0.04556 
 
 9.87107 
 
 
 0 
 
 d. c. 
 
 L.Tang 
 
 . L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 26 
 
 2.6 
 
 3.0 
 
 3.5 
 
 3.9 
 
 4.3 
 
 8.7 
 
 13.0 
 
 17.3 
 
 21.7 
 
 25 
 
 2.5 
 
 2.9 
 
 3.3 
 
 3.8 
 
 4.2 
 
 8.3 
 12.5 
 
 16.7 
 
 20.8 
 
 15 
 
 1.5 
 1 8 
 2.0 
 2.3 
 
 2.5 
 5.0 
 
 7.5 
 10.0 
 12.5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 14 
 
 1.4 
 
 1.6 
 
 1.9 
 
 2:1 
 
 2.3 
 4.7 
 7.0 
 
 9.3 
 11.7 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 12 
 
 1.2 
 
 1.4 
 
 1.6 
 
 1.8 
 
 2.0 
 
 4.0 
 
 6.0 
 8.0 
 
 10.0 
 
 P. P. 
 
 II 
 
 1.1 
 
 1.3 
 
 1.5 
 
 1.7 
 
 1.8 
 3.7 
 
 5.5 
 
 7.3 
 9.2 
 
 48 c 
 
534 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 42 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 
 P 
 
 . P. 
 
 
 0 
 
 9.82551 
 
 
 9.95444 
 
 
 0.04556 
 
 9.87107 
 
 
 60 
 
 
 
 
 
 1 
 
 9.82565 
 
 14 
 
 9.95469 
 
 2 5 
 
 0.04531 
 
 9.87096 
 
 11 
 
 59 
 
 
 
 
 
 2 
 
 9.82579 
 
 14 
 
 9.95495 
 
 26 
 
 0.04505 
 
 9.87085 
 
 11 
 
 58 
 
 
 
 0 c 
 
 3 
 
 9.82593 
 
 14 
 
 9.95520 
 
 25 
 
 0.04480 
 
 9.87073 
 
 12 
 
 57 
 
 
 R 
 
 CO 
 
 O R 
 
 4 
 
 9.82607 
 
 14 
 
 1 A 
 
 9.95545 
 
 25 
 
 26 
 
 0.04455 
 
 9.87062 
 
 11 
 
 12 
 
 56 
 
 
 0 
 
 7 
 
 Z.D 
 
 3.0 
 
 5 
 
 9.82621 
 
 
 9.95571 
 
 0.04429 
 
 ■9.87050 
 
 55 
 
 
 8 
 
 3.5 
 
 6 
 
 9.82635 
 
 14 
 
 9.95596 
 
 25 
 
 0.04404 
 
 9.87039 
 
 11 
 
 54 
 
 
 9 
 
 3.9 
 
 7 
 
 9.82649 
 
 14 
 
 9,95622 
 
 26 
 
 0.0437<S 
 
 9.87028 
 
 11 
 
 53 
 
 10 
 
 4.3 
 
 8 
 
 9.82663 
 
 14 
 
 9.95647 
 
 2i> 
 
 0.04353 
 
 9.87016 
 
 12 
 
 52 
 
 20 
 
 8.7 
 
 9 
 
 9.82677 
 
 14 
 r a 
 
 9.95672 
 
 25 
 
 26 
 
 0.04328 
 
 9.87005 
 
 11 
 
 12 
 
 51 
 
 30 
 
 13.0 
 
 10 
 
 9.82691 
 
 
 9.95698 
 
 0.04302 
 
 9.86993 
 
 50 
 
 40 
 
 17.3 
 
 11 
 
 9.82705 
 
 14 
 
 9.95723 
 
 25 
 
 0.04277 
 
 9.86982 
 
 11 
 
 49 
 
 50 
 
 21.7 
 
 12 
 
 9.82719 
 
 14 
 
 9.95748 
 
 25 
 
 0.04252 
 
 9.86970 
 
 12 
 
 48 
 
 
 
 
 
 13 
 
 9.82733 
 
 14 
 
 9.95774 
 
 26 
 
 0.04226 
 
 9.86959 
 
 11 
 
 47 
 
 
 
 
 
 14 
 
 9.82747 
 
 14 
 
 1, A 
 
 9.95799 
 
 25 
 
 0.04201 
 
 9.86947 
 
 12 
 
 11 
 
 46 
 
 
 
 25 
 
 15 
 
 9.82761 
 
 
 9.95825 
 
 
 0.04175 
 
 9.86936 
 
 45 
 
 
 6 
 
 2.5 
 
 16 
 
 9.82775 
 
 14 
 
 9.95850 
 
 25 
 
 0.04150 
 
 9.86924 
 
 12 
 
 44 
 
 
 7 
 
 2.9 
 
 17 - 
 
 9.82788 
 
 13 
 
 9.95875 
 
 25 
 
 0.04125 
 
 9.86913 
 
 11 
 
 43 
 
 
 8 
 
 3.3 
 
 18 
 
 9.82802 
 
 14 
 
 9.95901 
 
 26 
 
 0.04099 
 
 9.86902 
 
 11 
 
 42 
 
 
 9 
 
 3.8 
 
 19 
 
 9.82816 
 
 14 
 
 9.95926 
 
 25 
 
 0.04074 
 
 9.86890 
 
 12 
 
 41 
 
 10 
 
 4.2 
 
 
 
 
 
 26 
 
 
 
 11 
 
 
 
 
 
 
 20 
 
 9.82830 
 
 
 9.95952 
 
 0.04048 
 
 9.86879 
 
 40 
 
 ZO 
 
 S.3 
 
 21 
 
 9.82844 
 
 14 
 
 9.95977 
 
 25 
 
 0.04023 
 
 9.86867 
 
 12 
 
 39 
 
 30 
 
 12.5 
 
 22 
 
 9.82858 
 
 14 
 
 9.96002 
 
 25 
 
 0.03998 
 
 9.86855 
 
 12 
 
 38 
 
 40 
 
 16.7 
 
 23 
 
 9.82872 
 
 14 
 
 9.96028 
 
 26 
 
 0.03972 
 
 9.86844 
 
 11 
 
 37 
 
 50 
 
 20.8 
 
 24 
 
 9.82885 
 
 13 
 
 14 
 
 9.96053 
 
 25 
 
 25 
 
 0.03947 
 
 9.86832 
 
 12 
 
 11 
 
 36 
 
 
 
 
 
 25 
 
 9.82899 
 
 9.96078 
 
 0.03922 
 
 9.86821 
 
 35 
 
 
 
 
 
 26 
 
 9.82913 
 
 14 
 
 9.96104 
 
 26 
 
 0.03896 
 
 9.86809 
 
 12 
 
 34 
 
 
 
 14 
 
 27 
 
 9.82927 
 
 14 
 
 9.96129 
 
 25 
 
 0.03871 
 
 9.86798 
 
 11 
 
 33 
 
 
 6 
 
 1.4 
 
 28 
 
 9.82941 
 
 14 
 
 9.96155 
 
 26 
 
 0.03845 
 
 9.86786 
 
 12 
 
 32 
 
 
 7 
 
 1.6 
 
 29 
 
 9.82955 
 
 14 
 
 iq 
 
 9.96180 
 
 25 
 
 25 
 
 0.03820 
 
 9.86775 
 
 11 
 
 12 
 
 31 
 
 
 8 
 
 1.9 
 
 O 1 
 
 30 
 
 9.82968 
 
 AO 
 
 9.96205 
 
 0.03795 
 
 9.86763 
 
 30 
 
 i n 
 
 Z.A 
 
 9 ^ 
 
 31 
 
 9.82982 
 
 14 
 
 9.96231 
 
 26 
 
 0.03769 
 
 9.86752 
 
 11 
 
 29 
 
 ±u 
 
 on 
 
 A 7 
 
 32 
 
 9.82996 
 
 14 
 
 9.96256 
 
 25 
 
 0.03744 
 
 9.86740 
 
 12 
 
 28 
 
 
 4k. / 
 
 7 0 
 
 33 
 
 9.83010 
 
 14 
 
 9.96281 
 
 25 
 
 0.03719 
 
 9.86728 
 
 12 
 
 27 
 
 ou 
 
 d.0 
 
 / .u 
 Q ^ 
 
 34 
 
 9.83023 
 
 13 
 
 14 
 
 9.96307 
 
 26 
 
 25 
 
 OK 
 
 0.03693 
 
 9.86717 
 
 11 
 
 19 
 
 26 
 
 50 
 
 t/.O 
 
 11.7 
 
 35 
 
 9.83037 
 
 Art 
 
 i a 
 
 9.96332 
 
 0.03668 
 
 9.86705 
 
 A<£ 
 
 25 
 
 
 
 
 
 36 
 
 9.83051 
 
 14 
 
 9.96357 
 
 25 
 
 0.03643 
 
 9.86694 
 
 11 
 
 24 
 
 
 
 
 
 37 
 
 9.83065 
 
 14 
 
 9.96383 
 
 26 
 
 0.03617 
 
 9.86682 
 
 12 
 
 23 
 
 
 
 
 
 38 
 
 9.83078 
 
 13 
 
 9.96408 
 
 25 
 
 0.03592 
 
 9.86670 
 
 12 
 
 22 
 
 
 
 13 
 
 39 
 
 9.83092 
 
 14 
 
 14 
 
 9.96433 
 
 25 
 
 26 
 
 0.03567 
 
 9.86659 
 
 11 
 
 12 
 
 21 
 
 
 6 
 
 7 
 
 1.3 
 
 1.5 
 
 40 
 
 9.83106 
 
 9.96459 
 
 0.03541- 
 
 9.86647 
 
 20 
 
 
 8 
 
 L7 
 
 11 
 
 9.83120 
 
 14 
 
 9.96484 
 
 25 
 
 0.03516 
 
 9.86635 
 
 12 
 
 19 
 
 
 9 
 
 2.0 
 
 12 
 
 9.83133 
 
 13 
 
 9.96510 
 
 26 
 
 0.03490 
 
 9.86624 
 
 11 
 
 18 
 
 10 
 
 2.2 
 
 43 
 
 9.83147 
 
 14 
 
 9.96535 
 
 25 
 
 0.03465 
 
 9.86612 
 
 12 
 
 17 
 
 20 
 
 4.3 
 
 44 
 
 9.83161 
 
 14 
 
 -iq 
 
 9.96560 
 
 25 
 
 26 
 
 0.03440 
 
 9.86600 
 
 12 
 
 11 
 
 16 
 
 30 
 
 6.5 
 
 45 
 
 9.83174 
 
 _LO 
 
 9.96586 
 
 0.03414 
 
 9.86589 
 
 15 
 
 40 
 
 8.7 
 
 46 
 
 9.83188 
 
 14 
 
 9.96611 
 
 25 
 
 0.03389 
 
 9.86577 
 
 12 
 
 14 
 
 50 
 
 10.8 
 
 47 
 
 9.83202 
 
 14 
 
 9.96636 
 
 25 
 
 0.03364 
 
 9.86565 
 
 12 
 
 13 
 
 
 
 
 
 48 
 
 9.83215 
 
 13 
 
 9.96662 
 
 26 
 
 0.03338 
 
 9.86554 
 
 11 
 
 12 
 
 
 
 
 
 49 
 
 9.83229 
 
 14 
 
 IQ 
 
 9.96687 
 
 25 
 
 25 
 
 0.03313 
 
 9.86542 
 
 12 
 
 19 
 
 11 
 
 
 
 12 
 
 II 
 
 50 
 
 9.83242 
 
 i-O 
 
 i a 
 
 9.96712 
 
 0.03288 
 
 9.86530 
 
 AZi 
 
 10 
 
 6 
 
 1.2 
 
 1.1 
 
 51 
 
 9.83256 
 
 14 
 
 9.96738 
 
 26 
 
 0.03262 
 
 9.86518 
 
 12 
 
 9 
 
 7 
 
 1.4 
 
 1.3 
 
 52 
 
 9.83270 
 
 14 
 
 9.96763 
 
 25 
 
 0.03237 
 
 9.86507 
 
 11 
 
 8 
 
 8 
 
 1.6 
 
 1.5 
 
 53 
 
 9.83283 
 
 13 
 
 9.96788 
 
 25 
 
 0.03212 
 
 9.86495 
 
 12 
 
 7 
 
 9 
 
 1.8 
 
 1.7 
 
 54 
 
 9.83297 
 
 14 
 
 13 
 
 9.96814 
 
 26 
 
 25 
 
 0.03186 
 
 9.86483 
 
 12 
 
 11 
 
 6 
 
 10 
 
 2.0 
 
 1.8 
 
 55 
 
 9.83310 
 
 9.96839 
 
 0.03161 
 
 9.86472 
 
 5 
 
 20 
 
 4.0 
 
 3.7 
 
 56 
 
 9.83324 
 
 14 
 
 9.96864 
 
 25 
 
 0.03136 
 
 9.86460 
 
 12 
 
 4 
 
 30 
 
 6.0 
 
 5.5 
 
 57 
 
 9.83338 
 
 14 
 
 9.96890 
 
 26 
 
 0.03110 
 
 9.86448 
 
 12 
 
 3 
 
 40 
 
 8.0 
 
 7.3 
 
 58 
 
 9.83351 
 
 13 
 
 9.96915 
 
 25 
 
 0.03085 
 
 9.86436 
 
 12 
 
 2 
 
 50 
 
 10.0 
 
 9.2 
 
 59 
 
 9.83365 
 
 14 
 
 IQ 
 
 9.96940 
 
 25 
 
 26 
 
 0.03060 
 
 9.86425 
 
 11 
 
 12 
 
 1 
 
 
 
 
 
 60 
 
 9.83378 
 
 AO 
 
 9.96966 
 
 0.03034 
 
 9.86413 
 
 0 
 
 
 
 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 47 ° 
 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 535 
 
 / 
 
 L. Sin. - 
 
 0 ' 
 
 9.83378 
 
 1 ' 
 
 9.83392 
 
 2 1 
 
 9.83405 
 
 3 1 
 
 9.83419 
 
 4 1 
 
 9.83432 
 
 5 
 
 9.83446 
 
 6 
 
 9.83459 
 
 7 
 
 9.83473 
 
 8 
 
 9.83486 
 
 9 
 
 9.83500 
 
 10 
 
 9.83513 
 
 11 
 
 9.83527 
 
 12 
 
 9.83540 
 
 13 
 
 9.83554 
 
 14 
 
 9.83567 
 
 15 
 
 9.83581 
 
 16 
 
 9.83594 
 
 17 
 
 9.83608 
 
 18 
 
 9.83621 
 
 19 
 
 9.83634 
 
 20 
 
 9783648 
 
 21 
 
 9.83661 
 
 22 
 
 9.83674 
 
 23 
 
 9.83688 
 
 24 
 
 9.83701 
 
 25 
 
 9.83715 
 
 26 
 
 9.83728 
 
 27 
 
 9.83741 
 
 28 
 
 9.83755 
 
 29 
 
 9.83768 
 
 30 
 
 9.83781 
 
 31 
 
 9.83795 
 
 32 
 
 9.83808 
 
 33 
 
 9.83821 
 
 34 
 
 9.83834 
 
 35 
 
 9.83848 
 
 36 
 
 9.83861 
 
 37 
 
 9.83874 
 
 38 
 
 9.83887 
 
 39 
 
 9.83901 
 
 40 
 
 9.83914 
 
 41 
 
 9.83927 
 
 42 
 
 9.83940 
 
 43 
 
 9.83954 
 
 44 
 
 9.83967 
 
 45 
 
 9.83980 
 
 46 
 
 9.83993 
 
 47 
 
 9.84006 
 
 48 
 
 9.84020 
 
 49 
 
 9.84033 
 
 50 
 
 9.84046 
 
 51 
 
 9.84059 
 
 52 
 
 9.84072 
 
 53 
 
 9.84085 
 
 54 
 
 9.84098 
 
 55 
 
 9.84112 
 
 56 
 
 9.84125 
 
 57 
 
 9.84138 
 
 58 
 
 9.84151 
 
 59 
 
 i 9.84164 
 
 60 
 
 9.84177 
 
 
 L. Cos. 
 
 d. 
 
 L.Tang. 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 
 13 
 
 14 
 13 
 
 13 
 
 14 
 13 
 
 13 
 
 14 
 
 13 
 
 14 
 13 
 
 13 
 
 14 
 13 
 
 13 
 
 14 
 13 
 13 
 
 13 
 
 14 
 13 
 13 
 
 13 
 
 14 
 13 
 13 
 
 13 
 
 14 
 13 
 13 
 13 
 
 13 
 
 14 
 13 
 13 
 13 
 13 
 13 
 
 13 
 
 14 
 13 
 13 
 13 
 13 
 13 
 
 d7 
 
 9.97092 
 
 .97118 
 
 .97143 
 
 97168 
 
 .97193 
 
 96966 
 
 96991 
 
 97016 
 
 97042 
 
 .97067 
 
 .97219 
 
 .97244 
 
 97269 
 
 .97295 
 
 .97320 
 
 .97345 
 
 .97371 
 
 .97396 
 
 .97421 
 
 .97447 
 
 9.97472 
 
 9.97497 
 
 9.97523 
 
 9.97548 
 
 9.97573 
 
 d.c. I 
 
 j. cotg. ; 
 
 L. Cos. d 
 
 
 0.03034 
 
 9.86413 
 
 25 
 
 0.03009 
 
 9.86401 J; 
 
 25 
 
 0.02984 
 
 9.86389 j; 
 
 26 
 
 0.02958 
 
 9.86377 
 
 25 
 
 QC 
 
 0.02933 
 
 9.86366 
 
 ZD 
 
 0.02908 
 
 9.86354 
 
 26 
 
 0.02882 
 
 9.86342 }: 
 
 25 
 
 0.02857 
 
 9.86330 }: 
 
 25 
 
 0.02832 
 
 9.86318 J; 
 
 25 
 
 0.02807 
 
 9.86306 
 
 26 
 
 0.02781 
 
 9.86295 " 
 
 25 
 
 0.02756 
 
 9.86283 1; 
 
 25 
 
 0.02731 
 
 9.86271 } 
 
 26 
 
 0.02705 
 
 9.86259 } 
 
 25 
 
 ore 
 
 0.02680 
 
 9.86247 J 
 
 Zo 
 
 0.02655 
 
 9.86235 , 
 
 26 
 
 0.02629 
 
 9.86223 } 
 
 25 
 
 0.02604 
 
 9.86211 } 
 
 25 
 
 0.02579 
 
 9.86200 \ 
 
 26 
 - 25 
 
 0.02553 
 
 9.86188 | 
 
 0.02528 
 
 9.86176 . 
 
 25 
 
 0.02503 
 
 9.86164 { 
 
 26 
 
 0.02477 
 
 9.86152 } 
 
 25 
 
 0.02452 
 
 9.86140 } 
 
 25 
 
 ore 
 
 0.02427 
 
 9.86128 * 
 
 25 
 
 26 
 
 0.02402 
 
 9.86116 1 
 
 0.02376 
 
 9.86104 ] 
 
 ; 25 
 
 0.02351 
 
 9.86092 
 
 25 
 
 0.02326 
 
 9.86080 
 
 ; 26 
 ore 
 
 0.02300 
 
 9.86068 3 
 
 Zu 
 
 ; 25 
 
 0.02275 
 
 9.86056 ; 
 
 0.02250 
 
 9.86044 : 
 
 ; 26 
 
 0.02224 
 
 9.86032 : 
 
 25 
 
 0.02199 
 
 9.86020 : 
 
 1 t 
 
 0.02174 
 
 9.86008 : 
 
 L 25 
 1 
 
 0.02149 
 
 9.85996 : 
 
 0.02123 
 
 9.85984 
 
 l 25 
 
 0.02098 
 
 9.85972 
 
 j 25 
 
 0.02073 
 
 9.85960 
 
 5 25 
 
 0.02047 
 
 9.85948 
 
 25 
 
 0.02022 
 
 9.85936 
 
 0.01997 
 
 9.85924 
 
 
 0.01971 
 
 9.85912 
 
 4 25 
 
 0.01946 
 
 9.85900 
 
 9 95 
 
 0.01921 
 
 9.85888 
 
 4 25 
 
 o 25 
 
 0.01896 
 
 9.85876 
 
 0.01870 
 
 9.85864 
 
 5 25 
 
 0.01845 
 
 9.85851 
 
 0 25 
 
 0.01820 
 
 9.85839 
 
 16 25 
 
 0.01794 
 
 9.85827 
 
 i 1 
 
 0.01769 
 
 9.85815 
 
 0.01744 
 
 9.85803 
 
 d 25 
 
 0.01719 
 
 9.85791 
 
 )7 26 
 
 0.01693 
 
 9.85779 
 
 *2. |5 
 
 0.01668 
 
 9.85766 
 
 >7 25 
 
 S3 26 
 
 0.01643 
 
 . 9.85754 
 
 0.01617 
 
 9.85742 
 
 )8 25 
 
 0.01592 
 
 ! 9.85730 
 
 S3 25 
 
 0.01567 
 
 r 9.85718 
 
 58 26 
 
 0.01542 
 
 ! 9.85706 
 
 84 26 
 
 0.01516 
 
 I 9.85693 
 
 tg. d.c 
 
 . L.Tang 
 
 r. L. Sin. 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 d. 
 
 
 P. 
 
 P. 
 
 
 30 
 
 59 
 
 
 
 
 58 
 
 57 
 
 6 
 
 26 
 
 2.6 
 
 
 56 
 
 7 
 
 3.0 
 
 
 55 
 
 8 
 
 3.5 
 
 
 54 
 
 9 
 
 3.9 
 
 
 53 
 
 10 
 
 4.3 
 
 
 52 
 
 20 
 
 8.7 
 
 
 51 
 
 30 
 
 13.0 
 
 
 50 
 
 40 
 
 17.3 
 
 
 49 
 
 50 
 
 21.7 
 
 
 48 
 
 
 
 
 47 
 
 
 
 
 46 
 
 
 25 
 
 
 45 
 
 6 
 
 2.5 
 
 
 44 
 
 7 
 
 2.9 
 
 
 43 
 
 8 
 
 3.3 
 
 
 42 
 
 9 
 
 3.8 
 
 
 41 
 
 10 
 
 4.2 
 
 
 40 
 
 39 
 
 38 
 
 37 
 
 20 
 
 8.3 
 
 
 30 
 
 12.5 
 
 
 40 
 
 16.7 
 
 
 50 
 
 20.8 
 
 
 36 
 
 
 
 
 35 
 
 
 14 
 
 
 34 
 
 
 
 33 
 
 6 
 
 1.4 
 
 32 
 
 7 
 
 1.6 
 
 31 
 
 8 
 
 1.9 
 
 
 9 
 
 2.1 
 
 ~3cT 
 
 10 
 
 2.3 
 
 29 
 
 20 
 
 4.7 
 
 28 
 
 30 
 
 7.0 
 
 27 
 
 40 
 
 9.3 
 
 26 
 
 50 
 
 11.7 
 
 25 
 
 
 
 
 24 
 
 
 
 
 23 
 
 22 
 
 21 
 
 6 
 
 7 
 
 13 
 
 1.3 
 
 1.5 
 
 20 
 
 8 
 
 1.7 
 
 19 
 
 9 
 
 2.0 
 
 18 
 
 10 
 
 2.2 
 
 17 
 
 20 
 
 4.3 
 
 16 
 
 30 
 
 i 6.5 
 
 15 
 
 40 
 
 i 8.7 
 
 14 
 
 50 
 
 i 10.8 
 
 13 
 
 
 
 
 12 
 
 
 
 
 11 
 
 
 12 
 
 II 
 
 10 
 
 6 
 
 1.2 
 
 1.1 
 
 9 
 
 7 
 
 1.4 
 
 1.3 
 
 8 
 
 8 
 
 1.6 
 
 1.5 
 
 7 
 
 9 
 
 1.8 
 
 1.7 
 
 6 
 
 10 
 
 2.0 
 
 1.8 
 
 5 
 
 " 20 
 
 4.0 
 
 3.7 
 
 4 
 
 30 
 
 6.0 
 
 5.5 
 
 3 
 
 ; 40 
 
 8.0 
 
 7.3 
 
 2 
 
 1 
 
 1 50 
 
 10.0 
 
 9.2 
 
 0 
 
 i 
 
 
 
 t 
 
 P. P. 
 
 46 ° 
 
536 
 
 LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. 
 
 44 ° 
 
 / 
 
 L. Sin. 
 
 d. 
 
 L.Tang. 
 
 d. c. 
 
 L. Cotg. 
 
 L. Cos. 
 
 d. 
 
 
 P. P. 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 9.84177 
 
 9.84190 
 
 9.84203 
 
 9.84216 
 
 9.84229 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 13 
 
 14 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 12 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 13 
 12 
 13 
 13 
 
 13 
 
 13 
 
 13 
 
 33 
 
 12 
 
 13 
 
 13 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 12 
 
 13 
 
 13 
 
 9.98484 
 
 9.98509 
 
 9.98534 
 
 9.98560 
 
 9.98585 
 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 25 
 
 25 
 
 26 
 25 
 
 0.01516 
 
 0.01491 
 
 0.01466 
 
 0.01440 
 
 0.01415 
 
 9.85693 
 
 9.85681 
 
 9.85669 
 
 9.85657 
 
 9.85645 
 
 12 
 
 12 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 12 
 
 13 
 
 13 
 
 12 
 
 13 
 
 12 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 26 
 
 6 2.6 
 
 7 3.0 
 
 8 3.5 
 
 9 3.9 
 
 10 4.3 
 
 20 8.7 
 
 30 13.0 
 40 17.3 
 50 21.7 
 
 25 
 
 6 2.5 
 
 7 2.9 
 
 8 3.3 
 
 9 3.8 
 
 10 4.2 
 
 20 8.3 
 
 30 12.5 
 40 16.7 
 50 20.8 
 
 14 
 
 6 1.4 
 
 7 1.6 
 
 8 1.9 
 
 9 2.1 
 
 10 2.3 
 
 20 4.7 
 
 30 7.0 
 
 40 9.3 
 
 50 11.7 
 
 13 
 
 6 1.3 
 
 7 1.5 
 
 8 1.7 ! 
 
 9 2.0 
 
 10 2.2 
 20 4.3 
 
 30 6.5 
 
 40 8.7 
 
 50 10.8 
 
 12 
 
 6 1.2 
 
 7 1.4 
 
 8 1.6 
 
 9 1.8 
 
 10 2.0 | 
 20 4.0 
 
 30 6.0 
 
 40 8.0 
 
 50 10.0 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 9.84242 
 
 9.84255 
 
 9.84269 
 
 9.84282 
 
 9.84295 
 
 9.98610 
 
 9.98635 
 
 9.98661 
 
 9.98686 
 
 9.98711 
 
 0.01390 
 
 0.01365 
 
 0.01339 
 
 0.01314 
 
 0.01289 
 
 9.85632 
 
 9.85620 
 
 9.85608 
 
 9.85596 
 
 9.85583 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 9.84308 
 
 9.84321 
 
 9.84334 
 
 9.84347 
 
 9.84360 
 
 9.98737 
 
 9.98762 
 
 9.98787 
 
 9.98812 
 
 9.98838 
 
 0.01263 
 
 0.01238 
 
 0.01213 
 
 0.01188 
 
 0.01162 
 
 9.85571 
 
 9.85559 
 
 9.85547 
 
 9.85534 
 
 9.85522 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 
 9.84373 
 
 9.84385 
 
 9.84398 
 
 9.84411 
 
 9.84424 
 
 9.98863 
 
 9.98888 
 
 9.98913 
 
 9.98939 
 
 9.98964 
 
 0.01137 
 
 0.01112 
 
 0.01087 
 
 0.01061 
 
 0.01036 
 
 9.85510 
 
 9.85497 
 
 9.85485 
 
 9.85473 
 
 9.85460 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 9.84437 
 
 9.84450 
 
 9.84463 
 
 9.84476 
 
 9.84489 
 
 9.98989 
 
 9.99015 
 
 9.99040 
 
 9.99065 
 
 9.99090 
 
 0.01011 
 
 0.00985 
 
 0.00960 
 
 0.00935 
 
 0.00910 
 
 9.85448 
 
 9.85436 
 
 9.85423 
 
 9.85411 
 
 9.85399 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 25 
 
 26 
 
 27 
 
 28 
 29 
 
 9.84502 
 
 9.84515 
 
 9.84528 
 
 9.84540 
 
 9.84553 
 
 9.99116 
 
 9.99141 
 
 9.99166 
 
 9.99191 
 
 9.99217 
 
 0.00884 
 
 0.00859 
 
 0.00834 
 
 0.00809 
 
 0.00783 
 
 9.85386 
 
 9.85374 
 
 9.85361 
 
 9.85349 
 
 9.85337 
 
 35 
 
 34 
 
 33 
 
 32 
 
 31 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 9.84566 
 
 9.84579 
 
 9.84592 
 
 9.84605 
 
 9.84618 
 
 9.99242 
 
 9.99267 
 
 9.99293 
 
 9.99318 
 
 9.99343 
 
 0.00758 
 
 0.00733 
 
 0.00707 
 
 0.00682 
 
 0.00657 
 
 9.85324 
 
 9.85312 
 
 9.85299 
 
 9.85287 
 
 9.85274 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 9.84630 
 
 9.84643 
 
 9.84656 
 
 9.84669 
 
 9.84682 
 
 9.99368 
 
 9.99394 
 
 9.99419 
 
 9.99444 
 
 9.99469 
 
 0.00632 
 
 0.00606 
 
 0.00581 
 
 0.00556 
 
 0.00531 
 
 9.85262 
 
 9.85250 
 
 9.85237 
 
 9.85225 
 
 9.85212 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 9.84694 
 
 9.84707 
 
 9.84720 
 
 9.84733 
 
 9.84745 
 
 9.99495 
 
 9.99520 
 
 9.99545 
 
 9.99570 
 
 9.99596 
 
 0.00505 
 
 0.00480 
 
 0.00455 
 
 0.00430 
 
 0.00404 
 
 9.85200 
 
 9.85187 
 
 9.85175 
 
 9.85162 
 
 9.85150 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 9.84758 
 
 9.84771 
 
 9.84784 
 
 9.84796 
 
 9.84809 
 
 9.99621 
 
 9.99646 
 
 9.99672 
 
 9.99697 
 
 9.99722 
 
 0.00379 
 
 0.00354 
 
 0.00328 
 
 0.00303 
 
 0.00278 
 
 9.85137 
 
 9.85125 
 
 9.85112 
 
 9.85100 
 
 9.85087 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 9.84822 
 
 9.84835 
 
 9.84847 
 
 9.84860 
 
 9.84873 
 
 9.99747 
 
 9.99773 
 
 9.99798 
 
 9.99823 
 
 9.99848 
 
 0.00253 
 
 0.00227 
 
 0.00202 
 
 0.00177 
 
 0.00152 
 
 9.85074 
 
 9.85062 
 
 9.85049 
 
 9.85037 
 
 9.85024 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 9.84885 
 
 9.84898 
 
 9.84911 
 
 9.84923 
 
 9.84936 
 
 9.99874 
 
 9.99899 
 
 9.99924 
 
 9.99949 
 
 9.99975 
 
 0.00126 
 
 0.00101 
 
 0.00076 
 
 0.00051 
 
 0.00025 
 
 9.85012 
 
 9.84999 
 
 9.84986 
 
 9.84974 
 
 9.84961 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 60 
 
 9.84949 
 
 0.00000 
 
 0.00000 
 
 9.S4949 
 
 0 
 
 
 L. Cos. 
 
 d. 
 
 L. Cotg. 
 
 d. c. 
 
 L.Tang. 
 
 L. Sin. 
 
 d. 
 
 / 
 
 P. P. 
 
 45 ° 
 
TEA VERSE TABLES. 
 
 537 
 
 TRAVERSE TABLES. 
 
 To use the tables, find the number of degrees In the left-hand column if 
 thfA onp-lp hp Ipss than 45° and in the right-hand column if greater than 45 . 
 The numbers on the same line running across the page are the latitudes and 
 rlpnartures for that angle and for the respective distances, 1, 2, 3, 4, 5, 6, 7, 8, 9, 
 wlSch annear at the top and bottom of the pages. Thus, if the bearing of a 
 Hne be 10° and ^the distance 4, the latitude will be 3.939 and the departure 
 0 695‘ with the same bearing, and the distance 8, the latitude will be 7.878 and 
 ?i?e < departure^ 1 38? The latitude and departure for 80 is 10 times the latitude 
 and departure for 8, and is found by moving the decimal point one place to 
 the right* that for 500 is 100 times the latitude and departure ior 5, and is found 
 by two places to the right and so on , gy njo^g 
 the decimal point one, two, or more places to the right, the latitude ana 
 dpnarture mav be found for any multiple of any number given m the table. 
 Tn finding - the latitude and departure for any number such as 453, the number 
 is 1 re?olve¥tato^ “number!, viz.: 400, 50 3, and the latitude and departure 
 for each taken from the table and then added together. 
 
 and departure, neglecting 
 
 for the first figure of the given distance; write under them the latitude and depar- 
 ture for the second figure , setting them one place farther to the right; un ^J'J ie Jf 1 ) 
 rtlnce the latitude and departure for the third figure , setting them one place still 
 farther to thfrightand so continue until all the figures of tte given distance have 
 been used • add these latitudes and departures , and point off on the right of their 
 sums a number of decimal places equal to the number of 
 
 the tables being used are carried; the resulting numbers will be the latitude and 
 
 in feet, links , chains , or whatever unit of measure- 
 
 xImple^'a bearing is 16° and the distance 725 ft.; what is the latitude 
 and departure? 
 
 Distances. Latitudes. Departures. 
 
 700 6729 1 ®?? 1 
 
 20 °147S 
 
 5 4806 1378 
 
 7~25 6 9 6.9 3 6 1 9 9.7 8 8 
 
 Taking the nearest whole numbers and rejecting the decimals, we find 
 
 the V 7 hirfo a oc d cu^ r \hl given nuU“ next figure must he set fcro 
 
 P'The^elringfs* £ Tnf^d'isUnle^Tft "required, the latitude and 
 
 departure. Latitudes. Departures. 
 
 900 8345 3371 
 
 6490 
 
 2622 
 
 907 8 4 0.990 339.722 
 
 TTptp the nlace of 0 both in the distance column and in the latitude and 
 
 faffisTiM ^« h 7w^e|a|j goj Van 
 
 fhe&rr^ 
 
 and the departure of any bearing, as 30°, is the 1 ol i ts compimne , 
 
 60° Where the bearings are given m smaller fractions ot degrees tnan s 
 found in the table, the latitudes and departures can be found by inter- 
 polation. 
 
538 
 
 LATITUDES AND DEPARTURES. 
 
 Bearing. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 Bearing. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 0° 
 
 1.000 
 
 0.000 
 
 2.000 
 
 0.000 
 
 3.000 
 
 1 
 
 0.000 
 
 4.000 
 
 0.000 
 
 5.000 
 
 90° 
 
 0* 
 
 1.000 
 
 0.004 
 
 2.000 
 
 0.009 
 
 3.000 
 
 0.013 
 
 4.000 
 
 0.017 
 
 5.000 
 
 89f 
 
 0* 
 
 1.000 
 
 0.009 
 
 2.000 
 
 0.017 
 
 3.000 
 
 0.026 
 
 4.000 
 
 0.035 
 
 5.000 
 
 89* 
 
 Of 
 
 1.000 
 
 0.013 
 
 2.000 
 
 0.026 
 
 3.000 
 
 0.039 
 
 4.000 
 
 0.052 
 
 5.000 
 
 89* 
 
 1° 
 
 1.000 
 
 0.017 
 
 2.000 
 
 0.035 
 
 3.000 
 
 0.052 
 
 3.999 
 
 0.070 
 
 4.999 
 
 89° 
 
 l* 
 
 1.000 
 
 0.022 
 
 2.000 
 
 0.044 
 
 2.999 
 
 0.065 
 
 3.999 
 
 0.087 
 
 4.999 
 
 88f 
 
 l* 
 
 1.000 
 
 0.026 
 
 1.999 
 
 0.052 
 
 2.999 
 
 0.079 
 
 3.999 
 
 0.105 
 
 4.998 
 
 88* 
 
 if 
 
 1.000 
 
 0.031 
 
 1.999 
 
 0.061 
 
 2.999 
 
 0.092 
 
 3.998 
 
 0.122 
 
 4.998 
 
 88* 
 
 2° 
 
 0.999 
 
 0.035 
 
 1.999 
 
 0.070 
 
 2.998 
 
 0.105 
 
 3.998 
 
 0.140 
 
 4.997 
 
 88° 
 
 2* 
 
 0.999 
 
 0.039 
 
 1.998 
 
 0.079 
 
 2.998 
 
 0.118 
 
 3.997 
 
 0.157 
 
 4.996 
 
 87f 
 
 2* 
 
 0.999 
 
 0.044 
 
 1.998 
 
 0.087 
 
 2.997 
 
 0.131 
 
 3.996 
 
 0.174 
 
 4.995 
 
 87* 
 
 2f 
 
 0.999 
 
 0.048 
 
 1.998 
 
 0.096 
 
 2.997 
 
 0.144 
 
 3.995 
 
 0.192 
 
 4.994 
 
 87* 
 
 3° 
 
 0.999 
 
 0.052 
 
 1.997 
 
 0.105 
 
 2.996 
 
 0.157 
 
 3.995 
 
 0.209 
 
 4.993 
 
 87° 
 
 3* 
 
 0.998 
 
 0.057 
 
 1.997 
 
 0.113 
 
 2.995 
 
 0.170 
 
 3.994 
 
 0.227 
 
 4.992 
 
 86f 
 
 3* 
 
 0.998 
 
 0.061 
 
 1.996 
 
 0.122 
 
 2.994 
 
 0.183 
 
 3.993 
 
 0.244 
 
 4.991 
 
 86* 
 
 Bf 
 
 0.998 
 
 0.065 
 
 1.996 
 
 0.131 
 
 2.994 
 
 0.196 
 
 3.991 
 
 0.262 
 
 4.989 
 
 86* 
 
 4° 
 
 0.998 
 
 0.070 
 
 1.995 
 
 0.140 
 
 2.993 
 
 0.209 
 
 3.990 
 
 0.279 
 
 4.988 
 
 86° 
 
 4* 
 
 0.997 
 
 0.074 
 
 1.995 
 
 0.148 
 
 2.992 
 
 0.222 
 
 3.989 
 
 0.296 
 
 4.986 
 
 85f 
 
 4* 
 
 0.997 
 
 0.078 
 
 1.994 
 
 0.157 
 
 2.991 
 
 0.235 
 
 3.988 
 
 0.314 
 
 4.985 
 
 85* 
 
 4f 
 
 0.997 
 
 0.083 
 
 1.993 
 
 0.166 
 
 2.990 
 
 0.248 
 
 3.986 
 
 0.331 
 
 4.983 
 
 85* 
 
 5° 
 
 .0.996 
 
 0.087 
 
 1.992 
 
 0.174 
 
 2.989 
 
 0.261 
 
 3.985 
 
 0.349 
 
 4.981 
 
 85° 
 
 5* 
 
 0.996 
 
 0.092 
 
 1.992 
 
 0.183 
 
 2.987 
 
 0.275 
 
 3.983 
 
 0.366 
 
 4.979 
 
 84f 
 
 5* 
 
 0.995 
 
 0.096 
 
 1.991 
 
 0.192 
 
 2.986 
 
 0.288 
 
 3.982 
 
 0.383 
 
 4.977 
 
 84* 
 
 5f 
 
 0.995 
 
 0.100 
 
 1.990 
 
 0.200 
 
 2.985 
 
 0.301 
 
 3.980 
 
 0.401 
 
 4.975 
 
 84* 
 
 6° 
 
 0.995 
 
 0.105 
 
 1.989 
 
 0.209 
 
 2.984 
 
 0.314 
 
 3.978 
 
 0.418 
 
 4.973 
 
 84 a 
 
 6* 
 
 0.994 
 
 0.109 
 
 1.988 
 
 0.218 
 
 2.982 
 
 0.327 
 
 3.976 
 
 0.435 
 
 4.970 
 
 83f 
 
 6* 
 
 0.994 
 
 0.113 
 
 1.987 
 
 0.226 
 
 2.981 
 
 0.340 
 
 3.974 
 
 0.453 
 
 4.968 
 
 83* 
 
 6f 
 
 0.993 
 
 0.118 
 
 1.986 
 
 0.235 
 
 2.979 
 
 0.353 
 
 3.972 
 
 0.470 
 
 4.965 
 
 83* 
 
 7° 
 
 0.993 
 
 0.122 
 
 1.985 
 
 0.244 
 
 2.978 
 
 0.366 
 
 3.970 
 
 0.487 
 
 4.963 
 
 83° 
 
 7* 
 
 0.992 
 
 0.126 
 
 1.984 
 
 0.252 
 
 2.976 
 
 0.379 
 
 3.968 
 
 0.505 
 
 4.960 
 
 82f 
 
 n 
 
 0.991 
 
 0.131 
 
 1.983 
 
 0.261 
 
 2.974 
 
 0.392 
 
 3.966 
 
 0.522 
 
 4.957 
 
 82* 
 
 7f 
 
 0.991 
 
 0.135 
 
 1.982 
 
 0.270 
 
 2.973 
 
 0.405 
 
 3.963 
 
 0.539 
 
 4.954 
 
 82* 
 
 8° 
 
 0.990 
 
 0.139 
 
 1.981 
 
 0.278 
 
 2.971 
 
 0.418 
 
 3.961 
 
 0.557 
 
 4,951 
 
 82° 
 
 8* 
 
 0.990 
 
 0.143 
 
 1.979 
 
 0.287 
 
 2.969 
 
 0.430 
 
 3.959 
 
 0.574 
 
 4.948 
 
 81f 
 
 8* 
 
 0.989 
 
 0.148 
 
 1.978 
 
 0.296 
 
 2.967 
 
 0.443 
 
 3.956 
 
 0.591 
 
 4.945 
 
 81* 
 
 8f 
 
 0.988 
 
 0.152 
 
 1.977 
 
 0.304 
 
 2.965 
 
 0.456 
 
 3.953 
 
 0.608 
 
 4.942 
 
 81* 
 
 9° 
 
 0.988 
 
 0.156 
 
 1.975 
 
 0.313 
 
 2.963 
 
 0.469 
 
 3.951 
 
 0.626 
 
 4.938 
 
 81° 
 
 9* 
 
 0.987 
 
 0.161 
 
 1.974 
 
 0.321 
 
 2.961 
 
 0.482 
 
 3.948 
 
 0.643 
 
 4.935 
 
 80f 
 
 9* 
 
 0.986 
 
 0.165 
 
 1.973 
 
 0.330 
 
 2.959 
 
 0.495 
 
 3.945 
 
 0.660 
 
 4.931 
 
 80* 
 
 9f 
 
 0.986 
 
 0.169 
 
 1.971 
 
 0.339 
 
 2.957 
 
 0.508 
 
 3.942 
 
 0.677 
 
 4.928 
 
 80* 
 
 10° 
 
 0.985 
 
 0.174 
 
 1.970 
 
 0.347 
 
 2.954 
 
 0.521 
 
 3.939 
 
 0.695 
 
 4.924 
 
 80° 
 
 io* 
 
 0.984 
 
 0.178 
 
 1.968 
 
 0.356 
 
 2.952 
 
 0.534 
 
 3.936 
 
 0.712 
 
 4.920 
 
 79f 
 
 10| 
 
 0.983 
 
 0.182 
 
 1.967 
 
 0.364 
 
 2.950 
 
 0.547 
 
 3.933 
 
 0.729 
 
 4.916 
 
 79* 
 
 lOf 
 
 0.982 
 
 0.187 
 
 1.965 
 
 0.373 
 
 2.947 
 
 0.560 
 
 3.930 
 
 0.746 
 
 4.912 
 
 79* 
 
 ,,o 
 
 0.982 
 
 0.191 
 
 1.963 
 
 0.382 
 
 2.945 
 
 0.572 
 
 3.927 
 
 0.763 
 
 4.908 
 
 79° 
 
 1 H 
 
 0.981 
 
 0.195 
 
 1.962 
 
 0.390 
 
 2.942 
 
 0.585 
 
 3.923 
 
 0.780 
 
 4.904 
 
 78f 
 
 Hi 
 
 0.980 
 
 0.199 
 
 1.960 
 
 0.399 
 
 2.940 
 
 0.598 
 
 3.920 
 
 0.797 
 
 4.900 
 
 78* 
 
 -lif 
 
 0.979 
 
 0.204 
 
 1.958 
 
 0.407 
 
 2.937 
 
 0.611 
 
 3.916 
 
 0.815 
 
 4.895 
 
 78* 
 
 12° 
 
 0.978 
 
 0.208 
 
 1.956 
 
 0.416 
 
 2.934 
 
 0.624 
 
 3.913 
 
 0.832 
 
 4.891 
 
 78° 
 
 12* 
 
 0.977 
 
 0.212 
 
 1.954 
 
 0.424 
 
 2.932 
 
 0.637 
 
 3.909 
 
 0.849 
 
 4.886 
 
 77f 
 
 12* 
 
 0.976 
 
 0.216 
 
 1.953 
 
 0.433 
 
 2.929 
 
 0.649 
 
 3.905 
 
 0.866 
 
 4.881 
 
 77* 
 
 12f 
 
 0.975 
 
 0.221 
 
 1.951 
 
 0.441 
 
 2.926 
 
 0.662 
 
 3.901 
 
 0.883 
 
 4.877 
 
 77* 
 
 13° 
 
 0.974 
 
 0.225 
 
 1.949 
 
 0.450 
 
 2.923 
 
 0.675 
 
 3.897 
 
 0.900 
 
 4.872 
 
 77° 
 
 bub 
 
 c 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 bi 
 
 c 
 
 *lZ 
 
 co 
 
 69 
 
 as 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 CO 
 
 <L> 
 
 CD 
 
LATITUDES AND DEPARTURES. 
 
 539 
 
 feb 
 
 c 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 th 
 
 C 
 
 'u 
 
 1— 
 
 40 
 
 a> 
 
 CD 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 CO 
 
 0° 
 
 0.000 
 
 6.000 
 
 0.000 
 
 7.000 
 
 0.000 
 
 8.000 
 
 0.000 
 
 9.000 
 
 0.000 
 
 90° 
 
 0- 
 
 0 022 
 
 6.000 
 
 0.026 
 
 7.000 
 
 0.031 
 
 8.000 
 
 0.035 
 
 9.000 
 
 0.039 
 
 89$ 
 
 0 1 
 
 0 044 
 
 6.000 
 
 0.052 
 
 7.000 
 
 0.061 
 
 8.000 
 
 0.070 
 
 9.000 
 
 0.079 
 
 89$ 
 
 f n 
 
 0 065 
 
 5.999 
 
 0.079 
 
 6.999 
 
 0.092 
 
 7.999 
 
 0.105 
 
 8.999 
 
 0.118 
 
 89$ 
 
 1° 
 
 0 087 
 
 5.999 
 
 0.105 
 
 6.999 
 
 0.122 
 
 7.999 
 
 0.140 
 
 8.999 
 
 0.157 
 
 89° 
 
 1 1 
 
 0 109 
 
 5.999 
 
 0.131 
 
 6.998 
 
 0.153 
 
 7.998 
 
 0.175 
 
 8.998 
 
 0.196 
 
 88$ 
 
 1 1 
 
 0 131 
 
 5.998 
 
 0.157 
 
 6.998 
 
 0.183 
 
 7.997 
 
 0.209 
 
 8.997 
 
 0.236 
 
 88$ 
 
 13 
 
 0 153 
 
 5.997 
 
 0.183 
 
 6.997 
 
 0.214 
 
 7.996 
 
 0.244 
 
 8.996 
 
 0.275 
 
 88$ 
 
 2° 
 
 0 174 
 
 5.996 
 
 0.209 
 
 6.996 
 
 0.244 
 
 7.995 
 
 0.279 
 
 8.995 
 
 0.314 
 
 88° 
 
 2i 
 
 0 196 
 
 5.995 
 
 0.236 
 
 6.995 
 
 0.275 
 
 7.994 
 
 0.314 
 
 8.993 
 
 0.353 
 
 87$ 
 
 Oi 
 
 0 218 
 
 5.994 
 
 0.262 
 
 6.993 
 
 0.305 
 
 7.992 
 
 0.349 
 
 8.991 
 
 0.393 
 
 87i 
 
 91 
 
 0 240 
 
 5.993 
 
 0.288 
 
 6.992 
 
 0.336 
 
 7.991 
 
 0.384 
 
 8.990 
 
 0.432 
 
 87$ 
 
 3° 
 
 0 262 
 
 5.992 
 
 0.314 
 
 6.990 
 
 0.366 
 
 7.989 
 
 0.419 
 
 8.988 
 
 0.471 
 
 8 7° 
 
 3 1 
 
 0 283 
 
 5.990 
 
 0.340 
 
 6.989 
 
 0.397 
 
 7.987 
 
 0.454 
 
 8.986 
 
 0.510 
 
 86$ 
 
 31 
 
 0 305 
 
 5.989 
 
 0.366 
 
 6.987 
 
 0.427 
 
 7.985 
 
 0.488 
 
 8.983 
 
 0.549 
 
 86i 
 
 
 0 327 
 
 5.987 
 
 0.392 
 
 6.985 
 
 0.458 
 
 7.983 
 
 0.523 ■ 
 
 8.981 
 
 0.589 
 
 86$ 
 
 4° 
 
 0 349 
 
 5.985 
 
 0.419 
 
 6.983 
 
 0.488 
 
 7.981 
 
 0.558 
 
 8.978 
 
 0.628 
 
 86° 
 
 4 1 
 
 0 371 
 
 5.984 
 
 0.445 
 
 6.981 
 
 0.519 
 
 7.978 
 
 0.593 
 
 8.975 
 
 0.667 
 
 85$ 
 
 4 1 
 
 0 392 
 
 5.982 
 
 0.471 
 
 6.978 
 
 0.549 
 
 7.975 
 
 0.628 
 
 8.972 
 
 0.706 
 
 85$ 
 
 41 
 
 0.414 
 
 5.979 
 
 0.497 
 
 6.976 
 
 0.580 
 
 7.973 
 
 0.662 
 
 8.969 
 
 0.745 
 
 85$ 
 
 5° 
 
 0.436 
 
 5.977 
 
 0.523 
 
 6.973 
 
 0.610 
 
 7.970 
 
 0.697 
 
 8.966 
 
 0.784 
 
 85° 
 
 5 1 
 
 0.458 
 
 5.975 
 
 0.549 
 
 6.971 
 
 0.641 
 
 7.966 
 
 0.732 
 
 8.962 
 
 0.824 
 
 84$ 
 
 51 
 
 0.479 
 
 5.972 
 
 0.575 
 
 6.968 
 
 0.671 
 
 7.963 
 
 0.767 
 
 8.959 
 
 0.863 
 
 84$ 
 
 
 0.501 
 
 5.970 
 
 0.601 
 
 6.965 
 
 0.701 
 
 7.960 
 
 0.802 
 
 8.955 
 
 0.902 
 
 84$ 
 
 6° 
 
 0.523 
 
 5.967 
 
 0.627 
 
 6.962 
 
 0.732 
 
 7.956 
 
 0.836 
 
 8.951 
 
 0.941 
 
 84° 
 
 6i 
 
 0.544 
 
 5.964 
 
 0.653 
 
 6.958 
 
 0.762 
 
 7.952 
 
 0.871 
 
 8.947 
 
 0.980 
 
 83$ 
 
 
 0.566 
 
 5.961 
 
 0.679 
 
 6.955 
 
 0.792 
 
 7.949 
 
 0.906 
 
 8.942 
 
 1.019 
 
 83$ 
 
 u 9 
 
 A3. 
 
 0.588 
 
 5.958 
 
 0.705 
 
 6.951 
 
 0.823 
 
 7.945 
 
 0.940 
 
 8.938 
 
 1.058 
 
 83$ 
 
 V»4 
 
 7° 
 
 0.609 
 
 5.955 
 
 0.731 
 
 6.948 
 
 0.853 
 
 7.940 
 
 0.975 
 
 8.933 
 
 1.097 
 
 83° 
 
 7± 
 
 0.631 
 
 5.952 
 
 0.757 
 
 6.944 
 
 0.883 
 
 7.936 
 
 1.010 
 
 8.928 
 
 1.136 
 
 82$ 
 
 7 i 
 
 0.653 
 
 5.949 
 
 0.783 
 
 6.940 
 
 0.914 
 
 7.932 
 
 1.044 
 
 8.923 
 
 1.175 
 
 82$ 
 
 73- 
 
 0.674 
 
 5.945 
 
 0.809 
 
 6.936 
 
 0.944 
 
 7.927 
 
 1.079 
 
 8.918 
 
 1.214 
 
 82$ 
 
 8° 
 
 0.696 
 
 5.942 
 
 0.835 
 
 6.932 
 
 0.974 
 
 7.922 
 
 1.113 
 
 8.912 
 
 1.253 
 
 82° 
 
 84 
 
 0.717 
 
 5.938 
 
 0.861 
 
 6.928 
 
 1.004 
 
 7.917 
 
 1.148 
 
 8.907 
 
 1.291 
 
 81$ 
 
 °4 
 
 8$ 
 
 0.739 
 
 5.934 
 
 0.887 
 
 6.923 
 
 1.035 
 
 7.912 
 
 1.182 
 
 8.901 
 
 1.330 
 
 81$ 
 
 °a 
 
 8$ 
 
 0.761 
 
 5.930 
 
 0.913 
 
 6.919 
 
 1.065 
 
 7.907 
 
 1.217 
 
 8.895 
 
 1.369 
 
 81$ 
 
 °4 
 
 9° 
 
 0.782 
 
 5.926 
 
 0.939 
 
 6.914 
 
 1.095 
 
 7.902 
 
 1.251 
 
 8.889 
 
 1.408 
 
 8 1^ 
 
 9$ 
 
 0.804 
 
 5.922 
 
 0.964 
 
 6.909 
 
 1.125 
 
 7.896 
 
 1.286 
 
 8.883 
 
 1.447 
 
 80$ 
 
 gi- 
 
 0.825 
 
 5.918 
 
 0.990 
 
 6.904 
 
 1.155 
 
 7.890 
 
 1.320 
 
 8.877 
 
 1.485 
 
 80$ 
 
 “Q 
 
 9$ 
 
 0.847 
 
 5.913 
 
 1.016 
 
 6.899 
 
 1.185 
 
 7.884 
 
 1.355 
 
 8.870 
 
 1.524 
 
 • — - — ■ 
 
 80$ 
 
 10° 
 
 0.868 
 
 5.909 
 
 1.042 
 
 6.894 
 
 1.216 
 
 7.878 
 
 1.389 
 
 8.863 
 
 1.563 
 
 80° 
 
 10$ 
 
 0.890 
 
 5.904 
 
 1.068 
 
 6.888 
 
 1.246 
 
 7.872 
 
 1.424 
 
 8.856 
 
 1.601 
 
 79$ 
 
 10^ 
 
 0.911 
 
 5.900 
 
 1.093 
 
 6.883 
 
 1.276 
 
 7.866 
 
 1.458 
 
 8.849 
 
 1.640 
 
 79$ 
 
 10$ 
 
 0.933 
 
 5.895 
 
 1.119 
 
 6.877 
 
 1.306 
 
 7.860 
 
 1.492 
 
 8.842 
 
 1.679 
 
 79$ 
 
 1 1° 
 
 0.954 
 
 5 890 
 
 1.145 
 
 6.871 
 
 1.336 
 
 7.853 
 
 1.526 
 
 8.835 
 
 1.717 
 
 79° 
 
 111 
 
 0.975 
 
 5.885 
 
 1.171 
 
 6.866 
 
 1.366 
 
 7.846 
 
 1.561 
 
 8.827 
 
 1.756 
 
 78$ 
 
 Hi 
 
 0.997 
 
 5.880 
 
 1.196 
 
 6.859 
 
 1.396 
 
 7.839 
 
 1.595 
 
 8.819 
 
 1.794 
 
 78$ 
 
 111 
 
 1.018 
 
 5.874 
 
 1.222 
 
 6.853 
 
 1.425 
 
 7.832 
 
 1.629 
 
 8.811 
 
 1.833 
 
 
 12° 
 
 1.040 
 
 5.869 
 
 1.247 
 
 6.847 
 
 1.455 
 
 7.825 
 
 1.663 
 
 8.803 
 
 1.871 
 
 78° 
 
 12$ 
 
 1.061 
 
 5.863 
 
 1.273 
 
 6.841 
 
 1.485 
 
 7.818 
 
 1.697 
 
 8.795 
 
 1.910 
 
 77$ 
 
 12i 
 
 1.082 
 
 5.858 
 
 1.299 
 
 6.834 
 
 1.515 
 
 7.810 
 
 1.732 
 
 8.787 
 
 1.948 
 
 77$ 
 
 12$ 
 
 1.103 
 
 5.852 
 
 1.324 
 
 6.827 
 
 1.545 
 
 7.803 
 
 1.766 
 
 8.778 
 
 1.986 
 
 77$ 
 
 mm — ft 
 
 13° 
 
 1.125 
 
 5.846 
 
 1.350 
 
 6.821 
 
 1.575 
 
 7.795 
 
 1.800 
 
 8.769 
 
 2.025 
 
 77° 
 
 bh 
 
 c 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 1 Lat. 
 
 bb 
 
 c 
 
 _ 
 
 k_ 
 
 40 
 
 42 
 
 CD 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 a> 
 
 CD 
 
540 
 
 LATITUDES AND DEPARTURES. 
 
 tio 
 
 «= 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 bi 
 
 «D 
 
 JD 
 
 CO 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 to 
 
 0> 
 
 CO 
 
 13° 
 
 0.974 
 
 0.225 
 
 1.949 
 
 0.450 
 
 2.923 
 
 0.675 
 
 3.897 
 
 0.900 
 
 4.872 
 
 77° 
 
 131 
 
 0.973 
 
 0.229 
 
 1.947 
 
 0.458 
 
 2.920 
 
 0.688 
 
 3.894 
 
 0.917 
 
 4.867 
 
 76$ 
 
 13i 
 
 0.972 
 
 0.233 
 
 1.945 
 
 0.467 
 
 2.917 
 
 0.700 
 
 3.889 
 
 0.934 
 
 4.862 
 
 761 
 
 13$ 
 
 0.971 
 
 0.238 
 
 1.943 
 
 0.475 
 
 2.914 
 
 0.713 
 
 3.885 
 
 0.951 
 
 4.857 
 
 761 
 
 14° 
 
 0.970 
 
 0.242 
 
 1.941 
 
 0.484 
 
 2.911 
 
 0.726 
 
 3.881 
 
 0.968 
 
 4.851 
 
 76° 
 
 141 
 
 0.969 
 
 0.246 
 
 1.938 
 
 0.492 
 
 2.908 
 
 0.738 
 
 3.877 
 
 0.985 
 
 4.846 
 
 75$ 
 
 14i 
 
 0.968 
 
 0.250 
 
 1.936 
 
 0.501 
 
 2.904 
 
 0.751 
 
 3.873 
 
 1.002 
 
 4.841 
 
 751 
 
 14$ 
 
 0.967 
 
 0.255 
 
 1.934 
 
 0.509 
 
 2.901 
 
 0.764 
 
 3.868 
 
 1.018 
 
 4.835 
 
 751 
 
 15° 
 
 0.966 
 
 0.259 
 
 1.932 
 
 0.518 
 
 2.898 
 
 0.776 
 
 3.864 
 
 1.035 
 
 4.830 
 
 75° 
 
 151 
 
 0.965 
 
 0.263 
 
 1.930 
 
 0.526 
 
 2.894 
 
 0.789 
 
 3.859 
 
 1.052 
 
 4.824 
 
 74$ 
 
 151 
 
 0.964 
 
 0.267 
 
 1.927 
 
 0.534 
 
 2.891 
 
 0.802 
 
 3.855 
 
 1.069 
 
 4.818 
 
 741 
 
 15$ 
 
 0.962 
 
 0.271 
 
 1.925 
 
 0.543 
 
 2.887 
 
 0.814 
 
 3.850 ' 
 
 1.086 
 
 4.812 
 
 741 
 
 16° 
 
 0.961 
 
 0.276 
 
 1.923 
 
 0.551 
 
 2.884 
 
 0.827 
 
 3.845 
 
 1.103 
 
 4.806 
 
 74° 
 
 161 
 
 0.960 
 
 0.280 
 
 1.920 
 
 0.560 
 
 2.880 
 
 0.839 
 
 3.840 
 
 1.119 
 
 4.800 
 
 'CO 
 
 Mu 
 
 161 
 
 0.959 
 
 0.284 
 
 1.918 
 
 0.568 
 
 2.876 
 
 0.852 
 
 3.835 
 
 1.136 
 
 4.794 
 
 731 
 
 16$ 
 
 0.958 
 
 0.288 
 
 1.915 
 
 0.576 
 
 2.873 
 
 0.865 
 
 3.830 
 
 1.153 
 
 4.788 
 
 731 
 
 17° 
 
 0.956 
 
 0.292 
 
 1.913 
 
 0.585 
 
 2.869 
 
 0.877 
 
 3.825 
 
 1.169 
 
 4.782 
 
 73° 
 
 m 
 
 0.955 
 
 0.297 
 
 1.910 
 
 0.593 
 
 2.865 
 
 0.890 
 
 3.820 
 
 1.186 
 
 4.775 
 
 72$ 
 
 171 
 
 0.954 
 
 0.301 
 
 1.907 
 
 0.601 
 
 2.861 
 
 0.902 
 
 3.815 
 
 1.203 
 
 4.769 
 
 721 
 
 17$ 
 
 0.952 
 
 0.305 
 
 1.905 
 
 0.610 
 
 2.857 
 
 0.915 
 
 3.810 
 
 1.220 
 
 4.762 
 
 721 
 
 18° 
 
 0.951 
 
 0.309 
 
 1.902 
 
 0.618 
 
 2.853 
 
 0.927 
 
 3.804 
 
 1.236 
 
 4.755 
 
 72° 
 
 181 
 
 0.950 
 
 0.313 
 
 1.899 
 
 0.626 
 
 2.849 
 
 0.939 
 
 3.799 
 
 1.253 
 
 4.748 
 
 71$ 
 
 181 
 
 0.948 
 
 0.317 
 
 1.897 
 
 0.635 
 
 2.845 
 
 0.952 
 
 3.793 
 
 1.269 
 
 4.742 
 
 711 
 
 18| 
 
 0.947 
 
 0.321 
 
 1.894 
 
 0.643 
 
 2.841 
 
 0.964 
 
 3.788 
 
 1.286 
 
 4.735 
 
 711 
 
 19° 
 
 0.946 
 
 0.326 
 
 1.891 
 
 0.651 
 
 2.837 
 
 0.977 
 
 3.782 
 
 1.302 
 
 4.728 
 
 71° 
 
 191 
 
 0.944 
 
 0.330 
 
 1.888 
 
 0.659 
 
 2.832 
 
 0.989 
 
 3.776 
 
 1.319 
 
 4.720 
 
 70$ 
 
 m 
 
 0.943 
 
 0.334 
 
 1.885 
 
 0.668 
 
 2.828 
 
 1.001 
 
 3.771 
 
 1.335 
 
 4.713 
 
 701 
 
 19$ 
 
 0.941 
 
 0.338 
 
 1.882 
 
 0.676 
 
 2.824 
 
 1.014 
 
 3.765 
 
 1.352 
 
 4.706 
 
 701 
 
 20° 
 
 0.940 
 
 0.342 
 
 1.879 
 
 0.684 
 
 2.819 
 
 1.026 
 
 3.759 
 
 1.368 
 
 4.698 
 
 70° 
 
 201 
 
 0.938 
 
 0.346 
 
 1.876 
 
 0.692 
 
 2.815 
 
 1.038 
 
 3.753 
 
 1.384 
 
 4.691 
 
 69$ 
 
 20i 
 
 0.937 
 
 0.350 
 
 1.873 
 
 0.700 
 
 2.810 
 
 1.051 
 
 3.747 
 
 1.401 
 
 4.683 
 
 691 
 
 20f 
 
 0.935 
 
 0.354 
 
 1.870 
 
 0.709 
 
 2.805 
 
 1.063 
 
 3.741 
 
 1.417 
 
 4.676 
 
 691 
 
 21° 
 
 0.934 
 
 0.358 
 
 1.867 
 
 0.717 
 
 2.801 
 
 1.075 
 
 3.734 
 
 1.433 
 
 4.668 
 
 69° 
 
 2H 
 
 0.932 
 
 0.362 
 
 1.864 
 
 0.725 
 
 2.796 
 
 1.087 
 
 3.728 
 
 1.450 
 
 4.660 
 
 68$ 
 
 211 
 
 0.930 
 
 0.367 
 
 1.861 
 
 0.733 
 
 2.791 
 
 1.100 
 
 3.722 
 
 1.466 
 
 4.652 
 
 681 
 
 21$ 
 
 0.929 
 
 0.371 
 
 1.858 
 
 0.741 
 
 2.786 
 
 1.112 
 
 3.715 
 
 1.482 
 
 4.644 
 
 681 
 
 22° 
 
 0.927 
 
 0.375 
 
 1.854 
 
 0.749 
 
 2.782 
 
 1.124 
 
 3.709 
 
 1.498 
 
 4.636 
 
 68° 
 
 22i 
 
 0.926 
 
 0.379 
 
 1.851 
 
 0.757 
 
 2.777 
 
 1.136 
 
 3.702 
 
 1.515 
 
 4.628 
 
 67$ 
 
 22i 
 
 0.924 
 
 0.383 
 
 1.848 
 
 0.765 
 
 2.772 
 
 1.148 
 
 3.696 
 
 1.531 
 
 4.619 
 
 671 
 
 22$ 
 
 0.922 
 
 0.387 
 
 1.844 
 
 0.773 
 
 2.767 
 
 1.160 
 
 3.689 
 
 1.547 
 
 4.611 
 
 671 
 
 23° 
 
 0.921 
 
 0.391 
 
 1.841 
 
 0.781 
 
 2.762 
 
 1.172 
 
 3.682 
 
 1.563 
 
 4.603 
 
 67° 
 
 23i 
 
 0.919 
 
 0.395 
 
 1.838 
 
 0.789 
 
 2.756 
 
 1.184 
 
 3.675 
 
 1.579 
 
 4.594 
 
 66$ 
 
 23i 
 
 0.917 
 
 0.399 
 
 1.834 
 
 0.797 
 
 2.751 
 
 1.196 
 
 3.668 
 
 1.595 
 
 4.585 
 
 661 
 
 23$ 
 
 0.915 
 
 0.403 
 
 1.831 
 
 0.805 
 
 2.746 
 
 1.208 
 
 3.661 
 
 1.611 
 
 4.577 
 
 661 
 
 24° 
 
 0.914 
 
 0.407 
 
 1.827 
 
 0.813 
 
 2.741 
 
 1.220 
 
 3.654 
 
 1.627 
 
 4.568 
 
 66° 
 
 24i 
 
 0.912 
 
 0.411 
 
 1.824 
 
 0.821 
 
 2.735 
 
 1.232 
 
 3.647 
 
 1.643 
 
 4.559 
 
 65$ 
 
 24i 
 
 0.910 
 
 0.415 
 
 1.820 
 
 0.829 
 
 2.730 
 
 1.244 
 
 3.640 
 
 1.659 
 
 4.550 
 
 651 
 
 24$ 
 
 0.908 
 
 0.419 
 
 1.816 
 
 0.837 
 
 2.724 
 
 1.256 
 
 3.633 
 
 1.675 
 
 4.541 
 
 651 
 
 25° 
 
 0.906 
 
 0.423 
 
 1.813 
 
 0.845 
 
 2.719 
 
 1.268 
 
 3.625 
 
 1.690 
 
 4.532 
 
 65° 
 
 25i 
 
 0.904 
 
 0.427 
 
 1.809 
 
 0.853 
 
 2.713 
 
 1.280 
 
 3.618 
 
 1.706 
 
 4.522 
 
 64$ 
 
 25i 
 
 0.903 
 
 0.431 
 
 1.805 
 
 0.861 
 
 2.708 
 
 1.292 
 
 3.610 
 
 1.722 
 
 4.513 
 
 641 
 
 25$ 
 
 0.901 
 
 0.434 
 
 1.801 
 
 0.869 
 
 2.702 
 
 1.303 
 
 3.603 
 
 1.738 
 
 4.503 
 
 641 
 
 26° 
 
 0.899 
 
 0.438 
 
 1.798 
 
 0.877 
 
 2.696 
 
 1.315 
 
 3.595 
 
 1.753 
 
 4.494 
 
 64° 
 
 bfi 
 
 c 
 
 &_ 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 bi 
 
 c 
 
 u 
 
 CO 
 
 «> 
 
 CO 
 
 ' 
 
 2 
 
 3 
 
 4 
 
 5 
 
 <0 
 
 <D 
 
 CO 
 
LATITUDES AND DEPARTURES. 
 
 541 
 
 tub 
 
 c 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Bearing. 
 
 1— 
 
 to 
 
 03 
 
 00 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 13° 
 
 13* 
 
 13* 
 
 13* 
 
 14° 
 
 14* 
 
 14* 
 
 14* 
 
 1.125 
 
 1.146 
 
 1.167 
 
 1.188 
 
 1.210 
 
 1.231 
 
 1.252 
 
 1.273 
 
 5.846 
 
 5.840 
 
 5.834 
 
 5.828 
 
 5.822 
 
 5.815 
 
 5.809 
 
 5.802 
 
 1.350 
 
 1.375 
 
 1.401 
 
 1.426 
 
 1.452 
 
 1.477 
 
 1.502 
 
 1.528 
 
 6.821 
 
 6.814 
 
 6.807 
 
 6.799 
 
 6.792 
 
 6.785 
 
 6.777 
 
 6.769 
 
 1.575 
 
 1.604 
 
 1.634 
 
 1.664 
 
 1.693 
 
 1.723 
 
 1.753 
 
 1.782 
 
 7.795 
 
 7.787 
 
 7.779 
 
 7.771 
 
 7.762 
 
 7.754 
 
 7.745 
 
 7.736 
 
 1.800 
 
 1.834 
 
 1.868 
 
 1.902 
 
 1.935 
 
 1.969 
 
 2.003 
 
 2.037 
 
 8.769 
 
 8.760 
 
 8.751 
 
 8.742 
 
 8.733 
 
 8.723 
 
 8.713 
 
 8.703 
 
 2.025 
 
 2.063 
 
 2.101 
 
 2.139 
 
 2.177 
 
 2.215 
 
 2.253 
 
 2.291 
 
 77° 
 
 76* 
 
 76* 
 
 76* 
 
 76° 
 
 75* 
 
 75* 
 
 75* 
 
 15° 
 
 1§* 
 
 15* 
 
 15* 
 
 16° 
 
 16* 
 
 16* 
 
 16* 
 
 17° 
 
 17* 
 
 17* 
 
 17* 
 
 18° 
 
 18* 
 
 18* 
 
 18* 
 
 19° 
 
 19* 
 
 19* 
 
 19* 
 
 1.294 
 
 1.315 
 
 1.336 
 
 1.357 
 
 1.378 
 
 1.399 
 
 1.420 
 
 1.441 
 
 1.462 
 
 1.483 
 
 1.504 
 
 1.524 
 
 1.545 
 
 1.566 
 
 1.587 
 
 1.607 
 
 1.628 
 
 1.648 
 
 1.669 
 
 1.690 
 
 5.796 
 
 5.789 
 
 5.782 
 
 5.775 
 
 5.768 
 
 5.760 
 
 5.753 
 
 5.745 
 
 5.738 
 
 5.730 
 
 5.722 
 
 5.714 
 
 5.706 
 
 5.698 
 
 5.690 
 
 5.682 
 
 5.673 
 
 5.665 
 
 5.656 
 
 5.647 
 
 1.553 
 
 1.578 
 
 1.603 
 
 1.629 
 
 1.654 
 
 1.679 
 
 1.704 
 
 1.729 
 
 1.754 
 
 1.779 
 
 1.804 
 
 1.829 
 
 1.854 
 
 1.879 
 
 1.904 
 
 1.929 
 
 1.953 
 
 1.978 
 
 2.003 
 
 2.028 
 
 6.761 
 
 6.754 
 
 6.745 
 
 6.737 
 
 6.729 
 
 6.720 
 
 6.712 
 
 6.703 
 
 6.694 
 
 6.685 
 
 6.676 
 
 6.667 
 
 6.657 
 
 6.648 
 
 6.638 
 
 6.629 
 
 6.619 
 
 6.609 
 
 6.598 
 
 6.588 
 
 1.812 
 
 1.841 
 
 1.871 
 
 1.900 
 
 1.929 
 
 1.959 
 
 1.988 
 
 2.017 
 
 2.047 
 
 2.076 
 
 2.105 
 
 2.134 
 
 2.163 
 
 2.192 
 
 2.221 
 
 2.250 
 
 2.279 
 
 2.308 
 
 2.337 
 
 2.365 
 
 7.727 
 
 7.718 
 
 7.709 
 
 7.700 
 
 7.690 
 
 7.680 
 
 7.671 
 
 7.661 
 
 7.650 
 
 7.640 
 
 7.630 
 
 7.619 
 
 7.608 
 
 7.598 
 
 7.587 
 
 7.575 
 
 7.564 
 
 7.553 
 
 7.541 
 
 7.529 
 
 2.071 
 
 2.104 
 
 2.138 
 
 2.172 
 
 2.205 
 
 2.239 
 
 2.272 
 
 2.306 
 
 2.339 
 
 2.372 
 
 2.406 
 
 2.439 
 
 2.472 
 
 2.505 
 
 2.538 
 
 2.572 
 
 2.605 
 
 2.638 
 
 2.670 
 
 2.703 
 
 8.693 
 
 8.683 
 
 8.673 
 
 8.662 
 
 8.651 
 
 8.640 
 
 8.629 
 
 8.618 
 
 8.607 
 
 8.595 
 
 8.583 
 
 8.572 
 
 8.560 
 
 8.547 
 
 8.535 
 
 8.522 
 
 8.510 
 
 8.497 
 
 8.484 
 
 8.471 
 
 2.329 
 
 2.367 
 
 2.405 
 
 2.443 
 
 2.481 
 
 2.518 
 
 2.556 
 
 2.594 
 
 2.631 
 
 2.669 
 
 2.706 
 
 2.744 
 
 2.781 
 
 2.818 
 
 2.856 
 
 2.893 
 
 2.930 
 
 2.967 
 
 3.004 
 
 3.041 
 
 75° 
 
 74* 
 
 74* 
 
 74* 
 
 74° 
 
 73* 
 
 73* 
 
 73* 
 
 73° 
 
 72* 
 
 72* 
 
 72* 
 
 72° 
 
 71* 
 
 71* 
 
 71* 
 
 71° 
 
 70* 
 
 70* 
 
 70* 
 
 20° 
 
 1.710 
 
 5.638 
 
 2.052 
 
 6.578 
 
 2.394 
 
 7.518 
 
 2.736 
 
 8.457 
 
 3.078 
 
 7 0° 
 
 20* 
 
 1.731 
 
 5.629 
 
 2.077 
 
 6.567 
 
 2.423 
 
 7.506 
 
 2.769 
 
 8.444 
 
 3.115 
 
 69* 
 
 20* 
 
 1.751 
 
 5.620 
 
 2.101 
 
 6.557 
 
 2.451 
 
 7.493 
 
 2.802 
 
 8.430 
 
 3.152 
 
 69* 
 
 204 
 
 1.771 
 
 5.611 
 
 2.126 
 
 6.546 
 
 2.480 
 
 7.481 
 
 2.834 
 
 8.416 
 
 3.189 
 
 69* 
 
 ■“”4 
 
 21° 
 
 1.792 
 
 5.601 
 
 2.150 
 
 6.535 
 
 2.509 
 
 7.469 
 
 2.867 
 
 8.402 
 
 3.225 
 
 69° 
 
 214 
 
 1.812 
 
 5.592 
 
 2.175 
 
 6.524 
 
 2.537 
 
 7.456 
 
 2.900 
 
 8.388 
 
 3.262 
 
 68* 
 
 21 i 
 
 1.833 
 
 5.582 
 
 2.199 
 
 6.513 
 
 2.566 
 
 7.443 
 
 2.932 
 
 8.374 
 
 3.299 
 
 68* 
 
 214 
 
 1.853 
 
 5.573 
 
 2.223 
 
 6.502 
 
 2.594 
 
 7.430 
 
 2.964 
 
 8.359 
 
 3.335 
 
 68* 
 
 22° 
 
 1.873 
 
 5.563 
 
 2.248 
 
 6.490 
 
 2.622 
 
 7.417 
 
 2.997 
 
 8.345 
 
 3.371 
 
 68° 
 
 224 
 
 1.893 
 
 5.553 
 
 2.272 
 
 6.479 
 
 2.651 
 
 7.404 
 
 3.029 
 
 8.330 
 
 3.408 
 
 67* 
 
 22* 
 
 1.913 
 
 5.543 
 
 2.296 
 
 6.467 
 
 2.679 
 
 7.391 
 
 3.061 
 
 8.315 
 
 3.444 
 
 . 67* 
 
 22* 
 
 1.934 
 
 5.533 
 
 2.320 
 
 6.455 
 
 2.707 
 
 7.378 
 
 3.094 
 
 8.300 
 
 ! 3.480 
 
 67* 
 
 23°- 
 
 1.954 
 
 5.523 
 
 2.344 
 
 6.444 
 
 2. / o5 
 
 7.364 
 
 3.126 
 
 8.285 
 
 3.517 
 
 67° 
 
 234 
 
 1.974 
 
 5.513 
 
 2.368 
 
 6.432 
 
 2.763 
 
 7.350 
 
 3.158 
 
 8.269 
 
 3.553 
 
 66* 
 
 "'4 
 
 23* 
 
 1.994 
 
 5.502 
 
 2.392 
 
 6.419 
 
 2.791 
 
 7.336 
 
 3.190 
 
 8.254 
 
 3.589 
 
 66 * 
 
 23* 
 
 2.014 
 
 5.492 
 
 2.416 
 
 6.407 
 
 2.819 
 
 7.322 
 
 3.222 
 
 8.238 
 
 3.625 
 
 66* 
 
 24° 
 
 2.034 
 
 5.481 
 
 2.440 
 
 6.395 
 
 2.847 
 
 7.308 
 
 3.254 
 
 8.222 
 
 3.661 
 
 66° 
 
 24* 
 
 2.054 
 
 5.471 
 
 2.464 
 
 6.382 
 
 2.875 
 
 7.294 
 
 3.286 
 
 8.206 
 
 3.696 
 
 65* 
 
 24* 
 
 2.073 
 
 5.460 
 
 2.488 
 
 6.370 
 
 2.903 
 
 7.280 
 
 3.318 
 
 8.190 
 
 3.732 
 
 65 2 
 
 24* 
 
 2.093 
 
 5.449 
 
 2.512 
 
 6.357 
 
 2.931 
 
 7.265 
 
 3.349 
 
 8.173 
 
 3.768 
 
 65* 
 
 25° 
 
 2.113 
 
 5.438 
 
 2.536 
 
 6.344 
 
 2.958 
 
 7.250 
 
 3.381 
 
 8.157 
 
 3.804 
 
 65° 
 
 25* 
 
 2.133 
 
 5.427 
 
 2.559 
 
 6.331 
 
 2.986 
 
 7.236 
 
 3.413 
 
 8.140 
 
 3.839 
 
 64* 
 
 25* 
 
 2.153 
 
 5.416 
 
 2.583 
 
 6.318 
 
 3.014 
 
 7.221 
 
 3.444 
 
 8.123 
 
 3.8 / 5 
 
 64* 
 
 25* 
 
 2.172 
 
 5.404 
 
 2.607 
 
 6.305 
 
 3.041 
 
 7.206 
 
 3.476 
 
 8.106 
 
 3.910 
 
 64* 
 
 26° 
 
 2.192 
 
 5.393 
 
 2.630 
 
 6.292 
 
 3.069 
 
 7.190 
 
 3.507 
 
 8.089 
 
 3.945 
 
 64° 
 
 bi> 
 
 c 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 fci 
 
 c 
 
 u 
 
 k— 
 
 <0 
 
 03 
 
 CO 
 
 5 
 
 6 
 
 7 
 
 . 
 
 8 
 
 9 
 
 CO 
 
 0> 
 
 00 
 
542 
 
 LATITUDES AND DEPARTURES. 
 
 Bearing. 
 
 . 
 
 2 
 
 3 
 
 4 
 
 5 
 
 Bearing. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 26° 
 
 0.899 
 
 0.438 
 
 1.798 
 
 0.877 
 
 2.696 
 
 1.315 
 
 3.595 
 
 1.753 
 
 4.494 
 
 64° 
 
 26* 
 
 0.897 
 
 0,442 
 
 1.794 
 
 0.885 
 
 2.691 
 
 1.327 
 
 3.587 
 
 1.769 
 
 4.484 
 
 63* 
 
 26* 
 
 0.895 
 
 0.446 
 
 1.790 
 
 0.892 
 
 2.685 
 
 1.339 
 
 3.580 
 
 1.785 
 
 4.475 
 
 63* 
 
 26* 
 
 0.893 
 
 0.450 
 
 1.786 
 
 0.900 
 
 2.679 
 
 1.350 
 
 3.572 
 
 1.800 
 
 4.465 
 
 63* 
 
 27° 
 
 0.891 
 
 0.454 
 
 1.782 
 
 0.908 
 
 2.673 
 
 1.362 
 
 3.564 
 
 1.816 
 
 4.455 
 
 63° 
 
 27* 
 
 0.889 
 
 0.458 
 
 1.778 
 
 0.916 
 
 2.667 
 
 1.374 
 
 3.556 
 
 1.831 
 
 4.445 
 
 62* 
 
 27* 
 
 0.887 
 
 0.462 
 
 1.774 
 
 0.923 
 
 2.661 
 
 1.385 
 
 3.548 
 
 1.847 
 
 4.435 
 
 62* 
 
 27* 
 
 0.885 
 
 0.466 
 
 1.770 
 
 0.931 
 
 2.655 
 
 1.397 
 
 3.540 
 
 1.862 
 
 4.425 
 
 62* 
 
 28° 
 
 0.883 
 
 0.469 
 
 1.766 
 
 0.939 
 
 2.649 
 
 1.408 
 
 3.532 
 
 1.878 
 
 4.415 
 
 62° 
 
 28* 
 
 0.881 
 
 0.473 
 
 1.762 
 
 0.947 
 
 2.643 
 
 1.420 
 
 3.524 
 
 1.893 
 
 4.404 
 
 61* 
 
 28 g- 
 
 0.879 
 
 0.477 
 
 1.758 
 
 0.954 
 
 2.636 
 
 1.431 
 
 3.515 
 
 1.909 
 
 4.394 
 
 61* 
 
 28* 
 
 0,877 
 
 0.481 
 
 1.753 
 
 0.962 
 
 2.630 
 
 1.443 
 
 3.507 
 
 1.924 
 
 4.384 
 
 61* 
 
 29° 
 
 0.875 
 
 0.485 
 
 1.749 
 
 0.970 
 
 2.624 
 
 1.454 
 
 3.498 
 
 1.939 
 
 4.373 
 
 61° 
 
 29* 
 
 0.872 
 
 0.489 
 
 1.745 
 
 0.977 
 
 2.617 
 
 1.466 
 
 3.490 
 
 1.954 
 
 4.362 
 
 60* 
 
 29* 
 
 0.870 
 
 0.492 
 
 1.741 
 
 0.985 
 
 2.611 
 
 1.477 
 
 3.481 
 
 1.970 
 
 4.352 
 
 60* 
 
 29* 
 
 0.868 
 
 0.496 
 
 1.736 
 
 0.992 
 
 2.605 
 
 1.489 
 
 3.473 
 
 1.985 
 
 4.341 
 
 60* 
 
 30° 
 
 0.866 
 
 0.500 
 
 1.732 
 
 1.000 
 
 2.598 
 
 1.500 
 
 3.464 
 
 2.000 
 
 4.330 
 
 60° 
 
 30* 
 
 0.864 
 
 0.504 
 
 1.728 
 
 1.008 
 
 2.592 
 
 1.511 
 
 3.455 
 
 2.015 
 
 4.319 
 
 59* 
 
 30* 
 
 0.862 
 
 0.508 
 
 1.723 
 
 1.015 
 
 2.585 
 
 1.523 
 
 3.447 
 
 2.030 
 
 4.308 
 
 59* 
 
 30* 
 
 0.859 
 
 0.511 
 
 1.719 
 
 1.023 
 
 2.578 
 
 1.534 
 
 3.438 
 
 2.045 
 
 4.297 
 
 59* 
 
 31° 
 
 6.857 
 
 0.515 
 
 1.714 
 
 1.030 
 
 2.572 
 
 1.545 
 
 3.429 
 
 2.060 
 
 4.286 
 
 59° 
 
 31* 
 
 0.855 
 
 0.519 
 
 1.710 
 
 1.038 
 
 2.565 
 
 1.556 
 
 3.420 
 
 2.075 
 
 4.275 
 
 58* 
 
 31* 
 
 0.853 
 
 0.522 
 
 1.705 
 
 1.045 
 
 2.558 
 
 1.567 
 
 3.411 
 
 2.090 
 
 4.263 
 
 58* 
 
 31* 
 
 0.850 
 
 0.526 
 
 1.701 
 
 1.052 
 
 2.551 
 
 1.579 
 
 3.401 
 
 2.105 
 
 4.252 
 
 58* 
 
 32° 
 
 0.848 
 
 0.530 
 
 1.696 
 
 1.060 
 
 2.544 
 
 1.590 
 
 3.392 
 
 2.120 
 
 4.240 
 
 58° 
 
 32* 
 
 0.846 
 
 0.534 
 
 1.691 
 
 1.067 
 
 2.537 
 
 1.601 
 
 3.383 
 
 2.134 
 
 4.229 
 
 57* 
 
 32* 
 
 0.843 
 
 0.537 
 
 1.687 
 
 1.075 
 
 2.530 
 
 1.612 
 
 3.374 
 
 2.149 
 
 4.217 
 
 57* 
 
 32* 
 
 0.841 
 
 0.541 
 
 1.682 
 
 1.082 
 
 2.523 
 
 1.623 
 
 3.364 
 
 2.164 
 
 4.205 
 
 57* 
 
 33° 
 
 0.839 
 
 0.545 
 
 1.677 
 
 1.089 
 
 2.516 
 
 1.634 
 
 3.355 
 
 2.179 
 
 4.193 
 
 57° 
 
 33* 
 
 0.836 
 
 0.548 
 
 1.673 
 
 1.097 
 
 2.509 
 
 1.645 
 
 3.345 
 
 2.193 
 
 4.181 
 
 56* 
 
 33* 
 
 0.834 
 
 0.552 
 
 1.668 
 
 1.104 
 
 2.502 
 
 1.656 
 
 3.336 
 
 2.208 
 
 4.169 
 
 56* 
 
 33* 
 
 0.831 
 
 0.556 
 
 1.663 
 
 1.111 
 
 2.494 
 
 1.667 
 
 3.326 
 
 2.222 
 
 4.157 
 
 56* 
 
 34° 
 
 0.829 
 
 0.559 
 
 1.658 
 
 1.118 
 
 2.487 
 
 1.678 
 
 3.316 
 
 2.237 
 
 4.145 
 
 56° 
 
 34* 
 
 0.827 
 
 0.563 
 
 1.653 
 
 1.126 
 
 2.480 
 
 1.688 
 
 3.306 
 
 2.251 
 
 4.133 
 
 55* 
 
 34* 
 
 0.824 
 
 0.566 
 
 1.648 
 
 1.133 
 
 2.472 
 
 1.699 
 
 3.297 
 
 2.266 
 
 4.121 
 
 55* 
 
 34* 
 
 0.822 
 
 0.570 
 
 1.643 
 
 1.140 
 
 2.465 
 
 1.710 
 
 3.287 
 
 2.280 
 
 4.108 
 
 55* 
 
 35° 
 
 0.819 
 
 0.574 
 
 1.638 
 
 1.147 
 
 2.457 
 
 1.721 
 
 3.277 
 
 2.294 
 
 4.096 
 
 55° 
 
 35* 
 
 0.817 
 
 0.577 
 
 1.633 
 
 1.154 
 
 2.450 
 
 1.731 
 
 3.267 
 
 2.309 
 
 4.083 
 
 54* 
 
 35* 
 
 0.814 
 
 0.581 
 
 1.628 
 
 1.161 
 
 2.442 
 
 1.742 
 
 3.257 
 
 2.323 
 
 4.071 
 
 54* 
 
 35* 
 
 0.812 
 
 0.584 
 
 1.623 
 
 1.168 
 
 2.435 
 
 1.753 
 
 3.246 
 
 2.337 
 
 4.058 
 
 54* 
 
 36° 
 
 0.809 
 
 0.588 
 
 1.618 
 
 1.176 
 
 2,427 
 
 1.763 
 
 3.236 
 
 2.351 
 
 4.045 
 
 54° 
 
 36* 
 
 0.806 
 
 0.591 
 
 1.613 
 
 1.183 
 
 2.419 
 
 1.774 
 
 3.226 
 
 2.365 
 
 4.032 
 
 53* 
 
 36* 
 
 0.804 
 
 0.595 
 
 1.608 
 
 1.190 
 
 2.412 
 
 1.784 
 
 3.215 
 
 2.379 
 
 4.019 
 
 53* 
 
 36* 
 
 0.801 
 
 0.598 
 
 1.603 
 
 1.197 
 
 2.404 
 
 1.795 
 
 3.205 
 
 2.393 
 
 4.006 
 
 53* 
 
 37° 
 
 0.799 
 
 0.602 
 
 1.597 
 
 1.204 
 
 2.396 
 
 1.805 
 
 3.195 
 
 2.407 
 
 3.993 
 
 53° 
 
 37* 
 
 0.796 
 
 0.605 
 
 1.592 
 
 1.211 
 
 2,388 
 
 1.816 
 
 3.184 
 
 2.421 
 
 3.980 
 
 52* 
 
 37* 
 
 0.793 
 
 0.609 
 
 1.587 
 
 1.218 
 
 2.380 
 
 1.826 
 
 3.173 
 
 2.435 
 
 3.967 
 
 52* 
 
 37* 
 
 0.791 
 
 0.612 
 
 1.581 
 
 1.224 
 
 2.372 
 
 1.837 
 
 3.163 
 
 2.449 
 
 3.953 
 
 52* 
 
 38° 
 
 0.788 
 
 0.616 
 
 1.576 
 
 1.231 
 
 2.364 
 
 1.847 
 
 3.152 
 
 2.463 
 
 3.940 
 
 52° 
 
 38* 
 
 0.785 
 
 0.619 
 
 1.571 
 
 1.238 
 
 2.356 
 
 1.857 
 
 3.141 
 
 2.476 
 
 3.927 
 
 51* 
 
 38* 
 
 0.783 
 
 0.623 
 
 1.565 
 
 1.245 
 
 2.348 
 
 1.868 
 
 3.130 
 
 2.490 
 
 3.913 
 
 51* 
 
 38* 
 
 0.780 
 
 0.626 
 
 1.560 
 
 1.252 
 
 2.340 
 
 1.878 
 
 3.120 
 
 2.504 
 
 3.899 
 
 51* 
 
 39° 
 
 0.777 
 
 0.629 
 
 1.554 
 
 1.259 
 
 2.331 
 
 1.888 
 
 3.109 
 
 2.517 
 
 3.886 
 
 51° 
 
 eh 
 
 c 
 
 i_ 
 
 Dep. 
 
 Ld;t« 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 tab 
 
 c 
 
 CO 
 
 <o 
 
 CD 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 CO 
 
 <U 
 
 CQ 
 
LATITUDES AND DEPARTURES. 
 
 fa b 
 c 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Bearing. 
 
 hm 
 
 CO 
 
 £ 
 
 Dep. 
 
 Ls/t • 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 26° 
 
 26* 
 
 26* 
 
 26* 
 
 27° 
 
 27* 
 
 27* 
 
 27* 
 
 28° 
 
 28* 
 
 28* 
 
 28* 
 
 M° 
 
 29* 
 
 29* 
 
 29* 
 
 2.192 
 
 2.211 
 
 2.231 
 
 2.250 
 
 2.270 
 
 2.289 
 
 2.309 
 
 2.328 
 
 2.347 
 
 2.367 
 
 2.386 
 
 2.405 
 
 2.424 
 
 2.443 
 
 2.462 
 
 2.481 
 
 5.393 
 
 5.381 
 
 5.370 
 
 5.358 
 
 5.346 
 
 5.334 
 
 5.322 
 
 5.310 
 
 5.298 
 
 5.285 
 
 5.273 
 
 5.260 
 
 5.248 
 
 5.235 
 
 5.222 
 
 5.209 
 
 2.630 
 
 2.654 
 
 2.677 
 
 2.701 
 
 2.724 
 
 2.747 
 
 2.770 
 
 2.794 
 
 2.817 
 
 2.840 
 
 2.863 
 
 2.886 
 
 2.909 
 
 2.932 
 
 2.955 
 
 2.977 
 
 6.292 
 
 6.278 
 
 6.265 
 
 6.251 
 
 6.237 
 
 6.223 
 
 6.209 
 
 6.195 
 
 6.181 
 
 6.166 
 
 6.152 
 
 6.137 
 
 6.122 
 
 6.107 
 
 6.093 
 
 6.077 
 
 3.069 
 
 3.096 
 
 3.123 
 
 3.151 
 
 3.178 
 
 3.205 
 
 3.232 
 
 3.259 
 
 3.286 
 
 3.313 
 
 3.340 
 
 3.367 
 
 3.394 
 
 3.420 
 
 3.447 
 
 3.474 
 
 7.190 
 
 7.175 
 
 7.160 
 
 7.144 
 
 7.128 
 
 7.112 
 
 7.096 
 
 7.080 
 
 7.064 
 
 7.047 
 
 7.031 
 
 7.014 
 
 6.997 
 
 6.980 
 
 6.963 
 
 6.946 
 
 3.507 
 
 3.538 
 
 3.570 
 
 3.601 
 
 3.632 
 
 3.663 
 
 3.694 
 
 3.725 
 
 3.756 
 
 3.787 
 
 3.817 
 
 3.848 
 
 3.878 
 
 3.909 
 
 3.939 
 
 3.970 
 
 8.089 
 
 8.072 
 
 8.054 
 
 8.037 
 
 8.019 
 
 8.001 
 
 7.983 
 
 7.965 
 
 7.947 
 
 7.928 
 
 7.909 
 
 7.891 
 
 7.872 
 
 7.852 
 
 7.833 
 
 7.814 
 
 3.945 
 
 3.981 
 
 4.016 
 
 4.051 
 
 4.086 
 
 4.121 
 
 4.156 
 
 4.190 
 
 4.225 
 
 4.260 
 
 4.294 
 
 4.329 
 
 4.363 
 
 4.398 
 
 4.432 
 
 4.466 
 
 64° 
 
 63* 
 
 63* 
 
 63* 
 
 63° 
 
 62* 
 
 62* 
 
 62* 
 
 62° 
 
 61* 
 
 61* 
 
 61° 
 
 30° 
 
 30* 
 
 30* 
 
 30* 
 
 31° 
 
 31* 
 
 31* 
 
 31* 
 
 32° 
 
 | 
 
 33* 
 
 34° 
 
 3 
 
 34* 
 
 2.500 
 
 2.519 
 
 2.538 
 
 2.556 
 
 2.575 
 
 2.594 
 
 2.612 
 
 2.631 
 
 2.650 
 
 2.668 
 
 2.686 
 
 2.705 
 
 2.723 
 
 2.741 
 
 2.760 
 
 2.778 
 
 2.796 
 
 2.814 
 
 2.832 
 
 2.850 
 
 5.196 
 
 5.183 
 
 5.170 
 
 5.156 
 
 5.143 
 
 5.129 
 
 5.116 
 
 5.102 
 
 5.088 
 
 5.074 
 
 5.060 
 
 5.046 
 
 5.032 
 
 5.018 
 
 5.003 
 
 4.989 
 
 4.974 
 
 4.960 
 
 4.945 
 
 4.930 
 
 3.000 
 3.023 
 3.045 
 3.068 
 3.090 
 3.113 
 3.135 
 3.157 
 3.180 
 3.202 
 • 3.224 
 3.246 
 3.268 
 3.290 
 3.312 
 3.333 
 3.355 
 3.377 
 3.398 
 3.420 
 
 6.062 
 
 6.047 
 
 6.031 
 
 6.016 
 
 6.000 
 
 5.984 
 
 5.968 
 
 5.952 
 
 5.936 
 
 5.920 
 
 5.904 
 
 5.887 
 
 5.871 
 
 5.854 
 
 5.837 
 
 5.820 
 
 5.803 
 
 5.786 
 
 5.769 
 
 5.752 
 
 3.500 
 
 3.526 
 
 3.553 
 
 3.579 
 
 3.605 
 
 3.631 
 
 3.657 
 
 3.683 
 
 3.709 
 
 3.735 
 
 3.761 
 
 3.787 
 
 3.812 
 
 3.838 
 
 3.864 
 
 3.889 
 
 3.914 
 
 3.940 
 
 3.965 
 
 3.990 
 
 6.928 
 
 6.911 
 
 6.893 
 
 6.875 
 
 6.857 
 
 6.839 
 
 6.821 
 
 6.803 
 
 6.784 
 
 6.766 
 
 6.747 
 
 6.728 
 
 6.709 
 
 6.690 
 
 6.671 
 
 6.652 
 
 6.632 
 
 6.613 
 
 6.593 
 
 6.573 
 
 4.000 
 
 4.030 
 
 4.060 
 
 '4.090 
 
 4.120 
 
 4.150 
 
 4.180 
 
 4.210 
 
 4.239 
 
 4.269 
 
 4.298 
 
 4.328 
 
 4.357 
 
 4.386 
 
 4.416 
 
 4.445 
 
 4.474 
 
 4.502 
 
 4.531 
 
 4.560 
 
 7.794 
 
 7.775 
 
 7.755 
 
 7.735 
 
 7.715 
 
 7.694 
 
 7.674 
 
 7.653 
 
 7.632 
 
 7.612 
 
 7.591 
 
 7.569 
 
 7.548 
 
 7.527 
 
 7.505 
 
 7.483 
 
 7.461 
 
 7.439 
 
 7.417 
 
 7.395 
 
 4.500 
 
 4.534 
 
 4.568 
 
 4.602 
 
 4.635 
 
 4.669 
 
 4.702 
 
 4.736 
 
 4.769 
 
 4.802 
 
 4.836 
 
 4.869 
 
 4.902 
 
 4.935 
 
 4.967 
 
 5.000 
 
 5.033 
 
 5.065 
 
 5.098 
 
 5.130 
 
 60° 
 
 59* 
 
 59* 
 
 59* 
 
 59° 
 
 58* 
 
 If 
 
 Ooj 
 
 58° 
 
 57* 
 
 57* 
 
 57* 
 
 57° 
 
 56* 
 
 56* 
 
 56* 
 
 56° 
 
 55* 
 
 55* 
 
 55*__ 
 
 35° 
 
 38® 
 
 37° 
 
 37* 
 
 37* 
 
 37* 
 
 38° 
 
 it 
 
 38* 
 
 39° 
 
 2.868 
 
 2.886 
 
 2.904 
 
 2.921 
 
 2.939 
 
 2.957 
 
 2.974 
 
 2.992 
 
 3.009 
 
 3.026 
 
 3.044 
 
 3.061 
 
 3.078 
 
 3.095 
 
 3.113 
 
 3.130 
 
 3.147 
 
 4.915 
 
 4.900 
 
 4.885 
 
 4.869 
 
 4.854 
 
 4.839 
 
 4.823 
 
 4.808 
 
 4.792 
 
 4.776 
 
 4.760 
 
 4.744 
 
 4.728 
 
 4.712 
 
 4.696 
 
 4.679 
 
 4.663 
 
 3.441 
 
 3.463 
 
 3.484 
 
 3.505 
 
 3.527 
 
 3.548 
 
 3.569 
 
 3.590 
 
 3.611 
 
 3.632 
 
 3.653 
 
 3.673 
 
 3.694 
 
 3.715 
 
 3.735 
 
 3.756 
 
 3.776 
 
 5.734 
 
 5.716 
 
 5.699 
 
 5.681 
 
 5.663 
 
 5.645 
 
 5.627 
 
 5.609 
 
 5.590 
 
 5.572 
 
 5.554 
 
 5.535 
 
 5.516 
 
 5.497 
 
 5.478 
 
 5.459 
 
 5.440 
 
 4.015 
 
 4.040 
 
 4.065 
 
 4.090 
 
 4.115 
 
 4.139 
 
 4.164 
 
 4.188 
 
 4.213 
 
 4.237 
 
 4.261 
 
 4.286 
 
 4.310 
 
 4.334 
 
 4.358 
 
 4.381 
 
 4.405 
 
 6.553 
 
 6.533 
 
 6.513 
 
 6.493 
 
 6.472 
 
 6.452 
 
 6.431 
 
 6.410 
 
 6.389 
 
 6.368 
 
 6.347 
 
 6.326 
 
 6.304 
 
 6.283 
 
 6.261 
 
 6.239 
 
 6.217 
 
 4.589 
 
 4.617 
 
 4.646 
 
 4.674 
 
 4.702 
 
 4.730 
 
 4.759 
 
 4.787 
 
 4.815 
 
 4.842 
 
 4.870 
 
 4.898 
 
 4.925 
 
 4.953 
 
 4.980 
 
 5.007 
 
 5.035 
 
 7.372 
 
 7.350 
 
 7.327 
 
 7.304 
 
 7.281 
 
 7.258 
 
 7.235 
 
 7.211 
 
 7.188 
 
 7.164 
 
 7.140 
 
 7.116 
 
 7.092 
 
 7.068 
 
 7.043 
 
 7.019 
 
 6.994 
 
 5.162 
 
 5.194 
 
 5.226 
 
 5.258 
 
 5.290 
 
 5.322 
 
 5.353 
 
 5.385 
 
 5.416 
 
 5.448 
 
 5.479 
 
 5.510 
 
 5.541' 
 
 5.572 
 
 5.603 
 
 5.633 
 
 5.664 
 
 55° 
 
 54* 
 
 54* 
 
 54* 
 
 54° 
 
 53* 
 
 53* 
 
 53* 
 
 53° 
 
 52* 
 
 52* 
 
 52* 
 
 52° 
 
 51* 
 
 51* 
 
 51* 
 
 51° 
 
 fab 
 
 (0 
 
 <u 
 
 00 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 fab 
 
 c 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 <D 
 
 aa 
 
544 
 
 LATITUDES AND DEPARTURES. 
 
 da 
 
 c 
 
 \ 
 
 2 
 
 3 
 
 4 
 
 5 
 
 tut 
 
 c 
 
 “ 
 
 co 
 
 w> 
 
 CO 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 « 
 
 00 
 
 39° 
 
 0.777 
 
 0.629 
 
 1.554 
 
 1.259 
 
 2.331 
 
 1.888 
 
 3.109 
 
 2.517 
 
 3.886 
 
 51° 
 
 39£ 
 
 0.774 
 
 0.633 
 
 1.549 
 
 1.265 
 
 2.323 
 
 1.898 
 
 3.098 
 
 2.531 
 
 3.872 
 
 50} 
 
 m 
 
 0.772 
 
 0.636 
 
 1.543 
 
 1.272 
 
 2.315 
 
 1.908 
 
 3.086 
 
 2.544 
 
 3.858 
 
 50! 
 
 39} 
 
 0.769 
 
 0.639 
 
 1.538 
 
 1.279 
 
 2.307 
 
 1.918 
 
 3.075 
 
 2.558 
 
 3.844 
 
 50£ 
 
 40° 
 
 0.766 
 
 0.643 
 
 1.532 
 
 1.286 
 
 2.298 
 
 1.928 
 
 3.064 
 
 2.571 
 
 3.830 
 
 50° 
 
 40i 
 
 0.763 
 
 0.646 
 
 1.526 
 
 1.292 
 
 2.290 
 
 1.938 
 
 3.053 
 
 2.584 
 
 3.816 
 
 49} 
 
 40k 
 
 0.760 
 
 0.649 
 
 1.521 
 
 1.299 
 
 2.281 
 
 1.948 
 
 3.042 
 
 2.598 
 
 3.802 
 
 49k 
 
 40 } 
 
 0.758 
 
 0.653 
 
 1.515 
 
 1.306 
 
 2.273 
 
 1.958 
 
 3.030 
 
 2.611 
 
 3.788 
 
 49£ 
 
 41° 
 
 0.755 
 
 0.656 
 
 1.509 
 
 1.312 
 
 2.264 
 
 1.968 
 
 3.019 
 
 2.624 
 
 3.774 
 
 49° 
 
 41 £ 
 
 0.752 
 
 0.659 
 
 1.504 
 
 1.319 
 
 2.256 
 
 1.978 
 
 3.007 
 
 2.637 
 
 3.759 
 
 43} 
 
 4lJ 
 
 0.749 
 
 0.663 
 
 1.498 
 
 1.325 
 
 2.247 
 
 1.988 
 
 2.996 
 
 2.650 
 
 3.745 
 
 48! 
 
 41£ 
 
 0.746 
 
 0.666 
 
 1.492 
 
 1.332 
 
 2.238 
 
 1.998 
 
 2.984 
 
 2.664 
 
 3.730 
 
 48£ 
 
 42° 
 
 0.743 
 
 0.669 
 
 1.486 
 
 1*338 
 
 2.229 
 
 2.007 
 
 2.973 
 
 2.677 
 
 3.716 
 
 48° 
 
 42 £ 
 
 0.740 
 
 0.672 
 
 1.480 
 
 1.345 
 
 2.221 
 
 2.017 
 
 2.961 
 
 2.689 
 
 3.701 
 
 47} 
 
 42! 
 
 0.737 
 
 0.676 
 
 1.475 
 
 1.351 
 
 2.212 
 
 2.027 
 
 2.949 
 
 2.702 
 
 3.686 
 
 47k 
 
 42f 
 
 0.734 
 
 0.679 
 
 1.469 
 
 1.358 
 
 2.203 
 
 2.036 
 
 2.937 
 
 2.715 
 
 3.672 
 
 47£ 
 
 43° 
 
 0.731 
 
 0.682 
 
 1.463 
 
 1.364 
 
 2.194 
 
 2.046 
 
 2.925 
 
 2.728 
 
 3.657 
 
 47° 
 
 43£ 
 
 0.728 
 
 0.685 
 
 1.457 
 
 1.370 
 
 2.185 
 
 2.056 
 
 2.913 
 
 2.741 
 
 3.642 
 
 46£ 
 
 43! 
 
 0.725 
 
 0.688 
 
 1.451 
 
 1.377 
 
 2.176 
 
 2.065 
 
 2.901 
 
 2.753 
 
 3.627 
 
 46k 
 
 43} 
 
 0.722 
 
 0.692 
 
 1.445 
 
 1.383 
 
 2.167 
 
 2.075 
 
 2.889 
 
 2.766 
 
 3.612 
 
 46£ 
 
 44° 
 
 0.719 
 
 0.695 
 
 1.439 
 
 1.389 
 
 2.158 
 
 2.084 
 
 2.877 
 
 2.779 
 
 3.597 
 
 46° 
 
 44£ 
 
 0.716 
 
 0.698 
 
 1.433 
 
 1.396 
 
 2.149 
 
 2.093 
 
 2.865 
 
 2.791 
 
 3 582 
 
 45} 
 
 44! 
 
 0.713 
 
 0.701 
 
 1.427 
 
 1.402 
 
 2.140 
 
 2.103 
 
 2.853 
 
 2.804 
 
 3.566 
 
 45! 
 
 44£ 
 
 0.710 
 
 0.704 
 
 1.420 
 
 1.408 
 
 2.131 
 
 2.112 
 
 2.841 
 
 2.816 
 
 3.551 
 
 45£ 
 
 45° 
 
 0.707 
 
 0.707 
 
 1.414 
 
 1.414 
 
 2.121 
 
 2.121 
 
 2.828 
 
 2.828 
 
 3.536 
 
 45° 
 
 Bear- 
 
 ing. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Bear- 
 
 ing. 
 
 do 
 
 C 
 
 Z 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 bi 
 
 c 
 
 u 
 
 CO 
 
 03 
 
 00 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 CD 
 
 SO 
 
 39° 
 
 3.147 
 
 4.663 
 
 3.776 
 
 5.440 
 
 4.405 
 
 6.217 
 
 5.035 
 
 6.994 
 
 5.664 
 
 “Ti 5 " 
 
 39£ 
 
 3.164 
 
 4.646 
 
 3.796 
 
 5.421 
 
 4.429 
 
 6.195 
 
 5.062 
 
 6.970 
 
 5.694 
 
 50£ 
 
 39k 
 
 3.180 
 
 4.630 
 
 3.816 
 
 5.401 
 
 4.453 
 
 6.173 
 
 5.089 
 
 6.945 
 
 5.725 
 
 50| 
 
 39} 
 
 3.197 
 
 4.613 
 
 3.837 
 
 5.382 
 
 4.476 
 
 6.151 
 
 5.116 
 
 6.920 
 
 5.755 
 
 50£ 
 
 40° 
 
 3.214 
 
 4.596 
 
 3.857 
 
 5.362 
 
 4.500 
 
 6.128 
 
 5.142 
 
 6.894 
 
 5,785 
 
 50° 
 
 40£ 
 
 3.231 
 
 4.579 
 
 3.877 
 
 5.343 
 
 4.523 
 
 6.106 
 
 5.169 
 
 6.869 
 
 5.’815 
 
 49} 
 
 40| 
 
 3.247 
 
 4.562 
 
 3.897 
 
 5.323 
 
 4.546 
 
 6.083 
 
 5.196 
 
 6.844 
 
 5.845 
 
 49k 
 
 40£ 
 
 3.264 
 
 4.545 
 
 3.917 
 
 5.303 
 
 4.569 
 
 6.061 
 
 5.222 
 
 6.818 
 
 5.875 
 
 49£ 
 
 41° 
 
 3.280' 
 
 4.528 
 
 3.936 
 
 5.283 
 
 4.592 
 
 6.038 
 
 5.248 
 
 6.792 
 
 5.905 
 
 49° 
 
 41£ 
 
 3.297 
 
 4.511 
 
 3.956 
 
 5.263 
 
 4.615 
 
 6.015 
 
 5.275 
 
 6.767 
 
 5.934 
 
 43} 
 
 41! 
 
 3.313 
 
 4.494 
 
 3.976 
 
 5.243 
 
 4.638 
 
 5.992 
 
 5.301 
 
 6.741 
 
 5.964 
 
 43k 
 
 41£ 
 
 3.329 
 
 4.476 
 
 3.995 
 
 5.222 
 
 4.661 
 
 5.968 
 
 5.327 
 
 6.715 
 
 5.993 
 
 48£ 
 
 42° 
 
 3.346 
 
 4.459 
 
 4.015 
 
 5.202 
 
 4.684 
 
 5.945 
 
 5.353 
 
 6.688 
 
 6.022 
 
 48° 
 
 42£ 
 
 3.362 
 
 4.441 
 
 4.034 
 
 5.182 
 
 4.707 • 
 
 5.922 
 
 5.379 
 
 6.662 
 
 6.051 
 
 47} 
 
 42! 
 
 3.378 
 
 4.424 
 
 4.054 
 
 5.161 
 
 4.729 
 
 5.898 
 
 5.405 
 
 6.635 
 
 6.080 
 
 47 k 
 
 42f 
 
 3.394 
 
 4.406 
 
 4.073 
 
 5.140 
 
 4.752 
 
 5.875 
 
 5.430 
 
 6.609 
 
 6.109 
 
 47£ 
 
 43° 
 
 3.410 
 
 4.388 
 
 4.092 
 
 5.119 
 
 4.774 
 
 5.851 
 
 5.456 
 
 6.582 
 
 6.138 
 
 47° 
 
 43£ 
 
 3.426 
 
 4.370 
 
 4.111 
 
 5.099 
 
 4.796 
 
 5.827 
 
 5.481 
 
 6.555 
 
 6.167 
 
 46} 
 
 43! 
 
 3.442 
 
 4.352 
 
 4.130 
 
 5.078 
 
 4.818 
 
 5.803 
 
 5.507 
 
 6.528 
 
 6.195 
 
 46! 
 
 43£ 
 
 3.458 
 
 4.334 
 
 4.149 
 
 5.057 
 
 4.841 
 
 5.779 
 
 5.532 
 
 6.501 
 
 6.224 
 
 46£ 
 
 44° 
 
 3.473 
 
 4.316 
 
 4.168 
 
 5.035 
 
 4.863 
 
 5.755 
 
 5.557 
 
 6.474 
 
 6.252 
 
 46° 
 
 44£ 
 
 3.489 
 
 4.298 
 
 4.187 
 
 5.014 
 
 4.885 
 
 5.730 
 
 5.582 
 
 6.447 
 
 6.280 
 
 45} 
 
 44! 
 
 3.505 
 
 4.280 
 
 4.206 
 
 4.993 
 
 4.906 
 
 5.706 
 
 5.607 
 
 6.419 
 
 6.308 
 
 45! 
 
 44£ 
 
 3.520 
 
 4.261 
 
 4.224 
 
 4.971 
 
 4.928 
 
 5.681 
 
 5.632 
 
 6.392 
 
 6.336 
 
 45£ 
 
 45° 
 
 3.536 
 
 4.243 
 
 4.243 
 
 4.950 
 
 4.950 
 
 5.657 
 
 5.657 
 
 6.364 
 
 6.364 
 
 45° 
 
 Bear- 
 
 ing. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Bear- 
 
 ing. 
 
CIRCUMFERENCES, AND AREAS. 
 
 545 
 
 SQUARES, CUBES, SQUARE AND CUBE ROOTS, 
 CIRCUMFERENCES, AND AREAS. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 3q. Root. 1 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 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 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 54 
 
 55 
 
 1 
 
 4 
 
 9 
 
 16 
 
 25 
 
 36 
 
 49 
 
 64 
 
 81 
 
 100 
 
 121 
 
 144 
 
 169 
 
 196 
 
 225 
 
 256 
 
 289 
 
 324 
 
 361 
 
 400 
 
 441 
 
 484 
 
 529 
 
 576 
 
 625 
 
 676 
 
 729 
 
 784 
 
 841 
 
 900 
 
 961 
 
 1.024 
 1,089 
 1,156 
 1,225 
 1,296 
 1,369 
 1,444 
 1,521 
 1,600 
 1,681 
 1,764 
 1,849 
 1,936 
 
 2.025 
 2,116 
 2,209 
 2,304 
 2,401 
 2,500 
 2,601 
 2,704 
 2,809 
 2,916 
 
 3.025 
 
 1 
 
 8 
 
 27 
 
 64 
 
 125 
 
 216 
 
 343 
 
 512 
 
 729 
 
 1,000 
 
 1,331 
 
 1,728 
 
 2,197 
 
 2,744 
 
 3.375 
 4,096 
 4,913 
 5,832 
 6,859 
 8,000 
 9,261 
 
 10,648 
 
 12,167 
 
 13,824 
 
 15,625 
 
 17,576 
 
 19,683 
 
 21,952 
 
 24,389 
 
 27.000 
 29,791 
 32,768 
 35,937 
 39,304 
 42,875 
 46,656 
 50,653 
 54.872 
 59,319 
 
 64.000 
 68,921 
 74,088 
 79,507 
 85,184 
 91,125 
 97,336 
 
 103,823 
 
 110,592 
 
 117,649 
 
 125.000 
 132,651 
 140,608 
 148,877 
 157,464 
 
 166.375 
 
 1.0000 
 
 1.4142 
 
 1.7321 
 
 2.0000 
 
 2.2361 
 
 2.4495 
 
 2.6458 
 
 2.8284 
 
 3.0000 
 
 3.1623 
 
 3.3166 
 
 3.4641 
 
 3.6056 
 
 3.7417 
 
 3.8730 
 
 4.0000 
 4.1231 
 4.2426 
 4.3589 
 4.4721 
 4.5826 
 4.6904 
 4.7958 
 4.8990 
 
 5.0000 
 5.0990 
 5.1962 
 5.2915 
 5.3852 
 5.4772 
 5.5678 
 5.6569 
 5.7446 
 5.8310 
 5.9161 
 6.0000 
 6.0828 
 6.1644 
 6.2450 
 6.3246 
 6.4031 
 6.4807 
 6.5574 
 6.6332 
 6.7082 
 6.7823 
 6.8557 
 6.9282 
 7.0000 
 7.0711 
 7.1414 
 7.2111 
 7.2801 
 7.3485 
 7.4162 
 
 1.0000 ] 
 1.2599 
 1.4422 
 1.5874 
 1.7100 
 1.8171 
 1.9129 
 2.0000 
 2.0801 
 2.1544 
 2.2240 
 2.2894 
 2.3513 
 2.4101 
 2.4662 
 2.5198 
 2.5713 
 2.6207 
 2.6684 
 2.7144 
 2.7589 
 2.8020 
 ' 2.8439 
 2.8845 
 2.9240 
 2.9625 
 3.0000 
 3.0366 
 3.0723 
 3.1072 
 3.1414 
 3.1748 
 3.2075 
 3.2396 
 3.2717 
 3.3019 
 3.3322 
 3.3620 
 3.3912 
 3.4200 
 3.4482 
 3.4760 
 3.5034 
 3.5303 
 3.5569 
 3.5830 
 3.6088 
 3.6342 
 3.6593 
 3.6840 
 3.7084 
 3.7325 
 3.7563 
 3.7798 
 3.8030 
 
 L.000000000 
 
 .500000000 
 
 .333333333 
 
 .250000000 
 
 .200000000 
 
 .166666667 
 
 .142857143 
 
 .125000000 
 
 .111111111 
 
 .100000000 
 
 .090909091 
 
 .083333333 
 
 .076923077 
 
 .071428571 
 
 .066666667 
 
 .062500000 
 
 .058823529 
 
 .055555556 
 
 .052631579 
 
 .050000000 
 
 .047619048 
 
 .045454545 
 
 .043478261 
 
 .041666667 
 
 .040000000 
 
 .038461538 
 
 .037037037 
 
 .035714286 
 
 .034482759 
 
 .033333333 
 
 .032258065 
 
 .031250000 
 
 .030303030 
 
 .029411765 
 
 .028571429 
 
 .027777778 
 
 .027027027 
 
 .026315789 
 
 .025641026 
 
 .025000000 
 
 .024390244 
 
 .023809524 
 
 .023255814 
 
 .022727273 
 
 .022222222 
 
 .021739130 
 
 .021276600 
 
 .020833333 
 
 .020408163 
 
 .020000000 
 
 .019607843 
 
 .019230769 
 
 .018867925 
 
 .018518519 
 
 .018181818 
 
 3.1416 
 
 6.2832 
 
 9.4248 
 
 12.5664 
 
 15.7080 
 
 18.850 
 
 21.991 
 
 25.133 
 
 28.274 
 
 31.416 
 
 34.558 
 
 37-699 
 
 40.841 
 
 43.982 
 
 47.124 
 
 50.265 
 
 53.407 
 
 56.549 
 
 59.690 
 
 62.832 
 
 65.973 
 
 69.115 
 
 72.257 
 
 75.398 
 
 78.540 
 
 81.681 
 
 84.823 
 
 87.965 
 
 91.106 
 
 94.248 
 
 97.389 
 
 100.53 
 
 103.67 
 
 106.81 
 
 109.96 
 
 113.10 
 
 116.24 
 
 119.38 
 
 122.52 
 
 125.66 
 
 128.81 
 
 131.95 
 
 135.09 
 
 138.23 
 
 141.37 
 
 144.51 
 
 147.65 
 150.80 
 153.94 
 157.08 
 160.22 
 163.36 
 166.50 
 
 169.65 
 172.79 
 
 0.7854 
 
 3.1416 
 
 7.0686 
 
 12.5664 
 
 19.635 
 
 28.274 
 
 38.485 
 
 50.266 
 
 63.617 
 
 78.540 
 
 95.033 
 
 113.10 
 
 132.73 
 
 153.94 
 176.71 
 201.06 
 226.98 
 
 254.47 
 
 283.53 
 314.16 
 346.36 
 380.13 
 
 415.48 
 452.39 
 490.87 
 530.93 
 572.56 
 615.75 
 660.52 
 706.86 
 754.77 
 804.25 
 855.30 
 907.92 
 962.11 
 
 1,017.88 
 
 1.075.21 
 1,134.11 
 1,194.59 
 1,256.64 
 1,320.25 
 1,385.44 
 1,452.20 
 
 1.520.53 
 1,590.43 
 1,661.90 
 
 1.734.94 
 1,809.56 
 1,885.74 
 1,963.50 1 
 
 2.042.82 
 2,123.72 
 2,206.18 
 
 2.290.22 ! 
 
 2.375.83 
 
546 
 
 SQUARES, CUBES, SQUARE AND CUBE ROOTS , 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 56 
 
 3,136 
 
 175,616 
 
 7.4833 
 
 3.8259 
 
 .017857143 
 
 175.93 
 
 2,463.01 
 
 57 
 
 3,249 
 
 185,193 
 
 7.5498 
 
 3.8485 
 
 .017543860 
 
 179.07 
 
 2,551.76 
 
 58 
 
 3,364 
 
 195,112 
 
 7.6158 
 
 3.8709 
 
 .017241379 
 
 182.21 
 
 2,642.08 
 
 59 
 
 3,481 
 
 205,379 
 
 7.6811 
 
 3.8930 
 
 .016949153 
 
 185.35 
 
 2,733.97 
 
 60 
 
 3,600 
 
 216,000 
 
 7.7460 
 
 3.9149 
 
 .016666667 
 
 188.50 
 
 2,827.43 
 
 61 
 
 3,721 
 
 226,981 
 
 7.8102 
 
 3.9365 
 
 .016393443 
 
 191.64 
 
 2,922.47 
 
 62 
 
 3,844 
 
 238,328 
 
 7.8740 
 
 3.9579 
 
 .016129032 
 
 194.78 
 
 3,019.07 
 
 63 
 
 3,969 
 
 250,047 
 
 7.9373 
 
 3.9791 
 
 .015873016 
 
 197.92 
 
 3,117.25 
 
 64 
 
 4,096 
 
 262,144 
 
 8.0000 
 
 4.0000 
 
 .015625000 
 
 201.06 
 
 3,216.99 
 
 65 
 
 4,225 
 
 274,625 
 
 8.0623 
 
 4.0207 
 
 .015384615 
 
 204.20 
 
 3,318.31 
 
 66 
 
 4,356 
 
 287,496 
 
 8.1240 
 
 4.0412 
 
 .015151515 
 
 207.34 
 
 3,421.19 
 
 67 
 
 4,489 
 
 300,763 
 
 8.1854 
 
 4.0615 
 
 .014925373 
 
 210.49 
 
 3,525.65 
 
 68 
 
 4,624 
 
 314,432 
 
 8.2462 
 
 4.0817 
 
 .014705882 
 
 213.63 
 
 3,631.68 
 
 69 
 
 4^761 
 
 328,509 
 
 8.3066 
 
 4.1016 
 
 .014492754 
 
 216.77 
 
 3,739.28 
 
 70 
 
 4,900 
 
 343,000 
 
 8.3666 
 
 4.1213 
 
 .014285714 
 
 219.91 
 
 3,848.45 
 
 71 
 
 5,041 
 
 357,911 
 
 8.4261 
 
 4.1408 
 
 .014084517 
 
 223.05 
 
 3,959.19 
 
 72 
 
 5,184 
 
 373,248 
 
 8.4853 
 
 4.1602 
 
 .013888889 
 
 226.19 
 
 4,071.50 
 
 73 
 
 5,329 
 
 389,017 
 
 8.5440 
 
 4.1793 
 
 .013698630 
 
 229.34 
 
 4,185.39 
 
 74 
 
 5,476 
 
 405,224 
 
 8.6023 
 
 4.1983 
 
 .013513514 
 
 232.48 
 
 4,300.84 
 
 75 
 
 5,625 
 
 421,875 
 
 8.6603 
 
 4.2172 
 
 .013333333 
 
 235.62 
 
 4,417.86 
 
 76 
 
 5,776 
 
 438,976 
 
 8.7178 
 
 4.2358 
 
 .013157895 
 
 238.76 
 
 4,536.46 
 
 77 
 
 5*929 
 
 456,533 
 
 8.7750 
 
 4.2543 
 
 .012987013 
 
 241.90 
 
 4,656.63 
 
 78 
 
 6,084 
 
 474,552 
 
 8.8318 
 
 4.2727 
 
 .012820513 
 
 245.04 
 
 4,778.36 
 
 79 
 
 6,241 
 
 493,039 
 
 8.8882 
 
 4.2908 
 
 .012658228 
 
 248.19 
 
 4,901.67 
 
 80 
 
 6,400 
 
 512,000 
 
 8.9443 
 
 4.3089 
 
 .012500000 
 
 251.33 
 
 5,026.55 
 
 81 
 
 6,561 
 
 531,441 
 
 9.0000 
 
 4.3267 
 
 .012345679 
 
 254.47 
 
 5,153.00 
 
 82 
 
 6,724 
 
 551,368 
 
 9.0554 
 
 4.3445 
 
 .012195122 
 
 257.61 
 
 5,281.02 
 
 83 
 
 6 i 889 
 
 571,787 
 
 9.1104 
 
 4.3621 
 
 .012048193 
 
 260.75 
 
 5,410.61 
 
 84 
 
 7,056 
 
 592,704 
 
 9.1652 
 
 4.3795 
 
 .011904762 
 
 263.89 
 
 5,541.77 
 
 85 
 
 7,225 
 
 614,125 
 
 9.2195 
 
 4.3968 
 
 .011764706 
 
 267.04 
 
 5,674.50 
 
 86 
 
 7,396 
 
 636,056 
 
 9.2736 
 
 4.4140 
 
 .011627907 
 
 270.18 
 
 5,808.80 
 
 87 
 
 7,569 
 
 658,503 
 
 9.3274 
 
 4.4310 
 
 .011494253 
 
 273.32 
 
 5,944.68 
 
 88 
 
 7,744 
 
 681,472 
 
 9.3808 
 
 4.4480 
 
 .011363636 
 
 276.46 
 
 6,082.12 
 
 89 
 
 7.921 
 
 704,969 
 
 9.4340 
 
 4.4647 
 
 .011235955 
 
 279.60 
 
 6,221.14 
 
 90 
 
 8,100 
 
 729,000 
 
 9.4868 
 
 4.4814 
 
 .011111111 
 
 282.74 
 
 6,361.73 
 
 91 
 
 8,281 
 
 753,571 
 
 9.5394 
 
 4.4979 
 
 .010989011 
 
 285.88 
 
 6,503.88 
 
 92 
 
 8,464 
 
 778,688 
 
 9.5917 
 
 4.5144 
 
 .010869565 
 
 289.03 
 
 6,647.61 
 
 93 
 
 8,649 
 
 804,357 
 
 9.6437 
 
 4.5307 
 
 .010752688 
 
 292.17 
 
 6,792.91 
 
 94 
 
 8,836 
 
 830,584 
 
 9.6954 
 
 4.5468 
 
 .010638298 
 
 295.31 
 
 6,939.78 
 
 95 
 
 9,025 
 
 857,375 
 
 9; 7468 
 
 4.5629 
 
 .010526316 
 
 298.45 
 
 7,088.22 
 
 96 
 
 9,216 
 
 884,736 
 
 9.7980 
 
 4.5789 
 
 .410416667 
 
 301.59 
 
 7,238.23 
 
 97 
 
 9,409 
 
 912,673 
 
 9.8489 
 
 4.5947 
 
 .010309278 
 
 304.73 
 
 7,389.81 
 
 98 
 
 9,604 
 
 941,192 
 
 9.8995 
 
 4.6104 
 
 .010204082 
 
 307.88 
 
 7,542.96 
 
 99 
 
 9,801 
 
 970,299 
 
 9.9499 
 
 4.6261 
 
 .010101010 
 
 311.02 
 
 7,697.69 
 
 100 
 
 10,000 
 
 1,000,000 
 
 10.0000 
 
 4.6416 
 
 .010000000 
 
 314.16 
 
 7,853.98 
 
 101 
 
 10,201 
 
 1,030,301 
 
 10.0499 
 
 4.6570 
 
 .009900990 
 
 317.30 
 
 8,011.85 
 
 102 
 
 10,404 
 
 1,061,208 
 
 10.0995 
 
 4.6723 
 
 .009803922 
 
 320.44 
 
 8,171.28 
 
 103 
 
 10,609 
 
 1,092,727 
 
 10.1489 
 
 4.6875 
 
 .009708738 
 
 323.58 
 
 8,332.29 
 
 104 
 
 10,816 
 
 1,124,864 
 
 10.1980 
 
 4.7027 
 
 .009615385 
 
 326.73 
 
 8,494.87 
 
 105 
 
 11,025 
 
 1,157,625 
 
 10.2470 
 
 4.7177 
 
 .009523810 
 
 329.87 
 
 8,659.01 
 
 106 
 
 11,236 
 
 1,191,016 
 
 10.2956 
 
 4.7326 
 
 .009433962 
 
 333.01 
 
 8,824.73 
 
 107 
 
 11,449 
 
 1,225,043 
 
 10.3441 
 
 4.7475 
 
 .009345794 
 
 336.15 
 
 8.992.02 
 
 108 
 
 11,664 
 
 1,259,712 
 
 10.3923 
 
 4.7622 
 
 .009259259 
 
 339.29 
 
 9,160.88 
 
 109 
 
 11,881 
 
 1,295,029 
 
 10.4403 
 
 4.7769 
 
 .009174312 
 
 342.43 
 
 9,331.32 
 
 110 
 
 12,100 
 
 1,331,000 
 
 10.4881 
 
 4.7914 
 
 .009090909 
 
 345.58 
 
 9,503.32 
 
 111 
 
 12,321 
 
 1,367,631 
 
 10.5357 
 
 4.8059 
 
 .009009009 
 
 348.72 
 
 9,676.89 
 
 112 
 
 12,544 
 
 1,404,928 
 
 10.5830 
 
 4.8203 
 
 .008928571 
 
 351.86 
 
 9,852.03 
 
 113 
 
 12,769 
 
 1,442,897 
 
 10.6301 
 
 4.8346 
 
 .008849558 
 
 355.00 
 
 10,028.75 
 
 114 
 
 12,996 
 
 1,481,544 
 
 10.6771 
 
 4.8488 
 
 .008771930 
 
 358.14 
 
 10,207.03 
 
 115 
 
 13,225 
 
 1,520,875 
 
 10.7238 
 
 4.8629 
 
 .008695652 
 
 361.28 
 
 10,386.89 
 
 116 
 
 13,456 
 
 1,560,896 
 
 10.7703 
 
 4.8770 
 
 .008020690 
 
 364.42 
 
 10,568.32 
 
 117 
 
 13,689 
 
 1,601,613 
 
 10.8167 
 
 4.8910 
 
 .008547009 
 
 367.57 
 
 10,751.32 
 
 118 
 
 13,924 
 
 1,643,032 
 
 10.8628 
 
 4.9049 
 
 .008474576 
 
 370.71 
 
 10,935.88 
 
CIRCUMFERENCES, AND AREAS. 
 
 547 
 
 I 1 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 119 
 
 120 
 121 
 122 
 
 123 
 
 124 
 
 125 
 
 126 
 
 127 
 
 128 
 
 129 
 
 130 
 
 131 
 
 132 
 
 133 
 
 134 
 
 135 
 
 136 
 
 137 
 
 138 
 
 139 
 
 140 
 
 141 
 
 142 
 
 143 
 
 144 
 
 145 
 
 146 
 
 147 
 
 148 
 
 149 
 
 150 
 
 151 
 
 152 
 
 153 
 
 154 
 
 155 
 
 156 
 
 157 
 
 158 
 
 159 
 
 160 
 161 
 162 
 
 163 
 
 164 
 
 165 
 
 166 
 
 167 
 
 168 
 
 169 
 
 170 
 
 171 
 
 172 
 
 173 
 
 174 
 
 175 
 
 176 
 
 177 
 
 178 
 
 179 
 
 180 
 181 
 
 14.161 
 14,400 
 
 14.641 
 14,834 
 
 15.129 
 15,376 
 15,625 
 15,876 
 
 16.129 
 16,384 
 
 16.641 
 16,900 
 
 17.161 
 17,424 
 17,689 
 17,956 
 18,225 
 18,496 
 18,769 
 19,044 
 19,321 
 
 19.600 
 19,881 
 20,164 
 20,449 
 20,736 
 
 21.025 
 21,316 
 21,609 
 21,904 
 22,201 
 22,500 
 22,801 
 23,104 
 23,409 
 23,716 
 
 24.025 
 24,336 
 24,649 
 24,964 
 25,281 
 
 25.600 
 25,921 
 26,244 
 26,569 
 26,896 
 27,225 
 27,556 
 27,889 
 28,224 
 28,561 
 28,900 
 29,241 
 28,584 
 29,929 
 30,276 
 30,625 
 30,976 
 31,329 
 31,684 
 32,041 
 32,400 
 32,761 
 
 1,685,159 
 
 1.728.000 
 1,771,561 
 1,815,848 
 1,860,867 
 
 1.906.624 
 1,953,125 
 2,000,376 
 2,048,383 
 2,097,152 
 2,146,689 
 
 2.197.000 
 2,248,091 
 2,299,968 
 2,352,637 
 2,406,104 
 2,460,375 
 2,515,456 
 2,571,353 
 2,628,072 
 2,685,619 
 
 2.744.000 
 2,803,221 
 2,863,288 
 2,924,207 
 2,985,984 
 
 3.048.625 
 3,112,136 
 3,176,523 
 3,241,792 
 3,307,949 
 
 3.375.000 
 3,442,951 
 3,511,008 
 3,581,577 
 3,652,264 
 3,723,875 
 3,796,416 
 3,869,893 
 3,944,312 
 4,019,679 
 
 4.096.000 
 4,173,281 
 4,251,528 
 4,330,747 
 4,410,944 
 4,492,125 
 4,574,296 
 4,657,463 
 4,741,632 
 4,826,809 
 
 4.913.000 
 5,000,211 
 5,088,448 
 5,177,717 
 5,268,024 
 5,359,375 
 5,451,776 
 5,545,233 
 5,639,752 
 5,735,339 
 
 5.832.000 
 5,929,741 
 
 10.9087 
 
 10.9545 
 
 11.0000 
 
 11.0454 
 
 11.0905 
 
 11.1355 
 
 11.1803 
 
 11.2250 
 
 11.2694 
 
 11.3137 
 
 11.3578 
 
 11.4018 
 
 11.4455 
 
 11.4891 
 
 11.5326 
 
 11.5758 
 
 11.6190 
 
 11.6619 
 
 11.7047 
 
 11.7473 
 
 11.7898 
 
 11.8322 
 
 11.8743 
 
 11.9164 
 
 11.9583 
 
 12.0000 
 
 12.0416 
 
 12.0830 
 
 12.1244 
 
 12.1655 
 
 12.2066 
 
 12.2474 
 
 12.2882 
 
 12.3288 
 
 12.3693 
 
 12.4097 
 
 12.4499 
 
 12.4900 
 
 12.5300 
 
 12.5698 
 
 12.6095 
 
 12.6491 
 
 12.6886 
 
 12.7279 
 
 12.7671 
 
 12.8062 
 
 12.8452 
 
 12.8841 
 
 12.9228 
 
 12.9615 
 
 13.0000 
 
 13.9384 
 
 13.0767 
 
 13.1149 
 
 13.1529 
 
 13.1909 
 
 13.2288 
 
 13.2665 
 
 13.3041 
 
 13.3417 
 
 13.37 &L 
 
 13.4164 
 
 13.4536 
 
 4.9187 
 
 4.9324 
 
 4.9461 
 
 4.9597 
 
 4.9732 
 
 4.9866 
 
 5.0000 
 
 5.0133 
 
 5.0265 
 
 5.0397 
 
 5.0528 
 
 5.0658 
 
 5.0788 
 
 5.0916 
 
 5.1045 
 
 5.1172 
 
 5.1299 
 
 5.1426 
 
 5.1551 
 
 51676 
 
 5.1801 
 
 5.1925 
 
 5.2048 
 
 5.2171 
 
 5.2293 
 
 5.2415 
 
 5.2536 
 
 5.2656 
 
 5.2776 
 
 5.2896 
 
 5.3015 
 
 5.3133 
 
 5.3251 
 
 5.3368 
 
 5.3485 
 
 5.3601 
 
 5.3717 
 
 5.3832 
 
 5.3947 
 
 5.4061 
 
 5.4175 
 
 5.4288 
 
 5.4401 
 
 5.4514 
 
 5.4626 
 
 5.4737 
 
 5.4848 
 
 5.4959 
 
 5.5069 
 
 5.5178 
 
 5.5288 
 
 5.5397 
 
 5.5505 
 
 5.5613 
 
 5.5721 
 
 5.5828 
 
 5.5934 
 
 5.6041 
 
 5.6147 
 
 5.6252 
 
 5.6357 
 
 5.6462 
 
 5.6567 
 
 .008403361 
 
 .008333333 
 
 .008264463 
 
 .008196721 
 
 .008130081 
 
 .008064516 
 
 .008000000 
 
 .007936508 
 
 .007874016 
 
 .007812500 
 
 .007751938 
 
 .007692308 
 
 .007633588 
 
 .007575758 
 
 .007518797 
 
 .007462687 
 
 .007407407 
 
 .007352941 
 
 .007299270 
 
 .007246377 
 
 .007194245 
 
 .007142857 
 
 .007092199 
 
 .007042254 
 
 .006993007 
 
 .006944444 
 
 .006896552 
 
 .006849315 
 
 .006802721 
 
 .006756757 
 
 .006711409 
 
 .006666667 
 
 .006622517 
 
 .006578947 
 
 .006535948 
 
 .006493506 
 
 .006451613 
 
 .006410256 
 
 .006369427 
 
 .006329114 
 
 .006289308 
 
 .006250000 
 
 .006211180 
 
 .006172840 
 
 .006134969 
 
 .006097561 
 
 .006060606 
 
 .006024096 
 
 .005988024 
 
 .005952381 
 
 .005917160 
 
 .005882353 
 
 .005847953 
 
 .005813953 
 
 .005780347 
 
 .005747126 
 
 .005714286 
 
 .005681818 
 
 .005649718 
 
 .005617978 
 
 .005586592 
 
 .005555556 
 
 .005524862 
 
 373.85 
 
 376.99 
 
 380.13 
 
 383.27 
 386.42 
 389.56 
 392.70 
 395.84 
 398.98 
 
 402.12 
 
 405.27 
 408.41 
 411.55 
 414.69 
 417.83 
 420.97 
 
 424.12 
 427.26 
 430.40 
 433.54 
 436.68 
 439.82 
 
 442.96 
 446.11 
 449.25 
 452.39 
 455.53 
 458.67 
 
 461.81 
 
 464.96 
 468.10 
 471.24 
 474.38 
 477.52 
 480.66 
 
 483.81 
 486.95 
 490.09 
 493.23 
 496.37 
 499.51 
 
 502.65 
 505.80 
 508.94 
 512.08 
 515.22 
 518.36 
 
 521.50 
 
 524.65 
 527.79 
 530.93 
 534.07 
 537.21 
 
 540.35 
 
 543.50 
 546.64 
 549.78 
 552.92 
 556.06 
 559.20 
 
 562.35 
 565.49 
 568.63 
 
 11,122.02 
 
 11,309.73 
 
 11.499.01 
 
 11.689.87 
 11,882.29 
 12,076.28 
 
 12.271.85 
 12,468.98 
 12,667.69 
 12,867.96 
 13,069.81 
 13,273.23 
 13,478.22 
 13,684.78 
 13,892.91 
 
 14.102.61 
 
 14.313.88 
 14,526.72 
 14,741.14 
 14,957.12 
 15,174.68 
 15,393.80 
 
 15.614.50 
 15,836.77 
 
 16.060.61 
 
 16.286.02 
 
 16.513.00 
 
 16.741.55 
 
 16.971.67 
 17,203.36 
 17,436.62 
 17,671.46 
 
 17.907.86 
 18,145.84 
 18,385.39 
 
 18.626.50 
 
 18.869.19 
 
 19.113.45 
 19,359.28 
 
 19.606.68 
 19,855.65 
 
 20.106.19 
 20,358.31 
 20,611.99 
 20,867.24 
 
 21.124.07 
 
 21.382.46 
 
 21.642.43 
 21,903.97 
 
 22.167.08 
 22,431.76 
 
 22.698.01 
 22,965.83 
 23,235.22 
 23,506.18 
 23,778.71 
 24,052.82 
 24,328.49 
 24,605.74 
 
 24.884.56 
 25,164.94 
 25,446.90 
 
 25.730.43 
 
548 
 
 SQUARES, CUBES, SQUARE AND CUBE ROOTS, 
 
 No . 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 182 
 
 183 
 
 184 
 
 185 
 
 186 
 
 187 
 
 188 
 
 189 
 
 190 
 
 191 
 
 192 
 
 193 
 
 194 
 
 195 
 
 196 
 
 197 
 
 198 
 
 199 
 
 200 
 201 
 202 
 
 203 
 
 204 
 
 205 
 
 206 
 
 207 
 
 208 
 
 209 
 
 210 
 211 
 212 
 
 213 
 
 214 
 
 215 
 
 216 
 
 217 
 
 218 
 
 219 
 
 220 
 221 
 222 
 
 223 
 
 224 
 
 225 
 
 226 
 
 227 
 
 228 
 
 229 
 
 230 
 
 231 
 
 232 
 
 233 
 
 234 
 
 235 
 
 236 
 
 237 
 
 238 
 
 239 
 
 240 
 
 241 
 
 242 
 
 243 
 
 244 
 
 33,124 
 
 33,489 
 
 33,856 
 
 34.225 
 34,596 
 34,969 
 35,344 
 35,721 
 
 36.100 
 36,481 
 36,864 
 37,249 
 37,636 
 
 38.025 
 38,416 
 38,809 
 39,204 
 39,601 
 40,000 
 40,401 
 40,804 
 41,209 
 41,616 
 
 42.025 
 42,436 
 42,849 
 43,264 
 43,681 
 
 44.100 
 44,521 
 44,944 
 45,369 
 45,796 
 
 46.225 
 46,656 
 47,089 
 47,524 
 47,961 
 48,400 
 48,841 
 49,284 
 49,729 
 50,176 
 50,625 
 51,076 
 51,529 
 51,984 
 52,441 
 52,900 
 53,361 
 53,824 
 54.289 
 54,756 
 
 55.225 
 55,696 
 56,169 
 56,644 
 57,121 
 57,600 
 58,081 
 58,564 
 59,049 
 59,536 
 
 6,028,568 
 
 6,128,487 
 
 6,229,504 
 
 6.331.625 
 6,434,856 
 6,539,203 
 6,644,672 
 6,751,269 
 
 6.859.000 
 6,967,871 
 7,077,888 
 7,189,017 
 7,301,384 
 
 7.414.875 
 7,529,536 
 7,645,373 
 7,762,392 
 7,880,599 
 8,000,000 
 8,120,601 
 8,242,408 
 8,365,427 
 8,489,664 
 8,615,125 
 8,741,836 
 8,869,743 
 8,998,912 
 9,129,329 
 
 9.261.000 
 9,393,931 
 9,528.128 
 9,663,597 
 9,800,344 
 9,938,375 
 
 10,077,696 
 
 10,218,313 
 
 10,360,232 
 
 10,503,459 
 
 10.648.000 
 10,793,861 
 10,941,048 
 11,089,567 
 11,239,424 
 
 11.390.625 
 11,543,176 
 11,697,083 
 11,852,352 
 12,008,989 
 
 12.167.000 
 12,326,391 
 12,487,168 
 12,649,337 
 12,812,904 
 
 12.977.875 
 13,144,256 
 13,312,053 
 13,481,272 
 13,651,919 
 
 13.824.000 
 13,997,521 
 14,172,488 
 14,348,907 
 14,526,784 
 
 13.4907 
 
 13.5277 
 
 13.5647 
 
 13.6015 
 
 13.6382 
 
 13.6748 
 
 13.7113 
 
 13.7477 
 
 13.7840 
 
 13.8203 
 
 13.8564 
 
 13.8924 
 
 13.9284 
 
 13.9642 
 
 14.0000 
 14.0357 
 14.0712 
 14.1067 
 14.1421 
 14.1774 
 14.2127 
 14.2478 
 14.2829 
 14.3178 
 14.3527 
 14.3875 
 14.4222 
 14.4568 
 14.4914 
 14.5258 
 14.5602 
 14.5945 
 14.6287 
 14.6629 
 14.6969 
 14.7309 
 14.7648 
 14.7986 
 14.8324 
 14.8661 
 14.8997 
 14.9332 
 14.9666 
 
 15.0000 
 15.0333 
 15.0665 
 15.0997 
 15.1327 
 15.1658 
 15.1987 
 15.2315 
 15.2643 
 15.2971 
 15.3297 
 15.3623 
 15.3948 
 15.4272 
 15.4596 
 15.4919 
 15.5242 
 15.5563 
 15.5885 
 15.6205 
 
 5.6671 
 
 5.6774 
 
 5.6877 
 
 5.6980 
 
 5.7083 
 
 5.7185 
 
 5.7287 
 
 5.7388 
 
 5.7489 
 
 5.7590 
 
 5.7690 
 
 5.7790 
 
 5.7890 
 
 5.7989 
 
 5.8088 
 
 5.8186 
 
 5.8285 
 
 5.8383 
 
 5.8480 
 
 5.8578 
 
 5.8675 
 
 5.8771 
 
 5.8868 
 
 5.8964 
 
 5.9059 
 
 5.9155 
 
 5.9250 
 
 5.9345 
 
 5.9439 
 
 5.9533 
 
 5.9627 
 
 5.9721 
 
 5.9814 
 
 5.9907 
 
 6.0000 
 
 6.0092 
 
 6.0185 
 
 6.0277 
 
 6.0368 
 
 6.0459 
 
 6.0550 
 
 6.0641 
 
 6.0732 
 
 6.0822 
 
 6.0912 
 
 6.1002 
 
 6.1091 
 
 6.1180 
 
 6.1269 
 
 6.1358 
 
 6.1446 
 
 6.1534 
 
 6.1622 
 
 6.1710 
 
 6.1797 
 
 6.1885 
 
 6.1672 
 
 6.2058 
 
 6.2145 
 
 6.2231 
 
 6.2317 
 
 6.2403 
 
 6.2488 
 
 .005494505 
 
 .005464481 
 
 .005434783 
 
 .005405405 
 
 .005376344 
 
 .005347594 
 
 .005319149 
 
 .005291005 
 
 .005263158 
 
 .005235602 
 
 .005208333 
 
 .005181347 
 
 .005154639 
 
 .005128205 
 
 .005102041 
 
 .005076142 
 
 .005050505 
 
 .005025126 
 
 .005000000 
 
 .004975124 
 
 .004950495 
 
 .004926108 
 
 .004901961 
 
 .004878049 
 
 .004854369 
 
 .004830918 
 
 .004807692 
 
 .004784689 
 
 .004761905 
 
 .004739336 
 
 .004716981 
 
 .004694836 
 
 .004672897 
 
 .004651163 
 
 .004629630 
 
 .004608295 
 
 .004587156 
 
 .004566210 
 
 .004545455 
 
 .004524887 
 
 .004504505 
 
 .004484305 
 
 .004464286 
 
 .004444444 
 
 .004424779 
 
 .004405286 
 
 .004385965 
 
 .004366812 
 
 .004347826 
 
 .004329004 
 
 .004310345 
 
 .004291845 
 
 .004273504 
 
 .004255319 
 
 .004237288 
 
 .004219409 
 
 .004201681 
 
 .004184100 
 
 .004166667 
 
 .004149378 
 
 .004132231 
 
 .004115226 
 
 .004098361 
 
 571.77 
 
 574.91 
 
 578.05 
 
 581.19 
 584.34 
 587.48 
 590.62 
 593.76 
 596.90 
 
 600.04 
 
 603.19 
 606.33 
 609.47 
 612.61 
 615.75 
 618.89 
 
 622.04 
 625.18 
 628.32 
 631.46 
 634.60 
 637.74 
 
 640.88 
 644.03 
 647.17 
 650.31 
 653.45 
 656.59 
 
 659.73 
 
 662.88 
 666.02 
 669.16 
 672.30 
 675.44 
 
 678.58 
 
 681.73 
 684.87 
 688.01 
 691.15 
 694.29 
 697.43 
 
 700.58 
 703.72 
 706.86 
 710.00 
 713.14 
 716.28 
 
 719.42 
 722.57 
 725.71 
 728.85 
 731.99 
 735.13 
 
 738.27 
 
 741.42 
 744.56 
 747.70 
 750.84 
 753.98 
 757.12 
 
 760.27 
 763.41 
 766.55 
 
 26,015.53 
 
 26.302.20 
 26,590.44 
 
 26.880.25 
 
 27.171.63 
 27,464.59 
 
 27.759.11 
 
 28.055.21 
 28,352.87 
 
 28.652.11 
 
 28.952.92 
 29,255.30 
 
 29.559.25 
 
 29.864.77 
 
 30.171.86 
 
 30.480.52 
 30,790.75 
 
 31.102.55 
 
 31.415.93 
 
 31.730.87 
 32,047.39 
 
 32.365.47 
 
 32.685.13 
 33.006.36 
 33,329.16 
 
 33.653.53 
 
 33.979.47 
 
 34.306.98 
 
 34.636.06 
 
 34.966.71 
 
 35.298.94 
 
 35.632.73 
 
 35.968.09 
 
 36.305.03 
 
 36.643.54 
 
 36.983.61 
 
 37.325.26 
 
 37.668.48 
 
 38.013.27 
 
 38.359.63 
 
 38.707.56 
 
 39.057.07 
 
 39.408.14 
 
 39.760.78 
 40,115.00 
 
 40.470.78 
 
 40.828.14 
 
 41.187.07 
 
 41.547.56 
 41*909.63 
 
 42.273.27 
 
 42.638.48 
 43,005.26 
 
 43.373.61 
 
 43.743.54 
 
 44.115.03 
 
 44.488.09 
 
 44.862.73 
 45,238.93 
 
 45.616.71 
 45,996.06 
 
 46.376.98 
 46,759.47 
 
CIRCUMFERENCES , AND AREAS. 
 
 549 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq . Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 245 
 
 246 
 
 247 
 
 248 
 
 249 
 
 250 
 
 251 
 
 252 
 258 
 
 254 
 
 255 
 256. 
 
 257 
 
 258 
 
 259 
 
 260 
 261 
 262 
 
 263 
 
 264 
 
 265 
 
 266 
 
 267 
 
 268 
 
 269 
 
 270 
 
 271 
 
 272 
 
 273 
 
 274 
 
 275 
 
 276 
 
 277 
 
 278 
 
 279 
 
 280 
 281 
 282 
 
 283 
 
 284 
 
 285 
 
 286 
 
 287 
 
 288 
 
 289 
 
 290 
 
 291 
 
 292 
 
 293 
 
 294 
 
 295 
 
 296 
 
 297 
 
 298 
 
 299 
 
 300 
 
 301 
 
 302 
 
 303 
 
 304 
 
 305 
 
 306 
 
 307 
 
 60.025 
 
 60.516 
 
 61.009 
 
 61.504 
 62,001 
 62,500 
 63,001 
 
 63.504 
 
 64.009 
 
 64.516 
 
 65.025 
 65.536 
 66,049 
 66,564 
 67,081 
 67,600 
 68,121 
 68,644 
 69,169 
 69,696 
 
 70.225 
 70,756 
 71,289 
 71,824 
 72,361 
 72,900 
 73,441 
 73,984 
 74,529 
 75,076 
 75,625 
 76,176 
 76,729 
 77,284 
 77,841 
 
 78.400 
 78,961 
 79,524 
 80,089 
 80,656 
 
 81.225 
 81,796 
 82,369 
 82,944 
 83,521 
 84,100. 
 84,681 
 85.264 
 85', 849 
 86,436 
 
 87.025 
 87,616 
 88,209 
 88,804 
 
 89.401 
 90,000 
 90,601 
 91,204 
 91,809 
 92,416 
 
 93.025 
 93,636 
 94,249 
 
 14,706,125 
 
 14,886,936 
 
 15,069,223 
 
 15,252,992 
 
 15,438,249 
 
 15.625.000 
 15,813,251 
 16,003,008 
 16,194,277 
 16,387,064 
 16,581,375 
 16,777,216 
 16,974,593 
 17,173,512 
 17,373,979 
 
 17.576.000 
 17,779,581 
 17,984,728 
 18,191,447 
 18,399,744 
 18,609,625 
 18,821,096 
 19,034,163 
 19,248,832 
 19.465,109 
 19^683,000 
 19,902,511 
 20,123,643 
 20,346,417 
 20,570,824 
 20,796,875 
 21,024,576 
 21,253,933 
 21,484,952 
 21,717,639 
 
 21.952.000 
 22,188,041 
 22,425,768 
 22,665,187 
 22,906,304 
 23,149,125 
 23,393,656 
 23,639,903 
 23,887,872 
 24,137,569 
 
 24.389.000 
 24,642,171 
 24,897,088 
 25,153,757 
 25,412,184 
 25,672,375 
 25,934,836 
 26,198,073 
 26,463,592 
 26,730,899 
 27,000,000 
 27,270,901 
 27,543,608 
 27,818,127 
 28,094,464 
 28,372,625 
 28,652,616 
 28,934,443 
 
 15.6525 
 
 15.6844 
 
 15.7162 
 
 15.7480 
 
 15.7797 
 
 15.8114 
 
 15.8430 
 
 15.8745 
 
 15.9060 
 
 15.9374 
 
 15.9687 
 
 16.0000 
 
 16.0312 
 
 16.0624 
 
 16.0935 
 
 16.1245 
 
 16.1555 
 
 16.1864 
 
 16.2173 
 
 16.2481 
 
 16.2788 
 
 16.3095 
 
 16.3401 
 
 16.3707 
 
 16.4012 
 
 16.4317 
 
 16.4621 
 
 16.4924 
 
 16.5227 
 
 16.5529 
 
 16.5831 
 
 16.6182 
 
 16.6433 
 
 16.6783 
 
 16.7033 
 
 16.7332 
 
 16.7631 
 
 16.7929 
 
 16.8226 
 
 16.8523 
 
 16.8819 
 
 16.9115 
 
 16.9411 
 
 16.9706 
 
 17.0000 
 
 17.0294 
 
 17.0587 
 
 17.0880 
 
 17.1172 
 
 17.1464 
 
 17.1756 
 
 17.2047 
 
 17.2337 
 
 17.2627 
 
 17.2916 
 
 17.3205 
 
 17.3494 
 
 17.3781 
 
 17.4069 
 
 17.4356 
 
 17.4642 
 
 17.4929 
 
 17.5214 
 
 6.2573 
 
 6.2658 
 
 6.2743 
 
 6.2828 
 
 6.2912 
 
 6.2996 
 
 6.3080 
 
 6.3164 
 
 6.3247 
 
 6.3330 
 
 6.3413 
 
 6.3496 
 
 6.3579 
 
 6.3661 
 
 6.3743 
 
 6.3825 
 
 6.3907 
 
 6.3988 
 
 6.4070 
 
 6.4151 
 
 6.4232 
 
 6.4312 
 
 6.4393 
 
 6.4473 
 
 6.4553 
 
 6.4633 
 
 6.4713 
 
 6.4792 
 
 6.4872 
 
 6.4951 
 
 6.5030 
 
 6.5108 
 
 6.5187 
 
 6.5265 
 
 6.5343 
 
 6.5421 
 
 6.5499 
 
 6.5577 
 
 6.5654 
 
 6.5731 
 
 6.5808 
 
 6.5885 
 
 6.5962 
 
 6.6039 
 
 6.6115 
 
 6.6191 
 
 6.6267 
 
 6.6343 
 
 6.6419 
 
 6.6494 
 
 6.6569 
 
 6.6644 
 
 6.6719 
 
 6.6794 
 
 6.6869 
 
 6.6943 
 
 6.7018 
 
 6.7092 
 
 6.7166 
 
 6.7240 
 
 6.7313 
 
 6.7387 
 
 6.7460 
 
 .004081633 
 
 .004065041 
 
 .004048583 
 
 .004032258 
 
 .004016064 
 
 .004000000 
 
 .003984064 
 
 .003968254 
 
 .003952569 
 
 .003937008 
 
 .003921569 
 
 .003906250 
 
 .003891051 
 
 .003875969 
 
 .003861004 
 
 .003846154 
 
 .003831418 
 
 .003816794 
 
 .003802281 
 
 .003787879 
 
 .003773585 
 
 .003759398 
 
 .003745318 
 
 .003731343 
 
 .003717472 
 
 .003703704 
 
 .003690037 
 
 .003676471 
 
 .003663004 
 
 .003649635 
 
 .003636364 
 
 .003623188 
 
 .003610108 
 
 .003597122 
 
 .003584229 
 
 .003571429 
 
 .003558719 
 
 .003546099 
 
 .003533569 
 
 .003522127 
 
 .003508772 
 
 .003496503 
 
 .003484321 
 
 .003472222 
 
 .003460208 
 
 .003448276 
 
 .003436426 
 
 .003424658 
 
 .003412969 
 
 .003401361 
 
 .003389831 
 
 .003378378 
 
 .003367003 
 
 .003355705 
 
 .003344482 
 
 .003333333 
 
 .003322259 
 
 .003311258 
 
 .003301330 
 
 .003289474 
 
 .003278689 
 
 .003267974 
 
 .003257329 
 
 769.69 
 
 772.83 
 
 775.97 
 
 779.11 
 782.26 
 785.40 
 788.54 
 791.68 
 794.82 
 
 797.96 
 
 801.11 
 804.25 
 807.39 
 810.53 
 813.67 
 816.81 
 
 819.96 
 823.10 
 826.24 
 829.38 
 832.52 
 835.66 
 838.81 
 841.95 
 845.09 
 848.23 
 851.37 
 854.51 
 
 857.65 
 860.80 
 863.94 
 867.08 
 870.22 
 873.36 
 
 876.50 
 
 879.65 
 882.79 
 885.93 
 889.07 
 892.21 
 
 895.35 
 
 898.50 
 901.64 
 904.78 
 907.92 
 911.06 
 914.20 
 
 917.35 
 920.49 
 923.63 
 926.77 
 929.91 
 933.05 
 
 936.19 
 939.34 
 942.48 
 945.62 
 948.76 
 951.90 
 955.04 
 
 958.19 
 961.33 
 964.47 
 
 47.143.52 
 47,529.16 
 47,916.36 
 48,305.13 
 48,695.47 
 49,087.39 
 
 49.480.87 
 
 49.875.92 
 
 50.272.55 
 
 50.670.75 
 
 51.070.52 
 51,471.85 
 
 51.874.76 
 
 52.279.24 
 52,685.29 
 
 53.092.92 
 
 53.502.11 
 
 53.912.87 
 
 54.325.21 
 
 54.739.11 
 55,154.59 
 55,571.63 
 
 55.990.25 
 56,410.44 
 56,832.20 
 
 57.255.53 
 
 57.680.43 
 58,106.90 
 58,534.94 
 
 58.964.55 
 
 59.395.74 
 
 59.828.49 
 60,262.82 
 60,698.71 
 
 61.136.18 
 
 61.575.22 
 
 62.015.82 
 62,458.00 
 
 62.901.75 
 
 63.347.07 
 63,793.97 
 
 64.242.43 
 64,692.46 
 
 65.144.07 
 65,597.24 
 
 66.051.99 
 66,508.30 
 
 66.966.19 
 
 67.425.65 
 67,886.68 
 68,349.28 
 
 68.813.45 
 
 69.279.19 
 
 69.746.50 
 70,215.38 
 
 70.685.83 
 71,157.86 
 
 71.631.45 
 72,106.62 
 72,583.36 
 
 73.061.66 
 73,541.54 
 
 74.022.99 
 
550 
 
 SQUARES, CUBES , SQUARE AND CUBE ROOTS , 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq . Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 308 
 
 309 
 
 310 
 
 311 
 
 312 
 
 313 
 
 314 
 
 315 
 
 316 
 
 317 
 
 318 
 
 319 
 
 320 
 
 321 
 
 322 
 
 323 
 
 324 
 
 325 
 
 326 
 
 327 
 
 328 
 
 329 
 
 330 
 
 331 
 
 332 
 
 333 
 
 334 
 
 335 
 
 336 
 
 337 
 
 338 
 
 339 
 
 340 
 
 341 
 
 342 
 
 343 
 
 344 
 
 345 
 
 346 
 
 347 
 
 348 
 
 349 
 
 350 
 
 351 
 
 352 
 
 353 
 
 354 
 
 355 
 
 356 
 
 357 
 
 358 
 
 359 
 
 360 
 
 361 
 
 362 
 
 363 
 
 364 
 
 365 
 
 366 
 
 367 
 
 368 
 
 369 
 
 370 
 
 94,864 
 
 95,481 
 
 96,100 
 
 96,721 
 
 97,344 
 
 97,969 
 
 98,596 
 
 99,225 
 
 99,856 
 
 100,489 
 
 101,124 
 
 101,761 
 
 102,400 
 
 103,041 
 
 103,684 
 
 104,329 
 
 104,976 
 
 105,625 
 
 106,276 
 
 106,929 
 
 107,584 
 
 108,241 
 
 108.900 
 109,561 
 
 110.224 
 110,889 
 111,556 
 
 112.225 
 112,896 
 113,569 
 114,244 
 114,921 
 
 115.600 
 116,281 
 116,964 
 117,649 
 118,336 
 
 119.025 
 119,716 
 120,409 
 121,104 
 121,801 
 122,500 
 123,201 
 123,904 
 124,609 
 125,316 
 
 126.025 
 126,736 
 127,449 
 128,164 
 128,881 
 
 129.600 
 130,321 
 131,044 
 131,769 
 132,496 
 
 133.225 
 133,956 
 134,689 
 135,424 
 136,161 
 
 136.900 
 
 29,218,112 
 
 29,503,629 
 
 29.791.000 
 30,080,231 
 30,371,328 
 30,664,297 
 30,959,144 
 
 31.255.875 
 31,554,496 
 31,855,013 
 32,157,432 
 32,461,759 
 
 32.768.000 
 33,076,161 
 33,386,248 
 33,698,267 
 34,012,224 
 
 34.328.125 
 
 34.645.976 
 34,965,783 
 35,287,552 
 35,611,289 
 
 35.937.000 
 36,264,691 
 36,594,368 
 36,926,037 
 37,259,704 
 37,595,375 
 37,933,056 
 38,272,753 
 38,614,472 
 38,958,219 
 
 39.304.000 
 39,651,821 
 40,001,688 
 40,353,607 
 40,707,584 
 41,063,625 
 41,421,736 
 41,781,923 
 42,144,192 
 42,508,549 
 
 42.875.000 
 43,243,551 
 43,614,208 
 
 43.986.977 
 44,361,864 
 
 44.738.875 
 45,118,016 
 45,499,293 
 45,882,712 
 46,268,279 
 
 46.656.000 
 47,045,881 
 47,437,928 
 47,832,147 
 48,228,544 
 
 48.627.125 
 49,027,896 
 49,430,863 
 49,836,032 
 50,243,409 
 
 50.653.000 
 
 17.5499 
 
 17.5784 
 
 17.6068 
 
 17.6352 
 
 17.6635 
 
 17.6918 
 
 17.7200 
 
 17.7482 
 
 17.7764 
 
 17.8045 
 
 17.8326 
 
 17.8606 
 
 17.8885 
 
 17.9165 
 
 17.9444 
 
 17.9722 
 
 18.0000 
 
 18.0278 
 
 18.0555 
 
 18.0831 
 
 18.1108 
 
 18.1384 
 
 18.1659 
 
 18.1934 
 
 18.2209 
 
 18.2483 
 
 18.2757 
 
 18.3030 
 
 18.3303 
 
 18.3576 
 
 18.3848 
 
 18.4120 
 
 18.4391 
 
 18.4662 
 
 18.4932 
 
 18.5203 
 
 18.5472 
 
 18.5742 
 
 18.6011 
 
 18.6279 
 
 18.6548 
 
 18.6815 
 
 18.7083 
 
 18.7350 
 
 18.7617 
 
 18.7883 
 
 18.8149 
 
 18.8414 
 
 18.8680 
 
 18.8944 
 
 18.9209 
 
 18.9473 
 
 18.9737 
 
 19.0000 
 
 19.0263 
 
 19.0526 
 
 19.0788 
 
 19.1050 
 
 19.1311 
 
 19.1572 
 
 19.1833 
 
 19.2094 
 
 19.2354 
 
 6.7533 
 
 6.7606 
 
 6.7679 
 
 6.7752 
 
 6.7824 
 
 6.7897 
 
 6.7969 
 
 6.8041 
 
 6.8113 
 
 6.8185 
 
 6.8256 
 
 6.8328 
 
 6.8399 
 
 6.8470 
 
 6.8541 
 
 6.8612 
 
 6.8683 
 
 6.8753 
 
 6.8824 
 
 6.8894 
 
 6.8964 
 
 6.9034 
 
 6.9104 
 
 6.9174 
 
 6.9244 
 
 6.9313 
 
 6.9382 
 
 6.9451 
 
 6.9521 
 
 6.9589 
 
 6.9658 
 
 6.9727 
 
 6.9795 
 
 6.9864 
 
 6.9932 
 
 7.0000 
 
 7.0068 
 
 7.0136 
 
 7.0203 
 
 7.0271 
 
 7.0338 
 
 7.0406 
 
 7.0473 
 
 7.0540 
 
 7.0607 
 
 7.0674 
 
 7.0740 
 
 7.0807 
 
 7.0873 
 
 7.0940 
 
 7.1006 
 
 7.1072 
 
 7.1138 
 
 7.1204 
 
 7.1269 
 
 7.1335 
 
 7.1400 
 
 7.1466 
 
 7.1531 
 
 7.1596 
 
 7.1661 
 
 7.1726 
 
 7.1791 
 
 .003246753 
 
 .003236246 
 
 .003225806 
 
 .003215434 
 
 .003205128 
 
 .003194888 
 
 .003184713 
 
 .003174603 
 
 .003164557 
 
 .003154574 
 
 .003144654 
 
 .003134796 
 
 .003125000 
 
 .003115265 
 
 .003105590 
 
 .003095975 
 
 .003086420 
 
 .003076923 
 
 .003067485 
 
 .003058104 
 
 .003048780 
 
 .003039514 
 
 .003030303 
 
 .003021148 
 
 .003012048 
 
 .003003003 
 
 .002994012 
 
 .002985075 
 
 .002976190 
 
 .002967359 
 
 .002958580 
 
 .002949853 
 
 .002941176 
 
 .002932551 
 
 .002923977 
 
 .002915452 
 
 .002906977 
 
 .002898551 
 
 .002890173 
 
 .002881844 
 
 .002873563 
 
 .002865330 
 
 .002857143 
 
 .002849003 
 
 .002840909 
 
 .002832861 
 
 .002824859 
 
 .002816901 
 
 .002808989 
 
 .002801120 
 
 .002793296 
 
 .002785515 
 
 .002777778 
 
 .002770083 
 
 .002762431 
 
 .002754821 
 
 .002747253 
 
 .002739726 
 
 .002732240 
 
 .002724796 
 
 .002717391 
 
 .002710027 
 
 .002702703 
 
 967.61 
 
 970.75 
 
 973.89 
 
 977.04 
 
 980.18 
 
 983.32 
 
 986.46 
 
 989.60 
 
 992.74 
 
 995.88 
 
 999.03 
 
 1,002.17 
 
 1,005.31 
 
 1,008.45 
 
 1,011.59 
 
 1.014.73 
 1,017.88 
 1,021.02 
 1,024.16 
 1,027.30 
 1,030.44 
 
 1.033.58 
 
 1.036.73 
 1,039.87 
 1,043.01 
 1,046.15 
 1,049.29 
 1,052.43 
 
 1.055.58 
 1,058.72 
 1,061.86 
 1,065.00 
 1,068.14 
 1,071.28 
 
 1.074.42 
 1,077.57 
 1,080.71 
 1,083.85 
 1,086.99 
 1,090.13 
 
 1.093.27 
 
 1.096.42 
 1,099.56 
 1,102.70 
 1,105.84 
 1,108.98 
 1,112.12 
 
 1.115.27 
 1,118.41 
 1,121.55 
 1,124.69 
 1,127.83 
 1,130.97 
 
 1.134.11 
 1,137.26 
 1,140.40 
 1,143.54 
 1,146.68 
 1,149.82 
 1,152.96 
 
 1.156.11 
 1,159.25 
 1,162.39 
 
 74.506.01 
 
 74.990.60 
 
 75.476.76 
 75,964.50 
 
 76.453.80 
 
 76.944.67 
 
 77.437.12 
 
 77.931.13 
 
 78.426.72 
 
 78.923.88 
 
 79.422.60 
 
 79.922.90 
 
 80.424.77 
 
 80.928.21 
 
 81.433.22 
 
 81.939.80 
 
 82.447.96 
 
 82.957.68 
 83,468.98 
 
 83.981.84 
 
 84.496.28 
 
 85.012.28 
 
 85.529.86 
 
 86.049.01 
 
 86.569.73 
 
 87.092.02 
 
 87.615.88 
 
 88.141.31 
 
 88.668.31 
 
 89.196.88 
 
 89.727.03 
 
 90.258.74 
 
 90.792.03 
 
 91.326.88 
 
 91.863.31 
 
 92.401.31 
 
 92.940.88 
 93,482.02 
 94,024.73 
 
 94.569.01 
 
 95.114.86 
 
 95.662.28 
 
 96.211.28 
 
 96.761.84 
 
 97.313.97 
 
 97.867.68 
 98,422.96 
 
 98.979.80 
 
 99.538.22 
 100,098.21 
 
 100.659.77 
 
 101.222.90 
 
 101.787.60 
 102,353.87 
 102,921.72 
 
 103.491.13 
 104,062.12 
 104,634.67 
 105,208.80 
 105,784.49 
 106,361.76 
 
 106.940.60 
 
 107.521.01 
 
CIR C UMEERENCES, AND AREAS. 
 
 551 
 
 No. 
 
 Square . 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 371 
 
 372 
 
 373 
 
 374 
 
 375 
 
 376 
 
 377 
 
 378 
 
 379 
 
 380 
 
 381 
 
 382 
 
 383 
 
 384 
 
 385 
 
 386 
 
 387 
 
 388 
 
 389 
 
 390 
 
 391 
 
 392 
 
 393 
 
 394 
 
 395 
 
 396 
 
 397 
 
 398 
 
 399 
 
 400 
 
 401 
 
 402 
 
 403 
 
 404 
 
 405 
 
 406 
 
 407 
 
 408 
 
 409 
 
 410 
 
 411 
 
 412 
 ’413 
 
 414 
 
 415 
 
 416 
 
 417 
 
 418 
 
 419 
 
 420 
 
 421 
 
 422 
 
 423 
 
 424 
 
 425 
 
 426 
 
 427 
 
 428 
 
 429 
 
 430 
 
 431 
 
 432 
 
 433 
 
 137.641 
 138,384 
 
 139.129 
 139,876 
 140,625 
 141,376 
 
 142.129 
 142,884 
 
 143.641 
 144,400 
 145,161 
 145,924 
 146,689 
 147,456 
 
 148.225 
 148,996 
 149,769 
 150,544 
 151,321 
 
 152.100 
 152,881 
 153,664 
 154,449 
 155,236 
 
 156.025 
 156,816 
 157,609 
 158,404 
 159,201 
 160,000 
 160,801 
 161,604 
 162,409 
 163,216 
 
 164.025 
 164,836 
 165,649 
 166,464 
 167,281 
 
 168.100 
 168,921 
 169,744 
 170,569 
 171,396 
 
 172.225 
 173,056 
 173,889 
 174,724 
 175,561 
 176,400 
 177,241 
 178,084 
 178,929 
 179,776 
 180,625 
 181,476 
 182,329 
 183,184 
 184,041 
 184,900 
 185,761 
 186,624 
 187,489 
 
 51,064,811 
 
 51,478,848 
 
 51,895,117 
 
 52.313.624 
 
 52.734.375 
 
 53.157.376 
 53,582,633 
 54,010,152 
 54,439,939 
 
 54.872.000 
 55,306,341 
 55,742,968 
 56,181,887 
 56,623,104 
 
 57.066.625 
 57,512,456 
 57,960,603 
 58,411,072 
 58,863,869 
 
 59.319.000 
 59,776,471 
 •60,236,288 
 
 60,698,457- 
 
 61,162,984 
 
 61,629,875 
 
 62,099,136 
 
 62,570,773 
 
 63,044,792 
 
 63,521,199 
 
 64,000,000 
 
 64,481,201 
 
 64,964,808 
 
 65,450,827 
 
 65,939,264 
 
 66,430,125 
 
 66,923,416 
 
 67,419,143 
 
 67,917,312 
 
 68,417,929 
 
 68.921.000 
 69,426,531 
 69,934,528 
 70,444,997 
 70,957,944 
 71,473,375 
 71,991,296 
 72,511,713 
 73,034,632 
 73,560,059 
 
 74.088.000 
 74,618,461 
 75,151,448 
 75,686,967 
 76,225,024 
 76,765,625 
 77,308,776 
 77,854,483 
 78,402,752 
 78,953,589 
 
 79.507.000 
 80,062,991 
 80,621,568 
 81,182,737 
 
 19.2614 
 
 19.2873 
 
 19.3132 
 
 19.3391 
 
 19.3649 
 
 19.3907 
 
 19.4165 
 
 19.4422 
 
 19.4679 
 
 19.4936 
 
 19.5192 
 
 19.5448 
 
 19.5704 
 
 19.5959 
 
 19.6214 
 
 19.6469 
 
 19.6723 
 
 19.6977 
 
 19.7231 
 
 19.7484 
 
 19.7737 
 
 19.7990 
 
 19.8242 
 
 19.8494 
 
 19.8746 
 
 19.8997 
 
 19.9249 
 
 19.9499 
 
 19.9750 
 
 20.0000 
 
 20.0250 
 
 20.0499 
 
 20.0749 
 
 20.0998 
 
 20.1246 
 
 20.1494 
 
 20.1742 
 
 20.1990 
 
 20.2237 
 
 20.2485 
 
 20.2731 
 
 20.2978 
 
 20.3224 
 
 20.3470 
 
 20.3715 
 
 20.3961 
 
 20.4206 
 
 20.4450 
 
 20.4695 
 
 20.4939 
 
 20.5183 
 
 20.5426 
 
 20.5670 
 
 20.5913 
 
 20.6155 
 
 20.6398 
 
 20.6640 
 
 20.6882 
 
 20.7123 
 
 20.7364 
 
 20.7605 
 
 20.7846 
 
 20.8087 
 
 7.1855 
 
 7.1920 
 
 7.1984 
 
 7.2048 
 
 7.2112 
 
 7.2177 
 
 7.2240 
 
 7.2304 
 
 7.2368 
 
 7.2432 
 
 7.2495 
 
 7.2558 
 
 7.2622 
 
 7.2685 
 
 7.2748 
 
 7.2811 
 
 7.2874 
 
 7.2936 
 
 7.2999 
 
 7.3061 
 
 7.3124 
 
 7.3186 
 
 7.3248 
 
 7.3310 
 
 7.3372 
 
 7.3434 
 
 7.3496 
 
 7.3558 
 
 7.3619 
 
 7.3681 
 
 7.3742 
 
 7.3803 
 
 7.3864 
 
 7.3925 
 
 7.3986 
 
 7.4047 
 
 7.4108 
 
 7.4169 
 
 7.4229 
 
 7.4290 
 
 7.4350 
 
 7.4410 
 
 7.4470 
 
 7.4530 
 
 7.4590 
 
 7.4650 
 
 7.4710 
 
 7.4770 
 
 7.4829 
 
 7.4889 
 
 7.4948 
 
 7.5007 
 
 7.5067 
 
 7.5126 
 
 7.5185 
 
 7.5244 
 
 7.5302 
 
 7.5361 
 
 7.5420 
 
 7.5478 
 
 7.5537 
 
 7.5595 
 
 7.5654 
 
 .002695418 : 
 
 .002688172 : 
 
 .002680965 : 
 
 .002673797 : 
 
 .002666667 : 
 
 .002659574 : 
 
 .002652520 
 
 .002645503 
 
 .002638521 
 
 .002631579 
 
 .002624672 
 
 .002617801 
 
 .002610966 
 
 .002604167 
 
 .002597403 
 
 .002590674 
 
 .002583979 
 
 .002577320 
 
 .002570694 
 
 .002564103 
 
 .002557545 
 
 .002551020 
 
 .002544529 
 
 .002538071 
 
 .002531646 
 
 .002525253 
 
 .002518892 
 
 .002512563 
 
 .002506266 
 
 .002500000 
 
 .002493766 
 
 .002487562 
 
 .002481390 
 
 .002475248 
 
 .002469136 
 
 .002463054 
 
 .002457002 
 
 .002450980 
 
 .002444988 
 
 .002439024 
 
 .002433090 
 
 .002427184 
 
 .002421308 
 
 .002415459 
 
 .002409639 
 
 .002406846 
 
 .002398082 
 
 .002392344 
 
 .002386635 
 
 .002380952 
 
 .002375297 
 
 .002369668 
 
 .002364066 
 
 .002358491 
 
 .002352941 
 
 .002347418 
 
 .002341920 
 
 .002336449 
 
 .002331002 
 
 .002325581 
 
 .002320186 
 
 .002314815 
 
 .002309469 
 
 1,165.53 ! 
 1,168.67 : 
 
 1.171.81 : 
 1,174.96 : 
 1,178.10 : 
 1,181.24 : 
 1,184.38 : 
 1,187.52 : 
 1,190.66 : 
 
 1.193.81 : 
 1,196.95 : 
 1,200.09 
 1,203.23 
 1,206.37 
 1,209.51 
 
 1.212.65 
 1,215.80 
 1,218.94 
 1,222.08 
 1.225.22 
 1,228.36 
 
 1.231.50 
 
 1.234.65 
 1,237.79 
 1,240.93 
 1,244.07 
 1,247.21 
 
 1.250.35 
 
 1.253.50 
 1,256.64 
 1,259.78 
 1,262.92 
 1,266.06 
 1,269.20 
 
 1.272.35 
 1,275.49 
 1,278.63 
 1,281.77 
 1,284.91 
 1,288.05 
 
 1.291.19 
 1,294.34 
 1,297.48 
 1,300.62 
 1,303.76 
 1,306.90 
 
 1.310.04 
 
 1.313.19 
 1,316.33 
 1,319.47 
 1,322.61 
 1,325.75 
 1,328.89 
 
 1.332.04 
 1,335.18 
 1,338.32 
 1,341.46 
 1,344.60 
 1,347.74 
 1,350.88 
 1,354.03 
 1,357.17 
 1,360.31 
 
 L 08, 102.99 
 L 08, 686.54 
 
 109.271.66 
 109, 858.35 
 
 110.446.62 
 
 111.036.45 
 
 111.627.86 
 112,220.83 
 112,815.38 
 113,411.49 
 
 114.009.18 
 114,608.44 
 115,209.27 
 
 115.811.67 
 116,415.64 
 
 117.021.18 
 117,628.30 
 118,236.98 
 
 118.847.24 
 
 119.459.06 
 
 120.072.46 
 
 120.687.42 
 121,303.96 
 
 121.922.07 
 
 122.541.75 
 123,163.00 
 123,785.82 
 124,410.21 
 125,036.17 
 125,663.71 
 126,292.81 
 126,923.48 
 
 127.555.73 
 128,189.55 
 128,824.93 
 129,461.89 
 
 130.100.42 
 130,740.52 
 
 131.382.19 
 
 132.025.43 
 
 132.670.24 
 
 133.316.63 
 133,964.58 
 
 134.614.10 
 
 135.265.20 
 
 135.917.86 
 
 136.572.10 
 
 137.227.91 
 137,885.29 
 
 138.544.24 
 
 139.204.76 
 
 139.866.85 
 
 140.530.51 
 
 141.195.74 
 141,862.54 
 
 142.530.92 
 
 143.200.86 
 143,872.38 
 144,545.46 
 145,220.12 
 145.896.35 
 146,574.15 
 
 147.253.52 
 
552 
 
 SQUARES, CUBES, SQUARE AND CUBE ROOTS , 
 
 | No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 - 
 
 Area. 
 
 4 B 4 
 
 435 
 
 436 
 
 437 
 
 438 
 
 439 
 
 440 
 
 441 
 
 442 
 
 443 
 
 444 
 
 445 
 
 446 
 
 447 
 
 448 
 
 449 
 
 450 
 
 451 
 
 452 
 
 453 
 
 454 
 
 455 
 
 456 
 
 457 
 
 458 
 
 459 
 
 460 
 
 461 
 
 462 
 
 463 
 
 464 
 
 465 
 
 466 
 
 467 i 
 
 468 
 
 469 
 
 470 
 
 471 
 
 472 
 
 473 
 
 474 
 
 475 
 
 476 
 
 477 
 
 478 
 
 479 
 
 480 
 
 481 
 
 482 
 
 483 
 
 484 
 
 485 
 
 486 
 
 487 
 
 488 
 
 489 
 
 490 
 
 491 
 
 492 
 
 493 
 
 494 
 
 495 
 
 496 
 
 188,356 
 
 189,225 
 
 190,096 
 
 190,969 
 
 191,844 
 
 192,721 
 
 193.600 
 194,481 
 195,364 
 196,249 
 197,136 
 
 198.025 
 198,916 
 199,809 
 200,704 
 
 201.601 
 202,500 
 203,401 
 204,304 
 205,209 
 206,116 
 
 207.025 
 207,936 
 208,849 
 209,764 
 210,681 
 
 , 211,600 
 212,521 
 213,444 
 214,369 
 215,296 
 
 216.225 
 217,156 
 218,089 
 
 219.024 
 219,961 
 220,900 
 221,841 
 222,784 
 223,729 
 224,676 
 225,625 
 226,576 
 227,529 
 228,484 
 229,441 
 230,400 
 231,361 
 232,324 
 233,289 
 234,256 
 
 235.225 
 236,196 
 237,169 
 238,144 
 239,121 
 240,100 
 241,081 
 242,064 
 243,049 
 244,036 
 
 245.025 
 246,^016 
 
 81,746,504 
 
 82.312.875 
 82,881,856 
 83,453,453 
 84,027,672 
 84,604,519 
 
 85.184.000 
 85,766,121 
 86,350,888 
 86,938,307 
 87,528,384 
 
 88.121.125 
 88,716,536 
 89,314,623 
 89,915,392 
 90,518,849 
 
 91.125.000 
 91,733,851 
 92,345,408 
 92,959,677 
 93,576,664 
 
 94.196.375 
 94,818,816 
 95,443,993 
 96.071,912 
 96,702,579 
 
 97.336.000 
 97,972,181 
 98,611,128 
 99,252,847 
 99,897,344 
 
 100,544,625 
 
 101,194,696 
 
 101,847,563 
 
 102,503,232 
 
 103,161,709 
 
 103.823.000 
 104,487,111 
 105,154,048 
 105,823,817 
 106.496,424 
 
 107.171.875 
 107,850,176 
 108,531,333 
 109,215,352 
 109,902,239 
 
 110.592.000 
 111,284,641 
 
 111.980.168 
 112,678,587 
 113,379,904 
 
 114.084.125 
 114,791,256 
 115,501,303 
 116,214,272 
 
 116.930.169 
 
 117.649.000 
 118,370,771 
 119,095,488 
 119,823,157 
 120,553,784 
 
 121.287.375 
 122,023,936 
 
 20.8327 
 
 20.8567 
 
 20.8806 
 
 20.9045 
 
 20.9284 
 
 20.9523 
 
 20.9762 
 
 21.0000 
 
 21.0238 
 
 21.0476 
 
 21.0713 
 
 21.0950 
 
 21.1187 
 
 21.1424 
 
 21.1660 
 
 21.1896 
 
 21.2132 
 
 21.2368 
 
 21.2603 
 
 21.2838 
 
 21.3073 
 
 21.3307 
 
 21.3542 
 
 21.3776 
 
 21.4009 
 
 21.4243 
 
 21.4476 
 
 21.4709 
 
 21.4942 
 
 21.5174 
 
 21.5407 
 
 21.5639 
 
 21.5870 
 
 21.6102 
 
 21.6333 
 
 21.6564 
 
 21.6795 
 
 21.7025 
 
 21.7256 
 
 21.7486 
 
 21.7715 
 
 21.7945 
 
 21.8174 
 
 21.8403 
 
 21.8632 
 
 21.8861 
 
 21.9089 
 
 21.9317 
 
 21.9545 
 
 21.9775 
 
 22.0000 
 
 22.0227 
 
 22.0454 
 
 22.0681 
 
 22.0907 
 
 22.1133 
 
 22.1359 
 
 22.1585 
 
 22.1811 
 
 22.2036 
 
 22.2261 
 
 22.2486 
 
 22.2711 
 
 7.5712 
 
 7.5770 
 
 7.5828 
 
 7.5886 
 
 7.5944 
 
 7.6001 
 
 7.6059 
 
 7.6117 
 
 7.6174 
 
 7.6232 
 
 7.6289 
 
 7.6346 
 
 7.6403 
 
 7.6460 
 
 7.6517 
 
 7.6574 
 
 7.6631 
 
 7.6688 
 
 7.6744 
 
 7.6801 
 
 7.6857 
 
 7.6914 
 
 7.6970 
 
 7.7026 
 
 7.7082 
 
 7.7188 
 
 7.7194 
 
 7.7250 
 
 7.7306 
 
 7.7362 
 
 7.7418 
 
 7.7473 
 
 7.7529 
 
 7.7584 
 
 7.7639 
 
 7.7695 
 
 7.7750 
 
 7.7805 
 
 7.7860 
 
 7.7915 
 
 7.7970 
 
 7.8025 
 
 7.8079 
 
 7.8134 
 
 7.8188 
 
 7.8243 
 
 7.8297 
 
 7.8352 
 
 7.8406 
 
 7.8460 
 
 7.8514 
 
 7.8568 
 
 7.8622 
 
 7.8676 
 
 7.8730 
 
 7.8784 
 
 7.8837 
 
 7.8891 
 
 7.8944 
 
 7.8998 
 
 7.9051 
 
 7.9105 
 
 7.9158 
 
 .002304147 : 
 .002298851 : 
 .002293578 : 
 .002288330 : 
 .002283105 : 
 .002277904 : 
 .002272727 ! 
 .002267574 : 
 .002262443 
 .002257336 
 .002252252 
 .002247191 
 .002242152 
 .002237136 
 .002232143 
 .002227171 
 .002222222 
 .002217295 
 .002212389 
 .002207506 
 .002202643 
 .002197802 
 .002192982 
 .002188184 
 .002183406 
 .002178649 
 .002173913 
 .002169197 
 .002164502 
 .002159827 
 .002155172 
 .002150538 
 .002145923 
 .002141328 
 .002136752 
 .002132196 
 .002127660 
 .002123142 
 .002118644 
 .002114165 
 .002109705 
 .002105263 
 .002100840 
 .002096486 
 .002092050 
 .002087683 
 .002083333 
 .002079002 
 .002074689 
 .002070393 
 .002066116 
 .002061856 
 .002057613 
 .002053388 
 .002049180 
 .002044990 
 .002040816 
 .002036660 
 .002032520 
 .002028398 
 .002024291 
 .002020292 
 .002016129 
 
 1,363.45 : 
 1,366.59 : 
 
 1.369.73 : 
 1,372.88 : 
 1,376.02 : 
 1,379.16 : 
 1,382.30 : 
 1,385.44 
 
 1.388.58 : 
 
 1.391.73 
 1,394.87 
 1,398.01 
 1,401.15 
 1,404.29 
 1,407.43 
 
 1.410.58 
 1,413.72 
 1,416.86 
 1,420.00 
 1,423.14 
 1,426.28 
 
 1.429.42 
 1,432.57 
 1,435.71 
 1,438.85 
 1,441.99 
 1,445.13 
 
 1.448.27 
 
 1.451.42 
 1,454.56 
 1,457.70 
 1,460.84 
 1,463.98 
 1,467.12 
 
 1.470.27 
 1,473.41 
 1,476.55 
 1,479.69 
 1,482.83 
 1,485.97 
 
 1.489.11 
 1,492.26 
 1,495.40 
 1,498.54 
 1,501.68 
 1,504.82 
 
 1.507.96 
 
 1.511.11 
 1,514.25 
 1.517.39 
 1,520.53 
 1,523.67 
 
 1.526.81 
 
 1.529.96 
 1,533.10 
 1,536.24 
 1,539.38 
 1,542.52 
 1,545.66 
 
 1.548.81 
 1,551.95 
 1,555.09 
 1,558.23 
 
 L 47, 934.46 
 L 48, 616.97 
 L 49, 301.05 
 
 149.986.70 
 150,673.93 
 
 151.362.72 
 
 152.053.08 
 
 152.745.02 
 
 153.438.53 
 
 154.133.60 
 
 154.830.25 
 
 155.528.47 
 
 156.228.26 
 
 156.929.62 
 
 157.632.55 
 158,337.06 
 
 159.043.13 
 
 159.750.77 
 160,459.99 
 
 161.170.77 
 
 161.883.13 
 
 162.597.05 
 
 163.312.55 
 
 164.029.62 
 
 164.748.26 
 
 165.468.47 
 166,190.25 
 
 166.913.60 
 
 167.638.53 
 
 168.365.02 
 
 169.093.08 
 
 169.822.72 
 170,553.92 
 
 171.286.70 
 
 172.021.05 
 172,756.97 
 
 173.494.45 
 174,233.51 
 
 174.974.14 
 175,716.35 
 176,460.12 
 
 177.205.46 
 177,952.37 
 
 178.700.86 
 179,450.91 
 
 180.202.54 
 
 180.955.74 
 
 181.710.50 
 182,466.84 
 
 183.224.75 
 
 183.984.23 
 134,745.28 
 185,507.90 
 
 186.272.10 
 
 187.037.86 
 187,805.19 
 
 188.574.10 
 189, 344., 57 
 190,116.62 
 
 190.890.24 
 191 ,665.43 
 192,442.18 
 
 193.220.51 
 
CIRCUMFERENCES, AND AREAS. 
 
 553 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 497 
 
 498 
 
 499 
 
 500 
 
 501 
 
 502 
 508 
 
 504 
 
 505 
 
 506 
 
 507 
 
 508 
 
 509 
 
 510 
 
 511 
 
 512 
 
 513 
 
 514 
 
 515 
 
 516 
 
 517 
 
 518 
 
 519 
 
 520 
 
 521 
 
 522 
 
 523 
 
 524 
 
 525 
 
 526 
 
 527 
 
 528 
 
 529 
 
 530 
 
 531 
 
 532 
 
 533 
 
 534 
 
 535 
 
 536 
 
 537 
 
 538 
 
 539 
 
 540 
 
 541 
 
 542 
 
 543 
 
 544 
 
 545 
 
 546 
 
 547 
 
 548 
 
 549 
 
 550 
 
 551 
 
 552 
 
 553 
 
 554 
 
 555 
 
 556 
 
 557 
 
 558 
 
 559 
 
 247.009 
 
 248.004 
 249,001 
 
 250.000 
 
 251.001 
 
 252.004 
 
 253.009 
 254,016 
 255,025 
 256,036 
 257,049 
 258,064 
 259,081 
 260,100 
 261,121 
 262,144 
 263,169 
 264,196 
 
 265.225 
 266,256 
 267,289 
 268,324 
 269,361 
 270,400 
 271,411 
 272,484 
 273,529 
 274,576 
 275,625 
 276,676 
 277,729 
 278,784 
 279,841 
 280,900 
 281,961 
 
 283.024 
 284,089 
 285,156 
 
 286.225 
 287,296 
 288,369 
 289,444 
 290,521 
 
 291.600 
 292,681 
 293,764 
 294,849 
 295,936 
 
 297.025 
 298,116 
 299,209 
 300,304 
 301,401 
 
 ' 302,500 
 
 303.601 
 304,704 
 305,809 
 306,916 
 
 308.025 
 309,136 
 310,249 
 311,364 
 312,481 
 
 122,763,473 
 
 123,505,992 
 
 124,251,499 
 
 125,000,000 
 
 125,751,501 
 
 126,506,008 
 
 127,263,527 
 
 128,024,064 
 
 128,787,625 
 
 129,554,216 
 
 130,323,843 
 
 131,096,512 
 
 131,872,229 
 
 132.651.000 
 
 133.432.831 
 134,217,728 
 135,005,697 
 135,796,744 
 136,590,875 
 137,388,096 
 138,188,413 
 
 138.991.832 
 139,798,359 
 
 140.608.000 
 141,420,761 
 142,236,648 
 143,055,667 
 143,877,824 
 144,703,125 
 145,531,576 
 
 146.363.183 
 147,197,952 
 148,035,889 
 
 148.877.001 
 149,721,291 
 150,568,768 
 151,419,437 
 152,273,304 
 153,130,375 
 153,990,656 
 154,854,153 
 155,720,872 
 156,590,819 
 
 157.464.000 
 158,340,421 
 159,220,088 
 160,103,007 
 
 160.989.184 
 161,878,625 
 162,771,336 
 163,667,323 
 164,566,592 
 165,469,149 
 
 166.375.000 
 167,284,151 
 168,196,608 
 169,112,377 
 170,031,464 
 170,953,875 
 171,879,616 
 172,808,693 
 173,741.112 
 174,676,879 
 
 22.2935 
 
 22.3159 
 
 22.3383 
 
 22.3607 
 
 22.3830 
 
 22.4054 
 
 22.4277 
 
 22.4499 
 
 22.4722 
 
 22.4944 
 
 22.5167 
 
 22.5389 
 
 22.5610 
 
 22.5832 
 
 22.6053 
 
 22.6274 
 
 22.6495 
 
 22.6716 
 
 22.6936 
 
 22.7156 
 
 22.7376 
 
 22.7596 
 
 22.7816 
 
 22.8035 
 
 22.8254 
 
 22.8473 
 
 22.8692 
 
 22.8910 
 
 22.9129 
 
 22.9347 
 
 22.9565 
 
 22.9783 
 
 23.0000 
 
 23.0217 
 
 23.0434 
 
 23.0651 
 
 23.0868 
 
 23.1084 
 
 23.1301 
 
 23.1517 
 
 23.1733 
 
 23.1948 
 
 23.2164 
 
 23.2379 
 
 23.2594 
 
 23.2809 
 
 23.3024 
 
 23.3238 
 
 23.3452 
 
 23.3666 
 
 23.3880 
 
 23.4094 
 
 23.4307 
 
 23.4521 
 
 23.4734 
 
 23.4947 
 
 23.5160 
 
 23.5372 
 
 23.5584 
 
 23.5797 
 
 23.6008 
 
 23.6220 
 
 23.6432 
 
 7.9211 
 
 7.9264 
 
 7.9317 
 
 7.9370 
 
 7.9423 
 
 7.9476 
 
 7.9528 
 
 7.9581 
 
 7.9634 
 
 7.9686 
 
 7.9739 
 
 7.9791 
 
 7.9843 
 
 7.9895 
 
 7.9948 
 
 8.0000 
 
 8.0052 
 
 8.0104 
 
 8.0156 
 
 8.0208 
 
 8.0260 
 
 8.0311 
 
 8.0363 
 
 8.0415 
 
 8.0466 
 
 8.0517 
 
 8.0569 
 
 8.0620 
 
 8.0671 
 
 8.0723 
 
 8.0774 
 
 8.0825 
 
 8.0876 
 
 8.0927 
 
 8.0978 
 
 8.1028 
 
 8.1079 
 
 8.1130 
 
 8.1180 
 
 8.1231 
 
 8.1281 
 
 8.1332 
 
 8.1382 
 
 8.1433 
 
 8.1483 
 
 8.1533 
 
 8.1583 
 
 8.1633 
 
 8.1683 
 
 8.1733 
 
 8.1783 
 
 8.1833 
 
 8.1882 
 
 8.1932 
 
 8.1982 
 
 8.2031 
 
 8.2081 
 
 8.2130 
 
 8.2180 
 
 8.2229 
 
 8.2278 
 
 8.2327 
 
 8.2377 
 
 .002012072 • : 
 .002008032 : 
 .002004008 : 
 
 .002000000 : 
 
 .001996008 ; 
 .001992032 
 .001988072 : 
 .001984127 : 
 .001980198 
 .001976285 
 .001972387 
 .001968504 
 .001964637 
 .001960785 
 .00L956947 
 .001953125 
 .001949318 
 .001945525 
 .001941748 
 .001937984 
 .001934236 
 .001930502 
 .001926782 
 .001923077 
 .001919386 
 .001915709 
 .001912046 
 .001908397 
 .001904762 
 .001901141 
 .001897533 
 .001893939 
 .001890359 
 .001886792 
 .001883239 
 .001879699 
 .001876173 
 .001872659 
 .001869159 
 .001865672 
 .001862197 
 .001858736 
 .001855288 
 .001851852 
 .001848429 
 .001845018 
 .001841621 
 .001838235 
 .001834862 
 .001831502 
 .001828154 
 .001824818 
 .001821494 
 .041818182 
 .001814882 
 .041811594 
 .001808318 
 .001805054 
 .001801802 
 .001798561 
 .001795332 
 .001792115 
 .001788909 
 
 L, 561.37 1 
 L, 564.51 1 
 L, 567.65 1 
 1,570.80 : 
 1,573.94 : 
 1,577.08 : 
 1,580.22 : 
 1,583.36 : 
 
 1.586.50 ! 
 1,589.65 : 
 1,592.79 ! 
 1,595.93 : 
 1,599.07 : 
 1,602.21 
 1,605.35 
 
 1.608.50 
 1,611.64 
 1,614.78 
 1,617.92 
 1,621.06 
 1,624.20 
 
 1.627.34 
 1,630.49 
 1,633.63 
 1,636.77 
 1,639.91 
 1,643.05 
 
 1.646.19 
 
 1.649.34 
 1,652.48 
 1,655.62 
 1,658.76 
 1,661.90 
 
 1.665.04 
 
 1.668.19 
 1,671.33 
 1,674.47 
 1,677.61 
 1,680.75 
 1,683.89 
 
 1.687.04 
 1,690.18 
 1,693.32 
 1,696.46 
 1,699.60 
 1,702.74 
 
 1.705.88 
 1,709.03 
 1,712.17 
 1,715.31 
 1,718.45 
 1,721.59 
 
 1.724.73 
 
 1.727.88 
 1,731.02 
 1,734.16 
 1,737.30 
 1,740.44 
 1,743.58 
 
 1.746.73 
 1,749.87 
 1,753.01 
 1,756.15 
 
 L94, 000.41 
 L94, 781.89 
 L95, 564.93 
 L96, 349.54 
 L97, 135.72 
 L97, 923.48 
 L98, 712.80 
 199,503.70 
 
 200.296.17 
 201,090.20 
 
 201.885.81 
 202,682.99 
 
 203.481.74 
 204,282.06 
 205,083.95 
 
 205.887.42 
 206,692.45 
 207,499.05 
 208,307.23 
 
 209.116.97 
 209,928.29 
 
 210.741.18 
 211,555.63 
 212,371.66 
 213,189.26 
 
 214.008.43 
 214,829.17 
 215,651.49 
 216,475.37 
 
 217.300.82 
 
 218.127.85 
 
 218.956.44 
 219,786.61 
 220,618.34 
 
 221.451.65 
 222,286.53 
 
 223.122.98 
 
 223.961.00 
 
 224.800.59 
 
 225.641.75 
 226,484.48 
 
 227.328.79 
 
 228.174.66 
 229,022.10 
 229,871.12 
 
 230.721.71 
 
 231.573.86 
 
 232.427.59 
 233,282.89 
 
 234.139.76 
 
 234.998.20 
 
 235.858.21 
 
 236.719.79 
 237,582.94 
 
 238.447.67 
 239,313.96 
 
 240.181.83 
 
 241.051.26 
 
 241.922.27 
 242,794.85 
 
 243.668.99 
 
 244.544.71 
 
 245.422.00 
 
554 SQUARES, CUBES, SQUARE AND CUBE ROOTS, 
 
 No . 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 560 
 
 561 
 
 562 
 
 563 
 
 564 
 
 565 
 
 566 
 
 567 
 
 568 
 
 569 
 
 570 
 
 571 
 
 572 
 
 573 
 
 574 
 
 575 
 
 576 
 
 577 
 
 578 
 
 579 
 
 580 
 
 581 
 
 582 
 
 583 
 
 584 
 
 585 
 
 586 
 
 587 
 
 588 
 
 589 
 
 590 
 
 591 
 
 592 
 
 593 
 
 594 
 
 595 
 
 596 
 
 597 
 
 598 
 
 599 
 
 600 
 601 
 602 
 
 603 
 
 604 
 
 605 
 
 606 
 
 607 
 
 608 
 
 609 
 
 610 
 611 
 612 
 
 613 
 
 614 
 
 615 
 
 616 
 
 617 
 
 618 
 
 619 
 
 620 
 621 
 622 
 
 313,600 
 
 314,721 
 
 315,844 
 
 316,969 
 
 318,096 
 
 319.225 
 320,356 
 321,489 
 
 322.624 
 323,761 
 324,900 
 326,041 
 327,184 
 328,329 
 329,476 
 
 330.625 
 331,776 
 332,929 
 334,084 
 335,241 
 336,400 
 337,561 
 338,724 
 339,889 
 341,056 
 
 342.225 
 343,396 
 344,569 
 345,744 
 346,921 
 
 348.100 
 349,281 
 350,464 
 351,649 
 352,836 
 
 354.025 
 355,216 
 356,409 
 357,604 
 358,801 
 360,000 
 361,201 
 362,404 
 363,609 
 364,816 
 
 366.025 
 367,236 
 368,449 
 369,664 
 370,881 
 
 372.100 
 373,321 
 374,544 
 375.769 
 376,996 
 
 378.225 
 379,456 
 380,689 
 381,924 
 383,161 
 384,400 
 385,641 
 386-884 
 
 175.616.000 
 176,558,481 
 177,504,328 
 178,453,547 
 179,406,144 
 
 180.362.125 
 181,321,496 
 182,284,263 
 183,250,432 
 184,220,009 
 
 185.193.000 
 186,169,411 
 187,149,248 
 188,132,517 
 189,119,224 
 190,109,375 
 191,102,976 
 192,100,033 
 193,100,552 
 194,104,539 
 
 195.112.000 
 196,122,941 
 197,137,368 
 198,155,287 
 199,176,704 
 200,201,625 
 201,230,056 
 202,262,003 
 203,297,472 
 204,336,469 
 
 205.379.000 
 206,425,071 
 207,474,688 
 208,527,857 
 209,584,584 
 210,644,875 
 211,708,736 
 212,776,173 
 213,847,192 
 214,921,799 
 216,000,000 
 217,081,801 
 218,167,208 
 219,256,227 
 220,348,864 
 
 221.445.125 
 222,545,016 
 
 223.648.543 
 224,755,712 
 225,866,529 
 
 226.981.000 
 228,099,131 
 229,220,928 
 230,346,397 
 
 231.475.544 
 232,608,375 
 233,744,896 
 234,885,113 
 236,029,032 
 237,176,659 
 
 238.328.000 
 239,483,061 
 240,641,848 
 
 23.6643 
 
 23.6854 
 
 23.7065 
 
 23.7276 
 
 23.7487 
 
 23.7697 
 
 23.7908 
 
 23.8118 
 
 23.8328 
 
 23.8537 
 
 23.8747 
 
 23.8956 
 
 23.9165 
 
 23.^374 
 
 23.9583 
 
 23.9792 
 
 24.0000 
 
 24.0208 
 
 24.0416 
 
 24.0624 
 
 24.0832 
 
 24.1039 
 
 24.1247 
 
 24.1454 
 
 24.1661 
 
 24.1868 
 
 24.2074 
 
 24.2281 
 
 24.2487 
 
 24.2693 
 
 24.2899 
 
 24.3105 
 
 24.3311 
 
 24.3516 
 
 24.3721 
 
 24.3926 
 
 24.4131 
 
 24.4336 
 
 24.4540 
 
 24.4745 
 
 24.4949 
 
 24.5153 
 
 24.5357 
 
 24.5561 
 
 24.5764 
 
 24.5968 
 
 24.6171 
 
 24.6374 
 
 24.6577 
 
 24.6779 
 
 24.6982 
 
 24.7184 
 
 24.7386 
 
 24.7588 
 
 24.7790 
 
 24.7992 
 
 24.8193 
 
 24.8395 
 
 24.8596 
 
 24.8797 
 
 24.8998 
 
 24.9199 
 
 24.9399 
 
 8.2426 
 
 8.2475 
 
 8.2524 
 
 8.2573 
 
 8.2621 
 
 8.2670 
 
 8.2719 
 
 8.2768 
 
 8.2816 
 
 8.2865 
 
 8.2913 
 
 8.2962 
 
 8.3010 
 
 8.3059 
 
 8.3107 
 
 8.3155 
 
 8.3203 
 
 8.3251 
 
 8.3300 
 
 8.3348 
 
 8.3396 
 
 8.3443 
 
 8.3491 
 
 8.3539 
 
 8.3587 
 
 8.3634 
 
 8.3682 
 
 8.3730 
 
 8.3777 
 
 8.3825 
 
 8.3872 
 
 8.3919 
 
 8.3967 
 
 8.4014 
 
 8.4061 
 
 8.4108 
 
 8.4155 
 
 8.4202 
 
 8.4249 
 
 8.4296 
 
 8.4343 
 
 8.4390 
 
 8.4437 
 
 8.4484 
 
 8.4530 
 
 8.4577 
 
 8.4623 
 
 8.4670 
 
 8.4716 
 
 8.4763 
 
 8.4809 
 
 8.4856 
 
 8.4902 
 
 8.4948 
 
 8.4994 
 
 8.5040 
 
 8.5086 
 
 8.5132 
 
 8.5178 
 
 8.5224 
 
 8.5270 
 
 8.5316 
 
 8.5362 
 
 .001785714 
 
 .001782531 
 
 .001779359 
 
 .001776199 
 
 .001773050 
 
 .001769912 
 
 .001766784 
 
 .001763668 
 
 .001760563 
 
 .001757469 
 
 .001754386 
 
 .001751313 
 
 .001748252 
 
 .001745201 
 
 .001742164 
 
 .001739130 
 
 .001736111 
 
 .001733102 
 
 .001730104 
 
 .001727116 
 
 .001724138 
 
 .001721170 
 
 .001718213 
 
 .001715266 
 
 .001712329 
 
 .001709402 
 
 .001706485 
 
 .001703578 
 
 .001700680 
 
 .001697793 
 
 .001694915 
 
 .001692047 
 
 .001689189 
 
 .001686341 
 
 .001683502 
 
 .001680672 
 
 .001677852 
 
 .001675042 
 
 .001672241 
 
 .001669449 
 
 .001666667 
 
 .001663894 
 
 .001661130 
 
 .001658375 
 
 .001655629 
 
 .001652893 
 
 .001650165 
 
 .001647446 
 
 .001644737 
 
 .001642036 
 
 .001639344 
 
 .001636661 
 
 .001633987 
 
 .001631321 
 
 .001628664 
 
 .001626016 
 
 .001623377 
 
 .001620746 
 
 .001618123 
 
 .001615509 
 
 .001612903 
 
 .001610306 
 
 .001607717 
 
 1,759.29 
 
 1,762.43 
 
 1,765.58 
 
 1,768.72 
 
 1,771.86 
 
 1,775.00 
 
 1,778.14 
 
 1,781.28 
 
 1.784.42 
 1,787.57 
 1,790.71 
 1,793.85 
 1,796.99 
 1,800.13 
 
 1.803.27 
 
 1.806.42 
 1,809.56 
 1,812.70 
 1,815.84 
 1,818.98 
 1,822.12 
 
 1.825.27 
 1,828.41 
 1,831.55 
 1,834.69 
 1.837.83 
 1,840.97 
 
 1.844.11 
 1,847.26 
 1,850.40 
 1,853.54 
 1,856.68 
 1,859.82 
 
 1.862.96 
 
 1.866.11 
 1,869.25 
 1,872.39 
 1,875.53 
 1,878.67 
 1,881.81 
 
 1.884.96 
 1,888.10 
 1,891.24 
 1,894.38 
 1,897.52 
 1,900.66 
 1,903.81 
 1,906.95 
 1,910.09 
 1,913.23 
 1,916.37 
 1,919.51 
 
 1.922.65 
 1,925.80 
 1,928.94 
 1,932.08 
 1,935.22 
 1,938.36 
 1,941.50 
 
 1.944.65 
 1,947.79 
 1,950.93 
 1,954.07 
 
 246.300.86 
 
 247.181.30 
 
 248.063.30 
 
 248.946.87 
 
 249.832.01 
 
 250.718.73 
 
 251.607.01 
 
 252.496.87 
 
 253.388.30 
 254,281.29 
 255,175.86 
 256,072.00 
 256,969.71 
 
 257.868.99 
 
 258.769.85 
 
 259.672.27 
 
 260.576.26 
 
 261.481.83 
 
 262.388.96 
 263,297.67 
 
 264.207.94 
 265,119.79 
 
 266.033.21 
 266,948.20 
 267,864.76 
 268,782.89 
 
 269.702.59 
 
 270.623.86 
 271,546.70 
 272,471.12 
 273,397.10 
 274,324.66 
 275,253.78 
 
 276.184.48 
 277,116.75 
 278,050.58 
 
 278.985.99 
 
 279.922.97 
 280,861.52 
 
 281.801.65 
 282,743.34 
 
 283.686.60 
 284,631.44 
 
 285.577.84 
 286,525.82 
 287,475.36 
 
 288.426.48 
 
 289.379.17 
 
 290.333.43 
 
 291.289.26 
 
 292.246.66 
 293,205.63 
 
 294.166.17 
 
 295.128.28 
 
 296.091.97 
 
 297.057.22 
 
 298.024.05 
 
 298.992.44 
 299,962.41 
 
 300.933.95 
 
 301.907.05 
 
 302.881.73 
 
 303.857.98 
 
CIRCUMFERENCES , AND AREAS. 
 
 555 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu . Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 623 
 
 624 
 
 625 
 
 626 
 
 627 
 
 628 
 
 629 
 
 630 
 
 631 
 
 632 
 
 633 
 
 634 
 
 635 
 
 636 
 
 637 
 
 638 
 
 639 
 
 640 
 
 641 
 
 642 
 
 643 
 
 644 
 
 645 
 
 646 
 
 647 
 
 648 
 
 649 
 
 650 
 
 651 
 
 652 
 
 653 
 
 654 
 
 655 
 
 656 
 
 657 
 
 658 
 
 659 
 
 660 
 661 
 662 
 
 663 
 
 664 
 
 665 
 
 666 
 
 667 
 
 668 
 
 669 
 
 670 
 
 671 
 
 672 
 
 673 
 
 674 
 
 675 
 
 676 
 
 677 
 
 678 
 
 679 
 
 680 
 681 
 682 
 
 683 
 
 684 
 
 685 
 
 388.129 
 389,376 
 390,625 
 391,876 
 
 393.129 
 394,384 
 395,641 
 396,900 
 398,161 
 399,424 
 400,689 
 401,956 
 403,225 
 404,496 
 405,769 
 407,044 
 408,321 
 
 409.600 
 410,881 
 412,164 
 413,449 
 414,736 
 416,125 
 417,316 
 418,609 
 419,904 
 421,201 
 422,500 
 423,801 
 425,104 
 426,409 
 427,716 
 429,025 
 430,336 
 431,639 
 432,964 
 434,281 
 
 435.600 
 436,921 
 438,244 
 439,569 
 440,896 
 442,225 
 443,556 
 
 444.899 
 
 446.224 
 447,561 
 
 448.900 
 450,241 
 451,584 
 452,929 
 454,276 
 455,625 
 456,976 
 458,329 
 459,684 
 461,041 
 462,400 
 463,761 
 465,124 
 466,489 
 467,856 
 
 469.225 
 
 241,804,367 
 
 242.970.624 
 
 244.140.625 
 245,314,376 
 246,491,883 
 247,673,152 
 248,858,189 
 
 250.047.000 
 251,239,591 
 252,435,968 
 253,636,137 
 254,840,104 
 256,047,875 
 257,259,456 
 258,474,853 
 259,694,072 
 260,917,119 
 
 262.144.000 
 263,374,721 
 264,609,288 
 265,847,707 
 267,089,984 
 268,336,125 
 269,585,136 
 270,840,023 
 272,097,792 
 273,359,449 
 
 274.625.000 
 275,894,451 
 277,167,808 
 278,445,077 
 279,726,264 
 281,011,375 
 282,300,416 
 283,593,393 
 284,890,312 
 286,191,179 
 
 287.496.000 
 288,804,781 
 290,117,528 
 291,434,247 
 292,754,944 
 294,079,625 
 295,408,296 
 296,740,963 
 298,077,632 
 299,418,309 
 
 300.763.000 
 302,111,711 
 303,464,448 
 304,821,217 
 306,182,024 
 307,546,875 
 308,915,776 
 310,288,733 
 311,665,752 
 313,046,839 
 
 314.432.000 
 315,821,241 
 317,214,568 
 318,611,987 
 320,013,504 
 321,419,125 
 
 24.9600 
 
 24.9800 
 
 25.0000 
 
 25.0200 
 
 25.0400 
 
 25.0599 
 
 25.0799 
 
 25.0998 
 
 25.1197 
 
 25.1396 
 
 25.1595 
 
 25.1794 
 
 25.1992 
 
 25.2190 
 
 25.2389 
 
 25.2587 
 
 25.2784 
 
 25.2982 
 
 25.3180 
 
 25.3377 
 
 25.3574 
 
 25.3772 
 
 25.3969 
 
 25.4165 
 
 25.4362 
 
 25.4558 
 
 25.4755 
 
 25.4951 
 
 25.5147 
 
 25.5343 
 
 25.5539 
 
 25.5734 
 
 25.5930 
 
 25.6125 
 
 25.6320 
 
 25.6515 
 
 25.6710 
 
 25.6905 
 
 25.7099 
 
 25.7294 
 
 25.7488 
 
 25.7682 
 
 25.7876 
 
 25.8070 
 
 25.8263 
 
 25.8457 
 
 25.8650 
 
 25.8844 
 
 25.9037 
 
 25.9230 
 
 25.9422 
 
 25.9615 
 
 25.9808 
 
 26.0000 
 
 26.0192 
 
 26.0384 
 
 26.0576 
 
 26.0768 
 
 26.0960 
 
 26.1151 
 
 26.1343 
 
 26.1534 
 
 26.1725 
 
 8.5408 
 
 8.5453 
 
 8.5499 
 
 8.5544 
 
 8.5589 
 
 8.5635 
 
 8.5681 
 
 8.5726 
 
 8.5772 
 
 8.5817 
 
 8.5862 
 
 8.5907 
 
 8.5952 
 
 8.5997 
 
 8.6043 
 
 8.6088 
 
 8.6132 
 
 8.6177 
 
 8.6222 
 
 8.6267 
 
 8.6312 
 
 8.6357 
 
 8.6401 
 
 8.6446 
 
 8.6490 
 
 8.6535 
 
 8.6579 
 
 8.6624 
 
 8.6668 
 
 8.6713 
 
 8.6757 
 
 8.6801 
 
 8.6845 
 
 8.6890 
 
 8.6934 
 
 8.6978 
 
 8.7022 
 
 8.7066 
 
 8.7110 
 
 8.7154 
 
 8.7198 
 
 8.7241 
 
 8.7285 
 
 8.7329 
 
 8.7373 
 
 8.7416 
 
 8.7460 
 
 8.7503 
 
 8.7547 
 
 8.7590 
 
 8.7634 
 
 8.7677 
 
 8.7721 
 
 8.7764 
 
 8.7807 
 
 8.7850 
 
 8.7893 
 
 8.7937 
 
 8.7980 
 
 8.8023 
 
 8.8066 
 
 8.8109 
 
 8.8152 
 
 .001605136 
 
 .001602564 
 
 .001600000 
 
 .001597444 
 
 .001594896 
 
 .001592357 
 
 .001589825 
 
 .001587302 
 
 .001584786 
 
 .001582278 
 
 .001579779 
 
 .001577287 
 
 .001574803 
 
 .001572327 
 
 .001569859 
 
 .001567398 
 
 .001564945 
 
 .001562500 
 
 .001560062 
 
 .001557632 
 
 .001555210 
 
 .001552795 
 
 .001550388 
 
 .001547988 
 
 .001545595 
 
 .001543210 
 
 .001540832 
 
 .001538462 
 
 .001536098 
 
 .001533742 
 
 .001531394 
 
 .001529052 
 
 .001526718 
 
 .001524390 
 
 .001522070 
 
 .001519751 
 
 .001517451 
 
 .001515152 
 
 .001512859 
 
 .001510574 
 
 .001508296 
 
 .001506024 
 
 .001503759 
 
 .001501502 
 
 .001499250 
 
 .001497006 
 
 .001494768 
 
 .001492537 
 
 .001490313 
 
 .001488095 
 
 .001485884 
 
 .001483680 
 
 .001481481 
 
 .001479290 
 
 .001477105 
 
 .001474926 
 
 .001472754 
 
 .001470588 
 
 .001468429 
 
 .001466276 
 
 .001464129 
 
 .001461988 
 
 .001459854 
 
 1,957.21 
 
 1.960.35 
 1,963.50 
 1,966.64 
 1,969.78 
 1,972.92 
 1,976.06 
 1,979.20 
 
 1.982.35 
 1,985.49 
 1,988.63 
 1,991.77 
 1,994.91 
 1,998.05 
 
 2.001.19 
 2,004.34 
 2,007.48 
 2,010.62 
 2,013.76 
 2,016.90 
 
 2.020.04 
 
 2.023.19 
 2,026.33 
 2,029.47 
 2,032.61 
 2,035.75 
 2,038.89 
 
 2.042.04 
 2,045.18 
 2,048.32 
 2,051.46 
 2,054.60 
 2,057.74 
 2,060.88 
 2,064.03 
 2,067.17 
 2,070.31 
 2,073.45 
 2,076.59 
 
 2.079.73 
 2,082.88 
 2,086.02 
 2,089.16 
 2,092.30 
 2,095.44 
 
 2.098.58 
 
 2.101.73 
 2,104.87 
 2,108.01 
 2,111.15 
 2,114.29 
 2,117.43 
 
 2.120.58 
 2,123.72 
 2,126.86 
 2,130.00 
 2,133.14 
 2,136.28 
 2,139.42 
 2,142.57 
 2,145.71 
 2,148.85 
 2,151.99 
 
 304,835.80 
 
 305,815.20 
 
 306.796.16 
 
 307.778.69 
 308,762.79 
 309,748.47 
 
 310.735.71 
 
 311.724.53 
 312,714.92 
 
 313.706.88 
 314,700.40 
 315,695.50 
 
 316.692.17 
 317,690.42 
 
 318.690.23 
 319,691.61 
 320,694.56 
 
 321.699.09 
 
 322.705.18 
 
 323.712.85 
 
 324.722.09 
 
 325.732.89 
 326,745.27 
 327,759.22 
 
 328.774.74 
 329,791.83 
 
 330.810.49 
 
 331.830.72 
 
 332.852.53 
 
 333.875.90 
 
 334.900.85 
 335,927.36 
 336,955.45 
 
 337.985.10 
 339,016.33 
 340,049.13 
 
 341.083.50 
 
 342.119.44 
 343,156.95 
 
 344.196.03 
 
 345.236.69 
 
 346.278.91 
 
 347.322.70 
 348,368.07 
 349,415.00 
 
 350.463.51 
 
 351.513.59 
 
 352.565.24 
 
 353.618.45 
 
 354.673.24 
 
 355.729.60 
 
 356.787.54 
 
 357.847.04 
 
 358.908.11 
 
 359.970.75 
 361,034.97 
 
 362.100.75 
 
 363.168.11 
 
 364.237.04 
 
 365.307.54 
 
 366.379.60 
 
 367.453.24 
 
 368.528.45 
 
556 SQUARES, CUBES, SQUARE AND CUBE ROOTS, 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 686 
 
 687 
 
 688 
 
 689 
 
 690 
 
 691 
 
 692 
 
 693 
 
 694 
 
 695 
 
 696 
 
 697 
 
 698 
 
 699 
 
 700 
 
 701 
 
 702 
 
 703 
 
 704 
 
 705 
 
 706 
 
 707 
 
 708 
 
 709 
 
 710 
 
 711 
 
 712 
 
 713 
 
 714 
 
 715 
 
 716 
 
 717 
 
 718 
 
 719 
 
 720 
 
 721 
 
 722 
 
 723 
 
 724 
 
 725 
 
 726 
 
 727 
 
 728 
 
 729 
 
 730 
 
 731 
 
 732 
 
 733 
 
 734 
 
 735 
 
 736 
 
 737 
 
 738 
 
 739 
 
 740 
 
 741 
 
 742 
 
 743 
 
 744 
 
 745 
 
 746 
 
 747 
 
 748 
 
 470,596 
 
 471,969 
 
 473,344 
 
 474,721 
 
 476.100 
 477,481 
 478,864 
 480,249 
 481,636 
 
 483.025 
 484, 4 l 6 
 485,809 
 487,204 
 488,601 
 490,000 
 491,401 
 492,804 
 494,209 
 495,616 
 
 497.025 
 498,436 
 499,849 
 501,264 
 502,681 
 
 504.100 
 505,521 
 506,944 
 508,369 
 509,796 
 
 511.225 
 512,656 
 514,089 
 515,524 
 516,961 
 518,400 
 519,841 
 521,284 
 522,729 
 524,176 
 525,625 
 527.076 
 528,529 
 529,984 
 531,441 
 532,900 
 534,361 
 535,824 
 537,289 
 538,756 
 
 540.225 
 541,696 
 543,169 
 544,644 
 546,121 
 547,600 
 549,801 
 550,564 
 552,049 
 553,536 
 555,025 
 556,516 
 558,009 
 559,504 
 
 322,828,856 
 
 324,242,703 
 
 325,660,672 
 
 327,082,769 
 
 328.509.000 
 329,939,371 
 331,373,888 
 332,812,557 
 334,255,384 
 335,702,375 
 337,153,536 
 338,608,873 
 340,068,392 
 341,532,099 
 343,000,000' 
 344,472,101 
 345,948,408 
 347,428,927 
 348,913,664 
 350,402,625 
 351,895,816 
 353,393,243 
 354,894,912 
 356,400,829 
 
 357.911.000 
 359,425,431 
 360,944,128 
 362,467,097 
 363,994,344 
 365,525,875 
 367,061,696 
 368,601,813 
 370,146,232 
 371,694,959 
 
 373.248.000 
 374,805,361 
 376,367,048 
 377,933,067 
 379,503,424 
 381,078,125 
 382,657,176 
 384,240,583 
 385,828,352 
 387,420,489 
 
 389.017.000 
 390,617,891 
 392,223,168 
 393,832,837 
 395,446,904 
 397,065,375 
 398,688,256 
 400,315,553 
 401,947,272 
 403,583,419 
 
 405.224.000 
 406,869,021 
 408,518,488 
 410,172,407 
 411,830,784 
 413,493,625 
 415,160,936 
 416,832,723 
 418,508,992 
 
 26.1916 
 
 26.2107 
 
 26.2298 
 
 26.2488 
 
 26.2679 
 
 26.2869 
 
 26.3059 
 
 26.3249 
 
 26.3439 
 
 26.3629 
 
 26.3818 
 
 26.4008 
 
 26.4197 
 
 26.4386 
 
 26.4575 
 
 26.4764 
 
 26.4953 
 
 26.5141 
 
 26.5330 
 
 26.5518 
 
 26.5707 
 
 26.5895 
 
 26.6083 
 
 26.6271 
 
 26.6458 
 
 26.6646 
 
 26.6833 
 
 26.7021 
 
 26.7208 
 
 26.7395 
 
 26.7582 
 
 .26.7769 
 
 26.7955 
 
 26.8142 
 
 26.8328 
 
 26.8514 
 
 26.8701 
 
 26.8887 
 
 26.9072 
 
 26.9258 
 
 26.9444 
 
 26.9629 
 
 26.9815 
 
 27.0000 
 
 27.0185 
 
 27.0370 
 
 27.0555 
 
 27.0740 
 
 27.0924 
 
 27.1109 
 
 27.1293 
 
 27.1477 
 
 27.1662 
 
 27.1846 
 
 27.2029 
 
 27.2213 
 
 27.2397 
 
 27.2580 
 
 27.2764 
 
 27.2947 
 
 27.3130 
 
 27.3313 
 
 27.3496 
 
 8.8194 
 
 8.8237 
 
 8.8280 
 
 8.8323 
 
 8.8366 
 
 8.8408 
 
 8.8451 
 
 8.8493 
 
 8.8536 
 
 8.8578 
 
 8.8621 
 
 8.8663 
 
 8.8706 
 
 8.8748 
 
 8.8790 
 
 8.8833 
 
 8.8875 
 
 8.8917 
 
 8.8959 
 
 8.9001 
 
 8.9043 
 
 8.9085 
 
 8.9127 
 
 8.9169 
 
 8.9211 
 
 8.9253 
 
 8.9295 
 
 8.9337 
 
 8.9378 
 
 8.9420 
 
 8.9462 
 
 8.9503 
 
 8.9545 
 
 8.9587 
 
 8.9628 
 
 8.9670 
 
 8.9711 
 
 8.9752 
 
 8.9794 
 
 8.9835 
 
 8.9876 
 
 8.9918 
 
 8.9959 
 
 9.0000 
 
 9.0041 
 
 9.0082 
 
 9.0123 
 
 9.0164 
 
 9.0205 
 
 9.0246 
 
 9.0287 
 
 9.0328 
 
 9.0369 
 
 9.0410 
 
 9.0450 
 
 9.0491 
 
 9.0532 
 
 9.0572 
 
 9.0613 
 
 9.0654 
 
 9.0694 
 
 9.0735 
 
 9.0775 
 
 .001457726 ! 
 
 .001455604 ! 
 
 .001453488 ! 
 
 .001451379 ! 
 
 .001449275 ! 
 
 .001447178 : 
 
 .001445087 : 
 
 .001443001 : 
 
 .001440922 : 
 
 .001438849 : 
 
 .001436782 
 
 .001434720 
 
 .001432665 
 
 .001430615 
 
 .001428571 
 
 .001426534 
 
 .001424501 
 
 .001422475 
 
 .001420455 
 
 .001418440 
 
 .001416431 
 
 .001414427 
 
 .001412429 
 
 .001410437 
 
 .001408451 
 
 .001406470 
 
 .001404494 
 
 .001402525 
 
 .001400560 
 
 .001398601 
 
 .001396648 
 
 .001394700 
 
 .001392758 
 
 .001390821 
 
 .001388889 
 
 .001386963 
 
 .001385042 
 
 .001383126 
 
 .001381215 
 
 .001379310 
 
 .001377410 
 
 .001375516 
 
 .001373626 
 
 .001371742 
 
 .001369863 
 
 .001367989 
 
 .001366120 
 
 .001364256 
 
 .001362398 
 
 .001360544 
 
 .001358696 
 
 .001356852 
 
 .001355014 
 
 .001353180 
 
 .001351351 
 
 .001349528 
 
 .001347709 
 
 .001345895 
 
 .001344086 
 
 .001342282 
 
 .001340483 
 
 .001338688 
 
 .001336898 
 
 2,155.13 ! 
 
 2.158.27 : 
 2,161.42 I 
 2,164.56 : 
 2,167.70 : 
 2,170.84 : 
 2,173.98 : 
 2,177.12 : 
 
 2.180.27 : 
 2,183.41 
 2,186.55 
 2,189.69 
 2,192.83 
 2,195.97 
 
 2.199.11 
 2,202.26 
 2,205.40 
 2,208.54 
 2,211.68 
 2,214.82 
 
 2.217.96 
 
 2.221.11 
 2,224.25 
 2,227.39 
 2,230.53 
 2,233.67 
 
 2.236.81 
 
 2.239.96 
 2,243.10 
 2,246.24 
 2,249.38 
 2,252.52 
 2,255.66 
 
 2.258.81 
 2,261.95 
 2,265.09 
 2,268.23 
 2,271.37 
 2,274.51 
 
 2.277.65 
 2,280.80 
 2,283.94 
 2,287.08 
 2,290.22 
 2,293.36 
 
 2.296.50 
 
 2.299.65 
 2,302.79 
 2,305.93 
 2,309.07 
 2,312.21 
 2,315.35 
 
 2.318.50 
 2,321.64 
 2,324.78 
 2,327.92 
 2,331.06 
 2,334.20 
 2,337.34 
 2,340.49 
 2,343.63 
 2,346.77 
 2,349.91 
 
 369,605.23 
 
 370.683.59 
 
 371.763.51 
 372,845.00 
 
 373.928.07 
 
 375.012.70 
 376,098.91 
 377,186.68 
 378,276.03 
 379,366.95 
 
 380.459.44 
 381,553.50 
 382,649.13 
 383,746.33 
 384,845.10 
 
 385.945.44 
 387,047.36 
 
 388.150.84 
 
 389.255.90 
 
 390.362.52 
 
 391.470.72 
 
 392.580.49 
 393,691.82 
 
 394.804.73 
 
 395.919.21 
 
 397.035.26 
 398,152.89 
 
 399.272.08 
 
 400.392.84 
 
 401.515.18 
 
 402.639.08 
 403,764.56 
 
 404.891.60 
 
 406.020.22 
 407,150.41 
 408,282.17 
 
 409.415.50 
 
 410.550.40 
 411,686.87 
 
 412.824.91 
 
 413.964.52 
 
 415.105.71 
 416,248.46 
 
 417.392.79 
 418,538.68 
 
 419.686.15 
 
 420.835.19 
 
 421.985.79 
 423,137.97 
 
 424.291.72 
 425,447.04 
 
 426.603.94 
 
 427.762.40 
 428,922.43 
 430,084.03 
 431,247.21 
 
 432.411.95 
 
 433.578.27 
 
 434.746.16 
 435,915.62 
 437,086.64 
 438,259.24 
 
 439.433.41 
 
CIRCUMFERENCES , AND AREAS. 
 
 557 
 
 No . 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 749 
 
 750 
 
 751 
 
 752 
 
 753 
 
 754 
 
 755 
 
 756 
 
 757 
 
 758 
 
 759 
 
 760 
 
 761 
 
 762 
 
 763 
 
 764 
 
 765 
 
 766 
 
 767 
 
 768 
 
 769 
 
 770 
 
 771 
 
 772 
 
 773 
 
 774 
 
 775 
 
 776 
 
 777 
 
 778 
 
 779 
 
 780 
 
 781 
 
 782 
 
 783 
 
 784 
 
 785 
 
 786 
 
 787 
 
 788 
 
 789 
 
 790 
 
 791 
 
 792 
 
 793 
 
 794 
 
 795 
 
 796 
 
 797 
 
 798 
 
 799 
 
 800 
 801 
 802 
 
 803 
 
 804 
 
 805 
 
 806 
 
 807 
 
 808 
 
 809 
 
 810 
 811 
 
 561.001 
 562,500 
 
 564.001 
 565,504 
 567,009 
 568,516 
 570,025 
 571,536 
 573,049 
 574,564 
 576,081 
 577,600 
 579,121 
 580,644 
 582,169 
 583,696 
 
 585.225 
 586,756 
 588,289 
 589,824 
 591,361 
 592,900 
 594,441 
 595,984 
 597,529 
 599,076 
 600,625 
 602,176 
 603,729 
 605,284 
 606,841 
 
 608.400 
 609,961 
 611,524 
 613,089 
 614,656 
 
 616.225 
 617,796 
 619,369 
 620,944 
 622,521 
 
 624.100 
 625,681 
 627,264 
 628,849 
 630,436 
 
 632.025 
 633,616 
 635,209 
 636.804 
 
 638.401 
 640,000 
 641,601 
 643,204 
 644,809 
 646,416 
 
 648.025 
 649,636 
 651,249 
 652,864 
 654,481 
 
 656.100 
 657,721 
 
 420,189,749 
 . 421,875,000 
 423,564,751 
 425,259,008 
 426,957,777 
 428,661,064 
 430,368,875 
 432,081,216 
 433,798,093 
 435,519,512 
 437,245,479 
 
 438.976.000 
 440,711,081 
 442,450,728 
 444,194,947 
 445,943,744 
 447,697,125 
 449,455,096 1 
 451,217,663 
 452,984,832 
 454,756,609 
 
 456.533.000 
 458,314,011 
 460,099,648 
 461,889,917 
 463,684,824 
 465,484,375 
 467,288,576 
 469,097,433 
 470,910,952 
 472,729,139 
 
 474.552.000 
 476,379,541 
 478,211,768 
 480,048,687 
 481,890,304 
 483,736,625 
 485,587,656 
 487,443,403 
 489,303,872 
 491,169.069 
 
 493.039.000 
 494,913,671 
 496,793,088 
 498,677,257 
 500,566,184 
 502,459,875 
 504,358,336 
 506,261,573 
 508,169,592 
 510,082,399 
 512,000,000 
 513,922,401 
 515,849,608 
 517,781,627 
 519,718,464 
 521,660,125 
 523,606,616 
 525,557,943 
 527,514,112 
 529,475,129 
 
 531.441.000 
 533,411,731 
 
 27.3679 
 
 27.3861 
 
 27.4044 
 
 27.4226 
 
 27.4408 
 
 27.4591 
 
 27.4773 
 
 27.4955 
 
 27.5136 
 
 27.5318 
 
 27.5500 
 
 27.5681 
 
 27.5862 
 
 27.6043 
 
 27.6225 
 
 27.6405 
 
 27.6586 
 
 27.6767 
 
 27.6948 
 
 27.7128 
 
 27.7308 
 
 27.7489 
 
 27.7669 
 
 27.7849 
 
 27.8029 
 
 27.8209 
 
 27.8388 
 
 27.8568 
 
 27.8747 
 
 27.8927 
 
 27.9106 
 
 27.9285 
 
 27.9464 
 
 27.9643 
 
 27.9821 
 
 28.0000 
 
 28.0179 
 
 28.0357 
 
 28.0535 
 
 28.0713 
 
 28.0891 
 
 28.1069 
 
 28.1247 
 
 28.1425 
 
 28.1603 
 
 28.1780 
 
 28.1957 
 
 28.2135 
 
 28.2312 
 
 28.2489 
 
 28.2666 
 
 28.2843 
 
 28.3019 
 
 28.3196 
 
 28.3373 
 
 28.3549 
 
 28.3725 
 
 28.3901 
 
 28.4077 
 
 28.4253 
 
 28.4429 
 
 28.4605 
 
 28.4781 
 
 9.0816 
 
 9.0856 
 
 9.0896 
 
 9.0937 
 
 9.0977 
 
 9.1017 
 
 9.1057 
 
 9.1098 
 
 9.1138 
 
 9.1178 
 
 9.1218 
 
 9.1258 
 
 9.1298 
 
 9.1338 
 
 9.1378 
 
 9.1418 
 
 9.1458 
 
 9.1498 
 
 9.1537 
 
 9.1577 
 
 9.1617 
 
 9.1657 
 
 9.1696 
 
 9.1736 
 
 9.1775 
 
 9.1815 
 
 9.1855 
 
 9.1894 
 
 9.1933 
 
 9.1973 
 
 9.2012 
 
 9.2052 
 
 9.2091 
 
 9.2130 
 
 9.2170 
 
 9.2209 
 
 9.2248 
 
 9.2287 
 
 9.2326 
 
 9.2365 
 
 9.2404 
 
 9.2443 
 
 9.2482 
 
 9.2521 
 
 9.2560 
 
 9.2599 
 
 9.2638 
 
 9.2677 
 
 9.2716 
 
 9.2754 
 
 9.2793 
 
 9.2832 
 
 9.2870 
 
 9.2909 
 
 9.2948 
 
 9.2986 
 
 9.3025 
 
 9.3063 
 
 9.3102 
 
 9.3140 
 
 9.3179 
 
 9.3217 
 
 9.3255 
 
 .001335113 : 
 .001333333 ! 
 .001331558 ! 
 .001329787 ! 
 .001328021 ! 
 .001326260 ! 
 .001324503 : 
 .001322751 : 
 .001321004 : 
 .001319261 : 
 .001317523 
 .001315789 
 .001314060 
 .001312336 
 .001310616 
 .001308901 
 .001307190 
 .001305483 
 .001303781 
 .001302083 
 .001300390 
 .001298701 
 .001297017 
 .001295337 
 .001293661 
 .001291990 
 .001290323 
 .001288660 
 .001287001 
 .001285347 
 .001283697 
 .001282051 
 .001280410 
 .001278772 
 .001277139 
 .001275510 
 .001273885 
 .001272265 
 .001270648 
 .001269036 
 .001267427 
 .001265823 
 .001264223 
 .001262626 
 .001261034 
 .001259446 
 .001257862 
 .001256281 
 .001254705 
 .001253133 
 .001251364 
 .001250000 
 .001248439 
 .001246883 
 .001245330 
 .001243781 
 .001242236 
 .001240695 
 .001239157 
 .001237624 
 .001236094 
 .001234568 
 .001233046 
 
 2,353.05 < 
 
 2.356.19 < 
 2,359.34 < 
 2,362.48 < 
 2,365.62 - 
 2,368.76 < 
 2,371.90 - 
 
 2.375.04 • 
 
 2.378.19 ■ 
 2,381.33 • 
 2,384.47 ■ 
 2,387.61 
 2,390.75 
 2,393.89 
 
 2.397.04 
 2,400.18 
 2,403.32 
 2,406.46 
 2,409.60 
 2,412.74 
 
 2.415.88 
 2,419.03 
 2,422.17 
 2,425.31 
 2,428.45 
 2,431.59 
 
 2.434.73 
 
 2.437.88 
 2,441.02 
 2,444.16 
 2,447.30 
 2,450.44 
 
 2.453.58 
 
 2.456.73 
 2,459.87 
 2,463.01 
 2,466.15 
 2,469.29 
 2,472.43 
 
 2.475.58 
 2,478.72 
 2,481.86 
 2,485.00 
 2,488.14 
 2,491.28 
 
 2.494.42 
 2,497.57 
 2,500.71 
 2,503.85 
 2,506.99 
 2,510.13 
 
 2.513.27 
 
 2.516.42 
 2,519.56 
 2,522.70 
 2,525.84 
 2,528.98 
 2,532.12 
 
 2.535.27 
 2,538.41 
 2,541.55 
 2,544.69 
 2,547.83 
 
 140,609.16 
 
 141,786.47 
 
 142,965.35 
 
 144.145.80 
 
 445.327.83 
 446,511.42 
 447,696.59 
 448,883.32 
 
 450.071.63 
 451,261.51 
 
 452.452.96 
 453,645.98 
 
 454.840.57 
 
 456.036.73 
 457,234.46 
 
 458.433.77 
 
 459.634.64 
 460,837.08 
 462,041.10 
 
 463.246.69 
 
 464.453.84 
 
 465.662.57 
 466,872.87 
 
 468.084.74 
 
 469.298.18 
 
 470.513.19 
 
 471.729.77 
 472,947.92 
 
 474.167.65 
 
 475.388.94 
 
 476.611.81 
 
 477.836.24 
 
 479.062.25 
 
 480.289.83 
 
 481.518.97 
 
 482.749.69 
 
 483.981.98 
 
 485.215.84 
 
 486.451.28 
 
 487.688.28 
 
 488.926.85 
 
 490.166.99 
 491,408.71 
 
 492.651.99 
 
 493.896.85 
 
 495.143.28 
 496,391.27 
 497,640.84 
 498,891.98 
 
 500.144.69 
 501,398.97 
 
 502.654.82 
 
 503.912.25 
 505,171.24 
 506,431.80 
 
 507.693.94 
 508,957.64 
 510,222.92 
 511,489.77 
 5^2,758.19 
 514,028.18 
 515,299.74 
 516,572.87 
 
558 SQUARES , CUBES , SQUARE AND CUBE ROOTS, 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 812 
 
 813 
 
 814 
 
 815 
 
 816 
 
 817 
 
 818 
 
 819 
 
 820 
 821 
 822 
 
 823 
 
 824 
 
 825 
 
 826 
 
 827 
 
 828 
 
 829 
 
 830 
 
 831 
 
 832 
 
 833 
 
 834 
 
 835 
 
 836 
 
 837 
 
 838 
 
 839 
 
 840 
 
 841 
 
 842 
 
 843 
 
 844 
 
 845 
 
 846 
 
 847 
 
 848 
 
 849 
 
 850 
 
 851 
 
 852 
 
 853 
 
 854 
 
 855 
 
 856 
 
 857 
 
 858 
 
 859 
 
 860 
 861 
 862 
 
 863 
 
 864 
 
 865 
 
 866 
 
 867 
 
 868 
 
 869 
 
 870 
 
 871 
 
 872 
 
 873 
 
 874 
 
 659,344 
 
 660,969 
 
 662,596 
 
 664.225 
 
 665,856 
 
 667,489 
 
 669,124 
 
 670,761 
 
 672,400 
 
 674,041 
 
 675.584 
 677,329 
 678,976 
 680,625 
 682,276 
 683,929 
 
 685.584 
 687,241 
 
 688.900 
 690.561 
 
 692.224 
 693,889 
 695,556 
 
 697.225 
 698,896 
 700,569 
 702,244 
 703,921 
 
 705.600 
 707,281 
 708,964 
 710,649 
 712,336 
 
 714.025 
 715,716 
 717,409 
 719,104 
 720,801 
 722,500 
 724,201 
 725,904 
 727,609 
 729,316 
 
 731.025 
 732,736 
 734,449 
 736,164 
 737,881 
 
 739.600 
 741,321 * 
 743,044 
 744,769 
 746,496 
 748/225 
 749,956 
 751,689 
 753,424 
 755,161 
 
 756.900 
 758,641 
 760,384 
 762,129 
 763,876 
 
 1 
 
 535,387,328 
 
 537,367,797 
 
 539,353,144 
 
 541.343.375 
 543,338.496 
 545,338,513 
 547,343,432 
 549,353,259 
 
 551.368.000 
 553,387,661 
 555,412,248 
 557,441,767 
 559,476,224 
 
 561.515.625 
 563,559,976 
 565,609,283 
 567,663,552 
 569,722,789 
 
 571.787.000 
 
 573.856.191 
 575,930,368 
 578,009,537 
 580,093,704 
 582,182,875 
 584,277,056 
 586,376,253 
 588,480,472 
 590,589,719 
 
 592.704.000 
 594,823,321 
 596,947,688 
 599,077,107 
 601,211,584 
 603,351,125 
 605.495,736 
 607,645,423 
 
 609.800.192 
 611,960,049 
 
 614.125.000 
 616,295,051 
 618,470,208 
 620,650,477 
 622,835,864 
 
 625.026.375 
 627,222,016 
 629,422,793 
 631,628,712 
 633,839,779 
 
 636.056.000 
 638,277,381 
 640,503,928 
 642,735,647 
 644,972,544 
 
 647.214.625 
 649,461,896 
 651,714,363 
 653,972,032 
 656,234,909 
 
 658.503.000 
 660,776,311 
 663,054,848 
 665,338,617 
 667,627,624 
 
 28.4956 
 
 28.5132 
 
 28.5307 
 
 28.5482 
 
 28.5657 
 
 28.5832 
 
 28.6007 
 
 28.6182 
 
 28.6356 
 
 28.6531 
 
 28.6705 
 
 28.6880 
 
 28.7054 
 
 28.7228 
 
 28.7402 
 
 28.7576 
 
 28.7750 
 
 28.7924 
 
 28.8097 
 
 28.8271 
 
 28.8444 
 
 28.8617 
 
 28.8791 
 
 28.8964 
 
 28.9137 
 
 28.9310 
 
 28.9482 
 
 28.9655 
 
 28.9828 
 
 29.0000 
 
 29.0172 
 
 29.0345 
 
 29.0517 
 
 29.0689 
 
 29.0861 
 
 29.1033 
 
 29.1204 
 
 29.1376 
 
 29.1548 
 
 29.1719 
 
 29.1890 
 
 29.2062 
 
 29.2233 
 
 29.2404 
 
 29.2575 
 
 29.2746 
 
 29.2916 
 
 29.3087 
 
 29.3258 
 
 29.3428 
 
 29.3598 
 
 29.3769 
 
 29.3939 
 
 29.4109 
 
 29.4279 
 
 29.4449 
 
 29.4618 
 
 29.4788 
 
 29.4958 
 
 29.5127 
 
 29.5296 
 
 29.5466 
 
 29.5635 
 
 9.3294 
 9.3332 
 9.3370 
 9.3408 
 9.3447 
 9.3485 
 9.3523 
 9.3561 
 9.3599 
 9.3637 
 9.3675 
 9.3713 
 9.3751 
 9.3789 
 9.3827 
 9.3865 
 9 3902 
 9.3940 
 9.3978 
 9.4016 
 9.4053 
 9.4091 
 9.4129 
 9.4166 
 9.4204 
 9.4241 
 9.4279 
 9.4316 
 9.4354 
 9.4391 
 9.4429 
 9.4466 
 9.4503 
 9.4541 
 9.4578 
 9.4615 
 9.4652 
 9.4690 
 9.4727 
 9.4764 
 9.4801 
 9.4838 
 9.4875 
 9.4912 
 9.4949 
 9.4986 
 9.5023 
 9.5060 
 9.5097 
 9.5135 
 9.5171 
 9.5207 
 9.5244 
 9.5281 
 9.5317 
 9.5354 
 9.5391 
 9.5427 
 9.5464 
 9.5501 
 9.5537 
 9.5574 
 9.5610 
 
 .001231527 
 
 .001230012 
 
 .001228501 
 
 .001226994 
 
 .001225490 
 
 .001223990 
 
 .001222494 
 
 .001221001 
 
 .001219512 
 
 .001218027 
 
 .001216545 
 
 .001215067 
 
 .001213592 
 
 .001212121 
 
 .001210654 
 
 .001209190 
 
 .001207729 
 
 .001206273 
 
 .001204819 
 
 .001203369 
 
 .001201923 
 
 .001200480 
 
 .001199041 
 
 .001197605 
 
 .001196172 
 
 .001194743 
 
 .001193317 
 
 .001191895 
 
 .001190476 
 
 .001189061 
 
 .001187648 
 
 .001186240 
 
 .001184834 
 
 .001183432 
 
 .001182033 
 
 .001180638 
 
 .001179245 
 
 .001177856 
 
 .001176471 
 
 .001175088 
 
 .001173709 
 
 .001172333 
 
 .001170960 
 
 .001169591 
 
 .001168224 
 
 .001166861 
 
 .001165501 
 
 .001164144 
 
 .001162791 
 
 .001161440 
 
 .001160093 
 
 .001158749 
 
 .001157407* 
 
 .00115606SF 
 
 .001154734 
 
 .001153403 
 
 .001152074 
 
 .001150748 
 
 .001149425 
 
 .001148106 
 
 .001146789 
 
 .001145475 
 
 .001144165 
 
 2,550.97 
 
 2.554.11 
 2,557.26 
 2,560.40 
 2,563.54 
 2,566.68 
 2,569.82 
 
 2.572.96 
 
 2.576.11 
 2,579.25 
 2,582.39 
 2,585.53 
 2,588.67 
 
 2.591.81 
 
 2.594.96 
 2,598.10 
 2,601.24 
 2,604.38 
 2,607.52 
 2,610.66 
 
 2.613.81 
 2,616.95 
 2,620.09 
 2,623.23 
 2,626.37 
 2,629.51 
 
 2.632.65 
 2,635.80 
 2,638.94 
 2,642.08 
 2,615.22 
 2,648.36 
 
 2.651.50 
 
 2.654.65 
 2,657.79 
 2,660.93 
 2,664.07 
 2,667.21 
 2,670.35 
 
 2.673.50 
 2,676.64 
 2.679.78 
 2,682.92 
 2,686.06 
 2,689.20 
 
 2.692.34 
 2,695.49 
 2,698.63 
 2,701.77 
 2,704.91 
 2,708.05 
 
 2.711.19 
 
 2.714.34 
 2,717.48 
 2,720.62 
 2,723.76 
 2,726.90 
 2,730.04 
 
 2.733.19 
 2,736.33 
 2,739.47 
 2,742.61 
 2,745.75 
 
 517.847.57 
 
 519.123.84 
 
 520.401.68 
 521,681.10 
 522,962.08 
 524,244.63 
 
 525.528.76 
 
 526.814.46 
 528,101.73 
 529,390.56 
 530,680.97 
 
 531.972.95 
 
 533.266.50 
 
 534.561.62 
 535,858.32 
 
 537.156.58 
 
 538.456.41 
 
 539.757.82 
 541,060.79 
 542,365.34 
 
 543.671.46 
 
 544.979.15 
 546,288.40 
 547,599.23 
 
 548.911.63 
 550,225.61 
 
 551.541.15 
 552,858.26 
 554,176.94 
 
 555.497.20 
 
 556.819.02 
 
 558.142.42 
 559,467.39 
 560,793.92 
 
 562.122.03 
 
 563.451.71 
 
 564.782.96 
 566,115.78 
 567,450.17 
 
 568.786.14 
 570,123.67 
 
 571.462.77 
 572,803.45 
 
 574.145.69 
 
 575.489.51 
 576,834.90 
 
 578.181.85 
 
 579.530.38 
 580,880.48 
 
 582.232.15 
 
 583.585.39 
 
 584.940.20 
 
 586.296.59 
 587,654.54 
 589,014.07 
 
 590.375.16 
 
 591.737.83 
 593,102.06 
 594.467.87 
 595,835.25 
 
 597.204.20 
 
 598.574.72 
 599,946.81 
 
CIRCUMFERENCES, AND AREAS. 
 
 559 
 
 No. 
 
 Square. 
 
 Cube. 
 
 3q. Root. < 
 
 Du. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 875 
 
 876 
 
 877 
 
 878 
 
 879 
 
 880 
 881 
 882 
 
 883 
 
 884 
 
 885 
 
 886 
 
 887 
 
 888 
 
 889 
 
 890 
 
 891 
 
 892 
 
 893 
 
 894 
 
 895 
 
 896 
 
 897 
 
 898 
 
 899 
 
 900 
 
 901 
 
 902 
 
 903 
 
 904 
 
 905 
 
 906 
 
 907 
 
 908 
 
 909 
 
 910 
 
 911 
 
 912 
 
 913 
 
 914 
 
 915 
 
 916 
 
 917 
 
 918 
 
 919 
 
 920 
 
 921 
 
 922 
 
 923 
 
 924 
 
 925 
 
 926 
 
 927 
 
 928 
 
 929 
 
 930 
 
 931 
 
 932 
 
 933 
 
 934 
 
 935 
 
 936 
 
 937 
 
 765,625 
 
 767,376 
 
 769,129 
 
 770,884 
 
 772,641 
 
 774,400 
 
 776,161 
 
 777,924 
 
 779,689 
 
 781,456 
 
 783,225 
 
 784,996 
 
 786,769 
 
 788,544 
 
 790,321 
 
 792.100 
 793,881 
 795,664 
 797,449 
 799,236 
 
 801.025 
 802,816 
 804,609 
 806,404 
 808,201 
 810,000 
 811,801 
 813,604 
 815,409 
 817,216 
 
 819.025 
 820,836 
 822,649 
 824,464 
 826,281 
 
 828.100 
 829,921 
 831,744 
 833,569 
 835,396 
 
 837.225 
 839,056 
 840,889 
 842,724 
 844,561 
 846,400 
 848,241 
 850,084 
 851,929 
 853,776 
 855,625 
 857,476 
 859,329 
 861,184 
 863,041 
 884,900 
 866,761 
 868,624 
 870,489 
 872,356 
 
 874.225 
 876,096 
 877,969 
 
 669,921,875 
 
 672,221,376 
 
 674,526,133 
 
 676,836,152 
 
 679,151,439 
 
 681.472.000 
 683,797.841 
 686,128,968 
 688,465,387 
 690,807,104 
 693,154,125 
 695,506,456 
 697,864,103 
 700,227,072 
 702,595,369 
 
 704.969.000 
 707,347,971 
 707,932,288 
 712,121,957 
 714,516,984 
 716,917,375 
 719,323,136 
 721,734,273 
 724,150,792 
 726,572,699 
 729,000,000 
 731,432,701 
 733,870,808 
 736,314,327 
 738,763,264 
 741,217,625 
 743,677,416 
 746,142,643 
 748,613,312 
 751,089,429 
 
 753.571.000 
 756,058,031 
 758,550,825 
 761,048,497 
 763,551,944 
 766,060,875 
 768,575,296 
 771,095,213 
 773,620,632 
 776,151,559 
 
 778.688.000 
 781,229,961 
 783.777,448 
 786^330,467 
 788,889,024 
 791,453,125 
 794,022,776 
 796,597,983 
 799,178,752 
 801,765,089 
 
 804.357.000 
 806,954,491 
 809,557,568 
 812,166,237 
 814,780,504 
 817,400,375 
 820,025,856 
 822,656,953 
 
 29.5804 
 
 29.5973 
 
 29.6142 
 
 29.6311 
 
 29.6479 
 
 29.6648 
 
 29.6816 
 
 29.6985 
 
 29.7153 
 
 29.7321 
 
 29.7489 
 
 29.7658 
 
 29.7825 
 
 29.7993 
 
 29.8161 
 
 29.8329 
 
 29.8496 
 
 29.8664 
 
 29.8831 
 
 29.8998 
 
 29.9166 
 
 29.9333 
 
 29.9500 
 
 29.9666 
 
 29.9833 
 
 30.0000 
 
 30.0167 
 
 30.0333 
 
 30.0500 
 
 30.0666 
 
 30.0832 
 
 30.0998 
 
 30.1164 
 
 30.1330 
 
 30.1496 
 
 30.1662 
 
 30.1828 
 
 30.1993 
 
 30.2159 
 
 30.2324 
 
 30.2490 
 
 30.2655 
 
 30.2820 
 
 30.2985 
 
 30.3150 
 
 30.3315 
 
 30.3480 
 
 30.3645 
 
 30.3809 
 
 30.3974 
 
 30.4138 
 
 30.4302 
 
 30.4467 
 
 30.4631 
 
 30.4795 
 
 30.4959 
 
 30.5123 
 
 30.5287 
 
 30.5450 
 
 30.5614 
 
 30.5778 
 
 30.5941 
 
 30.6105 
 
 9.5647 
 9.5683 
 9.5719 
 9.5756 
 9.5792 
 9.5828 
 9.5865 
 9.5901 
 9.5937 
 9.5973 
 9.6010 
 9.6046 
 9.6082 
 9.6118 
 9.6154 
 9.6190 
 9.6226 
 9.6262 
 9.6298 
 9.6334 
 9.6370 
 9.6406 
 9.6442 
 9.6477 
 9.6513 
 9.6549 
 9.6585 
 9.6620 
 9.6656 
 9.6692 
 9.6727 
 9.6763 
 9.6799 
 9.6834 
 9.6870 
 9.6905 
 9.6941 
 9.6976 
 9.7012 
 9.7047 
 9.7082 
 9.7118 
 9.7153 
 9.7188 
 9.7224 
 9.7259 
 9.7294 
 9.7329 
 9.7364 
 9.7400 
 9.7435 
 9.7470 
 9.7505 
 ■ 9.7540 
 9.7575 
 9.7610 
 9.7645 
 9.7680 
 9.7715 
 9.7750 
 9.7785 
 9.7829 
 9.7854 
 
 .001142857 S 
 .001141553 5 
 .001140251 5 
 .001138952 5 
 .001137656 5 
 .001136364 i 
 .001135074 5 
 .001133787 5 
 .001132503 i 
 .001131222 ! 
 .001129944 i 
 .001128668 i 
 .001127396 ! 
 .001126126 : 
 .001124859 ! 
 .001123596 ! 
 .001122334 : 
 .001121076 
 .001119821 
 .001118568 
 .001117818 
 .001116071 
 .001114827 
 .001113586 
 .001112347 
 .001111111 
 .001109878 
 .001108647 
 .001107420 
 .001106195 
 .001104972 
 .001103753 
 .001102536 
 .001101322 
 .001100110 
 .001098901 
 .001091695 
 .001096491 
 .001095290 
 .001094092 
 .001092896 
 .001091703 
 .001090513 
 .001089325 
 .001088139 
 .001086957 
 .001085776 
 .001084599 
 .001083423 
 .001082251 
 .001081081 
 .001079914 
 .001078749 
 .001077586 
 .001076426 
 .001075269 
 .001074114 
 .001072961 
 .001071811 
 .001070664 
 .001069519 
 .001068376 
 .001067236 
 
 >,748.89 ( 
 5,752.04 e 
 5,755.18 ( 
 5,758.32 ( 
 5,761.46 ( 
 5,764.60 ( 
 5,767.74 ( 
 
 5.770.88 ( 
 5,774.03 ( 
 5,777.17 < 
 5,780.31 < 
 5,783.45 < 
 2,786.59 i 
 
 2.789.73 i 
 
 2.792.88 i 
 2,796.02 1 
 2,799.16 i 
 2,802.30 
 2,805.44 
 
 2.808.58 
 
 2.811.73 
 2,814.87 
 2,818.01 
 2,821.15 
 2,824.29 
 2,827.43 
 
 2.830.58 
 2,833.72 
 2,836.86 
 2,840.00 
 2,843.14 
 2,846.28 
 
 2.849.42 
 2,852.57 
 2,855.71 
 2,858.85 
 2,861.99 
 2,865.13 
 
 2.868.27 
 
 2.871.42 
 2,874.56 
 2,877.70 
 2,880.84 
 2,883.98 
 2.887.12 
 
 2.890.27 
 2,893.41 
 2,896.55 
 2,899.69 
 2,902.83 
 2,905.97 
 
 2.909.11 
 2,912.26 
 2,915.40 
 2,918.54 
 2,921.68 
 2,924.82 
 2,927.96 
 
 2.931.11 
 2,934.25 
 2,937.39 
 2,940.53 
 2,943.67 
 
 >01,320.47 
 
 >02,695.70 
 
 >04,072.50 
 
 >05,450.88 
 
 >06,830.82 
 
 >08,212.34 
 
 >09,595.42 
 
 >10,980.08 
 
 312,366.31 
 
 313,754.11 
 
 515.143.48 
 
 516.534.42 
 517,926.93 
 
 519.321.01 
 620,716.66 
 622,113.89 
 623,512.68 
 624,913.04 
 626,314.98 
 
 627.718.49 
 
 629.123.56 
 
 630.530.21 
 
 631.938.43 
 
 633.348.22 
 
 634.759.58 
 
 636.172.51 
 
 637.587.01 
 
 639.003.09 
 640,420.73 
 641,839.95 
 *643,260.73 
 
 644.683.09 
 
 646.107.01 
 
 647.532.51 
 
 648.959.58 
 
 650.388.22 
 
 651.818.43 
 653,250.21 
 
 654.683.56 
 656,118.48 
 657,554.98 
 658,993.04 
 
 660.432.68 
 661,873.88 
 663,316.66 
 
 664.761.01 
 666,206.92 
 
 667.654.41 
 669,103.47 
 
 670.554.10 
 672,006.30 
 673,460.08 
 
 674.915.42 
 676,372.33 
 677,830.82 
 679,290.87 
 680,752.50 
 
 682.215.69 
 683,680.46 
 685,146.80 
 686,614.71 
 
 , 688,084.19 
 689,555.24 
 
560 SQUARES , CUBES, SQUARE AND CUBE ROOTS , 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 Cu. Root. 
 
 Reciprocal. 
 
 Circum. 
 
 Area. 
 
 938 
 
 939 
 
 940 
 
 941 
 
 942 
 
 943 
 
 944 
 
 945 
 
 946 
 
 947 
 
 948 
 
 949 
 
 950 
 
 951 
 
 952 
 
 953 
 
 954 
 
 955 
 
 956 
 
 957 
 
 958 
 
 959 
 
 960 
 
 961 
 
 962 
 
 963 
 
 964 
 
 965 
 
 966 
 
 967 
 
 968 
 
 969 
 
 970 
 
 971 
 
 972 
 
 973 
 
 974 
 
 975 
 
 976 
 
 977 
 
 978 
 
 979 
 
 980 
 
 981 
 
 982 
 
 983 
 
 984 
 
 985 
 
 986 
 
 987 . 
 
 988 
 
 989 
 
 990 
 
 991 
 
 992 
 
 993 
 
 994 
 
 995 
 
 996 
 
 997 
 
 998 
 
 999 
 1000 
 
 879,844 
 
 881,721 
 
 883.600 
 885,481 
 887,364 
 889,249 
 891,136 
 
 893.025 
 894,916 
 896,808 
 898,704 
 
 900.601 
 902,500 
 904,401 
 906,304 
 908,209 
 910,116 
 
 912.025 
 913,936 
 915,849 
 917,764 
 919,681 
 921,600 
 923,521 
 925,444 
 927,369 
 929,296 
 
 931.225 
 933,156 
 935,089 
 
 937.024 
 938,961 
 940,900 
 942,841 
 944,784 
 946,729 
 948,676 
 950,625 
 952,576 
 954,529 
 956,484 
 958,441 
 960,400 
 962,361 
 964,324 
 966,289 
 968,256 
 
 970.225 
 972,196 
 974,169 
 976,144 
 978,121 
 980,100 
 982,081 
 984,064 
 986,049 
 988,036 
 
 990.025 
 992,016 
 994,009 
 996,004 
 998,001 
 
 1,000,000 
 
 825,293,672 
 
 827,936,019 
 
 830.584.000 
 833,237,621 
 835,896,888 
 838,561,807 
 841,232,384 
 
 843.908.625 
 846,590,536 
 849,278,123 
 851,971,392 
 854,670,349 
 
 857.375.000 
 
 860.085.351 
 862,801,408 
 865,523,177 
 868,250,664 
 
 870.983.875 
 873,722,816 
 876,467,493 
 879,217,912 
 881,974,079 
 
 884.736.000 
 887,503,681 
 890,277,128 
 893,056,347 
 895,841,344 
 898,632,125 
 901,428,696 
 904,231,063 
 907,039,232 
 909,853,209 
 
 912.673.000 
 915,498,611 
 918,330,048 
 921,167,317 
 924,010,424 
 926,859,375 
 929,714,176 
 932,574,833 
 
 935.441.352 
 938,313,739 
 
 941.192.000 
 944,076,141 
 946,966,168 
 949,862,087 
 952,763,904 
 
 955.671.625 
 958,585,256 
 961,504,803 
 964,430,272 
 967,361,669 
 
 970.299.000 
 973,242,271 
 976,191,488 
 979,146,657 
 982,107,784 
 
 985.074.875 
 988,047,936 
 991,026,973 
 994,011,992 
 997,002,999 
 
 1,000,000,000 
 
 30.6268 
 
 30.6431 
 
 30.6594' 
 
 30.6757 
 
 30.6920 
 
 30.7083 
 
 30.7246 
 
 30.7409 
 
 30.7571 
 
 30.7734 
 
 30.7896 
 
 30.8058 
 
 30.8221 
 
 30.8383 
 
 30.8545 
 
 30.8707 
 
 30.8869 
 
 30.9031 
 
 30.9192 
 
 30.9354 
 
 30.9516 
 
 30.9677 
 
 30.9839 
 
 31.0000 
 
 31.0161 
 
 31.0322 
 
 31.0483 
 
 31.0644 
 
 31.0805 
 
 31.0966 
 
 31.1127 
 
 31.1288 
 
 31.1448 
 
 31.1609 
 
 31.1769 
 
 31.1929 
 
 31.2090 
 
 31.2250 
 
 31.2410 
 
 31.2570 
 
 31.2730 
 
 31.2890 
 
 31.3050 
 
 31.3209 
 
 31.3369 
 
 31.3528 
 
 31.3688 
 
 31.3847 
 
 31.4006 
 
 31.4166 
 
 31.4325 
 
 31.4484 
 
 31.4643 
 
 31.4802 
 
 31.4960 
 
 31.5119 
 
 31.5278 
 
 31.5436 
 
 31.5595 
 
 31.5753 
 
 31.5911 
 
 31.6070 
 
 31.6228 
 
 9.7889 
 
 9.7924 
 
 9.7959 
 
 9.7993 
 
 9.8028 
 
 9.8063 
 
 9.8097 
 
 9.8132 
 
 9.8167 
 
 9.8201 
 
 9.8236 
 
 9.8270 
 
 9.8305 
 
 9.8339 
 
 9.8374 
 
 9.8408 
 
 9.8443 
 
 9.8477 
 
 9.8511 
 
 9.8546 
 
 9.8580 
 
 9.8614 
 
 9.8648 
 
 9.8683 
 
 9.8717 
 
 9.8751 
 
 9.8785 
 
 9.8819 
 
 9.8854 
 
 9.8888 
 
 9.8922 
 
 9.8956 
 
 9.8990 
 
 9.9024 
 
 9.9058 
 
 9.9092 
 
 9.9126 
 
 9.9160 
 
 9.9194 
 
 9.9228' 
 
 9.9261 
 
 9.9295 
 
 9.9329 
 
 9.9363 
 
 9.9396 
 
 9.9430 
 
 9.9464 
 
 9.9497 
 
 9.9531 
 
 9.9565 
 
 9.9598 
 
 9.9632 
 
 9.9666 
 
 9.9699 
 
 9.9733 
 
 9.9766 
 
 9.9800 
 
 9.9833 
 
 9.9866 
 
 9.9900 
 
 9.9933 
 
 9.9967 
 
 10.0000 
 
 .001066098 
 
 .001064963 
 
 .001063830 
 
 .001062699 
 
 .001061571 
 
 .001060445 
 
 .001059322 
 
 .001058201 
 
 .001057082 
 
 .001055966 
 
 .001054852 
 
 .001053741 
 
 .001052632 
 
 .001051525 
 
 .001050420 
 
 .001049318 
 
 .001048218 
 
 .001047120 
 
 .001046025 
 
 .001044932 
 
 .001043841 
 
 .001042753 
 
 .001041667 
 
 .001040583 
 
 .001039501 
 
 .001038422 
 
 .001037344 
 
 .001036269 
 
 .001035197 
 
 .001034126 
 
 .001033058 
 
 .001031992 
 
 .001030928 
 
 .001029866 
 
 .001028807 
 
 .001027749 
 
 .001026694 
 
 .001025641 
 
 .001024590 
 
 .001023541 
 
 .001022495 
 
 .001021450 
 
 .001020408 
 
 .001019168 
 
 .001018330 
 
 .001017294 
 
 .001016260 
 
 .001015228 
 
 .001014199 
 
 .001013171 
 
 .001012146 
 
 .001011122 
 
 .001010101 
 
 .001009082 
 
 .001008065 
 
 .001007049 
 
 .001006036 
 
 .001005025 
 
 .001004016 
 
 .001003009 
 
 .001002004 
 
 .001001001 
 
 .001000000 
 
 2.946.81 
 2,949.96 
 2,953.10 
 2,956.24 
 2,959.38 
 2,962.52 
 2,965.66 
 
 2.968.81 
 2,971.95 
 2,975.09 
 2,978.23 
 2,981.37 
 2,984.51 
 
 2.987.65 
 2,990.80 
 2,993.94 
 2,997.08 
 3,000.22 
 3,003.36 
 
 3.006.50 
 
 3.009.65 
 3,012.79 
 3,015.93 
 3,019.07 
 3,022.21 
 3,025.35 
 
 3.028.50 
 3,031.64 
 3,034.78 
 3,037.92 
 3,041.06 
 3,044.20 
 
 3.047.34 
 3,050.49 
 3,053.63 
 3,056.77 
 3,059.91 
 3,063.05 
 
 3.066.19 
 
 3.069.34 
 3,072.48 
 3,075.62 
 3,078.76 
 3,081.90 
 
 3.085.04 
 
 3.088.19 
 3,091.33 
 3,094.47 
 3,097.61 
 3,100.75 
 3,103.89 
 
 3.107.04 
 3,110.18 
 3,113.32 
 3,116.46 
 3,119.60 
 3,122.74 
 3,125.88 
 3,129.03 
 3,132.17 
 3,135.31 
 3,138.45 
 3,141.59 
 
 691,027.86 
 
 692.502.05 
 
 693.977.82 
 
 695.455.15 
 
 696.934.06 
 
 698.414.53 
 
 699.896.58 
 701,380.19 
 
 702.865.38 
 
 704.352.14 
 
 705.840.47 
 
 707.330.37 
 
 708.821.84 
 710,314.88 
 711,809.50 
 713,305.68 
 
 714.803.43 
 
 716.302.76 
 
 717.803.66 
 
 719.306.12 
 
 720.810.16 
 
 722.315.77 
 
 723.822.95 
 
 725.331.70 
 726,842.02 
 728,353.91 
 
 729.867.37 
 731,382.40 
 732,899.01 
 734,417.18 
 
 735.936.93 
 737,458.24 
 
 738.981.13 
 
 740.505.59 
 742,031.62 
 743,559.22 
 
 745.088.39 
 
 746.619.13 
 
 748.151.44 
 749,685.32 
 
 751.220.78 
 752,757.80 
 
 754.296.40 
 755,836.56 
 757,378.30 
 
 758.921.61 
 
 760.466.48 
 
 762.012.93 
 
 763.560.95 
 
 765.110.54 
 
 766.661.70 
 
 768.214.44 
 769,768.74 
 
 771.324.61 
 
 772.882.06 
 
 774.441.07 
 
 776.001.66 
 
 777.563.82 
 
 779.127.54 
 
 780.692.84 
 
 782.259.71 
 
 783.828.15 
 
 785.398.16 
 
CIRCUMFERENCES 
 
 AND AREAS OF CIRCLES. 
 
 561 
 
 CIRCUMFERENCES AND AREAS OF CIRCLES 
 FROM 1-64 TO 100. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. * 
 
 Circum. 
 
 Area. 
 
 i 
 
 0491 
 
 0002 
 
 6 
 
 18.8496 
 
 28.2744 
 
 13} 
 
 41.2335 
 
 135.297 
 
 6* 
 
 1 
 
 0982 
 
 0008 
 
 6? 
 
 19.2423 
 
 29.4648 
 
 13} 
 
 41.6262 
 
 137.887 
 
 3l 
 
 1 
 
 1963 
 
 0031 
 
 61 
 
 19.6350 
 
 30.6797 
 
 13} 
 
 42.0189 
 
 140.501 
 
 TS 
 
 1 
 
 3927 
 
 0123 
 
 6} 
 
 20.0277 
 
 31.9191 
 
 13} 
 
 42.4116 
 
 143.139 
 
 S 
 
 3 
 
 5890 
 
 0276 
 
 61 
 
 20.4204 
 
 33.1831 
 
 13} 
 
 42.8043 
 
 145.802 
 
 TS 
 
 1 
 
 7854 
 
 ]0491 
 
 6} 
 
 20.8131 
 
 34.4717 
 
 13} 
 
 43.1970 
 
 148.490 
 
 ¥ 
 
 5 
 
 9817 
 
 0767 
 
 6} 
 
 21.2058 
 
 35.7848 
 
 13} 
 
 43.5897 
 
 151.202 
 
 TS 
 
 3 
 
 1 1781 
 
 .1104 
 
 61 
 
 21.5985 
 
 37.1224 
 
 14 
 
 43.9824 
 
 153.938 
 
 l 
 
 1 
 
 1 3744 
 
 1503 
 
 7 
 
 21.9912 
 
 38.4846 
 
 14} 
 
 44.3751 
 
 156.700 
 
 1 5708 
 
 1963 
 
 7} 
 
 22.3839 
 
 39.8713 
 
 14} 
 
 44.7678 
 
 159.485 
 
 i 
 
 TS 
 
 6 
 
 1 7671 
 
 2485 
 
 71 
 
 22.7766 
 
 41.2826 
 
 14} 
 
 45.1605 
 
 162.296 
 
 1 9635 
 
 .3068 
 
 7| 
 
 23.1693 
 
 42.7184 
 
 14} 
 
 45.5532 
 
 165.130 
 
 S 
 
 n 
 
 3 
 
 2.1598 
 2 3562 
 
 .3712 
 
 .4418 
 
 71 
 
 71 
 
 23.5620 
 
 23.9547 
 
 44.1787 
 
 45.6636 
 
 14} 
 
 14} 
 
 45.9459 
 
 46.3386 
 
 167.990 
 
 170.874 
 
 ?3 
 
 2 5525 
 
 .5185 
 
 71 
 
 24.3474 
 
 47.1731 
 
 14} 
 
 46.7313 
 
 173.782 
 
 f r 5 
 
 2 7489 
 
 6013 
 
 71 
 
 24.7401 
 
 48.7071 
 
 15 
 
 47.1240 
 
 176.715 
 
 A 
 
 2 9452 
 
 6903 
 
 8 
 
 25.1328 
 
 50.2656 
 
 15} 
 
 47.5167 
 
 179.673 
 
 l 1 
 
 3 1416 
 
 .7854 
 
 81 
 
 25.5255 
 
 51.8487 
 
 15} 
 
 47.9094 
 
 182.655 
 
 1 1 
 
 3 5343 
 
 .9940 
 
 81 
 
 25.9182 
 
 53.4563 
 
 15} 
 
 48.3021 
 
 185.661 
 
 -*-8 
 
 1 1 
 
 3 9270 
 
 1.2272 
 
 81 
 
 26.3109 
 
 55.0884 
 
 15} 
 
 48.6948 
 
 188.692 
 
 X 4 
 1 3 
 
 4 3197 
 
 1.4849 
 
 81 
 
 26.7036 
 
 56.7451 
 
 15} 
 
 49.0875 
 
 191.748 
 
 -•-8 
 
 4.7124 
 
 1.7671 
 
 81 
 
 27.0963 
 
 58.4264 
 
 15} 
 
 49.4802 
 
 194.828 
 
 It 
 
 5.1051 
 
 2.0739 
 
 81 
 
 27.4890 
 
 60.1322 
 
 15} 
 
 49.8729 
 
 197.933 
 
 8 
 
 1 3 
 
 5.4978 
 
 2.4053 
 
 81 
 
 27.8817 
 
 61.8625 
 
 16 
 
 50.2656 
 
 201.062 
 
 1- 
 
 5.8905 
 
 2.7612 
 
 9 
 
 28.2744 
 
 63.6174 
 
 16} 
 
 50.6583 
 
 204.216 
 
 1 8 
 
 2 
 
 6.2832 
 
 3.1416 
 
 91 
 
 28.6671 
 
 65.3968 
 
 16} 
 
 51.0510 
 
 207.395 
 
 ! 2^ 
 
 6.6759 
 
 3.5466 
 
 91 
 
 29.0598 
 
 67.2008 
 
 16} 
 
 51.4437 
 
 210.598 
 
 21 
 
 7.0686 
 
 3.9761 
 
 91 
 
 29.4525 
 
 69.0293 
 
 16} 
 
 51.8364 
 
 213.825 
 
 
 7.4613 
 
 4.4301 
 
 9a - 
 
 29.8452 
 
 70.8823 
 
 16} 
 
 52.2291 
 
 217.077 
 
 "8 
 
 21 
 
 7.8540 
 
 4.9087 
 
 91 
 
 30.2379 
 
 72.7599 
 
 16} 
 
 52.6218 
 
 220.354 
 
 "2 
 
 21 
 
 8.2467 
 
 5.4119 
 
 91 
 
 30.6306 
 
 74.6621 
 
 16} 
 
 53.0145 
 
 223.655 
 
 21 
 
 8.6394 
 
 5.9396 
 
 91 
 
 31.0233 
 
 76.589 
 
 17 
 
 53.4072 
 
 226.981 
 
 21 
 
 9.0321 
 
 6.4918 
 
 10 
 
 31.4160 
 
 78.540 
 
 17} 
 
 53.7999 
 
 230.331 
 
 "8 
 
 3 
 
 9.4248 
 
 7.0686 
 
 101 
 
 31.8087 
 
 80.516 
 
 17} 
 
 54.1926 
 
 233.706 
 
 3i 
 
 9.8175 
 
 7.6699 
 
 101 
 
 32.2014 
 
 82.516 
 
 17} 
 
 54.5853 
 
 237.105 
 
 Br 
 
 10.2102 
 
 8.2958 
 
 101 
 
 32.5941 
 
 84.541 
 
 17} 
 
 54.9780 
 
 240.529 
 
 31 
 
 10.6029 
 
 8.9462 
 
 101 
 
 32.9868 
 
 86.590 
 
 17} 
 
 55.3707 
 
 243.977 
 
 31 
 
 10.9956 
 
 9.6211 
 
 101 
 
 33.3795 
 
 88.664 
 
 17} 
 
 55. / 634 
 
 247.450 
 
 '-'2 
 
 31 
 
 11.3883 
 
 10.3206 
 
 101 
 
 33.7722 
 
 90.763 
 
 17} 
 
 56.1561 
 
 250.948 
 
 u 8 
 
 31 
 
 11.7810 
 
 11.0447 
 
 101' 
 
 34.1649 
 
 92.886 
 
 18 
 
 56.5488 . 
 
 254.470 
 
 °4 
 
 31 
 
 12.1737 
 
 11.7933 
 
 11 
 
 34.5576 
 
 95.033 
 
 18} 
 
 56.9415 
 
 258.016 
 
 4 
 
 12.5664 
 
 12.5664 
 
 111 
 
 34.9503 
 
 97.205 
 
 18} 
 
 57.3342 
 
 261.587 
 
 41 
 
 12.9591 
 
 13.3641 
 
 111 
 
 35.3430 
 
 99.402 
 
 18} 
 
 57.7269 
 
 265.183 
 
 41 
 
 13.3518 
 
 14.1863 
 
 111 
 
 35.7357 
 
 101.623 
 
 18} 
 
 58.1196 
 
 268.803 
 
 A 4 
 
 41 
 
 13.7445 
 
 15.0330 
 
 111 
 
 36.1284 
 
 103.869 
 
 18} 
 
 58.5123 
 
 272.448 
 
 ^8 
 
 41 
 
 14.1372 
 
 15.9043 
 
 111 
 
 36.5211 
 
 106.139 
 
 18} 
 
 58.9050 
 
 276.117 
 
 41 
 
 14.5299 
 
 16.8002 
 
 111 
 
 36.9138 
 
 108.434 
 
 18} 
 
 59.2977 
 
 279.811 
 
 A 8 
 41 
 
 14.9226 
 
 17.7206 
 
 111 
 
 37.3065 
 
 110.754 
 
 19 
 
 59.6904 
 
 283.529 
 
 A 4 
 
 41 
 
 15.3153 
 
 18.6555 
 
 12 
 
 37.6992 
 
 113.098 
 
 19} 
 
 60.0831 
 
 287.272 
 
 5 
 
 15.7080 
 
 19.6350 
 
 121 
 
 38.0919 
 
 115.466 
 
 19} 
 
 60.4758 
 
 291.040 
 
 51 
 
 16.1007 
 
 20.6290 
 
 121 
 
 38.4846 
 
 117.859 
 
 19} 
 
 60.8685 
 
 294.832 
 
 '"'8 
 
 51 
 
 16.4934 
 
 21.6476 
 
 12f 
 
 38.8773 
 
 120.277 
 
 19} 
 
 61.2612 
 
 298.648 
 
 u 4 
 
 51 
 
 16.8861 
 
 22.6907 
 
 121 
 
 39.2700 
 
 122.719 
 
 19} 
 
 61.6539 
 
 302.489 
 
 “8 
 
 51 
 
 17.2788 
 
 23.7583 
 
 121 
 
 39.6627 
 
 125.185 
 
 19} 
 
 62.0466 
 
 306.355 
 
 51 
 
 17.6715 
 
 24.8505 
 
 121 
 
 40.0554 
 
 127.677 
 
 19} 
 
 62.4393 
 
 310.245 
 
 w 8 
 
 51 
 
 18.0642 
 
 25.9673 
 
 121 
 
 40.4481 
 
 130.192 
 
 20 
 
 62.8320 
 
 314.160 
 
 51 
 
 18.4569 
 
 27.1086 
 
 13 
 
 40.8408 
 
 132.733 
 
 20} 
 
 63.2247 
 
 318.099 
 
562 
 
 CIRCUMFERENCES AND AREAS OF CIRCLES. 
 
 Diam. 
 
 I 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 20} 
 
 63.6174 
 
 322.063 
 
 28} 
 
 88.3575 
 
 621.264 
 
 36 
 
 113.098 
 
 1,017.878 
 
 20f 
 
 64.0101 
 
 326.051 
 
 28} 
 
 88.7502 
 
 626.798 
 
 36} 
 
 113.490 
 
 1,024.960 
 
 20} 
 
 64.4028 
 
 330.064 
 
 28} 
 
 89.1429 
 
 632.357 
 
 36} 
 
 113.8S3 
 
 1,032.065 
 
 20| 
 
 64.7955 
 
 334.102 
 
 28} 
 
 89.5356 
 
 637.941 
 
 36} 
 
 114.276 
 
 1,039.195 
 
 m 
 
 65.1882 
 
 338.164 
 
 28} 
 
 89.9283 
 
 643.549 
 
 36} 
 
 114.668 
 
 1.046.349 
 
 20} 
 
 65.5809 
 
 342.250 
 
 28} 
 
 90.3210 
 
 649.182 
 
 36} 
 
 115.061 
 
 1,053.528 
 
 21 
 
 65.9736 
 
 346.361 
 
 28} 
 
 90.7137 
 
 654.840 
 
 36} 
 
 115.454 
 
 1.060.732 
 
 21} 
 
 66.3663 
 
 350.497 
 
 29 
 
 91.1064 
 
 660.521 
 
 36} 
 
 115.846 
 
 1,067.960 
 
 21} 
 
 66.7590 
 
 354.657 
 
 29} 
 
 91.4991 
 
 666.228 
 
 37 
 
 116.239 
 
 1,075.213 
 
 21f 
 
 67.1517 
 
 358.842 
 
 29} 
 
 91.8918 
 
 671.959 
 
 37} 
 
 116.632 
 
 1.082.490 
 
 21| 
 
 67.5444 
 
 363.051 
 
 29} 
 
 92.2845 
 
 677.714 
 
 37} 
 
 117.025 
 
 1,089.792 
 
 21} 
 
 67.9371 
 
 367.285 
 
 29} 
 
 92.6772 
 
 683.494 
 
 37} 
 
 117.417 
 
 1.097.118 
 
 21} 
 
 68.3298 
 
 371.543 
 
 29} 
 
 93.0699 
 
 689.299 
 
 37} 
 
 117.810 
 
 1,104.469 
 
 21} 
 
 68.7225 
 
 375.826 
 
 29} 
 
 93.4626 
 
 695.128 
 
 37} 
 
 118.203 
 
 1,111.844 
 
 22 
 
 69.1152 
 
 380.134 
 
 29} 
 
 93.8553 
 
 700.982 
 
 37} 
 
 118.595 
 
 1,119.244 
 
 22} 
 
 69.5079 
 
 384.466 
 
 30 
 
 94.2480 
 
 706.860 
 
 37} 
 
 118.988 
 
 1.126.669 
 
 22} 
 
 69.9006 
 
 388.822 
 
 30} 
 
 94.6407 
 
 712.763 
 
 38 
 
 119.381 
 
 1.134.118 
 
 22} 
 
 70.2933 
 
 393.203 
 
 30} 
 
 95.0334 
 
 718.690 
 
 38} 
 
 119.773 
 
 1.141.591 
 
 22} 
 
 70.6860 
 
 397.609 
 
 30} 
 
 95.4261 
 
 724.642 
 
 38} 
 
 120.166 
 
 1,149.089 
 
 22f 
 
 71.0787 
 
 402.038 
 
 30} 
 
 95.8188 
 
 730.618 
 
 38} 
 
 120.559 
 
 1.156.612 
 
 22} 
 
 71.4714 
 
 406.494 
 
 30} 
 
 96.2115 
 
 736.619 
 
 38} 
 
 120.952 
 
 1,164.159 
 
 22} 
 
 71.8641 
 
 410.973 
 
 30} 
 
 96.6042 
 
 742.645 
 
 38} 
 
 121.344 
 
 1,171.731 
 
 23 
 
 72.2568 
 
 415.477 
 
 30} 
 
 96.9969 
 
 748.695 
 
 38} 
 
 121.737 
 
 1,179.327 
 
 23} 
 
 72.6495 
 
 420.004 
 
 31 
 
 97.3896 
 
 754.769 
 
 38} 
 
 122.130 
 
 1,186.948 
 
 23} 
 
 73.0422 
 
 424.558 
 
 31} 
 
 97.7823 
 
 760.869 
 
 39 
 
 122.522 
 
 1,194.593 
 
 23| 
 
 73.4349 
 
 429.135 
 
 31} 
 
 98.1750 
 
 766.992 
 
 39} 
 
 122.915 
 
 1.202.263 
 
 23} 
 
 73.8276 
 
 433.737 
 
 31} 
 
 98.5677 
 
 773.140 
 
 39} 
 
 123.308 
 
 1.209.958 
 
 23f 
 
 74.2203 
 
 438.364 
 
 31} 
 
 98.9604 
 
 779.313 
 
 39} 
 
 123.700 
 
 1.217.677 
 
 23} 
 
 74.6130 
 
 443.015 
 
 31} 
 
 99.3531 
 
 785.510 
 
 39} 
 
 124.093 
 
 1,225.420 
 
 23} 
 
 75.0057 
 
 447.690 
 
 31} 
 
 99.7458 
 
 791.732 
 
 39} 
 
 124.486 
 
 1.233.1S8 
 
 24 
 
 75.3984 
 
 452.390 
 
 31} 
 
 100.1385 
 
 797.979 
 
 39} 
 
 124. 879 
 
 1.240.981 
 
 24} 
 
 75.7911 
 
 457.115 
 
 32 
 
 100.5312 
 
 804.250 
 
 39} 
 
 125.271 
 
 1.248.798 
 
 24} 
 
 76.1838 
 
 461.864 
 
 32} 
 
 100.9239 
 
 810.545 
 
 40 
 
 125.664 
 
 1,256.640 
 
 24f 
 
 76.5765 
 
 466.638 
 
 32} 
 
 101.3166 
 
 816.865 
 
 40} 
 
 126.057 
 
 1.264.510 
 
 24} 
 
 76.9692 
 
 471.436 
 
 32} 1 
 
 101.7093 
 
 823.210 
 
 40} 
 
 126.449 
 
 1,272.400 
 
 24} 
 
 77.3619 
 
 476.259 
 
 32} 
 
 102.1020 
 
 829.579 
 
 40} 
 
 126.842 
 
 1,280.310 
 
 24} 
 
 77.7546 
 
 481.107 
 
 32} 
 
 102.4947 
 
 835.972 
 
 40} 
 
 127.235 
 
 1.288.250 
 
 24} 
 
 78.1473 
 
 485.979 
 
 32} : 
 
 102.8874 
 
 842.391 
 
 40} 
 
 127.627 
 
 1.296.220 
 
 25 
 
 78.5400 
 
 490.875 
 
 32} 
 
 103.280 
 
 848.833 
 
 40} 
 
 128.020 
 
 1.304.210 
 
 25} 
 
 78.9327 
 
 495.796 
 
 33 
 
 103.673 
 
 855.301 
 
 40} 
 
 128.413 
 
 1.312.220 
 
 25} 
 
 79.3254 
 
 500.742 
 
 33} 
 
 104.065 
 
 861.792 
 
 41 
 
 128.806 
 
 1.320.260 
 
 25} 
 
 79.7181 
 
 505.712 
 
 33} 
 
 104.458 
 
 868.309 
 
 41} 
 
 129.198 
 
 1.328.320 
 
 25} 
 
 80.1108 
 
 510.706 
 
 33} 
 
 104.851 
 
 874.850 
 
 41} 
 
 129.591 
 
 1.336.410 
 
 25} 
 
 80.5035 
 
 515.726 
 
 33} 
 
 105.244 
 
 881.415 
 
 41} 
 
 129.984 
 
 1.344.520 
 
 25} 
 
 80.8962 
 
 520.769 
 
 33} 
 
 105.636 
 
 888.005 
 
 41} 
 
 130.376 
 
 1.352.660 
 
 25} 
 
 81.2889 
 
 525.838 
 
 33} 
 
 106.029 
 
 894.620 
 
 41} 
 
 130.769 
 
 1.360.820 
 
 26 
 
 81.6816 
 
 530.930 
 
 33} 
 
 106.422 
 
 901.259 
 
 41} 
 
 131.162 
 
 1,369.000 
 
 26} 
 
 82.0743 
 
 536.048 
 
 34 
 
 106.814 
 
 907.922 
 
 41} 
 
 131.554 
 
 1,377.210 
 
 26} 
 
 82.4670 
 
 541.190 
 
 34} 
 
 107.207 
 
 914.611 
 
 42 
 
 131.947 
 
 1,385.450 
 
 26} 
 
 82.8597 
 
 546.356 
 
 34} 
 
 107.600 
 
 921.323 
 
 42} 
 
 132.340 
 
 1,393.700 
 
 26} 
 
 83.2524 
 
 551.547 
 
 34} 
 
 107.992 
 
 928.061 
 
 42} 
 
 132.733 
 
 1,401.990 
 
 26} 
 
 83.6451 
 
 556.763 
 
 84} 
 
 108.385 
 
 934.822 
 
 42} 
 
 133.125 
 
 1,410.300 
 
 26} 
 
 84.0378 
 
 562.003 
 
 34} 
 
 108.778 
 
 941.609 
 
 42} 
 
 133.518 
 
 1,418.630 
 
 26} 
 
 84.4305 
 
 567.267 
 
 34} 
 
 109.171 
 
 948.420 
 
 42} 
 
 133.911 
 
 1.426.990 
 
 27 
 
 84.8232 
 
 572.557 
 
 34} 
 
 109.563 
 
 955.255 
 
 42} 
 
 134.303 
 
 1,435.370 
 
 27} 
 
 85.2159 
 
 577.870 
 
 35 
 
 109.956 
 
 962.115 
 
 42} 
 
 134.696 
 
 1.443.770 
 
 27} 
 
 85.6086 
 
 583.209 
 
 35} 
 
 110.349 
 
 969.000 
 
 43 
 
 135.089 
 
 1.452.200 
 
 27} 
 
 86.0013 
 
 588.571 
 
 35} 
 
 110.741 
 
 975.909 
 
 43} -! 
 
 135.481 
 
 1,460.660 
 
 27} 
 
 86.3940 
 
 593.959 
 
 35} 
 
 111.134 
 
 982.842 
 
 43} 
 
 135.874 
 
 1.469.140 
 
 27} 
 
 86.7867 
 
 599.371 
 
 35} 
 
 111.527 
 
 989.800 
 
 43} 
 
 136.267 
 
 1,477.640 
 
 27} 
 
 87.1794 
 
 604.807 
 
 35} 
 
 111.919 
 
 996.783 
 
 43} 
 
 136.660 
 
 1,486.170 
 
 27} 
 
 87.5721 
 
 610.268 
 
 35} 
 
 112.312 
 
 1,003.790 
 
 43} 
 
 137.052 
 
 1,494.730 
 
 28 
 
 87.9648 
 
 615.754 
 
 35} 
 
 112.705 
 
 1,010.822 
 
 43} 
 
 137.445 
 
 1,503.300 
 
CIRCUMFERENCES AND AREAS OF CIRCLES. 
 
 563 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 43! 
 
 44 
 44| 
 44! 
 44! 
 44! 
 44! 
 44! 
 44! 
 
 45 
 45! 
 45! 
 45f 
 45! 
 45| 
 45! 
 45! 
 
 46 
 46! 
 46! 
 46f 
 46! 
 46f 
 46! 
 46! 
 
 47 
 47! 
 47! 
 47! 
 47! 
 47! 
 47! 
 47! 
 
 48 
 48! 
 48! 
 48! 
 48! 
 48! 
 48! 
 48! 
 
 49 
 49! 
 49! 
 49! 
 49! 
 49! 
 49! 
 49! 
 
 50 
 50! 
 50! 
 50! 
 50! 
 50! 
 50! 
 50! 
 
 51 
 51! 
 51! 
 51! 
 51! 
 51| 
 
 137.838 
 
 138.230 
 
 138.623 
 
 139.016 
 
 139.408 
 
 139.801 
 
 140.194 
 
 140.587 
 
 140.979 
 
 141.372 
 
 141.765 
 
 142.157 
 
 142.550 
 
 142.943 
 
 143.335 
 
 143.728 
 
 144.121 
 
 144.514 
 
 144.906 
 
 145.299 
 
 145.692 
 
 146.084 
 
 146.477 
 
 146.870 
 
 147.262 
 
 147.655 
 
 148.048 
 
 148.441 
 
 148.833 
 
 149.226 
 
 149.619 
 
 150.011 
 
 150.404 
 
 150.797 
 
 151.189 
 
 151.582 
 
 151.975 
 
 152.368 
 
 152.760 
 
 153.153 
 
 153.546 
 
 153.938 
 
 154.331 
 
 154.724 
 
 155.116 
 
 155.509 
 
 155.902 
 
 156.295 
 
 156.687 
 
 157.080 
 
 157.473 
 
 157.865 
 
 158.258 
 
 158.651 
 
 159.043 
 
 159.436 
 
 159.829 
 
 160.222 
 
 160.614 
 
 161.007 
 
 161.400 
 
 161.792 
 
 162.185 
 
 1,511.910 
 
 1,520.530 
 
 1,529.190 
 
 1,537.860 ' 
 
 1,546.56 
 
 1,555.29 
 
 1.564.04 
 1,572.81 
 1,581.61 
 1,590.43 
 1,599.28 
 1,608.16 
 
 1.617.05 
 1,625.97 
 1,634.92 
 
 1.643.89 
 
 1.652.89 
 1,661.91 
 
 1.670.95 
 1,680.02 
 1,689.11 
 1,698.23 
 
 1.707.37 
 
 1.716.54 
 
 1.725.73 
 
 1.734.95 
 1,744.19 
 
 1.753.45 
 
 1.762.74 
 
 1.772.06 
 1,781.40 
 1,790.76 
 
 1.800.15 
 1,809.56 
 1,819.00 
 
 1.828.46 
 
 1.837.95 
 
 1.847.46 
 1,856.99 
 
 1.866.55 
 1,876.14 
 
 1.885.75 
 
 1.895.38 
 1,905.04 
 1,914.72 
 1,924.43 
 
 1.934.16 
 1,943.91 
 1,953.69 
 1,963.50 
 1,973.33 
 1,983.18 
 1,993.06 
 
 2.002.97 
 
 2.012.89 
 
 2.022.85 
 
 2.032.82 
 
 2.042.83 
 
 2.052.85 
 
 2.062.90 
 
 2.072.98 
 2,083.08 
 2,093.20 
 
 51! 
 
 51! 
 
 52 
 52! 
 
 52! 
 
 52! 
 
 52! 
 
 52! 
 
 52! 
 
 52! 
 
 53 
 53! 
 
 53! 
 
 53! 
 
 53! 
 
 53! 
 
 53! 
 
 53! 
 
 54 
 54! 
 
 54! 
 
 54! 
 
 54! 
 
 54! 
 
 54! 
 
 54! 
 
 55 
 55! 
 55! 
 55! 
 55! | 
 55! 
 55! 
 55! 
 
 56 
 56! 
 56! 
 56! 
 56! 
 56! 
 56! 
 56! 
 
 57 
 57! 
 57! 
 57! 
 57! 
 57! 
 57! 
 57! 
 
 58 
 58! 
 58! 
 58! 
 58! 
 58! 
 58! 
 58! 
 
 59 
 59! 
 59! 
 59! 
 59! 
 
 162.578 
 
 162.970 
 
 163.363 
 
 163.756 
 
 164.149 
 
 164.541 
 
 164.934 
 
 165.327 
 
 165.719 
 
 166.112 
 
 166.505 
 
 166.897 
 
 167.290 
 
 167.683 
 
 168.076 
 
 168.468 
 
 168.861 
 
 169.254 
 
 169.646 
 
 170.039 
 
 170.432 
 
 170.824 
 
 171.217 
 
 171.610 
 
 172.003 
 
 172.395 
 
 172.788 
 
 173.181 
 
 173.573 
 
 173.966 
 
 174.359 
 
 174.751 
 
 175.144 
 
 175.537 
 
 175.930 
 
 176.322 
 
 176.715 
 
 177.108 
 
 177.500 
 
 177.893 
 
 178.286 
 
 178.678 
 
 179.071 
 
 179.464 
 
 179.857 
 
 180.249 
 
 180.642 
 
 181.035 
 
 181.427 
 
 181.820 
 
 182.213 
 
 182.605 
 
 182.998 
 
 183.391 
 
 183.784 
 
 184.176 
 
 184.569 
 
 184.962 
 
 185.354 
 
 185.747 
 
 186.140 
 
 186.532 
 
 186.925 
 
 2,103.35 
 
 2.113.52 
 2,123.72 
 2,133.94 
 
 2.144.19 
 2,154.46 
 2,164.76 
 2,175.08 
 2,185.42 
 2,195.79 
 
 2.206.19 
 2,216.61 
 
 2.227.05 
 
 2.237.52 
 
 2.248.01 
 
 2.258.53 
 
 2.269.07 
 
 2.279.64 
 
 2.290.23 
 2,300.84 
 
 2.311.48 
 2,322.15 
 
 2.332.83 
 2,343.55 
 
 2.354.29 
 
 2.365.05 
 
 2.375.83 
 
 2.386.65 
 
 2.397.48 
 2,408.34 
 
 2.419.23 
 2,430.14 
 
 2.441.07 
 2,452.03 
 
 2.463.01 
 
 2.474.02 
 
 2.485.05 
 2,496.11 
 
 2.507.19 
 
 2.518.30 
 2,529.43 
 2,540.58 
 2,551.76 
 
 2.562.97 
 
 2.574.20 
 2,585.45 
 2,596.73 
 
 2.608.03 
 
 2.619.36 
 2,630.71 
 2,642.09 
 2,653.49 
 2,664.91 
 
 2.676.36 
 
 2.687.84 
 2,699.33 
 2,710.86 
 2,722.41 
 
 2.733.98 
 2,745.57 
 
 2.757.20 
 
 2.768.84 
 2,780.51 
 
 59! 
 
 59! 
 
 59! 
 
 60 
 
 60! 
 
 60! 
 
 60! 
 
 60! 
 
 60! 
 
 60! 
 
 60! 
 
 61 
 
 61! 
 
 61! 
 
 61! 
 
 61! 
 
 61! 
 
 61! 
 
 61! 
 
 62 
 
 62! 
 
 62! 
 
 62! 
 
 62! 
 
 62! 
 
 62! 
 
 62! 
 
 63 
 63! 
 63! 
 63! 
 63! 
 63! 
 63! 
 63! 
 
 64 
 64! 
 64! 
 64! 
 64! 
 64! 
 64! 
 64! 
 
 65 
 65! 
 65! 
 65! 
 65! 
 65! 
 65! 
 65! 
 
 66 
 66! 
 66! 
 66! 
 66! 
 66! 
 66! 
 66! 
 67 
 67! 
 67! 
 67! 
 
 187.318 
 
 187.711 
 
 188.103 
 
 188.496 
 
 188.889 
 
 189.281 
 
 189.674 
 
 190.067 
 
 190.459 
 
 190.852 
 
 191.245 
 
 191.638 
 
 192.030 
 
 192.423 
 
 192.816 
 
 193.208 
 
 193.601 
 
 193.994 
 
 194.386 
 
 194.779 
 
 195.172 
 
 195.565 
 
 195.957 
 
 196.350 
 
 196.743 
 
 197.135 
 
 197.528 
 
 197.921 
 
 198.313 
 
 198.706 
 
 199.099 
 
 199.492 
 
 199.884 
 
 200.277 
 
 200.670 
 
 201.062 
 
 201.455 
 
 201.848 
 
 202.240 
 
 202.633 
 
 203.026 
 
 203.419 
 
 203.811 
 
 204.204 
 
 204.597 
 
 204.989 
 
 205.382 
 
 205.775 
 
 206.167 
 
 206.560 
 
 206.953 
 
 207.346 
 
 207.738 
 
 208.131 
 
 208.524 
 
 208.916 
 
 209.309 
 
 209.702 
 
 210.094 
 
 210.487 
 
 210.880 
 
 211.273 
 
 211.665 
 
 2,792.21 
 
 2,803.93 
 
 2.815.67 
 2,827.44 
 2,839.23 
 2,851.05 
 
 2.862.89 
 2,874.76 
 2,886.65 
 
 2.898.57 
 
 2.910.51 
 
 2.922.47 
 
 2.934.46 
 
 2.946.48 
 
 2.958.52 
 
 2.970.58 
 
 2.982.67 
 2,994.78 
 
 3.006.92 
 3,019.08 
 3,031.26 
 
 3.043.47 
 3,055.71 
 3,067.97 
 
 3.080.25 
 
 3.092.56 
 
 3.104.89 
 
 3.117.25 
 3,129.64 
 3,142.04 
 
 3.154.47 
 
 3.166.93 
 3,179.41 
 3,191.91 
 
 3.204.44 
 3,217.00 
 
 3.229.58 
 
 3.242.18 
 3,254.81' 
 3,267.46 
 3,280.14 
 3,292.84 
 
 3.305.56 
 3,318.31 
 3,331.09 
 
 3.343.89 
 3,356.71 
 
 3.369.56 
 
 3.382.44 
 3,395.33 
 3,408.26 
 3,421.20 
 
 3.434.17 
 
 3.447.17 
 
 3.460.19 
 
 3.473.24 
 3,486.30 
 3,499.40 
 3,512.52 
 3,525.66 
 3,538.83 
 3,552.02 
 
 3.565.24 
 
564 
 
 CIRCUMFERENCES AND AREAS OF CIRCLES. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. | 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 m 
 
 1 
 
 212.058 
 
 3,578.48 
 
 75# 
 
 236.798 
 
 4,462.16 
 
 83# 
 
 261.538 
 
 5,443.26 
 
 67# 
 
 
 212.451 
 
 3,591.74 
 
 75# 
 
 237.191 
 
 4,476.98 
 
 83# 
 
 261.931 
 
 5,459.62 
 
 67# 
 
 
 212.843 
 
 3,605.04 
 
 75# 
 
 237.583 
 
 4,491.81 
 
 83# 
 
 262.324 
 
 5,476.01 
 
 67# 
 
 
 213.236 
 
 3,618.35 
 
 75# 
 
 237.976 
 
 4,506.67 
 
 83# 
 
 262.716 
 
 5,492.41 
 
 68 
 
 
 213.629 
 
 3,631.69 
 
 75# 
 
 238.369 
 
 4,521.56 
 
 83# 
 
 263.109 
 
 5,508.84 
 
 68# 
 
 
 214.021 
 
 3,645.05 
 
 76 
 
 238.762 
 
 4,536.47 
 
 83# 
 
 263.502 
 
 5,525.30 
 
 68# 
 
 
 214.414 
 
 3,658.44 
 
 76# 
 
 239.154 
 
 4,551.41 
 
 84 
 
 263,894 
 
 5,541.78 
 
 68# 
 
 214.807 
 
 3,671.86 
 
 76# 
 
 239.547 
 
 4,566.36 
 
 84# 
 
 264.287 
 
 5,558.29 
 
 68# 
 
 
 215.200 
 
 3,685.29 
 
 76# 
 
 239.940 
 
 4,581.35 
 
 84# 
 
 264.680 
 
 5,574.82 
 
 68 1 
 
 215.592 
 
 3,698.76 
 
 76# 
 
 240.332 
 
 4,596.36 
 
 84# 
 
 265.072 
 
 5,591.37 
 
 68# 
 
 215.985 
 
 3,712.24 
 
 76# 
 
 240.725 
 
 4,611.39 
 
 84# 
 
 265.465 
 
 5,607.95 
 
 68# 
 
 
 216.378 
 
 3,725.75 
 
 76# 
 
 241.118 
 
 4,626.45 
 
 84# 
 
 265.858 
 
 5,624.56 
 
 69 
 
 
 216.770 
 
 3,739.29 
 
 76# 
 
 241.510 
 
 4,641.53 
 
 84# 
 
 266.251 
 
 5,641.18 
 
 69# 
 
 
 217.163 
 
 3,752.85 
 
 77 
 
 241.903 
 
 4,656.64 
 
 84# 
 
 266.643 
 
 5,657.84 
 
 m 
 
 
 217.556 
 
 3,766.43 
 
 771 
 
 242.296 
 
 4,671.77 
 
 85 
 
 267.036 
 
 5,674.51 
 
 69# 
 
 217.948 
 
 3,780.04 
 
 77# 
 
 242.689 
 
 4,686.92 
 
 85# 
 
 267.429 
 
 5,691.22 
 
 69# 
 
 
 218.341 
 
 3,793.68 
 
 77# 
 
 243.081 
 
 4,702.10 
 
 85# 
 
 267.821 
 
 5,707.94 
 
 69# 
 
 218.734 
 
 3,807.34 
 
 77# 
 
 243.474 
 
 4,717.31 
 
 85# 
 
 268.214 
 
 5,724.69 
 
 69| 
 
 
 219.127 
 
 3,821.02 
 
 77# 
 
 243.867 
 
 4,732.54 
 
 85# 
 
 268.607 
 
 5,741.47 
 
 69# 
 
 
 219.519 
 
 3,834.73 
 
 77# 
 
 244.259 
 
 4,747.79 
 
 85# 
 
 268.999 
 
 5,758.27 
 
 70 
 
 
 219.912 
 
 3,848.46 
 
 77# 
 
 244.652 
 
 4,763.07 
 
 85# 
 
 269.392 
 
 5,775.10 
 
 70# 
 
 
 220.305 
 
 3,862.22 
 
 78 
 
 245.045 
 
 4,778.37 
 
 85# 
 
 269.785 
 
 5,791.94 
 
 70# 
 
 220.697 
 
 3,876.00 
 
 78# 
 
 245.437 
 
 4,793.70 
 
 86 
 
 270.178 
 
 5,808.82 
 
 701 
 
 
 221.090 
 
 3,889.80 
 
 78# 
 
 245.830 
 
 4,809.05 
 
 86# 
 
 270.570 
 
 5,825.72 
 
 70# 
 
 
 221.483 
 
 3,903.63 
 
 78# 
 
 246.223 
 
 4,824.43 
 
 86# 
 
 270.963 
 
 5,842.64 
 
 70# 
 
 221.875 
 
 3,917.49 
 
 78# 
 
 246.616 
 
 4,839.83 
 
 86# 
 
 271.356 
 
 5,859.59 
 
 70# 
 
 222.268 
 
 3,931.37 
 
 78# 
 
 247.008 
 
 4,855.26 
 
 86# 
 
 271.748 
 
 5,876.56 
 
 70# 
 
 222.661 
 
 3,945.27 
 
 78# 
 
 247.401 
 
 4,870.71 
 
 86# 
 
 272.141 
 
 5,893.55 
 
 71 
 
 
 223.054 
 
 3,959.20 
 
 78# 
 
 247.794 
 
 4,886.18 
 
 86# 
 
 272.534 
 
 5,910.58 
 
 71# 
 
 t 
 
 223.446 
 
 3,973.15 
 
 79 
 
 248.186 
 
 4,901.68 
 
 86# 
 
 272.926 
 
 5,927.62 
 
 71# 
 
 
 223.839 
 
 3,987.13 
 
 79# 
 
 248.579 
 
 4,917.21 
 
 87 
 
 273.319 
 
 5,944.69 
 
 71# 
 
 t 
 
 224.232 
 
 4,001.13 
 
 79# 
 
 248.972 
 
 4,932.75 
 
 87# 
 
 273.712 
 
 5,961.79 
 
 71# 
 
 \ 
 
 224.624 
 
 4,015.16 
 
 79# 
 
 249.364 
 
 4,948.33 
 
 87# 
 
 274.105 
 
 5,978.91 
 
 71# 
 
 \ 
 
 225.017 
 
 4,029.21 
 
 79# 
 
 249.757 
 
 4,963.92 
 
 87# 
 
 274.497 
 
 5,996.05 
 
 71# 
 
 \ 
 
 225.410 
 
 4,043.29 
 
 79# 
 
 250.150 
 
 4,979.55 
 
 87# 
 
 274.890 
 
 6,013.22 
 
 71# 
 
 225.802 
 
 4,057.39 
 
 79# 
 
 250.543 
 
 4,995.19 
 
 87# 
 
 275.283 
 
 6,030.41 
 
 72 
 
 
 226.195 
 
 4,071.51 
 
 79# 
 
 250.935 
 
 5,010.86 
 
 87# 
 
 275.675 
 
 6,047.63 
 
 72- 
 
 ■ 
 
 226.588 
 
 4,085.66 
 
 80 
 
 251.328 
 
 5,026.56 
 
 87# 
 
 276.068 
 
 6,064.87 
 
 72, 
 
 
 226.981 
 
 4,099.84 
 
 80# 
 
 251.721 
 
 5,042.28 
 
 88 
 
 276.461 
 
 6,082.14 
 
 72 
 
 ■ 
 
 227.373 
 
 4,114.04 
 
 80# 
 
 252.113 
 
 5,058.03 
 
 88# 
 
 276.853 
 
 6,099.43 
 
 72- 
 
 • 
 
 227.766 
 
 4,128.26 
 
 80# 
 
 252.506 
 
 5,073.79 
 
 88# 
 
 277.246 
 
 6,116.74 
 
 72 
 
 ■ 
 
 228.159 
 
 4,142.51 
 
 80# 
 
 252.899 
 
 5,089.59 
 
 88# 
 
 277.629 
 
 6,134.08 
 
 72; 
 
 . 
 
 228.551 
 
 4,156.78 
 
 80# 
 
 253.291 
 
 5,105.41 
 
 88# 
 
 278.032 
 
 6,151.45 
 
 72 
 
 
 228.944 
 
 4,171.08 
 
 80# 
 
 253.684 
 
 5,121.25 
 
 88# 
 
 278.424 
 
 6,168.84 
 
 73 
 
 
 229.337 
 
 4,185.40 
 
 80# 
 
 254.077 
 
 5,137.12 
 
 88# 
 
 278.817 
 
 6,186.25 
 
 73# 
 
 229.729 
 
 4,199.74 
 
 81 
 
 254.470 
 
 5,153.01 
 
 88# 
 
 279.210 
 
 6,203.69 
 
 73# 
 
 230.122 
 
 4,214.11 
 
 81# 
 
 254.862 
 
 5,168.93 
 
 89 
 
 279.602 
 
 6,221.15 
 
 73# 
 
 230.515 
 
 4,228.51 
 
 81# 
 
 255.255 
 
 5,184.87 
 
 89# 
 
 279.995 
 
 6,238.64 
 
 73; 
 
 
 230.908 
 
 4,242.93 
 
 81# 
 
 255.648 
 
 5,200.83 
 
 89# 
 
 280.388 
 
 6,256.15 
 
 73# 
 
 231.300 
 
 4,257.37 
 
 81# 
 
 256.040 
 
 5,216.82 
 
 89# 
 
 280.780 
 
 6,273.69 
 
 73# 
 
 231.693 
 
 4,271.84 
 
 81# 
 
 256.433 
 
 5,232.84 
 
 89# 
 
 281.173 
 
 6,291.25 
 
 73 
 
 i 
 
 232.086 
 
 4,286.33 
 
 81# 
 
 256.826 
 
 5,248.88 
 
 89| 
 
 281.566 
 
 6,308.84 
 
 74 
 
 
 232.478 
 
 4,300.85 
 
 81# 
 
 257.218 
 
 5,264.94 
 
 89# 
 
 281.959 
 
 6,326.45 
 
 74 
 
 L 
 
 232.871 
 
 4,315.39 
 
 82 
 
 257.611 
 
 5,281.03 
 
 89# 
 
 282.351 
 
 6,344.08 
 
 74; 
 
 
 233.264 
 
 4,329.96 
 
 82# 
 
 258.004 
 
 5,297.14 
 
 90 
 
 282.744 
 
 6,361.74 
 
 74# 
 
 233.656 
 
 4,344.55 
 
 82# 
 
 258.397 
 
 5,313.28 
 
 90# 
 
 283.137 
 
 6,379.42 
 
 74 
 
 
 234.049 
 
 4,359.17 
 
 82# 
 
 258.789 
 
 5,329.44 
 
 90# 
 
 283.529 
 
 6,397.13 
 
 74# 
 
 234.442 
 
 4,373.81 
 
 82# 
 
 259.182 
 
 5,345.63 
 
 90# 
 
 283.922 
 
 6,414.86 
 
 74# 
 
 234.835 
 
 4,388.47 
 
 82# 
 
 259.575 
 
 5,361.84 
 
 90# 
 
 284.315 
 
 6,432.62 
 
 74- 
 
 i 
 
 235.227 
 
 4,403.16 
 
 82# 
 
 259.967 
 
 5,378.08 
 
 90# 
 
 284.707 
 
 6,450.40 
 
 75 
 
 
 235.620 
 
 4,417.87 
 
 82# 
 
 260.360 
 
 5,394.34 
 
 90# 
 
 285.100 
 
 6,468.21 
 
 • 75 
 
 i 
 
 236.013 
 
 4,432.61 
 
 83 
 
 250.753 
 
 5,410.62 
 
 90# 
 
 285.493 
 
 6,486.04 
 
 75# 
 
 236.405 
 
 4,447.38 
 
 83# 
 
 261.145 
 
 5,426.93 
 
 91 
 
 285.886 
 
 6,503.90 
 
CIRCUMFERENCES AND AREAS OF CIRCLES. 
 
 565 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. 
 
 Diam. 
 
 Circum. 
 
 Area. J 
 
 91* 
 
 91* 
 
 91* 
 
 91* 
 
 91* 
 
 91* 
 
 91* 
 
 92 
 92* 
 92* 
 92* 
 92* 
 92* 
 92* 
 92* 
 
 93 
 93* 
 93£ 
 93* 
 93* 
 93* 
 93* 
 93* 
 
 94 
 
 286.278 
 
 286.671 
 
 287.064 
 
 287.456 
 
 287.849 
 
 288.242 
 
 288.634 
 
 289.027 
 
 289.420 
 
 289.813 
 
 290.205 
 
 290.598 
 
 290.991 
 
 291.383 
 
 291.776 
 
 292.169 
 
 292.562 
 
 292.954 
 
 293.347 
 
 293.740 
 
 294.132 
 
 294.525 
 
 294.918 
 
 295.310 
 
 6.521.78 
 
 6.539.68 
 6,557.61 
 
 6.575.56 
 
 6.593.54 
 
 6.611.55 
 
 6.629.57 
 6,647.63 
 6,665.70 
 6,683.80 
 6,701.93 
 6,720.08 
 6,738.25 
 6,756.45 
 
 6.774.68 
 
 6.792.92 
 6,811.20 
 6,829.49 
 6,847.82 
 6,866.16 
 6,884.53 
 
 6.902.93 
 6,921.35 
 
 6.939.79 
 
 94* 
 
 94* 
 
 94* 
 
 94* 
 
 94* 
 
 94* 
 
 94* 
 
 95 
 95* 
 95* 
 95* 
 95* 
 95* 
 95* 
 95* 
 
 96 
 96* 
 96* 
 96* 
 96* 
 96* 
 96* 
 96* 
 
 97 
 
 295.703 
 
 296.096 
 
 296.488 
 
 296.881 
 
 297.274 
 
 297.667 
 
 298.059 
 
 298.452 
 
 298.845 
 
 299.237 
 
 299.630 
 
 300.023 
 
 300.415 
 
 300.808 
 
 301.201 
 
 301.594 
 
 301.986 
 
 302.379 
 
 302.772 
 
 303.164 
 
 303.557 
 
 303.950 
 
 304.342 
 
 304.735 
 
 6,958.26 
 
 6,976.76 
 
 6,995.28 
 
 7.013.82 
 7,032.39 
 
 7.050.98 
 
 7.069.59 
 
 7.088.24 
 
 7.106.90 
 
 7.125.59 
 7,144.31 
 7,163.04 
 7,181.81 
 
 7.200.60 
 7,219.41 
 
 7.238.25 
 7,257.11 
 
 7.275.99 
 
 7.294.91 
 7,313.84 
 7,332.80 
 
 7.351.79 
 
 7.370.79 
 
 7.389.83 
 
 97* 
 
 97* 
 
 97* 
 
 97* 
 
 97* 
 
 97* 
 
 97* 
 
 98 
 98* 
 98* 
 98* 
 98* 
 98* 
 98* 
 98* 
 
 99 
 99* 
 99* 
 99* 
 99* 
 99* 
 
 • 99* 
 99* 
 100 
 
 305.128 
 
 305.521 
 
 305.913 
 
 306.306 
 
 306.699 
 
 307.091 
 
 307.484 
 
 307.877 
 
 308.270 
 
 308.662 
 
 309.055 
 
 309.448 
 
 309.840 
 
 310.233 
 
 310.626 
 
 311.018 
 
 311.411 
 
 311.804 
 
 312.196 
 
 312.589 
 
 312.982 
 
 313.375 
 
 313.767 
 
 314.160 
 
 7,408.89 
 
 7.427.97 
 7,447.08 
 
 7.466.21 
 
 7.485.37 
 7,504.55 
 7,523.75 
 
 7.542.98 
 7,562.24 
 7,581.52 
 7,600.82 
 
 7.620.15 
 7,639.50 
 7,658.88 
 7,678.28 
 7,697.71 
 
 7.717.16 
 7,736.63 
 7,756.13 
 7,775.66 
 
 7.795.21 
 7,814.78 
 
 7.834.38 
 7,854.00 
 
 The preceding table may be used to determine the diameter when 
 the circumference or area is known. Thus, the diameter of a circle haying 
 an area of 7,200 sq. in. is approximately 95* in. 
 
 A GLOSSARY OF MINING TERMS. 
 
 The present glossary is a combination of glossaries of mining terms con- 
 tained in the following works: Coal and Metal Miners’ Pocketbook, Fifth 
 Edition; Raymond’s Glossary of Mining and Metallurgical Terms; Powers 
 Pocketbook for Miners and Metallurgists; Locke’s Miners ? Pocketbook; 
 Vol AC, Second Pennsylvania Geological Survey; Ilhseng s Manual ot 
 Mining; Chism’s Encyclopedia of Mexican Mining Law; a Glossary of Terms 
 as Used in Coal Mining, by W. S. Gresley; 11th Annual Report of the State 
 Mine Inspector of Missouri; Bullman’s Colliery Working and Management 
 Reynolds’ Handbook of Mining Laws; Report of the Mine Inspector of 
 Tennessee for 1897; Smithsonian Report for 1886; together with a large num- 
 ber of words which have been added from various stray sources It is 
 impossible to quote the authority for each definition, as many of the defini- 
 tions are combinations from a number of authors. Where such different 
 definitions have given distinctly different meanings each one has been 
 included, but where there has been expressed merely a slight shade of 
 difference, the definition agreeing most closely with current American 
 practice has been taken, or else modified to suit such .practice. The foreign 
 words selected are those with which an American is most likely, to come 
 in contact, and this portion of the glossary is, of course, not exhaustive. For 
 the large number of purely local terms used m the several coal fields of Great 
 Britain, the reader is referred to Mr. Gresley’s glossary. 
 
566 
 
 Aba 
 
 GLOSSARY. 
 
 Air 
 
 GLOSSARY. 
 
 Abattis (Leicester).— Cross-packing of branches or rough wood, used to keep 
 roads open for ventilation. 
 
 Abra (Spanish).— Fissure in a lode, unfilled or only partially filled. 
 
 Abronziado ( Spanish ) .—Copper sulphides. 
 
 Absolute Pressure— The pressure reckoned from a vacuum. 
 
 Absolute Temperature.—' The temperature reckoned from the absolute zero, 
 —459.2° F. or —273° C. 
 
 Accompt (Cornish).— Settling day or place. 
 
 Achicar (Mexican).— To diminish the quantity of water in any gallery or 
 working, generally by carrying it out in buckets or in leather bags. 
 Achicadores — Laborers employed for said purpose. Achichinques.— Same 
 as Achicadores. Also applied to hangers-on about police courts, etc. 
 Such people as are generally called strikers in the United States. 
 
 Acreage Rent (English).— Royalty or rent for working minerals. 
 
 Adarme (Mexican).— A weight for gold, about 1.8 grams. 
 
 Addlings (North of England).— Earnings. 
 
 Ademador ( Spanish) .—Mine carpenter, or timberman. 
 
 Ademar (Spanish). — To timber. . 
 
 Adit— A nearly horizontal passage from the surface, by which a mine is 
 entered and unwatered with just sufficient slope to insure drainage. 
 In the United States, an adit driven across the measures is usually called 
 a tunnel, though the latter, strictly speaking, passes entirely through a 
 hill, and is open at both ends. 
 
 Adobe.— Sun-dried brick. 
 
 Adventurers— Original prospectors. 
 
 Adverse— To oppose the granting of a patent to mining claim. 
 
 Adze. — A curved cutting instrument for dressing timber. 
 
 Aerage (French).— Ventilation. 
 
 Aerometers.— The air pistons of a Struve ventilator. 
 
 Aerophore— The name given to an apparatus that will enable a man to enter 
 places in mines filled with explosive or other deadly gases, with safety. 
 
 Afinar (Mexican).— Refining gold and silver. 
 
 Afterdamp— The gaseous mixture resulting from an explosion of firedamp. 
 
 Agent— The manager of a mining property. 
 
 Agitator— A mechanical stirrer used in pan amalgamation. 
 
 Ahondar (Spanish).— To sink. 
 
 Air —The current of atmospheric air circulating through and ventilating the 
 workings of a mine. 
 
 Air Box. — Wooden tubes used to convey air for ventilating headings or 
 sinkings or other local ventilation. 
 
 Air Compartment— An air-tight portion of any shaft, winze, rise, or level, 
 used for improving ventilation. 
 
 Air-Course— See Airway. 
 
 Air Crossing— A bridge that carries one air-course over another. 
 
 Air Cushion— A spring caused by confined air. 
 
 Air Door.— A door for the regulation of currents of air through the workings 
 of a mine. „ , . , . , 
 
 Air-End Way (Locke).— Ventilation levels run parallel with mam level. 
 
 Air Furnace. — A reverberatory furnace in which to smelt lead. 
 
 Air Gates (Locke).— (l) Underground roadways, used principally for venti- 
 lating purposes. (2) An air regulator. 
 
 Air Head (Staff).— Ventilation ways. 
 
 Air Heading— An airway. . 
 
 Air Hole (Powers). — A hole drilled in advance to improve ventilation by 
 communication with other workings or the surface. 
 
 Airless End.— The extremity of a stall in longwall workings in which there 
 is no current of air, or circulation of ventilation, but which is kept pure 
 by diffusion and by the ingress and egress of cars, men, etc. 
 
 
AIR 
 
 GLOSS AR Y. 
 
 Aqu 
 
 567 
 
 Air Level— A. level or airway of former workings made use of in subsequent 
 deeper mining operations for ventilating purposes. 
 
 Air Oven —A heated chamber for drying samples of ore, etc. 
 
 Air Five —A pipe made of canvas or metal, or a wooden box used in con- 
 A veying air P to the workmen, or for rock drills or air locomotives. 
 Air-Shaft.— A shaft or pit used expressly for ventilation. 
 
 Air Slit (Yorks). — A short head between other air heads. . , 
 
 Air Sollar .-A brattice carried beneath the tram rails or road bed in a head- 
 
 Air ^Stnclf.—A Sack or chimney built over a shaft for ventilation. 
 
 Airway— Any passage through which air is carried. 
 
 Aitch Piece— Parts of a pump in which the valves are fixed. 
 
 Albanil (Spanish).— Mason. 
 
 AWerU^rnac^— ^continuous reverberatory for mercury ores. 
 
 Alcam (Wales).— Tin. 
 
 jllloy homogeneous mixture of two or more metals by fusion. 
 
 Alluvial Gold— Gold found associated with water- worn material. 
 
 Alluvium. — Gravel, sand, and mud deposited by streams. 
 
 Almadeneta (Spanish).— Stamp head. 
 
 AUern^m^^^nl—Y^^ down, or backward and forward motion. 
 
 Alto ( Mexican) —The hanging wall of a vein. See Respaldos. 
 
 Aludel (Spanish).— Earthen condenser for mercury 
 Amalgam.- An alloy of quicksilver with some other metal. 
 
 Amalgamation. — Absorption of gold and silver by mercury. 
 
 Amalgamator.— One that amalgamates gold and silver ores. 
 
 Ana^is?— The dttSminatio^of the original elements and the proportions 
 
 Anemometer^— An instrument used for measuring the velocity of a ventilating 
 
 An^Beam — A two-limbed beam used for turning angles in shafts, etc. 
 
 heating and then cooling 
 
 Anth^iU- Coal containing a small percentage of volatile matter 
 Anticline.- A flexure or fold in which the rock s on the ° f Pff )sl {| t f 
 
 fold dip awav from each other, like the two legs ot tne letter a. ±iie 
 inclination on one side may be much greater than on n & e n ^ P wSh sides 
 An anticlinal is said to be overturned when the rocks on both sides 
 
 a saddle in a mineral vein, or the line along 
 the summit of a vein, from which the vein dips in opposite directions. 
 Anticlinal Flexure: Anticlinal Fold. — An anticline. < . . 
 
 Antiauos Los (Mexican).— The Spanish or Indian miners of colonial times. 
 Antimony Star.— The metal antimony when crystallized, showing fern-like 
 
 that rewash or rework tailings from silver 
 
 ^pcros^exican) 8 -^!' kinds^miningiupplies in general. Aperador.-A 
 
 Apex™ Th e e eP linding point at the top of a slope or incHned plane the 
 knuckle; also, the top of an anticlinal. In the U. S. Revised statutes, 
 the end or edge of a vein nearest the surface. 
 
 A pique (Mexican).— Perpendicular. 
 
 Apolvillados (Spanish). — Superior ores. a «nir- 
 
 Anrnn (English) — (1) A covering of timber, stone, or metal, .to protect a sur 
 ^ Pr ?ace a^ainst the actionof water flowing’over it. (2) A hinged extension 
 to a loading chute. 
 
 Aprons . — Stamp-battery copper plates. 
 
 Aqua Fortis. — Nitric acid. * . _ .. . . . 
 
 Aqua Regia— A mixture of hydrochloric acid and nitric acid. 
 
 Aqueduct— An artificial elevated way for carrying water. 
 
568 
 
 Ara 
 
 GLOSSARY. 
 
 AZO 
 
 Arajo (Mexican).— See Hatajo. 
 
 Arch (Cornish).— Portion of lode left standing to support hanging wall, or 
 because too poor. 
 
 Archean.— An early period of geological time. 
 
 Arching— Brickwork or stonework forming the roof of any underground 
 roadway. 
 
 Arenaceous. — Sandy; rocks are arenaceous when they contain a considerable 
 percentage of sand. 
 
 Arends Tap— An inverted siphon for drawing molten lead from a crucible 
 or furnace. ) 
 
 Arenillas (Spanish). — Refuse earth. 
 
 Argentiferous. — Silver-bearing. 
 
 Argillaceous. — Clayey; rocks are argillaceous when they contain a consid- 
 erable percentage of clay, or have some of the characteristics of clay. 
 
 Argol. — Crude tartar deposited from wine. 
 
 Arian (Wales).— Silver. 
 
 Arm. — The inclined leg of a set of timber. 
 
 Arrage (North England).— Sharp corner. 
 
 Arrastre. — A circular trough in which drags are pulled round by being con- 
 nected with a central revolving shaft by an arm and chain. Used for 
 grinding and amalgamating ores. Arrastre de cuchara , spoon arrastre; 
 de marca, large arrastre; de mula, mule-power arrastre. 
 
 Arvastrar (Mexican).— To drag along the ground. Arrastrar el Agua.— To 
 almost completely exhaust the water in a sump or working. 
 
 Arroba (Mexican). — 25 lb. 
 
 Artesian Well. — An artificial channel of escape, made by a bore hole, for a 
 subterranean stream, subject to hydrostatic pressure. 
 
 Ascensional Ventilation. — The arrangement of the ventilating currents in 
 such a manner that the air shall continuously rise until reaching the 
 bottom of the upcast shaft. Particularly applicable to steep seams. 
 
 Ashlar.— A facing of cut stone applied to a backing of rubble or rough 
 masonry or brickwork. 
 
 Aspirail (French).— Opening for ventilation. 
 
 Assay.— The determination of the quality and quantity of any particular 
 substance in a mineral. Assayer. — One who performs assays. 
 
 Assessment Work. — The annual work necessary to hold a mining claim. 
 
 Astel. — Overhead boarding in a gallery. 
 
 Astyllen (Cornish).— Small dam in an adit; partition between ore and deads 
 on grass. 
 
 .itacador (Mexican).— A tamping bar or tamping stick. 
 
 Atecas (Mexican).— Same as Achicadores , etc. 
 
 Atierres (Spanish).— Refuse rock or dirt inside a mine. 
 
 Attle (Cornish). — Refuse rock. 
 
 Attle {Addle). — The waste of a mine. 
 
 Attrition. — The act of wearing away by friction. 
 
 Auger Stem.— The iron rod or bar to which the bit is attached in rope drilling. 
 
 Auget. — Priming tube. 
 
 Aur (Wales).— Gold. 
 
 Auriferous. — Gold-bearing. 
 
 Ausscharen (German).— Junction of lodes. 
 
 Auszimmern ( German) . — Timbering. 
 
 Average Produce (Cornish). — Percentage of fine copper in ore. 
 
 Average Standard (Cornish). — Price of pure copper in ore. 
 
 Aviador (Spanish). — One who provides the capital to work a mine. 
 Avio. — Money furnished to the proprietors of a mine to work the mine, 
 by another person, the Aviador. Avio Contract. — A contract between 
 two parties for working a mine by which one of the parties, the aviador , 
 furnishes the money to the proprietors for working the mine. 
 
 Axis. — An imaginary line passing through a body that may be supposed to 
 revolve around it. 
 
 Azimuth— The azimuth of a body is that arc of the horizon that is included 
 between the meridian circle at the given place and a vertical plane 
 passing through the body. It is always measured from due north around 
 to the right. 
 
 Azogue (Spanish). — Mercury. Azogueria. — Amalgamating works. Azoguero. 
 Amalgamator. The person in charge of a patio works. Azogues— Free 
 milling ores. 
 
 Azoic.— The age of rocks that were formed before animal life existed. 
 
Bac 
 
 GLOSSARY. 
 
 Ban 
 
 569 
 
 Back. — (1) A plane or cleavage in coal, etc., having frequently a smooth 
 parting and some sooty coal included in it. (2) The inner end of a 
 heading or gangway. (3) To throw back into the gob or waste the small 
 slack, dirt, etc. g (4) To roll large coals out of a waste for loading into cars. 
 Back Balance.— A self-acting incline m the mnie^wherea glance caranda 
 carriage in which the mine car is placed are used. The lo aded car upo A . 
 the carriage will hoist the balance car, and the balance car will hoist the 
 
 Backbye^WorL— Wor^done between the shaft and the working face, in 
 contradistinction to face work, or work done at the face. . f 
 
 Back Casing.— A wall or lining of dry bricks used m ®^ 1 ?f t thr ^f h Q £ 1 
 deposits, the permanent walling being built up within it. The use oi 
 timber cribs and planking serves the same purpose. 
 
 Back End (England). -The last portion of a jud. 
 
 Backina.—( 1) The rough masonry of a wall faced with finer work. (2) Eartn 
 deposited behind a retaining wall, etc. (3) Timbers let into notches m 
 the rock across the top of a level. , . , - Plirh „ fnr 
 
 Backing Deals— Deal boards or planking placed at the back of curbs tor 
 supporting the sides of a shaft that is liable to run. 
 
 Back Joint. — Joint plane more or less parallel to the strike of the cleavage, 
 
 BaddasA— {!) Backward suction of air-currents produced after an explosion 
 of firedamp. (2) Reentry of air into a fan. , . 
 
 Back of Ore.— The ore between two levels which has to be worked from the 
 
 Back Pressured— The loss, expressed in pounds per square inch, due to getting 
 the steam out of the cylinder after it has done its work. 
 
 Back Shift— Afternoon shift. , . . . 
 
 Back Skin (North of England). -A leather jacket for wet workings. 
 Backstay.— A wrought-iron forked bar attached to the back of pars when 
 ascending an inclined plane, which throws them off the rails if the rope 
 
 BaffEnds^— Long wooden edges for adjusting linings in sinking shafts dur- 
 ing the operation of fixing the lining. 
 
 Baffle— To brush out firedamp. 
 
 Bait.— Provisions. _ , _ 
 
 Bajo (Mexican).— The footwall of a vein. See Respaldo. 
 
 B alance.— 0*) The counterpoise or weights attached to the 
 
 engine, to assist the engine m lifting the load out of a shaft bottom and 
 in helping it to slacken speed when the cage reaches the surface, it 
 consists often of a bunch of heavy chains suspended in a shallow shaft, 
 the chains resting on the shaft bottom as unwound off the balance drum 
 attached to the main shaft of the engine. (2) Scales used in chemical 
 
 , Balance ^Bob .^A farge beam or lever attached to the main rods of a Cornisb 
 pumping engine, carrying on its outer end a counterpoise. 
 
 Balance Box.— A large box placed on one end of a balance bob and fillea 
 with old iron, rock, etc. to counterbalance the weight of pump rods 
 Balance Brow.— An inclined plane in steep seams on which a platform on 
 wheels travels and carries the cars of coal. 
 
 Balance Car.— A small weighted truck mounted upon a short inchnM track, 
 and carrying a sheave around which the rope of an endless haulage 
 system passes as it winds off the drum. 
 
 Balance Pit.— A pit or shaft in which a balance rises or falls. 
 
 Balanzon (Mexican). -The balance bob of a Cornish pump 
 Balk _(i) a more or less sudden thinning out of a seam of coal. (2) irregu- 
 lar-shaped masses of stone intruding into a coal seam, or bulgings out 
 of the stone roof into the seam. (3) A bar of timber supporting the roof 
 of a mine, or for carrying any heavy load. 
 
 Balland (Derbyshire). — Pulverulent lead ore. . , D 
 
 Ballast.— Broken stone, gravel, sand, etc. used for keeping railroad ties stead, . 
 Bancos (Spanish) .—Horses in a vein or cross-courses. # 
 
 Band.— A seam or thin stratum of stone or other refuse in a seam of coal. 
 Bank.—( 1) The top of the shaft, or out of the shaft. (2) The surface around 
 the mouth of a shaft. (3) To manipulate coals, etc. on the bant. 
 (4) The whole or sometimes only one side or one end of a worxm & 
 place underground. (5) A large heap of mineral on the surface. 
 
570 
 
 Ban 
 
 GLOSSARY. 
 
 Bas 
 
 Bank Chain— A chain that includes the bank of a river or creek. 
 
 Bank Claim (Australian).— Mining right on bank of stream. 
 
 Banket. — Auriferous conglomerate of South Africa. 
 
 Bank Head. — The upper end of an inclined plane, next to the engine or drum, 
 made nearly level. 
 
 Bank Right (Australian). — Right to divert water to bank claim. 
 
 Banksman.— The man in attendance at the top of the shaft, superintending 
 the work of banking. 
 
 Bankwork.—A system of working coal in South Yorkshire. 
 
 Bank to Bank. — A shift. 
 
 Bannocking. — See Kirving. 
 
 Bano (Spanish).— Excess of mercury used in torta. 
 
 Bar— A length of timber placed horizontally for supporting the roof. In 
 some cases, bars of wrought iron, about 3 in. X 1 in. X 5 ft. are used. 
 
 Bar Diggings. — (1) River placers subject to overflow. (2) Auriferous claims 
 on shallow streams. 
 
 Bargain.— Portion of mine worked by a gang on contract. 
 
 Barilla (Spanish).— Grains of native copper disseminated through ores. 
 
 Baring.— See Stripping. 
 
 Barmaster (Derbyshire) — Mine manager, agent, and engineer. 
 
 Bar Mining.— The mining of river bars, usually between low and high water, 
 although the stream is sometimes deflected and the bar worked below 
 water level. 
 
 Barney.— A small car, used on inclined planes and slopes to push the mine 
 car up the slope. Barney Pit.— A pit at the bottom of a slope or plane 
 into which the barney runs to allow the mine car to pass over it. 
 
 Barra (Mexican).— (1) A bar, as of gold, silver, iron, steel, etc. (2) A cer- 
 tain share in a mine. The ancient Spanish laws, from time immemorial, 
 considered a mine as divided into 24 parts, and each part was called a 
 “ barra.” 
 
 Barra Viuda, or Aviada (Mexican).— These are “ barras ” or shares that par- 
 ticipate in the profits, but not in the expenses, of mining concerns. 
 Their share of the expenses is paid by the other shares. Non-assessable 
 shares. 
 
 Barranca (Mexican).— A ravine, a gulch. What is improperly called in the 
 United States a canyon or canon. 
 
 Barrel Amalgamation. — Amalgamating ores in revolving barrels. 
 
 Barrel Work. — (1) Native copper that can be hand-sorted ready for smelt- 
 ing. (2) Barrel amalgamation. 
 
 Barrena (Mexican).— A hand drill for opening holes in rocks for blasting 
 purposes. 
 
 Barrenarse (Mexican).— When two mines or two workings (as a shaft or 
 winze, or a gallery) communicate with each other. 
 
 Barren Ground— Strata unproductive of seams of coal, etc. of a workable 
 thickness. 
 
 Barreno (Mexican) .—(1) A drill hole for blasting purposes. In mechanics, any 
 bored hole. (21 A communication between two mines or two workings. 
 
 Barretero (Mexican).— A miner of the first class; one that knows how to 
 point his holes, drill, and blast, or work with a gad. 
 
 Barrier Pillar— A solid block or rib of coal, etc., left unworked between two 
 collieries or mines for security against accidents arising from influx of 
 water. 
 
 Barrier System. — The method of working a colliery by pillar and stall, where 
 solid ribs or barriers of coal are left in between a set or series of working 
 places. 
 
 Barrow— (1) A box with two handles at one end and a wheel at the other. 
 (2) Heap of waste stuff raised from a mine; a dump. 
 
 Bar Timbering. — A system of supporting a tunnel roof by long top bars, 
 while the whole lower tunnel core is taken out, leaving an open space 
 for the masons to run up the arching. Under certain conditions, the 
 bars are withdrawn after the masonry is completed, otherwise they are 
 bricked in and not drawn. 
 
 Base Bullion. — Lead combined with precious metals. 
 
 Base Metal.— Metal not classed with the precious metals, gold, silver, plat- 
 inum, etc., that are not easily oxidized. 
 
 Basin. — (1) A coal field having some resemblance in form to a basin. (2) 
 The synclinal axis of a seam of coal or stratum of rock 
 
 Basket.— A measure of weight = 2 cwt. 
 
Bas 
 
 GLOSSARY. 
 
 Ben 
 
 571 
 
 Basque.— Crucible or furnace lining. 
 
 Bass (Derbyshire).— Indurated clay. 
 
 Basset— Outcrop of a lode or stratum. 
 
 Rastard A particularly hard massive rock or boulder. 
 
 Batch.— An assorted parcel of ore, sometimes called doles, when divided 
 into equal quantities. , , 
 
 rtnfpn A shallow wooden bowl used for washing out gold, etc. 
 
 Batt (English).— (1) A highly bituminous shale found in the coal measures. 
 (2) Hardened clay, but not fireclay. Same as Bend and Bind. 
 
 Ratten A niece of thin board less than 12 in. m width. . 
 
 Batter. — The inclination of a face of masonry or of any inclined portion of 
 
 BaUeriL—O - )° ^structure built to keep coal from sliding down a chute or 
 breast. (2) An embankment or platform on which miners work. (3) 
 
 i^a^An^pen^pace for waste between two packs in a longwall working. 
 
 I «£• ° r p h iati rT- 
 
 Beans (Nortl? of England) .—All coal that will pass through about 9 
 screen 
 
 Bean Shot.— Copper granulated by pouring into hot water. 
 
 jipar A denosit of iron at the bottom of a furnace. ^ . , , , 
 
 Bear; to Bear In— Underholding or undermining; driving m at the top or at 
 
 Be Jern -Pieces of’ttaber 3 or 4 ft. longer than the breadth of a shaft, 
 which are fixed into the solid rock at the sides at certain intervals apart, 
 
 B eaZg-d The span or length in the clear 
 between the points of support of a beam, etc. (3) The points of support 
 
 Bernina JoT-AdoOT placid for the purpose of directing and regulating the 
 amount If ventilation passing through an entire district of a mine. 
 Bearing In.— The depth or distance under of the undercut or holing. 
 BmHng-up Pulley.-k pulley wheel fixed in a frame and arranged to tighten 
 ud or take up the slack rope in endless-rope haulage. . 
 
 Bearmg-up Stop.— A partition of brattice or plank that serves to conduct air 
 
 to a face. 
 
 means of wedges and sledge hammers 
 Bed — (i) The level surface of a rock upon which a curb or crib is laid. (2) A 
 
 Bed^^^S^Austraii^)!— Ilciaim^th^includes the bed of a river or creek. 
 
 Bednlate —A large plate of iron used as a foundation for an engine. 
 
 Bed-Rock. — The f olid rock underlying the soil, drift, or alluvial deposits. 
 
 o f “ the 
 
 form, disconnected from the 
 
 5eid^-A°form of lead poisoning to which lead miners are subject. 
 rpJIv — a swelling mass of ore in a lode. „ , . _ 
 
 Ben,' Benhayl (Cornish).— Productive. The productive portion of 
 
 Bench^—i 1) A natural terrace marking the outcrop of any stratum. 
 
 stratum of coal forming a portion of the vein. 
 
 Bench Diaaings.— River placers not subject to overnow. , ,. . 
 
 Benching'.— Yo break up with wedges the bottom coals when the holing is 
 
 done in the middle of the seam. * , 
 
 Benching Up (North of England). -Working on top of coal. 
 
 Bench Mark.— A mark cut in a tree or rock whose elevation is known. Used 
 bv surveyors for reference in determining elevations. , ■» . 
 
 Bench Working.— 1 The system of working one or more seams or beds ot 
 mineral bv open working or stripping, m stages or steps. 
 
 Bend (Derbyshire).— Indurated clay. _ + i, 0 
 
 Beneficiar (Mexican).— To treat ores for the purpose of extracting the 
 
 metallic contents. 
 
 a tin 
 (2) A 
 
572 
 
 Ben 
 
 GLOSSARY. 
 
 Blo 
 
 Beneficio (Mexican).— Any metallurgical process. 
 
 Benheyl (Cornish).— Flowing tin stream. 
 
 Bessemer Steel— Steel made by the Bessemer process. 
 
 Beton (English).— Concrete of hydraulic cement with broken stone, bricks, 
 gravel, etc. 
 
 Bevel.— The slope formed by trimming away on edge. 
 
 Bevel Gear.— A gear-wheel whose teeth are inclined to the axis of the wheel. 
 
 Biche — A hollow-ended tool for recovering boring rods. 
 
 Billy Boy. — A boy who attends a Billy Playfair. 
 
 Billy Playfair.— A mechanical contrivance for weighing coal, consisting of 
 an iron trough with a sort of hopper bottom, into which all the small 
 coal passing through the screen is conducted and weighed off and 
 emptied from time to time. 
 
 Bin.— A box with cover, used for tools, stones, ore, etc. 
 
 Bind, or Binder— Indurated argillaceous shales or clay, very commonly 
 forming the roof of a coal seam and frequently containing clay iron- 
 stone. See Batt. 
 
 Binding.— Hiring men. 
 
 Bing (North of England).— 8 cwt. of ore. 
 
 Bing Hole (Derbyshire). — An ore shoot. 
 
 Bing Ore (Derbyshire).— Lead ore in lumps. 
 
 Bing Tale (North of England).— Ore given to the miner for his labor. 
 
 Bit— A piece of steel placed in the cutting edge of a drill or point of a pick. 
 
 Blackband — Carbonaceous ironstone in beds, mingled with coaly matter 
 sufficient for its own calcination. 
 
 Black Batt, or Black Stone— Black carbonaceous shale. 
 
 Black Copper. — Impure smelted copper. 
 
 Blackdamp.— Carbonic-acid gas. 
 
 Black Diamonds. — Coal. 
 
 Black Ends.— Refuse coke. 
 
 Black Flux. — Charcoal and potassium carbonate. 
 
 Black Jack.—( 1) Properly speaking, dark varieties of zinc blende, but many 
 miners apply it to any black mineral. (2) Crude black oil used to oil 
 mine cars. 
 
 Black Lead— Graphite. 
 
 Black Ore (English).— Partly decomposed pyrites containing copper. 
 
 Black Sand. — Dark minerals found with alluvial gold. 
 
 Black Stone.— A carbonaceous shale. 
 
 Black Tin.— Dressed cassiterite; oxide of tin. 
 
 Blanch.— (1) A piece of ore found isolated in the hard rock. (2) Lead ore 
 mixed with other minerals. 
 
 Blanched Copper.— Copper alloyed with arsenic. 
 
 Blanket Stroke (Australian).— Sloping tables or sluices lined with baize, for 
 catching gold. , , , 
 
 Blanket Tables.— Inclined planes covered with blankets, to catch the heavier 
 minerals passing over them. / 
 
 Blast— (1) The sudden rush of fire, gas, and dust of an explosion through the 
 workings and roadways of a mine. (2) To cut or bring down coal, 
 rocks, etc. by the explosion of gunpowder, dynamite, etc. 
 
 Blasting Barrel— A small pipe used for blasting in wet or gaseous places. 
 
 Blast Pipe— A pipe for supplying air to furnaces. 
 
 Blende— Sulphide of zinc; sphalerite. 
 
 Bliclz (Germany).— Iridescence on gold and silver at end of cupeling. 
 
 Blind Coal— Coal altered by the heat of a trap dike. 
 
 Blind Creek.— { 1) A creek in which water flows only in very wet weather. 
 (2) (Australasian) Dry watercourse. 
 
 Blind Drift.— A horizontal passage in the mine not yet connected with the 
 other workings. 
 
 Blind Joint.— Obscure bedding plane. 
 
 Blind Lead, or Blind Lode.— A vein having no visible outcrop. 
 
 Blind Level. — (1) An incomplete level. (2) A drainage level. 
 
 Blind Shaft, or Blind Pit— A shaft not coming to the surface. 
 
 Bloat— A hammer swelled at the eye. 
 
 Block Claim (Australian).— A square mining claim. 
 
 Block Coal— Coal that breaks in large rectangular lumps. 
 
 Blocking 0ut.—(l) Working deep leads in blocks; somewhat like horizontal 
 stoping. (2) (Australian! Washing gold gravel in sections. 
 
 Block Reefs— Reefs showing frequent contractions longitudinally. 
 
Blo 
 
 GLOSSARY. 
 
 Bon 
 
 573 
 
 Block Tin. — Cast tin. 
 
 J iWn, or any indicating 
 traces of a coal bed or mineral deposit. Blossom Rock.— (1) Colored 
 veln?tone a detached from an outcrop. (2) The rock detached from 
 a vein but which has not been transported. 
 
 Blow. — (1) * To blast with gunpowder, etc. (2) A dam or stopping is said to 
 blow when gas escapes through it. . . /r) >. A 
 
 Blower — (1) A sudden emission or outburst of gas m a mine. (2) Any 
 emission of gas from a coal seam similar to that from an ordinary gas 
 burner. (3) A 'type of centrifugal fan used largely to force air into 
 furnaces (4) A blowdown ventilating fan. . , 
 
 Blow Fan.— A. small centrifugal fan used to force air through canvas pipes 
 or wooden boxes to the workmen. 
 
 Blowdown Fan.— A force fan. 
 
 Blow In . — To commence a smelting process. . 
 
 Blown- Out Shot— A shot that has blown out the tamping, but not broken the 
 
 coal or rock. „ x _ ... 
 
 Blow Off — To let off excess of steam from a boiler. . 
 
 Blow Out — (1) To finish a smelting campaign. (2) A blown-out shot. 
 
 (3) The decomposed mineral exposure of a vein. . „ o 
 
 Blowpipe— An instrument for creating a blast whereby the heat of a flame 
 
 or lamp can be better utilized. 
 
 Blue Billv. — Residue of copper pyrites after roasting with salt. 
 
 Blue Cap!— 1 The blue halo of ignited gas (firedamp and air) on the top of the 
 flame in a safety lamp, in an explosive mixture. 
 
 Blue Elvan (Cornish).— Greenstone. 
 
 Blue J JLeak —A blue^Sained stratum of gravel of great extent and richness. 
 Blue Metal.— A local term for shale possessing a bluish color. 
 
 Blue Peach (Cornish). — A slate-blue fine-grained schorl. 
 
 Milestone —(1) Sulphate of copper. (2) Lapis lazuli. (S) Basalt. (4) Maryland, 
 3 a gray gneiss; in Ohio, a gray sandstone; in the District of Columbia, 
 a mica schist; in New York, a blue-gray sandstone; in Pennsylvania, a 
 blue-gray sandstone. (5) A popular term among stone men not suf- 
 ficiently definite to be of value. 
 
 'Board.— A wide heading usually from 3 to 5 yd. wide. 
 
 Board-and-Pillar .— A system of working coal where the first stage of exca- 
 vation is accomplished with the roof sustained by pillars of coal left 
 between the breasts; often called Breast-and-Pillar. 
 
 Bob —An oscillating bell-crank, or lever, tnrough which the motion of an 
 engine is transmitted to the pump rods in an engine or pumping pit. 
 There are x bobs, L bobs, and V bobs. . , 
 
 Boca or Boca Mina (Mexican).— Mouth or mine mouth. This is the name 
 applied to the principal or first opening of a mine, or to the one where 
 
 the miners are accustomed to descend. ... 
 
 Bochorno (Mexican).— Excessive heat, with want of ventilation, so that the 
 
 Bodm—\lf°A^ ore body, or pocket of mineral deposit. (2) The thickness of 
 a lubricating oil or other liquid; also the measure of that thickness 
 exnressed in the number of seconds in which a given quantity of the 
 oil at a given temperature flows through a given aperture. 
 
 Bog Iron Ore.— Loose earthy brown hematite recently formed in swampy 
 
 Boleo (Mexican) .—A dump pile for waste rock. 
 
 Boliche (Spanish.).— Concentrating bowl. 
 
 Bollos (Spanish).— Triangular blocks of amalgam. 
 
 Bolsa (Spanish).— Small bunch of ore.. 
 
 Bonanza. — An aggregation of rich ore m a mine. , , » « 
 
 Bond.—{ 1) The arrangement of blocks of stone or brickwork to form a firm 
 structure by a judicious overlapping of each other so as to break joint. 
 (2) An agreement for hiring men. 
 
 Bone.— Slatv coal or carbonaceous shale found m coal seams. 
 
 Bone Ash —Burnt bones pulverized and sifted. „ . 
 
 Bonnet— (1) The overhead cover of a cage. .(2) A cover for the gauze of 
 a safety lamp. (3) A cap piece for an upright timber. 
 
 Bonney (Cornish) .—An isolated body of ore. 
 
574 
 
 Bon 
 
 GLOSSARY. 
 
 Bre 
 
 Bonze. — U ndressed lead ore. 
 
 Booming.— Ground sluicing on a large scale by emptying the contents of a 
 reservoir at once on material collected below, thus removing boulders. 
 
 Bord (English).— A narrow breast. 
 
 Bord-and-Pillar (English).— See Pillar-and- Breast. 
 
 Bord Room. — The space excavated in driving a bord. The term is used in 
 connection with the “ridding” of the fallen stone in old bords when 
 driving roads across them in pillar working; thus, “ ridding across the 
 old bord room.” 
 
 Bord Ways Course.— The direction at right angles to the mam cleavage 
 planes. In some mining districts, it is termed “on face.” 
 
 Bore.— To drill. 
 
 Bore Hole. — A hole made with a drill, auger, or other tools, m coal, rock, or 
 other material. 
 
 Borrasca (Mexican). — The reverse of bonanza. When the mine has a vein, 
 but no ore, it is said to be “ en borrasca.” 
 
 Bort.— Amorphous dark diamond. 
 
 Bosh.— The plane in a blast furnace where the greatest diameter is reached. 
 
 Boss (English).— (1) An increase of the diameter at any part of the shaft. 
 (2) A person in charge of a piece of work. 
 
 Botas (Mexican). — Buckets made of an entire ox skin, to take out water. 
 
 Botryoidal.— Grape-like in appearance. 
 
 Bottle Jack (English).— An appliance for lifting heavy weights. 
 
 Bottom.— (1) The landing at the bottom of the shaft or slope. (2) The lowest 
 point of mining operations. (3) The floor, bottom rock, or stratum 
 underlying a coal bed. (4) In alluvial, the bed rock or reef. 
 
 Bottomer, Bottomman. — The person that loads the cages at the pit bottom 
 and gives the signal to bank. 
 
 Bottom Joint.— Joint or bedding plane, horizontal or nearly so. 
 
 Bottom Lift.—( 1) The deepest column of a pump. (2) The lowest or deepest 
 lift or level of a mine. 
 
 Bottom Pillars.— Large pillars left around the bottom of a shaft. 
 
 Bottoms. — Impure copper alloy below the matte in smelting. 
 
 Boulders. — Loose rounded masses of stone detached from the parent rock. 
 
 Bounds (Cornish).— A tract of tin ground. 
 
 Bout (Derbyshire).— Twenty-four dishes of lead ore. 
 
 .Bow.— The handle of a kibble. „ , _ , 
 
 Bowk. — An iron barrel or tub used for hoisting rock and other debris when 
 sinking a shaft. , , , 
 
 Bowke (Staffordshire).— A small wooden box for hauling ironstone under- 
 ground. 
 
 Bowl Metal.— The impure antimony obtained from doubling. 
 
 Bowse (Derbyshire).— Lead ore as cut from the lode. 
 
 Box.— (1) A 12' to 14' section of a sluice. (2) A mine car. 
 
 Box Bill.— Tool for recovering boring rods. 
 
 Boxing.— A. method of securing shafts solely by slabs and w<3oden pegs. 
 
 Brace. — (1) An inclined beam, bar, or strut for sustaining compression 
 or tension. See Tie-Brace , Sway-Brace. (2) A platform at the top of a 
 shaft on which miners stand to work the tackle. (3) (Cornish) Building 
 at pit mouth. . _ , , . ,, , 
 
 Brace Heads— Wooden handles or bars for raising and rotating the rods 
 
 when boring a deep hole. 
 
 Braize .— Charcoal dust. 
 
 Brake Seive— Hand jigger. 
 
 Brances .— Iron pyrites in coal. 
 
 Branch .— Small vein shooting off from main lode. 
 
 Brashy— Short and tender. _ . - , .. 
 
 Brasque . — A mixture of clay and coke or charcoal used for furnace bottoms. 
 Brass. — (1) Iron pyrites in coal. (2) An alloy of copper and zinc. 
 
 Brasses (English).— Fitting of brass in plummer blocks, etc., for diminishing 
 the friction of revolving journals that rest upon them. 
 
 Brat.— A thin bed of coal mixed with pyrites or limestone. 
 
 Brattice— A lining or partition. . , . 
 
 Brattice Cloth .— Ducking or canvas used for making a brattice. 
 
 Brazzil (North of England). — Iron pyrites in coal. _ . , , „ 
 
 Breaker . — In anthracite mining, the structure in which the coal is broken, 
 sized, and cleaned for market. Known also as Coal Breaker. 
 
 Breaker Bov.— A boy who works in a coal breaker. 
 
Bre 
 
 GLOSSARY. 
 
 BUL 
 
 575 
 
 Breakstaff.— The lever for blowing a Dlacksmiths’ bellows, or for working 
 
 Breakthrough^- A^arrow passage cut factor 
 
 Breas t— ( l) A stall, board, or room m which coal is mined. (2) I he lace or 
 
 wall of a auarrv is sometimes called by this name. 
 
 Breastand-Pttlar.—A system of working coal by boards or rooms with pillars 
 
 Breattinfor f-The orlTaken from the face or end of the tunnel. 
 
 breast Wall (English) —A wall built to prevent the falling of a \ertical face 
 
 Brecc?a.— A rock^omposed of angular fragments cemented together. 
 
 Breeding Fire. — See Gob L?ire. 
 
 Breese.— Fine slack. _ , . 
 
 Breeze— Small coke, probably same as braize or braise. 
 
 Brettis fDerbvshire) .— A timber crib filled with slacR. . 
 
 Bridge.— {1) A platform on wheels running on rails for covering 
 
 of a shaft or slope. (2) A track or platform passing over an inclined 
 haulage way and which can be raised out of the way of ascending and 
 
 BridtecTcdnt— Short chains by which a cage, . car, or attached to 
 
 a winding rope; of use in case the rope pulls out of socket. 
 
 Briquets. — Fuel made of slack or culm and pressed ^otaick mr ti a llv 
 
 Broaching Bit— A tool for reopening a bore hole that has been partially 
 closed by swelling of the walls. 
 
 Brob.—A spike to prevent timber slipping. 
 
 Broil (Cornish). — Traces of a vein in loose matter. ir, nrmtrfl- 
 
 Ttmken A district of coal pillars m process of removal, so called m contra 
 
 distinction^ to the tot working of a seam by bord-and-wall, or working 
 in the “whole.” See Whole Working. Q1 . 
 
 Broken Coal.— Anthracite coal that will pass through a mesh or bars 3 4 to 
 44- in. , and over a mesh 2f in. square. ( See page 434. ) 
 
 Bronce (Mexican).— In mining, copper or iron pyrites. 
 
 Brooch (Cornish).— Mixed ores. 
 
 Broochinq. — Smoothing. 
 
 Brood (Cornish).— Heavy waste from tin and copper ores. pithpr 
 
 Brow.— An underground roadway leading to a working place driven either 
 
 Brown^Coal— Lignite. 6 A fuel classed between peat and bituminous coal. 
 Brown Spar.— Dolomite containing carbonate of iron. 
 
 Brownstone.-iff) Decomposed iron pyrites. (2) Brown ^sandstone. 
 
 Browse . — Imperfectly smelted ore mixed with cinder and clay. 
 
 Brujula (Mexican).— A surveyors’ (or marine) magnetic 1 ^P^s. 
 
 Brush. — (1) To mix air with the gas m a mine working swinging a 
 iacket etc. which creates a current. (2) To brush the roof oi an 
 airway, is to take down some of the roof slate, to increase the height 
 
 or headroom. . . , 
 
 Brule (Cornish). — Traces of a vein in loose matter. 
 
 Bucket.— (1) An iron or wooden receptacle for hoisting ore, or for raising 
 rock in shaft sinking. (2) The top valve or clack of a pump. 
 
 Bucket Pump— A lifting pump, consisting of buckets fastened to an endless 
 
 Bii^etVword^—A wrought-iron rod to which the P um P^ c ^i\^ t e ached * 
 Bucket Tree —The pipe between the working barrel and the wind bore. 
 Bucking. — Breaking down ore with a very broad banmeii * : brelkfng 
 
 Bucking Hammer.— An iron disk, provided with a handle, used tor breaking 
 
 up minerals by hand. 
 
 Buck Quartz. — Hard non-auriferous quartz. o . 
 
 Buck Staff.— Uprights for bracing reverberatory furnaces together . 
 Buckwheat- Anthracite coal that will pass through a mesh * in. and over a 
 
 Buddie^- Aifinclined table, circular or oblong, on which ore is concentrated. 
 Buddling.— Washing. 
 
 Buggy.— A small mine car. „ ■ ... . , 
 
 Bug Hole.— A small cavity usually lined with crystals. f 
 
 Building.— A built-up block or pillar of stone or coal to support the roof. 
 Buitron (Spanish). — A silver furnace of peculiar form. 
 
 Bulkhead.— (1) A tight partition or stopping. (2) The end of a flume carry 
 ing water for hydraulicking. 
 
576 
 
 Bul 
 
 GLOSSARY. 
 
 Cal 
 
 Bulldog.— A. refractory furnace lining of calcined mill cinder, containing 
 iron and silica. 
 
 Bull Engine.— A single, direct-acting pumping engine, the pump rods form- 
 ing a continuation of the piston rod. 
 
 Buller Shot.— A second shot put in close to, and to do the work not done by, 
 a blown-out shot, loose powder being used. 
 
 Bud.— An iron rod used in ramming clay to line a shot hole. 
 
 Bulling. — Lining a shot hole with clay. 
 
 Bullion.— Uncoined gold and silver. 
 
 Bull Pump. — A single-acting pumping engine in which the steam cylinder is 
 placed over the shaft or slope and the pump rods are attached directly to 
 the piston rod. The steam enters below the piston and raises the pump 
 rods; the water is pumped on the down stroke by the weight of the rode. 
 
 Bull Pup.— A worthless claim. 
 
 Bull Wheel. — A wheel on which the rope carrying the boring rod is coiled 
 when boring by steam machinery. 
 
 Bully. — A miners’ hammer. 
 
 Bumping Table. — A concentrating table with a jolting motion. 
 
 Bunch.— A small rich deposit of ore. 
 
 Bunding— A staging in a level for carrying debris. 
 
 Bunkers— Steam coal consumed on board ship. 
 
 Bunney.—A nest of ore not lying in a regular vein. 
 
 Buntons. — Timbers placed horizontally across a shaft or slope to carry the 
 cage guides, pump rods, column pipe, etc.; also, to strengthen the shaft 
 timbering. 
 
 Burden. — (1) Earth overlying a bed of useful mineral. (2) The proportion 
 of ore and flux to fuel in the charge of a blast furnace. 
 
 Burr. — Solid rock. 
 
 Burrow. — Refuse heap. 
 
 Buscones (Spanish).— Prospectors, fossickers, tribute workers. 
 
 Bush.— To line a circular hole with a ring of metal, to prevent the hole from 
 wearing out. 
 
 Butt— (1) Coal surface exposed at right angles to the face; the “ends” of 
 the coal. (2) The butt of a slate quarry is where the overlying rock 
 comes in contact with an inclined stratum of slate rock. 
 
 Butt Entry.— A gallery driven at right angles with the butt joint (see page 285). 
 
 Butterfly Valve.— A circular valve that revolves on an axis passing through 
 its center. 
 
 Butt Heading. — See Butt Entry. 
 
 Button.— The globule of metal, the result of an assay. 
 
 Button Balance.— A small very delicate balance used for weighing assay 
 buttons. 
 
 Butty'— A partner in a contract for driving or mining; a comrade, crony. 
 Sometimes called “ Buddy.” 
 
 By Level . — A side level driven for some unusual but necessary purpose. 
 
 Cab. — The side parts of a lode, nearest the walls, which are generally hard 
 and deficient of ore. 
 
 Caballo (Mexican).— A “ horse ” or mass of barren rock in a vein. 
 
 Cabezuela (Spanish).— Rich gold and silver concentrates.. 
 
 Cabin . — (1) A miner’s house. (2) A small room in the mine for the use of the 
 officials. 
 
 Cable Drilling . — Rope drilling. 
 
 Cage . — A platform on which mine cars are raised to the surface. 
 
 Cage Guides.— Vertical rods of pine, iron, or steel, or wire rope, fixed in a 
 shaft, between which cages run, and whereby they are prevented from 
 striking one another, or against any portion of the shaft. 
 
 Cager . — The person that puts the cars on the cage at the bottom of the shaft. 
 
 Cage Seat . — Scaffolding, sometimes fitted v/ith strong springs, to take off the 
 shock, and on which the cage drops when reaching the pit bottom. 
 
 Cage Sheets . — Short props or catches on which cages stand during caging or 
 changing cars. 
 
 Caking Coal— Coal that agglomerates on the grate. 
 
 Cal— Wolfram. 
 
 Cala (Spanish).— Prospecting pit. 
 
 Calcareous . — Containing lime. # . 
 
 Calcine . — To heat a substance; not sufficiently to melt it, but enough to drive 
 off the volatile contents. 
 
Cal 
 
 GLOSSARY. 
 
 Cau 
 
 577 
 
 Calcining Furnace— A furnace used for roasting ore in order to drive off 
 certain impurities. 
 
 Wmnta^mp^-Itndevnmp made of a wooden box t h / ou ,fh which an 
 endless belfwith floats circulates; used for pumping water from shallow 
 
 CaKra°(€ornish). — Stratified rocks traversed by lodes. . . t 
 
 Cam.—( 1) A curved arm attached to a revolving shaft for raising stamps. 
 
 (2) Carbonate of lime and fluorspar, found on the joints of lodes 
 Camino (Mexican).— Any gallery, winze, or shaft, inside of a mine used tor 
 
 general transit. „ . n . Woct 
 
 Campaign.— The length of time a furnace remains m blast. 
 
 ' required to .be ^removed l to make 
 
 sloping sides and very narrow bottom. 
 
 Cancha (Spanish).— Space for drying slimes. 
 
 Cand ( Cornish ) —Fluorspar. 
 
 Cank (Derbyshire).— Whinstone. . 
 
 Canker . — The ocherous sediment m coal-pit waters. 
 
 a mine - " 
 
 Cante^Enghsh)!— Tlm'pieces^ornnng the ends of ^ckete of a waterwheel. 
 
 Capdlma{ Mexican).— An old-style retort for retorting silver amalgam. 
 
 Caple (Cornish).— Hard rock lining tin lodes. 
 
 worked by horizontal 
 
 arms or bars. . ~ 
 
 Captain. — Cornish name for manager or boss of a mine. o. aT1 o-x VflVS Q r 
 
 Car.— Any car used for the conveyance of coal along the gangways or 
 haulage roads of a mine. 
 
 Carat.— A weight nearly equal to 4 grains. . . thp comno- 
 
 Carbon.— A combustible elementary substance forming the largest compo 
 
 carh^-m Arichbunch of ore in the country rock connected with the 
 lodeby a mere thr" ad of mineral. (2) (Cornish) An irregullr deposit 
 of tin ore. . . . , 
 
 Carbonaceous.— Coaly, containing carbon or coal. 
 
 Carbonate.— Carbonic acid combined with a base. , f i d . 
 
 Carbonates . — Lead ore. The oxide and carbonic-acid compounds ot lead, 
 also applied to lead sulphate. 
 
 generally of 300 pounds, but 
 
 variable in different parts of Mexico. 
 
 case filled with gunpowder, 
 
 forming the charge for blasting. 
 
 Cascajo (Mexican).— Gravel. 
 
 Case— A fissure admitting water into a mine. . . steel bv 
 
 Case-Harden— To convert the outer surface of wrought iron into steei Dy 
 
 Tubing 1 in serf ed h? a? ^^"keep out water or to protect the 
 
 Casfn^n™. Sg^iro^thal contains carbon (up to 5$), silicon, sulphur, phos- 
 
 Cata (Spanish ).— A mine denounced but not worked. 
 
 Catches— {1) Iron levers or props at the top and bottom of a shaft. ( ) P 
 fitted on a cage to prevent cars from running ofl. 
 
 Catch Pit— A reservoir for saving tailings from reduction works. 
 
 Cauf (North of England).— A coal bucket or basket. , trunks of sieil- 
 
 Cauldron Bottoms.- The fossil remains or the casts of the trunks ot s g 
 laria that have remained vertical above or belowtheseam. 
 
 Caulk.— To fill seams or joints with something to prevent leaking. 
 
GLOSSARY. 
 
 Cho 
 
 \78 Cau 
 
 Cannier , or Cannier Lode (Cornish).— A vein running obliquely across the 
 regular veins of the district. 
 
 Cave , or Cave In— A caving-in of the roof strata of a mine, sometimes extend- 
 ing to the surface. 
 
 Cavils— Lots drawn by the hewers each quarter year to determine their 
 working places. 
 
 Cawk — Baryta sulphate. 
 
 Cazeador (Spanish) . — Amalgamator. 
 
 Cazo (Mexican).— A vessel for hot amalgamation. Any large copper or iron 
 vessel. 
 
 Cebar (Mexican).— (1) To melt rich ores, or lead bullion, etc. in a smelting 
 furnace. (2) To add small quantities of material, from time to time, to 
 the melted mass within a furnace. (3) Generally, to feed any kind or 
 metallurgical machinery or process. 
 
 Cement— (1) Auriferous gravel consolidated together. (2) A finely divided 
 metal obtained by precipitation. (3) A binding material. 
 
 Cementation— The process of converting wrought iron into steel by heating 
 it in contact with charcoal, or of treating cast iron in a bed of hema- 
 tite ore. 
 
 Cendrada (Mexican).— The cupel bottom of a furnace. 
 
 CendradUla (Mexican).— A small reverberatory furnace for smelting rich 
 silver ores in a rough way. Also called Galeme. 
 
 Center —A temporary support, serving at the same time as a guide to the 
 masons, placed under an arch during the progress of its construction. 
 
 Centrifugal Force.— A force drawing away from the center. 
 
 Centripetal Force.— A force drawing toward the center. 
 
 CH a — Marsh gas (see page 348). 
 
 Chain.— A measure 66 or 100 ft. long, divided into 100 links. 
 
 Chain-Brow Way— An underground inclined plane worked on the endless- 
 chain svstem of haulage. 
 
 Chain Pillar.— A pillar left to protect the gangway and air-course, and run- 
 ning parallel to these passages. 
 
 Chain Road— An underground wagomvay worked on the endless-chain 
 system of haulage. 
 
 Chair. — Sometimes applied to keeps. 
 
 Chamber. — See Breast. 
 
 Charco (Mexican).— A pool of water. 
 
 Charge. — (1) The amount of powder or other explosive used in one blast or 
 shot. (2) The amount of flux used in assaying. (3) The material fed 
 into a furnace at one time. 
 
 Charqnear (Mexican).— To dip out water from pools within the mine, 
 throwing it into gutters or pipes that will conduct it to the shaft. 
 
 Chats— (1) The gravel-like tailings derived from the concentration of ores. 
 (2) A low-grade ore, often too poor to handle; the refuse from concen- 
 tration works. (3) (North of England) Small pieces of stone with ore. 
 
 Check- Battei'y— A battery to close the lower part of a chute, acting as a 
 check to the flow of coal and as an air stopping. 
 
 Checker Coal— Anthracite coal that seems to be made up of rectangular 
 grains. 
 
 Check- Weighman.—A man appointed and paid by the miners to check the 
 weighing of the coal at the surface. 
 
 Cheek. — Wall. 
 
 Chert.— A silicious rock, often the gangue of lead and zinc. 
 
 Chestnut Coa l.— Anthracite coal that will pass through a mesh If in. square 
 and over a mesh f in. square (see page 434). 
 
 Chifion (Mexican).— A narrow drift directed obliquely downwards. Any 
 pipe from which issues water or air under pressure, or at high velocity. 
 
 Chile Bars.— Bars of impure copper, weighing about 200 lb., imported from 
 Chile, corresponding to the Welsh blister copper, containing 98# Cu. 
 
 Chilian Mill.— A roller mill for crushing ore. 
 
 Chill Hardening. — Giving a greater hardness to the outside of cast iron by 
 pouring it into iron molds, which causes the skin of the casting to cool 
 rapidly. 
 
 Chimney.— (1) An ore shoot. (2) A furnace or air stack. 
 
 Chinese Pump.— Like a California pump, but made entirely of wood. 
 
 Chock.— A square pillar for supporting the roof, constructed of prop timber 
 laid up in alternate cross-layers, in log-cabin style, the center being filled 
 with waste. \ * 
 
Cho 
 
 GLOSS AR Y. 
 
 Cof 579 
 
 Chun^rlli -A long iron bar with a cutting end of steel, used in quarrying, 
 andworked by faising and letting it fall. When worked by blows of a 
 
 CAMt^^l^^pelle^bVmfe)^— fl) 6 A narrow'lnclined passage in a mine, down 
 which coal or ore is either pushed or slides by gravity. (2) The load- 
 ing chute of a tipple. . 
 
 CT^f^^n?)-AcS^^r<*^ e ^ U &---<>i r erheadstoping. 
 
 cS-A^vrS i“opTned and closed by the force of the water. . 
 
 Clack Door — The opening into the valve chamber to facilitate repairs and 
 renewals without unseating the pump or breaking the connections. 
 
 Clack Piece.— The casting forming the valve chamber. 
 
 Clack Seal . — The receptacle for the valve to rest on. . , f 
 
 Clanav (North of England).— When coal is tightly joined to the roof. 
 C^m.—A portion of' ground staked out and held by virtue of a miner’s 
 
 ciminu^—A tvne of safety lamp invented by Dr. Clanny. . 
 
 Clastic— Constituted of rocks or minerals that are fragments derived from 
 
 Cla^ Coursed— A clay seam or gouge found at the sides of some veins. 
 
 Claying Bar.— For molding clay in a wet bore hole. mooonr , 0fi 
 
 Clay Band. — Argillaceous iron ore; common m many coal measures. 
 
 Clean-Up —Collecting the product of a period of work with battery or sluice. 
 Clearance— ( 1) The distance between the piston at the end of its stroke and 
 the end of the cylinder. (2) The volume or entire space filled with 
 steam at end of a stroke including the space between piston and 
 cylinder head, and the steam ducts to the valve seat. . . . 
 
 Cleat — ( 1) Vertical cleavage of coal seams, irrespective of dip or strike. (2) 
 A small piece of wood nailed to two planks to keep them together, or 
 nailed to any structure to make a support for something else. 
 
 Cleavage . — The property of splitting more readily in some directions than in 
 
 Clinometer — An instrument used to measure the angle of dip. 
 
 CY^-Soft and tough shale or slate forming the roof or floor of a coal seam. 
 Closed Season— When placers cannot be worked. 
 
 separate shafts by means of which 
 they catch into each other, so that both can revolve together. 
 
 Coal Breaker.— See Breaker. 
 
 Coni Cutter — A machine for holing or undercutting coal. 
 
 Coal Dust.— Very finely powdered coal suspended in the airways of a mine. 
 Coal Measures.— Strata of coal with the attendant rocks. 
 
 Coal Pipes (North of England). -Very thin irregular coal beds. 
 
 Coal Road— An underground roadway or heading in coai. 
 
 Co^ly^Rashings— Soft dark shale, in small pieces, containing much carbona- 
 
 CoarseTc “se) 1 - When lode stuff is not rich, the ore being only thinly dis- 
 
 CoaS^l^ol^f salting, the compound — the^ copper 
 concentrated in it after the first smelting to get rid of the bulk ot tne 
 gangue in the ore. 
 
 Coaster.— One that picks ore from the dump. 
 
 Cob (Cornish).— 1 To break up ore for sorting. 
 
 Cobbing Hammer.— A short double-ended hammer for breaking minerals to 
 sizes. 
 
 CocSrme^or 1 CoXers.— Timber used to hold coal face while it is being 
 
 undGrcut • 
 
 Cockle (Cornish).— Black tourmaline, often mistaken for tin. 
 
 Cod (North of England).— The bearing of an axle. ... 
 
 Cofer (Derbyshire).— To calk a shaft by ramming clay behind the lining. 
 Cofer.— Mortar box of a battery. . . 
 
 ' Coffer Dam.— An enclosure built in the water, and then pumped dry, so as to 
 permit masonry or other work to be carried on inside of it. 
 
 Coffin (Cornish).— An old pit. 
 
580 
 
 Cog 
 
 GLOSSARY. 
 
 Cor 
 
 Cog. A chock. 
 
 Coiiete (Mexican).— A rocket; applied to a blast within a mine or outside. 
 
 Co# Drag— A tool for picking pebbles, etc. from drill holes. 
 
 Coke . — The fixed carbon and ash of coal sintered together. 
 
 Colas (Spanish).— Tailings from a stamp mill or any wet process. 
 
 Collar.— (1) A flat ring surrounding anything closely. (2) Collar of a shaft 
 is the first wood frame of a shaft. (3) The bar or crosspiece of a framing 
 in entry timbering. 
 
 Colliery— The whole plant, including the mine and all adjuncts. 
 
 Colliery Warnings (English).— Telegraphic messages sent from signal-service 
 stations to the principal colliery centers to warn managers of mines 
 when sudden falls of the barometer occur. 
 
 Color ados (Spanish).— Decomposed ores stained with iron. 
 
 Colores (Mexican).— Metal-stained ground or rocks. 
 
 Colrake — A shovel for stirring lead ores while washing. 
 
 Color— Minute traces or individual specks of gold. 
 
 Column , or Column Pipe .— The pipe conveying the drainage water from the 
 mine to the surface. 
 
 Comer (Mexican).— To eat. Comerse los Pilares .— To take out the last vestiges 
 of mineral from the sides and rock pillars of a mine. 
 
 Conchoidal. — Shell-like, such as the curved fracture of flint. 
 
 Concrete .— Artificial stone, formed by mixing broken stone, gravel, etc. with 
 lime, cement, tar, or other binder. When hydraulic cement is used 
 instead of lime, the mixture is called beton (English). 
 
 Concretion.— A cemented aggregation of one or more kinds of minerals 
 around a nucleus. 
 
 Conduit.— (1) A covered waterway. (2) An airway. 
 
 Conduit Hole— A flat hole drilled for blasting up a thin piece in the bottom 
 of a level. 
 
 Conductors (English).— See Guides. 
 
 Conformable . — Strata are conformable when they lie one over the other with 
 the same dip. 
 
 Conglomerate .— The rock formation underlying the Coal Measures; a rock 
 containing or consisting of pebbles, or of fragments of other rocks 
 cemented together; English Pudding Rock or millstone grit. 
 
 Conical Drum .— The rope roll or drum of a winding engine, constructed in 
 the form of two truncated cones placed back to back, the outer ends 
 being usually the smaller in diameter. 
 
 Consumido (Mexican).— The amount of mercury that disappears by chem- 
 ical combination during the treatment of ore by any amalgamation 
 process. 
 
 Contact .— Union of different formations. 
 
 Contact Load or Vein — A vein lying between two differently constituted 
 rocks. 
 
 Contour.— (1) The line that bounds the figure of an object. (2) In survey- 
 ing, a contour line is a line eveiy point of which is at an equal elevation. 
 
 Contramina (Mexican).— Countermine. Any communication between two 
 or more mines. Also, a tunnel communicating with a shaft. 
 
 Cope (Derbyshire) .—Lead mining on contract. 
 
 Cope , or Coup .— An exchange of working places between hewers. 
 
 Copelilla ( Spanish ) .-Zinc-blende. 
 
 Copella (Spanish).— Dry amalgam. 
 
 Copper Plate.— A sheet of copper that, when coated with mercury, is used in 
 amalgamation. 
 
 Corbond . — An irregular mass from a lode. 
 
 Cord . — A cord weighs about 8 tons. 
 
 Cores. — Cylinder-shaped pieces of rock produced by the diamond-drill system 
 of boring. 
 
 Corf . — A mine wagon or tub. 
 
 Cornish Pumps.— A single-acting pump, in which the motion is transmitted 
 through a walking beam; in other respects similar to a Bull Pump. 
 
 Coro-Coro (South American).— Grains of native copper mixed with pyrite, 
 chalcopyrite, mispickel, etc. 
 
 Cortar Pillar (Mexican).— To form a rock support or pillar within a mine, 
 at the opening of a cross-cut or elsewhere. ' 
 
 Cortar Sogas (Mexican).— Literally, to cut the ropes. To abandon the mine, 
 taking away everything useful or movable. 
 
 Corve.—A mining wagon or tub. 
 
Cos 
 
 GLOSSARY. 
 
 Cro 
 
 581 
 
 Costean (Cornish).— To prospect a lode by sinking pits on its supposed 
 course. 
 
 Costeaning . — Trenching for a lode. 
 
 Cost Book (Cornish). — Mining accounts. . ^ 
 
 Cotton Rock.—( 1) Decomposed chert. (2) A variety of earthy limestone. 
 
 Coulee .— (1) A solified stream or sheet of lava extending down a volcano, 
 often forming a ridge or spur. (2) A deep gulch or water channel. 
 
 Count™— [if 1 !* cross-vein. (2) (English) An apparatus for recording the 
 number of strokes made by the Cornish pumping engine. (3) A second- 
 ary haulageway in a coal mine. . , x , 
 
 Counterchute.— A chute down which coal is dumped to a lower level or 
 ffftnfifw&v. 
 
 Counter gangway A level or gangway driven at a higher level than the 
 main one. 
 
 Country — The formation traversed by a lode. 
 
 Country Rock— The main rock of the region through which the veins cut, 
 or that surrounding the veins. 
 
 Course — The direction of a line in regard to the points of compass. 
 
 Coursing or Coursing the Air.— Conducting it through the different portions 
 of a mine by means of doors, stoppings, and brattices. 
 
 Cow— A self-acting brake. 
 
 Coyoting . — Irregular mining by small pits. A . 
 
 Crab —A variety of windlass or capstan consisting of a short shaft or axle, 
 either horizontal or vertical, which serves as a rope drum for raising 
 weights; it may be worked by a winch or handspikes. 
 
 Crab Holes— Holes often met with in the bed rock of alluvial. Also depres- 
 sions on the surface owing to unequal disintegration of the underlying 
 rock. 
 
 Cradle .— A box with a sieve mounted on rockers for washing auriferous 
 alluvial. „ _ _ 
 
 Cradle Dump . — A rocking tipple for dumping cars. See Dump. 
 
 Cramp (English).— (1) A short bar of metal having its two ends bent down- 
 wards at right angles for insertion into two adjoining pieces of stone, 
 wood, etc. to hold them together. (2) A pillar left for support in a mine. 
 
 Cranch— Part of a vein left by previous workers. 
 
 Crane (English).— A hoisting machine consisting of a revolving vertical 
 post or stalk, a projecting jib, and a stay for sustaining the outer end of 
 the jib; these do not change their relative positions as they do in a 
 derrick. There is also a rope drum with winding rope, etc. v ^ 
 
 Creaze (Cornish).— (1) Tin ore collected in the middle of the huddle. (2) The 
 middle of a huddle. „ . _ . . 
 
 Creep .—The gradual upheaval of the floor or sagging of the roof of mine 
 workings due to the weighting action of the roof and a tender floor. 
 
 Creston (Mexican).— The outcrop or apex of a vein or mineral deposit. 
 
 Crevice .— A fissure. . 
 
 C revicing . — Picking out the gold caught in cracks and crevices m the rocks 
 over which it has been washed. ' x . ... 
 
 Criadero (Mexican).— (1) A mineral deposit of irregular form, not vem-like. 
 (2) A chamber in a vein filled with ore of more or less richness. (3) Any 
 mineral deposit. This latter is the more modern sense, and the word is 
 so used in the mining laws at present in force in Mexico. 
 
 Crib —(1) A structure composed of horizontal timbers laid on one another, 
 or a framework built like a log cabin. See Chock. (2) A miner’s lunch- 
 eon. (3) See Curb. x . 
 
 Cribbing . — Close timbering, as the lining of a shaft, or the construction ot 
 cribs of timber, or timber and earth or rock to support a roof. 
 
 Cribble A sieve. 
 
 Crisol (Mexican).— A crucible of any kind. 
 
 Crop. — See Outcrop. , x 
 
 Crop Fall.— A caving in of the surface at or near the outcrop of a bed of coal. 
 
 Cropping Coal . — The leaving of a small thickness of coal at the bottom of the 
 seam in a working place, usually in order to keep back water. The coal 
 so left is termed “ Cropper Coal.” 
 
 Croppings .— Portions of a vein as seen exposed at the surface. 
 
 Cropping Out .— Appearing at the surface; outcropping. . 
 
 Cross-Course . — A vein lying more or less at right angles to the regular vein of 
 the district. 
 
582 
 
 Cro 
 
 GLOSSARY. 
 
 Deb 
 
 Crosscut.— (1) A tunnel driven through or across the measures from one 
 seam to another. (2) A small passageway driven at right angles to the 
 main gangway to connect it with a parallel gangway or air-course. 
 
 Crosses and Holes (Derbyshire).— Made in the ground by the discoverer of a 
 lode to tempoiarily secure possession. 
 
 Cross-Heading. — A passage driven for ventilation from the airway to the gang- 
 way, or from one breast through the pillar to the adjoining working. 
 
 Cross-Heading , or Cross-Gateway .—A road kept through goaf and cutting off 
 the gateways at right angles or diagonally. 
 
 Cross-Hole. — See Crosscut (2). 
 
 Cross- Latches. — See Latches. 
 
 Cross-Spur. — A vein of quartz that crosses the reef. 
 
 Cross- Vein. — An intersecting vein. 
 
 Crouan (Cornish).— Granite. 
 
 Crowbar. — A strong iron bar with a slightly curved and flattened end. 
 
 Crowfoot— A tool for drawing broken boring rods. 
 
 Crown Tree.— A piece of timber set on props to support the roof. 
 
 Crucero (Mexican). — A crosscut for ventilation to get around a horse, or to 
 prospect for the vein. 
 
 Crucible.— ( 1) The bottom of a cupola furnace in which the molten materials 
 collect. (2) Pots for smelting assays in. 
 
 Crush. — See Squeeze , Thrust. 
 
 Crusher— A machine used for crushing ores and rock. 
 
 Crushing— Reduction of mineral in size by machinery. 
 
 Crystal. — A solid of definite geometrical form, which mineral (or sometimes 
 organic) matter has assumed. 
 
 Culm.— Anthracite-coal dirt. 
 
 Culm Bank , or Culm Dump— Heaps of culm now generally kept separate 
 from the rock and slate dumps. 
 
 Cuna (Mexican). — Literally, a wedge. A short drill or picker generally 
 known in the United States as a “gad.” 
 
 Cupel— A cup made of bone ash for absorbing litharge. 
 
 Curb.— (1) A timber frame intended as a support or foundation for the lining 
 of a shaft. (2) The heavy frame or sill at the top of a shaft. 
 
 Curbing.— The wooden lining* of a shaft. 
 
 Cut. — (1) To strike or reach a vein. (2) To excavate in the side of a hill. 
 
 Cutter. — A term employed in speaking of any coal-cutting or rock-cutting 
 machines; the men operating them, or the men engaged in underholing 
 by pick or drill. 
 
 Cutting Down.— To cut down a shaft is to increase its sectional area. 
 
 Dam.— A timber bulkhead, or a masonry or brick stopping built to prevent 
 the water in old workings from flooding other workings, or to confine 
 the water in a mine flooded to drown out a mine fire. 
 
 Damp.— Mine gases and gaseous mixtures are called damps. See also After- 
 damp, Blackdamp , Firedamp , Stinkdamp. 
 
 Dan (North of England).— A truck without wheels. 
 
 Danger Board. — See Fireboard. 
 
 Dant (North of England). — Soft inferior coal. 
 
 Datum Water Level.— The level at which water is first struck in a shaft 
 sunk on a reef or gutter. 
 
 Davy.— A safety lamp invented by Sir Humphrey Davy. 
 
 Day. — Light seen at the top of a shaft. 
 
 Day Fall. — See Crop Fall. 
 
 Day Shift.— The relay of men working in the daytime. 
 
 Dead. — The air of a mine is said to be dead or heavy when it contains car- 
 bonic-acid gas, or when the ventilation is sluggish. 
 
 Dead. — (1) Unproductive. (2) Unventilated. 
 
 Dead Men’s Gi'aves (Australian). — Grave-like mounds in the basalt under- 
 lying auriferous gravels. 
 
 Dead Quartz.— Quartz carrying no mineral. 
 
 Dead Riches.— Lead carrying much bullion. 
 
 Dead Roast. — To completely drive off all volatile substances. 
 
 Deads. — Waste or rubbish from a mine. 
 
 Dead Work. — Exploratory or prospecting work that is not directly productive. 
 Brushing roof, lifting bottom, cleaning up falls, blowing rock, etc. 
 
 Dean (Cornish).— The end of a level. 
 
 Debris— Fragments from any kind of disintegration. 
 
GLOSSARY. 
 
 Dip 
 
 583 
 
 Dee 
 
 Beep (English).— “ To the deep,” toward the lower portion of a mine; hence, 
 
 Bella. — A triangularly shaped piece of alluvial land at the mouth of the 
 
 Yiver • 
 
 Bemasia (Mexican). — A piece of unoccupied ground between two mining 
 
 concessions. 
 
 Denudation The laving bare by water or other agency. 
 
 Benuncio (Mexican).— Denouncement. The act of applying for a mining 
 concession under the old mining laws. 
 
 Deposit.— { 1) Irregular ore bodies not veins. (2) A bed or any sedimentary 
 
 DenSvTEnglish).— (1) A man who fixes and withdraws % e ti m ker sup- 
 porting^!^ roof of a mine, and attends to the safety of the roof and 
 sides, builds stoppings, puts up bratticing, and looks after t^ safety of 
 the hewers, etc. (2) An underground othcial who sees to the general 
 safety of a certain number of stalls or of a district, but does not set 
 the timber himself, although he has to see that it is properly and suffi- 
 ciently done. (3) (American) A deputy sheriff. . 
 
 Berrick. — (1) A crane in which the rope or chain forming 
 
 let out or hauled in at pleasure, thus altering the inclination of Ihe jib 
 (2) The structure erected to sink a drill hole and the framework above 
 shafts are sometimes called by this name. . . - 0 
 
 Derrumbe , or Derrumbamineto (Mexican). — The caving m of the whole or a 
 portion of a mine. . . 
 
 Desaguador (Spanish). — A water pipe or dram. 
 
 Besague ( Mexican ) . — Drainage of a mine by any means. - 
 
 Bescargar (Mexican).— Literally, “to unload. Bescargar un Homo. To 
 
 Descubridoral Mexican^— The first mine opened in a new district or on a 
 new mineral deposit. 
 
 Besecho (Spanish).— Foul red mercury. 
 
 Besfrute (Mexican).— Taking out ore. Obras de Desfrute — Stopes, etc. 
 
 Besmontar (Mexican).— Literally, to clear away underbrush. In mining, to 
 take away useless and barren rocks; to remove rubbisn. 
 
 Besmontes (Spanish).— Poor ores. 
 
 Bespensa (Mexican) .—(1) A pantry or storeroom. (2) 
 up rich ore. 
 
 Bespoblado (Spanish).— Ore with much gangue. , 
 
 Bespoblar ( Mexican ) . — To suspend work m a mine. 
 
 Bessue (Cornish) —To cut away the ground beside 
 
 remove the latter whole. , . _ _ . . _ , „ 
 
 Bestajo (Mexican).— (1) A contract to do any kind of work in or ab 9 ut a 
 mine or elsewhere for a fixed price. (2) Piece work, as distinguished 
 from time work. Bestajero .— A contractor for piece work. 
 
 Detaching Hook.— A self-acting mechanical contrivance for setting free J 
 winding rope from a cage when the latter is raised beyond a certain 
 point in the head-gear; the rope being released, the cage remains 
 suspended in the frame. 
 
 Devil's Dice . — Cubes of limomte, pseudomorphs after pyrites. 
 
 Diagonal Joints.- Joints diagonal to the strike of the cleavage. 
 
 Dial (English). — An instrument similar to a surveyors compass, witfi 
 vernier attached. 
 
 Die.— The bottom iron block of a battery, or grinding pan on which the 
 sllOG 9/Ct)S 
 
 Digging. — Mining operations in coal or other minerals. 
 
 Diggings— Where gold and other minerals are dug out from shallow 
 alluvials. 
 
 Lillies, oT M^Short self-acting inclines where one or two tubs at a 
 time are run. . 
 
 Dillueing (Cornish). — Dressing tin slimes in a fine sieve. 
 
 Dip —(l) To slope downwards. (2) The inclination of strata with a hon 
 zontal plane. (3) The lower workings of a mine. 
 
 Dip Joint . — V ertical joints about parallel to the direction of the cleavage dip. 
 
 Dippa (Cornish). — A small catch-water pit. 
 
 Dipping 'Needle. — A magnetic needle suspended in a vertical plane, tor 
 locating iron deposits. 
 
 A secure room to lock 
 
 thin vein so as to 
 
584 
 
 Dir 
 
 GLOSSARY. 
 
 Dre 
 
 Dirt Fault —A confusion in a seam of coal, the top and bottom of the seam 
 being well defined, but the body of the vein being soft and dirty. 
 
 Dish (Cornish) —An ore measure; in lead mines, a trough 28 in. long, 4 in. 
 
 deep and 6 in. broad; sometimes 1 gallon, sometimes 14 to 16 pints. 
 Disintegration. — Separation by mechanical means, not by decomposition. 
 Ditch — (1) The drainage gutter in a mine. (2) A drainage gutter on the 
 surface. (3) An open conveyor of water for hydraulic or irrigation 
 purposes. , 
 
 Divide . — The top of a ridge, hill, or mountain. 
 
 Dividing Slate— A stratum of slate separating two benches of coal. See 
 PVLVtVflQ 
 
 Divining , or Dowsing , Rod— A small forked hazel twig that, when held 
 loosely in the hands, is supposed to dip downwards when passing over 
 water or metallic minerals. 
 
 Dizzue (Cornish).— See Dessue. . , , 
 
 Doa—(l) An iron bar, spiked at the ends, with which timbers are held 
 together or steadied. (2) A short heavy iron bar, used as a drag behind 
 a car or trip of cars when ascending a slope to prevent their running 
 back down the slope in case of accident. See Drag. 
 
 Dog Hole.— A little opening from one place in a mine to another, smaller 
 than a breakthrough. , „ . , , _ . . , 
 
 Doa Iron.— A short bar of iron with both ends pointed and bent down so as 
 to hoid together two pieces of wood into which the points are driven. 
 Or one end may be bent down and pointed, while the other is formed 
 into an eye, so that if the point be driven into a log, the other end may 
 be usedvto haul on. ^ _ 
 
 Doles. — Small piles of assorted or concentrated ore. . , 
 
 Dollv — (1) A machine for breaking up minerals, being a rough pestle and 
 mortar, the former being attached to a spring pole by a rope. (2) A 
 tool used to sharpen drills. . . , , . ^ „ 
 
 Dolly Tub (Cornish).— A tub in which ore is washed, being agitated by a 
 dollv or perforated boards. 
 
 Donk (North of Ehgland).— Soft mineral found m cross-veins. 
 
 Donkey Engine (English).— (1) A small steam engine attached to a large one, 
 and fed from the same boiler; used for pumping water into the boiler. 
 (2) A small steam engine. „ . , . , , , 
 
 Door Piece (English).— The portion of a lift of pumps in which the clack or 
 
 valve is situated. „ , . . , « . . 
 
 j) 00rs —Wooden doors in underground roads or airways to deflect the air- 
 
 current. . , 
 
 Door Tender— A boy whose duty it is to open and close a mine door before 
 and after the passage of a train of mine cars. , 
 
 Dope. — An absorbent for holding a thick liquid. The material that absorbs 
 the nitroglycerine in explosives. ...... 
 
 Double Shift.— When there are two sets of men at work, one set relieving the 
 
 Double *Tape Fuse— Fuse of superior quality, or having a heavier and stronger 
 
 Double timber. —Two props with a bar placed across the tops of them to sup- 
 port the roof and sides. „ , . . , „ , . . 
 
 Downcast. — The opening through which the fresh air is drawn or forced into 
 
 Dradge fCornish ) 6 ^ {if Inferior ore separated from the prill. (2) Pulverized 
 
 refuse ^ 
 
 Draftage.—A deduction made from the gross weight of ore when transported, 
 
 1 ) ' t he frictional resistance offered to a current of air in a mine. 
 
 Dra^-il) To^‘ draw ” the pillars; robbing the pillars after the breasts are 
 exhausted. (2) An effect of creep upon the pillars of a mine. 
 
 Draw a Charge.— To take a charge from a furnace. 
 
 Drawlift.—A pump that receives its water by suction and will not iorce it 
 
 Dr(m-Hrte*— Anaperture in a battery through which the coal is drawn. 
 Drawing an Entry. — Removing the last of the coal from an entry 
 Drawn.— The condition in which an entry or room is left after all the coal 
 has been removed. See Robbed. 
 
 Dresser (Staffordshire).— A large coal pick. 
 
L)RE 
 
 GLOSSARY. 
 
 Egg 
 
 585 
 
 Pressing— Preparing poor or mixed ores mechanically for metallurgical 
 
 nresSZa^Floors —The floors or places where ores are dressed. 
 
 Drift (1) A horizontal passage underground. A drift follows the vein, as 
 distinguished from a Crosscut, which intersects it, or a level or gahery, 
 which may do either. (2) In coal mining, a gangway above water 
 level driven from the surface in the seam. (3) Unstratified diluvium. 
 
 Drifting.— Winning pay dirt from the ground by means of drives. 
 
 Drill— An instrument used in boring holes. 
 
 Drive ( Drift). A horizontal passage in a lode. 
 
 Drive —To cut an opening through strata. . 
 
 Driving . — Excavating horizontal passages, in contradistinction to sinking or 
 
 Driving on' iine.-Keeping a heading or breast accurately on a given course 
 
 bv means of a compass or transit. . . n , 
 
 Dronper — (1) A spur dropping into the lode. (2) A feeder. (3) A branch 
 leaving the vein on the footwall side. (4) Water dropping from the roof 
 
 Drop Shaft— A monkey shaft down which earth and other matter are lowered 
 by means of a drop (i. e., a kind of pulley with break attached), the 
 empty bucket is brought up as the full one is lowered. 
 
 Druqqon (Staffordshire).— A vessel for carrying fresh water into a mine. 
 
 Drum.— The cylinder or pulley on which the winding ropes are coiled or 
 
 Drmn Rings.— Cast-iron rings with projections to which are bolted the 
 laggings forming the surface for the ropes to lap on. _ 
 
 Drummy.— Sounding loose, open, shaky, or dangerous when tested. 
 
 Druse.— A hollow cavity lined with small crystals. 
 
 Dry Amalgamation.— Treating ores with hot, dry mercury. 
 
 Dry Diggings— Placers never subject to overflow. 
 
 Dry Orel— Argentiferous ores that do not contain enough lead for smelting 
 
 DuckMuMne .— An arrangement of two boxes, one working within the 
 other, for forcing air into mines. 
 
 Duelas (Mexican).— Staves of a barrel or cask, etc. 
 
 Dumb’d. — Choked, of a sieve or grating. . 
 
 Dumb Drift.- A short tunnel or passage connecting the mam return airways 
 of a mine with the upcast shaft some distance above the furnace, m order 
 to prevent the return air laden with mine gases from passing through or 
 over the ventilating furnace. . « .. 
 
 Dump— (1) A pile or heap of ore, coal, culm, slate, or rock. (2) The tipple 
 by which the cars are dumped. (3) To unload a car by tipping it up. 
 
 (4) The pile of mullock as discharged from a mine. 
 
 Dumper.— A car so constructed that the body may be revolved to dump the 
 material in front or on either side of the track. 
 
 Durn (Cornish).— A timber frame. 
 
 Durr (German).— Barren ground. 
 
 Dust.— See Coal Dust. 
 
 Dust Gold. — Pieces under 2 to 3 dwt. . . _ , . 
 
 Duty. — The unit of measure of the work of a pumping engine expressed m 
 foot-pounds of work obtained from a bushel, or 100 lb., or other unit of 
 
 Dyke* or Dike.—{ 1) A wall of igneous rock passing through strata, with or 
 without accompanying dislocation of the strata. (2) A fissure filled with 
 igneous matter. (3) Barren rock. 
 
 Dzhu (Cornish).— See Dessue. 
 
 Ear.— The inlet or intake of a fan. . , . , , 
 
 Echadero (Mexican). — A level place near a mine where ore is cleaned, piled, 
 weighed, and loaded on mules or other conveyance. Also called patio ot 
 the mine. 
 
 Echado (Mexican).— The dip of the vein. . 
 
 Edge Coals (English).— Highly inclined seams of coal, or those having a dip 
 greater than 30^* 
 
 Efflorescence.— An incrustation by a secondary mineral, due to loss of water 
 of crystallization. 
 
 ¥gf Cool— Anthrac?^ 61 coal that will pass through a 2*" square mesh and 
 over a 2" square mesh (see page 434). 
 
586 
 
 Elb 
 
 GLOSSARY. 
 
 Fal 
 
 Elbow— A sharp bend, as in a lode or pipe. 
 
 Electric Blast— Instantaneous blasting of rock by means of electricity. 
 
 Elevator Pump— An endless band with buckets attached, running oyer two 
 drums for draining shallow ground. 
 
 Elvan — A Cornish name applied to most dike rocks of that county, irre- 
 spective of the mineral constitution, but in the present day restricted 
 to quartz porphyries. 
 
 Emborrascarse (Mexican).— To go barren by the vein terminating or pinching 
 out, etc. 
 
 Empties— Empty mine or railroad cars. 
 
 Encino (Mexican).— Live oak. 
 
 End Joint (End Cleat).— A joint or cleat in a seam about at right angles to 
 the principal or race cleats. 
 
 Endless Chain— A system of haulage or pumping by the moving of an 
 endless chain. 
 
 Endless Rope— A system of haulage same as endless chain, except that a 
 wire rope is used instead of chain. 
 
 End, or End-On— Working a seam of coal at right angles to the principal 
 or face cleats. 
 
 Engine Plane— An incline up which loaded cars are drawn by a rope 
 operated by an engine located at the top or bottom of the incline. The 
 empty cars descend by gravity, pulling the rope after them. 
 
 Engineer— { 1) One who has charge of the surveying or machinery about a 
 mine. (2) One who runs an engine. 
 
 Ensayes (Mexican).— Assays. 
 
 Entibar (Mexican).— To timber a mine or any part thereof. 
 
 Entry —A main haulage road, gangway, or airway. An underground passage 
 used for haulage or ventilation, or as a manway. 
 
 Entry Stops— Pillars of coal left in the mouths of abandoned rooms to 
 support the road, entry, or gangway till the entry pillars are drawn. 
 
 Erosion.— The wearing away of rocks by rains, etc. 
 
 Escaleras (Mexican).— Ladders, generally made of notched sticks. 
 
 Escarpment — A nearly vertical natural face of rock or soil. 
 
 Escoria (Mexican).— Slag or cinders. 
 
 Escorial. — Slag pile. 
 
 Escorificador (Mexican).— A scorifier, in assaying. 
 
 Espejuelo (Mexican).— A mineral gangue, with a faintly reflecting surface. 
 
 Espeton (Mexican).— The tapping bar of a smelting furnace. 
 
 Estano (Spanish). — Tin. 
 
 Estrujon (Mexican).— A second collection of amalgam, generally very pasty. 
 
 Exploder.— A chemical employed for the instantaneous explosion of powder. 
 
 Exploitation— The working of a mine, and similar undertakings; the exami- 
 nation instituted for that purpose. 
 
 Exploration— Development. 
 
 Explosion.— Sudden ignition of a body of firedamp. 
 
 Eye (English).— (1) A circular hole in a bar for receiving a pin and for 
 other purposes. (2) The eye of a shaft is the very beginning of a pit. 
 (3) The eye of a fan is the central or intake opening. 
 
 Face.— (1) The place at which the material is actually being WOTked, either 
 in a breast or heading or in longwall. (2) The end of a drift or tunnel. 
 
 Face-On. — When the face of the breast or entry is parallel to the face cleats 
 of the seam (see page 285). . 
 
 Face Wall.— A wall built to sustain a face cut into the natural earth, in 
 distinction to a retaining wall, which supports earth deposited behind it. 
 
 Faenas (Mexican).— Dead work, in the way of development. 
 
 Fahlband (German). — A course impregnated with metallic sulphides. 
 
 Faiscador (Spanish).— A gold washer. 
 
 Fall.—{ 1) A mass of roof or side which has fallen in any part of a mine. 
 (2) To blast or wedge down coal. 
 
 False Bedding.— Irregular lamination, wherein the laminae, though for 
 short distances parallel to each other, are oblique to the general strati- 
 fication of the mass at varying angles and directions. 
 
 False Bottom.— (1) A movable bottom in some apparatus. (2) A stratum on 
 which pay dirt lies, but which has other layers below it. 
 
 False Cleavage.— A secondary slip cleavage superinduced on slaty cleavage. 
 
 False Set.— A temporary set of timber used until work is far enough advanced 
 to put in a permanent set. 
 
Fam 
 
 GLOSSARY ; 
 
 Fla 
 
 587 
 
 Famp (North of England).—' Thin beds of soft tough shale. 
 
 Fan!— A machine for creating a circulation of air in a mine. 
 
 Fan Drift— A short tunnel or conduit leading from the top of the air-shaft to 
 the fan. , „ , A , . , 
 
 Fanega (Mexican).— A Spanish measure of about 2£ bushels. 
 
 Fang (Derbyshire). — An air-course. 
 
 Fascines (English).— Bunches of twigs and small branches for forming 
 foundations on soft ground. , 
 
 j past —(1) A road driven in a seam with the solid coal at each side. Fast 
 at an end,” or “ fast at one side,” implies that one side is solid coal and 
 the other open to the goaf or some previous excavation. (2) Bed rock. 
 Fast End— A n end of a breast of coal that requires cutting. 
 
 Fat Coals.— Those containing volatile oily matters. 
 
 jfaS?.— i^fracture or disturbance of the strata breaking the continuity of 
 
 the formation. , , 
 
 Feather —A slightly projecting narrow rib lengthwise on a shaft, arranged 
 to catch into a corresponding groove in anything that surrounds' and 
 slides along the shaft. , 
 
 Feather Edge — (1) A passage from false to true bottom. (2) The thin end 
 of a wedge-shaped piece of rock or coal. 
 
 Feather Ore. — Sulphide of lead and antimony. ^ 
 
 Feed.— Forward motion imparted to the cutters or drills of rock-drilling or 
 coal-cutting machinery, either hand or automatic. 
 
 Feeder. — (1) A runner of water. (2) A small blower of gas. 
 
 Feigh (North of England). — Ore refuse. ^ , 
 
 Fencing . — Fencing in a claim is to make a drive round the boundaries of 
 an alluvial claim, to prevent wash dirt from being worked out by 
 adjoining claim holders. , „ J , . , , 
 
 Fend-OJf (English).— A sort of bell-crank for turning a pump rod past the 
 angle of a crooked shaft. 
 
 Fierros (Mexican).— Iron matte. 
 
 Fiery.— Containing explosive gas. 1 
 
 Fines— Very small material produced in breaking up large lumps. 
 
 Fire— (1) A miners’ term for firedamp. (2) To blast with gunpowder or 
 other explosive. (3) A word shouted by miners to warn one another 
 when a shot is fired. , . , . . 
 
 Fire-Bars (English) .—The iron bars of a grate on which the fuel rests. 
 Fireboard.—A piece of board with the word fire painted upon it and sus- 
 pended to a prop, etc., in the workings, to caution men not to take a 
 naked light beyond it, or to pass it without the consent of the foreman 
 or his assistants. _ , 
 
 Fire Boss.— An underground official who examines the mine for gas and 
 inspects safety lamps taken into the mine. 
 
 Fireclay— Any clay that will withstand a great heat without vitrifying. 
 Firedamp.— (1) A mixture of light carburetted hydrogen ( CIL) and air m 
 explosive proportions; often applied to CH± alone or to any explosive 
 mixture of mine gases. 
 
 Fireman . — See Fire Boss. . 
 
 Fire-Setting.— The process of Exposing very hard rock to intense heat, ren- 
 dering it thereby easier for breaking down. 
 
 First Working . — See Whole Working. 
 
 Firsts— The best ore picked from a mine. 
 
 Fish.— 1 To join two beams, rails, etc. together by long pieces at their sides. 
 Fissure— An extensive crack. 
 
 Fissure Vein— Any mineralized crevice in the rock of very great depth. 
 Flags— Broad flat stones for paving. A , _ 
 
 Flagstone— Any kind of a stone that separates naturally into thm tabular 
 plates suitable for pavements and curbing. Especially applicable to 
 sandstone and schists. 
 
 Flang (Cornish).— A double-pointed pick. 
 
 Flange (English).— A projecting ledge or rim. 
 
 Flat.— (1) A district or set of workings separated by faults, old workings, or 
 barriers of solid coal. (2) The siding or station laid with two or more 
 lines of railway, to which the putters bring the full cars from the work- 
 ing face, and where they get the empty cars to take back. (3) The area 
 of working places, from which coal is brought to the same station, is also 
 called “flat.” 
 
588 
 
 Fla 
 
 GLOSSARY. 
 
 FrE 
 
 Flat Rod.— A horizontal rod for conveying power to a distance. 
 
 Flats— Narrow decomposed parts of limestones that are mineralized. 
 
 Flat Sheet— Sheet-iron flooring at landings and in the plats, chambers, and 
 junctions of drives, to facilitate the turning and management of trucks. 
 
 Flat Wall (Cornish).— Foot-wall. 
 
 Flintshire Furnace— A kind of reverberatory furnace used for smelting lead 
 
 jpfo^—Broken and transported particles or boulders of vein matter. 
 
 Float Gold— Gold in thin scales, which floats on water. 
 
 Float Ore.— A term applied by miners to ore found loose in the clay or soil. 
 
 Float Stowes— Loose boulders from lodes lying on or near the surface. 
 
 Flood Gate (English).— A gate to let off excess of water in flood or other 
 times. 
 
 Floor — (i) The stratum of rock upon which a seam of coal immediately 
 lies. (2) That part of a mine upon which you walk or upon which the 
 road bed is laid. 
 
 Floram (Cornish).— Very fine tin. 
 
 Flour Gold. — The finest alluvial gold. , 
 
 Flouring.— Mercury reduced to fine globules that are easily contaminated 
 and will not amalgamate. , . A ‘ . 
 
 Flucan.—A soft, greasy, clayey substance found m the joints of veins. 
 
 Fluke.— A rod for cleaning out drill holes. 
 
 Flume.— An artificial watercourse. „ . 
 
 Pluming — Lifting a river out of its bed with wooden launders or pipes, in 
 order to get at the bed for working. ^ . ,, 
 
 Flush .— (1) To clean out a line of pipes, gutters, etc. by letting m a sudden 
 rush of water. (2) The splitting of the edges of stone under pressure. 
 (3) Forming an even continuous line or surface. (4) To fill a mine 
 with fine material. 
 
 Fluthwerk (German). — Fiver prospecting. . 
 
 Flux— Iron ore, limestone, and sand, which are added m various propor- 
 tions to the charge in a furnace to make the gangue melt up and flow 
 off easily. „ , „ 
 
 Fodder (North of England).— 21 cwt. of lead. 
 
 Following Stone.— R oof stone that falls on the removal of the seam. 
 
 Foot (Cornish).— 2 gallons, or 60 lb., black tin. . 
 
 Foot-Hole. — Holes cut in the sides of shafts or winzes to enable miners to 
 ascend and descend. , , , , , ... 
 
 Foot- Piece.— (1) A wedge of wood or part of a slab placed on the toot-wail 
 against which a stull piece is jammed. (2) A piece of wood placed on 
 the floor of a drive to support a leg or prop of timber. 
 
 Foot- Wall. — The lower boundary of a lode. 
 
 Footway.— Ladders in mines. 
 
 Force Fan— See Blowdown Fan. 
 
 Force Piece— Diagonal timbering to secure the ground. 
 
 Force Pump— A pump that forces water above its valves. . 
 
 Forehay.— Penstock. The reservoir from which water passes directly to a 
 waterwheel. 
 
 Foreyolinq. — Driving the poles over the timbers so that their ends project 
 beyond the last set of timber, so as to protect the miner from roof falls; 
 used also in quicksand or other loose material. . , 
 
 Forewinninq— The first working of a seam in distinction from pillar drawing. 
 
 Fork .— (1) A deep receptacle in the rock, to enable a pump to extract the 
 bottom water. A pump is said to be “ going in fork ” when the water is 
 so low that air is sucked through the windbore. (2) (Cornish) Bottom 
 of sump. (3) (Derbyshire) Prop for soft ground. . 
 
 Formation.— A series of strata that belong to a single geological age. 
 
 Fossickers (Australian).— Grubbers for gold in the beach sand. 
 
 Fossicking— Overhauling old workings and refuse heaps for gold. 
 
 Fossil. — Organic remains or impressions of them found in mineral matter. 
 
 Fother (North of England). — i chaldron. .... , 
 
 Frame. — A table composed of boards, slightly inclined, over which water 
 runs to wash off waste from sluice tin. 
 
 Frame Set.— The legs and cap or collar arranged so as to support a passage 
 mined out of the rock or lode; also called Framing. . _ .. 
 
 Free— Coal is said to be “ free ” when it is loose and easily mined, or when it 
 will “run” without mining. 
 
 Free Milling . — Ores requiring no roasting or chemical treatment. 
 
FRE 
 
 GLOSSARY. 
 
 Gob 
 
 589 
 
 Free Miner.— Licensed miner. 
 
 Fresno (Mexican).— An ash tree. 
 
 Fronton (Mexican).— Any working face. 
 
 lT^oal fc or near the bottom of an upcast shaft, for pro- 
 dncing a current of air for ventilating the mine. 
 
 tridges. '2) To melt. 
 
 GaZap^^(MexiSn^^A S turtle-shaped pig of lead. 
 
 Gale.— A grant of mining ground. 
 
 Galiage.— Royalty. 
 
 &Mex"-Ki 0 rs^me^ of silver or gold ore, particularly those 
 -The* frame sufporiing a pulley over which the hoisting rope 
 Gam&lMexi^-A prospector for gold placers or ores. 
 
 Qanguet- Wasteniaterial from lodes. 
 
 GamS^-^hard^ompacL^^^ ® rec ^ a ^.' smelting 
 
 Gas Coal —Bituminous coal containing a large percentage of ga . 
 
 «5a aSSSSasif 48=*— 01 - 
 
 GaZX-l wo n o“lorTxed in an airway for regulatlng the supply 
 
 “ iS 
 
 <>octa— Large nodules of stone v 
 
 Geordie — A safety lamp invented by George btepnensou. 
 
 with a wedge-shaped key for holding pieces together. 
 RV«“-A vertical drum and framework by which the minerals 
 
 or waste rock when it arrives at the surface. degcribed as a p0 st roof 
 
 “thmlta? or^metal roof with post girdles, according as the 
 
 post or the metal predominates. 
 
 t», < 2SffidSffSS*<K‘»&4 S } satr 1 1 " l ”° *" 1 “ d 
 »qg- “H-nasi 
 
 , , v . 1 4. ^„„i T „f n Wo in thp mine. ( 
 
 Gob 
 
 that ' are*no t^marketabie hi the mine. (3) To stow or pack any useless 
 
 « f toe coal and slack in 
 
 the gob. 
 
590 
 
 Gob 
 
 GLOSSARY. 
 
 Gut 
 
 Gobbing Up— Filling with waste. 
 
 Gob Road,.— A roadway in a mine carried through the goaf. 
 
 Going Headways , or Going Bord.—A headway or bord laid with rails, and 
 used for conveying the coal tubs to and from the face. 
 
 Golpeador (Mexican).— A striker, in hand drilling. 
 
 Gossan.— A spongy ferruginous oxide, left after the soluble substances have 
 been dissolved out of a lode. 
 
 Goths (Staffordshire). — Sudden burstings of coal from the face, owing to 
 tension caused by unequal pressure. 
 
 Gouge— The layer of clay, or decomposed rock, that lies along the wall or 
 walls of a vein. It is not always valueless. 
 
 Grade— The amount of fall or inclination in ditches, flumes, roads, etc. 
 
 Grain— An obscure vertical cleavage usually more or less parallel to the end 
 or dip joints. 
 
 Granza (Mexican). — Metallic minerals from the size of rice to that of 
 hens’ eggs. 
 
 Grasa (Mexican). — Literally, grease. Slags. 
 
 Grass.— The surface of the ground. 
 
 Grassero (Spanish). — Slag heap. 
 
 Grate Coal. — See Broken Coal. 
 
 Grating.— A perforated iron sheet or wire gauze placed in front of reducing 
 machinery. 
 
 Gravel.— Water- worn stones about the size of marbles. 
 
 Gray Metal.— Shale of a grayish color. 
 
 Graywacke. — A compact gray sandstone frequently found in Paleozoic 
 formations. 
 
 Greenstone. — A general term employed to designate green-colored igneous 
 rocks, as diorite, dolerite, diabase, gabbro, etc. 
 
 Grena (Spanish).— Undressed ore. 
 
 Greta (Mexican). — Impure litharge formed in a reverberatory furnace. 
 
 Griddle— A coarse sieve used for sifting ores, clay, etc. 
 
 Grizzly.— A grating to throw out large stones from hydraulic gold 
 sluices. 
 
 Ground Rent.— Rent paid for surface occupied by the plant, etc. of a 
 colliery. 
 
 Groundsill.— A log laid on the floor of a drive on which the legs of a set of 
 timber rest. 
 
 Ground Sluicing. — Washing alluvial, loosened by pick and shovel, in trenches 
 cut out of the bed rock, using bars of rock as natural riffles. Used in 
 shallow placers, hill claims, bank claims, and stream diggings. 
 
 Grout (English).— Thin mortar poured into the interstices between stones 
 and bricks. 
 
 Grove (Derbyshire). — A mine. 
 
 Grub Stake.— The mining outfit or supplies furnished to a prospector on con- 
 dition of sharing in his finds. 
 
 Grueso (Mexican).— Lump ore. 
 
 Grundy.— Granulated pig iron. 
 
 Guag (Cornish).— Worked-out ground. 
 
 Gualdria (Mexican).— A long and stout beam, generally sustaining other 
 beams or some heavy weight. 
 
 Guano.— A brown, gray, or white, light powdery deposit, consisting mainly 
 of the excrement of sea fowl in rainless tracts, or of bats in caves. 
 
 Guarda Raya (Mexican). — A landmark; a monument. 
 
 Guardas (Mexican).— The country seat immediately enclosing any metal- 
 liferous vein or deposit. 
 
 Gubbin.— Ironstone. 
 
 Guia (Mexican). — Indications where to cut a pay streak or to find a vein. 
 
 Guides.— See Cage Guides. 
 
 Guija (Spanish). — Quartz. 
 
 Guijo (Mexican).— A pointed pivot, upon which turns the upright center 
 piece of an arrastre, of a door, etc. 
 
 Gunboat.— A self-dumping car, holding from 5 to 8 tons of coal, used upon 
 inclined planes or slopes. They are filled by emptying the mine cars 
 into them at the foot of the slope. 
 
 Gunnies ( Cornish ) .—3 ft. 
 
 Gurt (Cornish).— Water runnel from dressing floor. 
 
 Gutter— (1) A small water-draining channel. (2) The lowest part of a lead 
 that contains the most highly auriferous dirt. 
 
Hac 
 
 GLOSSAR Y. 
 
 Hea 
 
 591 
 
 Hacienda de Beneficio (Mexican).-In mining, a metaUnrgioal works; any 
 metallurgical works, usually an amalgamation works. 
 
 Hacienda de Fundicion (Mexican).— A smelting works. 
 
 Hacienda de Maquila (Mexican). — A custom mill. 
 
 Hade. — The inclination of a vein or fault, taking the vertical as zero. 
 
 At°an angle of 45° from general or previous course. (2) 
 Half on the level and half on the dip. 
 
 Half Set.— One leg piece and a cap. 
 
 Halvans Gangue containing a little ore. 
 
 Hammer-and-Plate. — A signaling apparatus. , 
 
 ft and Barrow — A long box with handles at each enu. 
 
 Hand Dog— A kind of spanner or wrench for screwing up and disconnecting 
 the joints of boring rods at the surface. . 
 
 Ha strike —A wooden lever for working a capstan or windlass. 
 
 Handwhip.— An apparatus used in shallow alluvial workings, consisting of 
 an unright at the top of which is balanced a long sapling, at the thick 
 end of the sapling, a bag of earth is fastened to counterbalance the 
 
 Th^man^thS'runs the^^ded cam on to the cages and gives 
 
 Hangin^Spmr JJod.-Wooden pump rods adjustable by screws, etc. by which 
 
 aSKSS" "theiiratum lying geologically 
 
 ffordtehi!— Residue from tin refining; contains much iron and arsenic. 
 Harrow.— Somewhat like an agricultural harrow; it is fixed to the pole of a 
 puddling machine and dragged around to break up and mix the aurifer- 
 ous clays with water. 
 
 ClWaped like a hat, revolving on a 
 vertical pin, for guiding inclined haulage ropes around curves. 
 
 TTntter A miner wor ki ng by himself on his own account. 
 
 Haulage Clip— Levers, jaws, wedges, etc. by which cars, singly or in trains, 
 
 Hai^in^^^e^rawing 6 or conveying of the product of the mine from the 
 working places to the bottom of the hoisting shaft, or slope. 
 Haunches.—' The parts of an arch from the keystone to the skew back. 
 
 Hazle (North of England). -Sandstone mixed with shale. 
 
 Head — (11 Pressure of water m pounds per square inch. (2) Any suoter 
 ranean passage driven in solid coal. (3) That part of a face nearest 
 
 Headfor Staiee Head (Australia and New Zealand) .-A supply of leu. ft. of 
 water per second, regardless of the head, pressure, or size of orifice. 
 
 He<M Block -( 1) A stop at the head of a slope or shaft to stop cars from 
 going down the shaft or slope. (2) A cap piece. . , . f 
 
 Heakboard.—A wedge of wood placed against the hanging wall, and against 
 which one end of the stull piece is jammed. 
 
 Header —(1) A rock that heads off or delays progress. (2) A blast hole at or 
 above the head. (3) A stone or brick laid lengthwise at right angles to 
 the face of the masonry. (4) The Stanley Header is an entry boring 
 machine that bores the entire section of the entry in one operation. 
 Head-Gear — The pullev frame erected over a shaft. . 
 
 Head-House.— When the head-frame is housed in, the structure is known by 
 
 Heading— (l)' A continuous passage for air °r for use as a ^nway; ^ Jgng- 
 way or entry. (2) A connecting passage between two rooms, breasts, or 
 other working places. 
 
 Head-Piece. — A cap; a collar. , ,, 
 
 Headrace— An aqueduct for bringing a supply of water on to the ground. 
 Headstocks. — Gallows frame; head-frame. . Tinrollol 
 
 Headways— (1) A road; usually 9 ft. wide, m a direction parallel to the 
 main-cleavage planes of the coal seams, "'hieh ^rection is Killed 
 “headways course,” and is generally about north and south m t e 
 Newcastle coal field. It is termed “on end .m other districts. (2) 
 
 He<we°— { The^shifting of rocks, seams, or lodes on the face of a cross-course, 
 
 etc. 
 
Hea 
 
 GLOSSARY. 
 
 Hyd 
 
 Heaving. — The rising of the thill (or floor) of a seam where the coal has been 
 removed. 
 
 Hechado (Spanish).— Dip. 
 
 Heel of Coal— A small body of coal left under a larger body as a support. 
 
 Heel of a Shot.— In blasting, the front of a shot, or the face of the shot farthest 
 from the charge. 
 
 Heep Stead (English).— The entire surface plant of a colliery. 
 
 Helper— A miner’s assistant, who works under the direction of the miner. 
 
 Helve.— A handle. 
 
 Hewer. — A collier that cuts coal; a digger. 
 
 High Reef— The bed rock or reef is frequently found to rise more abruptly 
 on one side of a gutter than on the other, and this abrupt reef is termed 
 a high reef. 
 
 Hijuelas (Mexican). — Literally, little children. A small-sized torta made up 
 as a sort of assay on a large scale, with from 1 to 5 kilograms of argentif- 
 erous mud. 
 
 Hill Diggings— Placers on hills. 
 
 Hilo (Spanish). — A thin metalliferous vein. 
 
 Hitch— { 1) A fault or dislocation of less throw than the thickness of the 
 seam in which it occurs. (2) Step cut in the rock or lode for holding 
 stay-beams, beams, or timber, etc. for various purposes. 
 
 Hoarding— A temporary close fence of boards placed around a work in 
 progress. 
 
 Hogback.— A roll occurring in the floor and not in the roof, the coal being 
 cut out or nearly so, for a distance. 
 
 Hoister. — A machine used in hoisting the product. It may be operated by 
 steampower or horsepower. 
 
 Hole. — (1) To undercut a seam of coal by hand or machine. (2) A bore 
 hole. (3) To make a communication from one part of a mine to 
 another. 
 
 Holing. — (1) The portion of the seam or underclay removed from beneath 
 tlie coal before it is broken down. (2) A short passage connecting two 
 roads. (3) See Kirving. 
 
 Holing Through. — Driving a passage through to make connection with 
 another part of the same workings, or with those in an adjacent mine. 
 
 Hood.— See Bonnet. 
 
 Hopper— A coal pocket; a funnel-shaped feeding trough. 
 
 Horn. — A piece of bullock’s horn about 8 in. in length, cut boat-shaped, for 
 concentrating by water on a small scale. 
 
 Horn Coal. — Coal worked partly end-on and partly face-on. 
 
 Horn Silver. — Chloride of silver. 
 
 Horse Gin.— A gearing for winding by horsepower. 
 
 Horsepower. — The power that will raise 33,000 lb. 1 ft. high per minute. 
 
 Horse, or Horsebacks. — (1) Natural channels cut or washed away by water in 
 a coal seam, and filled up with shale and sandstone. Sometimes a bank 
 or ridge of foreign matter in a coal seam. (2) A mass of country rock 
 lying within a vein or bed. (3) Any irregularity cutting out a portion 
 of the vein. See Dirt Fault and Rock Fault. 
 
 Horse Whim.— A vertical drum worked by a horse, for hauling or hoisting. 
 Called also Horse Gin. 
 
 Hose. — A strong flexible pipe made of leather, canvas, rubber, etc., and used 
 for the conveyance of water, steam, or air under pressure to any partic- 
 ular point. 
 
 H Piece— The portion of a column pipe containing the valves of the 
 pump. 
 
 Hueco (Mexican). — See Demasia. 
 
 Hulk (Cornish). — To pick out the soft portions of a lode. 
 
 Hundido (Mexican). — See Derrumbe. 
 
 Hungry. — Worthless looking. 
 
 Hurdy Gurdy. — A waterwheel that receives motion from the force of 
 traveling water. 
 
 Hushing— Prospecting by laying ground bare by sudden discharges of 
 pent-up water. 
 
 Hutch (Cornish).— (1) An ore- washing box. (2) (English) A mine car. 
 
 Hydraulic Cement. — A mixture of lime, magnesia, alumina, and silica that 
 solidifies beneath water. 
 
 Hydraulicking. — Working auriferous gravel beds by hydraulic power. 
 
 Hydrocarbons— Compounds of hydrogen and carbon. 
 
IGN 
 
 GLOSSARY. 
 
 Jig 
 
 593 
 
 n i, _ TVin<sp that, have been in a more or less fused state. 
 
 '!nbyl '-tea toe<!tion inward toward the face of the workings, or away from 
 
 Incline — bnort, C for inclined plane. Any inclined heading or slope road or 
 
 In -fort '-When a pump continues working after water has receded below 
 the holes of the wind bore. 
 
 T n aot — A lump of cast metal. ... . . . 
 
 Tn Place —A vein or deposit in its original position. _ , 
 
 mXpl-A a !^r 0 n°whfch coal is raised from a lower to a higher 
 Inspector^— A* government official whose duties are to enforce the laws regu- 
 Instmto— Tlfe^ight^o take coai from a royalty to the surface by a shaft in 
 
 Trp^tnne ( Cornish) . — Any hard tough stone. . , , 
 
 Iron Hat — Decomposed ferruginous mineral capping a lode. 
 
 Iron Man— A coal-cutting machine. 
 
 lahnnr'iiin (Mexican) — Decomposed talcose rock or hardened 
 
 Jab 7ound to a vtm and sometimes indicating the proximity of a rich strike. 
 
 took ‘-A ltnte?nS S ape“ made of tin, in which safety lamps are carried 
 
 -SSESSStfsa “oi suss 
 
 Jack-Lmap l -A Davy lamp, with the addition of a glass cylinder outside 
 
 Jacoiittf/cA'hrazilian).— Ferruginous ores associated with gold. 
 
 '(Mexica?).-Rlcli tailings or middlings from concentration 
 
 7or S 0 -lT?o!rdrillto'g, two long links > which take up the shock of impact 
 whpn the falling tools strike the bottom of the hole. . . , 
 
 JenMn - A road cut-in a pillar of coal in a bordways direction, that is, at 
 
 Jia -fl) 1 A^elf-actingfnffiSe. ^Y^^machine for separating ores or minerals 
 
 5 'from iorthlcsrfock by meaiis of their difference in specific gravity; 
 
 Jiane?- m A* Mnd'of Ijo^pling hook for connecting cars on an incline 
 99 W) An allowance of FiquSr sometimes issued to workmen (almost 
 
 Separatmg C hea*vy from light particles by agitation in water. 
 
594 
 
 Joe 
 
 GLOSSARY. 
 
 Lag 
 
 Jockey.— A self-acting apparatus carried on the front truck of a set for re- 
 leasing it from the hauling rope. 
 
 Joggle . — A joint of trusses or sets of timber for receiving pressure at right 
 angles, or nearly so. 
 
 Joints.— (1) Divisional planes that divide the rock in a quarry into natural 
 blocks. There are usually two or three nearly parallel series, called by 
 quarrymen end joints, back joints, and bottom joints, according to their 
 position. (2) In coal seams, the less pronounced cleats or vertical 
 cleavages in the coal. The shorter cleats, about at right angles to the 
 face cleats and the bedding plane of the coal. 
 
 Jud.—( 1) A portion of the working face loosened by “ kirving” underneath, 
 and “nicking” up one side. The operation of kirving and nicking is 
 spoken of as “making a jud.” (2) The term jud is also applied to a 
 working place, usually 6 to 8 yd. wide, driven in a pillar of coal. When 
 a jud has been driven the distance required, the timber and rails are 
 removed, and this is termed “ drawing a jud.” 
 
 Judge (Derbyshire and North of England) .—A measuring staff. 
 
 Jugglei's, or Jugulars .— Timbers set obliquely against the rib in a breast, to 
 form a triangular passage to be used as a manway, airway, or chute. 
 
 Jump .— An upthrow or a downthrow fault. 
 
 Jumper.— A hand drill used in boring holes in rock for blasting. 
 
 Kann (Cornish).— Fluorspar. 
 
 Kazen (Cornish).— A sieve. 
 
 Keckle-Meckle — Poorest lead ore. 
 
 Keeker— An official that superintends the screening and cleaning of the coal. 
 
 Keel Wedge . — A long iron wedge for driving over the top of a pick hilt. 
 
 Keeps , or Keps.— Wings, catches, or rests to hold the cage at rest when it 
 reaches any landing. 
 
 Keeve.—A large wooden tub used for the final concentration of tin oxide. 
 
 Kenner . — Time for quitting work. 
 
 Kerf .— The undercut made to assist the breaking of the coal. 
 
 Kerned (Cornish).— Pyrites hardened by exposure. 
 
 Kerve (North of England). — In coal mining, to cut under. 
 
 Kevil (Derbyshire).— Calcspar found in lead veins. 
 
 Key.—{ 1) An iron bar of suitable size and taper for filling the key ways of 
 shaft and pulley so as to keep both together. (2) A kind of spanner 
 used in deep boring by hand. 
 
 Kibble— See Bowk. Often made with a bow or handle, and carrying over a 
 ton of debris. 
 
 Kickup .— An apparatus for emptying trucks. 
 
 Kieve .— Tossing tub. 
 
 Killas (Cornish).— Clay slate. 
 
 Kiln.— A chamber built of stone or brick, or sunk in the ground, for burning 
 minerals in. 
 
 Kind. — (1) Tender, soft, easy. (2) Likely looking stone. 
 
 Kind-Chaudron.—A system of sinking shafts through water-bearing strata. 
 
 Kirving (North of England).— The cutting made beneath the coal seam. 
 
 Kist — The wooden box or chest in which the deputy keeps his tools. The 
 chest is always placed at the flat or lamp station, and this spot is often 
 referred to by the expression “at the kist.” 
 
 Kit .— Any workman’s necessary outfit, as tools, etc. 
 
 Kitty . — A squib made of a straw tube filled with powder. 
 
 Knee Piece . — A bent piece of piping. 
 
 Knocker.— A lever that strikes on a plate of iron at the mouth of a shaft, by 
 means of which miners below can signal to those on the top. 
 
 Knocker Line— Tine signal line extending down the shaft from the knocker. 
 
 Koepe System . — A system of hoisting without using drums, the rope being 
 endless and passing over pulleys instead of around a drum. 
 
 Labor (Mexican).— Mine workings in general. Specifically, a stope or any 
 other place where ore is being taken out. 
 
 Ladderway, Ladder Road . — The particular shaft, or compartment of a shaft, 
 used for ladders. 
 
 Lagging— (1) Small round timbers, slabs, or plank, driven in behind the 
 legs and over the collar, to prevent pieces of the sides or roof from falling 
 through. (2) Long pieces of timber closely fitted together and fastened 
 to the drum rings to form a surface for the rope to wind on. 
 
Lam 
 
 GLOSSARY. 
 
 Lid 
 
 595 ' 
 
 Lamas (Spanish).— (1) Slimes. (2) Argentiferous mud treated by amalga- 
 mation. 
 
 Lamer o (Mexican) .—Place of deposit for lamas. 
 
 Laminae. — Sheets not naturally separated, but which may be forced apart. 
 
 Lampazo (Mexican).— A sort of broom formed of green branches on the end 
 of a long stick, to dampen the flame in a reverberatory furnace. 
 
 Lamp Men. — Cleaners, repairers, and those having charge of the safety lamps 
 at a colliery. , . , „ • , 
 
 Lamp Stations.— Certain fixed stations m a mine at which safety lamps 
 are allowed to be opened and relighted by men appointed for that 
 purpose, or beyond which, on no pretense, is a naked light allowed 
 to be taken. 
 
 Lander.— The man that receives a load of ore at the mouth of a shaft. 
 
 Lander's Crook— A hook or tongs for upsetting the bucket of hoisted rock. 
 
 Landing— (1) A level stage for loading or unloading a cage or skip. 
 (2) The top or bottom of a slope, shaft, or inclined plane. 
 
 Land Sate.— The sale of coal loaded into carts or wagons for local consump- 
 tion. 
 
 Land-Sale Collieries— Those selling the entire product for local consumption, 
 and shipping none by rail or water. 
 
 Lap. — One coil of rope on a drum or pulley. 
 
 Lappior (Cornish). — An ore dresser. 
 
 Large. — The largest lumps of coal sent to the surface, or all coal that is hand 
 picked or does not pass over screens; also, the large coal that passes 
 over screens. , „ , . . _ 
 
 Larry— (1) A car to which an endless rope is attached, fixed at the inside 
 end of the road, forming part of the appliance for taking up slack rope. 
 See Balance Car. (2) See Barney. (3) A car with a hopper bottom and 
 adjustable chutes for feeding coke ovens. (4) A hopper-shaped car for 
 charging coke ovens. , , . 
 
 Latches. — (1) A synonym of switch. Applied to the split rail and hinged 
 switches. (2) Hinged switch points, or short pieces of rail that form 
 rail crossings. 
 
 Lateral.— From the side. 
 
 Lath.— A plank laid over a framed center or used in poling. 
 
 Launder. Water trough. 
 
 Laundry Box.— The box at the surface receiving the water pumped up from 
 below. 
 
 Lava.— A common term for all rock matter that has flowed from a volcano or 
 fissure. 
 
 Lavadero (Mexican).— A washer. A tank with a stirring arrangement to 
 loosen up the argentiferous mud from the patio, and dilute the same 
 with water, so that the silver amalgam may have a chance to precipitate. 
 An agitator. 
 
 Lazadores (Mexican).— Men formerly employed in recruiting Indians for 
 work in the mines by the gentle persuasion of a lasso. 
 
 Lazy Back (Staffordshire).— A coal stack, or pile of coal. 
 
 Leaching.— To dissolve out by some liquid. 
 
 Lead (pronounced teed).— ( 1) Ledge (America); reef (Australia); lode or 
 vein (England). A more or less vertical deposit of ore formed after the 
 rock in which it occurs. (2) A bed of alluvial pay dirt or an auriferous 
 gutter. (3) The distance to which earth is hauled or wheeled. 
 
 Leader.— A seam of coal too small to be worked profitably, but often being a 
 guide to larger seams lying in known proximity to it. 
 
 Leat.—A small water ditch. 
 
 Leavings (Cornish).— Hal vans. 
 
 Ledge.— See Lead. 
 
 Leg.— A wooden prop supporting one end of a collar. 
 
 Leg Piece.— An upright log placed against the side of a drive to support the 
 cap piece. 
 
 Lenador (Mexican).— One that cuts, carries, or furnishes wood for com- 
 bustible. 
 
 Level. — A road or gangway running parallel or nearly so with the strike 
 of the seam. 
 
 Ley (Mexican) .—Law. As applied in mining matters, it means the propor- 
 tion of precious or other metals contained in any mineral substance or 
 metallic alloy. 
 
 Lid .— A cap piece used in timbering. 
 
596 
 
 LIP 
 
 GLOSSARY. 
 
 Lum 
 
 Lift — rn The vertical height traveled by a cage in a shaft. (2) The lift of a 
 'Dump is theoretical height from the level of the water m the sump to the 
 noint of discharge. (3) The distance between the first level and the 
 surface, or between two levels. (4) The levels of a shaft or slope. 
 
 Liftinq Guards. — Fencing placed around the mouth of a shaft, which is 
 lifted out of the way by the ascending cage. 
 
 Lignite— A coal of a woody character containing about 66$ carbon and 
 y having a brown streak. . „ , , 
 
 Limadura ( Mexican) . — Filings. The mercurial globules seen when a piece 
 of argentiferous mud from a patio is washed m a spoon or saucer for 
 
 T irru^Cartridae — A charge or measured quantity of compressed dry caustic 
 lime made up into a cartridge and used instead of gunpowder for 
 breaking down coal. Water is applied to the cartridge, and the expan- 
 sion breaks down the coal without producing a flame. 
 
 Lime Coal.— Small coal suitable for lime burning. 
 
 Lines —Plumb-lines, not less than two m number, hung from hooks driven 
 in wooden plugs. A line drawn through the centers of the two strings 
 or wires, as the case may be, represents the bearing or course to be 
 
 driven on. , . . _ . 
 
 Lining.— The planks arranged against frame-sets. 
 
 Linnets (Derbyshire).— Oxidized lead ores. 
 
 Linternilla ( Mexican) .—The drum of a Horse Whim. 
 
 Lip Screen. — A small screen or screen bars, placed at the draw hole of a coal 
 pocket to take out the fine coal. 
 
 Lis (Mexican).—' The flouring of mercury. , , 
 
 Little Giant.— The name given to a special sort of hydraulic nozzle used for 
 
 Live^Quartz.—A variety of quartz usually associated with mineral. 
 
 AZa^es^Mexic^in ) .—Horizontal cross-beams in a shaft, or the upright pieces 
 that sustain the roof beams in a drift or tunnel. 
 
 Loaded Track.- Track used for loaded cars. . 
 
 Loader— One that fills the mine cars at the working places. 
 
 Loam. — Any natural mixture of sand and clay that is neither distinctly sandy 
 
 Locatiln^-The first approximate staking out or survey of a mining claim, in 
 distinction from a Patent Survey , or a Patented Claim. 
 
 in the country rock filled with mineral; 
 usually applied to metalliferous lodes. In general miners 
 vein or ledge is a tabular deposit of valuable minerals between definite 
 boundaries Whether it be a fissure formation or not is not always 
 known, and does not affect the legal title under the United States federal 
 and local statutes and customs relative to lodes. But it m ^ s t not be a 
 placer, i. e., it must consist of quartz or other rock in place, and bearing 
 
 Xodlstofor^ Magnetic iron ore. (2) Stone found in veins or lodes. 
 
 Logs. — Portions of trunks of trees cut to lengths and built 
 
 the mouth or collar of a shaft from the surface, in order to give the 
 reauisite space for the dumping of mullock and ore. 
 
 Lonq-Pillar Work— A system of working coal seams in three separate oper- 
 tions* (1) large pillars are left; (2) a number of parallel headings are 
 driven through the block; and (3) the ribs or narrow pillars are worked 
 
 LonTZt^-T woSdenltafce about 24 ft. long, 2 ft. wide, and 1 ft. high, for 
 washing auriferous gravel. 
 
 ^ngwall.— 't system of working a seam of coal in J?® 
 
 taken out and no pillars left, excepting the shaft pillars, and sometimes 
 the main-road pillars. 
 
 Loob (Cornish).— Sludge from tin dressing. 
 
 Loose End— ( 1) A portion of a seam worked on two sides. (2) A portion 
 that projects in the shape of a wedge between previous workings. 
 
 Lumber—' Timber cutto" the various sizes and shapes 
 
 Lumbreras (Mexican) .-Ventilating shafts in a mine or other underground 
 work. 
 
GLOSSARY. 
 
 Mer 
 
 597 
 
 IrtJM vr MJWK,**** - . 
 
 TnllYnrn r^ni All ooal (anthracite only) larger than broken coal, or, when 
 co^ is mXlumps larger than this size. (2) In soft coal, all 
 poai nassing over the nut-coal screen. . 
 
 Lute.— An adhesive clay used either to protect any iron vessel from too 
 strong a heat or for securing air- and gas-tight joints. 
 
 Lye (English).— A siding or turnout. 
 
 Marhote ( Mexican). — A stake or permanent bench mark fixed in an under- 
 
 M ground working, from which the length and progress thereof is 
 measured. _ 
 
 Magidr^ (Spanish) .—Roasted copper pyrites, copper sulphate, etc., used to 
 reduce silver ores. 
 
 the north end of the magnetic 
 
 Ma™eMc%eridian.-The line or great circle in which the magnetic needle 
 
 m i»%ad a -a'h^ V prin?ipal‘ haulage road of a mine from which the several 
 crossroads lead to the working face. 
 
 IT g&SfSSW hSgt the rope that draws the loaded cars 
 
 MakU^as (North of Englandl.-Small coal produced in kirving. Fines. 
 
 MaSte (Mexican).— A Horse Whim; now extended to any hoisting machine 
 used in mines. 
 
 KT-in^ and supervision of a mine, both 
 
 Man^Er^ncs— An°apparatus consisting of one or two reciprocating rods, to 
 which suitable stages are attached, used for lowering and raising men 
 
 A^^ tTelXof a gangway, tunnel, 
 
 or slope * (2) A hole in cylindrical boilers through which a man can 
 get into the boiler to examine and repair it. 
 
 which ore or waste 
 
 Maufeo a ( Mexican) .-The act of hoisting ore or waste from a mine. 
 
 MSaj/ M -AX2ll^a|eufeFS\ traveling way for the miner, and also 
 often used as an airway or chute, or both. 
 
 fScaXTo U work “re for its owner on shares or for a money 
 payment. 
 
 Marco (Mexican).— A weight of 8 oz. 
 
 tapered round iron, used in 
 
 splicing ropes. j 
 
 Marmajas (Spanish). — Concentrated sulphides. 
 
 mZZrLamp- ‘“'type of safety lamp whose chief characteristic is the 
 
 lKorsh U fito?— Cffn often used synonymously with Firedamp (see page 348). 
 
 Match.— (1) A charge of gunpowder put into a paper several inches long, and 
 used for igniting explosives. (2) The touch end of a squib. 
 
 >faffe S — A compound of iron and other metals, chiefly copper, with sulphur, 
 formed during smelting. , _ ,. . 
 
 Mattock— A kind of pick with broad ends for digging. 
 
 MaundHl^A pick with two shanks and points, used for getting coal, etc. 
 
 Mazo (Mexican). — A stamp. 
 
 Mear (Derbyshire). — 32 yd. along the vein. 
 
 Measures— Strata. 
 
 Mecha (Mexican).— A wick for a lamp or candle, a torch. 
 
 Merced (Mexican).— A gift, grant, or concession. 
 
 Meridian.— A north and south line, either true or approximate. 
 
598 
 
 Met 
 
 GLOSSARY ; 
 
 Mou 
 
 Metal.— {1) In coal mining, indurated clay or slate. (2) An element that 
 forms a base by combining with oxygen that is solid at ordinary tem- 
 perature (with exception of quicksilver), opaque (except in the thinnest 
 possible films), has a metallic luster, and is a good conductor of heat 
 and electricity, and, as a rule, of a higher specific gravity than the non- 
 metals. (3) (Mexican) All kinds of metalliferous minerals are called 
 “ metal” in Mexico. Metal de Ayuda.— Fluxing ore of any kind. Metal 
 de Cebo. — Very rich ore, usually treated in small reverberatory furnaces. 
 Metal Ordinario— Common ore. Metal Pepena.— The best class of selected 
 
 ore. 
 
 Metlapil (Mexican).— See Mano. 
 
 Mill. — Works for crushing and amalgamating gold and silver ores. 
 
 Mill Cinder.— The slag from the puddling furnace of a rolling mill. 
 
 Mill Hole.— An auxiliary shaft connecting a stope or other excavation with 
 the level below. 
 
 Mill Run.— The test of a given quantity of ore by actual treatment in a mill. 
 
 Mine.— A ny excavation made for the extraction of minerals. 
 
 Miner.— One who mines. a 
 
 Mineral— Any constituent of the earth’s crust that has a definite com- 
 position. 
 
 Mineral Oil.— Petroleum obtained from the earth, and its distillates. 
 
 Minero (Mexican).— A mine owner; a mining captain; an underground boss. 
 
 Mine Road.— Any mine track used for general haulage. 
 
 Mine Run— The entire unscreened output of a mine. 
 
 Minero Mayor (Mexican).— The head mining captain. A mining workman 
 is called Operario. 
 
 Miners' Dial.— An instrument used in surveying underground workings. 
 
 Miners' Inch.— A measure of water varying in different districts, being the 
 quantity of water that passes through a slit 1 in. high, of a certain width 
 under a given head (see page 136). 
 
 Miner's Right.— An annual permit from the Government to occupy and work 
 mineral land. . _ V. - , 
 
 Mining— In its broad sense, it embraces all that is concerned with the 
 extraction of minerals and their complete utilization. 
 
 Mining Engineer —A man having knowledge and experience in the many 
 departments of mining. , . , A , , 
 
 Mining Retreating.— A process of mining by which the vein is untouched 
 until after all the gangways, etc. are driven, when the mineral extraction 
 begins at the boundarv and progresses toward the shaft. 
 
 Mistress (North of England).— A miner’s lamp. 
 
 Mock Lead (Cornish).— Zinc blende. 
 
 Mogrollo (Mexican).— Same as Metal de Cebo. . „ „ 
 
 Moil— A short length of steel rod tapered to a point, used for cutting 
 hitches, etc. . „ , . , . 
 
 Molonque (Mexican).— A rich specimen of which one-half or more is native 
 
 silver. 
 
 Monitor. — See Gunboat. 
 
 Monkey.— The hammer or ram of a pile driver. 
 
 Monkey Drift.— A small drift driven in for prospecting purposes, or a crosscut 
 driven to an airway above the gangway. 
 
 Monkey Gangway.— A small gangway parallel with the mam gangway. 
 
 Monkey Rolls.—' The smaller rolls in an anthracite breaker. 
 
 Monkey Shaft.— A shaft rising from a lower to a higher level. 
 
 Monoclinal— Applied to an area in which the rocks all dip m the same 
 
 Mop— Some material surrounding a drill in the form of a disk, to prevent 
 water from splashing up. ,, 
 
 Mortar. — The vessel in which ore is placed to be pulverized by a pestle. 
 Mortise. — A hQle cut in one piece of timber, etc. to receive the tenon that 
 projects from another piece. .... , . 
 
 Mote (Moat).— A straw filled with gunpowder, for igniting a shot. 
 
 Mother Gate.— The main road of a district in longwall working. 
 
 Mother Lode (Main Lode).— The principal vein of any district. , 
 
 Motive Column.— The length of a column of air whose weight is equal to the 
 difference in weight of like columns of air in downcast and upcast shafts. 
 The ventilating pressure in furnace ventilation is measured by the differ- 
 ence of the weights of the air columns in the two shafts. 
 
 Mouth . — The top of a shaft or slope, or the entrance to a drift or tunnel. 
 
Moy 
 
 GLOSSARY 
 
 Ope 
 
 599 
 
 sharp steel point, for driving into clefts when 
 
 Moyle.— An iron with a 
 
 levering off rock. . _ 
 
 Muckle.— Soft clay overlying or underlying coal. 
 
 Mucks (Staffordshire).— Bad earthy coal. ^ , 
 
 Muescas (Mexican).— Notches in a stick; mortises; notches cut in a round 
 or square beam, for the purpose of using it as a ladder. 
 
 Mueseler Lamp.— A type of safety lamp invented and used m the collieries 
 of Belgium. Its chief characteristic is the inner sheet-iron chimney for 
 increasing the draft of the lamp. 
 
 Muffle. — A thin clay oven heated from the outside. 
 
 Muller— The upper grinding iron or rubbing shoe of amalgamating 
 T3RHS otc 
 
 Mullock.— Country rock and worthless minerals taken from a mine. 
 
 Mundic — Iron pyrites. 
 
 Naked Light.— A candle or any form cf lamp that is not a safety lamp. 
 
 Narrow Work.—{ 1) All work for which a price per yard of length driven is 
 paid, and which, therefore, must be measured. (2) Headings, chutes, 
 crosscuts, gangways, etc. 
 
 Natas (Mexican) —Same as Escoma or Grasa. 
 
 Native Metal.— A metal found naturally, in that state. 
 
 Natural Ventilation.— Ventilation of a mine without either furnace or otner 
 artificial means; the heat imparted to the air by the strata, men, 
 animals, and lights in the mine, causing it to flow in one direction, or 
 
 to ascend. • 
 
 Neck. — A cylindrical body of rock differing from the country around it. 
 
 Needle— (1) A sharp-pointed metal rod with which a small hole is made 
 througn the stemming to the cartridge in blasting operations. (2) A 
 hitch cut in the side rock to receive the end of a timber. 
 
 Negrillo (Mexican).— Black sulphide of silver. 
 
 Nick. — To cut or shear coal after holing. , fo . 
 
 Nicking— (1) A vertical cutting or shearing up one side of a face of coal. (2) 
 The chipping of the coal along the rib of an entry or room which is 
 usually the first indications of a squeeze. 
 
 Night Shift.—’ The set of men that work during the night. 
 
 Nip —When the roof and floor of a coal seam come close together, pinching 
 the coal between them. ... . . , • 
 
 Nip Out. — The disappearance of a coal seam by the thickening of the adjoin- 
 ing strata, which takes its place. 
 
 Nitro.—A corrupted abbreviation for nitroglycerine or dynamite. 
 
 Nittings. — Refuse of good ore. 
 
 Nodular. — Blistered or kidney-shaped ore. . 
 
 Nodules.— Concretions that are frequently found to enclose organic remains. 
 
 Nogs _ Logs of wood piled one on another to support the roof. See Chock. 
 
 Nook —The corner of a working place made by the face with one side. 
 
 Noria (Spanish). — An endless chain of buckets. . . 
 
 Nozzle. — The front nose piece of bellows of a blast pipe for a furnace, or of a 
 w'ater pipe. 
 
 Nugget.— A natural lump of gold or other metal, applied to any size above 2 
 
 to 3 dwt. , . . . 
 
 Nut Coal — A contraction of the term chestnut coal. 
 
 Nuts. — Small lumps of coal that will pass through a screen or bars, the spaces 
 between which vary in width from £ to 2£ in. 
 
 Ocote (Mexican). — Pitch pine. _ . . 
 
 Odd Work. — Work other than that done by contract, such as repairing 
 roads, constructing stoppings, dams, etc. * _ 
 
 Offtake —The raised portion of an upcast shaft ^bove the surface, for carrying 
 off smoke and steam, etc., produced by the furnaces and engines under- 
 
 Oil IS!— Shale containing such a proportion of hydrocarbons as to be 
 capable of yielding mineral oil on slow distillation. 
 
 Oil Smellers.— Men that profess to be able to indicate where petroleum oil is 
 to be found. 
 
 Old Man. — Old workings in a mine. . ... .. „„„ ~ 
 
 Oolitic.— A structure peculiar to certain rocks, resembling the roe of 
 a fish. 
 
 Open Cast— Workings having no roof. 
 
600 
 
 Ope 
 
 GLOSSARY. 
 
 Pan 
 
 Open Cutting— (1) An excavation made on the surface for the purpose of get* 
 ting a face wherein a tunnel can be driven. (2) Any surface excavation 
 
 Openings , An Opening— Any excavation on a coal or ore bed, or to reach the 
 same; a mine. * 
 
 Openwork— An open cut. 
 
 Operario ( Mexican ) .—A working miner. 
 
 Operator — The individual or company actually working a colliery. 
 
 Ore— A mineral of sufficient value (as to quality and quantity), to be mined 
 with profit. . , , ^ 
 
 Ores .—Minerals or mineral masses from which metals or metallic combina- 
 tions can be extracted on a large scale in an economic manner. 
 
 Ore Shoot— A large and usually rich aggregation of mineral in a vein. 
 Distinguished from pay streak in that it is a more or less vertical zone 
 or chimney of rich vein matter extending from wall to wall, and having 
 a definite width laterally. 
 
 Oro (Spanish).— Gold. , . % . . . 
 
 Oroche (Spanish).— (1) Retorted bullion. (2) (Mexican) Bullion containing 
 gold and silver. A . A . . _ _ _ 
 
 Outburst —A blower. A sudden emission of large quantities of occluded gas. 
 
 Outbye. — In the direction of the shaft or slope bottom, or toward the outside. 
 
 Outcrop. — The portion of a vein or bed, or any stratum appearing at the sur- 
 face, or occurring immediately below the soil or diluvial drift. 
 
 Outcropping.— See Cropping Out. . 
 
 Outlet.— A passage furnishing an outlet for air, for the miners, for water, or 
 for the mineral mined. , , , , , , . _ 
 
 Output.— The product of a mine sent to market, or the total product of a mine. 
 
 Outset.—' The walling of shafts built up above the original level of the ground. 
 
 Outstroke Rent.— The rent that the owner of a royalty receives on coal brought 
 into his royalty from adjacent properties. . _ , . 
 
 Outtake — The passage by which the ventilating current is taken out of the 
 mine; the upcast. 
 
 Overburden.— The covering of rock, earth, etc. overlying a mineral deposit 
 that must be removed before effective work can be performed. 
 
 Overcast.— A passage through which the ventilating current is conveyed over 
 a gangway or airway. , „ „ A . 
 
 Overhand Stoping— The ordinary method of stopmg upwards. 
 
 Overlap Fault— A fault in which the shifted strata double back over them- 
 
 SG1VGS. 
 
 Overman— One who has charge of the workings while the men are in the 
 mine. He takes his orders from the Underviewer. 
 
 Overwind.—' To hoist the cage into or over the top of the head-frame. 
 
 Oyamel (Mexican).— White pine. 
 
 Pack. — A rough wall or block of coal or stone built up to support the roof. 
 
 Packing. — The material placed in stuffingboxes, etc. to prevent leaks. 
 
 Pack Wall— A wall of stone or rubbish built on either side of a mine road, *o 
 carry the roof and keep the sides up. 
 
 Pacos (Spanish).— Ferruginous silver ores. . ... 
 
 Paddock. -( 1) An excavation made for procuring wash dirt in shallow 
 ground. (2) A place built near the mouth of a shaft where ore is stored. 
 
 Paint Gold.— The very finest films of gold coating other minerals. 
 
 Paleozoic— The oldest series of rocks in which fossils of animals occur. 
 
 Palero (Mexican). — A mine carpenter. „ ^ , , ,, 
 
 Palm.— A piece of stout leather fitting the palm of the hand, and secured by 
 a loop to the thumb; this has a flat indented plate for forcing the needle. 
 
 Palm Needle.— A straight triangular-sectioned needle, used for sewing canvas. 
 
 Palo (Mexican).— A stick; a piece of timber 
 
 Pan— A thin sheet-iron dish;16 in. across the top, and 10 in. at the bottom, 
 used for panning gold. 
 
 Panel— ( 1) A large rectangular block or pillar of coal measuring, say, 
 130 by 100 yd. (2) A group of breasts or rooms separated from the other 
 workings by large pillars. . . ... 
 
 Panel Working.— A system of working coal seams m which the colliery is 
 divided up into large squares or panels, isolated or surrounded by solid 
 ribs of coal, in each of which a separate set of breasts and pillars is 
 worked, and the ventilation is kept distinct, that is, every panel has its 
 own circulation, the air of one not passing into the adjoining one, but 
 being carried direct to the main return airway. 
 
Pan 
 
 GLOSSARY 
 
 Pic 
 
 601 
 
 Panino (Mexican).— 1 The peculiar appearance, form, or manner in which 
 the metalliferous minerals present themselves m any given distnc 
 
 Panning ot’ P anning Off.- Separating gold or tin from its accompanying 
 minerals by washing off the latter in a pan. 
 
 a chattering noise 
 
 ’ “s t" n a g c a eS 
 
 nart 5 the ores in place of wages. Usually, the mine owner provides 
 handles nowder and steel, and keeps the drills sharpened, and receive*, 
 in navment of royalty and supplies, two-thirds or more of the ore taken 
 ^This contract is renewed weekly or monthly, etc., and the proper - 
 Son ofoSe retained by the miners is more or less, according to the richness 
 of The stoves ^whlreUey wSrk This is a cheap way of getting ore as far 
 as ^labor^^c^neemed. But the miners must be constantly watched; 
 otherwise they will leave the mine in bad state. The proportion of ore 
 £3 to the Miners is generally bought from them by the mine 
 
 Parting!— (1 ^Any ’ toto “SK' bed of earthy material. (2) A side 
 track or turnout in a haulage road. 
 
 down ore to a lower level (21 A 
 pas^ge left irSd workings for men to travel in from one level to 
 
 PasSfy — A siding in which cars pass one another underground. A turnout 
 
 Pass- Into. — Wh en one mineral gradually passes into another without any 
 
 PaMpZl-imlu coal mixed with 8 to 10jt of pitch or tar, and compressed 
 
 Patofe^ta^-A^fato to which a patent right has been secured from the 
 
 i government, by compliance with the laws relating to such claims. 
 
 Afent Survey. — An^ accurate survey of a claim fj h ? t o^e cllim 7 
 required by law m order to secure a patent right to the claim. 
 
 Pa(*WMerfcan) fl -Aiay paved enclosure more or less surrounded by build- 
 toga ^ av ^ A floor Qr yard where argentlfer ous mud is 
 
 treated by amalgamation. 
 
 ZJrt . — That portion of an alluvial deposit that contains gold in payable 
 quantities. 
 
 Pay Out.— To slacken or let out rope. 
 
 Pay Rock — Mineralized rock. 
 
 Pay Streak— Mineralized part of rock. 
 
 Pea C Coal.—l smalTsize o^nthracite coal (see page 434). 
 
 Peat — T^decomposed partly carbonized organic matter ofbogs, swamps, eta 
 
 Pebble Jack.- Zinc blende in small crystals or pebble-like forms is not 
 attached to rock, but is found m clay openings m the rock. 
 
 Pee (Derbyshire).— A fragment of lead ore. 
 
 Pdla , or Plata Pella (Mexican).— Silver amalgam. 
 
 P^tHo'ji- -A^odfn covering for the protection of sinkers working in a 
 
 PerJice — A*few pieces of timber laid as a roof over men’s heads, to screen 
 them when working in dangerous places, e.g., at the bottom of shafts. 
 
 Pepenado ( Spanish ) .—Dressed ore. 
 
 P^ZZonTabU^r-kintof jolting table used in separating very tine ores 
 from slimes. . . , 
 
 fSVi-To^ “tonally decrease in thickness. 
 
 Petlanque (Mexican).— Ruby silver. 
 
 Petrifaction — Organic remains converted into stone. - 
 
 i) a tooffor cutting and holing coal. (2) To dress the sides Ox face 
 
 of an excavation with a pick. 
 
602 
 
 Pic 
 
 GLOSSARY. 
 
 Pla 
 
 Picker.— { 1) A small tool used to pull up the wick of a miner’s lamp. 
 (2) A person who picks the slate from the coal in an anthracite-coal 
 breaker. 
 
 Picking Chute. — A chute in an anthracite breaker along which boys are 
 stationed to pick the slate from coal. 
 
 Picking Table— (1) A flat or slightly inclined platform on which anthracite 
 coal is run to be picked free from slate. (2) A sorting table. 
 
 Pico (Mexican). — A striking or sledge hammer. 
 
 Picture. — A screen to keep off falling water from men at work. 
 
 Piedras de Mano (Mexican). — Hand specimens. 
 
 Pig— A piece of lead or iron cast into a long iron mold. 
 
 Pigsty Timbering. — Hollow pillows built up of logs of wood laid crosswise 
 for supporting heavy weights. 
 
 Pike. — A pick. 
 
 Pilar (Mexican). — A pillar of rock or ore left to sustain some portion of the 
 mine. 
 
 Pileta (Mexican).— (1) A sump. (2) The basin or pot where melted metal 
 is collected. 
 
 Piling.— Long pieces of timber driven into soft ground for the purpose of 
 securing a solid base on which to build any superstructure. 
 
 Pillar. — (1) A solid block of coal, etc. varying in area from a few square 
 yards to several acres. (2) Sometimes applied to a single timber 
 support. 
 
 Pillar -and-Room — A system of working coal by which solid blocks of coal 
 are left on either side of the rooms, entries, etc. to support the roof until 
 the rooms are driven up, after which they are drawn out. 
 
 Pillar-and-Stall. — See Breast-and-Pillar . 
 
 Pillar Roads— Working roads or inclines in pillars having a range of long- 
 wall faces on either side. 
 
 Pillion ( Cornish) .—Metal remaining in slag. 
 
 Pina (Mexican).— Same as Pella. 
 
 Pinch. — A contraction in the vein. 
 
 Pinch Out. — When a lode runs out to nothing. 
 
 Pinta (Mexican).— The color, weight, grain, etc. of ores, whereby it is pos- 
 sible to form some idea of their richness in the various metals. 
 
 Pipe— An elongated body of mineral. Also the name given to the fossil 
 trunks of trees found in coal veins. 
 
 Pipe Clay.— A soft white clay. 
 
 Piped Air. — Air carried into the working place by pipes or brattices. 
 
 Piping— Undercutting and washing away gravel before the water nozzle. 
 
 Pit.—( 1) A shaft. (2) The underground portion of a colliery, including all 
 workings. (3) A gravel pit. 
 
 Pit Bank.— The raised ground or platform where the coal is sorted and 
 screened at the surface. 
 
 Pit Bottom.— The portion of a mine immediately around the bottom of a shaft 
 or slope. See Shaft Bottom. 
 
 (1; Rise of a seam. (2) Grade of an incline. (3) Inclination. (Cor- 
 ’'ish) A part of a lode let out to be worked on shares, or by the piece. 
 
 Pit Coal. — Generally signifies the bituminous varieties of coal. 
 
 Pit Frame.— See Head-Frame. 
 
 Pit Headman— The man who has charge at the top of the shaft or slope. 
 
 Pitman. — A miner; also, one who looks after the pumps, etc. 
 
 Pit Prop.— A piece of timber used as a temporary support for the roof. 
 
 Pit Rails. — Mine rails for underground roads. 
 
 Pit Room.— The extent of underground workings in use or available for use. 
 
 Pit's Eye. — Pit bottom or entrance into a shaft. 
 
 Pit Top. — The mouth of a shaft or slope. 
 
 Place. — The portion of coal face allotted to a hewer is spoken of as his 
 “working place,” or simply “place.” 
 
 Placer. — A surface accumulation of mineral in the wash of streams. 
 
 Placer Mining. — Surface mining for gold where there is but little depth of 
 alluvial. 
 
 Plan.— (1) The system on which a colliery is worked as Longwall, Pillar- 
 and-Breast , etc. (2) A map or plan of the colliery showing outside 
 improvements and underground workings. (3) (Mexican) The very 
 lowest working in a mine. Trabajar de Plan.— To work to gain depth. 
 
 Plancha —A pig of lead, etc. A plate, thick sheet, or mass of any 
 
 meta 
 
Pla 
 
 GLOSSARY. 
 
 Pop 
 
 603 
 
 Planchera (Mexican).— A mold of sand, earth, or iron, to form pigs 
 of lead. , , . . . . . , 
 
 Plane— A main road, either level or inclined, along which coal is conveyed 
 by engine power or gravity. 
 
 Plane Table — A simple surveying instrument by means of which one can 
 plot on the field. , , x ... 
 
 Planilla (Mexican).— An inclined plane of mason work, wood, etc., on which 
 tailings are spread out, to be concentrated by jets of water, skilfully 
 applied. 
 
 Planillero (Mexican).— A workman who devotes himself to concentrating 
 tailings, etc. on the Planillas; always paid by weight, measure, or con- 
 centrates produced. . . , _ . 
 
 Plank Dam.— A water-tight stopping fixed m a heading constructed of 
 timber placed across the passage, one upon another, sidewise, and 
 tightly wedged. „ , , . _ 
 
 Plank Tubbing— Shaft lining of planks driven down vertically behind 
 wooden cribs all around the shaft, all joints being tightly wedged, to 
 keep back the water. 
 
 Plant .— The shafts or slope, tunnels, engine houses, railways, machinery, 
 workshops, etc. of a colliery or other mine. 
 
 Plat , or Map— A map of the surface and underground workings, or of 
 either, to draw such a map from survey. 
 
 Plata ( Spanish ) .—Silver. 
 
 Plata Blanca (Mexican).— Native silver. 
 
 Plata Cornea Amarillia (Spanish).— Iodyrite. 
 
 Plata Cornea Blanca (Spanish).— Cerargy rite. 
 
 Plata Cornea Verde (Spanish).— Embolite. 
 
 Plata Mixta (Spanish).— Gold and silver alloy. 
 
 Plata Negra (Spanish).— Argentite. 
 
 Plata Pasta (Spanish).— Spongy silver bars after retorting. 
 
 Plata Piha (Spanish).— Silver after retorting. 
 
 Plata Verde (Spanish) .— Bromyrite. 
 
 Plate (North of England).— Scaly shale in limestone beds. 
 
 Plates— Metal rails 4 ft. long. 
 
 Plenum— A mode of ventilating a mine or a heading by forcing fresh air 
 
 J.I/. 
 
 Plomada (Mexican).— A plumb-line or plumb-bob. 
 
 Plomb d' Oeuvre (French).— Dressed galena. 
 
 Plomillos (Mexican).— Shots of lead found in slags. 
 
 Plomo (Spanish).— Lead, galena. , , 
 
 Plugging— When drift water forces its way through the puddle clay into 
 the shaft, holes are bored through the slabs near the leakage point, and 
 plugs of clay forced into them until the leakage is stopped. 
 
 Plumb ^ j 0^1(33,1 
 
 Plummet.— (1) A heavy weight attached to a string or fine copper wire used 
 for determining the verticality of shaft timbering. (2) A plumb-bob for 
 setting a surveying instrument over a point. 
 
 Plunger— The solid ram of a force pump working in the plunger case. 
 
 Plunger Case .— The pump cylinder or barrel in which the plunger works. 
 
 Plush Copper.— Chalcotrichite. 
 
 Plwm (Welsh). — Lead. 
 
 Poblar (Mexican).— To set men at work in a mine. 
 
 Pocket .— (1) A thickening out of a seam of coal or other mineral over a small 
 area. (2) A hopper-shaped receptacle from which coal or ore is loaded 
 into cars or boats. 
 
 ' Podar (Cornish).— Copper pyrites. , . 
 
 Pole Tools . — Drilling tools used in drilling in the old fashion with rods, now 
 superseded by the rope-drilling method. 
 
 Polroz (Cornish).— Waterwheel pit. 
 
 Poling . — Refining metal, when in a molten condition, by stirring it up with a 
 green pole of wood. 
 
 Poll Pick . — A pick having the longer end pointed and the shorter ena tiam- 
 mer-shaped. 
 
 Polvillos (Spanish).— Rich ores or concentrates. 
 
 Polvoulla (Spanish).— Black silver. 
 
 Poppet Heads .— The pulley frame or hoisting gear over a shaft. 
 
 Poppet {Puppet).— (1) A pulley frame or the head-gear oyer a sliait. (2) A 
 valve that lifts bodily from its seat instead of being hinged. 
 
GLOSSARY. 
 
 Pun 
 
 004 
 
 Pos 
 
 Post _(1) Any upright timber; applied particularly to the timbers used for 
 propping. See Prop. (2) Local term for sandstone. Post stone may be 
 “strong,” “ framey,” “ short,” or “ broken.” 
 
 Post-and-Stall. — A system of working coal much the same as Pillar-and-Stall. 
 Post Ternary.— Strata younger than the Tertiary formation. 
 
 Pot Bottom. — A large boulder in the roof slate, having the appearance of the 
 rounded bottom of a pot, and which easily becomes detached. 
 
 Pot Growan (Cornish).— Decomposed granite. . 
 
 Pot Hole —A circular hole in the rock caused by the action of stones whirled 
 around by the water when the strata was covered by water. They are 
 
 generally filled with sand and drift. . . 
 
 Power Drill— A rock drill employing steam, air, or electricity as a motor. 
 Prian (Cornish). — Soft white clay. , , . ' . . 
 
 Pricker— ( 1) A thin brass rod for making a hole m the stemming when 
 blasting, for the insertion of a fuse. (2) A piece of bent wire by which 
 the size of the flame in a safety lamp is regulated without removing the 
 top of the lamp. x A _ , _ , . 
 
 p r iU — (l) An extra-rich stone of ore. (2) A bead of metal. 
 
 Prong (English).— The forked end of the bucket-pump rods for attachment to 
 the traveling valve and seat. _ A . . 
 
 Prop.— A wooden or cast-iron temporary support for the root. 
 
 Propping. — The timbering of a mine. , , 
 
 Prospect —The name given to underground workings whose value has not 
 yet been made manifest. A prospect is to a mine what mineral is 
 to ore. 
 
 Prospect Hole— Any shaft or drift hole put down for the purpose of prospect- 
 ing the ground. , _ , 
 
 Prospect Tunnel or Entry— A tunnel or entry driven through barren measures 
 or a fault to ascertain the character of strata beyond. 
 
 Prospecting.— Examining a tract of country in search ol minerals. 
 Prospector— One engaged in searching for minerals. 
 
 Protector Lamp— A safety lamp whose flame cannot be exposed to the out- 
 ward atmosphere, as the action of opening the lamp extinguishes the 
 
 Prove ?— (1) To ascertain, by boring, driving, etc., the position and character 
 of a coal seam, a fault, etc. (2) To examine a mine m search of fire- 
 damp, etc., known as “ proving the pit.” 
 
 Proving Hole.—{ 1) A bore hole driven for prospecting purposes. (2) A 
 small heading driven in to find a bed or vein lost by a dislocation of 
 the strata, or to prove the quality of the mineral in advance of the 
 other workings. 
 
 Pudding Machine— A circular machine for washing pay dirt. 
 
 Pudding Rock.— Conglomerate. , . . , . 
 
 Puddle.— (1) Earth well rammed into a trench, etc., to prevent leaking. 
 
 (2) A process for converting cast iron into wrought iron. 
 
 Pueble (Mexican).— The actual working of a mine; the aggregation ot 
 persons employed therein. . * T . 
 
 Puertas (Mexican).— Massive barren rocks, or “ horses,” occurring m a vein. 
 Pug Mill.— A mill for preparing clay for bricks, pottery, etc. 
 
 Pulley.— { 1) The wheel over which a winding rope passes at the top ot the 
 head-gear. (2) Small wooden cylinders over which a winding rope is 
 carried on the floor or sides of a plane. . ... „ 
 
 Pulleying . — Overwinding or drawing up a cage into the pulley trame. 
 
 Pulp.— Crushed ore, wet or dry. 
 
 Pump— Any mechanism for raising water. 
 
 Purnp Ring —A fiat iron ring that, when lapped with tarred baize or engine 
 shag, secures the joints of water columns. . . 
 
 Pump Rods. — Heavy timbers by which the motion of the engine is trans- 
 mitted to the pump. In Cornish and bull pumps, the weight of the rods 
 makes the effective (pumping) stroke, the engine merely lifting the rods 
 on the up stroke. 
 
 Pump Slope. — A slope used for pumping machinery. to 
 
 Pump Station.— An enlargement made in the shaft, slope, or gangway, to 
 
 Pump C Tree. -Cast-iron pipes, generally 9 ft. long, of which the column or set 
 is formed. , . . . 
 
 Punch-and- Thirl. — A kind of pillar-and-stall system of working. 
 
Pun 
 
 GLOSSARY. 
 
 Ree 
 
 605 
 
 Punch Prop— A short timber prop set on the top of a crown tree, or used in 
 
 Put^Stonfs.- !oft P pfeces of decomposed rock found in placer deposits. 
 
 Pyran (Cornish).— See Prian. 
 
 Pyrites— Sulphide of iron. . . , . , . 
 
 Pyrometer— An instrument for measuring high degrees of heat. 
 
 Marty — ftT-An ^op^^rllce 0 excavation for working valuable rocks or 
 minerals. (2) An underground excavation for obtaining stone for 
 stowage or pack walls. 
 
 Quartz Bucket— A bucket for hoisting quartz. 
 
 ^emadero ’ ( T^xfcmi ) !— A burning place; a retorting furnace for silver or 
 
 Qaem^dos '(MeScan ) .—Burnt stuff. Any dark cinder-like mineral encoun- 
 H tered in a vein or mineral deposit, generally mangamferous. 
 
 Queme (Mexican) .-A roast of ore; the process of roasting ore, 
 
 Quick (Adjective).— Soft, running ground; an ore or pay streak is said to be 
 Quickening when the associated minerals indicate richer mineral 
 ahead. Quick (Noun).— (1) Productive. (2) Mercury 
 Quicksand. — Soft watery strata easily moved, or readily yielding to pressure. 
 Quicksilver. — Mercury . 
 
 ^itapepena^ Mexican^— A watchman that searches the miners as they 
 come out at the mouth of a mine. 
 
 Rabban (Cornish).— Yellow dry gossan. 
 
 Rabbling. — Stirring up a charge of ore in a reverberatory furnace wit* 
 specially designed iron rods. „ , .. 
 
 Race — A channel for conducting water to or from the piace where it pei' 
 forms work. The former is termed the headrace, and the latter tiu 
 
 tailrace. . _ 
 
 Rack (Cornish).— A stationary buddle. . . 
 
 Paff.—The coarse ore after crushing by Cornish rolls. 
 
 Rat a Wheel°—A revolving wheel with side buckets for elevating the raff. 
 Rafter Timbering.— That in which the timbers appear like roof rafters. 
 
 Rag Burning (Cornish).— The first roasting of tin-witts. 
 
 Rag-Wheel— Sprocket wheel. A wheel with teeth or pins that catch into th e 
 
 links of chains. „ , A .. , 
 
 Rails —The iron or steel portion of the tramway or railroad. . 
 
 Rake (Cornish).— (1) A vein. (2) (Derbyshire) Fissure vein crossing strata. 
 Pam —(1) The plunger of a pump. (2) A device for raising water. 
 
 Ramal (Mexican). — A branch vein. . . 
 
 Ramalear (Mexican). — To branch off into various divisions. 
 
 Ramble.— Stone of little coherence above a seam that falls readily on tne 
 removal of the coal. See Following Stone. 
 
 Rcmwr -—A iewev with a hammer attached at one end, which signals by 
 striking a plate of metal, when the signaling wire to which it is attached 
 
 Pash —A term used to designate the bottom of a mine when soft and slaty. 
 Rastrillo (Mexican).— A rake; a stirrer for moving ore in a furnace. 
 
 Rastron (Mexican).— A Chilian mill. 
 
 Raw Ore— Not roasted or calcined. 
 
 Reacher. — A slim prop reaching from one wall to the other. 
 
 Reamer. — An enlarging tool. 
 
 Reaming. — Enlarging the diameter of a bore hole. . . . . 
 
 Receiving Pit— A shallow pit for containing material run into it. 
 
 Red-Ash CM— Coal that produces a reddish ash, when burnt. 
 
 Red Rab (Cornish).— Red slaty rock. . . .. . ., - 
 
 Reduced— When a metal is freed from its chemical associate it is said to be 
 reduced to the metallic state. . 
 
 Reduction Works— Works for reducing metals from their ores. 
 
 Peef.—i l) A vein of quartz. (2) Bed rock of alluvial claims. 
 
 Reef Drive.— In alluvial mines, drives made in the country rock or reef. 
 
606 
 
 Ref 
 
 GLOSSARY. 
 
 ' Riv 
 
 Refining.— The freeing of metals from impurities. 
 
 Refractory.— Rebellious ore, not easily treated by ordinary processes. 
 
 Refuge Hole.— A. place formed in the side of an underground plane in which 
 a man can take refuge during the passing of a train, or when shots are 
 fired. 
 
 Regulator.— A door in a mine, the opening or shutting of which regulates the 
 supply of ventilation to a district of the mine. 
 
 Regulus. — See Matte. 
 
 Relampago , or Relampaguear (Mexican).— The brightening of the silver 
 button during cupellation. 
 
 Reliz (Spanish).— Wall of lode. 
 
 Rendir (Mexican).— Is when all the silver has been amalgamated in a heap 
 of argentiferous mud on a patio. 
 
 Rendrock.— A variety of dynamite. 
 
 Repairman— A workman whose duty it is to repair tracks, doors, brattices, 
 or to reset timbers, etc., under the direction of the foreman. 
 
 Repaso-Repasar (Mexican) .—The art of mixing up the mud heaps in the 
 patio process of amalgamation by treading them over with horses or 
 mules. 
 
 Repos Adero (Mexican).— The bottom of a crucible or pot in an upright 
 smelting furnace. 
 
 Rescatadores (Mexican).— Ore buyers. 
 
 .Reserve.— Mineral already opened up by shafts, winzes, levels, etc., which 
 mav be broken at short notice for any emergency. 
 
 Reservoir— An artificially built, dammed, or excavated place for holding a 
 reserve of water. 
 
 Respaldos (Mexican).— The walls enclosing a vein. Respdldo Alto.— The 
 hanging wall. Respaldo Bajo. — The foot-wall. 
 
 Rests, Keeps, Wings.— Supports on which a cage rests when the loaded car 
 is being taken off and the empty one put on. 
 
 Resue See Stripping . 
 
 Retort— (1) A vessel with a long neck, used for distilling the quicksilver 
 from amalgam. (2) The vessel used in distilling zinc. 
 
 Return.— The air-course along which the vitiated air of a mine is returned 
 or conducted back to the upcast shaft. 
 
 Return Air.— The air that has been passed through the workings. 
 
 Reverberatory.— A class of furnaces in which the flame from the fire-grate is 
 made to beat down on the charge in the body of the furnace. 
 
 Reversed Fault— See Overlap Fault. 
 
 Rib.— The side of a pillar. 
 
 Rib-and-Pillar.—A system of working similar to Pillar-and-Stall. 
 
 Ribbon.— A line of bedding or a thin bed appearing on the cleavage surface 
 and sometimes of a different color. 
 
 Rick.— Open heap in which coal is coked. 
 
 Ridding. — Clearing away fallen stone and debris. 
 
 Riddle.— An oblong frame holding iron bars parallel to each other, used for 
 sifting material that is thrown against it. 
 
 Ride, Riding.— To be conveyed on a cage or mine car. 
 
 Rider.— (1) A guide frame for steadying a sinking bucket. (2) Boys that 
 ride on trips on mechanical haulage roads. (3) A thin seam of coal 
 overlying a thicker one. 
 
 Riffle, or Ripple.— Crosspieces placed on the bottom of a sluice to save gold; 
 or grooves cut across inclined tables. 
 
 Right Shore— The right shore of a river is on the right hand when descend- 
 ing the river. 
 
 Rill.— The coarse ore at the periphery of a pile. 
 
 Rim Rock.— Bed rock forming a boundary to gravel deposit. 
 
 Ring.—{1) A complete circle of tubbing plates placed round a circular shaft. 
 (2) Troughs placed in shafts to catch the falling water, and so arranged 
 as to convey it to a certain point. 
 
 Ripping. — Removing stone from its natural position above the seam. 
 
 Riscos (Mexican).— Sharp and precipitous rocks; amorphous quartz found 
 in veins or outcrops. . 
 
 Rise.— The inclination of the strata, when looking up the pitch. 
 
 Rise Workings.— Underground workings carried on to the rise or high side 
 of the shaft. 
 
 River Mining.— Working beds of existing rivers by deflecting their course 
 or by dredging. 
 
Roa 
 
 GLOSSARY ; 
 
 Saf 
 
 607 
 
 Road.—( 1) Any underground passageway or gallery. (2) The iron rails, 
 etc. of underground roads. 
 
 Roasting— Heating ores at a temperature sufficient to cause a chemical 
 change, but not enough to smelt them. 
 
 Rob.— To cut away or reduce the size of pillars of coal. 
 
 Robbing .—' The taking of mineral from pillars. 
 
 Robbing an Entry— See Drawing an Entry. 
 
 Rock— A mixture of different minerals in varying proportions. 
 
 Rock Breaker —A machine for reducing ore in size by crunching it between 
 powerful jaws. 
 
 Rock Chute.— See Slate Chute. 
 
 Rock Drill— A rock-boring machine worked by hand, compressed air, steam, 
 or electrical power. 
 
 Rocker . — See Cradle. 
 
 Rock Fault— A replacement of a coal seam over greater or less area, by some 
 other rock, usually sandstone. 
 
 Rodding — The operation of fixing or repairing wooden eye guides in 
 shafts. 
 
 Roll— An inequality in the roof or floor of a mine. 
 
 Roller.— A small steel, iron, or wooden wheel or cylinder upon which the 
 hauling rope is carried just above the floor. 
 
 Rolleyway . — A main haulage road. 
 
 Rolling Ground — When the surface is much varied by many small hills and 
 valleys. 
 
 Rolls -Cast-iron cylinders, either plain or fitted with steel teeth, used to 
 break coal and other materials into various sizes. 
 
 Roof .— The top of any subterranean passage. 
 
 Room . — Synonymous with Breast. 
 
 Room-and-Rance — A system of working coal similar to Pillar-and-Stall. 
 
 Rope Roll— The drum of a winding engine. 
 
 Rosiclara (Spanish).— Ruby silver ore. 
 
 Roughs (Cornish).— Second quality tin sands. 
 
 Round Coal .— Coal in large lumps, either hand-picked, or, after passing over 
 screens, to take out the small. 
 
 Royalty .— The price paid per ton to the owner of mineral land-by the lessee. 
 
 Rubbing Surface .— The total area of a given length of airway; that is, the 
 area of top, bottom, and sides added together, or the perimeter multi- 
 plied by the length. 
 
 Rubble .— Coarse pieces of rock. 
 
 Rumbo (Mexican).— The course or direction of a vein. 
 
 Run. — (1) The sliding and crushing of pillars of coal. (2) The length of a 
 lease or tract on the strike of the seam. 
 
 Run Coal.—Soit bituminous coal. 
 
 Rung , Rundle , or Round— A step or cross-bar of a ladder. 
 
 Runner.— A man or boy whose duty it is to run mine cars by gravity from 
 working places to the gangway. 
 
 Running Lift.— A sinking set of pumps constructed to lengthen or shorten 
 at will, by means of a sliding or telescoping wind bore. 
 
 Rush— An old-fashioned way of exploding blasts by filling a hollow stalk 
 with slow powder and then igniting it. 
 
 Rush Gold.—G old coated with oxide of iron or manganese. 
 
 Rush Together.— See Caved In. 
 
 Rusty— Stained by iron oxide. 
 
 Saca (Mexican).— A bagful of ore. A mine is said to be de buena saca when 
 it has large quantities of ore easy to get out. 
 
 Saddle .—An anticlinal, a hogback. 
 
 Saddleback —A depression in the strata. See Roll. 
 
 Saddle Reef . — A reef having the form of an- inverted V. 
 
 Safety Cage.— A cage fitted with an apparatus for arresting its motion in the 
 shaft in case the rope breaks. 
 
 Safety Car . — See Barney. 
 
 Safety Catches .— Appliances fitted to cages, to make them safety cages. 
 
 Safety Door . — A strongly constructed door, hinged to the roof, and always 
 kept open and hung near to the main door, for immediate use. when 
 main door is damaged by an explosion or otherwise. 
 
 Safety Fuse— A cord with slow-burning powder in the center for exploding 
 charged blast holes. 
 
608 
 
 Saf 
 
 GLOSSARY. 
 
 Seg 
 
 Safety Lamp— A miner’s lamp in which the flame is protected in such a 
 manner that an explosive mixture of air and firedamp can be detected 
 by the mixture burning inside the gauze. 
 
 Sag— A depression, e.g., in ropes, ranges of mountains, etc. 
 
 Sagre, or Seggar.—A local term for fireclay, often forming the floor (or thill,) 
 
 of coal seams. _ . 
 
 Salting.— (1) Changing the value of the ore m a mine or of ore samples 
 before they have been assayed, so that the assay will show much higher 
 values than it should. (2) Sprinkling salt on the floors of underground 
 passages in very dry mines, in order to lay the dust. 
 
 Sampler.— (1) An instrument or apparatus for taking samples. (2) One whose 
 duty it is to select the samples for an assay, or to prepare the mineral 
 to be assayed, by grinding and sampling. . . . , . . , 
 
 Sampling Wor ks.— Works for sampling and determining the values obtained 
 in ores; where ores are bought and sold. 
 
 Samson Post.— An upright supporting the working beam that communicates 
 oscillatory motion to pump or drill rod. 
 
 Sand Bag— A bag filled with sand for preventing a washout by obstructing 
 
 Sand^ Pump. ‘—A sludger; a cylinder provided with a stem (or other) valve, 
 lowered into a drill hole to remove the pulverized rock. 
 
 Scaffolding. — Incrustations on the inside of a blast furnace. 
 
 Scale. — (1) A small portion of the ventilating current in a mine passing 
 through a certain size of aperture. (2) The rate of wages to be paid, 
 which varies under certain contingencies. 
 
 Scale Door. — See Regulator. . 
 
 Scallop —To hew coal without kirving or nuking or shot firing. 
 
 Schist — Crystalline or metamorphic rocks having a slaty structure. 
 
 Scissors Fault— A fault of dislocation, in which two beds are thrown so as to 
 cross each other. , 
 
 Scoop— A large-sized shovel with a scoop-shaped blade. 
 
 Scoria. — Ashes. 
 
 Scorifier.— A small dish used in assaying. 
 
 Scovan (Cornish).— A tin lode showing no gossan at surface. 
 
 Scove (Cornish).— Purest tin ore. „ ,. 
 
 Scramming. — Cleaning up small bodies or patches of ore left in the ordinary 
 
 Scraper.— {!) T tool ' for cleaning the dust out of the bore hole. (2) A 
 mechanical contrivance used at colleries to scrape the culm or slack 
 along a trough to the place of deposit. . 
 
 Scrapper — A local name given to parties that pick up the ore left on dumps. 
 
 Screen. — (1) A mechanical apparatus for sizing materials. (2) A cloth brat- 
 tice or curtain hung across a road in a mine, to direct the ventilation. 
 
 Serin (Derbyshire). — A small vein. . 
 
 Scrowl (Cornish).— Loose ore where a vein is crossed. 
 
 Sculping. — Fracturing the slate along the grain, l. e., across the cleavage. 
 
 Scupper Nails.— Nails with broad heads, for nailing down canvas, etc. 
 
 Sea Coal. — That which is transported by sea. • . . ^ e 
 
 Sealing— Shutting off all air from a mine or a part of a mine by stoppings. 
 
 Seam.—{ 1) Synonymous with Bed , Vein , etc. (2) (Cornish) A horse load 
 
 Seam-Oid.— A term applied to a shot or blast that has simply blown i out a softer 
 stratum of the deposit in which it was placed, without dislodging the 
 other strata or layers of the seam. . 
 
 Second Outlet ( Second Opening).— A passageway out of a mine, for use in case 
 of accident to the main outlet. 
 
 Seconds.— 1 The second-class ore of a mine that requires dressing. 
 
 Second Working.— The operation of getting or working out the pillars .ormed 
 
 Section 1 — (LtoA verticaf or horizontal exposure of strata. (2) A drawing or 
 sketch representing the rock strata as cut by a vertical or a horizontal 
 
 Sed&ntary Rocks.— Rocks formed from deposits of sediment by wind or 
 
 SeeFbeup— A water-tight packing of flaxseed around the tube of a drill hole, 
 to prevent the influx into the hole of water from above. 
 
 Segregations.— Detached portions of veins in place. 
 
Sel 
 
 GLOSSARY. 
 
 Sho 
 
 609 
 
 Self-Acting Plane— An inclined plane upon which the weight or force of 
 gravity acting on the full cars is sufficient to overcome the resistance of 
 the empties; in other words, the full car, running down, pulls the other 
 car up. 
 
 Self-Detaching Hook.— A self-acting hook for setting free a hoisting rope m 
 case of overwinding. ■ 
 
 Self-Feeders— Automatic appliances for feeding ore-dressing machines. 
 Selvage— The clay seam on the walls of veins; gouge. 
 
 Separation Doors.— The main doors at or near the shaft or slope bottom, 
 which separate the intake from the return airways. 
 
 Separation Valve— A massive cast-iron plate suspended from the roof of a 
 return airway through which all the return air of a separate district 
 flows, allowing the air to always flow past or underneath it; but in 
 the event of an explosion of gas, the force of the blast closes it against 
 its frame or seating, and prevents a communication with other districts. 
 The blast being over, the weight of the valve allows it to return ^o its 
 normal position. 
 
 Set. — To fix in place a prop or sprag. ... , 
 
 Set Hammer.— The flat-faced hammer held on hot iron by a blacksmith when 
 shaping or smoothing a surface by aid of his striker’s sledge. 
 
 Set of Timber— The timbers which compose any framing, whether used in 
 a shaft, slope, level, or gangway. Thus, the four pieces fuming a single 
 course in the curbing of a shaft, or the three or four pieces forming 
 the legs and collar, and sometimes the sill of an entry framing are 
 together called a set of timber, or timber set. . 
 
 Shackle.— A U-shaped link in a chain closed by a pm; when the latter is with- 
 drawn the chain is severed at that point. 
 
 Shadd (Cornish).— Rounded fragments of ore overlying a vein. 
 
 Shaft.— A vertical or highly inclined pit or hole made through strata, through 
 which the product of the mine is hoisted, and through which the ventila- 
 tion is passed either into or out of the mine. A shaft sunk from one 
 seam to another is called a “blind shaft.” 
 
 Shaft Pillar.— Solid material left unworked beneath buildings and around 
 the shaft, to support them against subsidence. . 
 
 Shaking Table.— An inclined table for concentrating fine grains of ore, which 
 is rapidly shaken by a short motion 
 
 Shale— (1) Strictly Speaking, all argillaceous strata that split up or peel off 
 in thin laminae. (2) A laminated and stratified sedimentary deposit of 
 clay, often impregnated with bituminous matter. 
 
 Shank.— 1 The body portion of any tool, up from its cutting edge or bit. 
 Shearing— Cutting a vertical groove in a coal face or breast. The cutting of 
 a “ fast end” of coal. . 
 
 Shear Legs.— A high wooden frame placed over an engine or pumping shaft 
 fitted with small pulleys and rope for lifting heavy weights. 
 
 Shears , or Sheers (English).— Two tall poles, with their feet some distance 
 apart and their tops fastened together, for supporting hoisting tackle. 
 Shear Zone.— Hogback. 
 
 Sheave— A wheel with a grooved circumference over which a rope is turned 
 either for the transmission of power or for winding or hauling. 
 
 Sheet Pump.— See Sludger , . , _ . 
 
 Sheets .—Coarse cloth curtains or screens for directing the ventilating current 
 underground. _ _ _ _ _ 
 
 Shelly— A name applied to coal that has been so crushed and fractured that 
 it easily breaks up into small pieces. The term is also applied to a lami- 
 nated roof that sounds hollow and breaks into thin layers of slate or 
 shale. 
 
 Shet (Staffordshire). — Fallen roof of coal mine. . _ 
 
 Sheth. — An old term denoting a district of about eight or nine adjacent bords. 
 
 Thus, a “ sheth of bords,” or a “ sheth of pillars.” 
 
 Shift— {1) The number of hours worked without change. (2) A gang or 
 force of workmen employed at one time upon any work, as the day shift, 
 or the night shift. 
 
 Shoad (Cornish).— See Shadd. 
 
 Shoading (Cornish). — Prospecting. . , 
 
 Shoe. — (1) A steel or iron guide piece fixed to the ends or sides of cages, to 
 fit or run on the conductors. (2) The upper working face of a stamp or 
 grinding pan. (3) The lower capping of any post or pile, to protect its 
 end while driving. (4) A wooden or sheet-iron frame or muff arranged 
 
610 
 
 Sho 
 
 GLOSSARY. 
 
 Sin 
 
 at the bottom of a shaft while sinking through quicksand, to prevent the 
 inflow of sand while inserting the shaft lining. 
 
 Shoot, Chute, Shute .— (1) A run of rich material in a vein. (2) An inclined 
 or vertical trough or pipe for conveying materials from a higher to a 
 lower level. 
 
 Shoot.— 1 To break rock or coal by means of explosives. 
 
 Shooting— Blasting in a mine. 
 
 Shore (English).— A studdle or thrusting stay. 
 
 Shore Up. — To stay, prop up, or support by braces. 
 
 Shot.— (1) A charge or blast. (2) The firing of a blast. (3) Injured by a 
 blast. 
 
 Shot-Firer.— See Shot Lighter. 
 
 Shot Hole— The bore hole in which an explosive substance is placed for 
 blasting. 
 
 Shot Lighter, or Shot Firer.—A man specially appointed by the manager of 
 the mine to fire off every shot in a certain district, it, after he has 
 examined the immediate neighborhood of the shot, he finds it free from 
 gas, and otherwise safe. 
 
 Shotty Gold— Granular pieces like shot. 
 
 Show.— When the flame of a safety lamp becomes elongated or unsteady, 
 owing to the presence of firedamp in the air, it is said to show. 
 
 Showing . — The first appearance of float, indicating the approach to an out- 
 cropping vein or seam. Blossom. 
 
 Shroud.— A housing or jacket. 
 
 Shute. — See Chute, Shoot, and Schute. 
 
 Shutter.— (1) A movable sliding door, fitted within the outer casing of a 
 Guibal or other closed' fan, for regulating the size of the opening from 
 the fan, to suit the ventilation and economical working of the machine. 
 (2) A slide covering the opening in a door or brattice, and forming a 
 regulator for the proportionate division of the air-current between two 
 or more districts of a mine. 
 
 Sickening.— A coating of impurities on quicksilver that retards amalgama- 
 tion or the coalescence of the globules of quicksilver. 
 
 Siddle. — Inclination. 
 
 Sid e .—{ 1) The more or less vertical face or wall of coal or goaf forming one 
 side of an underground working place. (2) Rib. (3) A district. 
 
 Side Chain.— A chain hooked on to the sides of cars running on an incline or 
 along a gangway, to keep the cars together in case the coupling 
 breaks 
 
 Sidelong Reef.— An overhanging wall of bed rock in alluvial formations 
 running parallel with the course of the gutter; generally only on one 
 side of it. 
 
 Siding— A short piece of track parallel to the mam track, to serve as a 
 passing place. , . 
 
 Siding Over.— A short road driven in a pillar in a head wise direction. 
 
 Sight— (1) A bearing or angle taken with a compass or transit when making 
 a survey. (2) Any established point of a survey. 
 
 Sights . — Bobs or weighted strings hung from two or more established points 
 in the roof of a room or entry, to give direction to the men driving the 
 entry or room. , , . , 
 
 Sill.—(1) The floor piece of a timber set, or that on which the track rests; 
 the base of any framing or structure. (2) The floor of a seam. 
 
 Silver. — (1) A certain white ductile and valuable metal. (2) Short for quick- 
 silver. . 
 
 Sing . — The noise made by a feeder of gas issuing from the coal. 
 
 Singing Coal.— Coal from which gas is issuing with a hissing sound. 
 
 Singing Lamp . — A safety lamp, which, when placed in an atmosphere ol 
 explosive gas, gives out a peculiar sound or note, the strength of the 
 note varying in proportion to the percentage of firedamp present. 
 
 Single-Entry System.— A system of opening a mine by driving a single entry 
 only, in place of a pair of entries:" The air-current returns along the 
 face of the rooms, which must be kept open. 
 
 Single-Intake Fan— A ventilating fan that takes or receives its air upon one 
 side only. , . ... . . 
 
 Single-Rope Haulage.— A system of underground haulage m which a single 
 rope is used, the empty trip running in by gravity. This is engine-plane 
 haulage. , . . , 
 
 Sink— To excavate a shaft or slope; to bore or put down a bore hole. 
 
Sin 
 
 GLOSSARY. 
 
 Sli 
 
 611 
 
 Sinker.— A. man who works at the bottom of a shaft or face of a slope during 
 the course of sinking. 
 
 Sinker Bar.— In rope drilling, a heavy bar attached above the jars, to give 
 force to the up stroke, so as to dislodge the bit in the hole. 
 
 Sinking— The process of excavating a shaft or slope or boring a hole. 
 
 Siphon.— A simple, effective, and economical mode of conveying water over 
 a hill whose height is not greater than what the atmospheric pressure 
 will raise the water. Its form is that of an iron pipe, bent like an 
 inverted U; the vertical height between the surface of the water in the 
 upper basin and the top of the hill is called the lift of the siphon; while 
 the vertical height between the surfaces of the water in the upper and 
 lower basins is called the fall of the siphon. 
 
 Sizing.— To sort minerals into sizes. 
 
 Skew Back.— The beveled stone from which an arch springs, and upon which 
 it) rests 
 
 Skids. — Slides upon which heavy bodies are slid from place to place. 
 
 Skimpings (Cornish).— The poorest ore skimmed off the jigger. 
 
 Skip .— (1) A mine car. (2) A car for hoisting out of a slbpe. (3) A thin 
 slice taken off from a breast or pillar or rib along its entire length or 
 part of its length. 
 
 Skirting.— Road opened up or driven next a fall of stone, or an old fallen place. 
 
 Skit (Cornish). — A pump. 
 
 Slab.— Split pieces of timber from 2 in. to 3 in. thick, 4 ft. to 6 ft. long, an<J * 
 7 in. to 14 in. wide, placed behind sets or frames of timber in shafts or 
 levels. . 
 
 Slack.— (1) Fine coal that will pass through the smallest sized screen. The 
 fine coal and dust resulting from the handling of coal, and the disinte- 
 gration of soft coal. (2) The process by which soft coal disintegrates 
 when exposed to the air and weather. 
 
 Slag.— The liquid refuse from a smelting operation, which floats on top of 
 the metal. 
 
 Slant.— (1) An underground roadway driven at an angle between the full 
 rise or dip of the seam and the strike or level. (2) Any inclined road 
 in a seam. 
 
 Slant Chutes— Chutes driven diagonally across a pillar, to connect a breast 
 manway with a manway chute. 
 
 Slate.— (1) A hardened clay having a peculiar cleavage. (2) About coal 
 mines, slate is any shale accompanying the coal, also sometimes applied 
 to bony coal. 
 
 Slate Picker.— (1) A man or boy that picks the slate or bony coal from 
 anthracite coal. (2) A mechanical contrivance for separating slate and 
 coal. 
 
 Slate Chute.— (1) A chute for conveying slate or bony coal to a pocket from 
 which it is loaded into “ dumpers.” (2) A chute driven through slate. 
 
 Sleek (Derbyshire).— Mud in a mine. 
 
 Sled.— A drag used to convey coal along the face to the road head where it is 
 loaded, or to the chute. 
 
 Sledge.— A heavy double-handed hammer. 
 
 Sleeper (English). -The foundation pieces or cross-ties on which rails rest. 
 
 Sleeping Tab'e (Cornish). — A buddle. 
 
 Sleeve.— A hollow cylinder fitting over two pieces, to hold them together. 
 
 Slickensides— Polished surfaces of vein walls. 
 
 Slide— Loose deposit covering the outcrop of a seam. 
 
 Slides— See Guides. 
 
 Sliding Scale.— A mode of regulating the wages paid workingmen by taking 
 as a basis for calculation the market price of coal, the wages rising and 
 falling with the state of trade. 
 
 Sliding Wind Bore (English).— The bottom pipe or suction piece of a sinking 
 set of pumps having a lining made to slide like a telescope within it, to 
 give length without altering the adjustment of the whole column 
 of pipes. 
 
 Slime, Sludge. — (1) The pulp or fine mud from a mill or from a drill hole. 
 (2) Silt containing a very fine ore, which passes off in the water from 
 the jigs. 
 
 Slings.— Pieces of ropes or chains to be put around stones, etc. for raising 
 them. 
 
 Slip.— (1) A fault. (2) A smooth joint or crack where the strata have 
 moved upon each other. 
 
612 
 
 Sli 
 
 GLOSSAR Y. 
 
 Spi 
 
 Slip Cleavage.— Microscopic folding and fracture accompanied by slippage; 
 quarryinen’s “ false cleavage.” 
 
 Slit.— A short heading put through to connect two other headings. 
 
 "slope — A plane or inclined roadway, usually driven in the seam from the 
 surface. A rock slope is a slope driven across the strata, to connect two 
 seams; or a slope opening driven from the surface, to reach a seam below 
 that does not outcrop at an accessible point. 
 
 fludger Sludge Pump.— A cylinder having an upward opening valve at the 
 bottom, which is lowered into a bore hole, to pump out the sludge or fine 
 rock resulting from drillings. „ . , . _ , . , , , 
 
 Sluice. (1) A long channel in rock or built of timber, with checks to catch 
 
 gfold. (2) Anv overflow channel. 
 
 Sluice Box.— A trough with ripples or false bottom for catching gold. 
 
 Sluice Head, or Head (Australia and New Zealand).— A supply of 1 cu. ft. of 
 water per second, regardless of the head, pressure, or size of orifice. 
 Sluicing. — Ground sluicing is working gravel by excavating with pick and 
 shovel, and washing the debris m trenches with water not under 
 pressure. , 
 
 Slurry (North of Wales).— Half-smelted ore. 
 
 Small. — See Slack. 
 
 Smeddum — Lead-ore dust. . . . , .. 
 
 Smelting. — Method of extracting a precious metal from its ores 
 Smift Snifi —A. bit of touch paper, touch wood, etc. attached by a bit of 
 ciay or grease to the outside end of the train of gunpowder when 
 
 Smitton.— Fine gravel-like ore, occurring free in mud openings, or derived 
 from the breaking of the ore in blasting. 
 
 ^^Tn^Piece^- 1 The°holein the lower part of a sinking or Cornish pump, 
 
 Soa^^A^lnZ^^ea by the miner to any soft, unctuous 
 
 Socabon {Mexican) .— A mining tunnel; an adit. Socavon dhilo de vela. A 
 drift tunnel. Socavon crucero. — A crosscut tunnel or adit. 
 
 Socket. — (1) The innermost end of a shot hole, not blown away after finng. 
 (2) A wrought-iron contrivance by means of which a wire rope is 
 securely attached to a chain or block. 
 
 Sole Sole Plate.— A piece of timber set underneath a prop. 
 
 Sollar. — A wooden platform fixed in a shaft, for the ladders to rest on. 
 Sondear (Mexican).— To bore for prospecting purposes. 
 
 Sondeo (Mexican).— A boring for prospecting purposes. 
 
 Soplete (Mexican).— A blowpipe. Ensayeal Soplete — A blow pipe assay. 
 Sorting. — Separating valuable from worthless material. „ . , 
 
 Sounding.— (1) Knocking on a roof to see whether it is sound ^fe to work 
 under. (2) Rapping on a pillar so that a person on the other side of it 
 may be signaled to, or to enable him to estimate its width. „ 
 
 S 0 u\—{ 1) A tool used for sharpening drills. (2) Iron deposits at the bottom 
 
 Spall. — To break up rocks with a large hammer, for hand sorting. 
 
 Spalls.— The chips' and other waste material cut from a block of stone in 
 
 Spnr!-An\m/gfve™to certain white quartz-like minerals, e. g„ calcspar, 
 feldspar, fluorspar. 
 
 Spears. — Pump-rods. 
 
 I^rbal^fdeo'r anttmonide of iron, often containing nickek 
 P cobalt, lead, bismuth, copper, etc., having a metallic luster of high 
 specific gravity and a strong tendency tow ard crystallization. 
 
 but has not done its work, 
 
 Sp ew —The extension of mineral matter on the surface, past the ordinary 
 limits of the lode. 
 
 Spiders— See Drum Rings. 
 
 Svieaeleisen.— Manganiferous white cast iron. ... „ 
 
 Spiking Curbs.— A light ring of wood to which planks are spiked when 
 
 plank tubbing is used. 
 
Spi GLOSSARY. Sta 613 
 
 Spiles (Cornish).— A temporary lagging driven ahead on levels in loose 
 ground. Short pieces of planking sharpened flatways, and used for 
 driving into watery strata as sheath piling, to assist in checking the 
 flow; used much in sinking through quicksands. 
 
 Spiling— A process of timbering through soft ground. 
 
 Spiral brum.— See Conical Brum. 
 
 Splint , or Splent — A laminated, coarse, inferior, dull-looking, hard coal, pro- 
 ducing much white ash, intermediate between cannel and bituminous 
 coal. 
 
 Split.— (1) To divide an air-current into two or more separate currents. (2) 
 Any division or branch of the ventilating current. (3) The workings 
 ventilated by that branch. (4) Any member of a, coal bed split by thick 
 partings into two or more seams. (5) A bench separated by a consider- 
 able interval from the other benches of a coal bed. 
 
 Spoil.— D6bris from a coal mine. 
 
 Spoon. — A slender iron rod with a cup-shaped projection at right angles to 
 the rod, used for scraping drillings out of a bore hole. 
 
 Spout.— A short underground passage connecting a main road with an air- 
 course. 
 
 Sprag.— (1) A short wooden prop set in a slanting position for keeping up 
 the coal during the operation of holing. (2) A short round piece of hard 
 wood, pointed at both ends, to act as a brake when placed between the 
 spokes of mine-car wheels. (3) The horizontal member of a square set 
 of timber running longitudinally with the deposit. 
 
 Spragger.— One who attends to the spragging of cars. 
 
 Sprag Road— A mine road having such a sharp grade that sprags are needed 
 to control the speed of the car. 
 
 Spreader.— A timber stretched across a shaft or stope. 
 
 Spring Beams.— Two short parallel timber beams, built with a Cornish pump- 
 ing engine house, nearly on a level with the engine beam, for catching 
 the beam, etc., and preventing a smash in case of a breakdown. 
 
 Spring Latch— The latch or tongue of an automatic switch, operated by a 
 spring pole at the side of the track. 
 
 Spring Pole— An elastic wooden pole from which boring rods are suspended. 
 Used also to operate a spring latch. 
 
 Sprocket Wheel (English).— Rag wheel. A wheel with teeth or pins which 
 catch in the links of a chain. 
 
 Spud, Spad.—A horseshoe nail with a hole in the head, for driving into the 
 mine timbers, or into a wooden plug fitted into the roof, to mark a sur- 
 veying station. 
 
 Spur.—( 1) A short ridge or offsetting pointed branch from a main ridge or 
 mountain. (2) A short branch or feeder from the main lode of a vein. 
 
 Square Set— A variety of timbering for large excavations. 
 
 Squat (Cornish).— Tin ore mixed with spar. 
 
 Squeeze— See Creep. 
 
 Squib. — A straw, rush, paper, or quill tube filled with a priming of gun- 
 powder, with a slow match on one end. 
 
 Stage. — A platform on which mine cars stand. 
 
 Staging. — A temporary flooring or scaffold, or platform. 
 
 Stage Pumping— Draining a mine by means of two or more pumps placed at 
 different levels, each of which raises the water to the next pump above, 
 or to the surface. 
 
 Stage Working. — A system of working minerals by removing the strata above 
 the beds, after which the various beds are removed in steps or stages. 
 
 Stalactites.— Icicle-shaped formations of mineral matter depending from roof 
 strata. 
 
 Stalagmites.— Accumulations of mineral matter that form on the floor, caused 
 by the continual dripping of water impregnated with mineral matter. 
 
 Stall.— A narrow breast, or chamber. 
 
 Stall Gate— A road along which the mineral worked in a stall is conveyed to 
 the main road. 
 
 Stamp Mill , Stomps. —Machine for crushing ore. 
 
 Stanchion. — A vertical prop or strut. 
 
 Standage. — Pump reservoir. 
 
 Standing. — Not at work, not going forwards, idle. 
 
 Standing Gas.— A body of firedamp known to exist in a mine, but not in cir- 
 culation; sometimes fenced off. 
 
 Standing Sett (English).— A fixed lift of pumps in a sinking set. 
 
614 
 
 Sta 
 
 GLOSSARY. 
 
 STO 
 
 ^taple^—{ 1) ^shallow pit within a mine. (2) An underground shaft. 
 
 Starter. — A man who ascends a chute to the battery and starts the coal to 
 
 Starved (English), -When a pump is choked at the brass holes. 
 
 Station.— A plat or convenient resting place in a shaft or level. 
 
 Stavi Englfsh). — Props, struts, or ties for keeping anything in its place. 
 Steamboat Coal.- In anthracite only, coal small enough to pass through bars 
 set 6 to 8 in. apart, but too large to pass through bars from 3 9 to 5 in. 
 Comparatively few collieries make steamboat coal except to fill special 
 contracts or orders. . . 
 
 Steam Coal.— A hard, free-burning, non-caking coal. - 
 
 Steam Jet.— A system of ventilating a mine by means of a number °f jets of 
 steam, at high pressure, kept constantly blowing off from a series of pipes 
 in the bottom of the upcast shaft. . • 
 
 Steel Mill An apparatus for obtaining light in a fiery mine. It consisted of 
 
 a reviving steel wheel, to which a piece of flint was held, to produce 
 
 Steel^Needlef— An instrument used in preparing blasting holes, before the 
 safety fuse was invented. 
 
 Steening. ot Steining.— The brick or stone lining of a shaft. 
 
 stpmmer A copper or wooden bar used for stemming. , 
 
 Stemming —(1) Fine shale or dirt put into a shot hole after the powder, and 
 rammed hard. (2) Tamping a shot. . . ., . . __ 
 
 Stev (English) —(1) The cavity in a piece for receiving the pivot of an 
 uprifht sLft, or the end of an upright piece. (2) The shearing m a 
 
 Stird^ The amount of work to be done by a man in a specified time. 
 
 Stobb. A lqng steel wedge used in bringing down coal after it has been 
 
 Sioctoort'-A rock run through with a number of small veins close 
 together, the whole of which has to be worked when mining such 
 
 Stonfp. - A ‘short wooden plug fixed in the roof of a level, to serve as a bench 
 mark for surveys. , . .. v , 
 
 Stone Coal. — Anthracite ; also other hard varieties of coal. 
 
 TTpnd A heading^ or gangway driven m stone. A tunnel. ... 
 
 Stone Tubbing . — Water-tight stone walling of a shaft cemented at the 
 
 Stoo Jb-A pillar of coal about 4 yd. square being the last portion of a full- 
 
 sized pillar to be worked away in bord-and-pillar workings. 
 
 Stook-and-Feather.—A wedge for breaking down coal, worked by hydraulic 
 power, the pressure being applied at the extreme inner end of t e 
 drilled hole. 
 
 itoo?-a 7 id-Soom.— ^system of working coal very similar to pillar-and-stall. 
 Stop!— Any cleat or beam to check the descent of a cage, car, pu^ rods^e ^ 
 Stope. (1) To excavate mineral m a series of steps. (2) A place in a mine 
 
 Stop^.—WoCkh^ out ore between two levels or on the 
 
 steps. Stoping Overhand.— Mining a stope upwards, the flight of steps 
 being inverted. Stoping Underhand— Mining a stope in 
 
 such a series that it presents the appearance of a flight of steps. 
 Stovvinq.— An air-tight wall built across any passageway in a mine. 
 
 Stove Coal. — In anthracite only; two sizes of to 1 2^mesh 
 
 lar^e stove, known as No. 3, passes through a z 4 to z mesn 
 and over all" to 1£" mesh; small stove, known as No. 4 , passes through 
 a to 1-" mesh and over a l\" to 1" mesh. Only one size of stove coal 
 fs^ow usually made. It passes through a 2" square mesh and over H" 
 
 Stov%» r oriOTed.-Upset. When a rod of iron heated at one end is ham- 
 mered endwise the diameter of that end is enlarged, and it is said to be 
 
 Stow.— To pack away rubbish into goaves or old workings. 
 
 Th^ tack of a miner and which supports 
 
 the roof or hanging wall of the excavation. 
 
I 
 
 GLOSSARY. 
 
 Swi 
 
 615 
 
 Str 
 
 Straight Ends and Walls.— A system of working coal somewhat similar to 
 bord-and-pillar. Straight ends are headings from 4 ft. 6 in. to 6 ft. in 
 width. Walls are pillars 30 ft. wide. 
 
 Straight Work.— A system of getting coal by headings or narrow work. 
 
 Strake.— A slightly inclined table for separating heavier minerals from 
 lighter ones. 
 
 Stratification.— Arrangement in layers. 
 
 Stratum (plural, strata).— A layer or bed of rocks, or other deposit. 
 
 Streak.— The color of the mark made when a mineral is scratched against a 
 white surface. ' 
 
 Strett.— The system of getting coal by headings or narrow work. See 
 Bor d-and- Pillar. 
 
 Strike (of a seam or vein).— The intersection of an inclined seam or a vein 
 with a horizontal plane. A level course in the seam. The direction of 
 strike is always at right angles to the direction of the dip of the seam. 
 
 Strike Joints— Joints or cleavages that are parallel to the strike of the seam. 
 
 Striking Deal.— Planks fixed in a sloping direction just within the mouth of 
 a shaft, to guide the tub to the surface. 
 
 Stringer (English).— Any longitudinal timber or beam. 
 
 Stringpump.—A system' of pumping whereby the motion of the engine is 
 transmitted to the pump by timbers or stringers bolted together. 
 
 String Rods —A line of surface rods connected rigidly for the transmission 
 of power; used for operating small pumps in adjoining shafts from a 
 central station. 
 
 Strip.— (1) To remove the overlying strata of a bed or vein. (2) Mining a 
 deposit by first taking off the overlying material. 
 
 Strut {English). —A prop to sustain compression, whether vertical or inclined. 
 
 Struve Ventilator. — A pneumatic ventilating apparatus consisting of two 
 vessel-like gas holders, which are moved up and down in a tank of 
 water. By this means, the air is sucked out of the mine as required. 
 
 Studdle.—A piece of squared timber placed vertically between two sets of 
 timber in a shaft. 
 
 Stull.— A post for supporting the wall or roof in a mine; a prop timber. 
 
 Stump.— The pillar between the gangway and each room turned off the gang- 
 way. Sometimes the entry pillars are called stumps. 
 
 Stumping. — A kind of pillar-and-stall plan of getting coal. 
 
 Stup. — Powdered coke or coal mixed with clay. 
 
 Sturt.— A tribute bargain profitable to the miner. 
 
 Stuttle, or Sprag. — The horizontal member of a square set of timber running 
 longitudinally with the deposit. 
 
 Stythe. — Carbonic-acid gas (blackdamp). 
 
 Sucker Rod. — The pump rod of an oil or artesian well. 
 
 Suction Pump (English).— A pump wherein, by the movement of the piston, 
 water is drawn up into the vacuum caused. 
 
 Sulphur.— { 1) One of the elements. (2) Iron pyrites. 
 
 Sulphuret. — See Sulphide. 
 
 Sulphide. — A combination of sulphur and a base. 
 
 Sump, or Sumpt. — A catch basin into which the drainage of a mine flows and 
 from which it is pumped to the surface. 
 
 Surface Deposits.— Those that are exposed and can be mined from the surface. 
 
 Swab Stick. — A short wooden rod, bruised into a kind of stumpy brush at one 
 end, for cleaning out a drill hole. 
 
 Swally, or Swelly.—A trough, or syncline, in a coal seam. 
 
 Swamp —A depression or natural hollow in a seam. A basin. 
 
 Sweeping Table.— A stationary buddle. 
 
 Sweet— Free from deleterious gases. 
 
 Sweet Roast— To roast dead or completely. 
 
 Swing.— The arc or curve described by the point of an instrument, such as a 
 pick or hammer, when being used. 
 
 Swinging Plate. —Amalgamated copper plates hung in sluices, to catch float 
 gold. 
 
 Switch— (1) The movable tongue or rail by which a train is diverted from 
 one track to another. (2) The junction of two tracks. (3) A movable 
 arm for changing the course of an electrical current. 
 
 Switchboard.— A board where several electrical wires terminate, and where, 
 by means of switches, connection may be established between any of 
 these wires and the main wire. 
 
 Swither.—A crevice branching from a main-lead lode. 
 
616 
 
 Syn 
 
 GLOSSARY. 
 
 Tee 
 
 Sfw nrh’nal Axis — The line or course of a syncline. ... 
 
 Syncline —The point or axis of a basin toward which the strata upon either 
 ‘ side dip. An inverted anticline. , A basin. 
 
 Tarkle (English) —(1) Ropes, chain, detaching hooks, cages, and all other 
 appamtus for raising coal or ore in shafts. (2) Any rope for hoisting, as 
 
 Ta/ion^^eScan^— An arrastre moved by water-power. Tahonero.—' The 
 
 rail^alk^When the firedamp ignites and the flame is elongated or creeps 
 backwards against the current of air, it is said to tail-bacx. 
 
 d etritus from reduction or gold-washing machinery. 
 
 flows after it has done its work 
 
 Tail-Rope — (1) In a tail-rope system of haulage, the rope that is used l to 
 draw the empties back into the mine. (2) A wire rope attached beneath 
 
 TaU-^o^e Haulage —A haulage system in which the fulltnpis 
 
 drawn out by the main rope, and the empty trip is drawn m by the tail- 
 rope, these ropes being attached to the opposite ends of the trip (see 
 
 TaiLShmve 0 - The sheave at the inbye end of any haulage system. See Turn 
 
 Takethe V Air.-(l) To measure the ventilating current. (2) Applied to a 
 ventilating fan as working well, or working poorly. . _ 
 
 Taladn) (Mexican) .—A drill for mechanical or mining purposes. Taladrar .. 
 
 Talhi ^ b m e A r rnark or number placed by the miner on every car of coal sent 
 2 out of Ms place usually a tin ticket. By counting these, a tally is made 
 of all th(f (Eireof coaf he sends out. (2) Any numbering, or counting, or 
 
 TaT/S^T^fil^^bor e a holef after inserting the charge, with some substance 
 Thich is rammed hard ’as it is put into the hole Vertical holes are often 
 tamped with water, when blasting with dynamite. 
 
 Tamnina —The process of stemming or filling a bore hole. . 
 
 Taping Bar.-A copper-tipped bar, for ramming the tamping or stem- 
 
 Tanafesf Mexican) . — Leather, hide, or jute bags, to carry ore or waste rock 
 within or out of a mine. Tanatero .— A laborer or bag c ^ I ™ r - 
 Tap —(1) To cut or bore into old workings, for the purpose of liberating accu- 
 Emulations of gas or water. (2) To pierce or open any gas or water 
 feeder (3) To win coal in a new district. . „ 
 
 Tnneytle (Mexican) —A working platform or stage built up in a stope or any- 
 in Srie; a landing place between two flights of ladders. 
 
 TeZiHj' Tro«g r ft°-A P trougli into which the water from a mine is pumped 
 rSapft -Asheetdron trough-shaped chute, for conveying coal or slate 
 
 goM) containing tellurium. 
 
 Temesquitale (Mexican).— The earthy part of ground-up ore. 
 
 Temner— (1) To change the hardness of metals by first beating aim tnen 
 plunging them into water, oil, etc. (2) To mix mortar, or to prepare 
 
 The actlf reheating and properly cooling a bar of metal to any 
 to rope^riUing, a screw for gradually lowering the clamped 
 Tenon V — A 'pro^cttog^ongue 8 fitting into a corresponding cavity called a 
 
 white emd^ Siallo w^ssS^rf 6 any kin^^to^certa^n^e^amimnt^of 
 
 metallic portions of any mineral, and to cause the earthy portions to be 
 
 TepdSeAM exi can) .—An y rock or earth found in a mine, which does not 
 contain the metal sought for. 
 
Teq 
 
 GLOSSARY. 
 
 Tea 
 
 617 
 
 Tequio (Mexican).— A task set for a drillman or for any laborer in a mine, to 
 be regarded as a day’s work. 
 
 Terrace— A raised level bank, such as river terraces, lake terraces, etc. 
 
 Terrero (Mexican).— The dump of a mihe. 
 
 Test.— (1) A trial of an engine, fan, or other appliance or substance. (2) An 
 iron framework that is filled with bone ash for cupeling on a large scale. 
 
 Theodolite— An instrument used in surveying, for taking both vertical and 
 horizontal angular measurements. An engineer’s large transit, with 
 attachments. 
 
 Thill.— See Floor. 
 
 Thimble— (1) A short piece of tube slid over another piece, to strengthen a 
 joint, etc. (2) An iron ring with a groove around it on the outside, used 
 as an eye when a rope is doubled about it. 
 
 Thirl.— See Crosscut. 
 
 T hr ough-and- Through.— A system of getting bituminous coal, without regard 
 to the size of the lump. 
 
 Throw.— (1) A fault of dislocation. (2) The vertical distance between the 
 two ends of a faulted bed of coal. 
 
 Thrown— Faulted; broken by a fault. 
 
 Thrust— Creep or squeeze due to excessive weight, hard floor, and too 
 small pillars. 
 
 Thurl (Staffordshire).— To cut through from one working into another. 
 
 Ticketing— English periodical markets for the sale of ores. 
 
 Tie-Back.— ( 1) A beam serving a purpose similar to a fend-off beam, but 
 fixed at the opposite side of the shaft or inclined road. (2) The wire 
 ropes or stayrods which are sometimes used on the side of the tower 
 opposite the hoisting engine, in place of or to reenforce the engine 
 braces. 
 
 Tierras (Spanish).— Earth impregnated with mercury ore. 
 
 T terras de Labor (Mexican). — Dirt from a stope, mixed with particles of ore. 
 
 Tierras de Llunque (Mexican).— Chips made in breaking and sorting ore. 
 
 Tiff. — Calcite or carbonate of lime. 
 
 Timber— (1) Props, bars, collars, legs, laggings, etc. (2) To set or place 
 timber in a mine or shaft. 
 
 Timberer, Timberman.—A man who sets timber. 
 
 Time.— (1) Hours of work performed by workmen. (2) To count the 
 strokes of a pump or revolutions of an engine or fan. 
 
 Tin-Can Safety Lamp.— A Davy lamp placed inside a tin can or cylinder 
 having a glass in front, air holes near the bottom, and open-topped, 
 making the lamp safer in a rapid current of air. 
 
 Tin- Witts (Cornish).— Product of first dressing of tin ores, containing, also, 
 wolfram and sulphides. 
 
 Tip.— A dump. See Tipper , or Tipple. 
 
 Tipper , or Tipple— An apparatus for emptying cars of coal or ore, by turning 
 them upside down, and then bringing them back to original position, 
 with a minimum of manual labor. 
 
 Tipple— The dump trestle and tracks at the mouth of a shaft or slope, where 
 the output of a mine is dumped, screened, and loaded. 
 
 Tiro (Mexican).— A mining shaft. Tiro Vertical.— A vertical shaft. 
 
 Token.— (1) A mutually understood mark placed upon a bucket of ore when 
 it is hoisted or lowered into a shaft, to acquaint the lander or filler of 
 some important matter. (2) A piece of leather or metal stamped with 
 the hewer’s or putter’s number or distinctive mark, and fastened to the 
 tub he is filling or putting. 
 
 Ton.— A measure of weight. Long ton is 2,240 lb.; short ton is 2,000 lb.; 
 metric ton is 1,000 kilograms = 2,204.6 lb. 
 
 Top.— (1) See Roof. (2) Top of a shaft; surface over a mine. 
 
 Topit. — A kind of brace head screwed to the top of boring rods, when with- 
 drawing them from the hole. 
 
 Torta (Mexican).— A pie or cake; the heaps of argentiferous mud that are 
 treated in the patio process of amalgamation. 
 
 Tossing— Shaking powdered ore in water, to effect separation of heavy and 
 light particles. 
 
 Tovera (Mexican).— The tuyere of a smelting furnace. 
 
 Track.— Railways or tramways. 
 
 Tracking— Wooden rails. 
 
 Train Boy.— A boy that rides on a trip, to attend to rope attachments, signal 
 in case of derailment of cars, etc. Trip rider. 
 
618 
 
 Tra 
 
 GLOSSARY. 
 
 Tum 
 
 Train or Trip.— The cars taken at one time by mules, or by any motor, oi 
 run at one time on a slope, plane, or sprag road, always together. 
 
 Tram— A mine car, or the track on which it runs. 
 
 Trammer.— One who pushes cars along the track. 
 
 Tramroad. — A mine track or railroad. . 
 
 Tram’ Rone. — A hauling rope, to which the cars are attached by a clip or 
 chain, either singly or in trips. . 
 
 Tramway— A small, roughly constructed iron track for running wagons or 
 
 Transfer Carriage. — Movable platform or truck used to transfer mine cars 
 from one track to another. .... 
 
 Transome ( English ) —A heavy wooden bed or supporting piece. 
 
 Trap —( 1) A steep heading along which men travel. (2) A fault of dislo- 
 cation. ( 3 ) An eruptive rock. (4) A dangerous place. 
 
 Tran Door. — A small door, kept locked, fixed in a stopmg, for giving access 
 to firemen and certain others to the return airways, dams, or other 
 unused portions of the mine. . - . . . 
 
 Tran Dike— A fault (not necessarily accompanied by displacement of strata) 
 m which the spaces between the fractured edges of the beds are filled up 
 bv a thick wall of igneous rock. 
 
 Trapiche (Spanish).— A primitive grinding mill 
 
 Trapper.— A boy employed underground to tend doors. 
 
 Traveling Road. — An underground passage or way used expressly, though 
 not always exclusively, for men to travel along to and from their work- 
 
 TreenaiF— Along wooden pin for securing planks or beams together. 
 
 TrdooUng (Cornish).— Stirring tin slimes in water. 
 
 Trend.— The course of a vein, fault, or other feature. 
 
 Tribute. — A method of working mines by contract, whereby the miners 
 receive a certain share of the products won. Tribute) s. Miners paid 
 bv results. 
 
 Trig. — A sprag used to block or stop a wheel or any machinery. 
 
 Tritta (Mexican). — The same as Torta. . 
 
 Trip —The mine cars in one train or set. See Train. ^ Txnrnl1o1 
 
 Triple-Entry System.— A system of opening a mine by driving three parallel 
 entries for the main entries. 
 
 Triturate.— To grind or pulverize. , „ . , , , 
 
 Trolley. -{1) A small four-wheeled truck, used for carrying the ore bucket 
 underground. ( 2 ) An electric motor. (3) The sarm of a motor ^ con- 
 ducts the electric current from the wire above the track to the machine. 
 
 Trommel. — A drum, consisting of a cylinder- or cone-shaped sheet-iron 
 mantle (generally punched with holes) that revolves; used for washing 
 
 Trmnpal Mefican) — A funnel-shaped mouthpiece of cooled slag that forms 
 within a smelting furnace over the tuyere opemng. 
 
 Trompe. — An apparatus for producing ventilation by the fall of water down 
 
 SL slldift 
 
 Trouble. — A dislocation or fault; any irregularity in the bed. 
 
 Trouah Fault.— A wedge-shaped fault, or, more correctly, a mass of rock, 
 coal etc. let down in between two faults, which faults, however, are 
 
 through a pillar to connect two rooms. 
 
 Truck. — Used svnonymously with Barney. 
 
 Truck System. — Paying miners in food instead of money. 
 
 Trunnions. — Cylindrical projections or journals, attached to the sides of 
 a vessel, so that it can rotate in a vertical plane. 
 
 Truing the Lamp— The examination of the flame of a safety lamp for tne 
 purpose of forming a judgment as to the quantity of firedamp mi ed 
 
 Tulh—O.)* A 6 mine car. (2) An iron or wooden barrel used in a shaft, for 
 
 TuWi^—cSt iron, and sometimes timber, lining or walling of a circular 
 Wedges.— Small wooden wedges hammered between the joints of 
 
 TuMn^.— 1 Iron pipes or tubes used for lining bore holes, to prevent caving. 
 
 Tumbar (Mexican).— To knock down ore. etc. 
 
 Tumbe{ Mexican).— The act of knocking down and taking out ore. 
 
Tun 
 
 GLOSSARY. 
 
 Yen 
 
 619 
 
 Tunnel.— A. horizontal passage driven across the measures and open to day at 
 both ends; applied also to such passages open to day at only one end, or 
 not open to day at either end. 
 
 Turbary.— A peat bog. 
 
 Turbine.— A rapidly revolving waterwheel, impelled by the pressure of water 
 upon blades. 
 
 Turn.— (1) The hours during which coal, etc. is being raised from the mine. 
 (2) See Shift. (3) To open rooms, headings, or chutes off from an entry 
 or gangway. (4) The number of cars allowed each miner. 
 
 Turnout— A siding or passing on any tram or haulage road. 
 
 Turn Pulley.— A sheave fixed at the inside end of an endless- or tail-rope 
 hauling plane, around which the rope returns. See Tail-Sheave. 
 
 Turntable.— A revolving platform on which cars or locomotives are turned 
 around. 
 
 Tut Work. — Breaking ground at so much per foot or fathom. 
 
 Tuyere— The tubes through which air is forced into a furnace. • 
 
 Two-Throw.— When, in sinking, a depth of about 12 ft. has been reached, and 
 the debris has to be raised to the surface by two lifts or throws with the 
 shovel, one man working on staging above another. 
 
 Tye.— An inclined table used for dressing ores. 
 
 Unconformability.— When one layer of rock, resting on another layer, does not 
 correspond in its angle of bedding. 
 
 Undercast. — An air-course carried under another air-course or roadway. 
 
 Underclay.— A bed of fireclay or other less clayey stratum, lying immediately 
 beneath a seam of coal. 
 
 Undercut— To remove a small portion of the bottom of the bed or the under- 
 clay, so that the mass of coal or mineral can be wedged or blasted down. 
 
 Underhand Stoping. — See Stoping Underhand. 
 
 Underhand Work. — Picking or drilling downwards. 
 
 Underholing , Undermining.— To mine out a portion of the bottom of a seam 
 or the underclay, by pick or powder, thus leaving the top unsupported 
 and ready to be blown down by shots, broken down by wedges, or mined 
 with a pick or bar. 
 
 Underlie , or Underlay. — The inclination of a lode at right angles to its course, 
 or strike; the true dip. 
 
 Underviewer, or Underlooker.— An inside foreman. 
 
 Unit.—{ 1) The unit of metals is 1$ of whatever ton is used. Generally, the 
 20-cwt. ton, equal to 2,240 lb., is employed, but, when dealing with 
 copper ores, the 21-cwt. ton of 2,352 lb. is taken; therefore, the respective 
 units are 22.4 lb. and 23.52 lb. (F. Danvers Powers). (2) Ores are quoted 
 at a certain price per unit or per cent, of valuable material in the ore. 
 If an iron ore contains 40^ of metallic iron that is worth 5 cents per unit, 
 the value of the ore is $2 per ton. 
 
 Unwater.— To drain or pump the water from a mine, or shaft. 
 
 Upcast.— The shaft through which the return air ascends. 
 
 Upraise. — An auxiliary shaft, a mill hole, or heading carried from one level 
 up toward another. 
 
 Upthrow.— A fault in which the displacement has been upward. 
 
 Vapor (Mexican).— Steam; heated and stinking gas sometimes found in 
 mines, which causes candles to burn dimly and go out. 
 
 Vaso (Mexican).— A reverberatory furnace used for smelting rich ore, or for 
 cupeling silver. 
 
 Vat.— Large wooden tub used for leaching or precipitation. 
 
 Vein.— See Lode. Often applied incorrectly to a seam or bed of coal or other 
 mineral. 
 
 Veinstone. — The non-metallic portion of a vein associated with the ore. 
 
 Vena (Mexican).— A thin vein, not over 3 in. thick— a knife-blade vein. 
 
 Vend (North of England).— Total sales of coal from a mine. 
 
 Vent, or Vent Hole. — (1) A small passage made with a needle through the 
 tamping, which is used for admitting a squib, to enable the charge to be 
 lighted. (2) Any opening made into a confined space. 
 
 Ventilating Column.— See Motive Column. 
 
 Ventilating Pressure— The total pressure or force required to overcome the 
 friction of the air in mines; the unit of ventilating pressure or pressure 
 per sq. ft. of area multiplied by the area of the airway. 
 
 Ventilation. — Circulation. The atmospheric air circulating in a mine. 
 
620 
 
 Yen 
 
 GLOSSARY. 
 
 Wat 
 
 Ventilator.— Any means or apparatus for producing a current of air in mine 
 or other airways. 
 
 of rock; a true fissure vein Loosely, 
 
 direction or course. Veta Soda. — A vein that joins another. 1 eta Ramal. 
 —A branch vein. Veta Recostada.— An inclined vein. . 
 
 Viewer. — The general manager or mining engineer of one or more collieries, 
 who has control of the whole of the underground works, and also gen- 
 erally of those on the surface. 
 
 when all impurities have 
 
 been removed from the silver under treatment. 
 
 Vug, or Vugh (Cornish).— A cavity in the rock. 
 
 'iVagon Brea ft— A ' breast in which the mine cars are taken up to the 
 working face. „ „ 
 
 Wailing. — Picking stones and dirt from among coals. 
 
 Wale (North of England).— Hand-dressing coal. 
 
 Walking Beam.— See Working Beam. 
 
 Wall.—( 1) The face of a long wall working or breast. (2) A nb of solid coal 
 between two breasts. 
 
 Walling— See Steening. . . . u 
 
 Walling Cribs— Oak cribs or curbs upon which walling is built. 
 
 Walling Stage.— A movable wooden scaffold suspended from a crab °u the 
 surface, upon which the workmen stand when walling or lining a shaft. 
 
 Wall Plates.— The two longest pieces of timber in a set used in a rectan- 
 
 Warners — Apparatus consisting of a variety of delicately constructed 
 machines^ actuated by chemical, physical, electrical and 
 properties, for indicating the presence of small quantities of £ r edamp m 
 the mines. At present, most of these ingenious contrivances are more 
 suited to the laboratory than for practical application underground. 
 
 Warning Lamp. — A safety lamp fitted with certain delicate apparatus, for 
 indicating very small proportions of firedamp in the atmosphere of a 
 mine As small a quantity as 3$ can be determined by this means. 
 
 Wash— Drift, clay, stones, etc. overlying the strata. 
 
 wTs^Dirt— That portion of alluvial working in which most of the gold is 
 
 Wash U Fcwlt.— A portion of a seam of coal replaced by shale or sandstone. 
 
 Washing Apparatus, or Washer y.-(l) Machinery and applianceserected on 
 the surface at a colliery, generally in c^nectmn with °vens, for 
 extracting, by washing with water, the impurities mixed with the coa^ 
 dust or small slack. (2) Machinery for removing impurities from small 
 
 Washout^- The^r^^ appreciable extent of a coal seam by aqueous 
 
 Wash PZace.— A place where the ores are washed and separated from the 
 waste, usually applied to places where the hand jigs ap used. 
 
 Waste — m See Goaf. (2) Very small coal or slack. (3) The portion of a 
 mine occupied by the return airways. (4) A1 %^d bo „ d /?^i h g e 
 spaces between the pack walls m the gob of longwall working. (5) Refuse 
 material 
 
 Waste Gate (English). — A door for regulating discharge of surplus water. 
 
 Water Blast.— The sudden escape of air pent up in rise f ° ^ 
 
 siderable pressure from a head of water that has accumulated m a 
 
 Wat^Cartridge.—A waterproof cartridge surrounded by an ou |er case. 
 The snace between being filled with water, which is employed to 
 destroy the flame produced when the shot is fired, thereby lessens the 
 chance of an explosion should gas be present in the Place. 
 
 Water Gauge.— An instrument for measuring the pressure per square foot 
 
 Water JTammfr.— 1 Thlfhammering noise caused by the intermittent escape 
 of gas through water in pipes. 
 
Wat 
 
 GLOSSARY. 
 
 Won 
 
 621 
 
 Water-Jacket. — A jacket filled with water, to keep cool a cylinder or furnace. 
 
 Water Level— An underground passage or heading driven very nearly dead 
 level or with sufficient grade only to drain off the water. 
 
 Water Right. — The privilege of taking a certain quantity of water from a 
 watercourse. 
 
 Watershed.— The elevated land or ridge that divides drainage areas. 
 
 Waterwheel (English).— Overshot, undershot, breast wheels. A wheel pro- 
 vided with buckets, which is set in motion by the weight or impact of a 
 stream of water. 
 
 Weather. — To crumble by exposure to the atmosphere. 
 
 Weather Door— See Trap Door. 
 
 Web.— The face of a longwall stall in course of being holed and broken down 
 for removal. The length of breast or face brought down by one mining. 
 
 Wedging.— The material, moss or wood, used to render the shaft lining 
 tight. 
 
 Wedging Crib.— A curb or crib of wood or cast iron wedged tightly in place 
 and packed, in order to form a water-tight joint and upon which tubbing 
 is built. 
 
 Wedging Down.— Breaking down the coal at the face with hammers and 
 wedges instead of by blasting. 
 
 Weeldon.— Old ironstone workings. 
 
 Weigh Bridge (English).— A platform large enough to carry a wagon, resting 
 on a series of levers, by means of which heavy bodies are weighed. 
 
 Weize. — A band or ring of spun yarn, rope, rubber, lead, etc. put in between 
 the flanges of pipes before bolting them together,, in order to make a 
 water-tight joint. , „ 
 
 Well.—(1) The well of a furnace is the deepest lying portion or hollow in 
 which the metal collects. (2) A sump, or a branch from the sump. 
 
 Whim,.— A winding drum worked by a horse. 
 
 W.him Shaft— A shaft through which coal, ore, water, etc. are raised from a 
 mine by means of a whim. 
 
 Whin. — A hard, compact rock. 
 
 Whin Dike.— A fault or fissure filled with whin and the debris of other 
 rocks, sometimes accompanied by a dislocation of the strata. 
 
 Whip. — A hoisting appliance consisting of a pulley supporting the hoisting 
 rope to which the horse is directly attached. 
 
 W hitedamp . —Carbonic oxide (CO). A gas found in coal mines, generally 
 where ventilation is slack. A product of slow combustion in a limited 
 supply of air. It burns and will support combustion. It is extremely 
 poisonous. 
 
 White Tin— The commercial name for metallic tin. 
 
 Whits. See Tin- Witts. 
 
 Whole Working. — The first working of a seam, which divides it into pillars. 
 
 Wild Lead. — Zinc-blende. 
 
 Wild Rock.— Anv rock not fit for commercial slate. 
 
 Win.— To sink a' shaft or slope, or drive a drift to a workable seam of mineral 
 in such a manner as to permit its being successfully worked. 
 
 Winch , or Windlass —A hoisting machine consisting of a horizonal drum 
 operated by crank-arm and manual labor. 
 
 Wind Bore (England).— The bottom or suction pipe of a lift of pumps, which 
 has suitable brass holes or perforations for suction of water or air. 
 
 Wind Gauge— An anemometer, for testing the velocity of air in mines. 
 
 Winding. — The operation of raising or hauling by means of a steam engine 
 and ropes, the product of a mine. 
 
 Winding Engines— Hoisting or haulage engines. 
 
 Wind Method.— A system of separating coal into various sizes, and extracting 
 the dirt from it, whibh in principle depends on the specific gravity or 
 size of the coal and the strength of the current of air. 
 
 Wind Sail.—' Tim top part of canvas piping, which is used for conveying air 
 down shallow shafts. 
 
 Wing Bore.— A side or flank bore hole. 
 
 Wings. — See Rests and Keeps. 
 
 Winning.— A sinking shaft, a new coal, ironstone, clay, shale, or other mine 
 of stratified material. A working place in a mine. 
 
 Winnowing Gold. — Air-blowing. Tossing up dry powdered auriferous mate- 
 rial in the air, and catching the heavier particles not blown away. 
 
 Winze— Interior shaft connecting levels, sometimes used as an ore chute. 
 
 ITon.— Proved, sunk to, and tested. 
 
622 
 
 WOR 
 
 GLOSSARY. 
 
 ZON 
 
 Work.-( 1) To mine. (2) Applied to mine working when affected by squeeze 
 or creep. 
 
 Workable— Any seam that can be profitably mined. : p 
 
 Worked Out — When all available mineral has been extracted from a mine, it 
 
 is worked out. 
 
 Working . — Applied to mine workings when squeezing. 
 
 Wolunl imm^fiEngHshb— ^beam^aving'a^ertieal motion on a rock shaft 
 at ifs ceXr on g e end being connected with the piston rod and the othei 
 with a crank or pump rod, etc. . 
 
 Working Cost— The total cost of producing the mineral. 
 
 Working Face— See Face. , . frnm thp 
 
 Working Home .— Getting or working out a seam of coal, etc., from the 
 boundary or far end of the mine toward the shaft bottom. 
 
 Working on Air— A pump works on air when air is sucked up wit 
 
 Worting^but .— Working outwards or in the direction of the boundaries ol 
 
 WorUngFla^ 1 .— The actual place in a mine at wliich the coal is being mmed. 
 Workings.— The openings of a colliery, including all roads, ways, levels, 
 dips, airways, etc. 
 
 Work Lead— Base bullion, silver lead. . . 
 
 Wrought Iron .— Iron in its minimum state of carburization. 
 
 Wythern (Wales). -Lode. 
 
 Xacal (Mexican).— A miner’s cabin; a storehouse for mining goods; a shaft 
 house. 
 
 pricns),' paid for ?oads of certain widths, and driven in certain directions. 
 Yellow Ore (Cornish).— Chalcopyrite. 
 
 Yield — The proportion of a seam sent to market. 
 
 7 onp T n coal-mining phraseology, this word means a certain series of coal 
 
 Z ° ne seams wfthTheir IcSompanyifg shales, etc., which eonmn for exampie 
 much firedamp, called a fiery zone, or, if much water, a watery zone. 
 
INDEX 
 
 Absolute Pressure, 195, 345. 
 
 temperature and pressure, 344, 345. 
 Abuse of instruments, 80. 
 
 Abutments (dams), 154. 
 
 Acceleration in jigging, 440. 
 
 on inclined planes, 399. 
 
 Acid waters, Pumps for, 165. 
 
 Acres to square feet, 4. 
 
 Adiabatic compression, 195, 198. 
 Adjustable bars, 432. 
 
 Adjustment of Burt’s solar attach- 
 ment, 48. 
 of compass, 38. 
 of level, 53. 
 of transit, 41. 
 
 Afterdamp, 351. 
 
 Agonic line chart, 40. 
 
 Air and mercury columns, 347. 
 brattice, 394. 
 bridges, 393. 
 
 columns in furnace ventilation, 
 384. 
 
 Compressed, 194. 
 compression, Theory of, 194. 
 compression, Horsepower neces- 
 sary for, 406. 
 
 Air compressors, 194. 
 
 Classification of, 194. 
 Compound, 194. 
 
 Construction of, 194. 
 
 Cooling, 195. 
 
 Dry, 196. 
 
 Duplex, 194. 
 
 Horizontal, cross-compound, 
 194. 
 
 Rating of, 195. 
 
 Simple, 194. 
 
 Stage, 194. 
 
 Straight-line, 194. 
 
 Vertical, 194. 
 
 Wet, 196. 
 
 Air currents, Conducting, 393. 
 currents, Splitting o£ 373, 378. 
 leaks, 186. 
 lift pumps, 164. 
 required for ventilation, 362. 
 transmission, 196. 
 
 Airways, Similar, 372. 
 
 Alabama method of mining, 297. 
 Alternating current, 217. 
 current dynamos, 224. 
 current motors, 226. 
 
 Alternators, 225. 
 
 Multiphase, 225, 226. 
 
 Single-phase, 225. 
 
 Three-phase, 226. 
 
 Two-phase, 226. 
 
 Altitude, Effect of, on air compres- 
 sion, 195. 
 
 Aluminum, Electric properties of, 
 209. 
 
 American coals, 168. 
 gauge wire, 208. 
 measures of area, 4. 
 measures of length, 2. 
 measures of volume, 5. 
 
 Amount crushed by rolls, 424. 
 of charge, blasting, 331. 
 of gas liberated per ton of coal, 352. 
 Analyses of American coals, 168. 
 Analysis of coal, 173. 
 
 Andre’s formula for shaft pillars, 285. 
 Aneroid barometer, 339. 
 
 Angle of inertia, 398. 
 
 of rolling friction, 398. 
 reading, 45. 
 
 Annunciator system, 232. 
 
 Anthracite coal, 169. 
 
 Cubic feet in ton of, 449. 
 Handling, 443. 
 
 Sizes, of 434. 
 
 Specific gravity of, 109. 
 mining, Costs of, 323. 
 mining methods, 305. 
 
 Percentages of different sizes of, 
 323, 326. 
 
 Preparation of, 442. 
 screens, Duty of, 434. 
 
 Tests of compressive strength of, 
 290. 
 
 Apothecaries’ weight, 1. 
 
 Appliances in mine ventilation, 381. 
 Arc, Error in, 76. 
 
 lamps, 214. 
 
 Area, Measures of, 4. 
 of circle, 31. 
 of ellipse, 32. 
 
 of largest square inscribed in a 
 circle, 28. 
 
 of parallelogram, 28. 
 
 of polygons, 30. 
 
 of tract of land, To find, 52. 
 
 of trapezium, 30. 
 
 of trapezoid, 30. 
 
 of triangle, 28. 
 
 Areas, Calculation of, 77. 
 
 of circles, 1 to 1,000, Table of, 545. 
 of circles, to 100, Table of, 561. 
 Arithmetic, 15. 
 
 Arithmetical progression, 20. 
 Armature, 216. 
 
 Arrangement of drill holes, 244, 335. 
 of mine plan, 381. 
 of slope tracks, 413. 
 
 Ascensional ventilation, 381. 
 
 Ash (coal), 171. 
 
 Ashworth - Hepplewhite - Gray lamp, 
 358. 
 
 Asphyxiation, 451. 
 
 Atmosphere, Composition of, 337. 
 
 Weight of. 337. 
 
 Atmospheric pressure, 339. 
 
INDEX . 
 
 624 
 
 Atom, 341. 
 
 Atomic volume, 341. 
 weight, 344. 
 
 Austrian measures of length, 3. 
 Avoirdupois weight, 2. 
 
 Back of Ore, Caving a, 320. 
 
 Balancing a conical drum, 396. 
 
 Ball mills, 427. 
 
 Barometer, 339. 
 
 Barometric elevations, 340. 
 
 variations, 339. 
 
 Barrell, J., 62. 
 
 Barrier pillars, 287. 
 
 Bar signals, 236. 
 
 Base lines, 46. 
 
 Batteries, Electric, 229. 
 
 Battery ceils, Composition of, 231. 
 wa*ter, 430. 
 working, 307. 
 
 Beams, 102, 105. 
 
 Bearing in, 283. 
 
 value of masonry, 107. 
 
 Bedded deposits, 279. 
 
 materials, Prospecting for, 238. 
 Bell wiring, 230. 
 
 Belting and velocity of pulleys, 193. 
 Bending rails, 412. 
 
 Bends in pipes, 153. 
 
 Bent plumb-line method of slope sur- 
 veying, 73. 
 
 Biram’s ventilator, 387. 
 
 Bitumen, Prospecting for, 249. 
 Bituminous coal, 169. 
 
 Cubic feet in ton of, 449. 
 Handling, 443. 
 
 Blackdamp, 350. 
 
 Blacksmith coal, 173. 
 
 Blake crushers, 419. 
 
 Blasting by electricity, 332. 
 
 Bleeding from scalp wounds, 451. 
 Blossburg method of mining, 298. 
 Blower fans, 386. 
 
 Board measure, 13. 
 
 Boiler, 176. 
 
 Care of, 185. 
 capacity, 181. 
 
 Choice of, 179, 180. 
 
 cleaning, 186. 
 
 coverings, 183. 
 
 feed-pumps, 161. 
 
 iron, thickness required, 187. 
 
 scale, 181. 
 
 tests, 188. 
 
 Boltheads, Weight of, 116. 
 
 Bolts per mile of track, 117. 
 
 Bolts, Weight of, 116. 
 
 Bonnett, 268. 
 
 Bord-and-pillar, 280. 
 
 Bore-hole records, 244, 250. 
 holes, 242. 
 
 Bottoms, Slope, 413. 
 
 Boxes for electric fuses, 224. 
 
 Box regulator, 375, 377. 
 
 Boyle’s law, 195. 345. 
 
 Brass, Weight of, 111, 112, 114, 115. 
 Brattice, Air, ^94. 
 
 Breast boards, 270. 
 
 , wheels, 157. 
 
 Bridges, Air, 393. 
 
 Briqueting, 448. 
 
 British imperial measure (liquid and 
 dry), 5. 
 
 measures of area, 4. 
 measures of length, 2. 
 measures of volume, 5. 
 thermal unit, 168. 
 
 Brown coal, 170. 
 
 Brown’s method of mining anthra- 
 cite, 306. 
 
 Brown & Sharpe gauge, 208. 
 
 Buckets, Conveying, 446. 
 
 Buggy breasts, 291. 
 
 Buildings, 283. 
 
 Bulling, 330. 
 
 Buntons, 270. 
 
 Burt’s sdlar attachment, 47. 
 
 Butt cleats, 284. 
 
 Cables, Electric, 209. 
 
 Cableways in mining, 122b, 278. 
 
 Calculation of areas, 77. 
 of mine resistance, 366. 
 of wires for electric transmission, 
 210 . 
 
 California method of mining, 300. 
 stamps, 428. 
 
 Calorie, 168. 
 
 Calumet classifier, 435. 
 
 Cams for stamps, 429. 
 
 Cannel coal, 170. 
 
 Capacity of boilers, 181. 
 of electric cables, 209. 
 of mining machines, 337. 
 of pumps, 161. 
 of shaking screens, 432. 
 of standard steel buckets, 446. 
 
 Capell ventilator, 389. 
 
 Carat, 12. 
 
 Carbonic-acid gas, 350. 
 oxide gas, 349. 
 
 Castings, Weight of iron, copper, lead, 
 brass, or zinc, 111. 
 
 Caving a back of ore, 320. 
 methods, 320. 
 waste only, 320. 
 
 Cells, Various types of, 231. 
 
 Ppntprs! RQ 
 
 Centigrade to Fahrenheit, 366. 
 to Reaumur, 366. 
 
 Centrifugal fans, 386. 
 pumps, 164. 
 roller mills, 426. 
 
 Chain, Surveyor’s, 42. 
 cutter machines, 337. 
 pillars, 287. 
 
 Chains, Iron, 129. 
 
 Chamber-and-pillar method, 318. 
 
 Chambering, 330. 
 
 Channels, Flow in, 142. 
 
 Character of floor and roof on size of 
 pillars, Influence of, 280. 
 
 Charging a hole, 330. 
 
 Charles’ law, 345. 
 
 
INDEX. 
 
 625 
 
 Chemical compounds, 341. 
 equations, 341. 
 symbols, 341. 
 
 Chemistry of gases, 341. 
 
 Chimneys, 189. 
 
 Chinese measures of length, 4. 
 
 Chock, 268. 
 
 Choice of a boiler, 179, 180. 
 
 Chokedamp, 350. 
 
 Churn drills, 242. 
 
 Chute breasts, 292. 
 mining, 309. 
 
 Circles, 31. _ __ r 
 
 areas of, 1 to 1,000, Table of, 545. 
 areas of, & to 100, Table of, 561. 
 Circuit, Electric, 205. 
 
 Circular mil, 207. 
 
 Circulation of boilers, 181. 
 Circumferences, 1 to 1,000, Table of, 
 545. 
 
 to 100, Table of, 561. 
 
 Clanny lamp, 356. 
 
 Classifiers, Hydraulic, 434. 
 
 Classifying apparatus, 431. 
 
 C’eaning safety lamps, 358. 
 
 Clearfield method of mining, 295. 
 
 Cleats, 284. 
 
 Clinometer, 44. 
 
 Closed work, 279. . 
 
 Close workings, Shots in, 361. 
 
 Closing surveys, 68. 
 
 Coal, 169. 
 
 Analyses, 168. 
 analysis, 173. 
 
 Anthracite, 169. 
 
 Bituminous, 169. 
 
 Blacksmith, 173. 
 
 Brown, 170. 
 
 Calorific Power, 168. 
 
 Cannel, 170. 
 
 Coking, 170. 
 
 Composition of, 170. 
 dealers’ computing table, 452. 
 Domestic, 173. 
 dust, Effect of, 360. 
 
 Free-burning, 170. 
 
 Gas, 173. 
 
 Hardness of, 171. 
 
 Iron-making, 172. 
 
 Lignite, 170. 
 
 Preparation of, 418. 
 
 Prices of, 326, 327. 
 
 Production of U. S., 326. 
 Prospecting for, 238. 
 
 Running of, 312. 
 
 Semianthracite, 169. 
 Semibituminous, 169. 
 
 Sizes of, 173, 434. 
 
 Specific gravity of, 109, 171. 
 
 Splint, 170. 
 
 Steaming, 171. 
 
 Storage of, 291. 
 
 Strength of, 171. 
 washers, 436. 
 
 Weight of, 170. 
 
 Coals, Cubic feet in ton of various, 
 109, 170. 
 
 Cockermegs, 268. 
 
 Coefficient of contraction (water), 
 135. 
 
 of discharge (water), 135. 
 of discharge with weirs, 140. 
 of rolling friction, 398. 
 of roughness for water channels, 
 144. 
 
 of velocity (water), 135. 
 
 Coefficients of friction for various 
 materials, 95, 96. 
 of friction of air in mines, 367. 
 
 Cog, 268. 
 
 Coins, 10. 
 
 Coke, 172. 
 
 ovens, Cost of, 328. 
 
 Coking coals, 170. 
 
 coal, Cost of, 328. 
 
 Collar, 268. 
 
 Collimation, Line of, 53. 
 
 Colorado method of mining, 302. 
 
 Color of coal ash, 171. 
 
 Columns, Safe loads for, 106. 
 Combustibles, 166. 
 
 Combustion, Spontaneous, 291. 
 Common fractions, 15. 
 
 Commutator, 216. 
 
 Comparison of aluminum and copper 
 for electric use, 209. 
 of hydraulic formulas, 149. 
 of methods of shaft sinking, 262. 
 of vacuum and plenum systems of 
 ventilation, 386. 
 
 Compass, 38. 
 
 To adjust, 38. 
 
 To use, 39. 
 vernier, 39. 
 field notes, 44. 
 
 Complement of angle, 34. 
 
 Composition of atmosphere, 337. 
 of coals, 170. 
 of forces, 95. 
 of fuels, 169. 
 
 Compound, Chemical, 341. 
 lever, 92. 
 
 wound dynamos, 219. 
 
 Compressed air, 194. 
 
 haulage, 403. 
 haulage problems, 404. 
 Compressibility of liquids, 133. 
 Compressive strength of anthracite, 
 289. 
 
 Condensers, 176. . 
 
 Conducting power of various sub- 
 stances, 184. 
 
 Conductors (guides), 398. 
 
 (electrical), 207. 
 for electric-haulage plants, 214. 
 Cone, 33. 
 
 Conical drums, 394. 
 
 drum, To balance, 396. . 
 
 Connecting outside and inside sur- 
 veys, 68. . . 
 
 Connections for continuous-current 
 motors, 223. . . 
 
 Connellsville method of mining, 293. 
 Constant-current circuit, 206. 
 potential circuits, 206. 
 power, 372. 
 
626 
 
 INDEX. 
 
 Constant pressure , 372. 
 pressure circuits, 206. 
 quantity, 372. 
 velocity, 372. 
 
 Constants for mine gases, 349. 
 
 for wooden beams, 103. 
 Construction of air compressors, 194. 
 of dams, 133. 
 of a mine furnace, 383. 
 
 Contents of a coal seam, To find, 52. 
 of cylinders, 6. 
 
 Continuous-current motors, Connec- 
 tions for, 223. 
 vernier, 45. 
 
 Contraction, Coefficient of ( water) ,135. 
 Control of roof pressure, 284. 
 Conversion factors (hydraulic), 141. 
 of thermometer readings, 366. 
 tables, 7, 10. 
 
 Conveyors, Horsepower of, 445. 
 Coordinates, 51. 
 
 Copper, Electric properties of, 209. 
 Weight of, 111, 112, 114, 115. 
 
 Wire, 208. 
 
 Cornish pumps, 158. 
 rolls, 423. 
 
 underhand stoping, 316. 
 
 Corps, Mine, 66. 
 
 Corresponding mercury and air col- 
 umns, Table of, 347. 
 
 Corrugated rolls, 422. 
 
 Cosecant, 35, 454. 
 
 Cosine, 35, 453. 
 
 Cost of briqueting, 449. 
 of coke ovens, 328. 
 of coking coal, 328. 
 of drilling, 244-247. 
 of haulage, 409. 
 of mining anthracite, 322. 
 of sinking, 263. 
 of stamping, 430. 
 of unloading coal, 447. 
 of well drilling, 242. 
 
 Cotangent, 35, 453. 
 
 Cotangents, Natural, Table of, 464. 
 Coverings, Boiler and pipe, 183. 
 Coversed sine, 35. 
 
 Coyoting, 321. 
 
 Cracking rolls, 421. 
 
 Crib, 268. 
 
 Cribbing, 270. 
 
 Cross-section of electric wires, 207. 
 Cross-sections, Construction of, 249. 
 Crushing load of wood, 105. 
 machinery, 418, 431. 
 mills, 426. 
 rolls, 423. 
 
 rolls, Table of, 425, 426. 
 
 Cube, 33. 
 
 root, 19, 545. 
 roots, Table of, 545. 
 
 Cubes, Table of, 545. 
 
 Cubic feet occupied by ton of various 
 coals, 449. 
 
 Culm, Flushing of, 314. 
 
 Current estimates, 212. 
 
 motors, 157. 
 
 Curtains, 394. 
 
 Curves for mine roads, 411. 
 
 ' Railroad, 78. 
 
 on slopes, Vertical, 416. 
 
 Cylinder, 33. 
 
 Cylinders, Contents of, 6. 
 in a pump, Ratio of, 160. 
 of a hoisting engine, To find size 
 of, 397. 
 
 To find contents of, in U. S. gal- 
 lons or bushels, 5. 
 
 Cylindrical boiler, Maximum work 
 of, 178. 
 
 boilers, Economy of, 188. 
 drums, 394. 
 
 drum, To find period of winding, 
 397. 
 
 rings, 34. 
 
 Dams, 154. 
 
 Debris, 156. 
 
 Earth, 156. 
 in mines, 133. 
 
 Masonry, 156. 
 
 Stone, 155. 
 
 Wing, 156. 
 
 Danish measures of length, 3. 
 
 Darcy’s formulas (hydraulic), 148. 
 Davy lamp, 356. 
 
 Debris dams, 156. 
 
 Decimals, 16. 
 
 Decimals of a foot for each & inch , 3. 
 Declination, Magnetic, 39. 
 
 of Polaris, 46. 
 
 Deflected angle, 45. 
 
 Deflections in p o w e r - transmission 
 ropes, Table of, 122. 
 
 Density of a gas, 344. 
 
 Departures, 537. 
 
 Deposits over 8 ft. thick, 318. 
 
 Depth of shafts, Calculation of, 340. 
 of suction, 162. 
 
 Derangement of ventilating current, 
 359. 
 
 Detaching hooks, 398. 
 
 Detection of small percentages of gas, 
 356. 
 
 Detonation, 331. 
 
 Diagram for reporting on mineral 
 lands, 252. 
 
 Diameter of holes (blasting), 330. 
 Diamond drill, 243. 
 
 weight, 12. 
 
 Dies for stamps, 429. 
 
 Differential pulley, 94. 
 
 Diffusion of gases, 346, 348. 
 
 Dip and strike from bore-hole records, 
 250. 
 
 workings, Ventilation of, 382. 
 Direct-current circuits, 210. 
 dynamos, 215. 
 motors, 220. 
 
 Direction of face, 284. 
 
 Directions for blasting by electricity, 
 332. 
 
 Discharge, Coefficient of (water), 135. 
 gates (dams), 154. 
 through V notch, 138. 
 
INDEX . 
 
 627 
 
 Disintegrating rolls, 423. 
 
 Disk fans, 385. 
 
 Distance, Errors in, 77. 
 
 from center to center of breasts, 
 Table, 287. 
 
 Distribution of air in mine ventila- 
 tion, 373. 
 
 Ditches, 142. 
 
 Banks of, 143. 
 
 Capacity of, 143. 
 
 Grades of, 143. 
 
 Velocity in, 142. 
 
 Division of air-current, Proportional, 
 375. 
 
 D., L. & W. telephone system, 234. 
 Dodge crushers, 419. 
 
 Domestic coals, 173. 
 
 Door regulator, 375, 377. # 
 
 regulator, Size of opening for, 377. 
 Doors, 393. 
 
 Double-chute battery, 309. 
 cylindrical drums, 394. 
 entry, 284. 
 
 Draft, 190. . , . 
 
 Drainage in shaft sinking, 263. 
 Drawbar pull of electric locomotives, 
 407. 
 
 Drawing pillars, 289. 
 
 Dredge (centrifugal pump), 164. 
 Dredging, 279. 
 
 Dressing of ores, 418. 
 
 Drift of drill holes, 243. 
 
 Drill, Diamond, 243. 
 holes, 243. 
 
 holes, Arrangement of, 335. 
 records, 244, 250. 
 
 Drilling, 242, 330. 
 
 Driving the gangway, 264. 
 
 Dron’s formula tor shaft pillars, 285. 
 Drop for stamps, 428. 
 
 Drums, Conical, 394. 
 
 Double-cylindrical, 394. 
 for wire rope, 12,21. 
 
 Dry measure (U. S.), 5. 
 
 Dunn’s table of size of pillars, 286. 
 Duplex pumps, 158. 
 
 Dust briquets, 448. 
 
 Effect of, 360. 
 
 Duty of anthracite screens, 434. 
 of miners’ inch, 137. 
 of stamps, 430. 
 
 Dynamite, Thawing, 329. 
 
 Dynamos, 215. 
 
 Alternating-current, 224. 
 Compound-wound, 219. 
 Series-wound, 219. 
 
 Shunt-wound, 219. 
 
 Earth Augers, 242. 
 
 dams, 156. 
 
 Economizers, 185. 
 
 Effect of altitude on air compression, 
 195. 
 
 Efficiency, Manometrical, 390. 
 Mechanical, 390. 
 of water-power, 156. . 
 
 Electrical Expressions and Equiva 
 lents, 205. 
 
 Electric blasting, 332. 
 
 Electric circuit, 205. 
 exploder, 332. 
 haulage, 406. 
 
 haulage plants, Conductors for, 
 214. 
 
 haulage problem, 407. 
 locomotives, Drawbar pull of, 407. 
 locomotives, Hauling capacity of, 
 408. 
 
 power, 204. 
 
 pumps, 162. . 
 
 resistance, Estimation of, 209. 
 
 signaling, 229-235. 
 
 units, 203. 
 
 wiring, 207. 
 
 Electricity, 203. 
 
 Electromotive force, 203, 218. 
 
 Elements, 341. 
 
 in ventilation, 363. 
 of mechanics, 91. 
 
 Table of, 342. 
 
 Elevating capacity of buckets, 446. 
 Elevation of rails for mine roads, 412. 
 Elevations, Barometric, 340. 
 
 Elevators, Water, 164. 
 
 Ellipse, 32. 
 
 End cleats, 284. 
 on, 285. 
 plates, 270. 
 
 Endless-rope haulage, 401. 
 
 Engine drivers, Rules for, 191. 
 
 planes, 399. 
 
 Engines, Sinking, 263. 
 
 Steam, 190. 
 
 Entries, Number of, 284. 
 
 Equal settling factors, Table of, 439. 
 settling particles, 439. 
 shadows, 47. 
 splits of air, 374. 
 
 Equations, Chemical, 341. 
 
 Equilibrium of liquids, 130. 
 Equivalent orifice, 367. 
 
 Errors, 76. 
 in arc, 76. 
 in distance, 77. 
 in measurement, 77. 
 in surveying, 66, 76. 
 
 Eschka’s method of analysis for sul- 
 phur, 174. 
 
 Establishing a meridian wi*h solar 
 attachment, 47. 
 
 Estimates of current, 212. 
 
 Evolution, 19, 545. 
 
 Excitation of dynamos, 218. 
 
 Exhaust fans, 386. 
 
 Expansion of gases, 344. 
 
 Exploder, Electric, 332. 
 
 Exploring workings, 361. 
 
 Explosions, Exploring workings after 
 361. 
 
 in boilers, 179. 
 
 Explosive conditions in mines, 359. 
 Explosives, 329. 
 
 Pressures developed by, 334. 
 Relative strengths of various 
 brands of, 330. 
 
 Values of, 335. 
 
 Eytelwein’s formulas (hydraulic), 148 
 
628 
 
 INDEX. 
 
 Face Cleats, 284. 
 
 Direction of, 284. 
 on, 285. 
 
 Factor of a mine, Potential, 367. 
 Factors, Conversion (hydraulic), 141. 
 Fahrenheit to centigrade, 366. 
 
 Fan construction, Principles of, 391. 
 tests, 392. 
 
 ventilation, 373, 385. « 
 
 Fans, Ventilating, 386. 
 
 Fastenings for wire rope, 126. 
 
 Feeders of gas, 352. 
 
 Feed-pumps, 161, 186. 
 
 Feedwater for boilers, 186. 
 
 Field excitation of dynamos, 218, 
 magnet, 216. 
 
 notes for outside compass survey* 
 44. 
 
 Filling methods, 319. 
 
 Finger bars, 432. 
 
 Firedamp, 351. 
 
 Fires, Gob, 291. 
 
 Firing a boiler, 186. 
 blasts, 331. 
 by detonation, 331. 
 
 Fixed carbon in coal, 171. 174. 
 
 Flash signals, 234. 
 
 Flat deposits, 318. 
 
 Flat ropes, 119, 394. 
 
 Flights, Capacity of, 446. 
 
 Flow of water in channels, 142, 144. 
 in rivers, 145. 
 through flumes, 146. 
 through orifices, 135. 
 through pipes, 147, 150. 
 Flumes, 145, 443. 
 
 Flow of water through, 146. 
 
 Grade of, 145. 
 
 Flushing of culm, 314. 
 
 Foaming (boilers), 186. 
 
 Forced draft, 190. 
 
 Force fans, 386. 
 
 Forces, Composition of, 95. 
 
 Foreign coins, Values of, 11. 
 Forepoling, 260, 270. 
 
 Form of roll teeth, 422, 
 
 Forms of mine timbering, 267. 
 Formulas for air splitting, 378. 
 for inclined planes, 399. 
 in ventilation, 370. 
 
 Foster’s formula for shaft pillars, 286. 
 Foundations (dams), 154. 
 
 Fractions, 15. 
 
 Common, 15. 
 
 Decimal, 16. 
 
 Table of equivalent decimal, 16. 
 Fragmental deposits, Mining of, 278. 
 Free-burning coals, 170. 
 
 gold per ton of ore, Value of, 241 
 Freezing process, 260. 
 
 Friction, 95. 
 
 coefficients for air in mines, 367. 
 coefficients for various materials, 
 95, 96. 
 
 in knees and bends, 153. 
 of air in pipes, 201. 
 of mine cars, 96. 
 of water in pipes, 151. 
 
 Friction pull on endless rope. 402. 
 Frictional resistance of shafting, 96. 
 Frozen ground, Mining of, 322. 
 Frustums. 34. 
 
 Fuel dust briquets, 448. 
 
 Fuels, 166. 
 
 Composition of, 169, 
 
 Furnace stack, 385. 
 
 Construction of ventilating, 383. 
 ventilation, 373. 
 
 ventilation, Air columns in, 384. 
 Fuse boxes, 224. 
 
 Fuses, 224. 
 
 Fusible plugs, 186. 
 
 Galvanic Action Around Boilers, 187. 
 
 Gangway driving, 264. 
 
 timbers, 268. 
 
 Gas coals, 173. 
 feeders, 352. 
 
 Testing for, 354. 
 
 Outbursts of, 352, 360. 
 
 Gases, Chemistry of, 341. 
 
 Diffusion of, 346 ; 348. 
 enclosed in the pores of coal, 353. 
 found in mines, 348. 
 
 Pressure of, 345. 
 
 Properties of, 344. 
 
 Transpiration of, 346, 348. 
 
 Gate for jig, 438. 
 
 Gates crusher, Table of, 422. 
 
 Gauge cocks, 186. 
 
 of mine tracks, 411. 
 
 Pressure, 185. 
 
 Water, 365. 
 
 Wire, 122k, 207, 208. 
 
 Gauging by weirs, 138. 
 water, 136. 
 
 Gay-Lussac’s law, 195, 345, 
 
 Gears, Train of, 92. 
 
 Gems and precious stones, Prospect- 
 ing for, 241. 
 
 General mathematical principles, 14. 
 
 remarks on surveying, 49. 
 Geological maps, Construction of, 249. 
 periods, 237. 
 
 Geometrical problems, 25. 
 
 progression, 21. 
 
 Geometry, 24. 
 
 Geordy lamp. 356. 
 
 George’s creek method of mining, 297. 
 Gilpin county stamps, 428. 
 
 Glossary of mining terms, 565. 
 
 Gob fires, 291. 
 
 Gold coins, 10. 
 
 mine, Opening of, 258. 
 
 ; Gonda type of cell, 230. 
 
 Gophering, 321. 
 
 Gould’s formulas (hydraulic), 148. 
 Grade of mine road, 410. 
 
 Gradient, Hydraulic, 147. 
 
 Graham’s law, 346. 
 
 Gravity planes, 398. 
 
 Specific, 107. 
 specific, Table of, 108. 
 stamps, 427. 
 i Gray lamp, 358. 
 
INDEX. 
 
 629 
 
 Grizzlies, 431. 
 
 Guibal ventilator, 388. 
 Guides, 398. 
 
 Gyratory crushers, 420. 
 
 Half on, 285. 
 
 Hammer crushers, 423. 
 
 Handling of material, 443. 
 
 Hard coal, 169, 313. 
 
 Hardness of coal, 171. 
 
 Haulage, 398. . 
 
 by compressed air, 403. 
 
 Cost of, 409. 
 
 Electric, 214, 406. 
 
 Motor, 402. 
 problems, 399, 407. 
 road signals, 235. 
 rope. 120, 122, 400. 
 
 Speed of, 408. 
 
 Hauling capacity of electric locomo- 
 tives, 408. . i 
 
 Ha wksley’s formula (hydraulic), 148. 
 
 Head-bars, 431. 
 block, 268. 
 frames, 275, 397. 
 frames, Sinking, 262. 
 gears, 275. 
 sheaves, 397. 
 
 Headboard, 268. 
 
 Heating surface of a boiler, 1/7. 
 values of American coals, 168. 
 
 Heberle gate, 438. 
 
 High explosives, 329. 
 
 Hill, E., 198. . 
 
 Hints for mining small seams, 313. 
 to beginners in surveying, 80. 
 
 Hoisting, 394. 
 
 engine cylinders, 397. 
 engine, To find size of, 396. 
 problems, 396. 
 
 ropes, Starting strain on, 124, 126. 
 ropes, Stress in, 1221. 
 
 Hooks, Detaching, 398. 
 
 Horizontal distances, 50, 87, 537. 
 distances, Stadia table of, 87. 
 
 Horsepower for box regulators, 377. 
 for bucket elevators, 445. 
 for door regulators, 377. 
 necessary to compress air, 406. 
 of air-currents, 363. 
 of boilers, 177. 
 of coal conveyor, 445. 
 of engine, 190. 
 of manila ropes, 126. 
 of stamps, 430. 
 of a stream, 157. 
 required to raise water, 161. 
 
 Hughes’s formula for shaft pillars, 
 286. 
 
 Hydraulic classifiers, 434. 
 gradient, 147. 
 placer mining, 278. 
 
 Hydraulics, 135. 
 
 Hydrocarbons, 349. 
 
 Hydrogen disulphide H^S, 350. 
 Sulphuretted, 350. 
 
 Hydrostatics, 130. 
 
 I Beams, Safe Loads for, 104 
 
 Illuminating power of safety lamps, 
 359. 
 
 Impulse wheels, 158. 
 
 Incandescent lamps, 213. 
 
 Inch, Miners’, 136. 
 
 Inclined plane, 93. 
 
 plane, Stress in hoisting ropes on, 
 1221. 
 
 roads, Calculation of power for, 
 402. 
 
 roads, Haulage on, 398. 
 shafts, Surveying, 73. 
 
 Included angle, 45. 
 
 Incrustation and scale, 182. 
 
 Indiana method of mining, 298. 
 Individual angle, 45. 
 
 Induction motors, 227. 
 
 Inertia, Angle of, 398. 
 
 Influence of furnace stack on venti- 
 lation, 385. 
 
 of seasons on ventilation, 382. 
 Injector, 186. . 
 
 Area of nozzle of, 187. 
 
 Delivery in gallons per hour oi, 
 187. 
 
 Inside surveys, 56, 67. 
 
 Inspection of boilers, 181. 
 
 Insulated wires, 212. 
 
 Interstitial currents, 439. 
 
 factors, 440. 
 
 Involution, 19, 545. 
 
 Iowa method of mining, 299. 
 
 Iron beams, 103. 
 for track, 117. 
 pipe, Weight of, 113. 
 supports, 272. 
 
 Weight of flat, 115. 
 
 Weight of wrought, 114, 
 
 Irregular mining deposits, 321. 
 Isogonic chart, 40. . 
 
 Isothermal compression, 195, 198. 
 
 Jaw Crushers, 419. 
 
 Jeffrey-Robinson coal washer, 436. 
 Joints in mine timbering, 267. 
 
 Kind-Chaudron Method of Sinking, 262. 
 
 Knees in pipes, 153. 
 
 ICoepe system of hoisting, 395. 
 
 Lamp Tests for Gas, 354. 
 
 Lamps, Arc, 214. 
 
 Incandescent, 213. 
 
 Safety, 355. 
 
 Testing, 355. 
 
 Lancashire boiler, 177. 
 
 Landings, Timbering of, 2/2. 
 
 Large deposits over 8 ft. thick, Min- 
 ing, 318. 
 
 Latches, 413. 
 
 Latitude, 50, 537. 
 
 with Burt’s solar, 48. 
 
 Launders, 443. 
 
630 
 
 INDEX. 
 
 Law of settling particles, 439. 
 
 Laws in regard to air splitting, 373. 
 in regard to quantity of air, 363. 
 of volume, 341. 
 
 Leaching methods of mining, 322. 
 Lead, Weight of, 111, 113, 114, 115. 
 Leclanche cell, 229. 
 
 Lehigh region, Costs of mining in, 
 323. 
 
 Length, Measures of, 2. 
 
 of steep pitch on inclined plane, 
 399. 
 
 Leveling, 53. 
 
 Trigonometric, 56. 
 
 Level notes, 55, 62. 
 
 roads, Haulage on, 398. 
 timbers, 268. 
 
 Levels (gangways), 316, 
 in metal mines, 264. 
 
 Levers, 91. 
 
 Lid, 268. 
 
 Life of shoes and dies, 429. 
 
 of wire rope, 1221 
 Lignite, 170. 
 
 Line shafting, 110. 
 
 Liquid measure (U. S.), 5. 
 
 Liquids, Compressibility of, 133. 
 
 Load for wire rope, 125. 
 
 that a hoisting engine will start, 
 To find, 396. 
 
 Locating errors, 76. 
 
 special work, 77. 
 
 Locks for lamps, 358. 
 
 Locomotive haulage, 402. 
 
 Logarithmic functions, Table of, 492. 
 Logarithms, 22, 473. 
 
 of numbers, Table, 473. 
 of trigonometric functions, Table 
 of, 492. 
 
 Log washer, 436. 
 
 Long-hole process of shaft sinking, 
 262. 
 
 horn, 285. 
 section, 64. 
 splice, 128. 
 
 Longitudinal back stoping with fill- 
 ing, 319. 
 
 Longwall method, 281, 302. 
 Modifications of, 302. 
 
 Timbering, 283. 
 
 Loss in transmitting air, 198. 
 of blood, 449. 
 
 of head in pipe by friction, 151. 
 of heat from steam pipes, 184. 
 of pressure of air in pipes, 202. 
 Low explosives, 329. 
 
 Lubricants for different purposes, 102. 
 Lubrication, 100, 124b. 
 
 Machine Mining, 336. 
 
 Magnetic prospecting, 248. 
 variation, 39. 
 
 Malleable-iron buckets, 446. 
 
 Manila ropes, Power transmitted by, 
 126. 
 
 Manometrical efficiency, 390. 
 Mapping, 74. 
 
 Maps, Geological, 249. 
 
 Mariotte’s law, 345. 
 
 Marsaut lamp, 358. 
 
 Marsh gas, 348. 
 
 Masonry, Bearing value of, 107. 
 dams, 156. 
 supports, 272. 
 
 Material, Handling of, 443. 
 Mathematical signs, 14. 
 
 Mathematics, 14. 
 
 Measurement, Error jn, 77. 
 of temperature, 366. 
 of ventilating currents, 364. 
 Measures of area, 4. 
 
 American, 4. 
 
 British, 4. 
 
 Metric, 4. 
 
 Measures of Length, 2. 
 
 American, 2. 
 
 Austrian, 3. 
 
 British, 2. 
 
 Chinese, 4. 
 
 Danish, 3. 
 
 Metric, 3. 
 
 Norwegian, 3. 
 
 Prussian, 3. 
 
 Russian, 3. 
 
 Swedish, 4. 
 
 Measures of volume, 5. 
 
 American, 5 
 British, 5. 
 
 Metric, 5. 
 
 Mechanical efficiency, 390. 
 mixture, 341. 
 ventilators, 385. 
 
 Mechanics, 91. 
 
 Mensuration, 28. 
 of solids, 33. 
 of surfaces, 28. 
 
 Mercurial barometer, 339. 
 
 Mercury and air columns, 347. 
 
 column corresponding to water 
 column, 340. 
 
 Meridians, 46. 
 
 Merivale’s formula for shaft pillars, 
 285. 
 
 Mesh for shaking screens, 433. 
 
 of revolving screens for anthra* 
 cite, 434. 
 
 Size of, 433. 
 
 Metal linings for shafts, 260. 
 
 Methods and appliances in mine ven- 
 tilation, 381. 
 
 Methods of mining, 277. 
 
 Alabama, 297. 
 anthracite, 305. 
 
 Blossburg, 298. 
 
 Brown’s, 306. 
 
 California, 300. 
 
 Clearfield, 295. 
 
 Colorado, 302. 
 
 Connellsville, 293. 
 
 George’s creek, 297. 
 
 Indiana, 298. 
 
 Iowa, 299. 
 
 mineral deposits, 316. 
 Newcastle, Colo., 302. 
 Pittsburg, 295. 
 
INDEX . 
 
 631 
 
 Methods of Mining, Reynoldsville, 
 295 
 
 Tesla, Cal., 300. 
 
 West Virginia, 296. 
 
 Williams’, 312. 
 
 Methods of surveying, 67. 
 
 Metric conversion tables, 7-10. 
 measures of area, 4. 
 measures of length, 3. 
 measures of volume, 5, 
 system, 1, 2, 3, 9, 10. 
 weight, 2. 
 
 Mil, 207. 
 
 Milling system, 278. 
 
 Mine-car friction tests, 98. 
 cars, Friction of, 96. 
 corps, 66. 
 dams, 133. 
 explosions, 361. 
 gases, 348. 
 gases, Table of, 349. 
 
 Opening a, 257. 
 plan, Arrangement of, 381. 
 resistances, 364, 366. 
 roads, 410b. 
 sampling, 174. 
 telephones, 233. 
 timber and timbering, 265. 
 tracks 410b. 
 
 Mineral available in a prospect, 251. 
 deposits, Methods of mining, 316. 
 lands, Report on, 252. 
 
 Miners’ inch, 136. 
 
 Mining terms, Glossary of, 565. 
 
 Mixture, Mechanical, 341. 
 
 Moisture in coal, 171, 174. 
 
 Molecule, 341. 
 
 Money, Tables of, 10. 
 
 Morris, W. H., 127. 
 
 Motor haulage, 402. 
 
 Motors, 214, 215. 
 
 Current (water), 157. 
 
 Induction, 227. 
 
 Regulation of speed of, 222. 
 
 Synchronous, 226. 
 
 Movable bars, 432. 
 
 pulley, 94. 
 
 Mueseler lamp, 358. 
 
 Multiphase alternators, 225. 
 
 Murphy ventilator, 389. 
 
 Nails, Sizes, Etc. of, 114. 
 
 Nasmyth fan, 387. 
 
 Natural division of air-currents, 374. 
 functions, Tables of, 453. 
 gas, Prospecting for, 249. 
 splitting, Calculation of, 374. 
 ventilation, 381. 
 
 Needling, 268. 
 
 Neville’s formula (hydraulic), 148. 
 Newcastle, Colorado, method of 
 mining, 302. 
 
 Nitrogen, 348. 
 
 Non-conductors, Relative values of, 
 185. 
 
 Norwegian measures of length, 3. 
 Notes (survey), 60. 
 
 Notes, Compass field, 44. 
 
 for outside compass survey, 44. 
 Level, 55, 62. 
 on mapping, 74. 
 
 Side, 60, 61. 
 
 Stope-book, 62. 
 
 Transit, 60. 
 
 Nozzle, Injector, 187. 
 
 Number of cars in a trip on a self- 
 acting incline, 399. 
 
 Nuts, Weight of, 116. 
 
 Occluded Gases, 352. _ 
 
 Occurrence of gases in mines, 351. 
 Ohm’s law, 204. 
 
 Oils for safety lamps, 356. 
 
 Lubricating, 100, 102. 
 
 Open channels, 142. 
 work, 277. 
 
 Opening a mine, 257. 
 
 in box regulators, 376. 
 
 Order of drop of stamps, 428. 
 
 Ore deposits, 238. 
 dressing, 418. 
 
 Handling of, 443. 
 
 Orifice, Equivalent, 367. 
 Oscillating bars, 432. 
 
 Outbursts of gas, 352, 360. 
 
 Outside surveys, 67. 
 
 Overcasts, 393. 
 
 Overhand stoping, 304, 317. 
 Overshot wheels, 158. 
 
 Oxygen, 348. 
 
 Packing for Pumps, 159. 
 
 Pack walls, 283. 
 
 Paint, 59. 
 
 Pamely’s formula for shaft pillars, 286. 
 Panel system, 283. 
 
 Parallel circuits, 206. 
 
 Parallelograms, 28. 
 
 Parallelopiped, 33. 
 
 Peele, Robert, 194. 
 
 Percentage, 20. 
 
 Percussion drills, 242. . 
 
 Period of winding on a cylindrical 
 drum, To find, 397. 
 
 Permitted explosives, 329. 
 
 Petroleum, Fuel value of 167. 
 
 Prospecting for, 249 
 Pick machines, 336. 
 
 Pillar timber, 268. 
 and chamber, 280. 
 and stall, 281, 292. 
 drawing, 289. 
 
 Pillars, 285. , , 
 
 Weight on, at different depths, 28/. 
 Wooden, 105. 
 
 Pins, 43. 
 
 Pipes, Flow through, 147, 150. 
 
 Friction in, 151. 
 
 Pressure of water in, 132. 
 Thickness of, 133. 
 used for compressed-air haulage, 
 Table of, 406. 
 
 Piston speed of pumps, 161. 
 
632 
 
 INDEX. 
 
 Pitch at which anthracite will run, 
 Table of, 443. 
 distance, 51. 
 
 Pitching work surveys, 68. 
 
 Pittsburg method of mining, 295. 
 Placer deposits, Prospecting for, 240. 
 
 mining, 278. 
 
 Plane trigonometry, 34. 
 
 Planes, Engine, 399. 
 
 Gravity, 398. 
 
 Plates, metal, Weight of. 112. 
 
 Platform bars, 431. 
 
 Plats, Timbering of, 272. 
 
 Plenum system of ventilation, 386. 
 Plotting, 49. 
 
 by coordinates, 51. 
 
 Plow-steel rope, 120. 
 
 Plugs, 186. 
 
 Plumb-bob, 44. 
 
 Plumbing of shafts, 69. 
 
 Pneumatic method of shaft sinking 
 260. 
 
 stamps, 430. 
 
 Pockets of gas, 352. 
 
 Poetsch-Sooy smith process (freezing 
 method), 260. 
 
 Polaris observation, 46. 
 
 Polygons, 30. 
 
 Post and breast cap, 268. 
 
 Potential factor of a mine, 367. 
 
 Pound calorie, 168. 
 
 Power, Electrical, 204. 
 for hoisting, 395. 
 in mine ventilation, 367. 
 of air-currents, 363. 
 of a hoisting engine, To find, 396. 
 of an explosive, 334. 
 of waterfall, 157. 
 pumps, 162. 
 
 required for inclined roads, 402. 
 stamps, 431. 
 
 Water, 156. 
 
 Practical examples in the solution of 
 triangles, 35. 
 
 splitting of air-currents, 373. 
 Precious stones, Prospecting for, 241. 
 Preliminary work, 257. 
 
 Preparation' of anthracite, Diagram 
 of, 442. 
 
 of coal and ore, 418. 
 
 Preservation of timber, 265. 
 
 Pressure. Absolute, 345. / 
 
 as affecting explosive conditions, 
 361. 
 
 developed by explosives, 334. 
 for box regulators, 376. 
 gauge, 185. 
 
 of anthracite coal against walls, 
 445. 
 
 of bituminous coal against walls, 
 444. 
 
 of gases, 345. 
 
 of liquids on surfaces, 130. 
 of occluded gases, 352. 
 of steam at different tempera- 
 tures, 188. 
 
 of water in pipes, 132. 
 on heading, 130. 
 
 Pressure of water on plane surface, 
 132. 
 
 Prices per ton of coal at mines, Table, 
 327. 
 
 Primary splits, 374. 
 
 Prism, 33. 
 
 Prismoidal formula, 34. 
 
 Problem in compressed-air haulage, 
 404. 
 
 Problems in geometrical construc- 
 j tion, 25. 
 
 in haulage, 399. 
 
 I in hoisting, 396. 
 
 j Progression, Arithmetical, 20. 
 
 | Geometrical, 21. 
 
 Properties of copper wire, 208. 
 
 of materials, 107. 
 
 Proportion, or Pule of Three, 18. 
 Proportional division of the air-cur- 
 rent, 375. 
 
 Props, Undersetting of, 276, 277. 
 
 Wooden, 105. 
 
 Prospecting, 235. 
 for bitumen, 249. 
 for natural gas, 249. 
 for petroleum, 249. 
 
 Magnetic, 248. 
 
 Prussian measures of length, 3. 
 Pulley, 94. 
 
 Pulleys and belting, 193. 
 
 Pulverizers, 423. 
 
 Pump machinery, 158. 
 memoranda, 163. 
 packing, 159. 
 valves, 162. 
 
 Pumps, 158. 
 
 Air-lift, 164. 
 
 Centrifugal, 164. 
 
 Cornish, 158. 
 
 Electrical, 162. 
 for acid waters, 165. 
 
 , Power, 162. 
 
 Simple and duplex, 158. 
 
 Sinking, 165. 
 
 Vacuum, 164. 
 
 Push button, 234. 
 
 Pyramid, 33. 
 
 Quadrant, 34. 
 
 Quantity of air required by State 
 Laws, 363. 
 
 for dilution of mine gases, 363. 
 for ventilation, 362. 
 to produce necessary velocity 
 at face, 363. 
 
 Radial Roller Mills, 426. 
 
 Radii of curves, 79. 
 
 Rail bending for mine roads, 412. 
 
 elevation for mine roads, 412. 
 Railroad curves, 78. 
 
 Rails for mine roads, 411. 
 
 per mile of track, 117. 
 
 Raises, 316. 
 
 j Rapid firing of boilers, 187. 
 
 method of splicing a wire rope, 
 127. 
 
 I Rate of diffusion, 346. 
 
INDEX. 
 
 633 
 
 Rating of compressors, 195. 
 
 Ratio of steam and water cylinders in 
 pump, 160. 
 
 Reaumur to Fahrenheit, 366. 
 Reciprocals, 545. . 
 
 Recoil of an explosion, 361. 
 
 Reduction of inches to decimals of a 
 foot, 2. 
 
 Regular polygon, 30. 
 
 Regulation of motors for speed, 222. 
 Regulators, 375. , 
 
 Relation of power, pressure, and ve- 
 locity, 364. 
 
 Relative volume of gases, 343. 
 Relighting stations, 359. 
 
 Removal of sulphur from coal, 441. 
 Repair of boiler coverings, 187. 
 Repairs to boilers, 180. 
 
 Reporting on mineral lands, 252. 
 Requirements of law as to splitting, 
 373. 
 
 Reservoirs, 154. 
 
 Resistance, Electric, 203. 
 
 Estimation of (electric) , 209. 
 in electric lines, 206. 
 of soils to erosion, 143. 
 
 Mine, 364. 
 
 Return-call system, 232. 
 
 Revolving screen mesh for anthracite, 
 
 Reynoldsville method of mining, 295. 
 Right-angled V notch, 137 . 
 
 Rise workings, Ventilation of, 382. 
 Rivers, Flow of water in, 145. 
 
 Roads, Haulage on inclined, 398. 
 Level, 398. 
 
 Mine, 410. 
 
 Rock-chute mining, 310. 
 drills, 263. 
 
 Handling of, 443. 
 
 Roller mills, 426. 
 
 Rollers, 412. . 
 
 Rolling friction, Coefficient of, 398. 
 Roll-jaw crushers, 420. 
 
 Rolls, 421. 
 
 Amount crushed by, 424. 
 Crushing, 423. 
 
 Speeds of, 424. 
 
 Teeth of, 422. 
 
 Roof pressure, 280. 
 
 Control of, 284. 
 
 Room-and-pillar, 280. 
 modifications of, 291. 
 openings, 281. 
 pillars, 286. 
 
 Rooming with filling, 319. 
 
 Rope haulage, 122, 400. 
 
 Ropes, 118. 
 
 fastenings, 126. 
 
 Flat, 394. 
 
 Manila. 120, 126. 
 
 Rotary crushers, 420, 422. 
 
 Roughness, Coefficient of, 144. 
 
 Rule of Three, 18. . 
 
 Rules for engine drivers, 191. 
 Running of coal, 312. 
 
 Russian measures of length, 3. 
 
 Safe Loads for Cast-Iron Columns, 106. 
 
 Safe loads for I beams, 104. 
 
 Safety catches, 398. 
 explosives, 329. 
 lamp oils, 356. 
 lamps, 355, 356. 
 
 lamps, Locks for, 358. • 
 
 lamps, Illuminating power of, 3o9. 
 valves, 185. . 
 
 Sampling available mineral, 251. 
 
 of coal, 173. • 
 
 Scaife trough washer, 437. 
 
 Scale, Removal of, from boilers, 181. 
 Scalp wounds, 451. 
 
 Schiele ventilator, 388. 
 
 Schmidt’s law of faults, 239. 
 
 Screens, 431, 432. 
 
 Screw, 93. „ , , 
 
 Diameter and number of, 113. 
 
 Seam blasting, 331. 
 
 Seasons, Influence of, 382. 
 
 Secant, 35, 454. 
 
 Secondary splits, 374. 
 
 Sederholm, E. T., 125. 
 
 Semianthracite coal, 169. 
 Semibituminous coal', 169. 
 Series-circuits, 205. 
 
 parallel method of regulation, 222- 
 wound dynamos, 219. 
 
 Settling boxes, 435. 
 
 particles, Law of, 439. 
 
 Shaft bottoms, Steel, 276. 
 bottom tracks, 416. 
 pillars, 285. 
 
 S n ?iraina g e for, 263. 
 sinking, Ventilation, 263. 
 timbering, 270. 
 
 Shafting, Frictional resistance of, 96. 
 
 Strength of, 110. 
 
 Shafts, 259. # ^ . 
 
 Calculation of depth of, 340. 
 
 Form of, 259. 
 
 Methods of sinking, 259. 
 
 Size of, 259, 
 
 Surface tracks at, 417. 
 
 Table of depths of, etc., 261. 
 Shaking screens, 432. 
 
 Shanty, 268. 
 
 Shearing machines, 337. 
 
 Sheaves for wire rope, 1221. 
 Sheet-metal gauges, 122k. 
 
 Shoes for stamps, 429. 
 
 Short horn, 285. 
 
 Shots in close workings, 361. 
 Shunt-wound dynamos, 219. 
 
 Side notes, 60, 61. 
 
 •Signal for haulage roads, 235. 
 Signaling, Electric, 229-235. 
 
 Silver coins, 10. 
 
 Similar airways, 372. 
 
 Simple bell circuit, 230. 
 
 Sine, 34, 453. 
 
 Sines, Natural, Table ot, 453. 
 Single-chute battery, 309. 
 entry, 284. 
 
 phase alternators, 225. . 
 
 wire method of slope surveying, 74 
 
634 
 
 INDEX. 
 
 Sinking a shaft, 259. 
 
 Cost of, 263. 
 engines, 263. 
 head-frames, 262. 
 pumps, 165. 
 
 Slope, 263. 
 
 Speed of, 263. 
 
 Siphons, 149. 
 
 Size of engine for engine-plane haul- 
 age, 399. 
 
 of hoisting engine, To find, 396. 
 of mesh for screens, 433. 
 of opening for door regulator, 377. 
 of opening in box regulator, 376. 
 of pillars, 285. 
 of timber, 267, 
 
 Sizes of anthracite, percentages of 
 each, 323, 326. 
 coal, 173, 434. 
 
 Sizing apparatus, 431. 
 
 Slack, 167. 
 
 Slicing method, 319. 
 
 Slightly inclined deposits, 318. 
 
 Slope bottoms, 413. 
 level, 44. 
 sinking, 263. 
 surveying, 73. 
 tracks, 413. * 
 
 Surface tracks at, 417. 
 
 Small percentages of gas, 356. 
 
 seams, mining of, 313. 
 
 Soft coal, 169. 
 
 Soils, Resistance of, to erosion, 143. 
 Solar attachment, 47. 
 
 Sole, 268. 
 
 Space required to store coins, 11. 
 Special forms of supports, 272. 
 mining methods, 322. 
 work, 77. 
 
 Specific gravity, 107. 
 of coal, 171. 
 of gases, 344. 
 
 of various substances, Table 
 of, 108, 109. 
 volume, 341. 
 
 Speed of crushing rolls, 426. 
 of drilling, 244. 
 of haulage, 408. 
 of revolving screens, 434. 
 of rolls, 423. 
 of sinking, 263. 
 of stamps, 429. 
 
 of water through pump passages, 
 160. 
 
 regulation of motors, 222. 
 
 Sphere, 33. 
 
 Spikes, Railroad, 117. 
 
 Sizes, etc., 114. 
 
 Spiling, 270. 
 
 Spillways, 155. 
 
 Spitzkasten, 435. 
 
 Spitzlutten, 435. 
 
 Splices per mile of track, 117. 
 
 Splicing a wire rope, 127. 
 
 Splint coal, 170. 
 
 Splitting formulas, 378. 
 
 of air-currents, 373. 
 
 Spontaneous combustion, 291. 
 
 Sprags, 270. 
 
 Square feet to acres, 4. 
 root, 19, 545. 
 roots, Table of, 545. 
 sets, 270. 
 set system, 321. 
 work, 318. 
 
 Squares, Table of, 545. 
 
 Squibbing, 330. 
 
 Stack, Furnace, 385. 
 
 Stadia measurements, 81. 
 
 table, 86. 
 
 Stamps, 427. 
 heads, 429. 
 
 Pneumatic, 430. 
 
 Power, 431. 
 
 Speed of, 429. 
 
 Steam, 431. 
 
 Standard steel buckets, Weights and 
 capacities of, 446. 
 
 Starting strain on hoisting rope, 126. 
 Stationary screen jigs, 437. 
 screens, 431. 
 
 Stations, Distinguishing, 59. 
 Establishment of, 57. 
 
 Kinds of, 57. 
 
 Marking, 58. 
 
 Relighting, 359. 
 
 Timbering of, 272. 
 
 Steam, 175. 
 engine, 190. 
 pipe coverings, 183. 
 pressure at different tempera- 
 tures, 188. 
 shovel mines, 278. 
 stamps, 431. 
 
 Steaming coals, 171. 
 
 Steel beams, 103. 
 
 shaft bottoms, 276. 
 supports, 272. 
 tape, 43. 
 
 Stems for stamps, 429. 
 
 Stinkdamp, 350. 
 
 Stone dams, 155. 
 
 Stope books, 62. 
 
 Stoping, 316. 
 
 Overhand, 304. 
 with filling, 319. 
 
 Stoppings, 393. 
 
 Storage, Coal, 291. 
 
 Stowing, 283. 
 
 Stream horsepower, 157. 
 
 Strength of anthracite, 289. 
 of electric current, 203. 
 of materials, 102. 
 of metals, 115. 
 of roof, 280. 
 of shafting, 110. 
 of wire ropes, 119. 
 
 Stress in hoisting ropes, 1221. 
 
 Strike from bore-hole records, 250. 
 Stulls, 268. 
 
 Stuttles, 270. 
 
 Suction, 162. 
 
 in jigging, 440. 
 
 Sulphureted hydrogen, 350. 
 
 Sulphur in coal, 171, 174. 
 
 Removal of, from coal, 441. 
 
INDEX. 
 
 635 
 
 Sump, 264. 
 
 Supplement of angle, 34. 
 
 Surface tracks for shafts and slopes, 
 417. 
 
 Surveying, 38. 
 
 drill holes, 243. 
 methods, 67. 
 
 Underground, 56, 67. 
 
 Susquehanna Coal Co. (friction of 
 mine cars), 98. 
 
 Swedish measures of length, 4. 
 Switches, 413. 
 
 Symbols, Chemical, 341. 
 
 Synchronous motors, 226. 
 
 Systems of working coal, 280. 
 
 Table, Coal Dealers’ Computing, 452. 
 Tables of Barrier Pillars, 288. 
 batteries, 231. 
 
 circles, 545. . 
 
 circumferences and areas of cir- 
 cles, ^ to 100, 561. 
 combustibles, 166. 
 elements, 342. 
 hydraulic, 135-154. 
 logarithms of numbers, 473. 
 logarithms of trigonometric func- 
 tions, 492. 
 mine gases, 349. 
 natural sines and cosines, 453. 
 natural tangents and cotangents, 
 464. 
 
 rail elevations, 412. 
 reciprocals, 545. 
 
 squares, cubes, square roots, cube 
 roots, circumferences and areas, 
 545-560. 
 
 Stadia, 88. 
 
 strength of materials, 102-106. 
 traverse (latitudes and depar- 
 tures), 537. 
 
 well-known shafts, 261. 
 
 ( For tables not enumerated above see the 
 various subjects.) 
 
 Tail-rope haulage, 400. 
 
 Tamping, 331. 
 
 Tangent, 35, 453. 
 
 Tangents, Natural, Table of, 464. 
 Tappets for stamps, 429. 
 
 Teeth for rolls, 422. 
 
 Telephones in mines, 233. 
 
 Telpherage, 278. 
 
 Temperature, Absolute, 344, 345. 
 
 Measurement of, 366. 
 
 Tension on hauling rope, Calculation 
 of, 401. 
 
 Tertiary splits, 374. 
 
 Tesla, Cal., method of mining, 300. 
 Test of mine-car frictions, 98. 
 
 Testing for gas by lamp flame, 354. 
 
 lamps, 355. 
 
 Tests, Boiler, 188. 
 
 Fan, 392. 
 
 of compressive strength of an- 
 thracite, 290. 
 
 Thawing dynamite, 329. 
 
 Theory of air compression, 194. 
 jigging, 439. 
 
 Theory of stadia measurements, 81. 
 Thermal unit, 168. 
 
 Thermometer readings, Conversion 
 of, 366. 
 
 Thermometers, 366. 
 
 Thickness of boiler iron, 187. 
 pipe, 133. 
 
 Thin seams, Mining, 313. ♦ 
 
 Three-phase alternators, 226. 
 
 Thurston, table of lubricants, 102. 
 
 Ties for mine roads, 410. 
 
 Timber, Crushing load of, 105. 
 Gangway, 268. 
 joints, 267. 
 
 Level, 268. 
 measure, 12. 
 
 Placing of, 266. 
 
 Preservation of, 265. 
 
 Size of, 267. 
 
 Timbering, 265. 
 
 Forms of, 267. 
 longwall face, 284. 
 
 Tools for sinking, 263. 
 
 Torque, 220. 
 
 T't’ppIt iron 117 
 
 Tracks for shaft bottoms, 416. 
 
 Mine, 410. 
 
 Tractive efforts of compressed-air 
 locomotives, 404. 
 
 Train of gears, 92. 
 
 Transformers. 228. 
 
 Transit, 40. 
 
 adjustment, 41. 
 notes, 60. 
 surveying, 45. 
 
 Transmission of air in pipes, 196, 198. 
 Electric, 210. 
 
 pressure through water, 132. 
 
 Rope, 120, 122, 
 
 Transpiration of gases, 346, 348. 
 Transporting a wounded person, 
 450. 
 
 Transverse rooming with filling, 
 319. 
 
 Trapeziums, 30. k 
 
 Trapezoids, 30. 
 
 Traverse tables, 537. 
 
 Traversing a survey, 51. 
 
 Treatment of injured persons, 449. 
 
 persons overcome by gas, 451. 
 Tremain stamp, 431. 
 
 Trestles, 274. 
 
 Triangles, 28, 35. 
 
 Trigonometric leveling, 56. 
 Trigonometry, 34. 
 
 Triple entry, 284. 
 
 Trommels, 433. 
 
 Trough washer, 437. 
 
 Troy weight, 1. 
 
 True north from Polaris, 46. 
 
 with the Burt solar, 48. 
 
 T-square method of plumbing shafts, 
 73. 
 
 Tunnels, 265. 
 
 . Flow of water through, 147. 
 Turbines, 158. 
 
 Turnouts, 413. 
 
 Two-phase alternators, 226. 
 
636 
 
 INDEX. 
 
 Undercasts, 393. 
 
 Undercutting, 283. 
 
 Underground prospecting, 239. 
 supports, 267. 
 surveying, 56, 67. 
 
 Underhand stoping, 316. 
 
 Undersetting of props, 267, 276, 277. 
 Undershot wheels, 157. 
 
 Unequal splits of air. 374. 
 
 Units of electricity, 203. 
 
 of resistance, Electric, 203. 
 of work, Electric, 205. 
 
 Unloading coal, Cost of, 447. 
 
 Useful horsepower during winding, 
 To find, 397. 
 
 Use of compass, 39. 
 
 Vacuum Pumps, 164. 
 
 V acuum system of ventilation, 386. 
 Value of a'fuel, 166. 
 
 Values of explosives, 335. 
 
 Valves, Pump, 162. 
 
 Safety, 185. 
 
 Variation, Barometric, 339. 
 
 of ventilation elements, 372. 
 
 To turn off the, 39. 
 
 Velocity, Coefficient of (water), 135. 
 of air-current, 364. 
 of a water jet, 135. 
 of water through pump passages, 
 ±60. 
 
 Ventilating currents, Measurements 
 of, 364. 
 
 Ventilation elements, 363. 
 
 elements, Variation of, 372. 
 formulas, 370. 
 in shaft sinking, 263. 
 methods and appliances, 381. 
 of Mines, 337. 
 
 of rise and dip workings, 382. 
 Ventilators, Mechanical, 385. 
 Verniers, Reading, 39. 
 
 Versed sine, 35. 
 
 Vertical angle, 51. 
 
 curves on slopes, 416. 
 distances, 50, 87, 537. 
 distances, stadia table, 87. 
 
 V Notch, 137. 
 
 Volatile combustible matter of coal, 
 171, 174. 
 
 Volume and absolute pressure, 345. 
 and absolute temperature, Rela- 
 tion of, 345. 
 
 Atomic, 341. 
 
 Measures of, 5. 
 
 Specific, 341. 
 
 Waddle Ventilator, 387. 
 
 Wall plates, 270. 
 
 Wardle’s formula for shaft pillars, 2S5. 
 Washers, Ore and coal, 436. 
 
 Weight of, 116. 
 
 Waste gates, 146. 
 ways, 155. 
 
 Water, Battery, 430. 
 buckets, 164. 
 
 column corresponding to any 
 mercury column, 340. 
 
 Water elevators, 164. 
 
 flow through orifices, 135. 
 gauge. 186, 365. 
 
 Gauging, 136. 
 level, 186. 
 memoranda, 163. 
 power, 156. 
 
 raised by single-acting lift pump. 
 162. 
 
 weight of, 107, 130. 
 
 Waterproof push button, 234. 
 Waterwheels, 157. 
 
 See Wheels. 
 
 Watt-hour, 205. 
 
 Weather-proof wire, 212. 
 
 Wedge, 93. 
 
 Weight, Apothecaries’, 1. 
 
 Atomic, 344. 
 
 Avoirdupois, 2. 
 
 Metric, 2. 
 
 Troy, 1. 
 
 Weight of atmosphere, 337. 
 
 boltheads, nuts, and washers, 116. 
 
 bolts, 116. 
 
 castings, 111. 
 
 chains, 130. 
 
 coal, 170. 
 
 flat wrought iron, 115. 
 gases, 344. 
 iron pipe, 113. 
 materials, 102. 
 
 plates of steel, wrought iron, etc., 
 112 . 
 
 standard steel buckets, 446. 
 water, 107, 130. 
 wire ropes. 119. 
 
 ■wrought iron, 114. 
 
 Weight on pillars at different depths, 
 Table of, 287. 
 
 Weights and measures, 7. 
 
 Weir discharge, 140. 
 
 Weirs, 138. 
 
 Well drilling. Cost of. 242. 
 
 West Virginia method of mining. 
 296. 
 
 West Vulcan telephone system, 234. 
 Wheel and axle, 92. 
 
 Wheels, Water, 157. 
 
 Breast, 157. 
 
 Impulse, 158. 
 
 Overshot, 158. 
 
 Turbines, 158. 
 
 Undershot, 157. 
 
 Whitedamp, 349. 
 
 Whiting system of hoisting, 395. 
 Williams’ 'method of mining, 312. 
 Wing dams, 156. 
 
 Winslow, Arthur, 81. 
 
 Winzes, 316. 
 
 Wire gauge 122k, 207. 
 
 nails, 114. 
 
 Wire rope, 118. 
 
 calculations, 124. 
 fastenings. 126. 
 flattened-strand rope, 122. 
 glossary, Z22h. 
 
 Inspection of, 124b. 
 
 Life of, 1221. 
 
INDEX 
 
 637 
 
 Wire rope lubrication, 124b. 
 
 size of drums for, 1221. 
 splicing, 127. 
 
 suspension cableways, 122b. 
 tramways, 122c. 
 transmission of power by, 122d. 
 
 Wear of, 124b. 
 
 Weight and strength of, 119. 
 Wiring, Bell, 230. 
 
 Electric, 207. 
 
 Wolf lamp, 358. 
 
 Wood, Crushing load of, 105. 
 
 Wood fuel, Value of, 166. 
 
 screws, 113. 
 
 Wooden beams, 102. 
 
 Constants for, 103. 
 dams, 154. 
 
 Working load for wire rope, 120. 
 
 Methods of, 277. 
 
 Wrought iron. Fl^t, 115. 
 
 Wyoming region, Costs of mining m 
 325. 
 
 Zinc, Weight of, 111. 
 
 
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Practical Promotion 
 For Mine Employes 
 
 The use of labor-saving machinery and the State mine-exam- 
 ination laws have shaped conditions so that the only practical 
 way for ambitious mine employes to advance is by securing 
 theoretical technical training for the position to which they 
 desire to attain. The surest, most practical, most thorough, 
 cheapest, and simplest way for mine workers to gain the train- 
 ing necessary to qualify them for the highest positions in the 
 profession is through the Courses offered by the International 
 Correspondence Schools, of Scranton, Pa. 
 
 These Courses have been written by the best mining authorities 
 in the country and edited into Courses of Instruction by the 
 Schools’ own experts. They are taught by a system of cor- 
 respondence instruction that has been tried, proved, and per- 
 fected for 16 years, during which time it has been the means of 
 advancing hundreds upon hundreds of ambitious miners from 
 low-paying positions to places of responsibility and high salary. 
 You need not leave your present work. You can study in your 
 own home, in your spare time, and the cost is insignificant 
 compared with the results achieved. 
 
 The I. C. S. has in its curriculum 208 different Courses of 
 Instruction covering all the leading trades and professions. 
 The Courses especially adapted to the needs of miners are those 
 that will fit them for positions such as the following: 
 
 Mining Engineer 
 Mine Inspector 
 Mine Manager 
 Mine Superintendent 
 Mine Foreman 
 
 Mine Surveyor 
 Mine Fire Boss 
 Metallurgist 
 Assayer 
 
 Electrical Engineer 
 
 Electrician 
 
 Write for special catalog on the position you desire to secure. 
 
 International Correspondence Schools 
 
 Drawer G, SCRANTON, PA. 
 
 1 
 
Better Earnings 
 For Coal Miners 
 
 The I. C. S. Courses in Coal Mining have been remarkably suc- 
 cessful in enabling coal miners to secure the knowledge of their 
 professions in both the anthracite, and bituminous fields that 
 has enabled them to pass State Examinations and secure higher 
 positions and better wages. Sixteen of the Mine Inspectors of 
 the State of Pennsylvania are I. C. S. students. Concerning 
 these Schools, John Mitchell says: “It gives me pleasure to 
 
 recommend the International Correspondence Schools, ; of 
 Scranton, Pa. During my connection with the miners’ organiza- 
 tion I have formed the acquaintance of a vast number of men 
 and boys who have taken advantage of the opportunities offered 
 by these Schools, to obtain, with little cost to themselves, 
 a practical education that fitted them to hold lucrative and 
 responsible positions.” . 
 
 The Complete Coal Mining Course contains about 4,500 pages 
 of instruction matter and 2,500 illustrations. The following 
 subjects are taught: 
 
 Arithmetic 
 Formulas 
 Logarithms 
 
 Geometry and Trigonometry 
 Geometrical Drawing 
 Mine Surveying 
 Properties of Gases 
 Mine Gases 
 Mine Ventilation 
 Geology of Coal 
 Examination of Coal Properties 
 Rock Boring 
 Rock Drilling 
 Explosives and Blasting 
 Drifts, Slopes, and Shafts 
 Methods of Working 
 Mechanics 
 Fuels 
 
 Steam and Steam Boilers 
 Steam Engines 
 Air Compression 
 Elements of Electricity and 
 
 Magnetism 
 
 Direct-Current Dynamos and 
 
 Motors 
 
 We also have a Course for Mine Foremen and one for Fire 
 Bosses. Special catalogs on any of these Courses will be sent 
 on request. 
 
 International Correspondence Schools 
 
 Drawer G» SCRANTON* PA. 
 
 Alternating - Current Machin- 
 ery 
 
 Operation of Dynamo-Electric 
 Machinery 
 
 Transmission, Lighting, and 
 Signaling 
 
 Coal-Cutting Machinery 
 
 Trackwork 
 
 Timbering 
 
 Timber Trees 
 
 Hoisting 
 
 Haulage 
 
 Hydromechanics 
 Mine Drainage 
 Mine Pumps 
 
 Surface Arrangements at Bitu- 
 minous Mines 
 
 Surface Arrangements at An- 
 thracite Mines 
 Preparation of Anthracite 
 Coal Washing 
 Principles of Coking 
 Coking in the Beehive Oven 
 By-Product Coking 
 First Aid to the Injured 
 
 2 
 
Advancement 
 For Metal Miners 
 
 As in coal mining, the man who wishes to rise in the metal- 
 mining profession must secure a knowledge of the theory and 
 science of modern metal mining. A large amount of money 
 was expended in making the I. C. S. Metal Mining Course, m 
 many ways the most valuable source of knowledge on mining 
 that is in existence, and it contains the results of years of study 
 by our best mining experts. The Metal Mining Course com- 
 prises about 2,800 pages of instruction and 1,238 illustrations. 
 The following subjects are included: 
 
 Arithmetic 
 
 Formulas 
 
 Geometry and Trigonometry 
 Logarithms (Optional) 
 Geometrical Drawing 
 Mechanics 
 Fuels 
 
 Mine Surveying 
 Metal-Mine Surveying 
 Mineral-Land Surveying 
 Steam and Steam Boilers 
 Steam Engines 
 Air Compression 
 Elements of Electricity and 
 Magnetism 
 
 Direct-Current Dynamos and 
 Motors 
 
 Alternating-Current Machinery 
 Operation of Dynamo-Electric 
 Machinery 
 
 Transmission, Lighting, and 
 Signaling 
 Hydromechanics 
 Mine Pumps 
 Hoisting 
 
 Haulage 
 Mine Drainage 
 Rock Boring 
 Rock Drilling 
 Explosives and Blasting 
 Timber Trees 
 Gas and Oil Engines 
 Management of Stationary Gas 
 Engines 
 
 Ore Dressing and Milling 
 Theoretical Chemistry 
 Practical Chemistry 
 Elementary Inorganic Chemis- 
 ' try 
 
 Blowpiping 
 
 Mineralogy 
 
 Geology 
 
 Prospecting 
 
 Assaying 
 
 Placer Mining 
 
 Surface Arrangements at Ore 
 Mines 
 
 Preliminary Operations 
 Ore Mining 
 
 Supporting Excavations 
 
 For those who do not wish to take up the complete Course, 
 a shorter Course — the Metal Prospectors’ Course — may be 
 obtained. 
 
 Write for special circulars on either Course. 
 
 International Correspondence Schools 
 
 Drawer G, SCRANTON, PA. 
 
 3 
 
Salary -Raising 
 Training 
 
 Mechanical Engineering 
 Mechanical Drawing 
 Stationary Engineering 
 
 Those who are held back in their promotion by lack of a 
 theoretical knowledge of the above subjects can find no means 
 equal to I. C. S. Courses to overcome the obstacles to. their 
 advancement. These Courses are thorough and concise in 
 every respect and are remarkably easy to learn, remember, and 
 * apply to the daily work of the ambitious mechanic.. They have 
 been the means of advancing hundreds of ambitious workers 
 and there is no reason why they should not be of the same 
 benefit to you. Professor F. R. Hutton, Dean of the School of 
 Engineering, Columbia University, and Secretary of the Ameri- 
 can Society of Mechanical Engineers, recommends the I. C.. S. 
 as follows. “I take pleasure in falling in with your suggestion 
 which asks for a line of comment on the work of the Correspond- 
 ence Schools, at Scranton, the result of a recent inspection. 
 ... I regard therefore the I. C. S. as meeting a widespread 
 need and as serving a very useful purpose. It seems to me.it 
 opens a door of hope and of opportunity which would otherwise 
 be almost of necessity closed to a large number of persons.” 
 
 Our Complete Mechanical Engineering Course embodies two 
 divisions, each of which treats of its own particular branches. 
 The Mechanical Division contains 2,543 pages and 1,241 illus- 
 trations. Starting with the simple subject of Arithmetic it 
 includes Drawing, Electricity, Mechanics, Steam and Steam 
 Engines, Applied Mechanics, etc. 
 
 The Shop Practice Division contains 3,166 pages and 2,142 
 illustrations, and covers all those subjects that are ordinarily 
 met with in connection with the work in machine shops, 
 foundries, and allied industries. Write for special catalog on 
 the Course that interests you. 
 
 International Correspondence Schools 
 
 Drawer G* SCRANTON* PA. 
 
 4 
 

 
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