LIBRARY UNIVEJtiSITY OF CAUFOSHIM PAVIS Digitized by tine Internet Arciiive in 2007 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/civilenginpocketOOtrauricli THE CIVIL ENGINEER'S POCKET-BOOK BY JOHN C. TRAUTWINE CIVIL ENGINEER EEVISED BY JOHN C. TRAUTWINE, Je. AND JOHN C. TRAUTWINE, 3d. CIVIL ENGINEERS EIGHTEENTH EDITION, EIGHTY-SIXTH THOUSAND LIBRARY UkiM^lTY OF CALIFORNIA iJAVIS - NEW YORK M JOHN WILEY & SONS London: CHAPMAN & HALL, Limited 1906 UNIVERSITY OF CALIFORNIA LIBRARY BRANCH OF THE COLLEGE OF AGRICULTURE CONTENTS. XXVll Air. Atmospbere. page Properties 320 Pressure in Diving Bells, etc. . . 321 Dew Point 321 Heat and Cold, Records of 321 Wind. Velocity and Pressure. Table. 321 Rain and Snow. Precipitation. Average 322 Effect of Climate on — 322 and Stream-flow 323 Maximum Rates of — 323 Weight of Snow 323 Rain Gauge . 324 Precipitation, Details of — in U.S., Table 325 IVater. Composition, Properties 326 Ice 326 Effects of Water on Metals, etc. 327 Tides 328 Evaporation, Filtration, lieakag-e 329 MECHAXICS, FORCE IN RIOII> BODIES. Definitions 330 Matter; Body 330 Dynamics. Motion, Velocity 331 Force 332 Action and Reaction 333 Acceleration 334 Mass 336 Impulse 337 Density; Inertia 338 Opposite Forces 339 Work 341 Power 342 Kinetic Energy 343 Momentum 345 Potential Energy 346 Impact 347 Gravity, Falling Bodies 348 Descent on Inclined Planes . . . 349 Pendulums 350 Center of Oscillation 351 Center of Percussion 351 Angular Velocity 351 Moment of Inertia 351 Radius of Gyration 352 Centrifugal Force «... 354 Statics.^ PAQB Forces 358 Line of Action 359 Stress 359 Moments 360 Classification of Forces 361 Composition and Resolution of Forces 362 Force Parallelogram 364 Force Triangle 367 Rectangular Components 369 Inclined Plane 369 Stress Components 371 Applied and Imparted Forces. . 372 Resolution, etc., by means of Co-ordinates 372 Force Polygon 374 ^on-concurrentCoplanar Forces 375 Equilibrium of Moments 376 Cord Polygon 377 Concurrent Non - coplanar Forces 380 Non-concurrent Non-coplanar Forces 381 Parallel Forces 382 Coplanar 382 Non-coplanar 385 Center of Gravity 386 Stable, Unstable, and Indif- ferent Equilibrium 387 General Rules 387 Special Rules 391 Line of Pressure. Center of Force or of Pressure 399 Position of Resultant 399 Distribution of Pressure .... 400 "Middle Third" 402 Couples 404 Friction 407 Coefficient 408 Morin's Laws 410 Table of Coefficients 411 Other Experiments 412 Rolling Friction 414 Lubricated Surfaces 415 Friction Rollers 417 Resistance of Trains 417 WorkofOvercomingFriction 418 Natural Slope 419 Friction of Revolving Shaft 419 Levers 419 Stability 422 Work of Overturning 422 On Inclined Planes 424 The Cord 425 Funicular Machine 427 Toggle Joint 427 Pulley 428 Loaded Cord or Chain 428 Arches, Dams, etc. Thrust and Resistance Linec; .... 430 Arches 430 Graphic Method 430 Practical Considerations . . 432 Masonry Dam 433 Graphic Method 435 Practical Considerations . . 436 The Screw 436 XXVIU CONTENTS. PAGE Forces Acting upon Beams and Trusses 437 Conditions of Equilibrium . . 437 End Reactions 439 Moments 440 In Cantilevers 442 In Beams 443 Inclined Beams 445 Curved Beams 446 Shear 446 Influence Diagrams 449 For Moments 449 For Shear 450 Relation between Moment and Shear,. 452 STRENOTH OF MATE- RIAIiS. Oeneral Principles. 454 Stretch, Stress and Strain .... 455 Modulus of Elasticity 456 Limit of Elasticity 458 Yield Point 459 Resilience 460 Suddenly Applied Loads ..... 460 Elastic Ratio 461 Strengths of Sections 462 Fatigue of Materials 465 Transverse Stren^tb Conditions of Equilibrium .... 466 Neutral Axis , 466 Resisting Moment 467 Modulus of Rupture . 468 Moment of Inertia 468 Table 469 Section Modulus 473 Loading. Strength 473 Table 474 Beam of Unit Dimensions. . . . 475 Coefficients, Table 476 Weight of Beam as Load 477 Comparison of Similar Beams. 478 Horizontal Shear 478 Deflections 480 Elastic Limit 482 Elastic Curve 482 Deflection CoeSicient 483 Eccentric Loads 484 Uniform Loads 485 Inclined Beams 485 Cylindrical Beams 485 Maximum Permissible — . . . . 485 Suddenly Applied Loads . . . 486 Uniform Strength 486 Cantilevers. Table 487 Beams. Table 488 Continuous Beams 489 Table 490 Cross-shaped Beam 492 Plates 492 Transverse and Longitudinal Combined 493 PAGE Streng^tli of Pillars. 495 Radius of Gyration 496 Table 496 Remarks 498 Sbearinsr Streng^tb . 499 Torsional Streng^tb. 499 Vj / HTDROSTATICSt. Principles 501 Center of Pressure 501 Air Pressure 502 Horizontal and Vertical Components 503 Pressure in Vessels 503 Opposite Pressures 503 Rules o 504 Transmission of Pressure 506 Center of Pressure 506 Walls to Resist Pressure 508 Thickness at Base 509 Stability 510 Contents 510 Liability to Crush 510 Thickness for Cylinders 511 Iron Pipes 512 Lead Pipes 513 Buoyancy 513 Flotation. Metacenter 514 Draught of Vessels 515 \/ HYBRAUIilCS. Flow of Water tbroug-b Pipes 516 Head of Water 516 Velocity Head 516 Entry Head 516 Friction Head 516 Pressure Head 518 Piezometers 518 Hydraulic Grade Line , . 519 Siphon 520 Velocity Formulae 522 Kutter's Formula 523 Weight of Water in Pipes 525 Areas and Contents of Pipes . . . 526 Total Head Required 527 Table of Velocity and Friction Heads and Discharge 528 Compound Pipe 531 Venturi Meter. Theory 532 Tube 534 Register 535 Ferris-Pitot Meter 535 Curves and Bends ........... 537 CONTENTS. PAGE Flow tbron^b Orifices Theoretical Velocities 539 With Short Tubes 540 Through Thin Partition . . 541 Discharge from One Reservoir to Another 543 Rectangular Openings 544 Time of Emptying Pond .... 545 Miner's Inch 546 Flow over Weirs End Contractions 547 Measurement of Head 648 Formulae 649 Francis 550 Table of Discharges 551 Bazin 552 Values of m .^ 553 Submerged Weirs 554 Velocity of Approach 556 Inclined Weirs 558 Broad-crested Overfall 559 Triangular Notch 559 Trapezoidal Notch 559 Flow in Open Channels Relations of Velocities 560 Steam Gauging 560 Pitot Tube, etc 561 Wheel Meter 662 Abrasion of Channel 563 Theory of Flow 563 Kutter's Formula 564 Coefficient of Roughness. . . . 564 Coeffs of Roughness. Table 565 Coefficient, c, Table 666 To Draw Kutter Diagram. 570 Flow in Sewers 574 Flow to Sewers 575 Flow in Drain-pipes 575 Constriction of Channel 576 Scour 577 Obstructions in Streams 577 Power of Falling Water 578 Water Wheels 678 Hydraulic Ram 578 Power of Running Stream .... 578 CONSTRUCTIONS, ETC. I>redgingr. Cost of Dredging 580 Horse Dredges 581 Weight of Material 581 Foundations. Foundations 582 Borings in Common Soils 582 Unreliable Soils 583 Resistance of Soils 583 PAGE Rip-rap 583 Protection from Scour 583 Timber Cribs 584 Caissons 585 Coffer-dams 586 Earth Banks 586 Crib Coffer-dams 587 Mooring Caissons or Cribs 589 Sinking through Soft Soil 589 Piles 589 Sheet Piles 590 Grillage 590 Pile Drivers 590 Resistance of Piles 592 Penetrability of Soils 593 Driving 593 Screw Piles • 694 Driving by Water Jet 595 Hollow Iron Cylinders 696 Pneumatic Process 696 Timber Caisson 598 Masonry Cylinders 699 Fascines 699 Sand-Piles 699 Stonework. Cost, etc 600 Retaining Walls. General Remarks 603 Theory 606 Surcharged Walls. 609 Wharf Walls 611 Transformation of profile 611 Sliding, etc 612 Stone Bridges. Definitions 613 Depth of Keystone 613 Pressures on Arch-stones 614 Table of Arches 615 Abutments 617 Abutment Piers 619 Inclination of Courses 620 Culverts 622 Wing Walls 624 Foundations 627 Drains 627 Drainage of Roadway 628 Contents of Piers 628 Brick Arches 629 Centers 631 Timber Dams. Primary Requisites 642 Examples 642 Abutments, Sluices, Ground Plan, Cost 645 Measuring Weirs 646 Trembling 648 Thickness of Planking Re- quired 648 XXX CONTENTS. WATER SUPPI.Y, PAGE Consumption, Use and Waste. 649 Waste Restriction ; Water Meters 649 Water for Fire Protection . . . 650 Reservoirs 650 Leakage through — , Mud in — 651 Storage Reservoirs 652 Valve Towers, etc 652 Compensation 653 Distributing Reservoirs . 653 Water Pipes 653 Concretions in — , preven- tion of — 655 Weights of Cast Iron Pipes . . 656 Wrought Iron Pipes 656 Wooden and Other Pipes . . . 657 Costs of Pipes and Laying. . 658 Pipe Joints 660 Pipe Jointer 660 Flexible Joints 661 Special Castings 661 Repairs and Connections. . . 662 Air Valves 662 Air Vessels, Stand-pipes 663 Service Pipes 664 Tapping Machines 664 Anti-bursting Device 665 Valves, Gates 666 Fire Hydrants 668 TEST AND WEIali BORIHTG. Test Boring Tools 670 Artesian Well Drilling 671 ROCK DRI1.I.S. Diamond Drills 675 Percussion Drills 676 Hand Drills 681 Channeling 681 Air Compressors 681 TRACTIOX, ANIItlAI. POWER. On Roads, Canals, etc 683 TRUSSES. Introduction. General Principles 689 Loading, Counterbracing 690 Cross bracing 691 Types of Trusses 691 Camber .• . . 696 Cantilevers 696 Movable Bridges 696 Skew Bridges 697 Roof Trusses 698 Stresses in Trass Mem- bers General Principles 698 Method by Sections 700 Chord Stresses, Moments, Chord Increments 701 PAGE Shear 702 Influence Diagram 702 Dead Load Stresses . , . *. 703 Live Load Stresses 705 Typical Wheel Loads 705 Cooper's 706 Live Load Web Stresses .... 706 Live Load Chord Stresses. . . 709 Wind Loads 710 Impact, etc. . '. 711 Maximum and Minimum Stresses 712 Effect of Curves 712 Counterbracing 713 Stresses in Roof Trusses. 713 Weights and Loads 713 Wind Pressures 714 Graphic Method 715 Timber Roof Trusses 716 Deflections , 718 Redundant Members , . 720 Bridge Details and Con- struction General Principles 720 Floor System and Bearings . . 720 Design 721 Flexible and Rigid Tension Members 721 Compression Members 721 Pin and Riveted Connec- tions 721 Floor Beam Connections. . . . 721 Tension Members, Detail . . . 722 Compression Members, De- tail 722 End Post and Portal Bracing 723 Joints 724 Pin Plates 724 Pins 725 Expansion Bearings 725 Loads, Clearance, etc., for Highway Bridges 726 Camber 726 Examples 726 Weights of Steel Railroad Bridges 731 List of Large Bridges 732 Timber Ti;usses 732 Joints 733 Howe Truss Bridges 736 Examples 738 Metal Roof Trusses 740 Broad Street Station, Phila. . 740 List of Large Arched Roofs. 742 Timber Roof Trusses 742 Transportation and Erection . . 743 Digests of Specifications for Bridges and Buildings. For Steel Railroad and Highway Bridges. General Design 745 Material . 751 Loads , . , 755 CONTENTS. PAGE Stresses and Dimensions 759 Protection 763 Erection 763 For Combination Railroad Bridg-es. General Design 763 Material 763 Loads 764 Stresses and Dimensions 764 Protection 764 For Roofs, Buildings, etc. General Design, Material, etc.. 764 SUSPENSIOX BRIDGES. Data Required 765 Formulas 766 Anchorages 770 RIVETS AHri> RIVETING. Rules and Tables 772 RAIEROADS. Curves. Deianitions 780 Tables, etc 784 Earthwork. Table of Level Cuttings 790 Shrinkage of Embankment .... 799 Cost of Earthwork 800 Tunnels. Construction 812 Trestles. Construction 813 Track. Ballast 815 Ties 815 Tie Plates 816 Rails 817 Spikes 818 Rail Joints 819 Turnouts 824 Equipment. Turntables 845 Water Stations 851 Track Tanks 853 Track Scales, Fences, etc 854 Cost of Mile of Track 855 Rolling: Stock. page Locomotives. Dimensions, Weights, etc... 856 Performance 860 Tonnage Rating 862 Fast Runs 863 Running Expenses 864 Cars 865 Statistics. Earnings, Expenses, etc 867 MATERIAES. Metals. Iron and Steel. Requirements. International Ass'n for Testing Materials. 870 Cast Iron .* 874 Weight 875 Weight of Cast Iron Pipes . . 876 Weight of. Wrought Iron and Steel 877 Roofing Iron 880 Corrugated Iron 881 Wrought Iron Pipes and Fit- tings 882 Screw Threads, Bolts, Nuts and Washers 883 Lock-nut Washers 885 Buckle Plates 885 Bolts. Weight and Strength, Table 886 Wire Gauges 887 Circular Measure 889 Wire, Table 891 Structural Shapes. I Beams 892 Channels 894 Angles and T Shapes 896 Separators for I Beams 900 Z-Bar Columns 901 Phoenix Segment Columns . . 904 Gray Column 905 Strengths of Iron Pillars, Tables 907 Floor Sections 914 Chains 916 Other Metals. Tin and Zinc 916 Copper, Lead, etc 918 Tensile Strengths, Table 920 Compressive Strengths, Table. 921 Stone, etc. Tensile Strengths, Table 922 Compressive Strengths, Table 923 Transverse Strength, Table... 924 XXXll CONTENTS. Mortar, Bricks, etc. page Lime Mortar 925 Bricks 927 Cement 930 Cement Mortar 931 Sand 935 Effects on Metals 936 Efflorescence 936 Silica Cement 937 Rec&mmendations, Am. Soc. C. E 937 Tests 938 Report of Board of U. S. A. Engineer Officers 940 Tests 941 Requirements 942 Concrete 943 Properties 943 Handling 946 Explosives. Nitro-glycerin^ and Dynamite. 948 Blasting Powders 951 Firing 952 Gunpowder 953 Timber. Decay and Preservation 954 Tensile Strength 957 Compressive Strength 958 Transverse Strength 959 Strength as Pillars 963 Biiilding: Materials and Operations. pagb Plastering 968 Slating 969 Shingles 971 Painting 971 Glass and Glazing 973 Sundry Materials. Rope 975 Wire Ropes 976 Paper 978 Blue Prints, etc 979 Price liist and Business Di- rectory. Price List 984 Business Directory 996 Bibliog'rapby. List of Engineering Books. .1008 GLOSSARY 1025 INDEX 1039 PREFACE. XV phia and Reading Railway; Paul L. Wolf el, Chief Engineer, American Bridge Co.; J. Sterling Deans, Chief Engineer, and Moritz G. Lippert, Assistant Engineer, Phoenix Bridge Co. ; Ralph Modjeski, Northern Pacific Railway; D. J. Whittemore, Chief Engineer, and C. F. Loweth, Engineer and Superintendent of Bridges and Buildings, Chicago, Milwaukee and St. Paul Railway. Specifications for Bridges and Buildings, Messrs. C. C. Schnei- der, Vice President, American Bridge Company; J. E. Greiner, Engineer of Bridges and Buildings, Baltimore and Ohio Railroad ; Theodore Cooper; W. K. McFarlin, Chief Engineer, Delaware, Lackawanna and Western Railway; Mason B. Strong, Bridge Engineer, Erie Railroad; F. C. Osborn, President, Osborn En- gineering Co. ; Wm. A. Pratt, Engineer of Bridges, Pennsylvania Railroad ; W. B. Riegner, Engineer of Bridges, Philadelphia and Reading Railway; W. J. Wilgus, Chief Engineer, New York Central Railroad. Locomotives, Baldwin Locomotive Work's; Messrs. Wilson Miller, President, Pittsburgh Locomotive and Car Works ; Theo. N. Ely, Chief of Motive Power, Pennsylvania Railroad; A. E. Mitchell, C. W. Buchholz and A. Mordecai, of the Erie Rail- road; Edwin F. Smith, Wm. Hunter, A. T. Dice and Samuel F. Prince, Jr., of the Philadelphia and Reading Railway; and Thomas Tait, Manager, Canadian Pacific Railway; and Major E. T. D. Myers, of the Richmond, Fredericksburg and Potomac Railroad. Cars, Alhson Manufacturing Co., Harlan & Hollingsworth Co., and Mr. Jos. W. Taylor, Secretary, Master Car Builders' Associa- tion. Railroad Statistics, Mr. Edward A. Moseley, Secretary, Inter- state Commerce Commission. Iron and Steel, Mr. Wm. R. Webster. Cement, Mr. Richard L. Humphrey. Concrete Beams, Mr. Howard A. Carson, Chief Engineer, Bos- ton Transit Commission. Preservation of Timber, Mr. O. Chanute. Building Material, Mr. John T. WilHs. John C. Trautwine, Jr., John C. Trautwine, 3d. Philadelphia, October, 1902, Folios xvi to xxiv inclusive are left blank, to provide for future additions to prefaces. • CONTENTS. MATHEMATICS. Mathematical Symbols 33 Greek Alphabet 34 Aritbinetie. Factors and Multiples 35 Fractions 35 Decimals 37 Ratio and Proportion 38 Progression 39 Permutation, Combination, Al- ligation 40 Percentage, Interest, Annuities 40 Simple Interest . , . . 41 Equation of Payments 42 Compound Interest 42 Annuity, Sinking Fund, De- preciation, etc 43 Equations and Tables . . . 44-46 Duodenal Notation 47 Reciprocals 48-52 Roots and Powers. Square and cube. Tables 54 ,4 Rules 66 Fifth Roots and Powers 67 ' Logarithms 70 Rules 70 Logarithmic Chart and Slide Rule 73 Two-page Table 78 Twelve-page Table 80 Geometry, Mensuration, and Trig-onouietry. liines. Definitions 92 Angles. Definitions 92 Construction 93 Bisection 94 Inscribed 94 Complement and Supplement. 94 In a Parallelogram 95 Minutes ai;id Seconds in Deci- mals of a Degree, Table of — 95 Approximate Measurement of Angles 96 Sine, Tangent, etc 97 Definitions 97 Table 98 Chords. Table of — 143 Surfaces. paob Polygons. Regular — , Tables, etc.. of — 148 Triangles. Definitions. Properties .... 148 Right-angled — 150 Trigonometrical Problems . . 150 Parallelogram 157 Trapezoid. Trapezium 158 Polygons 159 Regular 159 Reduction of Figures. . .159, 160 Circle 161 Radius, Diameter 161 Area, Center, to Find — ... 161 Problems 161, 162 Tables of — . Diameter in Units, Eighths, etc 163 Diameters in Units and Tenths 166 Diameters in Units and Twelfths 172 Arc. Circular. Chord, Length 179 Radius, Rise, and Ordinates. 180 Of Large Radius, to Draw — 181 Tables of — 182-185 Circular Sector, Ring, Zone, and Lune 186 Circular Segment. Area of — ; to Find 186 Area of — ; Table 187 Ellipse. Properties of — 189 Ordinates and Circumference of — ; to Find — 189 Elliptic Arc 189 Tables of Lengths of — . . . 190 Area of; to Find — 190 Construction. Tangents. . . 190 Oval or False — 191 Cyma Recta, Cyma Reversa, Ogee 191 Parabola, Properties of — 192 Parabolic Curve. Length of — 192 Area 192 Parabolic Zone or Frustum . 192 Construction 193 Cycloid 194 Solids. Regular Bodies, Tetiahedron, Hexahedron, etc 194 Guldinus Theorem 194 Paralleiopiped, Properties .... 195 XXV CONTENTS. PAGE Prism 195 Frustum 195 Cylinder. Volume and Surface of — . . 196 Volume. Table of — , in Cu. Ft. andU. S. Gals 197 Wells; Contents of — and Masonry in Walls of — . . . 198 Cylindric Ungula 199 Pyramid and Cone 200 Frustums of . . . .' 201 Prismoid 202 Wedge 203 Sphere. Properties 204 Volume, Surface, etc. Formulas for — 204 Tables of — 205-207 Segment and Zone of — . . . . 208 Spherical Shell 208 Spheroid or Ellipsoid 208 Paraboloid 209 Frustum of — 209 Circular Spindle 209 Circular Ring 209 Specific Gravity. Principles 210 Table 212-215 Weigiits and Measures. U. S., British and Metric — , Units of — 216 Coins; Foreign and U. S. — . . . . 218 Gold and Silver 219 Weights; Troy, Apothecaries' and Avoirdupois — 220 Long Measure 220 Degrees of Longitude. Length. 221 Inches Reduced to Decimals of a Foot. Table 221 Square or Land Measure 222 Cubic or Solid Measure 222 Liquid Measures 223 Dry Measure 223 British Imperial Measures ..... 224 Volumes and Weights of Water 224 Metric Units 225 Systeme Usuel, — Ancien 226 Russian 227 Spanish . 227 Conversion Tables 228 Introduction and Explana- tion 228 List of Tables 229 Fundamental Equivalents . . 230 Abbreviations 230 Equivalents and Numbers in Common Use 231 Metric Prefixes 231 Tables 232 Acres per Mile and per 100 feet. Table 254 PAGE Grades, Tables of — 365-257 Heads and Pressures of Water; Tables of— 258-260 Discharges in . Gals, per Day and Cu. Ft. per Second; Tables 261-265 Time. Definitions, etc 265 Standard Railway — 267 Dialing 268 Board Measure. Table 269 Surveying. Tests of Accuracy, Distribution of Error, etc 274 Chaining 282 Location of Meridian 284 By Circumpolar Stars 284 Definitions 284 By Means of Polaris 285 By Means of Any Star at Equal Altitudes 287 Times of Elongation and Cul- mination of Polaris 288 Azimuths of Polaris, Table. . 289 Polar Distances and Azi- muths of Polaris, Table . . 290 Engineer's Transit 291 Adjustment and Repairs. .. . 294 Vernier 296 Cross-hairs ; to Replace 296 Bubble Glass; to Replace. . . 296 Theodolite 296 Pocket Sextant 297 Compass. Adjustment 298 Magnetic Declination and Variation. Isogenic Chart of U. S 300 Declination 301 Variation 301 Demagnetization 302 Leveling'. Contour Lines 302 Y Level 306 Adjustrhent 3(J7 Forms for Notes 309 Hand Level, Adjustment ... 310 Builder's Plumb Level 311 Clinometer or Slope Inst. ... 311 Leveling by the Barometer or Boiling Point .''12 Table 315 ]«^ATURAI. PHENOMEIirA. Sound. Velocity of 316 Heat. Expansion and Melting Points. Table 317 Thermometer. Conversion of Scales 318 Tables >18, 319 THE AUTHOR DEDICATES THIS BOOK TO THE MEMORY OF HIS FRIEND, THE LATE BENJAMIN H.,LATROBE,Esq., CIVIL ENGINEER. z ZS<^S No pains have been spared to maintain tlie position of this as the foremost Civil Engineer's Pocket-book, not only in the United States, but in the English language. JOHN -WILEY & SONS, Scientific Publishers, 43 East Nineteenth Street, New Tork City. PREFACE TO FIRST EDITION, 1872. SHOULD experts in engineering complain that they do not find anything of interest in this volume, the writer would merely remind them that it was not his intention that they should. The book has been prepared for young members of the profession ; and one of the leading objects has been to elucidate, in plain English, a few important elementary principles which the savants have envel- oped in such a haze of mystery as to render pursuit hopeless to any but a confirmed mathematician. Comparatively few engineers are good mathematicians ; and in the writer's opinion, it is fortunate that such is the case ; for nature rarely combines high mathematical talent, with that practical tact, and observation of outward things, so essential to a successful engineer. There have been, it is true, brilliant exceptions ; but they are very rare. ^ But few even of those who have been tolerable mathe- maticians when young, can, as they advance in years, and become engaged in business, spare the time necessary for retaining such accomplishments. Nearly all the scientific principles which constitute the founda- tion of civil engineering are susceptible of complete and satis- factory explanation to any person who really possesses only so much elementary knowledge of arithmetic and natural philosophy as is supposed to be taught to boys of twelve or fourteen in our public schools."^ * Let two little boys weigh each other on a platform scale. Then when they balance each other on their board see-saw, let them see (and measure for them- ' selves) that the lighter one is farther from the fence-rail on which their board is placed, in the same proportion as the heavier boy outweighs the lighter one. They will then have learned the grand principle of the lever. Then let them measure and see that the light one see-saws farther than the heavy one, in the same proportion ; and they will have acquired the principle of virtual velocities. Explain to them that equality of moments means nothing more than that when V VI PREFACE. The little tliat is beyond this, might safely be intrusted to the savants. Let them work out the results, and give them to the engi- neer in intelligible language. We could afford to take their words for it, because such things are their specialty ; and because we know that they are the best qualified to investigate them. On the same principle we intrust our lives to our physician, or to the captain of the vessel at sea. Medicine and seamanship are their respective specialties. If there is any point in which the writer may hope to meet the approbation of proficients, it is in the accuracy of the tables. The pains taken in this respect have been very great. Most of the tables have been entirely recalculated expressly for this book ; and one of the results has been the detection of a great many errors in those in common use. He trusts that none will be found exceed- ing one, or sometimes two, in the last figure of any table in which great accuracy is required. There are many errors to that amount, they seat themselves at their measured distances on their see-saw, they balance each other. Let them see that the weight of the heavy boy, when multiplied by his distance in feet from the fence-rail amounts to just as much as the weight of the light one when multiplied* by his distance. Explain to them that each of the amounts is in foot-pounds. Tell them that the lightest one, because he see- saws so much faster than the other, will bump against the ground just as hard as the heavy one ; and that this means that their momentums are equal. The boys may then go in to dinner, and probably puzzle their big lout of a brother who has just passed through college with high honors. They will not forget what they have learned, for they learned it as play, withoutany ear-pulling, spanking, or keeping in. Let their bats and balls, their marbles, their swings, &c, once become their philosophical apparatus, and children may be taught {really taught) many of the most important principles of engineering before they can read or write. It is the ignorance of these principles, so easily taught even to children, that constitutes what is popularly called ** The Practical Engineer ; " which, in the great majority of cases, means simply an ignoramus, who blunders along without knowing any other reason for what he does, than ttiat he has seen it done so before. And it is this same ignorance that causes employers to prefer this practical man to one who is conversant with principles. They, themselves, were spanked, kept in, &c, when boys, because they could not master leverage, equality of moments, and virtual velocities, enveloped in x's, p's, Greek letters, square- roots, cube-roots, &c, and they naturally set down any man as a fool who could. They turn up their noses at science, not dreaming that the word means simply, knowing why. And it must be confessed that they are not altogether without reason ; for the savants appear to prepare their books with the express object of preventing purchasers, (they have but few readers,) from learning why. PREFACE. Vll especially where the recalculation was very tedious, and where, consequently, interpolation was resorted to. They are too small to be of practical importance. He knows, however, the almost impos- sibility of avoiding larger errors entirely ; and will be glad to be informed of any that may be detected, except the final ones alluded to, that they may be corrected in case another edition should be called for. Tables which are absolutely reliable, possess an in- trinsic value that is not to be measured by money alone. "With this consideration the volume has been made a trifle larger than would otherwise have been necessary, in order to admit the stereotyped sines and tangents from his book on railroad curves. These have been so thoroughly compared with standards prepared independ- ently of each other, that the writer believes them to be absolutely correct. In order to reduce the volume to pocket-size, smaller type has been used than would otherwise have been desirable. Many abbreviations of common words in frequent use have been introduced, such as abut, cen, diag, hor, vert, pres, &c, instead of abutment, center, diagonal, horizontal, vertical, pressure, &c. They can in no case lead to doubt ; while they appreciably reduce the thickness of the volume. Where prices have been added, they are placed in footnotes. They are intended merely to give an approximate or comparative idea of value ; for constant fluctuations prevent anything farther. The addresses of a few manufacturing establishments have also been inserted in notes, in the belief that they might at times be found convenient. They have been given without the knowledge of the proprietors. The writer is frequently asked to name good elementary books on civil engineering ; but regrets to say that there are very few such in our language. "Civil Engineering," by Prof. Mahan of West Point ; " Roads and Railroads," by the late Prof. Gillespie ; and the " Handbook of Railroad Construction," by Mr. George L. Vose, Civ. Eng. of Boston, are the best. The writer has reason to know that a new edition of the last, now in press, will be far Vlll PREFACE. superior to all predecessors ; and better adapted to the wants of the young engineer than any book that has appeared. Many of Weale's series are excellent. Some few of them are behind the times ; but it is to be hoped that this may be rectified in future editions. Among pocket-books, Haswell, Hamilton's Useful Information, Henck, Molesworth, Nystrom, Weale, &c, abound in valuable matter. The writer does not include Rankine, Moseley, and Weisbach, because, although their books are the productions of master-minds, and exhibit a profundity of knowledge beyond the reach of ordi- nary men, yet their language also is so profound that very few engineers can read them. The writer himself, having long since forgotten the little higher mathematics he once knew, cannot. To him they are but little more than striking instances of how com- pletely the most simple facts may be buried out of sight under heaps of mathematical rubbish. Where the word ' ' ton ' ' is used in this volume, it always means 2240 lbs. There is no table of errata, because no errors are known to exist expept two or three of a single letter in spelling ; and which will probably escape notice. John C. Trautwine. Philadelphia, November 13th, 1871. PREFACE TO NINTH EDITION. TWENTY-SECOND THOUSAND, 1885. QJINCE the appearance of its last edition (the twentieth thousand) ^ in 1883, the *' Pocket-Book " has been thoroughly revised, and many important additions and other alterations have been made. These necessitated considerable change in the places of the former matter, and it was deemed best to turn this necessity to advantage, and to make a thorough re-arrangement, putting all of the articles, as far as possible, in a rational order. The list of new matter and of revisions and extensions is condensed as follows, 1902 : New articles on the steam-hammer pile driver, machine rock drills, air com- pressors, high explosives, cost of earthwork by drag and wheel scrapers and by steam excavators, iron trestles, track tanks, artesian well-boring and standard time, and new tables of railroad curves in metric measure, circumferences and areas of circles, thermometric scales, and fractions with their decimal equivalents. Articles revised and extended, on circular arcs, thermometers, flotation, flow in pipes, waterworks appliances, velocities, &c, of falling bodies, centrifugal force, strength of timber, strength of beams, riveting, riveted girders, trusses, suspension bridges, rail joints, turnouts, turntables, locomotives, cars, railroad statistics and manufactured articles, including columns, beams,' channels, angles and tees. Most of the new matter is in nonpareil, the larger of the two types heretofore used. Boldfaced type has been freely used ; but only for the purpose of guiding the reader rapidly to a desired division of a subject. For emphasis, italics have been employed. Illustrations which were lacking in clearness or neatness have been te-touched and re-lettered, or replaced with new and better cuts. The new matter is very freely illustrated. New rules have been put in the shai^e of formulae, and many of the old rules have been re-cast into the same form. ix X , PREFACE. The addition of new matter, and a number of blank spaces necessarily left in making the re-arrangement, have increased the number of pages about one-fifth. The new index is in stricter alphabetical order than that of former editions, and contains more than twice as many entries, although much repetition has been avoided by the free use of cross- references, without which this part of the work might have been indefinitely extended. The selection of articles of manufacture or merchandise for illus- tratioD, has been guided by no other consideration than their fitness for the purpose, and the courtesy of the parties representing them, in supplying information. The writer gratefully acknowledges the kindness of those who have assisted in furnishing and arranging data. Philadelphia, January, 1885. J. C. T., Jr. PREFACE TO EIGHTEENTH EDITION. (SEVENTIETH THOUSAND, 1902.) IN preparation for its eighteenth edition, The Civil Engineer's Pocket Book, the first edition of which appeared thirty years ago, has undergone a far more extensive revision than at any other time. More than 370 pages of new matter have been added; and the new edition is larger, by about 100 pages, than its recent predecessors. Among the new matter in this edition will be found; Pages 43- 46 Annuities, Depreciation, etc. 70- 72 Logarithms. 73- 77 Logarithmic Chart and SHde Rule. 80- 91 New Table of Logarithms. 228- 253 Conversion Table of Units of Measiu-ement. 300- 301 Isogonic Chart. 532- 535 Venturi Meter. 536 Ferris-Pitot Meter. 546 Miner's Inch. 649 Water Consumption in Cities. 658- 659 Cost of Water Pipe and Laying. 745- 764 Digests of Specifications for Bridges and Buildings. 816 Tie Plates.. 870- 873 Digest of Specification for Iron and Steel. 905- 906 Gray Column. 914 Trough Floor Sections. 983- 995 Price List of Manufactured Articles. 996-1007 Business Directory. 1008-1023 Bibhography. The following articles have been almost or entirely rewritten: New Pages Old Pages 35- 47 Arithmetic. , 33-37 210-211 Specific Gravity .380-381 265-266 Time : 395 282-283 Chains and Chaining 176 284-290 Location of the Meridian 177-179 322-325 Rain and Snow 220-221 358-453 Statics „ 318 f-361, 370-375 xi Xll PREFACE. New Pages Old Pages 466-494 Strength of Beams 478-520, 528-536 499 Shearing Strength 476 499-500 Torsional Strength 476-477 501-503 Opening Remarks on Hydrostatics . ........ 222-224 537-538 Effect of Curves and Bends on Flow in Pipes 255-256 689-744 Trusses . 547-614 856-864 Locomotives 805-810 865-866 Cars 811-813 867-869 Railroad Statistics 814-818 892-899 I Beams, Channels, Angles and T Shapes 521-527 930-942 Cement 673-678 943-947 Concrete 678-682 954-956 Timber Preservation , 425-425 a The articles on arithmetic are considerably extended, notably by the addition of new matter relating to interest, annuities, depreciation, etc., including several tables. The new and greatly enlarged table of five-place logarithms is arranged in a somewhat novel form. In constructing this table, the effort has been to obviate the difficulty, present in all tables where the difference between successive numbers is constant throughout, that the differences between successive logarithms of the lower numbers are relatively very great. In the new table the differences between logarithms are much more nearly con- stant. For convenience in rough calculations, the old table of five-place logarithms, on two facing pages, is retained. The Conversion Tables contain the equivalents of both English and metric units, and of each of these in terms of the other ; but, owing to the extreme ease with which one metric unit may be converted into others of the same system, it has been unnecessary to burden the table with many of the metric units. The tables have been separately calculated by at least two persons, and their results compared and corrected. One of these results has then been used by the compositor in setting the type, and the proofs have been compared with the other. The new article on the location of the meridian is much more complete than its predecessors, and a new table of azimuths of Polaris, corresponding to different hour-angles, has been added. Perhaps the most radical and extensive of all the changes in this edition are those in the articles on Statics, on Beams and on Trusses. These have been almost entirely rewritten and com- pletely modernized. Under Trusses, modern methods of cal- culating the stresses in and the dimensions of the several PREFACE. Xlll members, and modern methods of construction, are explained, and several modern roofs and bridges are described and illus- trated. One of the most notable features in the new article is the digest of prominent modern specifications for bridges for steam and electric railroads and for highways. The articles on the strength of beams are greatly sirnplified and brought into harmony with modern methods of dealing with that subject. In preparing the digests of specifications for iron and steel, use has been made of the specifications recently adopted by the American Section of the International Association for Testing Materials; while those of the American Society of Civil Engineers and of the recent report of a Board of United States Army engineer officers have been similarly used in con- nection with cement. The price list of engineering materials and appliances has been prepared merely as a useful guide in roughly estimating the ap- proximate costs of work, and it is not to be supposed that it can, in any important case, take the place of personal inquiry and correspondence with manufacturers or their agents, nearly 700 of whom are named in the accompanying list of names and addresses of manufacturers, etc. From its first appearance, the Pocket Book has undertaken to give prices of certain manufac- tured articles, and addresses of those from whom they may be obtained; but these, scattered as they were throughout the volume, were necessarily desultory, and limited in their extent and usefulness. It is hoped that the present articles will be found at least an acceptable substitute for them. As in preceding editions, all new work and all revisions have been the subject of our personal attention, and " scissors-and- paste " methods have been scrupulously avoided. Even in using lists of manufactured articles, etc., although their statements have in general been left unchanged, the matter has in most or all cases been rearranged and classified, to suit the requirements of this work. For instance, the " digests " of specifications for Cement, for Steel and Iron, for Railroad and Highway Bridges and for Steel Buildings, are by no means mere quotations from the originals; but, as their name implies, the result of careful digesting of the contents of the specifications selected for the purpose; their several provisions being carefully studied, in nearly all cases re- worded or reduced to figures, and tabulated in form convenient XIV PREFACE. for reference, the whole being arranged in such logical order as to facilitate reference. As in all cases heretofore, every rule or formula and every description of methods, etc., can be readily understood and ap- plied by any one, engineer or layman, understanding the use of common and decimal fractions, of roots and powers, of loga- rithms, and of sines, tangents, etc., of angles. On the other hand, one who is not possessed of this very meager stock of mathemati- , cal knowledge will hardly approach engineering problems, even as an amateur; and we have therefore followed the precedent, established seventeen years ago, of putting rules in the shape of formulas, which have " the great advantage of showing the whole operation at a glance, of making its principle more apparent, and of being much more convenient for reference" (From. Preface to ninth edition, 1885). The new matter is very fully illustrated. As heretofore, all cuts have been engraved expressly for this work. As in preparing for the ninth edition (1885), all the matter of the book has been rearranged. This has necessitated a new paging; and, in making this, the lettering of pages, introduced from time to time as new editions have appeared in the past, has been eliminated. The rearrangement and the addition of so much new matter have of course necessitated the preparation of a new table of contents and a new index. In this, as in all previous editions since the eighth (1883), practically all new matter has been set in nonpareil, the larger of the two types hitherto used, and much of the old matter retained has been reset in the larger type. We take pleasure in acknowledging our indebtedness to many who have kindly assisted us in our work, notably to Messrs. Otis E. Hovey and Wm. M. White, of the American Bridge Co., for painstaking examination of the article on Trusses; to Mr. C. Robert Grimm and Professor E. J. McCaustland for similar as- sistance in connection with.the article on Statics ; to Misses Laura Agnes Whyte and Louise C. Hazen for suggestions respecting mathematics and astronomy ; and to the following gentlemen for valuable information respecting the subjects named : Isogonic Chart, Mr. O. H. Tittmann, Sup't, U. S. Coast and Geodetic Survey. Trusses, Messrs. Wm. A. Pratt, Engineer of Bridges, Pennsyl- vania Railroad ; W. B. Riegner, Engineer of Bridges, Philadel- MATHEMATICS. MATHEMATICAI. STMBOIiS. + PiUs, p'jsitive, add. 1.414+ means 1.414 + other decimals. — Minus, negative, subtract. ± Plus or minus, positive or negative. Thus, ^cfi = ± a. =f Minus or plus. X Multiplied by, times. Thus, x X 7 = x.y =xy;3X4=12. : y Divided by. Thus, a -=- b = a : b = a/b = -r-- : : : Proportion. Thus, a : 6 : : c : d, as a is to &, so is c to d. = Equals, is equal to. > Is greater than. Thus, 6 > 5. < Is less than. Thus, 5 < 6. =f^ Is not equal to. i:^ Is greater or less than. :^ Is not greater than. 5^ Is not less than. ^ Is equal to or greater than. ^ Is equal to or less than, oc Is proportional to, varies with, oo Infinity. J. Is perpendicular to. ^ j Angle. 'N^ Is similar to. II Is parallel to. V V'^Root of. Thus, j/oor j/a'-= square root of a,y''a = 3d or cube root of a, ^4 a == nih root of a. Parenthesis. "| Brackets. ! Quantities enclosed or covered by the symbol are to be I taken together. ■Vinculum, j •.' Since, because. .'. Hence, therefore. ° Degrees. ' Minutes of arc,* feet. " Seconds of arc,* inches. ' " '" etc. Prime, second, third, etc. Distinguishing accents. Thus, a\ a prime ; a", a second, etc. TT Circumference _ 3 14159255+ arc of semicircle, or 180°. Diameter E, Modulus of elasticity. e e, Base of Napierian, natural or hyperbolic logarithms = 2.718281828. g, Acceleration of gravity = approximately 32.2 feet per second per second =■ approximately 9.81 meters per second per second. ♦ Minutes and seconds of time, formerly also denoted by ' and ", are now de- noted by m and s, or by min and sec, respectively. ^ 88 il 34 GREEK ALPHABET. THE ORSEK AI.PHABET. This alphabet is inserted for the benefit of those who have occasion to consult scientific works in which Greek letters are used, and who find it inconvenient to memorize the letters. Greek letters. Name. Approximate equivalent. Commonly used to designate Capital. Small. A a Alpha a Angles, Coefficients. B iS Beta b " « r V Gamma g " " Specific gravity. A 6 Delta d " " Density, Variation. / Base of hyperbolic logarithms = £ e Epsilon e (short) J 2.7182818. '^Eccentricity in conic sections. Z c Zeta z Co-ordinates, Coefficients. « V Eta e (long) " " © e^ Theta th Angles. I I Iota i K K Kappa k A \ Lambda 1 Angles, Coefficients, Latitude. M M Mu m U i. N V Nu n " H i Xi X Co-ordinates. O o Omicron o (short) n IT Pi P Circumference -^ radius.* p P Rho r Radius, Ratio. 2 Phi ph Angles, Coefficients. X X Chi ch * ^ Psi ps Angles. n (a Omega (lon.p;) Angular velocities. * The small letter ir {pi) is universally employed to designate the number of times (= 3.14159265 . . ,) the diameter of a circle is contained in the circum- ference, or the radius in the semi-circumference. In the circular measure of angles, an angle is designated by the number of times the radius of any circle is contained in an arc of the same circle subtending that angle, n then stands for an angle of 180° (= two right angles), because, in any circle, tt X radius = the semi-circumference. The capital letter IT (pi) is used by some mathematical writers to indicate the product obtained by multiplying together the numbers 1, 2, 3, 4, 5 . . . etc., up to any given point. Thus, n4=lX2X3X4 = 24. t The capital letter 2 (sigma) is used to designate a sum. Thus, in a system of parallel forces, if we call each of the forces (irrespective of their amounts) F, then their resultant, which is equal to the (algebraic) sum of the forces, may be written R = 2 F. ARITHMETIC. 35 AEITHMETIO. FACTORS AND MUIiTIPIiES. (1) Factors of any number, n, are numbers whose product is = n. Thus, 17 and 4 are factors of 68 ; so also are 34 and 2 ; also 17, 2, and 2. (2) A prime number, or prime, is a number which has no factors, except itself and 1 ; as 2, 3, 5, 19, 233. (3) A coiiimoii factor, common divisor or common measure, of two or more numbers, is a number which exactly divides each of them. Thus, 3 is a common divisor of 6, 12, and 18. (4) The hig-hest common factor or g-reatest common divisor, of two or more numbers, is called their H. C. F. or their O. C I>. Thus, 6 is the H. C. F. of 6, 12, and 18. (5) To find the H. C. F. of two or more numbers ; find the prime factors of each, and multiply together those factors which are common to all, taking each factor only once. Thus, required the H. C. F. of 78, 126, and 234. 78 = 2 X 3 X 13 126 = 2X3X3X7 234 = 2X3X3X13 and H. C. F. = 2 X 3 = 6. (6) To find the H. C F. of two large numbers ; divide the greater by the less ; then the less by the remainder, A ; A by the second remainder, B ; B by the third remainder, C ; and so on until there is no remainder. The last divisor is the Ho C. F. Thus, required the H. C. F. of 575 and 782. 575)782(1 575 A 207)575(2 414 C 46)161(3 138 D 23)46(2 H. C. F. = D = 23. 46 (7) A common multiple of two or more numbers is a number which is exactly divisible by each of them. (8) The least common multiple of two or more numbers is called their I*. €. M. (9) To find the L.. C. M. of two or more numbers ; find the prime factors of each. Multiply the factors together, taking each as many times as it is con- tained in that number in which it is oftenest repeated. Thus, required the L. C. M. of 7, 30, and 48. 7 = 7 30 = 2 X 3 X 5 48 = 2X2X2X2X3 L. C. M. =7X2X2X2X2X3X5 = 1680. (10) To find the li. C. M. of two large numbers; find the H. C. F., as above ; and, by means of it, find the other factors. Then find the product of the factors, as before. Thus, required the L. C. M. of 575 and 782. As above, H. C. F. = 23 ; ^ = 25 ; and —■ = 34. Hence, 575 = 23 X 25 782 = 23 X 34 and L. C. Mo = 23 X 25 X 34 = 19,550. FRACTIONS. (1) A common denominator of two or more fractions is a common multiple of their denominators. (2) The least common denominator, or Li. C. D., of two or more fractions is the L. C. M. of their denominators. 36 ARITHMETIC. (3) To reduce to a common denominator. Let N = the new numerator of any fraction n = its old numerator d = its old denominator C = the common denominator Then .^ C N = n - d Thus, J, 1^, |-. C = L. C. M. of denominators = 24. 24 3 _ ^4 ^ 3X6 _ 18 . 5 _ 5X4 _ 20 . 7 _ 7X3 ^ 21 4 "" 24 ~ 4 X 6 ~" 24' 6 "" 6 X 4 ~ 24' 8 "" 8 X 3 24* If none of the denominators have a common factor, then C=the product of all the n denominators, - = the product, P, of all the other denominators, and N = P n. Thus, |, 1., |-. C = 84 2 _ 2X4X7 _ 56 . 1 _ 1X3X7 _ 2 i . 5 _ 5X3X4 _ go 3 84 8 4 ' 4 84 8 4 ' 7 84 8 4' (4) Addition and (Subtraction. If necessary, reduce the fractions to a common denominator, the lower the better. Add or subtract the numerators. Thus, JL-i-i— 2_i.3.,l._4._i.3.,A=2.7.,2.0._j47:_i 1.1. 2"^2 2 ^'4^4 4 ^'4'^9 36^36 36 ■^36» 3,7_6,7_13_i5 4 + 8- - ¥ + ¥ - "8" - ^ 8-- 3_1 _2_1.3 o_27_20__7.7_3_7 6_1 4 4~'4~2'4 9~3 6' 36 36'8 4"~8 8~"8" (5) Multiplication. Multiply together the numerators, also the denomi- nators, cancelling where possible. Thus, 4 "" Te" ' t ^ t ^ t ^ T^ q4vi2_25v'^— 125_r20. 2v3_2. 3y X I3 - -y- X -3 - -^1 &2"T' 3" X 5" - -5-1 Inf-LofAofl — -^viv-^v-I — 1- _ 01 2- ot -3 01 5- - ^ X 2 X 3 X 3- - g- . (6) Division. Invert the divisor and multiply. Thus, l^l_lv2_2_i. 3.^i_3.v4_3_,. 2 • 2~2^T~2"~-^' 4- 4~4^1~T~'' 3 . 5_3v9 — 27_iJ7 . Ivl — 1- 3vl 3_. 3.v'5v2__5 4 . i2_2 5 . 5_25v/3_5v3_15_ol. y--l-3--y--v-3---y-X-5--yXY--y--2y, 7- — 1 "s T~ ~ 3" ~ ~T~ X X — T ^ 5 ^ I- = 5 X 4 _ .5 "7 ^T- (7) A fraction is said to be in its lowest terms, or to be simplified^ when its numerator and denominator have no common factor. Thus, -|-|- simplified = -I-. (8) To reduce to lonrest terms. Divide numerator and denominator by their H. C. F. Thus, required the lowest terms of —. 85 H. C. F. of 34 and 85 = 17 ; and f| = f f "^ ]l = \. 85 85 -T- 17 5 ARITHMETIC. 37 ]>£€IMAIiS. (9) Multiplication. The product has as many decimal places as the factors combined. Thus, Factors : 100 X 3 X 3.5 X 0.004 X 465.21 = 1953.882000 Number of decimal places: + 0+1+ 3+ 2= 6 (10) Division. The number of decimal places in the quotient = those in the dividend minus those in the divisor. Thus 5.125 = 1.25; = 200; 5 _ 5^ _ . 3 _ 3.00 ^ Q 7^5 . 0-42 _ 0.4200 4.1 ^ '4~4~'"^'4~'4~' ' 0.0021 "" 0.0021 '' When the divisor is a fraction or a mixed number, we may multiply both divisor and dividend by the least power of 10 which will make the divisor a whole number. Thus, 2.679454 26,794.54 =-432.17. 0.0062 62 (11) To reduce a common fraction to decimal form ; divid« the numerator by the denominator. Thus, ^-^ = 0.8 ; 1-|- = -f- = 1.6. Table 1. Decimal equivalents of common fractions. 8th8 16th8 32ds 64th8 8ths leths 32ds 61ths 1 .015625 33 .515625 1 2 3 .03125 .046875 17 34 35 .53125 .546875 1 2 4 5 .0625 .078125 9 18 36 37 .5625 .578125 3 6 7 .09375 .109375 19 38 39 .59375 .609375 1 2 4 8 9 .125 .140625 5 10 20 40 41 .625 .640625 5 10 11 • .15625 .171875 21 42 43 ' .65625 .671875 3 6 12 13 .1875 .203125 11 22 44 45 .6875 .703125 7 14 15 .21875 .234375 23 46 47 .71875 .734375 2 4 8 16 17 .25 .265625 6 12 24 48 49 .75 .765625 9 18 19 .28125 .296875 25 50 51 .78125 .796875 5 10 20 21 .3125 .328125 13 26 52 53 .8125 .828125 11 22 23 .34375 .359375 27 54 55 , .84375 .859375 3 6 12 24 25 .375 .390625 7 14 28 56 57 .875 .890625 13 26 27 .40625 .421875 29 58 59 .90625 .921875 7 14 28 29 .4375 .453125 15 30 60 61 .9375 .953125 15 30 31 .46875 .484375 31 62 63 .96875 .984375 4 8 16 32 .5 8 16 32 64 1. (12) To reduce a dec imal fractioi 1 to commo n form 1. Supply the denominator (1), and red uce the resulting fraction to its 1( iwest ter ns. Thus : 0.25 25 1 7.1 .3 890625 57 0.25 -= = = : 0.75 = - == - ; 0.890625 = 1 .uo luO 4 ' 1 )0 4 1 OOOOOO "" ' 64' 38 ARITHMETIC. (13) Recntrring, circulating-, or repeating* decimals are those in which certain digits, or series of digits, recur indefinitely. Thus, i =^ 0.3333...., and so on ; -^-^ = 1.428571428571 , and so on. Recurring decimals may be in- dicated thus : 0.3, 1.428571 ; or thus : 0.*3, l.*428571. RATIO AXD PROPORTION. (1) Ratio. The ratio of two quantities, as A and B, is expressed by their quotient, - or — . Thus, the ratio of 10 to 5 is = = 2 ; the ratio of 5 to 10 B 5 10 -1- - 0.5. Thus, A2 B2 (2) Duplicate ratio is the ratio of the squares of numbers, is the duplicate ratio of A and B. (3) Proportion is equality of ratios. Thus, -LQ- = -2-2. ^ _1^2^6^6 ^ 2. In the figure, which represents segments, A, B, C, and D, between parallel lines ; A:B::C:D,or| = 5 (4) The first'and fourth terijos, A and T>, are called the extremes, and the second and third, B and C, are called the means. The first term, A or C, of each ratio, is called the antecedent, and the second term, B or D, Is called the consequent. I) is called the fourth proportional of A, B, and C. (5) In a proportion, A : B = C : D, we have : Product of extremes = product of means. A D = B C. A B C "" D* A~C' D^C' B~A' ^ ... A + B C+DA + B Composition. — - — = — j^ — ; • — :^ — Alternation. ^ = f^> Inversion. -D D Division. A— B C— D A— B Composition and division. We have, also : m B ~ B ~ T) ~ nB ~ tTIj* ' ' B A + B A — B m A nB '' C — D D C -hP C-D* m C ^ n b'' = C = m, then A : m - (6) If, in the proportion, A : B = C : D, we have B w : B, or — = 7- or m 2 = A D, or m = t/a D. 'ml) (7) In such cases, m is called the mean proportional between A and D, and D is called the third proportional of A and m,. A continued proportion is a series of equal ratios, as A:B = C:D = E:F, etc. ■■ In continued proportion, = ^' ^" B = D = F'^*"-=^ A -1- C + E + etc. B + D + F -f etc. A ~ B ' = ^ = |etc, = R A B ^ C " D' A' C . A" ' b' ~ D' ' B'' ^, A A' A'' C C C If (8) Let A, B, and C be any three numbers. Then A_AB AAC ^ -- ^ . -, and g - ^ . g. etc. * 0.*3, l.*428571, etc., standing for 0.3 1.428571428571...., etc. ARITHMETIC. 39 (9) Reciprocal or inverse proportion. Two quantities are said to be reciprocally or inversely proportional, when the ratio - of two values, A and B, of the one, is = the reciprocal, — , of the ratio of the two corresponding ralues of the other. Thus, let A = a velocity of 2 miles per hour, and B = 3 miles per hour. Then the hours required per mile are respectively, A' = — — -l-' A ^ andB' = i = l. HereA:B = B':A',or^ = |'„or| = | = i = l-^.A;. (10) If two variable numbers, A and B, are reciprocally proportional, so that A' : ly = B" : A", the product, A' A'', of any two values of one of the numbers is equal to the product, B' B" of the two corresponding values of the other. (11) The application of proportion to practical problems is sometimes called the rule of three. Thus : sing^le rule of threes If 3 men lay 10,000 bricks in a certain time, how many could 6 men lay in the same time? As 3 men are to 6 men, so are 10,000 bricks to 20,000 bricks; or, 10,000 bricks X -| = 20,000 bricks. If 3 men require 10 hours to lay a certain number of bricks, how many hours would 6 men require to lay the same number? As 6 men are to 3 men, so are 10 hours to 5 hours ; or, 10 hours X ^ = 5 hours. (12) l>ouble rule of three. If 3 men can lay 4,000 bricks in 2 days, how many men can lay 12,000 bricks in 3 days? Here 4,000 bricks requires men 2 days, or 6 man-days, and 12,000 bricks will require 6 X -jT^r^?. = 6 X 3 = 18 man-days ; and, as the work is to be done in 3 days, -i^ = 6 men will be required. PROORESI^ilON. (1) Arithmetical Progression. A series of numbers is said to be in arithmetical progression when each number differs from the preceding one by the same amount. Thus, —2, —1, 0, 1, 2, 3, 4, etc., where difference = 1 ; or 4, 3, 2, 1, 0, —1, —2, etc., where difference = —1 ; or — i, —2, 0, 2, 4, 6, 8, 10, where difference = 2 ; or \%, V%, 1, %, 3^, ^, 0, —%, — 3^, etc., where difference = —%. (2) In any such series the numbers are called terms. Let a be the first term, I the last term, d the common difference, n the number of terms, and s the sum of the terms. Then l = a -{- {n — 1) d Required I Given a d n I ads s a d 71 a d I s I = — ^ d ± \/2 d s + (a — ^ d)i 5 = -i- w [2 a + (w — 1) tZ] a=^d±\/{l-\-^d)^ — 2ds ads d — 2 a ± y{2 a — d)^ + S d * 2d W7. „ 2l + d ± V{2l + dr~-8ds n a I s n = ^7-3 • 2 d (3) Geometrical Progression. A series of numbers is said to be in geometrical progression when each number stands to the preceding one in the same ratio. Thus: -i, -i, 1, 3, 9, 27, 81, etc., where ratio = 3; or 48, 24, 12, 6, 3, li, 1^, |-, etc., where ratio = -J-; or |-^, lii, 3-|, 6|, 13^, 27, etc., where 40 ARITHMETIC. (4) Let a be the first term, I the last term, r the constant ratio, n the numbel of terms, and s the sum of the terms. Then ; [Hired Given I a r n I a r s I r 71 s 1 = a + (r — 1)4 r ,. «— 1 -1 n-\. — s — -n-l yi - I W-l -/ n~l a n I y I -h a PERMUTATION, Etc. (1) Permutation shows in how many positions any number of things can be arranged in a row. To do this, multiply together all the numbers used in counting the things. Thus, in how many positions in a row can 9 things be placed ? Here, 1X2X3X4X5X6X7X8X9 = 362880 positions. Ans. (2) Combination shows how many combinations of .a few things can be made out of a greater number of things. To do this, first set down that number which indicates the greater number of things ; and after it a series of numbers, diminishing by 1, until there are in all as many as the number of the few things that are to form each combination. Then beginning under the last one, set down said number of few things; and going backward, set down another series, also diminishing by 1, until arriving under the first of the upper numbers. Multiply together all the upper numbers to form one product ; and all the lower ones to form another. Divide the upper product by the lower one. Ex. How many combinations of 4 figures each, can be made from the 9 figures 1, 2, 3, 4, 5, 6, 7, 8, 9, or from 9 any things? 9X8X7X6 3024 ,^^ , . ,. . (3) Allig^ation shows the value of a mixture of different ingredients, when the quantity and value of each of these last is known. Ex. What is the value of a pound of a mixture of 20 lbs of sugar worth 15 cti per ft) ; with 30 S)s worth 25 cts per ft)? ft)S. cts. cts. 20 X 15 = 300 ^^ ^ 1050 „, , 30X25 = 750 Therefore, -^ = 21 cts. Ans. 50 ft)s. 1050 cts. PERCENTAGE, INTEREST, ANNUITIES. Pereentag-e. (1) Ratio is often expressed by means of the word " per," Thus, we speak of a grade of 105.6 feet per mile, i. e., per 5280 feet. When the two numbers in the ratio refer to quantities of the same kind and denomination, the ratio is often expressed as a percentage {'per hundredage). Thus, a grade of 105.6 feet per mile, * Equations involving powers and roots are conveniently solved by means of logarithms. ARITHMETIC. 41 or per 5280 feet, is equivalent to a grade of 0.02 foot per foot,* or 2 feet per 100 feet, or simply (since both dimensions are in feet) 2 per 100, or 2 per " cent." (2) Oiie-tiftieth, or 1 per 50, is plainly equal to two hundredths, or 2 per hun' dred, or 2 per cent. Similarly, H^'lo per cent., % = 3 X 25 per cent. = 75 per cent., etc. Hence, to reduce a ratio to the form of percentage, divide 100 times* the first terra by the second. Thus, in a concrete of 1 part cement to 2 of sand and 5 of broken stone, there are 8 parts in all, and we have, by weight— f Cement = 4- = 0.125 = 12.5 per cent, of the whole. Sand ^-2 =0.250== 25.0 " " Stone = f = 0.625 = 62.5 " " (3) Percentage is of very wide application in money matters, payment for service in such matters being often based upon the amount of money involved. Thus, a purchasing or selling agent may be paid a brokerage or commission which forms a certain percentage of the money value of the goods bought or sold ; the premium paid for insurance is a percentage upon the value of the goods insured; etc. Interest. (4) Interest is hire or rental paid for the loan of money. The sum loaned is called the principal, and the number of cents paid annually for the loan of each dollar, or of dollars per hundred dollars, is called the rate of interest. The rate is always stated as a percentage. (5) If the interest is paid to the lender as it accrues, the money is said to be at simple interest; but if the interest is periodically added to the princi- pal, so that it also earns interest, the money is said to be at compound interest, and the interest is said to be compounded. Simple Interest. (6) At the end of a year, the interest on the principal, P, at the rate, r, is =■ P r, and the amount. A, or sum of principal and interest, is A = P + P r = P (1 -i- r). (7) At the end of a number, w, of years, the interest is = V r n (see right- hand side of Fig. 1), and A = P + P r 71 = P (1 + r n). Thus, let P = $865.32, r = 3 per cent., or 0.03, n = l year, 3 months and 10 days = 1 year and 100 days = ^^^-^ years = 1.274 years. Then A = P (1 -\- rn) = $865.32 X (1 + 0.03 X 1.274) = $865.32 X 1.03822 = $898.39. (8) For the present ^(vorth, principal, or capitalization, P, of the amount, A, we have 1 -\- rn Thus, for the sum, P, which, in 1 year, 3 months, 10 days, at 3 per cent* 89839 simple interest, will amount to $898.39, we have P = -^-^ — r-^r^ = $865.32. (9) In commercial business, interest is commonly calculated approxi- mately by taking the year as consisting of 12 months of 30 days each. Then, at 6 per cent., the interest for 2 months, or 60 days, = 1 per cent.; 1 inonth, or 30 days, = 3^ per cent.; 6 days = 0.1 per cent. Thus, required the interest on $1264.35 for 5 months, 28 days, at 5 per cent. *K fraction, as -i-, ■^, etc., or its decimal equivalent, as 0.125, 0.3125, etc., is compared with wm/y or one; but in percentage the first term of the ratio is compared with one hundred units of the second terra. Mistakes often occur through neglect of this distinction. Thus, 0.06 (six per cent, or six per hundred) is sometimes mis-read six one-hundredths of one per cent, or six oue-hun- dredths per cent, fFor proportions by volume, see pp 935 and 943. 42 ARITHMETIC. Principal .S1264.35 Interest, 2 mos, 1 per cent 12.64 2mos, 1 " 12.64 " 1 mo, ^ " 6.32 " 20 days, ^ " 4.21 " 6 days, 0.1 " 1.26 " 2 days, ^ " 0.42 Interest at 6 per cent ^37.49 Deduct one-sixth 6.25 Interest at 5 per cent ^31.24 Equation of Payments. (10) A owes B $1200; of wliicli $400 are to be paid in 3 months; $500 in 4 months ; and $300 in 6 months ; all bearing interest until paid ; but it has been agreed to pay all at once. Now, at what time must this payment be made so that neither party shall lose any interest? $ months. 400 X 3 - 1200 . ,. 5000 ,,^ 500 X 4 = 2000 Average time = —- = 4% months. Ans. 300 X 6 = 1800 1200 5000 Compound Interest. (11) Interest is usually compounded annually, semi-annually, or quarterly. If it is compounded annually, then (see left side of Fig. 1) at the end of 1 vear A = P (1 + r) " " 2 years A - P (1 -f (1 + r) = P (1 + r)2 " " n years A = P (1 + r)^; and A P = (1 + r'r P = (1 + r)« = A (1 + r)- (12) If the interest is compounded q times per year, we have (13) The principal, P, is sometimes called the present ivortli or present value of the amount, A. Thus, in the following table, $1.00 is the present worth of $2,191 due in 20 years at 4 per cent, compound interest, etc, etc. ^ ^ "T I>(l + r)n ^ ^ ?> .^ 1 Of. I ^ ^ '' ••* '^ f^ 1^^ '8 1 ^ S A t - ? i C) \ 4 J O 7 Years Fig. 1. ARITHMETIC. 43 Table 3. Compound Interest. Amount of $1 at Compound Interest. 3 3K 4 43^ 5 5K 6 6>^ Years. per per per per per per per per ceut. cent. cent. cent cent. cent. cent. cent. 1 1.030 1.035 1.040 1.045 1.050 1.055 1.060 1.065 2 1.061 1.071 1.082 1.092 1.103 1.113 1.124 1.134 3 1.093 1.109 1.125 1.141 1.158 1.174 1.191 1.208 4 1.126 1.148 1.170 1.193 1.216 1.239 1.262 1.286 5 1.159 1.188 1.217 1.246 1.276 1.307 1.338 1.370 6 1.194 1.229 1.265 1.302 1.340 1.379 1.419 1.459 7 1.230 1.272 1.316 1.361 1.407 1.455 1.504 1.554 8 1.267 1.317 1.369 1.422 1.477 1.535 1.594 1.655 9 1.305 1.363 1.423 1.486 1.551 1.619 1.689 1.763 10 1.344 1.411 1.480 1.553 1.629 1.708 1.791 1.877 11 1.384 1.460 1.539 1.623 1.710 1.802 1.898 1.999 12 1.426 1.511 1.601 1.696 1.796 1.901 2.012 2.129 13 1.469 1.564 1.665 1.772 1.886 2.006 2.133 2.267 14 1.513 1.619 1.732 1.852 1.980 2.116 2.261 2.415 15 1.558 1.675 1.801 1.935 2.079 2.232 2.397 2.572 16 1.605 1.734 1.873 2.022 2.183 2.355 2.540 2.739 17 1.653 1.795 1.948 2.113 2.292 2.485 2.693 2.917 18 1.702 1.858 2.026 2.208 2.407 2.621 2.854 3.107 19 1.754 1.923 2.107 2.308 2.527 2.766 3.026 8.309 20 1.806 1.990 2.191 2.412 2.653 2.918 3.207 3.524 21 1.860 2.059 2.279 2.520 2.786 3.078 3.400 3.753 22 1.916 2.1.32 2.370 2.634 2.925 3.248 3.604 3.997 23 1.974 2.206 2.485 2.752 3.072 3.426 3.820 4.256 24 2.033 2.283 2.563 2.876 3.225 3.615 4.049 4.533 25 2.094 2.363 2.666 3.005 3.386 3.813 4.292 4.828 26 2.157 2.446 2.772 3.141 3.556 4.023 4.549 5.141 27 2.221 2.532 2.883 3.282 3.733 4.244 4.822 5.476 28 2.288 2.620 2.999 3.430 3.920 4.478 5.112 5.832 29 2.357 2.712 3.119 ■ 3.584 4.116 4.724 5.418 6.211 30 2.427 2.807 3.243 3.745 4.322 4.984 5.743 6.614 31 2.500 2.905 3.373 3.914 4.538 5.258 6.088 7.044 32 2.575 3.007 3.508 4.090 4.765 5.547 6.453 7.502 33 2.652 3.112 3.648 4.274 5.003 5.852 6.841 7.990 34 2.732 3.221 3.794 4.466 5.253 6.174 7.251 8.509 35 2.814 3.334 3.946 4.667 5.516 6.514 7.686 9.062 36 2.898 3.450 4.104 4.877 5.792 6.872 8.147 9.651 37 2.985 3.571 4.268 5.097 6.081 7.250 8.636 10.279 38 3.075 3.696 4.439 5.326 6.385 7.649 9.154 10.947 39 3.167 3.825 4.616 5.566 6.705 8.069 9.704 11.658 40 3.262 3.959 4.801 5.816 7.040 8.513 10.286 12.416 Compound interest on M dollars, at any rate r for n years = interest on %\ at same rate, r, and for n years. : M X compound Annuity, Sinking Fund, Amortization, Depreciation. (14) Under "Interest" we deal with cases where a certain sum or "prin- cipal," P, paid once for all, is allowed to accumulate either simple or compound interest ; but in many cases equal periodical payments or appropriations, called annuities, are allowed to accumulate, each earning its own interest, usually compound. 44 ARITHMETIC. (15) Thus, a sum of money is set aside annually to accumulate compound interest and thus form a sinking- fund, in order to extinguish a debt. In this way, the cost of engineering works is frequently paid virtually in instal- ments. This process is called amortization. (16) In estimating the operating expenses of engineering works, an allowance is made for depreciation. In calculating this allowance, we estimate or assume the life-time, n, of the plant, and find that annuity, p, which, at an assumed rate, r, of compound interest, will, in the time n, amount to the cost of the plant, and thus provide a fund by means of which the plant may be replaced when worn out or superseded. (17) The present wortb, present value, or capitalization, W, Fig. 2, of an annuity,/), for a given number, 7i, of years, is that sum which, if now placed at compound interest at the assumed rate, ?% will, at the end of that time, reach the same amount, A, as will be reached by that annuity. 4 5 6 t 8 9 n Years Fig. 1. 012S4JG789n Years Fig. 2. (18) Equations for Compound Interest and Annuities. (See Figs. 1 and 2.) P = principal ; r = rate of interest ; n = number of years ; A = amount ; p = annuity ; W = present worth. The interest is supposed to be compounded, and the annuities to be set aside, at the end of each year. Compound Interest. (1) The amount. A, of $l,at the end of w years, see (11), isA = (l +r)**. (2) Since the present worth of (1 + ?•)'*» 3651919 15.4596 6.2058 175 30625 5359375 13.2288 5.5934 240 67600 13824000 15.4919 6.214& 176 30976 5451776 13.2665 5.6041 241 ;,8081 13997521 " 15.5242 6.2231 177 31329 5545233 13.3041 5.6147 242 58564 14172488 15.5563 6.2317 178 31684 5639752 13.3417 5.6252 243 59049 14348907 15.5885 6.2403 179 32041 5735339 13.3791 5.6357 244 59536 14526784 15.6205 6.2488 180 32400 5832000 13.4164 5.6462 245 60025 14706125 15.6525 6.257a 181 32761 5929741 13.4536 5.6567 246 60516 14886936 15.6844 6.2658 182 33124 6028568 13.4907 5.6671 247 61009 15069223 15.7162 «.2743 183 33489 6128487 13.5277 5.6774 248 61504 15252992 15.7480 6.2828 184 33856 6229504 13.5647 5.6877 249 62001 15438249 15.7797 6.2912 185 34225 6331625 13.6015 6.6980 250 62500 15625000 15.8114 6.29^6 SQUARES, CUBES, AND BOOTS. 67 TABIi£ of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1000 — (Continued.) No. Square. Cube. Sq. Rt. C. Rt. No. Square. Cube. Sq. Rt. C. Rt. 251 63001 15813251 15.8430 6.3080 316 99856 31554496 17.7764 6.8113 252 63504 16003008 15.8745 6.3164 317 100489 31855013 17.8045 6.8185 25a 64009 16194277 15.9060 6.3247 318 101124 32157432 17.8326 6.8256 254 64516 16387064 15.9374 6.3330 319 101761 324617.59 17.8606 6.8328 255 65025 16581375 15.9687 6.3413 320 102400 32768000 17.8885 6.839& 256 65536 16777216 16. 6.3496 321 103041 33076161 17.9165 6.847a 257 66049 16974593 16.0312 6.3579 322 103684 33386248 17.9444 6.8541 258 66564 17173512 16.0624 6.3661 323 104329 33698267 17.9722 6.8612: 259 67081 17373979 16.0935 6.3743 324 104976 34012224 18. 6.868S 260 67600 17576000 16.1245 6.3825 325 105625 34328125 18.0278 6.875a 261 68121 17779581 16.1555 6.3907 326 106276 34645976 18.0555 6.8824 262 68644 17984728 16.1864 6.3988 327 106929 34965783 18.0831 6.8894 263 69169 18191447 16.2173 6.4070 328 107584 35287552 18.1108 6.8964 264 69696 18399744 16.2481 6.4151 329 108241 35611289 18.1384 6.9034 265 70225 18609625 16.2788 6.4232 330 108900 35937000 18.1659 6.9104 266 70756 18821096 16.3095 6.4312 331 109561 36264691 18.19.34 6.9174 267 71289 i:j034163 16.3401 6.4393 332 110224 36594368 18.2209 6.9244 268 71824 19248832 16.3707 6.4473 333 110889 36926037 18.2483 6.9313 269 72361 i;)4o5109 16.4012 6.4553 334 111556 37259704 18.2757 6.9382 270 72900 19683000 16.4317 6.4633 335 112225 37595375 18.3030 6.9451 271 73441 19902511 16.4621 6.4713 336* 11-2896 37933056 18.3303 6.9521 272 73JW4 20123648 16.4924 6.4792 337 113569 38272753 18.3576 6.958» 273 74529 20346417 16.5227 6.4872 338 114244 38614472 18.3848 6.9658 274 75076 20570824 16.5529 6.4951 339 114921 38958219 18.41':0 6.972T 275 75625 20796875 16.5831 6.5030 340 115600 39304000 18.4391 6.979& 276 76176 21024576 16.6132 6.5108 341 116281 .39651821 18.4662 6.9864 277 76729 21253933 16.6433 6.5187 342 116964 40001688 18.4932 6.»93S 278 77284 21484952 16.6733 6.5265 343 117649 40353607 18.5203 7. 279 77841 217176:^9 16.7033 6.5343 344 118336 40707584 18.5472 7.0068 280 78400 21952000 16.7332 6.5421 345 119025 41063625 18.5742 7.0136 281 78961 22188041 16.7631 6.5499 346 119716 41421736 18.6011 7.020$ 282 79524 22425768 16.7929 6.5577 347 120409 41781923 1S.6279 7.0271 283 80089 22665187 16.8226 6.5654 348 121104 42144192 18.6548 7.0338 284 80656 22906304 16.8523 6.5731 349 121801 42508549 18.6815 7.0406 285 81225 23149125 16.8819 fi.5808 350 122500 42875000 18.7083 7.047a 286 81796 23393656 16.9115 6.5885 351 123201 43243551 18.7350 7.0540 287 82369 23639903 16.9411 6.5962 352 123904 43614208 18.7617 7.0607 288 829U 23887872 16.9706 6.6039 353 124609 43986977 18.7883 7.0674 289 83521 24137569 17. 6.6115 354 125316 44361864 18.8149 7.0740 290 84100 24389000 17.0294 6.6191 355 126025 44738875 18.8414 7.0807 291 84681 24642171 17.0587 6.6267 356 126736 45118016 18.8680 7.087a 292 85264 24897088 17.0880 6.6343 357 127449 45499293 18.8944 7.0940 293 85849 25153757 17.1172 6.6419 358 128164 45882712 18.9209 7.1006 294 86436 25412184 17.1464 6.3494 359 128881 46268279 18.9473 7.1072 295 87025 25672375 17.1756 6.6569 360 129600 18.9737 7.1138 2% 87616 25934336 17.2047 6.6644 .361 130321 47045881 19. 7.1204 297 88209 26198073 17.2337 6.6719 362 131044 47437928 19.0263 7.126^ 298 88804 26463592 17.2627 6.6794 363 131769 47832147 19.0526 7.1335 299 89401 26730899 17.2916 6.6869 364 132496 48228544 '19.0788 7.1400 300 90000 27000000 17.3205 6.6943 365 133225 48627125 19.1050 7.1466 801 90601 27270901 17.3494 6.7018 366 133956 49027896 19.1311 7.1531 302 91204 27543608 17.3781 6.7092 367 134689 49430863 19.1572 7.1596 303 91809 27818127 17.4069 6.7166 368 135424 49836032 19.1833 7.1661 804 92416 28094464 17.4356 6.7240 369 136161 5024.3409 19.2094 7.1726 805 93025 28372625 17.4642 6.7313 370 136900 50653000 19.2354 7.1791 306 93636 28652616 17.4929 6.7387 371 137641 51064811 19.2614 7.1855 307 94249 28934443 175214 6.7460 372 138384 51478848 19.2873 7.1920 308 94864 29218112 17.5499 6.7533 373 139129 51895117 19.31.32 7.1984 309 95481 29503629 17.5784 6.7606 374 139876 52313624 19.3.391 7.2048 310 96100 29791000 17.6068 6.7679 375 140625 52734375 19.3649 7.2112 311 96721 30080231 17.6.352 6.7752 376 141.376 53157376 19.3907 7.2177 312 97344 30371328 17.6635 6.7824 377 142129 53582633 19.4165 7.2240 313 97969 30664297 17.6918 6.7897 378 142S84 54010152 19.4422 7.2304 314 98596 30959144 17.7200 6.7969 379 143641 54439939 19.4679 7.2368 315 99225 31255875 17.7482 6.fc041 380 144400 54872000 19.4936 7.2432 58 SQUARES, CUBES, AND ROOTS. TABIi£ of Squares, Cubes, Square Roots, and Cube Roots, of JSTumbers from 1 to 1000 — (Continued.) No. Square. Cube. . Sq. Rt. C. Rt. No. Square. Cube. Sq. Rt. C.Rt. 381 145161 55306341 19.5192 7.2495 446 198916 88716536 21.1187 7.6403 382 145924 55742968 19.5448 7.2558 447 199809 89314623 21.1424 7.6460 383 146689 56181887 19.5704 7.2622 448 200704 89915392 21.1660 7.6517 384 147456 56623104 19.5959 7.2685 449 201601 90518849 21.1896 7.6574 385 148225 57066625 19.6214 7.2748 450 202500 91125000 21.2132 7.6631 386 148996 57512456 19.6469 7.2811 451 203401 91733851 21.2368 7.6688 387 149769 57960603 19.6723 7.2874 452 204304 92345408 21.2603 7.6744 388 150544 58411072 19.6977 7.2936 453 205209 92959677 21.2838 7.6801 389 151321 58863869 19.7231 7.2999 454 206116 93576664 21.3073 7.6857 390 152100 59319000 19.7484 7.3061 455 207025 94196375 21.3307 7.6914 391 152881 59776471 19.7737 7.3124 456 207936 94818816 21.3542 7.6970 392 153664 60236288 19.7990 7.3186 457 208849 95443993 21.3776 7.7026 393 154448 60698457 19.8242 7.3248 458 209764 96071912 21.4009 7.7082 394 155236 61162984 19.8494 7.3310 459 210681 96702579 21.4243 7.7138 395 156025 61629875 19.8746 7.3372 460 211600 97336000 21.4476 7.7194 396 156816 62099136 19.8997 7.3434 461 212521 97972181 21.4709 7.7250 397 157609 62570773 19.9249 7.3496 462 213444 98611128 21.4042 7.7306 398 158404 63044792 19.9499 7.3558 463 214369 99252847 21.5174 7.7362 399 159201 63521199 19.9750 7.3619 464 215296 99897344 21.5407 7.7418 400 160000 64000000 20. 7.3681 465 216225 100544625 21.5639 7.7473 401 160801 64481201 20.0250 7.3742 466 217156 101194696 21.5870 7.7529 402 161604 64964808 20.0499 7.3803 467 218089 101847563 21.6102 7.7584 403 162409 65450827 20.0749 7.3864 468 219024 102503232 21.6333 7.7639 404 163216 65939264 20.0998 7.3925 469 219961 103161709 21.6564 7.7695 405 164025 66430125 20.1246 7.3986 470 220900 103823000 21.67^5 7.7750 406 164836 66923416 20.1494 7.4047 471 221841 104487111 21.7025 7.7805 407 165649 67419143 20.1742 7.4108 472 222784 105154048 21.7256 7.7869 408 166464 67917312 20.1990 7.4169 473 223729 105823817 21.7486 7.7915 409 167281 68417929 20.2237 7.4229 474 224676 106496424 21.7715 7.7970 410 168100 68921000 20.2485 7.4290 475 225625 107171875 21.7945 7.8025 411 168921 69426531 20.2731 7.4350 476 226576 107850176 21.8174 7.8079 412 169744 69934528 20.2978 7.4410 477 227529 108531333 21.8403 7.8134 413 170569 70444997 20.3224 7.4470 478 228484 109215352 21.8632 7.8188 414 171396 70957944 20.3470 7.4530 479 229441 109902239 21.8861 7.8243 415 172225 71473375 20.3715 7.4590 480 230400 110592000 21.9089 7.829T 416 173056 71991296 20.3961 7.4650 481 231361 111284641 21.9317 7.8362 417 173889 72511713 20.4206 7.4710 482 232324 111980168 21.9545 7.840« 418 174724 73034632 20.4450 7.4770 483 233289 112678587 21.9773 7.8460 419 175561 73560059 20.4695 7.4829 484 234256 113379904 22. 7.8514 420 176400 74088000 20.4939 7.4889 485 235225 114084125 22.0227 7.8568 421 177241 74618461 20.5ia3 7.4948 486 236196 114791256 22.0454 7.8622 422 178084 75151448 20.5426 7.5007 487 237169 115501303 22.0681 7.8676 423 178929 75686967 20.5670 7.5067 488 238144 116214272 22.0907 7.8730 424 179776 76225024 20.5913 7.5126 489 239121 116930169 22.1133 7.8784 425 180625 76765625 20.6155 7.5185 490 240100 117649000 22.1359 7.8837 426 181476 77308776 20.6398 7.5244 491 241081 118370771 22.1585 7.8891 427 182329 77854483 20.6640 7.5302 492 242064 119095488 22.1811 7.8944 428 183184 78402752 20.6882 7.5361 493 243049 119823157 22.2036 7.8998 429 184041 78953589 20.7123 7.5420 494 244036 120553784 22.2261 7.9051 430 184900 79507000 20.7364 7.5478 495 245025 121287376 22.2486 7.9105 431 185761 80062991 20.7605 7.5537 496 246016 122023935 22.2711 7.9158 432 186824 80621568 20.7846 7.5595 497 247009 122763473 22.2935 7.9211 433 187489 ' 81182737 20.8087 7.5654 498 248004 123505992 22.3159 7.9264 434 188356 81746504 20.8327 7.5712 499 249001 124251499 22.3383 7.9317 435 189225 82312875 20.8567 7.5770 500 250000 125000000 22.3607 7.9370 436 190096 82881856 20.8806 7.5828 501 251001 125751501 22.3830 7.9423 437 190969 83453453 20.9045 7.5S86 502 252004 126506008 22.4054 7.9476 438 191844 84027672 20.9284 7.5944 503 253009 127263527 22.4277 7.9528 439 192721 84604519 20.9523 7.6001 504 254016 128024064 22.4499 7.9581 440 193600 85184000 20.9762 7.6059 505 255025 128787625 22.4722 7.9634 441 194481 85766121 21. 7.6117 506 256036 129554216 22.4944 7.9686 442 195364 86350888 21.0238 7.6174 507 257049 130323843 22.5167 7.9739 443 196249 86938307 21.0476 7.6232 508 258064 131096512 22.5389 7.9791 444 197136 87528384 21.0713 7.6289 509 259081 131872229 22.5610 7.9843 445 198025 88121125 21.0950 7.6346 510 260100 132651000 22.5832 7.9896 59 TABIi£ of Sqnares, Cubes, ISquare Roots, and Cube Roots, ofKi " " ' ^ SQUARES, CUBES, AND ROOTS. nares. Cubes, Square Roots, am Bi^umbers from 1 to 1000 — (Continued.) No. Square. Cube. Sq. Rt. C. Rt. No. Square. Cube. Sq. Rt. C. Bt. 511 261121 133432831 22.6053 7.9948 576 331776 191102976 24. 8.3203 512 262144 134217728 22.6274 8. 577 332929 192100033 24.0208 8.3251 613 263169 135005697 22.'6495 8.0052 578 334084 193100552 24.0416 8.3300 614 264196 135796744 22.6716 8.0104 579 335241 194104539 24.0624 8.3348 515 265225 136590875 22.6936 8.0156 580 336400 195112000 24.0832 «.3396 616 266256 137388096 22.7156 8.0208 581 337561 196122941 24.1039 8.3443 617 267289 138188413 22.7376 8.0260 582 338724 197137368 24.1247 8.3491 518 268324 138991832 22.7596 8.0311 583 339889 198155287 24.1454 8.3539 519 269361 139798359 22.7816 8.0363 584 341056 199176704 24.1661 8.3587 620 270400 140608000 22.8035 8.0415 585 342225 200201625 24.1868 8.3634 521 271441 141420761 22.8254 8.0466 586 343396 2012.30056 24.2074 8.3682 §22 272484 142236648 22.8473 8.0517 587 344569 202262003 24.2281 B.3730 523 273529 143055G67 22.8692 8.0569 588 345744 203297472 24.2487 8.877T 524 274576 143877824 22.8910 8.0620 589 346921 204336469 24.2693 8.382& 525 275625 144703125 22.9129 8.0671 590 348100 205379000 24.2899 8.3872 526 276676 145531576 22.9347 8.0723 591 349281 206425071 24.3105 8.391& 627 277729 146363183 22.9565 8.0774 592 350464 207474688 24.3311 8.3967 528 278784 147197952 22.9783 8.0825 593 851649 208527857 24.3516 8.4014 529 279841 148035889 23. 8.0876 594 352836 209584584 24.3721 8.4061 S30 280900 148877000 23.0217 8.0927 595 354025 210644875 24.3926 8.4108 631 281961 149721291 23.0434 8.0978 596 355216 211708736 24.4131 8.4155 632 283024 150568768 23.0651 8.1028 597 356409 212776173 24.4336 8.4202 533 284089 151419437 23.0868 8.1079 598 357604 213847192 24.4540 8.4249 634 285156 152273304 23.1084 8.1130 599 358801 214921799 24.4745 8.4296 636 286226 153130375 23.1301 8.1180 600 360000 216000000 24.4949 8.4343 536 287296 153990656 23.1517 8.1231 601 361201 217061801 24.5153 8.4390 637 288369 154854153 23.1733 8.1281 602 362404 218167208 24.5357 8.443? 638 289444 155720872 2:j.l948 8.1332 603 363609 219256227 24.5561 8.448i 539 290521 156590819 23.2164 8.1382 604 364816 220348864 24.5764 8.453(1 640 291600 157464000 23.2379 8.1433 605 366025 221446125 24.5967 8.4677 641 292681 158340421 23.2594 8.1483 606 367236 222545016 24.6171 8.4623 542 293764 159220088 23.2809 8.1533 607 368449 223648543 24.6374 8.4670 643 294849 160103007 23.3024 8.1583 608 369664 224755712 24.6577 8.471« 644 295936 160989184 23.3238 8.1633 609 370881 225866529 24.6779 8.47G$ 646 297025 161878625 23.3452 8.1683 610 372100 226981000 24.6982 8.4809 546 298116 162771336 23.3666 8.1733 611 373321 228099131 24.7184 8.4856 547 299209 163667323 23.3880 8.1783 612 374544 229220928 24.7386 8.4002 548 300304 164566592 23.4094 8.1833 613 375769 230346397 24.7588 8.4948 549 301401 165469149 23.4307 8.1882 614 376996 231475544 :a<.7790 8.4994 650 302500 166375000 23.4521 8.1932 615 378225 232608375 24.:992 8.6040 661 303601 167284151 23.4734 8.1982 616 379456 233744896 24.8193 8.6066 652 304704 168196608 23.4947 8.2031 617 380689 234885113 24.8395 8.5132 553 305809 169112377 23.5160 8.2081 618 381924 236029032 24.8596 8.5178 554 306916 170031464 23.5372 8.2130 619 .383161 237176659 24.8797 8.5224 555 308025 170953875 23.5584 8.2180 620 384400 238328000 24.8998 8.5270 656 309136 171879616 23.5797 8.2229 621 385641 239483061 24.9199 8.5316 557 310249 172808693 23.6008 8.2278 622 386884 240641848 24.9399 8.5362 558 311364 173741112 23.6220 8.2327 623 388129 241804367 24.9600 8.5408 559 312481 174676879 23.6432 8.2377 624 389376 242970624 24.9800 8.5453 560 313600 175616000 23.6643 8.2426 625 390625 244140625 25. 8.6499 561 314721 176558481 23.6854 8.2475 626 391876 245314376 25.0200 8.6544 562 315844 177504328 23.7065 8.2524 627 393129 246491883 25.0400 8.5590 563 316969 178453547 23.7276 8.2573 628 394384 247673152 25.0599 8.5635 564 318096 179406144 23.7487 8 2621 629 395641 248858189 25.0799 8.5681 565 319225 180362125 23.7697 8.2670 630 396900 250047000 25.0998 8.5726 566 320356 181321496 23.7908 8.2719 an 398161 251239591 25.1197 8.5772 667 321489 182284263 23.8118 8.2768 632 399424 252435968 25.1396 8.5817 668 322624 1832504:^2 23.8328 8.2816 633 400689 253636137 25.1595 8.5862 669 323761 184220009 23.8537 8.2865 634 401956 2548*0104 25.1794 8.5907 670 324900 185193000 23.8747 8.2913 635 403225 256047875 25.1992 8.5952 571 326041 186169411 23.8956 8.2962 636 404496 2572594.56 25.2190 8.5997 572 327184 187149248 23.9165 8.3010 637 405769 258474853 25.2389 8.6043 573 328329 18813251? 23.9374 8.3059 638 407044 259694072 25.2587 8.6088 674 329476 189119224 23.9583 8.3107 639 408321 260917119 25.2784 8.6132 J75 330625 190109375 23.9792 8.3155 640 409600 262144000 25.2982 8.6171 60 SQUARES, CUBES, AND ROOTS. TABIi£ of Squares, Cnbes, Square Roots, and Cube Root»» of JK^umbers from 1 to 1000 — (Continued.) No. Square. Cube. Sq. Bt. C. Rt. No. Square. Cube. Sq. Rt. C. Rt. 641 410881 263374721 25.3180 8.6222 706 498436 351895816 26.5707 8.9043 642 41216*4 264609288 25.3377 8.6267 707 499849 353393243 26.5895 8.908& 643 413449 265847707 25.3574 8.6312 708 501264 354894912 26.6083 8.912T 644 414736 267089984 25.3772 8.6357 709 502681 356400829 26.6271 8.916* 645 416025 268336125 25.3969 8.6401 710 504100 357911000 26.6458 8.9211 646 417316 269586136 25.4165 8.6446 711 505521 359425431 26.6646 8.925a 647 418609 270840023 25.4362 8.6490 712 506944 360944128 26.6833 8.9295 648 419904 272097792 25.4558 8.6535 713 508369 362467097 26.7021 8.9337 649 421201 273359449 25.4755 8.6579 714 509796 363994344 26.7208 8.9378 660 422500 274625000 25.4951 8.6624 715 511225 365525875 26.7395 8.9420 651 423801 275894451 25.5147 8.6668 716 512656 367061696 26.7582 8.9462 652 425104 277167808 25.5343 8.6713 717 514089 368601813 26.7769 8.950a 653 426409 278445077 25.5539 8.6757 718 515524 370146232 26.7955 8.954S 654 427716 279726264 25.5734 8.6801 719 516961 371694959 26.8142 8.9587 655 429025 281011375 25.5930 8.6845 720 518400 373248000 26.8328 8.9628 656 430336 282300416 25.6125 8.6890 721 519841 374805361 26.8514 8.9670 657 431649 283593393 25.6320 8.6934 722 521284 376367048 26.8701 8.9711 658 432964 284890312 25.6515 8.6978 723 522729 377933067 26.8887 8.975? 659 434281 286191179 25.6710 8,7022 724 524176 379503424 26.9072 8.9794 660 435600 287496000 25.6905 8.7066 725 525625 381078125 26.9258 8.9835 661 436921 288804781 25.7099 8.7110 726 527076 382657176 26.9444 8.9876 662 438244 290117528 25.7294 8.7154 727 528529 384240583 26.9629 8.9918 663 439369 291434247 25.7488 8.7198 728 529984 385828352 26.9815 8.995» 664 440896 292754944 25.7682 8.7241 • 729 531441 387420489 27. 9. 665 442225 294079625 25.7876 8.7285 730 532900 389017000 27.0185 9.0041 666 443556 295408296 25.8070 8.7329 731 534361 390617891 27.0370 9.008* 667 444889 296740963 25.8263 8.7373 732 535824 392223168 27.0555 9.0123 668 446224 298077632 25.8457 8.7416 733 537289 393832837 27.0740 9.0164' 669 447*61 299418309 25.8650 8.7460 734 538756 395446904 27.0924 9.0205 mo 448900 600763000 25.8844 8.7503 735 540225 397065375 27.1109 9.0246 671 450241 302111711 25.9037 8.7547 736 541696 398688256 27.1293 9.0287 672 451584 303464448 25.9230 8.7590 737 543169 400315553 27.1477 9.0328 673 452929 304821217 25.9422 8.7634 738 544644 401947272 27.1662 9.036* 674 454276 306182024 25.9615 8.7677 739 546121 403583419 27.1816 9.0410 676 455625 307546875 25.9808 8.7721 740 547600 405224000 27.2029 9.0450 676 456976 308915776 26. 8.7764 741 549081 406869021 27.2213 9.0491 677 458329 310288733 26.0192 8.7807 742 550564 408518488 27.2397 9.0632 678 459684 311665752 26.0384 8.7850 743 552049 410172407 27.2580 9.057? 679. 461041 313046839 26.0576 8.7893 744 553536 411830784 27.2764 9.0613 680 462400 314432000 26.0768 8.7937 745 555025 413493625 27.2947 9.0654 681 463761 315821241 26.0960 8.7980 746 556516 415160936 27.3130 9.0694 682 465124 317214568 26.1151 8.8023 747 558009 416832723 27.3313 9.073* 683 466489 318611987 26.1343 8.8066 748 559504 418508992 27.3496 9.0775 684 467856 320013504 26.1534 8.8109 749 ^1001 562500 420189749 27.3679 9.0816 685 469225 321419125 26.1725 8.8152 750 421875000 27.3861 9.0866 686 470596 322828856 26.1916 8.8194 751 564001 423564751 27.4044 9.0896 687 471969 324242703 26.2107 8.8237 752 565504 425259008 27.4226 9.0937 688 473344 325660672 26.2298 8.8280 753 567009 426957777 27.4408 9.097T 689 474721 327082769 26.2488 8.8323 754 568516 428661064 27.4591 9.1017 690 476100 328509000 26.2679 8.8366 755 570025 430368875 27.4773 9.1057 891 477481 329939371 26.2869 8.8408 756 571536 432081216 27.4955 9.1098 692 478864 331-573888 26.3059 8.8451 757 573049 433798093 27.5136 9.1138 693 480249 332812557 26.3249 8.8493 758 574564 435519512 27.5318 9.1178 694 481636 334255384 26.3439 8.8536 759 576081 437245479 27.5500 9.1218 695 483025 S35702375 26.3629 8.8578 760 577600 438976000 27.5681 9.1268 696 484416 337153536 26.3818 8.8621 761 579121 440711081 27.5862 9.1298 697 485809 338608878 26.4008 8.8663 762 580644 442450728 27.6043 9.1338 698 487204 340068392 26.4197 8.8706 763 582169 444194947 27.6225 9.1378 699 488601 341532099 26.4386 8.8748 764 583696 445943744 27.6405 9.1418 700 490000 343000000 26.4575 8.8790 765 585225 447697125 27.6586 9.1458 701 491401 344472101 26.4761 8.8833 766 586756 449455096 27.6767 9.1498 702 492804 345948408 26.4953 8.8875 767 588289 451217663 27.6948 9.1537 703 494209 347428927 26.5141 8.8917 768 589824 452984832 27.7128 9.157T t04 495616 348913664 26.5330 8.8959 769 591361 454756609 27.7308 9.161T t05 497025 350402625 26.5518 8.9001 770 592900 456533000 27.7489 9.1657 SQUARES, CUBES, AND BOOTS. 61 TABL.E of Squares, Cubes, Square Roots, and €nbe Roots, of J^umbers from 1 to 1000 — (Continued.) Square. Sq. Ht. Square. Sq. Rt. 594441 595984 597529 599076 600625 602176 603729 605284 606841 608400 617796 619369 620944 622521 624100 627264 628849 630436 632025 633616 635209 636804 638401 640000 641601 643204 644809 646416 648025 649636 651249 652864 654481 656100 657721 659344 660969 662596 664225 669124 670761 672400 674041 675684 677329 678976 680625 682276 683929 685584 687241 690561 692224 693889 695556 697225 458314011 460099648 461889917 463684824 465484375 467288576 469097433 470910952 472729139 474552000 476379541 478211768 480048687 481890304 483736625 485587656 487443403 489303872 491169069 493039000 494913671 496793088 498677257 500566184 502459875 504358336 506261573 508169592 512000000 513922401 515849608 517781627 519718464 521660125 525557943 527514112 529475129 531441000 533411731 535387328 537367797 539353144 54134:3375 543338496 545338513 547343432 549353259 551368000 553387661 555412248 557441767 559476224 561515625 563559976 565609283 567663552 569722789 571787000 573856191 575930368 578009537 580093704 582182875 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 28.0357 28.0535 28.0713 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 28.4956 28.5182 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 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 9.27ie 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 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 9.4016 9.4053 9.4091 9.4129 9.4166 700569 702244 703921 705600 707281 708964 710649 712336 714025 715716 717409 719104 720801 722500 724201 725904 727609 729316 731025 732736 734449 736164 737881 739600 741321 743044 744769 746496 748225 749956 751689 753424 755161 756900 758641 760384 7621 29 763876 765625 767376 584277056 58637S253 588480472 770884 772641 774400 776161 777924 779689 781456 783225 784996 786769 788544 790321 792100 793881 795664 797449 799236 801025 802816 804609 806404 808201 810000 599077107 601211584 603351125 605495736 607645423 609800192 611960049 614125000 616295051 618470208 620650477 622835864 625026375 627222016 629422793 631628712 633839779 636056000 638277381 640503928 642735647 644972544 647214625 649461896 651714363 653972032 656234909 658503000 660776311 663054848 6653.38617 667627624 672221376 674526133 676836152 6791514.39 681472000 683797841 686128968 688465387 690807104 693154125 695506456 697864103 700227072 702595369 704969000 707347971 709732288 712121957 714516984 716917375 719323136 721734273 724150792 726572699 729000000 28.9137 28.9310 28.9482 28.9655 28.9828 29.0172 29.0345 29.0517 29.0689 S9.0861 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 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 29.9500 29.9666 29.9833 30. 62 SQUARES, CUBES, AND ROOTS. TABIiE of Squares, Cnbes, Sqnare Roots, and Cube Roots, of 3f^uiiiii>ers from 1 to 1000 — (Continued.) No. Square. Cube. Sq. Rt. cut. No. Square. Cube. Sq. Rt. C.Rt. 901 811801 731432701 30.0167 9.6585 951 904401 860085351 30.8383 9.8339 902 813604 73387080*: 30.0333 9.6620 952 906304 862801408 30.8545 9.8374 903 815409 736314327 30.0500 9.6656 953 908209 865523177 30.8707 9.8408 904 817216 738763264 30.0666 9.6692 954 910116 868250664 30.8869 9.8443 905 819025 741217625 30.0832 9.6727 955 912025 870983875 30.9031 9.847T 906 820836 743677416 30.0998 9.6763 956 913936 873722816 30.9192 9.8511 907 822649 . 746142643 30.1164 9.6799 957 915849 876467493 30.9354 9.8546 908 824464 748613312 30.1330 9.6834 958 917764 879217912 30.9516 9.8580 909 826281 751089429 30.1496 9.6870 959 919681 881974079 30.9677 9.8614 910 828100 753571000 30.1662 9.6905 960 921600 884736000 30.9839 9.8648 911 829921 756058031 30.1828 9.6941 961 923521 887503681 31. 9.8683 912 831744 758550528 30.1993 9.6976 962 925444 890277128 31.0161 9.871T 913 833569 761048497 30.2159 9.7012 963 927369 893056347 31.0322 9.8751 914 835396 763551944 30.2324 9.7047 964 929296 895841344 31.0483 9.8785 915 837225 766060875 30.2490 9.7082 965 931225 898632125 31.0644 9.8819 916 839056 768575296 30.2655 9.7118 966 933156 901428696 31.0805 9.8854 917 840889 771095213 30.2820 9.7153 967 935089 904231063 31.0966 9.8888 918 842724 773620632 30.2985 9.7188 968 937024 907039232 31.1127 9.8922 919 844561 776151559 30.3150 9.7224 969 938961 909853209 31.1288 9.8956 920 846400 778688000 30.3315 9.7259 970 940900 912673000 31.1448 9.889C 921 848241 781229961 30.3480 9.7294 971 942841 915498611 31.1609 9.9024 922 850084 783777448 30.3645 9.7329 972 944784 918330048 31.1769 9.9058 923 851929 786330467 30.3809 9.7364 973 946729 921167317 31.1929 9.909e 924 853776 788889024 30.3974 9.7400 974 948676 924010424 31.2090 9.9126 925 855625 791453125 30.4138 9.7435 975 950625 926859375 31.2250 9.9160 926 857476 794022776 30.4302 9.7470 976 952576 929714176 31.2410 9.9194 927 859329 796597983 30.4467 9.7505 977 954529 932574833 31.2570 9.922T 928 861184 799178752 30.4631 9.7540 978 956484 935441.352 31.2730 9.9261 929 863041 801765089 30.4795 9.7575 979 958441 938313739 31.2890 9.9295 930 864900 804357000 30.4959 9.7610 980 960400 941192000 31.3050 9.9329 931 866761 806954491 30.5123 9.76i5 981 962361 944076141 31.3209 9.9363 932 868624 809557568 30.5287 9.7680 982 964324 946966168 31.3369 9.9396 933 870489 812166237 30.5450 9.7715 983 966289 949862087 31.3528 9.9430 934 872356 814780504 30.5614 9.7750 984 968256 952763904 31.3688 9.9464 935 874225 817400375 30.5778 9.7785 985 970225 955671625 31.3847 9.9497 936 876096 820025856 30.5941 9.7819 986 972196 958585256 31.4006 9.9531 937 877969 822656953 30.6105 9.7854 987 974169 961504803 31.4166 9.9665 938 879844 825293672 30.6268 9.7889 988 976144 964430272 31.4325 9.9598 939 881721 827936019 30.6431 9.7924 989 978121 967361669 31.4484 9.9632 940 S83600 830584000 30.6594 9.7959 990 980100 970299000 31.4643 9.9666 941 885481 833237621 30.6757 9.7993 991 982081 973242271 31.4802 9.9699 942 887364 835896888 30.6920 9.8028 992 984064 976191488 31.4960 9.9733 943 889249 838561807 30.7083 9.8063 993 986049 979146657 31.5119 9.9766 944 891136 841232384 30.7246 9.8097 994 988036 982107784 31.5278 9.9800 945 893025 843908625 30.7409 9.8132 995 990025 985074875 31.5436 9.9833 946 894916 846590536 30.7571 9.8167 996 992016 988047936 31.5595 9.9866 947 896809 849278123 30.7734 9.8201 997 994009 991026973 31.5753 9.9900 948 898704 851971392 30.7896 9.8236 998 996004 994011992 31.5911 9.993S 949 900601 854670349 30.8058 9.8270 999 998001 997002999 31.6070 9.9967 950 902500 857375000 30.8221 9.8305 1000 1000000 1000000000 31.6228 10. To find tlie square or cube of any \¥hole number ending; with cipliers. First, omit all the final ciphers. Take from the table the square or cube (as the case may be) of the rest of the number. To this square add twice as many ciphers as there were final ciphers in the original number. To the cube add three times as many as m the ori^ginal number. Thus, for 905002; 9052 = 819025. Add twice 2 ciphers, obtaining 819025C0OO, For 905003, 9053 =: 741217625. Add 3 times 2 ciphers, obtaining 741217625000000. SQUARE AND CUBE ROOTS. 63 Square Roots aiie numbers not con- tained in tbe column of numbers of tbe table. Such roots may sometimes be taken at once from the table, by merely regarding the columns of powers as being columns of numbers; and those of numbers as being those of roots. Thus, if the gq rt of 25281 is reqd, first find that number in the column of squares ; and opposite to it, in the column of numbers, is its sq rt 159. For the cube rt of 857375, find that number in the column of cubes ; and opposite to it, in the col of numbers, is its cube rt 95. When the exact number is not con- tained in the column of squares, or cubes, as the case may be, we may use instead the number nearest to it, if no great accuracy is reqd. But when a considerable degree of accuracy is necessary, tb« following very correct methods may be used. For tbe square root. This rule applies both to whole numbers, and to those which are partly (not wholly) decimal. First, i> the foregoing manner, take out the tabular number, which is nearest to the given one ; and also itt tabular sq rt. Mult this tabular number by 3 ; to the prod add the given number. Call the sum Ju Then mult the given number by 3 ; to the prod add the tabular number. Call the sum B. Then A : B : : Tabular root : Reqd root. Ex. Let the given number be 946.53. Here we find the nearest tabular number to be 947 ; and Itl tabular sq rt 30.7734. Hence, 947 = tab num 'J r 946.53 = given num. 3 3 2841 946.53 = giTen num. J787,53 = A. and J, 28.39.59 947 =r tab num. I3786.59 = B. A. B. Tab root. Reqd root. Then 3787.53 : 3786.59 : : 30.7734 : 30.7657 -[-■ The root as found by actual mathematical process is also 30.7657 -f-. For the cube root. This rule applies both to whole numbers, and to those which ure partly decimal. First take out ttJe tabular number which is nearest to the given one; and also its tabular cube rt. Mult this tabular number by 2 ; and to the prod add the given number. Call the sum A. Then mult the given number by 2 ; and to the prod add the tabular number. Call the sum B. Then A : B : : Tabular root : Reqd root. Ex. Let the given number be 7368. Here we fiuu tne nearest tabular number (in the column ot cu6e«) to be 6859; and its tabular cube rt 19. Hence, 6859 = tab num. > f 7368 = given num. — — 13718 y and ■{ 147.36 7368 = given num. 6859 = tab num. A. B. Tab Root. Reqd Rt. Then, as 21086 : 21595 : : 19 ; 19.4585 The root as found by correct mathematical process is 19.4588. The engineer rarely require* « SQTJAEE AND CUBE BOOTS. 67 this degree of accuracy ; for his purposes, therefore, this process is greatly preferable to the ordinary laborious one. To find the square root of a number mrbieli is wholly decimal. Very simple, and correct to the third numeral figure inclusive. If the number does not contain at least five figures, counting from the first numeral, and including it, add one or more ciphers to make five. If, after thai, the whole number is not separable into twos, add another cipher to make it bo. Then beginning at the first numeral figure, and including it, assume the number to be a whole one. In the table find the number nearest to this assumed one ; take out its tabular sq rt; move the deci- mal point of this tabular root to the left, half a,a many places as the finally modified decimal number has figures. Ex. vvbat is the sq rt of the decimal .002? Here, in order to have at least five decimal figures, counting from the first numeral (2), and including it. add ciphers thus, .00.20.00.0. But. as it is not now separable into twos, add ano.ther cipher, thus, .00,20,00,00. Then beginning at the first numeral (2), assume this decimal to be the whole number 200000. The nearest to this in the table is 199809; and the sq rt of this is 447. Now. the decimal number as finally modihed, namely, .00.20,00.00, has eight figures ; one half of which is 4; therefore, move the decimal point of the root 447, four places to the left; making it .0447. This is the reqd sq rt of .002, correct to the third numeral 7 included. To find the cube root of a number nrhich is wholly decimal. Very simple, and correct to the third numeral inclusive. If the number does not contain at least five figures, counting from the first numeral, and including It, add one or more ciphers to make five. If. after that, the number is not separable into threes, add one or more ciphers to make it so. Then beginning at the first numeral, and including it. assume the number to be a whole one. In the table find the number nearest to this assumed one, and take out its tabular cub rt. Move the decimal point of this rt to the left, one-third as many places as the finally modified decimal number has figures. Ex. What is the cube rt of the decimal .002 ? Here, in order to have at least five figures, counting from the first numeral (2), and including it, add ciphers thus. .002.000,0. But as it is not now separ able into, threes, add two more ciphers to make it so: thus, .002,000.000. Then beginning witn the first numeral (2), assume the decimal to he the whole number 2000000. The nearest cube to this in the table in the column of cubes, is 2000.^76; and its tabular cube rt as found in the col of numbers, is 126. Now, the decimal number as finally modified, namely, .002 000 000. has nine figures ; one-third of which is 3; therefore, move the decimal point of the root 126, three places to the left, making it . 126. This i» the reqd cube rt of the decimal .002, correct to the third numeral 6 included. Fifth roots and fifth powers. Power. No. or Root. Power. No. or Root. Power. No. or Root. Power. No. or Root. Power. No. or Root. Power. No. or Root. .0000100 .1 .000142 .170 .004219 .335 .077760 .60 .695688 .93 8.11368 1.52 (J00164 .175 .004544 .340 .084460 .61 .733904 .94 8.66171 1.54 .0000110 .102 .000189 .180 .004888 .345 .091613 .62 .773781 .95 9.23896 1.56 .000217 .185 .005252 .350 .099244 .63 .815373 .96 9.8465« 1.58 .0000122 .104 .000248 .190 .005638 .355 .107374 .64 .858734 .97 10.4858 1.60 .000282 .195 .006047 .360 .116029 .65 .903921 .98 11.1577 1.62 .0000134 .106 .000:520 .200 .006478 .365 .125233 .66 .950990 .99 11.8637- 1.64 .0003o2 .205 .006934 .370 .135012 .67 1. 1. 12.6049 1.66 .0000147 .108 000408 .210 .007416 .375 .145393 .68 1.10408 1.02 13.. 3828 1.68 .0000161 .110 .000459 .215 .007924 .3S0 .156403 .69 1.21665 1.04 14.1986 l.TO .0000176 .112 .000515 .220 .008459 .385 .168070 .70 1.3.3823 1.06 15.0537 1.72 .0000193 .114 .000577 .225 .009022 .390 .180423 .71 1.46933 1.08 15.9495 1.74 .0000210 .116 .000644 .230 .009616 .395 .193492 .72 1.61051 1.10 16.8874 1.76 .0000229 .118 .000717 .235 .010240 .400 .207307 .73 1.76234 1.12 17.8690 1.78 .0000249 .120 .000796 .240 .011586 .41 .221901 .74 1.92541 1.14 18.8957 1.80 .0000270 .122 .000883 .245 .013069 .42 .237.305 .75 2.10034 1.16 19.9690 1.82 .0000293 .124 .000977 .250 .014701 .43 .253553 .76 2.28775 1.18 21.0906 1.84 .0000318 .126 .001078 .255 .016492 .44 .270678 .77 2.48832 1.20 22.2620 1.86 .0000344 .128 .001188 .260 .018453 .45 .288717 .78 2.70271 1.22 23.4849 1.88 .0000371 .130 .001307 .265 .020596 .46 .307706 .79 2.93163 1.24 24.7610 1.90 .0000401 .132 .001435 .270 .022935 .47 .327680 .80 3.17580 1.26 260919 1.92 .0000432 .134 .001573 .275 •025480 .48 .348678 .81 3.43597 1.28 27.4795 1.94 .0000465 .136 .001721 .280 .028248 .49 .370740 .82 3.71293 1.30 28.9255 1.96 .0000500 .138 .001880 .285 .031250 .50 .393904 .83 4.00746 1.32 30.4317 1.98 .0000538 .140 .002051 .290 .a34.503 .51 .418212 .84 4.32040 1.34 32.0000 2.00 .0000577 .142 .002234 .295 .0.38020 .52 .443705 .85 4.65259 1.36 36.2051 2.05 .0000619 .144 0024.30 .300 .041820 .53 .470427 .86 5.00490 1.38 40.8410 2.10 .0000663 .146 .0026.'i9 .305 .045917 .54 .498421 .87 5.37824 1.40 45.9401 2.15 .0000710 .148 .002863 .310 .050328 .55 ..527732 .88 5.77353 1.42 51.5363 2.20 .0000754 .150 .003101 .315 .055073 .56 .558406 .89 6.19174 1.44 57.6650 2.25 .0000895 .155 .00.3355 .320 .060169 .57 ..590490 .90 6.6.3383 1.46 64.3634 2.30 .000105 .160 .003626 .325 .06.56.36 .58 .624032 .91 7.10082 1.48 71.6703 2.35 .000122 .165 .003914 .330 .071492 .59 .659082 .92 7.59375 1.50 79.6262 2.40 ROOTS AND POWERS. Fiftb roots and fiftb powers— (Continued). 88.2735 97.6562 107.820 118.814 130.686 143.489 157.276 172.104 188.029 205.111 223.414 243.000 263.936 286.292 310.136 335.544 362.591 391.354 421.419 454.354 488.760 525.219 563.822 604.662 647.835 693.440 741.577 792.352 845.870 902.242 961.580 1024.00 1089.62 1158.56 1230.95 1306.91 1386.58 1470.08 1557.57 1649.16 1745.02 1845.28 1950.10 2059.63 2174.03 2293.45 2418.07 2548.04 2683.54 No. or Root. 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 No. or Root. 2824.75 2971.84 3125.00 3450.25 3802.04 4181.95 4591.65 5032.84 5507.32 6016.92 6563.57 7149.24 7776.00 8445.96 9161.33 9924.37 10737 11603 12523 l;5501 14539 15640 16807 18042 19349 20731 2190 23730 25355 27068 28872 30771 32768 34868 37074 ;59390 41821 44371 47043 49842 52773 55841 59049 62403 65908 69569 73390 7378 81537 85873 90392 95099 100000 110408 121665 133823 146933 161051 176234 192541 210034 228776 248832 270271 2931G3 317580 343597 371293 400746 432040 465259 500490 537824 577353 619174 663383 710082 759375 811368 866171 923896 984658 1048576 1115771 1186367 1260493 1338278 1419857 1505366 1594947 1688742 1 786899 1889568 1996903 2109061 2226203 2348493 2476099 No. or Root. 2609193 2747949 2892547 3043168 3200000 3363232 3533059 3709677 3893289 4084101 4282322 4488166 4701850 4923597 5153632 5392186 5639493 5895793 6161327 6436343 6721093 7015834 7320825 7636332 7962624 8299976 9008978 9381200 9765625 10162.550 18572278 10995116 11431377 11881376 12345437 12823886 13317055 13825281 14348907 14888280 15443752 16015681 16604430 17210368 17833868 18475309 19135075 19813557 No. or Boot. No. or Root. 20511149 21228253 21965275 22722628 23500728 24300000 26393634 28629151 31013642 33554432 36259082 39135393 42191410 45435424 48875980 52521875 5638216: 604661 7( 6478348^ 6934395': 74157715 79235168 84587005 90224199 96158012 102400000 108962013 115856201 123095020 130691232 138657910 147008443 155756538 164916224 174501858 184528125 195010045 205962976 217402615 229345007 241806543) 254803968 268354383 282475249 297184391 312500000 345025251 380204032 418195493 29.0 29.2 29.4 29.6 29.8 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34,0 34.5 35.0 35.5 36 36 5 37.0 37.5 38.0 38.5 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 440 44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5 50.0 459165024 503284375 550731776 601692057 656356768 714924299 777600000 844596301 916132832 992436543 1073741824 1160290625 1252332576 1350125107 1453933568 1564031349 1680700000 1804229351 1934917632 2073071593 2219006624 2373046875 25355253' 2706784157 2887174368 3077056399 3276800000 3486784401 07398432 3939040643 4182119424 4437053125 4704270176 4984209207 (319168 5584059449 5904900000 6240321451 6590815232 6956883693 7339040224 7-73780J 8153726976 8587340257 9039207968 9509900499 Square roots of fifth powers of numbers, l/n^, or % powers of numbers, n%. See table, page 69. The column headed "12 n" facilitates the use of the table in cases where, for instance, the quantity is given in inches^ atid where it is desired to obtain the % power of the same quantity in feet. Thus, suppose we have a i,^ inch pipe, and we require the % power of the diameter in feet. Find i^ (the diameter, in inches) in the column headed " 12 n," opposite which, in the column headed "n," is 0.041666 (the diameter, in feet), and, in column headed " n%," 0.00035 (the % power of the diameter, 0.041666, in feet). Values of n, ending in or in 5, are exact values. All others end in repeat- ing decimals. Thus: n = 0.052083 signifies n = 0.052083333 ROOTS AND POWERS. 69 Square roots of fifth powers of numbers. See page 6 12 11 n n^ 12 u n ni 12 n n nt 12 n n 39 ni Va 0.020833 0.00006 22 1.8333 4.551 84 7.000 129.64 468 9499 3|0.031250 0.00017 23 1.9166 5.086 85 7.083 133.53 480 40 10119 y' 0.041666 0.00035 24 2.0000 5.657 86 7.166 137,50 492 41 10764 0.052083 0.00062 25 2.0833 6.265 87 7.250 141.53 504 42 11432 0.06250010.00098 26 2.1666 6.910 88 7,333 145.63 516 43 12125 0.07291610.00144 27 2.2500 7.594 89 7.416 149.80 528 44 12842 1 0.083333 0.0020 28 2.3333 8.317 90 7.500 154.05 540 45 13584 Ys 0.093750 0.0027 29 2.4166 9.079 91 7.583 158.36 552 46 14351 i 0.104166 0.0035 30 2.5000 9.882 92 7.666 162.75 564 47 15144 0.114583 0.0044 31 2.5833 10.73 93 7.750 167.21 576 48 1596a 0.125000 0.0055 32 2.6666 11.61 94 7.833 171.74 588 49 16807 ^'0.135416 0.0067 33 2.7500 12.54 95 7.916 176.34 600 50 17678 % 0.145833 0.0081 34 2.8333 13.51 96 8.000 181.02 612 51 18575 % 0.156250 0.0097 35 2.9166 14.53 97 8.083 185.77 624 52 19499 2 0.166666 0.0113 36 3.0000 15.59 98 8.166 190.60 636 53 20450 ^ 0.187500 0.0152 37 3.0833 16.69 99 8.250 195.49 648 64 21428 0.208333 0.0198 38 3.1666 17.85 100 8.333 200.47 660 55 22434 0.229166 0.0251 39 3.2500 19.04 102 8.50 210.64 672 56 23468 3^ 0.250000 0.0313 40 3.3333 20.29 105 8.75 226.47 684 57 24529 M 0.270833 0.0382 41 3.4166 21.58 108 9.00 243.00 696 58 25619 H 0.291666 0.0459 42 3.5000 22.92 111 9.25 260.23 708 59 26738 0.312500 0.0546 43 3.5833 24.31 114 9.50 278.17 720 60 27885 4 0.333333 0.0642 44 3.6666 25.74 117 9.75 296.83 732 61 29062 0.354166 0.0746 45 3.7500 27.23 120 10.0 316.23 744 62 30268 1^ 0.375000 0.0861 46 3.8333 28.77 126 10.5 357.25 756 63 31508 % 0.395833 0.0986 47 3.9166 30.36 132 11.0 401.31 768 64 32768 5 0.416666 0.1121 48 4.0000 32.00 138 11.5 448.48 780 65 34068 ^ 0.437500 0.1266 49 4.0833 33.69 144 12.0 498.83 792 66 35388 0.458333 0.1422 50 4.1666 35.44 150 12.5 552.43 804 67 36744 3^0.479166 0.1589 51 4.2500 37.24 156 13.0 609.34 816 68 3813() 6^^ 0.500000 0.1768 52 4.3333 39.09 162 13.5 669.63 828 69 39548 y^ 0.541666 0.2159 53 4.4166 41.00 168 14.0 733.36 840 70 4099& 7 0.583333 0.2599 54 4.5000 42.96 174 14.5 800.61 852 71 42476 3^ 0.625000 0.3088 55 4.5833 44.97 180 15.0 871.42 864 72 43988 8 0.666666 0.3629 56 4.6666 47.05 186 15.5 945.87 876 73 45531 y-i 0.708333 0.4223 57 4.7ri00 49.17 192 16.0 1024.00 888 74 47106 9 0.750000 0.4871 58 4.8333 51.36 198 16.5 1105.9 900 75 48714 K 0.791666 0.5576 59 4.9166 53.60 204 17.0 1191.6 912 76 50354 10^ 0.833333 0.6339 60 5.0000 55.90 210 17.5 1281.1 924 77 52027 y'27 = 3, (log = 0.47 712) and 1^270 = 6.46 . . , (log = 0.81 023). The chart or rule gives all such possible roots, and care must be taken to select the proper one. Most operations exceed the limits of one scale, and facility in using either instrument depends largely upon the ability to pass readily and correctly from one scale to another. This ability is best gained by prac- tice, aided by a thorough grasp of the principles involved. Where several successive operations are to be performed, a sliding runner or marker (furnished with each slide rule) is used, in order to avoid error in shifting the slide. Petailed instructions are usually furnished with the slide rule. (*) A common form of chart has four or more similar squares joined together. See (4). Our figure represents one complete square, with por- tions of adjoining squares. For actual use, both charts and slide rules are, of course, much more finely subdivided than in our figures, which are given merely to illustrate the principles. Carefully engraved charts are published by Mr. John R. Freeman, Providence. R. I. (t) Other forms embodying the same principle are : The " Reaction Scale and General Slide Rule," by W. H. Breithaupt, M. Am. Soc. C. E. ; Sexton's Omnimeter or Circular Slide Rule, bv Thaddeus Norris : The Goodchild Computing Chart; The Thacher Calculating Machine or Cylindrical Slide Rule ; The Cox Computers, designed for special formulas ; and the Pocket Calculator, issued by " The Mechanical Engineer," London. LOGARITHMIC CHART AND SLIDE RULE. 75 (5) Multiplicatioii and division. For example, 2 X 1.5=- ? On 1-X, in the chart, or on C or D, in the sUde rule, the distance 1-1.5 repre- sents by scale the logarithm (0.17 609) of 1.5, and 1-2 represents the logarithm (0.30 103) of 2. If now we add these two distances together, by laying off 1-2 from 1.5 on 1-X of the chart, or by placing the slide as in the figure, we obtain the distance 1-3 = .47 712 = the mantissa of log 3 or of log (2 X 1.5).* Conversely, to divide 3 by 2, we graphically or mechani- cally subtract 1-2 from 1-3. 1.1- Logs. 1.0- 0.0- 0.8- 0.7- o.o- 0.4- 0.3- 0.2- o.r 0.0- d 1 — "T 1 1 1 \ 1 1 1 1 1 r Logs. 1.9 O.O O.l 0.2 0.3 0.4 O.J O.G 0.7 0.8 O/J l.O 1.1 . (6) In tlie lo^arittimie chart, the scales of both axes, 1-X and 1-Y, being equal, a line 1-H, marked x, bisecting the square and form- ing an angle of 45° witR each axis (tan 45° = l),t will bisect also the inter- sections of all equal co-ordinates. Thus, points in the line x, immediately over 2, 3, 4, etc., in 1-X, are also opposite 2, 3, 4, etc., respectively, on 1-Y. See (4). (7) If lines 2- A, 3-K, etc. (marked 2a:, 3a:, etc.\ parallel to and above 1-H, be drawn through 2, 3, etc., on 1-Y, then points in such lines, im- mediately over any number, x, in 1-X, will be respectively opposite the (*) In the slide rule, with the slide as shown, each number on D is = 1.5 X the coinciding number on C. (t) In discussing tangents of angles on log chart, we refer to the actual measured distances, as shown on the equally divided scales of logs in our figures, and not to the numbers, which, for mere convenience, are marked /"< T> -| n in on the chart. Thus, in line 1-B, tan C 1 B = = r-^, not 2.15 76 LOGARITHMIC CHART AND SLIDE RULE. numbers giving the products 22;, Zz, etc., on 1-Y; while similar lines, drawn helow 1-H and through 2, 3, etc., on 1-X, give values of ^, ? etc., respectively. If these lines |, |, etc., be produced downward, they will ^^/U~\iP^°^^^^^) ^i ^-^ (= ^)' 0-^3 • • (= ^)' etc., respectively * See (4) (8) Powers and roots. If a line x2 be drawn through 1, at an angle S2 1-X, whose tangent, "^2 jg 2^ jt will give values of x''. Thus, the ver- tical through 3, on 1-X, cuts the line x^ ppposite 9 (= 3^) on 1-Y. Simi- larly, line x^ (tangent = 3) gives values of x^ ; and line i>x (tangent = y^) gives values of x^ or ^^ g^^ (4-)^ (9) Any equation of the form y = Q.x^ in which log w = log- C + w loe x (such as : area of circle = -n radius^), is represented, on a logarithmic chirt' by a straight line so drawn that the tangent T of its angle with 1-X is = n and intersecting 1-Y at that point which represents the value C Thus' th?miSh ^^^1 ''^^I f ' ^^T^"^} ^ ^^ ^' f ^"'^ ?^ squares, and, being drawn i«H?,\^o ^r 5,-^- • \^'^}r^^ '^ ^'^^^ ^'^'l^es of TT a;2. Thus, for a circle of radius 2, we find m the line tt x^ over 2, a point L opposite E, or 12 57 the t'hTd^Vram, ?adts^=^'"''"^^^ ' ^"^^"^ "^^" ^ ^'•^^- ' ' ' "« obtain," from (10) If a chart is to be used for solving many equations of a single kind, such as y= C a:« where C is a variable coefficient, and n a constant exponent parallel lines, forming the proper angle with 1-X, should be perma- nently ruled across the sheet at short intervals. ^Vl ^^J ^^y ^o^' ^,^ ^"^ (= ^^S 3), we may substitute its equal, M-N or 3-N, extending to the central diagonal line 1-H, marked x- and then Ja^'^f'^fT "^^t^PC^' 1-1.2 ==N-Q, 1-3 = N-K, etc., we mav add anv log (as 1-3 by moving upward from line x (as from N to K) or to the "nW and subtract any log (as 1-1.2) by moving downward (as from N to Q) or to the left. This facilitates the performance of a series of operations Thus : To multiply 1.5 by 2 (- 3), by 3 (= 9), and divide by 2 (= 4.5). F-G = 1-F = log 1.5. Add G-J = 1-2 = log 2 ; sum = F-J = log 3 = 1-3 = M-N. Add N-K -= 1-3 = log 3; sum = M-K = log 9 = 1-9 = 9-R. Subtract R-T = 1-2 == log 2 ; remainder = 9-T = log 4.5. For an example of the application of this principle to engineering prob- lems, see " Diagrams for proportioning wooden beams and posts," by Carl S. Fogh, " Engineering News ", Sept. 27, 1894. (13) Negative exponents. If ar is in the dmsor, the line will slope in the opposite direction, or downward from left to right. Thus, line 4-2 leaving 1-Y, at 4, and forming, with 1-X, the angle X, 2, 4, with tangent = — — ' ' ' . = — 2, represents the equation : y = - ^ = 4 xr'. (13) If the lines of products, powers, and roots, C x, x^, and -//^ etc., be drawn at angles whose tangents are less by 1 than those of the angles formed by the correspondinfi: lines in our figure, the results may be read directly from oblique lines drawn parallel to 2-2. Lines (C a;) giving multi- . pies and sub-multiples of the first power of x then become horizontal lines (tan = 0).t (14) Powers and roots bT the slide rule. Scales C and D being twice as large as scales A and B, these scales, with their ends coinciding, form a table of squares and of square roots. See (3). By moving the slide we solve equations of the forms y = {C x)^ and 2/ = C a^^. Thus, with the (*) In each of these lines, the product of the two numbers at its ends is = 10. Thus, in line 2-A. 2 X 5 = 10 ; in 3-K, 3 X 3.33 . . . = 10, etc. The chart thus furnishes a table of reciprocals. (f) Even with full-size charts and slide rules for actual use, accuracy is not to be expected beyond the third or fourth significant figure. (1) A chart of this kind, prepared by Major Wm. H. Bixby, U. S. A., after, the method of Leon Lnlanne. Corps de Fonts et Chauss^es, France, is published by Messrs. John Wiley & Sons, New York. Price, 25 cents. LOGARITHMIC CHART AND SLIDE RULE. 77 slide as shown, each number on A is = the square of (1.5 X the coinciding number on C) ; while, with 1 on B opposite 1.5 on A, each number on A is = 1.5 X the square of the coinciding number on C. (15) Since x'-^ = x^ X x, we find cubes or third powers by placing the slide with 1 on B opposite x^ on A {i. e., opposite x on D), see (3), and read- ing x'^ from A opposite x on B. Thus, 1.5"^ = ?. Place 1 on B opposite 1.5 on D; i, e., opposite 1.5^ (= 2.25) on A. Then, on A, opposite 1.5 on B, find 3.375 = 1.5=^. Or, turn the slide end for end. Place 1.5 on B opposite 1.5 on D, i. e., opposite 1.5* = 2.25 on A. Then, adding log 1.5 (on B) to log 2.25 on A, we find 3.375 (= 1.5') on A opposite 1 on B. (16) Conversely, to find i/x7 we shift the slide (in its normal position) until we find, on B, opposite x on A, the same number as we have on D op- posite 1 on C, and this number will be = \/x^ . Or, turn the slide end for end,* place 1 on C opposite x on A, and find, on B, a number which coincides with its equal on D. This number is= i/'xT See also (17), (18). (17) On the back of the slide is usually placed a scale of logs (see scale shown below the rule in figure) and two scales of angles, marked " S " and " T " respectively, for finding sines of angles greater than 0° 34' . . . ", and tangents of angles between 5° 42' . . ." and 45°. (18) Placing 1 on C opposite any number a: on D (with slide in its normal Eosition), log x is read from the scale of logs by means of an index on the ack of the rule. The logs may be used in finding powers and roots. Logs. 1.8 O.O 0.2 0.4 O.e 0.8 l.O 1.2 1.4 1.0 1.8 2.0 2.2 :sos. kf 2 3 1 1 4 5 erSOl 2 345 07SVlJ\ 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 Bl i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 7891 2 3 4 5G7S91S A^ f 3 4 5 G 7 8 9 lc\ Nos. U'i 1.5 2 3 4 5 G 7 8 9 ID] 1.9 O.O O.l 0.2 0.3 0.4 0.5 O.G 0.7 0:8 0.9 l.O 1.1 Logs. * (19) To find the sine or tang:eiit of an angle a ; bring a, on scale S or T, as the case may be, opposite the index on back, and read the natural (not logarithmic) sine or tangent opposite 10 at the end of A or D : sines on B, and tangents on C. Or, invert the slide, placing S under A, and T over D, with the ends of the scales coinciding. Then the numbers on A and D are the sines and tangents, respectively, of the angles on S and T. Caution. Sines of angles less than 5° 45' . . . " are less than 0.1. " 90° " " " 1.0. Tangents '* " betw.5° 42' . . . " and45°arebetw. O.land 1.0. (20) On the back of the rule is usually printed a table of ratios of num- bers in common use, for convenience in operating with thesliderule. Thus: diameter 113 U. S. gallons 3 ,. . ,., ^ ^ , ~: 7. = i^ ; ^~, — = ^^ (in a given quantity of water). circumference 355 pounds 20®^ ^ ' (31) Soaping the edges of the slide and the groove in which it runs, will often cure sticking, which is apt to be very annoying. If the slide is too loose, the groove may be deepened, and small springs, cut from narrow steel tape, inserted between it and the edge of the slide. (*) With the slide thus reversed, and with the ends of the scales coin- ciding, the numbers on A and B are reciprocals (page 52), as are also those on C and D, 78 TABLE OF LOGARITHMS. Common or Brig^g^s I,og^aritbms , Base = 10. No. 1 2 3 4 5 6 7 8 9 Prop. 00000 30103 47712 60206 69897 77815 84510 90309 95424 10 00000 00432 00860 01283 01703 02118 02530 02938 03342 03742 415 11 04139 04532 04921 05307 05690 06069 06445 06818 07188 t>75o4 379 12 07918 08278 08636 08990 09342 09691 10037 10380 10721 11059 349 V.i 11394 11727 12057 12385 12710 13033 13353 13672 13987 14301 323 U 14613 14921 15228 15533 15836 16136 16435 16731 17026 17318 300 15 17609 17897 18184 18469 18752 19033 19312 19590 19865 20139 281 16 20412 20682 20951 21218 21484 21748 22010 22271 22530 22788 264 17 23045 23299 23552 23804 24054 24303 24551 24797 25042 25285 249 18 25527 25767 26007 26245 26481 26717 26951 27184 27415 27646 236 19 27875 28103 28330 28555 28780 29003 •-'9225 29446 29666 29885 223 20 30103 30319 30535 30749 30963 31175 31386 31597 31806 32014 212 21 32222 32428 32633 32S38 33041 33243 3:5445 33646 33845 34044 202 22 34242 34439 34635 34830 35024 35218 35410 35602 35793 35983 194 23 36173 36361 36548 36735 36921 37106 37291 37474 37657 37839 185 24 38021 38201 38381 38560 38739 38916 39093 39269 39445 39619 177 25 39794 39967 40140 40312 40483 40654 40824 40993 41162 41330 171 26 41497 41664 41830 41995 42160 42324 4248 S 42651 42813 42975 164 27 43136 43296 43456 43616 43775 43933 44090 44248 44404 44560 158 28 44716 44870 45024 45178 45331 45484 45636 45788 45939 46089 153 29 46240 46389 46538 46686 46834 46982 47129 47275 47421 47567 148 30 47712 47856 48000 48144 48287 48430 48572 48713 48855 48995 143 31 49136 49276 49415 49554 49693 49831 49968 50105 50242 50379 138 32 50515 50650 50785 50920 51054 51188 51321 51454 51587 51719 134 33 61851 51982 52113 52244 52374 52504 52633 52763 52891 53020 130 34 53148 53275 53402 53529 53656 53781 53907 54033 54157 54282 126 35 54407 54530 54654 54777 54900 55022 55145 55266 55388 55509 122 36 55630 55750 55S70 55990 56110 56229 56348 56466 56584 56702 119 37 56S20 56937 57054 57170 57287 57403 57518 57634 57749 57863 lie 38 57978 58092 58206 58319 58433 58546 58658 58771 58883 58995 113 39 59106 59217 59328 59439 59549 59659 59769 59879 59988 60097 110 40 60206 60314 60422 60530 60638 60745 60852 60959 61066 61172 107 41 61278 61384 61489 61595 61700 61804 61909 62013 62118 62221 104 43 62325 62428 62531 62634 62736 62838 62941 63042 63144 63245 102 43 63347 63447 63648 63648 63749 63848 63948 64048 64147 64246 99 44 64345 64443 64542 64640 64738 64836 64933 65030 65127 65224 98 4& 65321 65417 65513 65609 65705 65801 65896 65991 660% 66181 96 46 66276 66370 66t64 66558 66651 66745 66838 66931 670 -.'4 67117 94 47 67210 67302 67394 67486 67577 67669 67760 67851 67942 68033 92 48 68124 68214 68304 68394 68484 68574 68663 68752 68842 68930 90 49 69020 69108 69196 69284 69372 69460 69548 69635 69722 69810 88 50 69897 69983 70070 70156 70243 70329 70415 70500 70586 70671 86 61 70757 70842 70927 71011 71096 71180 71265 71349 71433 71516 84 52 71600 71683 71767 71850 71933 72015 72098 72181 72263 72345 82 53 72428 72509 72591 72672 72754 72835 72916 72997 73078 73158 81 64 73239 73319 73399 73480 73559 73639 73719 73798 73878 73957 80 55 74036 74115 74193 74272 74351 74429 74507 74585 74663 74741 78 66 74818 74896 74973 750o0 75127 75204 75281 75358 75434 75511 77 57 75587 75663 75739 75815 75891 75966 76042 76117 76192 76267 75 58 76342 76417 76492 76666 76641 76715 76789 76863 76937 77011 74 59 77085 77158 77232 77305 77378 77451 77524 77597 77670 77742 73 60 77815 77887 77959 78031 78103 78175 78247 78318 78390 78461 72 61 78533 78604 78675 78746 78816 78887 78958 79028 79098 79169 71 62 79239 79309 79379 79448 79518 79588 79657 79726 79796 79865 70 63 79934 80002 80071 80140 80208 80277 80345 80413 80482 80550 69 54 80618 80685 80753 80821 80888' 80956 81023 81090 81157 81224 68 65 81291 81358 81424 81491 81557 81624 816VK) 81756 81822 SI 883 6? TABLE OF LOGARITHMS. 79 ComiKion or Bri^^s I.og:aritliins . Base =10. No. 1 2 3 4 5 6 7 8 9 Prop. 66 81954 82020 82085 82151 82216 82282 82347 82412 82477 82542 66 67 82607 82672 82736 82801 82866 82930 82994 83068 83123 83187 66 68 83250 83314 83378 83442 83506 83569 83632 83696 83768 83821 64 69 83884 83947 84010 84073 84136 84198 84260 84323 84385 84447 63 70 84609 84671 84633 84695 84757 84818 84880 84941 85003 85064 62 71 85125 85187 85248 86309 85369 86430 85491 85551 85612 85672 61 72 86733 85793 85853 85913 85973 86033 86093 86153 86213 86272 60 73 86332 86391 86461 86610 86569 86628 86687 86746 86805 86864 69 74 86923 86981 87040 87098 87157 87216 87273 87332 87390 87448 58 75 87506 87564 87621 87679 87737 87794 87862 87909 87966 88024 57 76 88081 88138 88195 88252 88309 88366 88422 88479 88536 88592 66 77 88649 88705 88761 88818 88874 88930 88986 89042 89098 89153 56 78 89209 89265 89320 89376 89431 89487 89542 89597 89652 89707 55 79 89762 89817 89872 89927 89982 90036 90091 90145 90200 90254 64 80 90309 90363 90417 90471 90525 90579 90633 90687 90741 90794 54 81 90848 90902 90955 91009 91062 91115 91169 91222 91275 91328 53 82 91381 91434 91487 91640 91692 91645 91698 91750 91803 91856 53 83 91907 91960 92012 92064 92116 92168 92220 92272 92324 92376 52 84 92427 92479 92531 92582 92634 92685 92737 92788 92839 92890 51 85 92941 92993 93044 93096 93146 93196 93247 93298 93348 93399 51 86 93449 93500 93550 93601 93661 93701 93761 93802 93852 93902 50 87 93951 94001 94051 94101 94161 94200 94250 94300 94349 94398 49 88 94448 94497 94546 94696 94fi45 94694 94743 94792 94841 94890 4» 89 94939 94987 95036 95085 95133 95182 95230 95279 96327 95376 48 90 95424 95472 95520 96568 95616 96664 95712 96760 95808 95856 48 91 95904 95951 95999 96047 96094 96142 96189 96236 96284 96331 48 92 96378 96426 96473 96520 96667 96614 96661 96708 96764 96801 47 93 96848 96895 96941 96988 97034 97081 97127 97174 97220 97266 47 94 97312 97359 97405 97451 97497 97643 97589 97635 97680 97726 46 95 97772 97818 97863 97909 97964 98000 98046 98091 98136 98181 46 96 98227 98272 98317 98362 98407 98462 98497 98542 98587 98632 45 97 98677 98721 98766 98811 98855 98900 98946 98989 99033 99078 45 98 99122 99166 99211 99265 99299 99343 99387 99431 99475 99519 44 99 99563 99607 99661 99694 99738 99782 99825 99869 99913 99956 44 For extended table of logarithms see pages 80-91. The table above, being given on two opposite pages, avoids the necessity of turning leaves. It contains no error as great as 1 in the final figure. The proportional parts, in the last column, give merely the average difference for each line. Hence, when dealing with small numbers, and using 5-place logs, it is better to find diflfer- encesby subtraction : but where a two-page table is used, interpolation is often unnecessary. Indeed, the first four, or even the first three, places of the man- tissas here given will often be found sufiicient. If the first number dropped is 5 or more, increase by 1 the last figure retained. Thus, for log 660, mantissa = 81954, or 8195, or 820. Multiplication. Log a 6 = log a + log h. Division. Log - = log a — log b. Involution (Powers). Log «»» = n. log a. _ l?g_? Log^/a- n Evolution (Roots). Characteristics. Log 2870 = 3.45788 " 287 = 2.45788 " 28.7 = 1.45788 " 2.87 == 0.45788 Log 0.287 = 0.45788 — 1 = 1.45788 '• 0.0287 = 0.45788 — 2 = 2.45788 " 0.00287 = 0.45788 — 3 = 3.45788 ♦'. 0.000287 -- 0.45788 — 4 = 4.45788 80 LOGARITHMS. Common or Brig^gs ItO^aritlims. Base = 10. No. Log. 2 -087,-^ — 260:^ -303^ 346 4321 475 i 518 —561 —604 —647 1000 1 00000 ;, 01 02 ' 03 04 05 06 07 08 09 loio 11 12 13 14 15 16 17 18 19 1030 21 22 23 24' 25 26 27 28 29 1030 31 32 33 34 35 36 37 38 1040 41 42 43 44 45 46 47 48 732^^ -7/5 .^ 860'43 —903 42 945|43 —988 1 42 01030J- 072^2 15742 -242 43 42 -284 —326 •368 410 452 494 —536 — 57 620 —662 703 745 —787 828 870 912 953 995 02036 49 1—078 No. Log. 2 1050 51 52 53 54 55 56 57 58 59 1060 61 62 63 64 65 66 67 68 69 1070 71 72 73 74 75 76 77 78 79 1080 81 82 83 84 85 86 87 88 89 1090 91 92 93 94 95 96 97 ~~ 02119- 160 —202 —243 284 325 366 407 —449 —490 41 42 41 41 ;?4i 41 41 42 —531 —572 612 653 694 -735 —776 816 857 ■979 03019 —060 100 —141 181 —222 —262 302 342 -383 —423 —463 —503 —543 —583 623 -663 —703 -743 782 822 862 -902 941 981 04021 060 100 No. 1100 01 02 03 04 05 06 07 08 09 1110 11 12 13 14 15 16 • 17 18 19 Log. 2 04139 —1797 218 ': -258:^ -297 1 ^ 336 :; —376 —415 —454 493 532 571 610 —650 -689 727 766 805 844 1120 21 23 —922 ■961 999 05038 24 1—077 25 115 26 —154 27 28 29 1130 31 32 33 34 35 36 37 38 39 1140 41 42 43 j 44! 45 46 47 48 192 -231 269 346 —385 —423 461 -^00 —538 576 614 652 690 —729 767 ■805 918 956 994 49 106032 No. Log. 2 No. Log. g 1150 06070 51 !— 108 52 ! 145 53 1—183 54 —221 55 56 57 58 59 1160 61 62 63 64 65 66 67 68 69 1170 71 72 73 74 75 76 77 78 79 1180 81 82 258 —296 333 —371 408 —446 —558 7441' 1190 91 . 92 -856 -893 -930 -967 07004 041 —078 115 151 188 ■225 —262 298 335 —37-2 408 445 —482 518 — 555 591 —628 93 664 94 700 95 —737 96 773 97 809 98 —846 99 —882 1200 01 02 03 04 05 06 07 08 09 1210 11 12 13 14 15 16 17 18 19 1220 21 22 23 24 25 26 27 28 29 1230 31 32 33 34 35 36 37 1240 41 42 43 44 45 46 47 48 49 07918 36 954:36 990I37 080271^ -063;^^ —0991^^ -1351^5 -171 !^^ ■243l?2 —279' 314*; 3507 386' 42*2; ^p;c!36 35 ^36 .36 4937 529 ;: —565): 600^ -6361 —6727 7077 —7437 778 —814 849 884 —920 955 —991 09026 061 096 132 —167 202 —237 272 307 342 377 412 447 482 —517 -552 r. —587 621 656 i 35 35 35 35 35 35 35 34 35 35 Example: To find Log. 11826: Log. 11830 = 07298 Dif. = 10 36 Log. 11820 = 07262 11826— 11820 = 6 Dif. for 6 under 36 = 22 Log. 11826 = 07262 -h 22 = 07284 44 43 42 41 4 4 4 4 9 9 8 8 13 13 18 12 18 17 17 16 22 22 21 21 26 26 25 25 31 30 29 29 35 34' 34 33 40 39 38 37 38 37 36 4 4 4 • 8 7 7 11 11 11 15 16 14 19 19 18 23 22' 22 27 26 25 30 30 29 34 33 32 35 4 7 11 14 18 21 25 28 32 LOGARITHMS. 81 Common or Bri^grs liOgarithms. Base = 10. No. 1250 51 52 63 54 55 56 57 58 59 1260 61 62 63 64 65 66 67 1270 71 72 73 74 75 76 77 78 79 1280 ■ 81 82 83 84 85 86 87 1290 91 92 93 94 95 96 97 98 99 Log.|§ 09691 —726 760 795 —830 864 —899 —934 968 10003 03: —072 —106 140 175 209 243 •27 —312 346 —415 —449 —483 —517 551 585 619 653 687 -721 -755 —789 ■823 —857 890 924 —958 -992 11025 ■059 •093 126 —160 193 -227 ■261 —294 327 361 34 33 34 34 34 33 33 34 33 No. I Log. 2 1300 01 02 03 04 05 06 07 08 09 1310 11 12 13 14 15 16 17 18 19 1320 21 22 23 24 25 26 27 28 29 1330 31 32 33 34 35 36 37 38 39 1340 41 11394 —428 461 494 —528 501 594 —628 661 —694 727 760 793 826 860 : —893 926 —959 992 12024 057 090 123 156 -189 222 254 287 —320 352 385 —418 450 483 —510 5481 —581 613 646 678 710 743 775 42 43 44 45 46 —905 47 -937 48 —969 49 13001 •808 ^S, 840 loo No. 1350 51 52 53 54 55 56 57 58 59 1360 61 62 63 64 65 66 67 68 69 1370 71 72 73 74 75 76 77 78 79 1380 81 82 83 84 85 86 87 Log. § 13033 -066 —098 —130 162 194 —226 -258 290 —322 -354 -386 —418 —450 481 513 545 —577 —609 640 672 704 735 767 —799 830 —862 893 -925 956 14019 —051 082 —114 -145 176 •208 —239 89 270 1390 91 92 93 94 95 96 97 98 99 301 —333 —364 395 426 457 —489 —520 —551 —582 No. 1400 01 02 03 04 05 06 07 08 09 1410 11 12 13 14 15 16 17 18 19 1420 21 22 23 24 25 26 27 28 29 1430 31 32 Log.l g —675 —706 —737 —768 —799 829 860 891 —922 —953 983 15014 —045 -076 106 —137 168 198 —229 259 —290 320 —351 381 —412 442 -473 503 534 —564 594 33 —625 34 —655 35 685 715 -746 -776 806 1440 41 42 43 44 45 46 16017 47 —047 48 —077 49 —107 866 —897 927 —957 No. 1450 51 52 53 54 55 56 . 57 58 1460 61 62 63 64 65 66 67 68 69 1470 71 72 73 74 75 76 77 78 79 1480 81 82 84 88 89 1490 91 92 93 94 95 96 97 98 99 Log. § 16137 —167 —197 —227 256 286 316 —346 376 —406 435 465 —495 524 554 —584 613 643 -673 702 -732 761 -791 820 -850 879 -909 938 967 17026 -056 —085 114 143 -173 -202 231 260 289 —319 348 377 —406 435 464 493 522 551 580 Example: To find Log. 12605 : Log. 12610 = 10072 Dif. = 10 35 Log. 12600 = 10037 12605 — 12600 = 5 Dif. for 5 under 35 = 18 Log. 12605 = 10037 + 18 = 10055 6 35 34 33 32 31 30 29 1 4 3 3 3 3 3 3 1 2 7 7 7 6 6 6 6 2 3 11 10 10 10 9 9 9 3 4 14 14 13 13 12 12 12 4 5 18 17 17 16 16 15 15 5 6 21 20 20 19 19 18 17 6 7 25 24 23 22 22 21 20 7 8 28 27 26 26 25 24 23 8 9 32 31 30 29 28 27 26 9 A —217 ■248 279 311 —342 373 404 .J 4361 —4671 498 529! 560 1 . — 592p —623'^ — 654i^ 6851^ 32 ^'3 '31 2800— 7161 1 02 —747^1 31 31 31 31 30 31 31 31 04 —778 06 —809] 08 —840 2810 —871 12 -902 14 932 16 963 18 994 2820 45025 22 —056 24 086 26 117 28 —148 2830 -179 32 209 34 —240 36 —271 38 301 2840 -332 42 362 44 —393 46 423 48 —454 No. 2850 52 54 56 58 2860 62 64 66 • 68 2870 72 74 76 78 2880 82 84 86 i 88 I 2890 92 94 96 98 2900 02 04 06 08 2910 12 14 16 18 2920 22 24 26 28 2930 32 34 36 38 2940 42 44 46 48 Log. § No. 45484 —515 545 —576 606 —637 -667 697 ■728 —758 788 818 ■849 —879 909 939 969 46000 —030 —060 —090 —120 —150 —180 —210 —240 —270 —300 419 —449 —479 —509 568 —598 627 657 —687 716 746 —776 805 —835 864 -894 i'^ 923!; —953!" 2950 52 54 56 58 2960 62 64 66 68 2970 72 74 76 78 2980 82 84 86 88 2990 92 94 96 98 3000 02 04 06 08 3010 12 14 16 18 3020 22 24 26 28 3030 32 34 36 38 3040 42 44 46 48 46982 47012 041 070 —100 129 —159 -188 217 246 —276 ■305 334 363 392 422 —451 ■480 —509 Log. s 567 596 625 654 683 712 741 -770 —799 •828 •857 885 914 943 —972 48001 029 058 —087 116 144 173 —202 230 ■259 287 316 344 —373 401 No. 3050 52 54 56 58 3060 62 64 66 68 3070 72 74 76 78 3080 82 84 86 88 3090 92 94 96 98 3100 02 04 06 08 3110 12 14 16 18 3120 22 24 26 28 3130 32 34 48430 458 —487 515 —544 572 —601 ■629 657 —686 —714 742 770 —799 —827 855 883 911 -940 -968 -996 49024 052 3140 42 44 46 48 Log. g No. 136 164 192 220 248 276 —304 —332 -360 •388 415 443 471 —499 —527 554 582 610 ■638 665 —693 721 748 —776 3150 52 54 56 58 3160 62 64 66 . 68 3170 72 74 76 78 3180 82 84 86 88 3190 92 94 96 98 3200 02 04 06 08 3210 12 14 16 18 3220 22 24 26 28 3230 32 34 36 38 3240 42 44 46 Log.g 49831 —859 886 —914 941 —969 996 50024 051 —079 ■106 133 161 188 215 —243 270 297 —325 —352 379 406 433 —4611 -488 -515 542 569 596 623 —651 678 —705 ■732 ■759 —786 —813 —840 866 893 920 947 974 51001 —028 —055 081 108 —135 —162 To find Log. 29019 : Log. 29020 = 46270 Dif. 20 .30 Log. 29000 = 46240 29019 — 29000 -=19 Under 30 Dif. for 10 = 15 «' " 9 =J14 " " 19 = 29 Log, 29019 = 46240 + 29 = 46269. 32 31 30 29 28 1 2 2 2 1 1 2 3 3 3 3 3 3 5 5 5 4 4 4 6 6 6 6 6 5 8 8 8 7 7 6 10 9 9 9 8 7 11 11 11 10 10 8 13 12 12 12 11 9 14 14 14 13 13 10 16 16 15 15 14 A dash before or after a log. de- notes that its true value is less than the tabular value by less than half a unit in the last place. Thus : Log. 3128=4952667 ** 3130=4955445 86 LOGARITHMS. Comiiioii or Brig-g^s liOg-arittims. Base = 10. No. Log. § No. Log. ^ No. 3»50 52 54 56 58 3^60 62 64 •r ^ Log.lS 51188 .-,., —242 f' 268 ^^ 295 f' 68 3970 72 74 76 78 3^80 82 84 86 88 3^90 92 94 96 98 3300 02 04 06 08 3310 12 14 16 18 3320 22 24 26 28 3330 32 34 -322! 348' 875 66 —402 428: —455 481 —508 534 —561 587 —614 640 —6671 693; — 720! —746! 772! —799! 825 1 851! — 9o7i ■983! 52009 035 061 —114 —140 166 192 218! 244 1 270! -297! —323 —3491 — 375! —4011 —427 1 —453, —4791 3350 52 54 56 58 3360 62 64 6t) 68 3370 - 72. 74 76 78 3380 82 84 3390 92 94 3400 02 04 06 08 3410 12 14 16 18 3430 22 24 26 28 3430 32 34 36 38 3440 42 44 46 52504 530 556 582 608 ■634 —660 686 711 737 ■763 —789 ■815 840 ■866 ■892 917 943 969 994 53020 046 071 097 122 —148 173 —199 224 —260 275 ■301 326 —352 377 403 428 453 —479 504 529 555 580 605 -631 656 681 706 —732 757 3450 52 54 56 58 3460 62 64 66 54008 3470 —033 53782 807 832 857 882 -908 72 74 76 78 3480 82 84 86 88 3490 92 94 96 98 3500 02 04 06 08 3510 12 14 16 18 3520 22 24 26 28 3530 32 34 36 38 3540 42 44 46 48 —058 —083 —108 —133 —158 —183 —208 —233 —258 —283 307 332 357 —382 —407 —432 4o6 481 —506 —531 555 580 —605 —630 654 —679 —704 728 —753 777 802 —827 851 —876 900 —925 949 —974 ■^ No. 3550 52 54 56 58 3560 62 64 66 68 3570 72 74 76 78 3580 82 84 86 88 3590 92 94 96 98 3600 02 04 06 08 3610 12 14 16 18 3620 22 24 26 28 3630 32 34 36 38 3640 42 44 46 48 Log. § 55023 - 047 24 —072 096 —121 —145 169 —194 218 242 —267 291 315 —340 364 388 —413 —437 461 485 509 —534 25 —558 24 -582 i 24 606 24 24 630 654 678 —703 ■727 —751! ■775 —799 823 ■847 —871 —895 919 —943 —967 —991 56015 038 23 062 24 086 24 24 24 24 24 110 , —134 158 —182 oo 205 ^^ No. Log. § 3650 52 54 56 58 3660 62 64 66 68 3670 72 74 76 78 3680 82 84 86 88 3690 92 94 96 98 3700 02 04 06 08 3710 12 14 16 18 3720 22 24 26 28 3730 32 34 36 38 3740 42 44 46 48 56229 253 —277 —301 324 348 —372 ■396 419 —443 —467 490 —514 —538 561 —685 608 —632 656 679 —703 726 —750 773 —797 820 -844 867 -891 -914 937 —961 984 57008 —031 054 ■078 —101 124 148 —171 194 217 —241 -264 287 310 -334 -357 -380 To find Log. 36114: Log. 36120 = 55775 Log. 36100 = 55751 Dif. 20 24 36114 — 36100 = 14 Under 24 Dif. for 10 = 12 '' " 4 =5 *' " 14 = 17 Log. 36114 = 65751 + 17 = 65768. 27 26 25 24 23 1 1 1 1 1 1 1 2 3 3 3 2 2 2 3 4 4 4 4 3 3 4 5 5 6 5 5 4 5 7 7 6 6 6 5 6 8 8 8 7 7 6 7 10 9 9 8 8 7 8 11 10 10 10 9 8 9 12 12 11 11 10 9 10 14 13 13 12 12 10 A dasb before or after a log. de- notes that its true value is less than the tabular value by less than half a unit in the last place. Thus : Log. 3490 = 5428254 " 3492 = 5430742 LOGAEITHMS. 87 Commoii or Brig-g-s liOg^arithms. Base = 10. Log. 57403 -^61 519 576 634 692 749 —807 -864 921 978 58035 092 149 206 263 ■320 —377 433 —490 546 602 659 —715 771 827 883 939 995 59051 106 162 —218 273 -329 -384 439 494 —550 —605 —660 715 —770 824 879 —934 988 60043 097 —152 No. 4000 05 10 15 20 25 30 35 40 45 4050 55 60 65 70 75 80 85 90 95 4100 05 10 15 20 25 30 35 40 45 4150 55 60 65 70 75 80 85 90 95 4300 05 10 15 20 25 30 35 40 45 hog. 60206 260 314 —369 —423 —477 ■531 584 638 —692 —746 799 —853 906 959 61013 066 119 172 225 278 331 384 —437 490 542 595 —648 700 752 •805 857 909 962 62014 066 —118 •170 221 273 —325 —377 428 —480 531 ■583 634 685 ■737 —788 No. 4250 55 60 65 70 75 80 85 90 95 4300 05 10 15 20 25 30 35 40 45 4350 55 60 65 70 75 80 85 90 95 4400 05 10 15 20 25 30 35 40 45 4450 55 60 65 70 75 80 85 90 95 Log. 62839 —890 —941 —992 63043 —094 144 195 —246 296 —347 397 —448 498 548 —599 —649 —699 —749 •799 —849 899 —949 998 64048 •098 147 -197 246 •296 345 —395 —444 493 542 591 640 689 738 787 836 —885 933 982 65031 079 ■128 176 ■225 —273 No. 4500 05 10 15 20 25 30 35 40 45 4550 55 60 65 70 75 80 85 90 95 4600 05 . 10 15 20 25 30 35 40 45 4650 55 60 65 70 75 80 85 90 95 4700 05 10 15 20 25 30 35 40 45 Log. 2 No. Log. 2 65321 369 —418 —466 —514 —562 —610 —658 —706 753 801 —849 896 944 —992 66039 —087 —134 181 —229 —276 -323 370 417 464 511 558 605 —652 •699 745 —792 ■839 885 ■932 978 67025 —071 117 —164 •210 —256 302 348 394 440 486 —532 78 —624 4750 55 60 65 70 75 80 85 90 95 4800 05 10 15 20 25 30 35 40 45 4850 55 60 65 70 75 80 85 90 95 4900 05 10 15 20 25 30 35 40 45 4950 55 60 65 70 75 80 85 90 95 67669 715 —761 806 —852 897 —943 988 68034 —079 124 169 —215 —260 —305 —350 395 440 485 529 574 —619 —664 708 -753 797 -842 886 -931 975 69020 —064 108 152 —197 ■241 ■285 —329 —373 -417 —461 504 548 —592 •636 679 723 —767 810 •854 1 2 3 4 5 6 7 8 9 10 58 57 56 55 54 53 53 51 50 49 48 47 46 45 44 43 1.2 1.1 1.1 1.1 1.1 1.1 1.0 1.0 1.0 1.0 1.0 0.9 0.9 0.9 0.9 0.9 2.3 2.3 2.2 2.2 2.2 2.1 2.1 2.0 2.0 2.0 1.9 1.9 1.8 1.8 1.8 1.7' 3.0 3.4 3.4 3.3 3.2 3.2 3.1 3.1 3.0 2.9 2.9 2.8 2.8 2.7 2.6, 2.6 4.6 4.6 4.5 4.4 4.3 4.2 4.2 4:1 4.0 3.9 3.8 3.8 3.7 3.6 3.5; 3.4 5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5 4.4! 4.3 70 6.8 6.7 6.6 6.5 6.4 6.2 6.1 6.0 5.9 5.8 5.6 5.5 5.4 5.3 5.2 8.1 8.0 7.8 7.7 7.6 7.4 7.3 7.1 7.0 6.9 6.7 6.6 6.4 6.3 6.2 6.0 9.3 9.1 9.0 8.8 8.6 8.5 8.3 8.2 8.0 7.8 7.7 7.5 7.4 7.2 7.0: 6.9 10.4 10.3 10.1 9.9 9.7 9.5 9.4 9.2 9.0 8.8 8.6 8.5 8.3 8.1 7.9| 7.7 11.6 11.4 11.2 11.0 10.8 10.6 10.4 10.2 10.0 9.8 9.6 9.4 9.2 9.0 8.8 ' 8.6 1 2 3 4 5 6 7 8 9 10 88 LOGARITHMS. Commoii or Brig^g^s liOg-arithms. Base = 10* 39 38 37 36 0.8 0.8 0.7 0.7 1.6 1.5 1.5 1.4 2.3 2.3 2.2 2.2 3.1 3.0 3.0 2.9 3.9 ' 3.8 3.7 3.6 4.7 i 4.6 4.4 4.3 5.5 5.3 5.2 5.0 6.2 , 6.1 5.9 5.8 7.0 ; 6.8 6.7 6.5 7.8 7.6 7.4 7.2 To find Log. 58636 : Log. 58650 = 76827 Log. 58600 = 76790 Dif. 50 37 58636 — 58600 = 36 Under 37 Dif. for 10 X 3 = 22.0 " '' 6 = 4.4 " " 36 = 26.4 Log. 58636 = 76790 4- 26 = 76816 LOGARITHMS. 89 Commoii or Brig^g^s liOg^aritliins. Base = 10. Log. § No. JLog. g No. Log. g 6350 55 60 65 70 75 80 85 90 95 6300 05 10 15 20 25 30 35 40 45 6350 55 60 65 70 75 80 85 90 95 6400 05 10 15 20 25 30 35 40 45 6450 55 60 65 70 75 80 85 90 79588 ■623 657 692 —727 761 ■796 —831 865 —900 934 —969 80003 037 —072 106 140 175 —209 243 277 312 —346 ■380 —414 448 482 516 550 584 — 618l —652 1 ■686 —720 ■754 787 821 —855 -889 922 —956 —990 81023 —057 090 —124 —158 —191 224 95 —258 No. 6500 05 10 15 20 25 30 35 40 45 6550 55 60 65 70 75 80 85 90 95 6600 05 10 15 20 25 30 35 40 45 6650 55 60 65 70 75 80 85 90 95 6700 05 10 15 20 25 30 35 40 45 81291 —325 358 391 —425 458 491 —525 —558 -591 624 657 690 723 —757 ■790 —823 —856 921 954 987 82020 —053 —086 119 151 184 —217 249 282 ■315 347 380 -413 445 —478 510 —543 575 607 —640 672 705 —737 769 ■802 —834 6750 55 60 65 .70 75 80 85 90 95 6800 05 10 15 20 25 30 35 40 45 6850 55 60 65 70 75 80 85 90 95 6900 05 10 15 20 25 30 35 40 45 6950 55 60 65 70 75 80 85 90 95 —963 —995 83027 —059 —091 123 155 ■187 219 251 —283 315 —347 378 410 442 —474 —506 537 569 —601 632 664 —696 727 -759 790 -822 853 -885 916 -948 979 84011 —042 073 —105 136 167 198 —230 —261 292 323 354 —386 —417 —448 ^479 No. Log. 2 No, 7000 84510! 05 1—541 1 10 — 572i 15 — 603i 20 25 30 35 40 45 7050 55 60 65 70 75 80 85 90 95 7100 05 10 15 20 25 30 35 40 45 7150 55 60 65 70 75 80 85 90 95 7200 05 10 15 20 25 30 35 40 45 ■634 — 665i ■696! 726 i 757 788 —819 -850 880 911 —942 -973 85003 034 —065 095 —126 156 —187 217 —248 278 —309 339 —370 400 —431 —461 4911 —522 —5521 582; 612 —643 —673 703 733 763 —794 -824 —854 —884 —914 —944 —974 86004 7250 55 60 65 70 75 80 85 90 95 7300 05 10 15 20 25 30 35 40 45 7350 55 60 65 70 75 80 85 90 95 7400 05 10 15 20 25 30 35 40 45 7450 55 60 65 70 75 80 85 90 95 Log. g 86034 —064 —094 •124 153 183 213 —243 273 —303 332 362 —392 421 451 —481 510 540 -570 .599 —629 658 —688 717 —747 776 —806 835 864 —894 923 —953 982 87011 040 —070 —099 128 157 186 216 —245 —274 —303 332 361 390 419 448 477 To find Log. 63023 : Log. 63050 = 79969 Log. 63000 = 79934 Dif. 50 35 63023 — 63000 = 23 Under 35 Dif. for 10 X 2 = 14.0 " " 3 = 2J " " 23 = 16.1 Log. 63023 = '9934 -H 16 = 79950. 32 31 0.6 0.6 1.2 1.9 2.5 3.1 3.8 ' 3.7 4.5 4.3 5.1 5.0 5.8 1 5.6 6.4 6.2 A dash before or after a log. de- notes that its true value is less than the tabular value by less than half a unit in the last place. Thus: Log.7400 = 8692317 " 7405 = 8695251 90 LOGARITHMS. Common or Brigg-s liOg^arithms. Base = 10. 7500 05 10 15 20 25 30 35 40 45 87506 585 — 564 —593 —622 —651 679 708 737 —766 —795 823 852 —881 —910 938 —967 —996 88024 —053 7650 55 60 65 70 75 80 7700 05 10 15 20 25 30 35 40 45 Log.12 081 -110 138 -167 ^ 195 1 on -224' ^^ 252 649 677 705; —734 —762 —790 1 I 29 —423 ; —508 28 536 98 564 2^ —593 —621 874] 902! No. 7750 55 60 65 70 75 80 85 90 95 7800 05 10 15 20 25 30 35 40 45 7850 55 60 65 70 75 80 85 90 95 7900 05 10 15 20 25 30 35 40 45 7950 55 60 65 70 75 80 85 90 95 Log. 958 986 89014 042 070 098 —126 —154 —182 209 237 265 —293 —321 348 376 —404 —432 459 —487 —515 542 —570 597 625 —653 680 —708 735 —763 790 -818 845 —873 -900 927 ■955 982 90009 —037 064 091 —119 —146 173 200 227 —255 —282 No. 8000 05 10 15 20 25 30 35 40 45 8050 55 60 65 70 75 80 85 90 95 8100 05 10 15 20 25 30 35 40 45 8150 55 60 65 70 75 80 85 8^00 05 10 15 20 25 30 35 40 45 I^og. s 90309 336 363 390 417 445 —472 —499 —526 —553 —580 —607 —634 660 687 714 741 768 795 —822 —849 875 902 —929 —956 982 91009 —036 062 089 -116 142 169 —196 222 —249 275 —302 328 —355 381 —408 434 —461 487 — 514j 540 566 ;' -593 ;, 6191 o No. 8^50 55 60 65 70 75 80 85 90 95 830(I^ 05 10 15 20 25 30 35 40 45 8350 55 60 65 70 75 80 85 90 95 8400 05 10 15 20 25 30 35 40 45 8450 55 60 65 70 75 80 85 90 95 Log. 91645 —672 698 724 —751 -777 803 829 855 -882 -908 -934 960 986 92012 038 —065 —091 —117 143 169 —195 221 —247 -273 298 324 350 376 402 —428 —454 —480 505 531 -557 -583 —609 634 —660 •686 711 737 763 788 814 —840 865 —891 916 No. 8500 05 10 15 20 25 30 35 40 45 8550 55 60 65 70 75 80 85 90 95 8600 05 10 15 20 25 30 35 40 45 8650 55 60 65 70 75 80 85 90 95 8700 05 10 15 20 25 30 35 40 45 92942 967 —993 93018 —044 069 ■095 120 —146 171 —197 222 247 —273 298 323 —349 374 —952 —977 94002 —027 —052 —077 101 126 151 —176 To find Log. 83678 : Log. 83700 = 92273 Log. 83650 = 92247 Dif. 50 36 83678 — 83650 = 2S Under 36 Dif. for 10 X 2 = 10.0 " " 8= 4.2 « " 38 = 14.2 Log. 83678 = 92247 -f- 14 = 92261. 39 38 37 36 35 24 1 0.6 0.6 0.5 0.5 0.5 0.5 1 2 1.2 1.1 1.1 1.0 1.0 1.0 2 3 1.7 1.7 1.6 •1.6 1.5 1.4 3 4 2.3 2 2 2.2 2.1 2.0 1.9 4 5 2.9 2.8 2.7 2.6 2.5 2.4 5 6 3.5 3.4 3.2 3.1 3.0 2.9 6 7 4.1 3.9 3.8 3.6 3.5 3.4 7 8 4.6 4.5 4.3 4.2 4.0 3.8 8 9 5.2 5.0 4.9 4.7 4.5 4.3 9 10 5.8 5.6 6.4 5.2 5.0 4.8 110 A dash before or after a log. de- notes that its true value is less than the tabular value by less than half a unit in the last place. Thus : Log. 8325 = 9203842 " 8330 = 9206450 LOGARITHMS. 91 Commoii or Brig^g^s liOg^arithms. Base = 10. Loif. 94201 5 25 No. Log. ft 24 24 25 No. Log. 96614 24 No. 9500 Log. s No. Log. 9000 95424 9350 97772 23 9750 98900 -226 24 05 448 55 —638 23 05 795 23 55 -923 250 25 10 472 60 661 24 10 818 23 60 —945 275 25 15 —497 24 24 65 —685 23 15 —841 23 65 967 —300 25 20 —521 70 -708 23 20 —864 22 70 989 -325 24 25 —545 24 75 731 24 25 886 23 75 99012 349 25 30 —569 24 24 80 —755 23 30 909 23 80 —034 374 25 35 —593 85 778 24 35 932 23 85 056 —399 25 40 —617 24 90 —802 23 40 -955 23 90 078 -424 24 45 —641 24 95 —825 23 45 —978 22 95 100 448 25 9050 —665 24 9300 848 24 9550 98000 23 9800 —123 —473 25 55 —689 24 05 —872 23 55 023 23 05 —145 —498 24 60 —713 24 10 -895 23 60 -046 22 10 —167 522 25 65 -737 24 15 918 24 65 068 2o 15 189 -547 24 70 —761 24 20 —942 23 70 091 23 20 211 571 25 75 —785 24 25 —965 23 75 —114 23 25 233 696 26 80 —809 23 30 988 23 80 —137 22 30 255 —621 24 85 832 24 35 97011 24 85 159 23 35 277 645 25 90 856 24 40 —035 23 90 —182 22 40 —300 —670 24 95 880 24 45 —058 23 95 204 23 45 —322 694 .,- 9100 904 24 9350 081 23 9600 227 23 9850 —344 —719 24 05 —928 24 55 104 24 05 —250 22 55 -366 743 25 10 —952 24 23 60 —128 23 10 272 23 60 —388 -768 24 15 -976 65 —151 23 15 —295 23 65 -410 792 25 20 999 24 70 —174 23 20 —318 22 70 —432 —817 24 25 96023 24 75 197 23 25 340 23 75 —454 841 25 30 047 24 80 220 23 30 -363 22 80 —476 —866 24 35 —071 24 85 243 24 35 385 23 85 —498 890 25 40 -095 23 90 —267 23 40 —408 22 90 -520 —915 24 45 118 24 95 —290 23 45 430 23 95 —542 939 24 9150 142 94 9400 —313 23 9650 —453 22 9900 —564 963 25 55 —166! ;; 05 -336 23 55 475 23 05 585 —988 24 60 —190 23 10 —359 23 60 —498 22 10 607 95012 24 65 213 24 15 382 23 65 520 23 15 629 036 25 70 —237 24 20 405 23 70 —543 22 20 651 —061 24 75 —261 23 25 428 23 75 565 23 25 673 085 24 80 284 24 30 451 23 80 —588 22 30 —695 109 25 85 -308 24 35 474 23 85 —610 22 35 —717 —134 24 90 —332 23 40 497 23 90 632 23 40 —739 158 24 95 355 24 45 520 23 95 -655 22 45 760 182 25 24 24 24 24 25 24 24 24 24 9300 —379 23 24 24 23 9450 543 23 23 23 23 23 23 23 23 22 23 9700 677 23 22 22 23 22 22 23 22 22 22 9950 782 —207 05 402 55 566 05 -700 55 804 —231 10 —426 60 689 10 -722 60 —826 255 15 —450 65 612 15 744 65 —848 279 20 473 70 -635 20 —767 70 —870 303 25 -497,^^ 520i 11 -544' 5^ 75 —658 25 —789 75 891 —328 30 80 —681 30 811 80 913 —352 35 85 —704 35 —834 85 -935 -376 40 90 —727 40 —856 90 —957 400 45 —591 23 95 749 45 878 95 978 To tiud Log. 95544 : Log. 95550 = 98023 Log. 95500 = 98000 Dif. 50 33 95544 — 95500 = 44 Under 23 Dif. for 10 X 4 = 18.0 " " 4= 1.8 " " 44 = 19.8 Log. 95544 = 98000 + 20 = 98020. 25 34 23 23 21 1 0.5 0.5 0.5 0.4 0.4 1 2 1.0 1.0 0.9 0.9 0.8 2 3 1.5 1.4 1.4 1.3 1.3 3 4 2.0 1.9 1.8 1.8 1.7 4 5 2.5 2.4 2.3 2.2 2.1 5 6 3.0 2.9 2.8 2.6 2.5 6 7 8.5 3.4 3.2 3.1 2.9 7- 8 4.0 3.8 3.7 3.5 3.4 8 9 4.5 4.3 4.1 4.0 3.8 9 10 5.0 4.8 4.6 4.4 4.2 10 A dash before or after a log. de- notes that its true value is less than the tabular value by less than half a unit in the last place. Thus : Log. 9600 = 9822712 " 9605 = 9824974 GEOMETRY. 6E0METM. liines, Fig-nres, Solids, defined. Strictly speaking a geometrical line is simply length, or distance. Ttie lines we draw on paper have not only length, but breadth and thickness ; still they are the most convenient symbol we can employ for denoting a geometrical line. Sltraisirht lines are also called rig-lit lines. A vertical line is one that points toward the center of the earth ; and a horizontal one is at right angles to a vert one. A plane figure is merely any flat surface or area entirely enclosed by lines either straight or curved ; which are called its outline, boundary, circumf, or periphery. We often confound the outline with the fig itself as when we speak of drawing circles, squares, ic ; for we actually draw only their outlines. Geometrically speaking, a fig has length and breadth only ; no thickness. A solid is any body ; it has length, breadth, and thickness. Geometrically similar figs or solids, are not necessarily of the same sf 2se ; but only of precisely the same shape. Thus, any two squares are, scien- tifically speaking, similar to each other ; so also any two circles, cubes, &c, no matter how different they niay be in size. When they are not only of the same shape, but of the same size, they are said to be similar, and equal. The quantities of lines are to each other simply as their lengths; but the quantities, or areas, or surfaces of similar figures, are as, or in proportion to, the squares of any one of the corresponding lines or sides which enclose the figures, or which may be drawn upon them ; and the quantities, or solidities of similar solids, are as the cubes of any of the corresponding lines which form their edges, or the figures by which thej are enclosed. Kem. — Simple aS the following operations appear, it is only by care, and good instruments, that they are made to give accurate results. SeveraJ of them can be much better performed by means of a metallic triangle having one perfectly accurate right angle. In the field, the tape-line, chain, or a measuring- rod will take the place of the dividers and ruler used indoors. To divide a given line, a b, into two eqnal parts. From its ends a and h as centers, and with any rad greater than one-half of a 5, ff describe the arcs c and d, and join ef. If the'line a & is very long, first lay off g equal dists a o and b g, each way from the ends, so as to approach conveniently near to each other ; and then proceed as if o fir were the line to be divided. Or measure a & by a scale, and thus ascertain its center. To divide a given line, m n, into any^ given number of equal parts. From m and n draw any two parallel lines m o and n ee, to an indefinite dist ; and on them, from m and n step off the reqd number of equal parts of any convenient length : final- ly, join the corresponding points thus stepped off. Or only one line, as mo, may be drawn and stepped off. as to«; then join sn; and draw the other short lines parallel to it. To divide a given line, m n, into two parts wliicli sliall have a given proportion to each other. This is done on the same principle as the last ; thus, let the proportion be as I to 3. First draw any line mo; and with any convenient opening of the dividers, make mx equal to one step ; and »g equal to three steps. Join s n ; and parallel to it draw x c. Then m c is to c « as 1 is to 3. Angles. 'fVTien two straight, or right lines meet each other at any Inclina- tion, the inclination is called an angle; and is measured by the degrees con- tained in the arc of a circle described from the point of meeting as a center. Since all circles, whether large or small, are supposed to be divided into 360 degrees, it follows that any number of degrees of a small circle will measure the same degree of inclination as will the same number of a large one. When two straight lines, as o w and a b, meet in such a manner that the inclination o n ais equal to the inclination o nb, then the two lines are said to be perpendicular to each other; and the angles o n a and nb, are called right angles ; and are each measd by, or are equal to, 90°, or one-fourth part of the circumf of a circle. Any angle, asced, smaller than a right angle, is called acute or sharp ; and one c ef, larger than a right angle, is called obtuse, or blunt. When one line meets another, as in the first Fig on opposite page, the two angles on the same side of either line are called contiguous, or adjacent. Thus, vus and vu w are adjacent ; also tus and tuw ; sut and s uv ; wut and w u v. The sum of two adjacent, anglys is always equal to two right angles; or to IbO'^. Therefore, if we know the number of de- grees contained in one of them, and subtract it from 18CP, we obtain the other. 4 GEOMETRY. 93 When two straight lines cross each other, forming four angles, either pair of those angles which point in exactly opposite directions are called opposite, or vertical ang^les ; thus, the pair s u t and v u w are opposite an- gles ; also the pair suv and t u ijy. The opposite angles of any pair are always equal to each other. When a straight line a h crosses two parallel lines c d, ef, the alternate angles which form a kind of Z are equal to each other. Thus, the angles don and o nf are equal : as are also con and one. Also the sum of the two internal angles on the same side of a b, is equal to two right angles, or 180°; thus, c o w + o n/= 180°; also don -\- one = 180°. An interior ang^le. In any fig, is any angle formed inside of that fig, by the meet- ing of two of its sides, as the angles c ab, ab c, b c a, of tins triangle. All the interior angles of any straight-lined figure of any number of sides whatever, are together equal to twice as many right angles minus four, as the figure has sides. Thus, a triangle has 3 sides ; twice that number is 6 ; and 6 right angles, or6 X 90^=540°; from which take 4 right angles, or 360°; and there remain 180^, which is the number of degrees in every plane, or straight-lined triangle. This principle furnishes an easy means of testing our measurements of the angles of any fig; for if the sum of all our measurements does not agree with Ihe sum given by the ruie, it is a proof that we have committed some error. An exterior ang-le Of any straight-lined figure, is any angle, as a b d, formed by the meeting of any side, as a b, with the prolongation of an adjacent side, as c 6; so likewise the angles c a s and b c w. All the exterior angles of any straight-lined fig, no matter how many sides it may have, amount to 360°; but, in the case of a re-entering angle, as y ij, the interior angle, g ij, exceeds 180°, and the "exterior" angle, g i x, being = 180° — interior angle, is negative. Thus ab d -\- b cw -\- c a 8 = 360° ; and yhj-\-zji— gix-\-igw = 360°. Angles, as a, b, c, g, h, arndj, which point outward, are called salient. From any ^iven point, p, on a line s t, to draw a perp, p a. ' From p, -with any convenient opening of the dividers, step off the equals po,pg. From o and g as centers, with any opening greater than half o g, describe the two short arcs b and c; and join a p. Or still better, describe four arcs, and join a y. Or from p with any convenient scale describe two short arcs g and c either one of them with a radius 3, and the other with a rad 4. Then from g with rad 5 describe the arc b. Join p a. If the point p is at one end of the line, or very near it. Extend the line, if possible, and proceed as above. But if this «annot be done, then from any convenient point, w, open the divid- ers \,o p, and describe the semicircle, s p oj through o w draw o w a; joinp s. Or use the last foreg^oing; process with rads 3, 4, and 5. From a g-iven point, o, to let fall a perp o «, to a g^iven line, tn n. From o, measure to the line m n, any two equal dists, o c, o e ; and from c and e as centers, with any opening greater than half of c e, describe the two arcs a and b ; join o t. Or frorh any point, as d on the line, opon the dividers to o, and describe the arc o g ; make i x equal to j" o ; and join o x. 94 GEOMETRY, If tlie line, a b, is on f lie grronnd. And a perp is reqd to be drawn from c, first measure oflF any two equal dists, cm, en. At m aud n. hold the ends of a piece of string, tape-line, or chain, man; then tighten out the string, &c, as shown hymsn; s being its center. Then will « c be the reqd perp. Or if the perp a; z is to be drawn from the end of the line w x, first measure x y upon the line, and equal to three feet; then holding the end of a tape- line at X, aud its nine feet mark at y, hold the four feet mark at z, keep- ing zx&nd zy equally stretched. Then z x will be the reqd perp, because 8, 4, and 5, make the sides of a right-angled triangle. Instead of 3, 4, and 5, any multiples of those numbers may be used, such as 6, 8, and 10 ; OP 9, 12, 15, &o : also instead of feet, we may use yards, chains, &c. Throuf^b a g-iven point, a, to draur a line, a c, parallel to another line, e f. "With the perp dist, a e, from any point, n, in ef, describe aa arc, t; draw a c just touching the arc. "7"^^ At any point, a, in a line a &, to make an an^le c« ft^eqnal to a K'iven ang-le, ni n o. From n and a, with any convenient rad, describe She arcs s t, d e; measure s t, and make e d equal k> it ; through a d draw a c. To bisect, or divide any ang-le, taocy, into two equal parts. From X set oCf any two equal dists, xr, x a. From r and a with any rad describe two arcs intersecting, as at o ; and join o x. If the two sides of the angle do not meet, as c / and g h, either first extend them until they do meet ; or else draw lines x w, and x y, parallel to them, and at eqoiJ dista from them, so as to meet ; then proceed as before. All angles, as n a m, n o m. at the ciroumf of a semicircle, and stand ing on its diam n m, are right angles ; or, as it is usually expressed, all ansfles in a semicircle are right angles. An angle n s x at the centre of a circle, is twice as great as an angle h m X at the circumf, when both stand upon the same arc n x. All angles, as y d p, y e p, y g p, &t the circumf of a circle, and otanding upon the same arc, as y p, are equal to each other ; or, as usually expressed, all angles in the same segment of a circle are equal. The complement of an angle is what it lacks of 90°. Thus, the com< piemen t of 80° is 90° — 80° = 10° ; and that of 210° is 90° — 210° — — 120°. The snpplement of an angle is what it lacks of 180°. Thus, the supple, ment of 80° is 180° — 80° = 100° ; and that of 210° is 180° — 210° = — 30°. But ordinarily we may neglect the signs -J- and — , before complements and supplements, and call the complement of an angle its diff from 90°* aa4 the supplement its diff from 180°. ANGLIB. 95 Angeles in a Parallelogram. A parallelogram is any four-sided straight-lined fig-< ure whose opposite sides are equal, as ah c d; or a square, &c. Any line drawn across a parallelogram between 2 opposite angles, is called a diagonal, as a c, OT b d. A diag divides a parallelogram into two equal parts ; as does also any line m n drawn through the center of either diag ; and moreover, the line m n itself is div into two equal parts by the diag. Two diags bisect each other ; they also divide the parallel- ogram into four triangles of equjtl areas. The sum of the two angles at the ends of any one side is = 180° ; thus, dab + abc — abc-j- b c d — 180° ; and the sum of the four angles, d a b, a b c, b c d, c d a = 360°. The sum of the squares of the four sides, is equal to the sum of the squares of the two diags. To reduce Minutes and SSeconds to Degrees and decimals of a Deg-ree, etc. In any given angle — Humber of degrees = Number of minutes -f- 60. = Number of seconds -j- 3600." Number of minutes = Number of degrees X 60. = Number of seconds -^ 60. Number of seconds = Number of degrees X 3600. = Number of minutes X 60. Table of Minutes antl Seconds in Decimals of a Degree, and of i^ieconds in Decimals of a Minute. (The columns of Mins and Degs answer equally for Sees and Mins.) Mins. Deg. Mins. Deg. Mins. Deg. i Sees. Deg. Sees. Deg. | Sees. Deg. In each equivalent, the last < digit repeats inde fini itely. See * below : 1 0.016 21 0.350 41 0.683 1 0.00027 21 0.00583 41 0.01138 2 0.033 22 0.366 42 0.700 2 0.00055 22 0.00611 42 0.01166 3 0.050 23 0.383 43 0.716 3 0.00083 23 0.00638 43 0.01194 4 0.066 24 0.400 44 0.733 4 0.00111 24 0.0U666 1 44 0.01222 5 0.083 25 0.416 45 0.750 5 0.00138 25 0.00694 1 45 0.01250 6 0.100 26 0.433 46 0.766 6 0.00166 26 0.00722 46 0.01277 7 0.116 27 0.450 47 0.783 7 0.00194 27 0.00750 47 0.01305 8 (U33 28 0.466 48 0.800 8 0.00222 28 O.U0777 ' 48 0.01333 9 0.150 29 0.483 49 0.816 9 0.00250 29 0.00805 49 0.01361 10 0.166 30 ■ 0.500 50 0.833 10 0.00277 30 0.00833 1 60 0.01388 11 0.183 31 0.516 51 0.850 11 0.00305 31 0.00861 1 51 0.01416 12 0.200 32 0.533 52 0.866 12 0.00333 32 0.00888 1 52 0.01444 13 0.216 33 0.550 53 0.883 13 0.00361 33 0.00916 1 53 0.01472 14 0.233 34 0.566 54 0.900 14 0.00388 34 0.00944 54 0.01500 15 0.250 35 0.583 55 0.916 15 0.00416 35 0.00972 55 0.01527 16 0.266 36 0.600 56 0.933 16 0.00444 36 0.01000 56 0.01555 17 0.283 37 0.616 57 0.950 17 0.00472 37 0.01027 57 0.01583 18 0.300 38 0.633 58 0.966 18 0.005('0 38 0.01055 58 0.01611 19 0.316 39 0.650 59 0.983 19 0.00527 39 0.01083 59 0.01638 20 0.333 40 0.666 60 1.000 20 0.00555 40 0.01111 60 0.01666 Sees Min. Sees. Min. Sees Min. Sees . Deg. Sees. Deg. Sees. Deg. * Each equivalent is a repeating decimal, thus : 2 minutes == 0.0333333 .... degree 12 seconds = 0.2000000 7 " -= 0.1166666 .... " 1 second = 0.0002777 12 " = 0.2000000 .... " 50 seconds = 0.0138888 . minute . degree 96 ANGLES. Approximate Measurement of Angles. (1) The four fingrers of the hand, held at right angles to the arm and at arm's length from the eye, cover about 7 degrees. And an angle of 7° corre- sponds to about 12.2 feet in 100 feet ; or to 36.6 feet in 100 yards ; or to 645 feet in a mile. (2) By means of a two-foot rnle, either on a drawing or between dis- tant objects in the field. If the inner edges of a common two-foot rule be opened to the extent shown in the column of inches, they will be inclined to each other at the angles shown in the column of angles. Since an opening of ]/^ inch (up to 19 inches or about 105°) corresponds to from about 3^° to 1°, no great accuracy is to be expected, and beyond 105° still less; for the liability to error then in- creases very rapidly as the opening becomes greater. Thus, the last }/^ inch cor- responds to about 12° Angles for openings intermediate of those given may be calculated to the nearest minute or two, by simple proportion, up to 23 inches of opening, or about 147°. Table of Ang-les corresponding to openin^^s of a 2-foot rule. (Original). Correct. Ins. Deg. min. Ins. Deg. minJ Tns. Deg. min. Ins. Deg.min. Ins. Deg.min. Ins. Deg. min K 1 12 4M 20 24 8M 40 IS 12>i 61 23 16>i 85 14 20>i 115 5 1 48 21 40 51 62 5 86 3 116 12 H 2 24 H 21 37 ]4 41 29 J4 62 47 14 86 52 H 117 20 3 GO 22 13 42 7 63 28 87 41 118 30 H 3 36 H 22 50 H 42 46 H 64 11 H 88 31 H 119 40 4 11 23 27 43 24 64 53 89 21 120 52 1 4 47 5 24 3 9 44 3 13 65 35 17 90 12 21 122 6 5 23 24 39 44 42 66 18 91 3 123 20 H 5 58 H 25 16 H 45 21 U. 67 1 H 91 54 H 124 S6 6 34 25 53 45 59 67 44 92 46 125 54 K 7 10 ^ 26 30 H 46 38 H 6a 28 H 93 38 H 127 14 7 46 27 7 47 17 69 12 94 31 128 35 H 8 22 % 27 44 H 47 56 H 69 55 H 95 24 H 129 59 8 58 28 21 48 35 70 38 96 17 131 25 2 9 34 6 28 58 10 49 15 14 71 22 18 97 11 22 132 53 10 10 29 35 49 54 72 6 98 5 134 24 H 10 46 J^ 30 11 ^ 50 34 H 72 51 H 99 00 34 135 58 11 22 30 49 51 13 73 86 99 55 137 35 H 11 58 yi 31 26 H 51 53 H 74 21 M 100 51 H 139 16 12 34 32 3 52 33 75 6 101 48 Ul 1 H IS 10 H 32 40 H 53 13 H 75 51 H 102 45 H 142' 51 13 46 33 17 53 53 76 36 103 43 144 46 3 14 22 7 33 54 11 54 34 15 77 22 19 104 41 23 146 48 14 58 34 33 55 14 78 8 105 40* 148 56 M 15 34 H 35 10 H 55 55 H 78 54 H 106 39 H 151 If 16 10 35 47 56 35 79 40 107 40 153 41 ^ 16 46 14 36 25 H 57 16 H 80 27 H 108 41 H 156 3fc 17 22 37 3 57 57 81 14 109 43 159 48 H 17 59 H 37 41 H 5b 38 H 82 2 H 110 46 H 163 2T 18 35 38 19 59 19 82 49 111 49 168 18 4 19 12 8 38 57 12 60 00 16 83 37 20 112 53 24 180 00 19 48 39 35 60 41 84 26 113 58 (3) With the same table, usin^ feet instead of inches. From the given point measure 12 feet toward* each object, and place marks. Measure the distance in feet between these marks. Suppose the first column in the table to be feet instead of inches. Then opposite the distance in feet will be the angle. "% foot =• 1.5 inches. 1 in. = .083 ft. I 4 ins. = .333 ft. I 7 ins. = .583 ft. I 10 ins. = .833 ft. 2 ins. = .167 ft. 5 ins. = .416 ft. 8 ins. = .607 ft. 11 ins. = .917 ft. 3 ins. = .25 ft. 1 6 ins. -= .5 ft. | 9 ins. = .75 ft, I 12 ins. = 1.0 ft. (4) Or, measure toward * each object 100 or any other number of feet, and place marks. Measure the distance in feet between the marks. Then Sine of half _ half the distance between the marks the angle ~ the distance measured toward one of the objects* Find this sine in the table pp. 98, etc. ; take out the corresponding angle and multiply it by 2 (5) See last paragraph of foot-note, pp 152 and 153. * If it is inconvenient to measure toward the objects, measure directly /rom thenu SIKES, TANGENTS, ETC. 97 Sines, Tangents, Ac. $iine, a 8, of any angle, a ch, or -which is the same thins, the sine of any circular arc, a 6, which subtends or measures the angle, is a straight line drawn from one end, as a, of the arc, at right fciidjles to, and terminating at, the rad c b, drawn to the other end b of the arc. It is, therefore, equal to half the chord a n, of the arc abn, which is equal to twice the arc a6 ; or. the sine of an angle ia always squal to half the chord of twice that angle ; and vice versa, the chord of an angle is alwayj equal to twice the sine of half the angle. f he sine < c of an angle tcb, or of an arc in tab, of 90°, is equal to the rad of the arc or of the circle ; and this sine of 90° is greater than that of any other angle. Cosine c * of an angle a cb, l8 that part of the rad which lies between the sine and the center of the circle. It is always equal to the sine y a of the complement t c aof a c b; or of what a c b wants of being 90°. The prefix co be- fore sines, &c, meaus complement ; thus, cosine means sine of the complement. Versed sine 5 & of any angle o c 6, is that part of the diam which lies between the sine, and the outer end 6. It is very common, but erroneous, when speaking of bridges, &c, to call the rise or height s 6 of a circular arch abn, its rersed sine; while it is actually the versed sineofonly half lAie arch. Thisabsurdity should cease ; for the word rise or height is not only more expressive,but is correct. Tan8-eiit6z<;orarf,ofanyangle a c 2), is a line drawn from, and at right angles to, the end 6 or a of either rad c b, or c a, which forms one of the legs of the angle ; and terminating as at w, or d, in the prolongation of the rad which forms the other leg. This last rad thus pro- longed, that is. c w, or c d, as the case may lie, is the secant of the angle « c 6. The angle tcb being supposed to be equal to 90°, the angle tea becomes the complement of the angle a e &, or what aeb want! of being 90° ; and the sine y a of this complement ; its versed sine f y ; its tangent t o ; and its seoant c 0, are respectively the co-sine, co-versed sine; co-tangent; and co-secant, of the angle a c b. Or, vice versa, the sine, &c, of a c b, are the cosine, &c, of tea; because the angle a c 6 is the comple- ment of the angle t c a. When the rad c 6, c a. or c t, is assumed to be equal to unity, or 1, the cor- responding sines, tangents, &c, are called natwrai ones; and their several lengths"for ditT angles, for said rad of unity, have been calculated: constituting the well-known tables of nat sines, &c. 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— < iC CD lO CO o o CO CO 00 o C^ CO CO CD ^ o O OS CO C CO iC CO !> 00 OS CO CO CO CO CO —I OS t^ OS OS 00 00 (M i> t^ t^ CO O CD o ^ --^ t^ t- J> to CO ^ 00 OO 00 (Tt C^ C* l^ J> 1> C^ 00 (?* CO 0< (N CO CO Ir^ 1- t^ t^ OS 1> lO CO 1> J> 1> t- ©< C* C<» (M t^ i> 1> 1^ OOCOOO•^OCOCOO^-tO■^(^»^|— t^i— i^C<»T;t(?DOO CO'^.-HOSi>.T^(?^Ot^ir5CO'-i 3Sl>iCC0— OSl>iOC0 C0t^.-'-^t0X(MXiOTt*fc^<-H!O0S (^^^OOSOSOOQOt^CDCOUO'^Tjf^l>l>'CDCDCOCDCOCOCOCDCDCOCCCDXtOCDCOiO oooooooooooooooooooo© lOOSCOt-O^OOCOCTSUO— iOOOWO{^CD"«*COC<»i— ^ <-llOOr:J-C<00OS— iC^COr!^iO}>OOOSO-HCOT*< OS (M'T^ CDOOO(MTfCOOr^ CO ©< -H o ^" lO 00 — rrCOt-t-i-t-CO-^-tOOiO^ CO O T^ 00 fl-i Ti*0>'::* CO 00 (^^ CO fi O CS CT5 OS 0000i>i-COCOiOiOTtiC0C0C IQ CO rHCDl>iOC0— OSt^iOCO'-iOS i^ lO CO -^ fl o o o OOSOSOSOSCSQOOOOOGOOO^^ t- I- t- i> GO — lOOOOOOOOOOO O O O O J> i^ t^ 1> l>t>.l>J>01^i^J>l>l>l:^J> I- J> t- t- iO t^ OS (^» Tt^QO.-iiOO'^OSiOOCOCOOS CO ^ — O _ bJo (M CO iO coi>c:soc^co^coooc;^(M "■^ CO 00 o C lO OS CO t^r-tUOOr^OOC^COOT^J^OiCO 1> — ' iC o bl) « o OS OS 00003>J>COiOiOTtT^C0(NC< r-H i-H O O c o O oooooooooooooooo eb 9 o o o oooooooooooo o o o o h CO (^^ O) rj* .-HOOiiOC^005i>COiOiOtOUO CO 1> 00 o CS bn o 00 O CO '-HOOcoThNosi-ioco.-HOst^ lO CO -^ o c o o — t^ COOOr^OCO.-^l--COOSi(OOCO C^ 00 "^ o «s 05 OS o o ^^C^COCO'sH'<*«OilOCOt^t^ GO 00 OS o ^ 00 00 C3S OS 05OSO5OSO5OSOSOSOSOS0S0S OS OS OS o o Oi OS CS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS ^ o C5 {>. ^ -H t^cooo.c<»coosc<('>^LQcoco»r: T^ CO ^ 00 <^ ^ -^ 00 ^^{^^OCOCOOSiO^I^CO OS »0 '—' CO (U V 00 OS o cp '-•C^^(MC0'^'^U0iOCOt^l>00 00 OS o o a s CO CO CiD OCs!!':tiCOaOO(Mr*HCOOOO(r^ T^ CO OS I— • G(Q CO CO CO CO '^'^^^Tf^iOiOuOOUOCOCO CO CO CO {> n o o o o oooooooooooo o o o o o t^ i^ i> l^ J>J>l>i--f^f^i^l>.J>J>t^i^ £- t^ t^ J> jzr (M CO T^ iOCOi>GOOSO'-H(^^CO'^tOCO C- 00 OS o ^ Tf ^ -* rf ■^^■^'^t^'^iOiOUOOiOiOiO lO UO lO CO o: 00 {> CO uoTrlcO(^^'-lOosooi>•colOr^ CO C CO 00 fi ao t^ t^ J> COCOCOtOtOO'^rfCOCOCOC^ N r-. r^ O S o 00 CO T:tH l^ t- i^i>i>i>i>i^r^;.^i>i^i>i> l> t- l^ t- , o lO O CO (MOOUONOSb-mCOC^^r-t—i -^ ^ (M CO bl) lO lO CO CO t>-J>000500'-HC^COrJ 00 OS o G 05 CO J> r-1 iOCJSCOJ>^COOr^OOC^COO r:t^ 00 (M J> be C!3 (rt (?* .-1 ,-1 oosc:sooooi>J>couoiOT^'=^ CO (M C^ -• s o O c^ c< (^» c^ C^^p-,^^^^^^^^^ oooooooooooocpooo o o o o bH Tt^ ?3 (7* (M (^^(^^coT:Ho^>os.-^r+^^>oco l>- (?< CO ^ pi bo CO coot'-'^—'OOiOiMcrscO'^f-ioococo O 00 UO CO fi lO -H t^ C^ OO'Tj^OSiO.-iCOC^fOO-^OSiO.-i 1^ 0^ 00 ^ cS t- 00 00 OS 0SOO-H(^l(^{C0C0T:t^r^^0C0 CO i^ J> 00 ■^ f^ t>- 1^ t- 1>OOCOOOQOOOOOQOOOOOOOOO 00 00 00 00 o 9 OS OS OS OS OS OS OS vJS OS OS OS OS OS OS OS OS OS OS OS o CO CO lo CO ^OSCOCM00C01>r-.rf5>OS^ .-H C^ ,--. 1-1 05 J> UO CO ^00CO'*r-iO:>COr:t<-^00u0C0 O I- T^ -^ V o CO ^ lO CO ^-!>ooc3soor-.(^:^cocO'«^lO CO CO 1> 00 (3 a o C>< Tt CO 00OC^T:tOS— iC0iOi>OS— < CO lO 1> OS QQ OS crsooooo — .-*— i,-,^C^ (M (M C<» (M 05 OS OS OS OSOOOOOOOOOOO o o o o CO CO CO CO co^>^'^>i>l^^>^>^-i^^>^-. j> t^ J> 1> ^ ,_4 (M CO -^ uoco^>ooosO'-'!^^coTj^uoco f- 00 OS o Oi (M (^( c^ (^^(^^(^^(^^(^*cocococooococo CO CO CO rj^ o OS 00 i> COiO^COC UO CO Ot^COQOCOi^— ^rfCOOOOSOS OS OS i> lo CO <» Oi j> lO CO '-'00COC0r-i00CDC0OJ>'«d^^ 00 lO C~? CTi CO c CO CO CO CO C0CMC^(?^C^-H^^,-HOOO OS OS OS 00 CD (U CO ^ OS i> U0C0r-^OSJ>iOC0-HC:st^iOC0 O GO CO ^ f>f & n OS OS 00 GO 000000J>3>l>i>t^COwCOCO CO lO iO lO lO OD O J>- J> J> 5> J>^-^-^>^:^i>i>^-^>^>^-^- i^ 1^ J> J> t- o i^ O CO (MO0S0S0S0SO>-iC^^CD00 ^ ^ OD ^ CO br CO C r-HtOOSCOh-'-HiOOSCOt^'— iiO c:s CO J> r^ lO CJ lO Tfi T:t< CO C7fj<,^^OOOOOOOl^J>CO lO lO T:i< rt^ 00 fl O CO CO CO CO COCOCOCOCOCOC^(M.-Hco-^o— ti>cooco T^ r-- OS 1> lO bo be on o -H r^ COOCOTOCJSCO(MOSiOC^OSU:) C<( OS iC C J>J>J>J>t^ f- I- t- i> ^ o OS OS OS OS osososososososososososos OS OS OS OS crs o '^ CD J> 00 OSOS00I^iC(?iO^CO— lUO 00 — 1 CO Tj* o on i^ CO uo T^cotN.— oost^coLOco(^«o 00 t^ O CO 00 OSOr-«C<»COCOrjHiOCO?^OOOS OS O — 1 c^ CO a fl CO 00 o c^ ■^t^OS— HCOiOt^OS—iCOUOt^ OS (T? ^ CO GO ■^ r:^ lO UO lOlOtncocococcco^>^-^-^- J> 00 X 00 00 ^J OS OSOSOSC7SO:OSOSOSOSOSOSC3S OS OS OS OS OS CO CO CO CO cococo cococococococococo CO CO CO CD CO - o ^ W CO T^i0C0i>Q0i^O'^(??C0rf40 CO t* 00 OS o ^^ TABLE OF CHORDS. 143 The table of chords, below, furnishes the means of laying down angles on paper more accurately than by an ordinary protx'actor. To do this, after having drawn and measured the first side (say ac) of the figure that is to be plotted; from its end c as a center, describe an arc ny of a circle of suflacient extent to subtend the angle at that point. The rad en with which the arc is described should be as great as convenience will permit ; and it is to be assumed as unity or 1 ; and must be decimally divided, and subdivided, to be used as a scale for laying down the chords taken from the table, in which their lengths are given in parts of said rad 1. Having described the arc, find in the table the length of the chord n t corresponding to the angle act. Let us suppose this angle to be 45°; then we find that the tabular chord is .7654 of our rad 1. There- fore from n we lay off the chord nt, equal to .7654 of our radius-scale ; and the lint cs drawn through the point t will form the reqd angle act oi 45^^. And so at each angle. The degree of accuracy attained will evidently depend on the length of the rad, and the neatness of the drafting. The method becomes preferable to the com- mon protractor in proportion as the lengths of the sides of the angles exceed the rad of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not much exceeding the same limits, the protractor will usually be preferable. The di- viders in boxes of instruments are rarely fit for accurate arcs of more than about 6 ins diam. In practice it is not necessary to actually describe the whole arc, but merely the portion near t, as well as can be judged by eye. We thus avoid much use of the India-rubber, and dulling of the pencil-point. For larger radii we may dis- pense with the dividers, and use a straight strip of paper with the length of the rad marked on one edge ; and by laying it from c toward s, and at the same time placing another strip (with one edge divided to a radius-scale) from n toward t, we can by trial find their exact point of intersection at the required point t. In such mat- ters, practice and some iu<^enuity are very essential to satisfactory results. We can- not devote more space to the subject. CHORDS TO A EADIUS 1. M. 0° 1° ,o 3° 40 5° 6° 7° 8° 9° 10° M. 0' .0000 .0175 .0349 .0524 .0698 .0872 .1047 .1221 .1395 .1569 .1743 0' 2 .0006 .0180 .0355 .0529 .0704 .0878 .1053 .1227 .1401 .1575 .1749 2 4 .0012 .0186 .0361 .0535 .0710 .0884 .1058 .1233 .1407 .1581 .1755 4 (> .0017 .0192 .0366 .0541 .0715 .0890 .1064 .1238 .1413 .1587 .1761 6 8 .0023 .0198 .0372 .0547 .0721 .0896 .1070 .1244 .1418 .1592 .1766 8 10 .0029 .0204 .0378 .0553 .0727 .0901 .1076 .1250 .1424 .1598 .1772 10 12 .0035 .0209 .0384 .0558 .0733 .0907 .1082 .1256 .1430 .1604 .1778 12 14 .0041 .0215 .0390 .0564 .0739 .0913 .1087 .1262 .1436 .1610 .1784 14 16 .0047 .0221 .0396 .0570 ..0745 .0919 .1093 .1267 .1442 .1616 .1789 16 18 .0052 .0227 .0401 .0576 .0750 .0925 .1099 .1273 .1447 .1621 .1795 18 20 .0058 .0233 .0407 .0582 .0756 .0931 .1105 .1279 .1453 .1627 .1801 20 22 .0064 .0239 .0413 .0588 .0762 .0;-»36 .1111 .1285 .1459 .1633 .1807 22 24 .0070 .0244 .0419 .0593 .0768 .0942 .1116 .1291 .1465 .1639 .1813 24 26 .0076 .0250 .0425 .0599 .0774 .0948 .1122 .1296 .1471 .1645 .1818 26 28 .0081 .0256 .0430 .0605 .0779 .0954 .1128 .1302 .1476 .1650 .1824 28 30 .0087 .0262 .0268 .0436 .0611 .0785 .0960 .1134 .1.308 .1482 .1656 .1830 30 32 .0093 .0442 .0617 .0791 .0965 .1140 .1314 .1488 .1662 .1836 32 .0448 .0622 .0797 .0971 .1145 .1320 .1494 .1668 .1842 34 .0454 .0628 .0803 .0977 .1151 .1.325 ,1,500 .1674 .1847 36 .0460 .0634 .0808 .0983 .11.57 .1.331 ,1.505 .1679 .1853 38 40 .0116 .0291 .0297 .0465 .0640 .0814 .0989 .1163 .1.337 .1511 .1685 .1859 40 42 .0122 .0471 .0646 .0820 .0994 .1169 .1343 .1517 .1691 .1865 42 44 .0128 .0303 .0477 .0651 .0826 .1000 .1175 .1349 .1.523 ,1697 .1871 44 46 .0134 .0308 .0483 .0657 .0832 .1006 .1180 .1.355 .1.529 ,1703 .1876 46 48 50 .0140 .0314 .0489 .0663 .0838 .1012 .1186 .1.360 .1534 ,1708 .1882 48 .0145 .0320 .0494 .0669 .0843 .1018 .1192 .1366 .1540 ,1714 .1888 50 52 .0151 .0326 .0,500 .0675 .0849 .1023 .1198 .1.372 ,1546 .1720 .1894 52 .0332 .0506 .0681 .0855 .1029 .1204 .1378 ,15.52 .1726 .1900 54 .0.337 .051 2 .0686 .0861 .10.35 .1209 .1384 .1558 ,1732 .1905 .1911 56 58 .0169 .0343 .0518 .0692 .0867 .1041 .1215 .1389 .1563 ,1737 58 SO .0175 .0349 .0524 .0698 .0872 .1047 .1221 .1395 .1569 .1743 .1917 60 144 TABLE OF CHORDS. Table of Cbords, io parts of a rad 1; for protraeting*— Continued.! M. 11° 12° 13° 14° 15° 16° 17° 18° 19° 20° M. 0' .1917 .2091 .2264 .2437 .2611 .2783 .2956 .3129 .3301 .3473 0' 2 .1923 .2096 .2270 .2443 .2616 .2789 .2962 .3134 .3307 .3479 2 4 .1928 .2102 .2276 .2449 .2622 .2795 .2968 .3140 .3312 .3484 4 6 .1934 .2108 .2281 .2455 .2628 .2801 .2973 .3146 .3318 .3490 6 8 .1940 .2114 .2287 .2460 .2634 .2807 .2979 .3152 .3324 .3496 8 10 .1946 .2119 .2293 .2466 .2639 .2812 .2985 .3157 .3330 .3502 10 J2 .1952 .2125 .2299 .2472 .2645 .2818 .2991 .3163 .3335 .3507 12 U .1957 .2131 .2305 .2478 .2651 .2824 .2996 .3169 .3341 .3513 14 16 .1963 .2137 .2310 .2484 .2657 .2830 .3002 .3175 .3347 .3519 16 18 .1969 .2143 .2316 .2489 .2662 .2835 .3008 .3180 .3353 .3525 18 20 .1975 .2148 .2322 .2495 .2668 .2841 .3014 .3186 .3358 .3530 20 22 .1981 .2154 .2328 .2501 .2674 .2847 .3019 .3192 .8364 .3536 22 M .1986 .2160 .2333 .2507 .2680 .2853 .3025 .3198 .3370 .3542 24 26 .1992 .2166 .2339 .2512 .2686 .2858 .3031 .3203 .3376 .3547 26 28 .1998 .2172 .2345 .2518 .2691 .2864 .3037 .3209 .3381 .3553 28 30 .2004 .2177 .2.351 .2524 .2697 .2870 .3042 .3215 .3387 .3559 30 32 .2010 .2183 .2357 .2530 .2703 .2876 .3048 .3221 .3393 .3565 32 U .2015 .2189 .2362 .2536 .2709 .2881 .3054 .3226 .3398 .3570 34 36 .2021 .2195 .2368 .2541 .2714 .2887 .3060 .3232 .3404 .3576 36 38 .2027 .2200 .2374 .2547 .2720 .2893 .3065 .3238 .3410 .3582 38 40 .2033 .2206 .2380 .2553 .2726 .2899 .3071 .3244 .3416 .3587 40 42 .2038 .2212 .2385 .2559 .2732 .2904 .3077 .3249 .3421 .3593 42 44 .2044 .2218 .2391 .2564 .2737 .2910 .3083 .3255 .3427 .3599 44 46 .2050 .2224 .2397 .2570 .2743 .2916 .3088 .3261 .3433 .3605 46 48 .2056 .2229 .2403 .2576 .2749 .2922 .3094 .3267 .3439 .3610 48 50 .2062 .2235 .2409 .2582 .2755 .2927 .3100 .3272 .3444 .3616 60 52 .2067 .2241 .2414 .2587 .2760 .2933 .3106 .3278 .3450 .3622 52 54 .2073 .2247 .2420 .2593 .2766 .2939 .3111 .3284 .3456 .3628 54 56 .2079 .2253 .2426 .2599 .2772 .2945 .3117 .3289 .3462 .3633 .56 58 .2085 .2258 .2432 .2605 .2778 .2950 .3123 .3295 .3467 ..3639 58 60 .2091 .2264 .2437 .2611 .2783 .2956 .3129 .3301 .3473 .3645 60 M. 21° 22° 23° 24° 25° 26° 27° 28° 29° 30° M. 0' .3645 .3816 .3987 .4158 .4329 .4499 .4669 .4838 .5008 .5176 0' 2 .3650 .3822 .3993 .4164 .4334 .4505 .4675 .4844 .5013 .5182 4 .3656 .3828 .3999 .4170 .4340 .4510 .4680 .4850 .5019 .5188 6 .3662 .3833 .4004 .4175 .4346 .4516 .4686 .4855 .5024 .5198 8 .3668 .3839 .4010 .4181 .4352 .4522 .4692 .4861 .5030 .5199 10 .3673 .3845 .4016 .4187 .4357 .4527 .4697 .4867 .5036 .5204 12 .3679 .3850 .4022 .4192 .4363 .4533 .4703 .4872 .5041 .5210 14 .3685 .3856 .4027 .4198 .4369 .4539 .4708 .4878 .5047 .5216 16 .3690 .3862 .4033 .4204 .4374 .4544 .4714 .4884 .5053 .5221 18 .3696 .3868 .4039 .4209 .4380 .4550 .4720 .4889 .5058 .5227 20 .3702 .3873 .4044 .4215 .4386 .4556 .4725 .4895 .5064 .5233 20 22 .3708 .3879 .4050 .4221 .4391 .4561 .4731 .4901 .5070 .5238 22 24 .3713 .3885 .4056 .4226 .4397 .4567 .4737 .4906 .5075 .5244 24 26 .3719 .3890 .4061 .4232 .4403 .4573 .4742 .4912 .5081 .5249 26 28 .3725 .3896 .4067 .4238 .4408 .4578 .4748 .4917 .5086 .5255 28 30 .3730 .3902 .4073 .4244 .4414 .4584 .4754 .4923 .5092 .5261 30 32 .3736 .3908 .407« .4249 .4420 .4590 .4759 .4929 .5098 .5266 82 34 .3742 ..3913 .4084 .4255 .4425 .4595 .4765 .4934 .5103 .5272 34 36 .3748 .3919 .4090 .4261 .4431 .4601 .4771 .4940 .5109 .5277 36 88 .3753 ..3925 .4096 .4266 .4437 .4607 .4776 .4946 .5115 .5283 38 40 .3759 .3930 .4101 .4272 .4442 .4612 .4782 .4951 .5120 .5289 40 42 .3765 .3936 .4107 .4278 .4448 .4618 .4788 .4957 .5126 .5294 42 44 .8770 .3942 .4113 .4283 .4454 .4624 .4793 .4963 .5131 .5300 44 46 .3776 ..3947 .4118 .4289 .44.S9 .4629 .4799 .4968 .5137 .5306 46 48 .3782 .3953 .4124 .4295 .4465 .4635 .4805 .4974 .5143 .5311 48 50 .3788 .3959 .4130 .4300 .4471 .4641 .4810 .4979 .5148 .5317 50 52 .3793 .3965 .4135 .4.306 .4476 .4646 .4816 .4985 ..5154 .5322 52 54 .3799 ..3970 .4141 .4312 .4482 .4652 .4822 .4991 ..5160 .5328 54 56 .3805 ..3976 .4147 .4317 .4488 .4658 .4827 .4996 .5165 .5334 bS 58 .3810 ..3982 .4153 .4323 .4493 .4663 .4833 .5002 .5171 .53.39 58 eo .3816 .3987 .4158 .4329 .4499 .4669 .4838 .5008 .5176 .5345 «0 TABLE OF CHORDS. 145 Table of cliords,iii parts of a rad 1; for protracting— Continued M. 31° .5345 .5350 .5356 .5378 .5384 .5390 .5395 .5401 .5406 .5412 .5418 .1423 .5429 32° .5513 .5518 .5524 .5530 .5535 .5541 .5546 .5552 .5557 .5563 .5569 .5574 .5580 .5585 .5591 .5597 .5691 .5697 .5703 .5708 .5714 .5719 .5725 .5730 .5736 .5742 .5747 .5753 .5758 .5764 34° .5847 .5853 .5859 .5864 .5870 .5875 .5881 .5886 .5897 .5903 35° .6014 .6020 .6025 .6031 .6036 .6042 .5909 .5914 .5920 .5925 .5931 .6075 .6081 .(i086 .6032 .6097 37° .6180 .6186 .6191 .6197 .6202 .6208 .6214 .6219 .6225 .6230 .6236 .6346 ,6352 .6357 .6363 .6368 .6374 .6379 .6385 .6390 .6396 .6401 .6241 .6407 .6247 .6252 .6258 .6263 ,6412 .6118 .6423 .6429 38° .6511 .6517 .6522 .6528 .6533 .6539 .6544 .6550 .6555 .6561 .6572 .6577 .6583 .6588 .6594 39° .66«2 .6687 .6693 .6698 .6704 .6709 .6715 .6720 .6725 .6731 .6736 .6742 .6747 .6753 .6758 40° M. .6840 .6846 .6851 .6857 .6862 .6868 .6873 .6879 .6884 .6906 .6911 .6917 .6922 .5434 .5440 .5446 .5451 .5457 .5602 .5608 .5613 .5619 .5625 .5769 .5775 .5781 .5786 .5792 .5936 .5942 .5947 .5953 .5959 .6103 .6108 .6114 .6119 .6125 .6274 .6280 .6285 .6291 ,6434 ,6440 .6445 .6451 .6156 .6599 .6605 6610 .6616 .6621 .6764 .6769 .6775 .6780 .6786 .6928 .6933 .6939 .o944 .6950 .5462 .5468 .5474 .5479 .5485 .5490 .5496 .5502 .5507 .5513 .5630 .5636 .5641 .5647 .5652 .5797 .5803 .5808 .5814 .5820 .5964 .5970 .5975 .5981 .5986 .6130 .6136 .6142 .6147 .6153 .0236 .6302 .6307 .6313 .6318 .6462 I .6467 I .6473 I .6478 ! .6484 .6627 .6632 .6638 .6643 .6649 .6791 ,6797 .6802 .6808 .6813 .6955 .6961 .,5664 .5669 .5675 .5825 .5831 .5836 .5842 .5847 .5992 .5997 .6003 .6158 .6164 .6175 .6180 .6324 .6189 .6+95 .6335 .6341 .6346 .6500 .6506 .6511 .6654 .6660 .6665 .6819 .6824 .6829 .6835 ,6840 .6988 .6993 .6999 .70t4 M, 41° .7004 42° 43° 44° 45° 46° 47° 48° 49° 50° M. 0' .7167 .7330 .7492 .7654 .7815 .7975 .8135 .8294 .8452 0' 2 .7010 .7173 .7335 .7498 .7659 .7820 .7980 .8140 .8299 ,8458 2 4 .7015 .7178 .7341 ,7503 .7664 .7825 .7986 .8145 .8304 .8463 4 6 .7020 .7184 .7346 ,7508 .7670 .78;ii .7991 .8151 .8;^10 .8468 6 8 .7026 .7189 .7352 .7514 .7675 .7836 .7996 .8156 .8315 ,8473 8 10 .7031 .7195 .7.357 .7519 .7681 .7841 ,8002 .8161 .8320 ,8479 10 12 .7037 .7200 .7362 .7524 .7686 .7847 .8007 ,8167 .8326 .8484 12 14 .7042 .7205 .7368 .7530 .7691 .7852 .8012 .8172 .8331 ,8489 14 16 .7048 .7211 .7373 .7535 .7697 .7857 .8018 ,8177 .8336 ,8495 16 18 .7053 .7216 .7379 .7541 .7702 .7863 .8023 .8183 .8341 ,8500 18 20 .7059 .7222 .7384 .7546 .7707 .7868 .8028 .8188 .8347 ,8505 20 22 .7064 .7227 .7390 .7551 .7713 .7873 .8034 .8193 .8352 ,8510 22 24 .7069 .7232 .7395 .7557 .7718 .7879 .8039 8198 .8357 .8516 24 26 .7075 ,7238 .7400 .7562 .7723 .7884 .8044 .8204 .8363 .8521 26 28 .7080 .7243 .7406 .7.568 .7729 .7890 .8050 .8209 .8368 .8526 28 IK) .7086 .7249 .7411 .7573 ,7734 .7895 .8055 .8060 .8214 .8373 .8531 30 32 .7091 .7254 .7417 .7578 .7740 .7900 .8220 .8378 .8537 32 34 .7097 .7260 .7422 .7.584 .7745 .7906 .8066 .8225 .8384 .8542 54 36 .7102 .7265 .7427 .7589 .7750 .7911 .8071 .8230 .8389 .85+7 36 38 .7108 .7270 .7433 .7595 .7756 .7916 .8076 ,82.36 .8394 .8552 3d 40 ,7113 .7276 .7438 .7600 .7761 .7922 .8082 .8241 .8400 .8558 40 42 .7118 .7281 .7443 .7605 .7766 .7927 .8087 .8246 .8405 .8.563 42 44 .7124 .7287 .7449 .7611 .7772 .7932 .8092 .8251 .8410 .85«? 44 46 .7129 .7292 .7454 .7616 .7777 .7938 .8098 .8257 .8415 .8.573 46 48 .7135 .7298 .7460 .7621 .7782 .7943 .8103 .8262 .8421 .8579 48 50 .7140 .7303 .7465 .7627 .7788 .7948 .8108 .8267 .8426 .8584 50 52 .7146 .7308 .7471 .7632 .7793 .7954 .8113 ,8273 .8431 .8589 52 54 .7151 .7314 .7476 .7638 .7799 ..7959 .8119 .8278 .8437 .8594 54 56 .7156 .7319 .7481 .7643 .7804 .7964 .8124 .8283 .8442 .8600 56 58 ,7162 .7325 .7487 .7648 .7809 .7970 .8129 .8289 .8447 .8605 ,5« 30 .7167 .7330 .7492 ,7654 ,7815 .7975 ,8135 .829A .8452 .8610 6C« lu 146 TABLE OF CHORDS. Table of ebor€ls,iii parts of a rad 1 ; for protracting: — C M. 51° 52° 53° 54° 55° 56° 57° 58° 59° 60° M. 0' .8610 .8767 .8924 .9080 .9235 .9389 .9543 .9696 .9848 1.0000 0* 2 .8615 .8773 .8929 .9085 .9240 .9.395 .9548 .9701 .9854 1.0005 t 4 .8621 .8778 .8934 .9090 .9245 .9400 .9553 .9706 .9859 1.0010 4 6 .8626 .8783 .8940 .9095 .9250 .9405 .9559 .9711 .9864 1.0015 6 8 .8631 .8788 .8945 .9101 .9256 .9410 .9564 .9717 .9869 1.0020 8 10 .8636 .8794 .8950 .9106 .9261 .9415 .9569 .9722 .9874 1.0025 10 12 .8642 8799 .8955 .9111 .9266 .9420 .9574 .9727 .9879 1.0030 12 U .8647 .8804 .8960 .9116 .9271 .9425 .9579 .9732 .9884 1.0035 14 16 16652 .8809 .8966 .9121 .9276 .9430 .9584 .9737 .9889 1.0040 16 18 .8657 .8814 .8971 .9126 .9281 .9436 .9589 .9742 .9894 1.0045 18 20 .8663 .8820 .8976 .9132 .9287 .9441 .9594 .9747 .9899 1.0050 20 22 .8668 .8825 .8981 .9137 .9292 .9446 .9599 .9752 .9904 1.0055 22 24 .8673 .8830 .8986 .9142 .9297 .9451 .9604 .9757 .9909 1.0060 24 26 .8678 .8835 .8992 .9147 .9302 .9456 .9610 .9762 .9914 1.0065 26 28 .8684 .8841 .8997 .9152 .9307 .9461 .9615 .9767 .9919 1.0070 28 S9 .8689 .8846 .9002 .9157 .9312 .9466 .9620 .9772 .9924 1.0075 30 32 .8694 .8851 .9007 .9163 .9317 .9472 .9625 .9778 .9929 1.0080 32 34 .8699 .8856 .9012 .9168 .9323 .9477 .9630 .9783 .99.34 1.0086 34 36 .8705 .8861 .9018 .9173 .9328 .9482 .9635 .9788 .9939 1.0091 36 38 .8710 .8867 .9023 .9178 .9333 .9487 .9640 .9793 .9945 1.0096 38 40 .8715 .8872 .9028 .9183 .9338 .9492 .9645 .9798 .9950 1.0101 40 42 .8720 .8877 .9033 .9188 .9343 .9497 .9650 .9803 .9955 1.0106 42 44 .8726 .8882 .9038 .9194 .9348 .9502 .9655 .9808 .9960 1.0111 44 46 .8731 .8887 .9044 .9199 .9353 .9507 .9661 .9813 .9965 1.0116 46 48 .8736 .8893 .9049 .9204 .9359 .9512 .9666 .9818 .9970 1.0121 48 50 .8741 .8898 .9054 .9209 .9364 .9518 .9671 - .9823 .9975 1.0126 50 52 .8747 .8903 .9059 .9214 .9369 .9523 .9676 .9828 .9980 1.0131 52 54 .8752 .8908 .9064 .9219 .9374 .9528 .9681 .9833 .9985 1.0136 54 56 .8757 .8914 .9069 .9225 .9379 .9533 .9686 .9838 .9990 1.0141 56 58 .8762 .8919 .9075 .9230 .9384 .9538 .9691 .9843 .9995 1.0146 58 60 .8767 .8924 .9080 .9235 .9389 .9543 .9696 .9848 1.0000 1.0151 60 M. 6J° 62° 63° 64° 65° 66° 67° 68° 69° 70° M. 0' 1.0151 1.0301 1.0450 1.0598 1.0746 1.0893 1.1039 1.1184 1.1328 1.1472 0- 2 1.0156 1.0306 1.0455 1.0603 1.0751 1.0898 1.1044 1.1189 1.1333 1.1476 2 4 1.0161 1.0311 1.0460 1.0608 1.0756 1.0903 1.1048 1.1194 1.1338 1.1481 4 6 1.0166 1.0316 1.0465 1.0613 1.0761 1.0907 1.1053 1.1198 1.1342 1.1486 6 8 1.0171 1.0321 1.0470 1.0618 1.0766 1.0912 1.1058 1.1203 1.1347 1.1491 H 10 1.0176 1.0326 1.0475 1.0623 1.0771 1.0917 1.1063 1.1208 1.1352 1.1495 10 12 . 0181 1.0331 1.0480 1.0628 1.0775 1.0922 1.1068 1.1213 1.1357 1.1500 12 14 1.0186 1.0336 1.0485 1.0633 1.0780 1.0927 1.1073 1.1218 1.1362 1.1505 14 16 1.0191 1.0341 1.0490 1.0638 1.0785 1.0932 1.1078 1.1222 1.1366 1.1510 16 18 1.0196 1.0346 1.0495 1.0643 1.0790 1.0937 1.1082 1.1227 1.1371 1.1514 la 20 1.0201 1.0351 1.0500 1.0648 1.0795 1.0942 1.1087 1.1232 1.1376 1.1519 20 22 1.0206 1.0356 1.0504 1.0653 1.0800 1.0946 1.1092 1.1237 1.1381 1.1524 22 24 1.0211 1.0361 1.0509 1.0658 1 .0805 1.0951 1.1097 1.1242 1.1386 1.1529 24 26 1.0216 1.0366 1.0514 1.0662 1.0810 1.0956 1.1102 1.1246 1.1390 1.1533 26 28 1.0221 1.0370 1.0519 1.0667 1.0815 1.0961 1.1107 1.1251 1.1395 1.1538 28 30 1.0226 1.0375 1.0524 1.0672 1.0820 1.0966 1.1111 1.1256 1.1400 1.1543 30 32 1 .0231 1.0380 1.0529 1.0677 1.0824 1.0971 1.1116 1.1261 1.1405 1.1548 32 34 1.0236 1.0385 1.0534 1.0682 1.0829 1.0976 1.1121 1.1266 1.1409 1.1552 34 .% 1.0241 1.0390 1.0539 1.0687 1.0834 1.0980 1.1126 1.1271 1.1414 1.1557 36 88 1.0246 1 .0395 1.0544 1.0692 1.0839 1.0985 1.1131 1.1275 1.1419 1.1562 3S 40 1.0251 1.0400 1.0549 1.0697 1.0844 1.0990 1.1136 1.1280 1.1424 1.1567 40 42 1.0256 1 .0405 1.0554 1.0702 1 .0849 1.0995 1.1140 1.1285 1.1429 1.1571 42 44 1.0261 1.0410 1.0559 1.0707 1.0854 1.1000 1.1145 1.1?<'0 1.1433 1.1576 44 46 1.0266 1.0415 10564 1.0712 1.0859 1.1005 1.1150 1.12^.. 1.1438 1.1581 46 4fi 1.0271 1.0420 1.0569 1.0717 1.0863 1.1010 1.11.55 1.1299 1.1443 1.1586 48 50 1.0276 1.0425 1.0574 1 .0721 1.0868 1.1014 1.1160 1.1304 1.1448 1.1590 50 52 1.0281 1.0430 1.0579 1.0726 1.0873 1.1019 1.1165 1.1309 1.1452 1.1595 52 54 1.0286 1.0435 1.0584 1.0731 1.0878 1.1024 1.1169 1.1314 1.1457 1.1600 54 56 1.0291 1.0440 1.0589 1.0736 1.0883 1.1029 1.1174 1.1319 1.1462 1.1605 56 58 , 1.0296 1.0445 1.0593 1.0741 1.0888 1.1034 1.1179 1.1.323 1.1467 1.1609 58 60 1.0301 1.0460 1.0698 1.0746 1.0893 1.1039 1.1184 1.1S2S 1.1472 1.1j614 «U TABLE OF CHORDS. 147 Table of Cliovds, in parto of a rad 1 ; for protracting: — -Contiuued M. 71° 72° 73° 74° 75° 76° 77° 78° 79° 80° M, 0' 1.1614 1.1756 1.1896 1.2036 1.2175 1.2313 1.2450 1.2586 1.2722 1.2856 2 1.1619 1.1760 1.1901 1.2041 1.2180 1.2318 1.2455 1.2591 1.2726 1.2860 2 4 1.1624 1.1765 1.1906 1.2046 1.2184' 1.2322 1.2459 1.2595 1.2731 1.2865 ♦ fi 1.1623 1.177e 1.1910 1.2050 1.2189 1.2327 1.2464 1.2600 1.2735 1.2869 6 8 1.1633 1.1775 1.1915 1.2055 1.2194 1.2332 1.2468 1.2604 1.2740 1.2874 ft 10 1.1638 1.1779 1.1920 1.2060 1.2198 1.2336 1.2473 1.2609 1.2744 1.2^78 10 Vf. 1.1642 1.1784 1.1924 1.2064 1.2203 1.2341 1.2478 1.2614 1.2748 1.2882 12 14 1.1647 1.1789 1.1929 1.-2069 1.2208 1.2345 1.2482 1.2618 1.2753 1.2887 U 1« 1.1652 1.1793 1.1934 1.2073 1.2212 1.2350 1.2487 1.2623 1.2757 1.2891 IB 1« 1.1657 1.1798 1.1938 1.2078 1.2217 1.2354 1.2491 1.2627 1.2762 1.2896 1« 20 1.1661 1.1803 1.1943 1.2083 1.2221 1.2359 1.2496 1.2632 1.2766 1.2900 20 22 1.1666 1.1807 1.1948 1.2087 1.2226 1.2364 1.2500 1.2636 1.2771 1.2305 22 24 1.1671 l.lPi2 1.1952 1.2092 1.2231 1.2368 1.2505 1.2641 1.2775 1.2909 24 26 1.1676 1.1817 1.1957 1.2097 1.2235 1.2373 1.2509 1.2645 12780 1.2914 26 28 1.1680 J-1821 11962 1.2101 1.2240 1.2377 1.2514 1.2650 1.2784 1.2918 28 30 1.1685 1.1826 1.1966 1.2108 1.2244 1.2382 1.2518 1.2654 1.2789 1.2922 30 32 1.1690 1.1831 1.1971 1.2111 1.2249 1.2386 1.2523 1.2659 1.2793 1.2927 32 34 1.1694 1.1836 1.1976 1.2115 1.2254 1.2391 1.2528 1.2663 1.2798 1.2931 34 36 1.1699 1.1840 1.1980 1.2120 1.2258 1.2;i96 1.2532 1.2668 1.2802 1.2936 3« 38 1.1704 1.1845 1.1985 1.2124 1.2263 1.2400 1.2537 1.2672 1.2807 1.2940 38 40 1.170!* 1.1850 1.1990 1.2129 1.2267 1.2405 1.2541 1.2677 1.2811 1.2945 40 42 1.171S 1.1854 1.1994 1.2134 1.2272 1.2409 1.2546 1.2681 1.2816 1.2949 42 44 i.nis 1.1859 1.1999 1.2138 1.2277 1.2114 1.2550 1.2686 1.2820 1.2954 44 46 1.1723 1.1864 1.2004 1.2143 1.2281 1.2418 1.2555 1.2690 1.2825 1.2958 46 48 1.1727 1.1868 1.2008 1.2148 1.22K6 1.2423 1.2559 1.2695 1.2829 1.2962 48 50 1.1732 1.1873 1.2013 1.2152 1.2-290 1.2428 1.2564 1.2699 1.2833 1.2967 50 52 1.17.37 1.1878 1.2018 1.2157 1.2295 1.2432 1.2568 1.2704 1.2838 1.2971 52 54 1.1742 1.1882 1.2022 1.2161 1.2299 1.2437 1.2573 1.2708 1.2842 1.2976 54 56 1.1746 1.1887 1.2027 1.2166 1.2304 1.2441 1.2577 1.2713 1.2847 1.2980 56 58 1.1751 1. 1892 1.2032 1.2171 1.2309 1.2446 1.2.582 1.2717 1.2851 1.2985 58 60 1.1756 1.1896 1.2036 1.2175 1.2313 1.2450 1.2586 1.2722 1.2856 1.2989 60 K. 81° 82° 83° 84° 85° 86° 87° 88° 89° M. 0' 1.2;)89 1.3121 1.3252 1.3.383 1.3512 1.3640 1.3767 1.3893 1.4018 0' 2 1.2993 1.3126 1.3257 1.3387 1.3516 1.3614 1.3771 1.3897 1.40-22 2 4 1.2998 1.3130 1.3261 1.3391 1.35-20 1.3648 1.3776 1.3902 1.4026 4 6 1.3002 1.3134 1.3265 1-3396 1.35-25 1.3653 1.3780 1.3906 1.4031 • 8 1.3007 1.3139 1.3270 1.3400 1.352;) 1.3657 1.3784 1.3910 14035 8 10 1.3011 1.3143 1.3274 1.3404 1.3533 1.3538 1.3661 1.3665 1.3788 1.3914 1.4039 10 12 1.3015 1.3147 1.3279 1.3409 1.3792 1.3918 1.4043 12 14 1.3020 1.3152 1.3283 1.3413 1.3542 1.3670 1.3797 1.3922 1.4047 14 16 1.3024 1.3156 1.3287 1.3417 1.3546 1.3674 1.3501 1.3927 1.4051 16 18 1.3029 1.3161 1.3292 1.3421 1.3550 1.3678 1.3805 1.3931 1.4055 18 20 1.3033 1.3165 1.3296 1.3426 1.3555 1.3682 1.3809 1.3935 1.4060 SO 22 1.3038 1.3169 1.3300 1.3430 1.3559 1.3687 1.3813 1.3939 1.4064 22 24 1.3042 1.3174 1.3.305 1.31.34 1.3563 1.3691 1.3818 1.3943 1.4068 24 26 1.3016 1.3178 1.3309 1.3439 1.3567 1.3695 1.3822 1 .3947 1.4072 26 28 1.3051 1.3183 1.3313 1.3443 1.3572 1.3699 1.3826 1.3952 1.4076 28 30 1.3055 1.3187 1.3318 1.3447 1.3576 1.3704 1.3830 1.3956 1.4080 30 32 1.3060 1.3191 1.3322 1.3452 1.3580 I. .3708 1.3834 1.3960 1.4084 32 34 1 3064 1.3196 1.3323 1.3456 1.3585 1.3712 1.3839 1.3964 1.4089 34 S6 1.3068 1.3200 1.3331 1.3460 1.3589 1 3716 1.3843 1.3968 1.4093 36 38 1.3073 1.3204 1.3335 1.3465 1.3593 1.3721 1 .3847 1.3972 1.4097 38 40 1.3077 1.3209 1.3339 1.3169 1.3597 1.3725 1.3851 1.3977 1.4101 40 42 1.3082 1.3213 1.3344 1.3473 1.3602 1.3729 1.3855 1.3981 1.4105 42 44 1.3086 1.3218 1.3348 1.3477 1.3606 1.3733 1.3860 1.3985 1.4109 44 46 1.3090 1.3222 1.3352 1.3482 1.3610 1.3738 1.3864 1.3989 1,4113 46 48 1.3095 1..3226 1.3357 1..3486 1.3614 1.3742 1.3868 1.3993 1.4117 48 50 1.3099 1.3231 1.3361 1.3490 1.3619 1.3746 1.3872 1.3997 1.4122 50 52 1.3104 1.3235 1.3365 1.3495 1.3623 1.3750 1.3876 1.4002 1.4126 52 64 1.3108 1 .3239 1.3.370 1.3499 1.3627 1.3754 1.3881 i.40on 1.4130 64 66 1.S112 1.3244 1.3374 1.3503 1.3631 1.3759 1.3885 1.4010 1.4134 66 58 1.3117 1.3248 1.3378 1.3508 1..3636 1.3763 1..3ft89 1.4014 1.4138 58 60 1.3121 1.3252 1.3383 1.3512 1.3640 1.3767 1.3893 1.4018 1.4142 60 148 POLYGOKS. POIiYOONS. Pentagon. 5 siUtA. Any straight-sided fig is called a polygon. If all tne sides and angles are equal, it is a resHlat polygon ; if not, it is irregular. Of course the number of polygons is infinite. Table of Regular Polyg'ons. Number Name Area = Radius of cir- Interior angle Angle at cen* of of (square of one subtended Sides. Polygon. side) mult by mult by sides. by a side. M Equilateral triangle. 1 .433013 .577350 60° 120° 4 Square. 1.000000 .707107 90° 90° 5 Pentagon. 1.720477 .850651 108° 72° 6 Hexagon. 2.598076 1.000000 120° 60° 7 Heptagon. 3.638912 1.152382 128° 34.2857' 51° 25.7143' 8 Octagon. 4.828427 1.306563 135° 45° 9 Nonagon. fi.l81S24 1.461902 140° 40° 10 Decagon. 7.694209 1.618034 144° 36° 11 Undecagon. 9.365640 1774733 147° 16.3636' 32° 43.6364' 12 Dodecagon. 11.196152 1.931854 150° 30° Area of any regular polygon = length of one side, ab X perp p drawn from cen of fig t« cen of side X hulf the number of sides. Sum of interior angles, a b c, etc, of any polygon, regular or irregular = 180° X (number of side.s — 2). Angle at cen subtended by a side, in any regular polygon = 360° -^ number of aides* TRIA?rOI.ES. We speak here of plane triangles only ; or those having straight sides. A triangle is equilateral when all its sides aie equal, as a ; isosceles when only two sides are equal, as B; scalene when all the sides are unequal, as C. D. and E ; ucutc-angled when all its angles are acute, or each less than 90°, as A, B, and C ; right-angled when it contains a right angle, as D ; obtuse-angled when it contains an obtuse augle. ur one gi eater than 90^. as E. All the three angles of any triangle are equal to two right angles, or 180 -' ; tlieretore, it we know two or them, we can tind the third by ^ subtracting their sum from 180°. All triangles which liavc equal bases, ^___„^ J? and equal perp heights, have also equal iirciis; tliusthe areas ot awe, aw d.and a w c, are equal to each other. The area of any triangle is equal to half that of any parallelogram which has an equal base, and unequal [..erp height. The areas of triangles which have equal bases, but diff perp heights, are to each other as, or in proportion to. their perp heights; thus the triangle awn, •with a perp height s n, equal to but one-half that (s e) of the three other trian- gles, but with the same base a w, has also but half the area of either of those "" others. Area of any triangle. Figs A, B, C, D, E, = half the base, S. X the height, or perp dist p to the opposite angle. Any side may be talsen as the base of a triangle ; but the perp height must always be measured from thoside so assumed; to do which, the side must sometimes be prolonged, as in Fig E ; but the prolongation is not to be considered as a part of the base. Area of any equilateral triangle = .433013 X square of one side. To find area, having the three sides. Add them opether ; div the sum by 2 ; from the -lalf sum, subtract each side separately; mult the half sum and the three remainders continuously together ; take the sq rt of the prod. Ex.— The three Bides = 20, 30, 40 ft. Here 20 + 30 + 40 =90; and — = 45. And 45 — 20 = 25; 45 — 30 = 15; and 45 — 40 = 5. And 45 X 25 X 15x5 = 84375' and the sq rt of 84375 is 290.47 sq ft, «rea reqd. TRIANGLES. 149 To find area, liaTing: one side and tbe 2 angrles at its ends. Add the 2 angles together: take the sum from 180°; the rem will be the angle opp the given side. Find the nat sine of this angle ; also find the nat sines of the other angles, and mult them together. Then as the nat sine of the single angle, is to the prod of the nat sines of the other 2 angles, so is the square of the given side to double the reqd area. To find area, having: two sides, and the included ang^le. Mult together the two sides, and the nat sine of the included angle ; div by 2. Ex.— Sides 650 ft and 980 ft ; included angle 69"^ 20'. By the table we find the nat sine .93561 ^^ , 650 X 980 X .9356 „„„„„, therefore, =: 297988.6 square ft area. To find area, having- the three ang^les and the perp heig-ht, a b. Find the nat sines of the three angles ; mult together the sines of the angles d and o ; div the sine of the angle b by the prod ; mult the quot by the square of the perp height a b; div by 2. To find any side, as a o, having the three angles, d, b and o, and the area. (Sinp of rf X sine of o) j sine of ft t » twice the area : square of d o. The perp height of an equilateral triangle is e(|ual to one side X .866025. Hence one of its sides is equal to the perp height div by . 866025 or to perp height X M547. Or, to find a Klde« mult the sq rt of its area by 1.51967. The side of an equilateral triangle, mult by .658037 — side of $ square of the same area ; or mult by .742517 it gives the diam of a circle oif the same area. The following apply to any plane triangle, whether oblique or right-angled : 1, The three angles amount to 180^, or two right angles. 8. Any exterior angle, as A C 7i, is equal to the two interior and opposite •nes, A and B. 8. The greater side is opposite the greater angle, 4. The sides are as the sines of the, opposite angles. Thus, the side a is to the side b as the sine of A is to the sine of B. 5. If any angle as s be bisected by a line s o, the two parts mo, on of the opposite side m n will be to each other as the other two sides am, sn ; or, m : o n :: s m : 8 n. 6. If lines be drawn from each angle r s t to the center of the opposite side, they will cross each other at one point, a, and the short part of each of the lines will be the third-part of the whole Hue. Also, a is the een of grav of the triangle. 7. If lines be drawn bisecting the three angles, they will meet at a point ^V"~.^^^\^ perpendicularly equidistant from each side, and consequently the centev f l^^ \ ^^^-<:^>. 1 of the greatest circle that can be drawn in the tria/nglo. ^ 8. If a line s n be drawn parallel to any side c a, the two triangles ran^rca, will be similar. 9. To divide any triangle a c r into two equal parts by a line a n parallel to any one of its sides c o. On either one of the other sides, as o r, as a diam, iescribe a semicircle a o r; and find its middle o. From r (opposite c a), with radius r o, describe the arc o n. Prom n draw n s. par- allel to c a. 10. To find the greatest parallelogram that can be drawn in any given triangle o nb. Bisect the three sides at a c e, and join a c, a e, e c. Then either a e b c, a e c o, or a c en, each equal to half the triangle, will be the reqd parallelogram. Any of these parallelograms can plainly be converted into a rectangle of equal area, and the greatest that can be drawn in the triangle. „ , 10^4. If a line a c bisects any two sides o 5, o n, of a triangle, it will be par- allel to the third side n b. and half as long as it. 11. To find the greatest square that can be drawn in any triangle axr. From an angle as a draw a perp a n to the opposite side xr, and find its length. Then n, or a side v t oi the square will — ; xr-{- an Rem.— If the triangle is such that two or three such perps can be drawn, then two or three equal squares may be found. mo n 150 PLANE TRIGONOMETRY. Rig:Iit-aii$>:led Triaiig^les. A.11 the foregoing apply also to right-angled triangles; but what follow apply to them only. Call the right angle A, and the others B and C ; and call the sides respaotivelj opposite to them a, b, and c. Then is B n = -~'—^ ~ " '^ Sine IV ** - "Sin( Xa 6=aX = c X Sec E p^ = 6X See 0=1/524. A Cosine ( Sine B = a X Cos = c X Cot C = c X Tang B, Sine C = aXCosB-6X Tang C. e & e Also Sine of C = - ; Cos C = - / Tang C = j" . And Sine of B = r Cos B =■ ~ ; Tang B = -. And Sine of A or 90° = 1. Cos A = 0. Tang A =: infinity. Sec A = Infinity. I If from the right angle o a line w be drawn perp to the hypothenuse or long side A g, then th€ two small triangles to h, o tv g, and the large one ohg, will be similar. Or g w:w o',:w o: IV h; and gw X w h = w 0^. 5. A line drawn from the right angle to the center of the long sida will be half as long as sa'd side. 3. If on the three sides oh, o g, gh we draw three squares t, u, v, or three circles, or triangles, or any other three figs that are similar, then the area of the largest one is equal to the sum of the areas of the two others. 4« In a triangle whose sides are as .3, 4, and 5 fas are those of the tri> angle A B C), the angles are very approximately 90"; 53" 7' 48.38"; and 360 52' 11,62'/, Their Sines, 1. ; .8; and .6. Their Tangs, infinity ; 1.3333 ; and .75. 6, One whose sides are as 7, 7, and 9.9, has very approv. one angle of 9(fi and two of 45° each, near enough for all practical purposes. PLANE TEIGONOMETEY. fzAtra trigonometry teaches how to find certain unknown parts of plane, or straight - sided' irt* angles, by means of other parts which are known ; and thus enables us to measure inaccessible dis- tances, &c. A triangle consists of six parts, namely, three sides, and three angles; and if we know any three of these, (except the three angles, and in the ambiguous case under " Case 2,") we can find the other three. The following four cases include the whole subject; the student shotLJ commit them to memory, Case 1. Having any two ang^les, and one side, to find tlie otlier sides and an$?Ie. Add the two angles together ; and subtract their sum from ISO©; the rem ♦ill be the third ang'.e. And for the sides, as Sine of the angle , Sine of the angle opp the given side • opp the reqd side Use the side thus found, as the given one ; and in the same manner the third side. Fig-. W given side : reqd side. Case 2. b a Figr. X. Having two sides, 6«, ac. Fig X, and the angle fihe^ opposite to one of them, to find the other side and anglei* Side a c opp The other the given an- * given side ; gle ab c b a Sine of the given angle ab c Sine of angle 6 d a or bca opposite the other given side b a. Having found the sine, take out the corresponding angle from the table of nat sines, but, in doing so, if the side o c opp the given angle iff shorter than the other given side b a, bear in mind that an angle and its sup- plement have the same sine. Thus, in Fig X, the sine, as found above, is opp the angle 6 c a in the table. But a c, if shorter than b a, can evidently b» laid ofiF in the opp direction, a d, in which case b da ia the supplement of 6 c a. If a c is as long as, or longer than, b a, there can be no doubt ; for in that case it cannot be drawn toward b, but only toward n. and the angle ic a will be found «t once iu the table, oop the sine as found above. PLAKE TRIGONOMETR'X. 151 When the two angles, ah c, b e a, have been found, find the remaintng side by Case I. Vor the remaining angle, bac, add together the angle ab c first given, and the one, 6 c a, foam aa above. Deduct their sum from 180°. Case 3. Having: two sidesi, and tbe ang^le included between tliem. Take the angle rrom 180°; the rem will be the sum of the two unknown angles. Div this sum bj S; and find tbe nat tang of the quot. Theu as The sum of the , ipu-j- Ji» . . Tang of half the sum of , Tang of half two given sides • ^neirain , , the two unknown angles • their diflF. Take from the table of nat tang, the angle opposite this last tang. Add this angle to the half sum •f the two unknown angles, and it will give the angle opp the longest given side ; and subtract it from the same half sum, for the angle opp the shortest given side. Having thus found the angles, Bad the third side bj Case 1. As a practical example of the use of Case 3, we can ascertain the dist n m, across a deep pond, by measuring two lines n o and m n; and the angle n o m. Prom these data we may calculate nm ; or by drawing the two sides, and the angle on paper, by a scale, we can afterward measure n m on the drawing. The base ! Case 4. Having: the three sides, T» find the three anglea ; upon one side a & as a base, draw (or suppose to be drawn) a perp c g troru fche opposite angle c. Find the diff between the other two sides, a c and c b ; also their sum. Then, as Sum of the , . Difif of other . Diff of the two other two sides • • two aides • parts ag and bg, of the base. Add half this diff of the parts, to half the base a 6; the sum will be the longest part ag; which taken from the whole base, gives the shortest part g b. By this means we get in each of the small tri- angles a c. g and cgb, two sides, (namely, a e and a g; and c b and g b;) and an augie (namely, the right angle c g a, or c g b) opposite to one of the given aides. Therefore, use Case 2 for finding the aagles a and 6. When that is done, take their sum from 180<^. for the angle a c b. Or« Sd mode t call half the sum of the three sides, s; and call the two sides which form either angle, m and n. Then the nat sine of half that angle will be equal to \ / (« — "*) X J^«r^>. V mXn Ex. 1. To find the dist from n to an inac- cessible object c. Measure a line ab ; and from its ends measure the angles cab and cb a. Thus having found one side and two angles of the triangle a be, calculate a c by means of Case 1. Or if extreme accuracy is not reqd, draw the line a 6 on paper to any convenient scale ; then by means of a protractor lay off the angles c ab, cb a; and draw a c and c b ; then measure' a c by the same scale. Ex. 2. To find the heigrht of a vertieta object, n a. Place the instrument for measuring angles, at any conve. nient spot o ; also meas Che dist oa\ or if o a cannot be actually measd in consequence of some obstacle, calculate it by the same process as a c in Pig 1. Then, first directing the instru- ment horizontally,* as o 8, measure the angle of depression, s o a, say 12^ ; also the angle son, say 30^^. These two angles added together, give the angle a o n, 42^. Now. in the small triangle o s a we have the angle os a equal to 90°. because on is vert, and o s hor ; and since the three angles of any triangle are equal to 180°, if we subtract the angles osa (90°). and soa (12°) from 180°, the rem (78^) will be the angle o a s or o a n. Therefore, in the triangle on a, we have one side o a; and twa angles a on, and o a n, to calculate the side a n by Case 1. " Angeles and dists on sloping: ground must be measured hor- izon tally. The graduated hor circle of the instrument evidently meaa< t'"" "-."."' "V"-V----'--'^0 \ »res the angle between two objects bori ' ■---" zontally, no matter how much higher one -iQ \ of them may be than the other ; onepei- \ haps requiring the telescope of the instru- ment to be directed upward toward it; and the other downward. If, therefore, the sides of triangles lying upon sloping C \ ground, are not also measd hor, there can be no aocordance between the two. Tbaa 152 PLANE TRIGONOMETRY. Rbm. If, as in Fig 3, it should be necessary to ascertain the vert height an from a point o, entirely Bbovo it, then both the angles measd at o, namelv, son, and a o a, will be angles of depression, or below the hor line o s assumed to measure them from. In this case we have the side o a as before: the angle noa = 8oa — 8on; and the angle o an = l&)°— (oaa (90°), and $ o a;^ to calcolatt «n by Case 1. Fig. 3. Figr.4. Orlf, as in Fig 4, the observations are to be taken from a point o, entirely below the object an, thep botk the angles g o a, s o n, will be angles of elevation, or above the assumed hor line ot. Here wd have in the triangle o n a, the given side o a as before ; the angle a o n~ a on — «oo; and the angle ona^lSOC-— (o an (90°), and no «, ^ to calculate a n by Case 1, If the object a n, as in Fig 5, instead of being vert, is inclined; and instead of its vert height, we wish to tind its length a n, we must first as- certain its angle y ti of inclination to the hori ton ; to which angle each of the angles oan will be equal. To tind this angle yti, suspend a plumb- line t y, of any convenieut kuown length, from the object an; and measure also y t horizontally. Then say as y t : iji '. : I '. nat tang of angle y t i. From the table of nat tangs take out the angle yti found opposite this nat tang; and use it for the angles o « n or o«o; instead of the 90° of Figs S and 4. Also when the object inclines, the side a o of the triangle must be measd in line, or in range with the inclination. If the object, as the rock a n, Fig 6, is curved or irregular, a pole o a may be planted sloping in the direction a n ; and Fig. 5. in the triangle ab c, upon sloping ground, the instrument at o, measures the hor angle ton; and not the angle bac. Therefore, the side which corresponds with this hor angle i o n. is the hor dist i n ; and not the sloping dist 6 c. In other words, when sides and angles are on sloping ground, we do not seek their actual measures; but their ftor ones. This remark applies to all surveying for farms, railroads, triangulations of countries, &c, &c ; and the want of a strict attention to it, is one cause of the small errors, almost unavoidable, (and fortunately, of but triding consequence in practice), which oocur in all ordinary field operations. When a sextant is nsed* angles between objects at difF altitudes, asp and q, may be measd hor, by first planting two vert rods and 8. in range with the objects; and then taking the hor angle o n «, subtended by the rods. Ang-les may be measd without any inst, thus: Measure 100 ft toward each object, and drive stakes j measure the dist across from one stake to the other. Half this dist will be thesineof Tioi/theangletoaradof 100: and if we move the decimal pdint two places to the left, we get the nat aineoT this one half of the angle to arad of 1. as in the tables. Thus, suppose the dist to be 80.64 feet ; then 40.32 is the sine of half the angle ; and ,4032 will be the nat sine, opposite to which in the table of nat ■ines we find the angle 23° 47' ; which mult bv 2 gives il° 34', the reqd angle. If obstacles prevent measuring toward the objects, we may measure directly from them ; because, when two lines intersect, the opposite angles are equal. A rough measurement may be made bv sticking three pins vert, and a few ins apart, into a small piece of board, nailed hor to the top of a post. The pins would occupy the positions n o «, of the last figure. Pencil-linof= may then be drawn, connecting the pin-holes ; and the angle be measd with a protractor. By nailing a piece of board vert to a tree, and then drawing upon it a short hor line, by means of a pocket car- penters' spirit-level, vert angles of elevation and depression mav be taken roughly in the same way. In this way the writer has at times availed himself of the outer door of a house, by opening it until i» pointed toward some mountain-peak, the dist of which he knew approximately ; but of the height of PLANE TRIGONOMETRY. 153 its angle yti of inclination with the horizon found as before; in which case the dist aw is calculated. Or if the vert height en is sought, the point c may first be found by sighting upward along a plumb-line held above the head. Ex. 3. To find tbe approximate tieight, « a?^ of a iiiountaiii. Of which, perhaps, only the very summit, x, is visible above interposing forests, or other obstacles ; but the dist, mi, of which is known. In this case, first direct tbe instrument bor, as m h', and then measure the anglb i m x. Then in the triangle imx we have one side mi; the measd angle imx, and the angle mix (90°), to find im by Case 1. But to this i x we must add i o, equal to the height ym of the instrument above the ground; and also o 8. Now, o s is apparently due • entirely to the curvature of the earth, which is equal to very nearly 8 ins, or .667 ft in one mile ; and increases as the squares of the dists ; being 4 times 8 ins in 2 miles ; 9 times 8 ins in 3 miles, &c. But this is somewhat diminished by the refraction of the atmosphere ; which varies with temperature, moisture, &c ; but alwayi tends to make the object x appear higher than it actually is. At an average, this deceptive elevation amounts to about-— th part of the curvature of the earth ; and like the latter, it varies with the aquares of the dists. Consequently if we subtract — - part from 8 ins, or .667 ft, we have at once the combined eflFect of curvature and refraction for one mile, equal to 6.857 ins, or .6714 ft ; and for other dists, as shown in the following table, bj the use of which we avoid the necessity of making separate allowances for curvature and refraction. ¥ig,7. Table of allowances to be added for curvature of the eartb: and for refraction ; combined. Dist. Allow. Dist. Allow. Dist. Allow. Dist. Allow. in yards. feet. in miles. feet. in miles. feet. in miles. feet. 100 .002 K .036 6 20.6 20 229 150 .004 x| .143 7 28.0 22 277 200 .007 /x ,321 8 36.6 25 357 300 .017 1 .572 9 46.3 30 614 400 .030 IK .893 10 57.2 35 700 500 .046 W2, 1.29 11 69.2 40 915 600 .066 m 1.75 12 82.3 45 1158 700 .090 2 2.29 13 96.6 50 1429 800 .118 2^ 3.57 14 112 55 1729 900 .149 3^ 5.14 15 129 60 2058 . 1000 .185 33^ • 7.00 16 146 70 2801 1200 .266 4 9.15 17 165 80 3659 1500 .415 4^ 11.6 18 185 90 4631 2000 .738 /^ 14.3 19 206 100 5717 Hence, if a person whose eye is 5.14 ft, or 112 ft above the sea, sees an object just at the sea's horizon, that object will be about 3 miles, or 14 miles distant from him. A horizontal line is not a level one, for a straight line cannot be a level one. The curve of the earth, as exemplified in an expanse of quiet water, is level. In Fig 7, if we suppose the curved line ty a g to represent the surface of the sea, then the points ty$ and g are on a level with each other. They need not be equidistant from the center of the earth, for the sea at the poles is about 13 miles nearer it than at the equator; yet its surface is everywhere on a level. Up, and down, refer to sea level, l^evel means parallel to the curvature of the sea; and horizontal means tangential to a level. Ex. 4. If the inaccessible vert height c d, Fig^ 8, Is 80 situated that we cannot reach it at all, then place the instrument for measuring angles, at any convenient spot n ; and in range betwe'en n and d, plant two staffs, whose tops o and i shall range precisely with n, though they need not be on the same level or hor line with it. Measure n o: also from n measure the angles on d and o n c. Then move the instrument to the precise spot previously which he had no idea. For allowance for curvature and refraction see above Table, A triang-le whose sides are as 3, 4, and 5, is right angled; and one whose sides are as 7 ; 7 ; and 9. 9; contains 1 right angle; and 2 angles of 45° each. As it is fre- quently necessary to lay down angles of 45" and 90° on the ground, these proportions may be used for the purpose, by shaping a portion of a tape-line or chain into such a triangle, and driving a stake at each angle. 154 PLANE TRIGONOMETKY. •ooupied by the top o of the staff; and from o measure the angles iod and doc. This being doiie, sub tract the augle i o c from 180° , the rem will be the angle c o n. Consequent- ly in the triangle noc, we have one side n o, and two angles, en o and con, to 5* find by Case 1 the side t Again, take the angle iod - from 180°; the remainder "will be the angle n o d, so that in the triangle dn o we have one side n o, and the two angles dn o and nod, to And by Case 1 the side od. Finally, in • the triangle cod, we have two sides c o and o d, and their included angle cod, to find c d, the reqd vert height. Rkm. If c d were in a valley, or on a hill, and the observations reqd to be made from either higher or lower ground, the operation would be precisely the same. £x. 5. See Ex 10. To find the dist ao, ¥!§ 9, between two entirely inaccessible objects. Fiff. 8. Fig-. 9. m measure the angles o Fig 9, and the angles a Tig. 10. Measure a side n m: at w measure the angles anm and onm; also at amn. This being done, we have in the triangle a n m, one side n m, nma; hence, by Case 1, we can calculate the side a 7i. Again, in the triangle o m n we have one side n m, and the two angles c mn, and mno; hence, by Case 1, we can calculate the side n o. This being done, we have in the triangle ano, two sides an, and no; and their included angle ano; hence, by Case 3, we can calculate the side ao, which is the reqd dist. It is plain that in this manner we may obtain also the position or direction of the inacces- sible line ao ; for we can calculate, the angle nao ; and can therefrom deduce that of ao; and thus be enabled to run a line parallel to it, if required. By drawing n m on pa- per by a scale, and laying down the four measd angles, the dist a o may be measd upon the drawing by the same scale. If the position of the inaccessible dist c n. Fig 10, be such that ^e can place a stakes in line with it, we may proceed thus : Place the instrument at any suitable point s, and take the angles p s c andc«n. Also find the angle ops, and measure the distj's. Then in the triangle ^ s c fi:;d s c by Case 1 ; agnin, the exterior angle n c «, being equal to the two interior and opposite angles c p s, and p s c, we have in the triangle c s n, one side and two angles to fiad c n by Case 1. Ex.6. To find a dist a &^Fi^ 11. of which the ends only are accessible. Prom a and h, measure any two lines ache, meeting at c ; also measure the angle ach. Then in the triangle a ft c we have two sides, and the included angle, to find the third side a h by Case 3. Ex. 7. To find the vert hei$£ht o m, of a hill, above a ^iven point i. Place the instrument at t; measure a m. Directing the instrument hor, as an, take the angle nam. Then, since a n m is 90° Fig 12, we have one side a m, and two angles, nam and anm, to find n m by Case I. Add no, equal to at, the height of the instrument. Also, if the hill is a long one, add for curvature of the earth, and for refraction, as explained in Example 3, ^i --' Pig 7. Or the instrument may be placed at the top of the hill ; and an angle of depression measured j instead of the angle of elevation nam. Rem. 1. It is plain, that if the height om be previously known, and we wish to ascertain the dist from its sum- mit m to any point i, the same measurement as before, of the angle nam, will enable us to calculate a m by Case 1. So in Ex. 2, if the height na be known, the angles measd in that examplfe, will enable as. to oompate the dist a o ; so also in Figs 3, 4, 5, and 7 j in all of which the procen is so plain as te require no furtker explanation. Rem. 2. The height of a vert object by mcanS Of itS ShadoW. Plant one end of a straight stick vert in the ground ; and measure ts shadow ; also measure the length of the shadow of the object. Then, as the length of the shadow of the stick is to the length of the stick above Fig. 11. ¥1^.12. J PLANE TRIGONOMETRY. 155 Fig. 12^^ round, so is the length of the shadow of the object, to its height. If the object is inclined, the stick lUst be equally inclined. Rem. 3. Or the lieig^ht of a vert object mw, Fig 12^^, whose distance r m is known, may be found by its reflection in a vessel of water, or in a piece of looking glass placed perfectly horizontal at r; for as r a is to the height - a i of the eye above the reflector r, so is r m to^ xji* ^ I Ql/ -'CI the height mn of the object above r. /t/j-^^ X ic* -*-*'Z3 Rem. 4. Or let o c, Fig 12}4, be I planted pole, or a rod held vf rt by an assistant. Then taud at a proper dist back from it, and keeping the eyes steady, let marks be lade at o and c, where the lines of sight i n and t m strike the rod. Then c is to c o, so 18 t m to m n. The following examples may be regarded as substitutes for strict trigonom*. try : and will at times be useful, in case a table of sines, &c, is not at hand for making trigonometrical calculations. Ex. 8. To find the dist ab, of which one end only is accessible. Drive a stake at any convenient point a ; from a lay off any angle b a e. In the line a c, at any convenient point c, drive a stake ; and from c lay off an angle acd, equal to the angle 6 a c. In the line c d, at any convenient point, as cJ, drive a stake. Then, standing at d, and looking at b, place a stake o in range withdh; and at the same time in the line a c. Measure a o, o c, and erf j than, from the principle of similar triangles, as oe:ed:ino:ab. *, and Or thus. At a lay off angle o a c = 5° 43'. Lay ^ off at right angles to ao. Measure oc. Then flo = 10 c, too long on^y 1 part in 935.6, or 5.643 feet in a mile, or .1069 foot (full U inches) in 100 feet. 156 PLANE TRIGONOMETRY. Fig. 17. Ex. 9. To find the dist a &, of wliicb th« ends only are accessible. Prom a lay off the angle h ac; and from 5, the angle ah d, each 90°. Make a c and h d equal to each other. Then, cd = afe. Or a h may be considered as the dist across the river in Figs 15, 13, or 14 ; and be ascertained in the same way. Or measure any dist, Fig 17, a o; and make on in line and equal to it. Also treasure ho\ and piake om in line and equal to it. Then will mn be both paral. lei to a 5, and equal to it. Fig. 18. Ex. 10. See Ex. 4. To find tlie entirely inaccessible dist y z, and also its direction. At any two convenient points a and 6, from each of which y and z can be seen, drive stakes. Then we have the four corners of a four-sided figure, in which are given the directions of three of its sides, and of its two diags. These data enable us to lay out on the ground, the small four-sided fig a c o t, exactly similar to the large one. Thus, in the line a b place a stake C.: and make co parallel to 62:; o being at the same time in range of the diag a z. Also, from c make c i parallel to 6 y; i being at the same time in range of a y. Then will i o be in the same direction as y z, or parallel to it. Measure ac, ab, and io; then evidently, from the principle of similar figures, as a c : ab : '. i o '. y z. If y z were a visible line, such as a fence or road, we could from a divide it into any required portions. Thus, if we wish to place a stake halfway between y and z, first place one half- way between % and o ; then standing at a, by means of signals, place a person in range on y z. Or, to find along ab, a. point t perp to y « at y, first make oi s'= 90° ; and measure a 8. Then, as - oiiasiiyziat. Ex. 11. To find the position of a point, n. Tig 19, By meana of two angles a n b and b n c. taken from it tlie three objects a b c, whose positions and dists apart are known. The use of this problem is more frequent in marine than in land surveying. It is chiefly employed for determining the position n of a boat from which soundings are being taken along a coast. As the boat moves from point to point to take fresh soundings, it becomes necessary to make a fresh observation at each point, in order to define its position on the chart. An observation consists in the measurement by a sextant of the two angles anb, bn c. to the sig- nals ab c, previously arranged on the shore. When practicable, this method shoald be, rejected ; and the observa- tions taken to the boat at the same instant, by two observers on shore, at ' two of the stations. The boat to show a signal at the proper moment. The most expeditious mode of fixing the point n upon the map, is to draw three lines, forminfit the two angles, and ex- tended indefinitely, on a piece of trans- parent paper. Place the paper upon the map, and move it about until the three lines pass through the three stations ; then prick through the point n wherever it happens to come. Instead of the transparent paper, an instrument called a station pointer may be used when there are many points to be fixed. But the position of the point n can be found more correctly by describing two circles, as in Fig 19, each of which shall pass through n and two of the station points. The question is to find the centers o and » of two such circle?. This is very simple. We know that the angle a o 6 at the center of a circle is twice as great as any angle « n ft at the circumf of the same circle, when both are subtended by the same chord a b. Consequently, if the angle anb, observed from the boat, is say, 50°, the angle aob must be 100°. And, since the three angles of every plane triangle are equal to" 180°, the two angles o ab and o b a are together equal to 180° — 100° = 80°. And, since the two sides a o and b o are equal (being radii of the same circle), therefore, the angles oab and oba are equal ; and each equal to 80° —- = 40°. Consequently, on the map we have only to lay down at a and b, two angles of 40°; the r>int of intersection will be the center of the circle a b n. Proceed in the same way with the angle f» c, to find the center at. Then the intersection of the two eircles at n will be the point sought. Fig. 19. PARALLELOGRAMS. 157 PARAIil^EI^OORAMS. Square. Rectangle. Rhombus. Rhomboid. a P or P\ "fm P\ --4 A PARALLELOGRAM is any figure of four straight sides, the opposite ones of which ire paral?el. There are but four, as in the above figs. The rhombus, like the rhom- }ohedron, Fig 3, p 195, is sometimes called "rhomb." In the square and rhombus ill the four sides are equal ; in the rectangle and rhomboid only the opposite ones ire equal. In any parallelogram the four angles amount to four right angles, or J60° ; and any two diagonally opposite angles are equal to each other ; hence, having me angle given, the other three can readily be found. In a square, or a rhombus, a liag divides each of two angles into two equal parts ; but in the two other parallel- )grams it does not. To find the area of any parallelogram. Multiply any side, as S, by the perp height, or dist p to the opposite side. Or, multiply togethtf wo sides and nat sine of their included angle. The dlas a b of any eauare is equal to one side mult by 1.41421 ; and a side is equal to ^■^^^^ ; or, to diag mult by .707107. The side of a square equal in area to a gtvex circle* is equal to diam X .886227. The side of the greatest square, t/iat can be inscribed in I criven circle, is equal to diam X .707107. The side of a square mult by 1.51967 gives the side of an equl- Bteral triangle of the same area. All parallelograms as A nd C, which have equal bases, « c, and equal perp heights n , have aiso equal areas ; aud the area of each is twice that of a tri- ngle having the same base, and perp height. The area of a quare Inscribed in a circle is equal to twice the square of the ad. In every parallelogram, the 4 squares drawn on its sides have a united area equal to that of lie two squares drawn on its 2 diags. If a larger square be drawn on the diag a i of a smaller ijuare. its area will be twice that of said smaller square. Either diag of any parallelogram ivides it into two equal triangles, and the 2 diags div it into 4 triangles of equal areas. Tlie two Hags of any parallelogram divide each other into two equal parts. Any line drawn through he center of a diag divides the parallelogram into two equal parts. Remark 1.— The area of any fig whatever as B that is enclosed by four straight [nes, may be found thus : Mult together the two diags am, n b; and the nat sine of the least angle o 6 ; orn o wi, formed by their intersection. Div the product by 2. This is useful in land surveying, 'hen obstacles, as is often the case, make it difficult to measure the sides of the 6g or held ; while it lay be easy to measure the diags ; and after finding their point of intersection o, to measure the re- Hired angle. Sut if the fig is to be drawn, the parts o a, o &, o n, o m of the diags must also e measd. Rem. 3. — The sides of a parallelogram, triangle, and many other figs may be bund, when only the area and angles are given, thus : Assume some particular one of its ides to be ot the length 1 ; and calculate what its area would be if that were the case. Then as the q rt of the area thus found is to this side 1, so is the sq rt of the actual given area, to the corre* ponding actual side of the fig. On a ^tven line wx^ta draiF a square* From w and x, with rad w x, describe the arcs xry and wre. Prom their intersection r,and with rad equal to 3^ of w x, describe 8 8 s. From to and x draw w n and xm tangential to a s «, and ending at the other arcs ; join nm. 158 TRAPEZOIDS AND TRAPEZIUMS. TRAPEZOIDS. n t . rn n m is any fig with four straight sides, of which no two are parallel. c a 8 c a 8 c A trapezoid a c n m, is any figure with four straight aides, only two of which, as ac and n i)t, ar€ parallel. To find tlie area of any trapezoid. Add together the two parallel sides, a c and m n ; mult the sum by the perp dist s t b«f fen. them ; div the prod by 2. See the following rules for trapeziums, which are all equally appL *)ie to trapezoids ; also see Remarks after Parallelograms. TRAPEZIUMS. A trapezium To find the area of any trapezinm, having" g-iven the diag* bOf or a c, between either pair of opposite ang-les; and also the two perps, n., n, from the other two an^rles. Add together these two perps ; mult the sum by the diag; div the prod by 2. HaTing; the four sides ; and either pair of opposite ang^les, as (the, o the same with the other given side, and its two adjacent angles, (or their rems, as the case may be.) Subtract the least of the areas thus found, from the greatest; theYem will be the reqd area. Having- three sides; and the tivo included angles. Mult together the middle side, and one of the adjacent sides ; mult the prod by the i^at sine of their included angle ; call the result a. Do the same with the middle side and its other adjacent side, and the nat sine of the other included angle ; call the result b. Add the two angles together ; find the diff between their sum and 180°, whether greater or less ; find the nat sine of this diff; molt together the two given sides which are opposite one another ; mult the prod by the nat sine just fomnd ; call the result c. Add together the results a and b ; then, if the sum of the two given angles is leas than 180°, subtract c from the sum of a and b ; half the rem will be the area of the trapezium. But if the sum of the two given angles be greater than 180°, add together the three results a, 6, and c; half their sum will be the area. Having^ the two diagonals, and either angle formed by their intersection. See Remarks after Parallelograms. In railroad measurements Of excavation and embankment, the trapezium tm n frequently occurs ; as well as the two 5-sided figures im n t and I mn o s; in all of which m n represents the roadway ; rs.rc, and r t the center- depths or heights ; I u and o v the side-depths er heights, as given bv the level; Im and n o the side- slopes. The same general rule for area applies to all three of these figs ; namely, mult the extreme hor width uvhj half the center depth r s, re. or r t, as the case may be. Also mult one fourth of the width of roadway m n, by the sum, of the two side-depths I u and o V. Add the two prods together ; the sum is the reqd area. This rule applies whether the two side- slopes m I and n o have the same asgle of inclination or not. In railroad work, etc., the mid- way hor width, center depth, and side depths of a prismoid are respectively = The half sams of toe corresponding end ones, a'nd thus can be found without actual measurement. POLYGONS. 159 To draw a hexagron, each side of which shall be equal to a g^iven line, a b. Prom a and h, with rad a h, describe the two arcs; from their intersection, », with the same rad, describe a circle; around the circumf of which step off the same rad. ' Side of a hexag^on = nn X .57735. To draTT an octag'on, w^ith each side equal to a g^iven line, c e. From cand e draw two perps, cp, ep. Also prolong cc toward / and g; and from c and c, with rad equal c c, draw the two quadrants; and find their centers ft A; join c A, and e ft; ^'^^ h s and ft < parallel to cp; and make each of tliem equal to xuake c o, and e o, each equal to ft ft ; join o 0,0 a, and o t. draw to ce; Side of an octag^on = w n X .41421354. To draw an ocfag^on in a g^iven square. From each corner of the square, and with a rad equal to half its diag, describe the four arcs; and join the points at which they cut the sides of the square. , To draw any regrular polygon, with each side equal to m n. Div 360 degrees by the number of sides; take the quot from 180°; div the rem by 2. Thi» will give the angle cmn, or cum. htm and n lay down these angles by a protractor: the sides of these augles will meet at«i point, c. from , which describe the circle mny; and around its circumf step oft dists equal to mn. In any circle, m n y^ to draw any regular polygon. DivSGO^ by the number of sides; thequot will be the angle men. at the center. Lay off this angle by a protractor ; and its chord m n will be one side ; which step off Around tl* circumf. To reduce any polygon, as abed efa, to a triangle of the same area. If- we produce the side /a toward ro; and draw 6 g parallel to a c, and join g c. we get equal trl. angles a oft, and a c g, both on the same base a c ; and both of ibe same perp height, inasmuch aa they are between the two parallels a c and g b. But the part a c i forms a portion of both these trl* angles, or in other words, is common to both. Therefore, if it be taken away from both triangles, the remaining parts, t c 6 of one of them, and i g a o( the other, are also equal. Therefore, if the part icbbe left off from the polygon, and the part ig ab& taken into it, the polygon g fed cig will have the same area as afe d c b a; but it will have but five sides, while the other has six. Again, if e s be drawn parallel to d /, and ds joined, we have upon the same base e«, and between the same parallels e a and df, the two equal triangles e s d, and e s/, with the part eos common to both ; and consequently the remaining part e o d ot one. and o sf of the other, are equal. Therefore, if o s/ be left off from the polygon, and e o oJ be taken into it, the new polygon gad eg, Fig 2, will have the same area as gfe d c g; but it has but four sides, while the other has five. Finally, if g s, Fig 2, be extended toward n ; and d n drawn parallel to c s ; and c n joined, we have on the same base c s, and between the same parallels c s and d n, the two equal triangles e a n, and cad, with the part cat common to both. Therefore, if we leave out cdt, and take itx.a t n, we have the triangle gnc equal to tliepolygon ^ | In a rectang-ular fig", g h s d, Representing an open panel, to find the points o o o o in its sides ; and at equal dists from the angles .9, and s ; for inserting a diag piece o o o o, of a given width I I, measured at right angles to its length. From g and s as centers, describe several concentric arcs, as in the Fig>. Draw upon transparent paper, two parallel lines a a, c c, at a distance apart equal toll: and placing these line.s on top of the panel, move them about until it is shown by the arcs that the four dists g o, go, s o, s o. are equal. Instead of the transparent paper, a strip of common paper, of the width 1 1 may be used. Rem. Many problems which would otherwise be very diflBcult, may be thus solved with an accuracy sufficient for practical purposes, by means of transparent paper. To find the area of any irreg^ular poly* gpon, anb c m. Div it into triangles, as anh, am c, and 06 c; in each of which find the perp dist 0, between its base a 6, a c, or b c; and the opposite angle n, m, or a; mult each base by its perp dist; add all the prods together ; div by 2, To find approx tlie area of a long* ir« reg^ular fig, ».Habcd. Between its ends ah,cA, space off equal dists, (the shorter they are the more accurate will be the result,) through which draw the intermediate parallel lines 1, 2, 3, &c, across the breadth of the fig. Measure the lengths of these intermediate lines ; add them together ; to the sum add half the sum of the two end breadths a 6 and c d. Mult the entire sum by one of the equal spaces between the parallel lines. The prod will be the area This rule answers as well if either one or both the ends terminate in points, as at m and n. In tne last of these cases, both a b and c d will be included in tne iniermediate lines; and half the two end breadths will be 0, or nothing. To find the area of any irregular figure. Draw around it lines which shall enclose within them (as nearly as can be judged by the eye) as much space not belonging to the figure as they exclude space belonging to it. The area of the simplified figure thus formed, being in this manner rendered equal to that of the com- plicated one, may be calculated by dividing it into triangles, Ac. By using a piece of fine thread, the proper position for the new boundary lines may be found, before drawing them in. Areas of irregular figures may be found from a drawing, by laying upon it a piece of transparent paper carefully ruled into small squares, each of a given area, say 10 20, or 100 sq. ft. each ; and by first counting the whole squares, and then adding the fractions of squares. ^=^^-s^ CIRCLES. 161 A circle is th* area Included within a curved line of such a character that every point In it Is •qually distant from a certain point within it, called its center. The curved line itself is called the •iroumference, or periphery of the circle ; or very commonly it is called the circle. To find tlie circumtereiice. If ult diam by S.1416, which gives too much by only .148 of an inch in a mile. Or» as US is to 366 «o is diam to circumf ; too great 1 inch in 186 miles. Or, mult diam by 3-^ ; too great by about 1 part in 2485. Or, mult area by 12.566, and take sq root of prod. To find the diam. Div the oircumf by 3.1416 ; or, as 355 is to 113, so is circumf to diam ; or, mult the ciroumf. by 7 ; and div the prod by 22, which gives the diam too small by oojy about one part in 2485 ; or, mult the area by 1.2732; and take the sq rt of the prod. The diam is to the circumf more exactly as 1 to 3.14159265. To find the area of a circle. Square the diam; mult this square by .7854; or more accurately by .78589816; or square the cir- cumf ; mult this square by .07958 : or more accurately by .07957747 ; or mult half the diam by half the oircumf; or refer to the following table of areas of circles. Also area = sq of rad X 3.1416. The area of a circle is to the area of any circumscribed straight-sided fig, as the circumf of the circle is to the circumf or periphery of the fig. The area of a square inscribed in a circle, is equal to twice the square of the rad. Of a circle in a square, — square X .7854. It is convenient to remember, in rounding off a square corner a & c, by a quarter of ^ ^ a circle that the shaded area «'>''>■ omml tn »hniit. 1 nnrr. CnnrrpnMv -914.fi'i rtf t.hi^ **|- whole square abed. To find the diam of a circle equal in area to a ^iven sqnare. Mult one side of the square by 1.12838. To find the rad of a circle to circnmscribe a g^iven square. Mult one side by .7071 ; or take ^ the diag. To find the side of a square equal in area to a g^iven circle. Mult the diam by .88623. To find the side of the grreatest square in a §:iven circle. Mult diam by .7071. The area of the greatest square that can be inscribed in a circle is equal t* twrice the square of the rad. The diam X by 1.3468 gives the side of an equilateral triangle of equal aroa. To find the center c, of a given circle. Draw any chord a b ; and from the middle of it o, draw at rght angles !• it, a diam d g ; find the center c of this diam. To describe a circle througrh any three points, ah Cf not in a straig-ht line. Join the points by the lines ab, he; from the centers of these lines dravf the dotted perps meeting, as at o, which will be the center of the circle. Or from b, with any convenient rad, draw the arc m n; and from a and c, with the same rad, draw arcs y and z; then two lines drawn through the intersections of these arcs, will meet at the center o. To describe a circle to touch the three ang^les of a triang^le is plainly the same as this. To inscribe a circle in a triang-ledraw two lines bisecting any two of the angles. Where these lines meet is the center of the circle. 162 CIRCLES. To ibrwL-w a tang-ent, iei, to a circle, from an^ driven point, e^ in its circumf. Through the center n, and the giren point e, draw n o; make e o equal i« e 71 ; from n and o, with any rad greater than half of o n, describe the tw« pairs of arc i i; join their intersections i i. Here, and in the following three figs, the tangents are ordinary or geo- metrical ones ; and may end where we please. But the trigonometricai *" tangent of a given angle, must end in a secant. Or from e lay off two equal distances e c, e /; and draw i i parallel to c t. To draiir a tan^, ash, to a circle, from a point, a, iFlilcli is ontside of tlie circle. Draw a c, and on it describe a semicircle ; through the intersection, $, drsp«9 asb. Here c is the center of the circle. To drai¥ a tangr, g Ji* from a circular arc, c. Diam. ' Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. ArMi. 1-6 1 .omsi .00019 3.~^ 10.9956 9.621l| 103^ 31.8086 80.516 1934 60.4757 291.04 1-32 .0:*,S175 .00077 9-16 11.1919 9.9678J U 32,2013 82.516 % 60.8684 294.8S 3-641 ,U72fi2 .00173 % 11.3883 10.321 % 32.5940 84.541 Yz 61.2611 298.6J 1-16 .196350 .00307 11-16 11.5846 10.680 H 32.9867 86.590 % 61.6538 302.49 3-32 .294524 .00690 H 11.7810 11.045 % 33.3794 88.664 H 62.0435 306.35 H .392699 .01227 13-16 11.9773 11.416 % 33.7721 90.763 y» 62.4392 310.24 5-32 .490874 .01917 % 12.1737 11.793 % 34.1648 92.886 20 62.8319 314.1S 3-16 589049 .02761 15-16 12.3700 12.177 11. 34.5575 95.033 % 63.2246 318.10 7-32 .687223 .03758 4. 12.5664 12.566 H 34.9502 97.205 Yi 63.6173 322.0S H .78.3398 .04909 1-16 12.7627 12.962 35.3429 99.402 % 64.0100 326.06 9-32 .88.^573 .06213 % 12 9591 13.364 % 35.7356 101.62 H 64.4026 330.06 5-16 .981748 .07670 3-16 13.1554 13.772 }4 36.1283 103.87 % 64.7953 334.10 11-32 1.07;)92 .09281 34 13.3518 14.186 Yt 36.5210 106.14 % 65.1880 338.16 Vs 1.17810 .11045 5-16 13.5481 14.607 % 36.9137 108.43 Ys 65.5807 342.25 13-32 1.27627 .12962 % 13.7445 15.033 % 37.3064 110.75 21 05.9734 346.38 7-16 1.37445 .15033 7-16 13.9408 15.466 12. 37.6991 113.10 H 66.3661 350.50 15-32 1.47262 .17257 }4 14.1372 15.904 % 38.0918 115.47 H 66.7588 354.66 H 1.57080 .19635 9-16 14.3335 16.349 Yi 38.4845 117.86 % 67.1515 358.84 17-32 1.66897 .22166 % 14.5299 16.800 % 38.8772 120.28 3^ 67.5442 363.05 "9-16 1.76716 .24850 11-16 14.7262 17.257 Ji 39.2699 122.72 % 67.9369 367.28 19-32 1.86532 .27688 H 14.9226 17.721 % 39.6626 125.19 % 68.3296 371.54 % 1.96350 .30680 13-16 15.1189 18.190 % 40.0553 127.68 Y» 68.7223 375.88 21-32 2.06167 .33824 Vs 15.3153 18.666 % 40.4480 130.19 22. 69.1150 380.13 11-16 2.15984 .37122 15-16 15.5116 19.147 13. 40.8407 132.73 % 69.5077 384.46 23-32 2.25802 .40574 5. 15.7080 19.635 H 41.2334 135.30 69.9004 388.82 H 2.35619 .44179 i-16 15.9043 J0.129 H 41.6261 137.89 % 70.2931 393.20 25-32 2.46437 .47937 H 16.1007 20.629 % 42.0188 140.50 70.6858 397.61 13-16 2.55254 .51849 3-16 16.2970 ai.l35 14 42.4115 143.14 Y% 71.0785 402.04 27-32 2.65072 .55914 }4 16.4934 21.648 42.8042 145.80 71.4712 406.49 % 2.74889 .60132 5-16 16.6897 22.166 H 43.1969 148.49 Y» 71.8639 410.97 29-32 2.84707 .64504 % 16.8861 22.691 y% 43.5896 151.20 23. 72.2566 415.48 1516 2.94524 .69029 7-16 17.0824 23.221 14. 43.9823 153.94 H 72.6493 420.00 31-32 3.04342 .73708 ^ 17.2788 23.758 % 44.3750 156.70 34 73.0420 424.56 1. 3.14159 .78540 9-16 17.4751 24.301 y< 44.7677 159.48 Ys 73.4347 429.13 1-16 3.33794 .88664 % 17.6715 24.850 % 45.1604 162.30 }4 73.8274 433.74 14 3.53429 .99402 11-16 17.8678 25.406 H 45.5531 165.13 % 74.2201 438.36 3-16! 3.73064 1.1075 H 18.06 i2 25.967 % 45.9458 167.99 74.6128 443.01 34 1 3.92699 1.2272 13-16 18.2605 26.535 46.3385 170.87 Yt 75.0055 447.69 5-16 i 4.12334 1.3530 % 18.4569 27.109 Yi 46.7312 173.78 24. 75.3982 452.39 ^1 4 31969 1.4849 15-16 18.6532 27.688 15. 47.1239 176.71 H 75.7909 457.11 7-16 4.51604 1.6230 6. 18.8496 28.274 ^ 47.5166 179.67 34 76.1836 461.86 H 4.71239 1.7671 H 19.2423 29.465 Y 47.9093 182.65 f» 76.5763 466.64 9-16 4.90874 1.9175 y< 19.6350 80.680 % 48.3020 185.66 76.9690 471.44 %\ 5.10509 2.0739 % 20.0277 31.919 Yi. 48.6947 188.69 % 77.3617 476.26 11-16 5.30144 2.2365 H 20.4204 83.183 % 49.0874 191.75 % 77.7544 481.11 H' 5.49779 2.4053 % 20.8131 84.472 % 49.4801 194.83 Yi 78.1471 485.91 1»-16 5.69414 2.5802 % 21.2058 35.785 y» 49.8728 197.93 2b 78.5398 490.8T y, 5.89049 2.7612 y» 21.5984 37.122 16. 50.2655 201.06 H 78.9325 495.79 15-16 6.08684 2.9483 7. 21.9911 38.485 H 50.6582 204.22 Q, 79.3252 500.74 2. i 6.28319 3.1416 H 22.3838 39.871 H 51.0509 207.39 % 79.7179 505.71 1-16 6.47953 3.3410 Va. 22.7765 41.282 H 51.4436 210.60 H 80.1106 510.71 H 6.67588 3.5466 Va 23.1692 42.718 H 51.8363 213.82 Y% 80.5033 515.72 3-16; 6.87223 3.7583 y% 23.5619 44.179 H 52.2290 217.08 Vi 80.8960 520.77 J4! 7.06858 3.9761 % 23.9546 45.664 H 52.6217 220.35 Yt 81.2887 525.84 5 16 7.26493 4.2000 % 24.3473 47.173 % 53.0144 223.65 26.' 81.6814 530.93 %' 7.46128 4.4301 % 24.7400 48.707 17. 53.4071 226.98 H 82.0741 536.05 7-16 7.65763* 4.6664 8. 25.1327 50.265 H 53.7998 230.33 % 82.4668 541.19 }4\ 7.85398 4.9087 H 25.5254 51.849 .54.1925 233.71 % 82.8595 546.35 9-16 8.05033 5.1572 yi 25 9181 53.45G % 54.5852 237.10 ^ 83.2522 551.55 %\ 8.24668 5.4119 % 26.3108 55.088 M 54.9779 240.53 Yt 83.6449 556.76 11-16 8.44;W3 5.6727 }4 26.7035 56.745 % 55.3706 243.98 Y< 84.0376 562.00 ^l 8.63938 IS-lS 8.83573 5.9396 27.0962 58.4?« H 55.7633 247.45 Yt 84.4303 567.27 6.2126 H 27.4889 60.132 y% 56.1560 250.95 27. 84.8230 572.56 %' 9.03208 6.4918 % 27.8816 61.862 18. 56.5487 254.47 ■ H 85.2157 577.87 15-16 9.22843 6.7771 9. 28.2743 63.617 ^ 66.9414 258.02 Yi 85.6084 583.21 g. i 9.42478 7.0686 H 28.6670 65.397 34 57.3341 261.59 % 86.0011 588.57 1-16 9.62113 7.3662 y* 29.0597 67.201 % 57.7268 265.18 u, 86 39.38 593.98 %\ 9.81748 7.6699 % 29.4524 69.029 3t 58.1195 268.80 % 86.7865 599..37 3-16 10.0138 7.9798 34 29.8451 70.882 % 58.5122 272.45 % 87.1792 '504.81 J4 110.2102 8.2958 30.2378 72.760 H 58.9049 276.12 % 87.5719 610.27 5-16! 10.4065 8.6179 H 30.6305 74.662 % 59.2976 279.81 28 87.9646 615.75 %i 10.6029 8.9462 H 31.0232 76.589 19. 59.6903 283.53 % 88.3573 621.26 M6 10.7992 9.2806 10. 31.4159 78.540 H 60.0830 287.27 Y4, 88.7500 626.86 164 CIRCLES. TABIiE 1 OF CIRCLES— (Continued). Diameters in units and ei^litlis, Ac. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Aretu asrs 89.1427 632.36 38. 119.381 1134.1 47^ 149.618 1781.4 57^ 179.856 2574.1 H 89.5354 637.94 H 119.773 1141.6 H 150.011 1790.8 % 180.249 2585.4 % 89.9281 643.55 H 120.166 1149:1 Vb 150.404 1800.1 }i 180.642 2596.7 H 90.3208 649.18 % 120.559 1156.6 48. 150.796 1809.6 .H 181.034 3608.0 % 90.7135 654.84 H 120.951 1164.2 H 151.189 1819.0 % 181.427 2619.4 29. 91.1062 660.52 % 121.344 1171.7 151.582 1828.5 y% 181.820 2630.7 H 91.4989 666.23 % 121.737 1179.3 % 151.975 1837.9 58. 182.212 2642.1 91.8916 671.96 % 122.129 1186.9 Ml 152.367 1847.5 H 182.605 2653.5 y» 92.2843 677.71 39. 122.522 1194.6 H 152.760 1857.0 Ya 182.998 2664.9 14 92.6770 683.49 H 122.915 1202.3 % 153.153 1866.5 % 183.390 2676.4 % 93.0697 689.30 123.308 1210.0 y% 153.545 1876.1 M 183.783 2687.8 H 93.4624 695.13 % 123.700 1217.7 49. 153.938 1885.7 V% 184.176 2699.3 % 93.8551 700.98 )4 124.093 1225.4 H 154.331 1895.4 H 184.569 2710.9 SO. 94.2478 706.86 % 124.486 1233.2 H 154.723 1905.0 % 184.961 2722.4 H 94.6405 712.76 % 124.878 1241.0 % 155.116 1914.7 59 185.354 2734.0 H 95.0332 718.69 y» 125.271 1248.8 H 155.509 1924.4 H 185.747 2745.6 % 95.4259 724.64 40. 125.664 1256.6 H 155.902 1934.2 h 186.139 2757.2 95.8186 730.62 H 126.056 1264.5 H 156.294 1943.9 % 186.532 2768.8 % 96.2113 736.62 126.449 1272.4 % 156.687 1953.7 14 186.925 2780.5 % 96.6040, 742.64 % 126.842 1280.3 50." 157.080 1963.5 % 187.317 2792.2 ^"^ 96.9967 748.69 H 127.235 1288.2 H 157.472 1973.3 % 187.710 2803.9 M. 97.3894 754.77 % 127.627 1296.2 157.865 1983.2 % 188.103 2815.7 % 97.7821 760.87 % 128.020 1304.2 % 158.258 1993.1 60. 188.496 2827.4 98.1748 766.99 % 128.413 1312.2 14 158.650 2003.0 % 188.888 2839.2 % 98.5675 773.14 41. 128.805 ' 1320.3 % 159.043 2012.9 Ya 189.281 2851.0 14 98.9602 779.31 H 129.198 1328.3 % 159.436 2022.8 % 189.674 2862.9 % 99.3529 785.51 129.591 1336.4 K • 159.829 2032.8 y. 190.066 2874.8 % 99.7456 791.73 % 129.983 1344.5 51. 160.221 2042.8 % 190.459 2886.6 % 100.138 797.98 ^ 130.376 1352.7 H 160.614 2052.8 % 190.852 2898.6 B2 100.531 804.25 % 130.769 1360.8 M 161.007 2062.9 K 191.244 2910.5 H 100.924 810.54 % 131.161 1369.0 % 161.399 2078.0 61. 191.637 2922.5 101.316 816.86 % 131.554 1377.2 }4 161.792 2083.1 % 192.0.50 2934.5 % 101.709 823.21 42 131.947 1385.4 % 162.185 2093.2 H 192.423 2946.5 H 102.102 829.58 M 132.340 1393.7 H 162.577 2103.3 Va 192.815 2958.5 102.494 835.97 132.732 1402.0 % 162.970 2113.5 }4 193.208 2970.6 8/ 102.887 842.39 y 133.125 1410.3 52. 163.363 2123.7 % 193.601 2982.7 yi 103.280 848.83 14 133.518 1418.6 ^ 163.756 2133.9 % 193.993 2994.8 18. 103.673 855.30 133.910 1427.0 164.148 2144.2 % 194.386 3006.9 H 104.065 861.79 H 134.303 1435.4 % 164.541 2154.5 62. 194.779 3019.1 H 104.458 868.31 % 134.696 1443.8 % 164.934 2164.8 "A 195.171 3031.3 % 104.851 874.85 43. 135.088 1452.2 % 165.326 2175.1 % 195.564 3043.5 14 105.243 881.41 H 135.481 1460.7 H 165.719 2185.4 % 195.957 3055.7 % 105.636 888.00 135.874 1469,1 % 166.112 2195.8 % 196.350 3068.0 Vi 106.029 894.62 y 136.267 1477.6 53. 166.504 2206.2 % 196.742 3080.8 y» 106.421 901.26 rz 136.659 1486.2 H 166.897 2216.8 % 197.135 3092.6 14. 106.814 907.92 % 137.052 1.37.445 1494.7 y* 167.290 2227.0 % 197.528 3104.9 H 107.207 914.61 % 1503.3 % 167.683 2237.5 63. 197.920 3117.2 107.600 921.32 y% 137.837 1511.9 }4 168.075 2248.0 % 198.313 312«.« % 107.992 928.06 44. 1.38.230 1520.5 % 168.468 2258.5 Ya 198.706 3142.0 /^ 108.385 934.82 M 138.623 1529.2 % 168.861 2269.1 % 199.098 3154.fi % 108.778 941.61 M 139.015 1537.9 % 169.253 2279.6 M 199.491 3166.t % 109.170 948.42 % 139.4C8 1546.6 54. 169.646 2290.2 % 199.884 3179.4* H 109.563 955.25 3^ 139.801 1555.3 % 170.039 230<3.8 % 200.277 3191.9 ii. 109.956 962.11 ^ 140.194 1564.0 170.431 2311.5 % 200.669 3204.4 H 110,348 969.00 % 140.586 1572.8 % 170.824 2322.1 64. 201.062 3217.0 110.741 975.91 % 140.979 1581.6 H 171.217 2332.8 H 201.455 3229.6 % 111.134 982.84 45. 141.372 1590.4 % 171.609 2343.5 Ya 201.847 3242.2 K 111.527 989.80 M 141.764 1599.3 % 172.002 2354.3 % 202.240 3254.8 H 111.919 996.78 142.157 1608.2 y% 172..395 2365.0 K 202.633 3267.5 a/ 112.312 1003.8 y 142.550 1617.0 65. 172.788 2375.8 % 203.025 3280.1 r/ 112.705 1010.8 14 142.942 1626.0 H 173.180 2386.6 % 203.418 3202.8 S6. 113.097 1017.9 % 143.335 1634.9 M 173.573 2397.5 % 203.811 3305.6 H 113.490 1025.0 H 143.728 1643.9 % 173.966 2408.3 65. 204.204 3318.3 H 113.883 1032.1 % 144.121 1652.9 H 174.358 2419.2 H 204.596 3331.1 % 114.275 1039.2 46.' 144.513 1661.9 % 174.751 2430.1 204.989 3343.9 H 114.668 1046.3 H 144.906 1670.9 % 175.144 2441.1 y» 205.382 3356.7 H 115.061 1053.6 H 145.299 1680.0 K 175.536 2452.0 Y2 205.774 3369.6 S 115.454 1060.7 % 145.691 1689.1 56. 175.929 2463.0 % 206.167 3882.4 % 115.846 1068.0 A 146.084 1698.2 H 176.322 2474.0 % 206.560 3396.8 n. 116.239 1075.2 % 146.477 1707.4 H 176.715 2485.0 % 206.952 3406.2 H 116.632 1082.5 % 146.869 1716.5 % 177.107 2496.1 66. 207.345 3421.2 ^ 117.024 1089.8 % 147.262 1725.7 % 177.500 2507.2 % 207.738 3484.2 % 117.417 1097.1 47 147.655 1734.9 % 177.893 2518.3 u 208.131 .3447.2 H 117.810 1104.5 % 148.048 1744.2 H 178.285 2529.4 % 208.523 3460.2 118.202 1111.8 148.440 1753.5 % 178.678 2540.6 ^ 208.916 3473.2 9i 118.596 1119.2 % lis.ass 1762.7 57. 179.071 2551.8 % 209.309 3486.3 H 118.988 U26.7 H 149.226 1772.1 H 179.463 2568.0 H 209.701 3499.4 CIRCIiBS. 165 TABIiE 1 OF CIRCIiES— (Continued). Diameters in units and eigbtlis, Ae, Mam. Ciraamf. Area. Diam. Circumf. Area. Olam. Circamf. Area. Diam. Gireumf. ArM. .,.'* 210.094 3512.5 7534 236.405 4447.4 83)^ 262.716 5492.4 92. 289.027 6647.« 210.487 3525.7 H 236.798 4462.2 % 268.108 5508.8 yi 289.419 6666.7 H 210.879 3538.8 4 237.190 4477.0 Y» 263.501 5525.3 yi 289.812 6683.8 }4 211.272 3552.0 % 237.583 4491.8 84. 263.894 5541.8 % 290.205 6701.9 211.665 3565.2 H 237.976 4506.7 H 264.286 5558.3 yi 290.597 6720.1 212.058 3578.5 % 238.368 4521.5 Y* 264.679 5574.8 Ys 290.990 6738.2 g 212.450 3591.7 76. 238.761 4536.5 Yt 265.072 5591.4 Yi 291.383 6756.4 212.843 3605.0 yi 239.154 4551.4 yi 265.465 5607.9 Ys 291.775 6774.7 K 213.236 3618.3 yi 239.546 4566.4 % 265.857 5624.5 93. 292.168 6792.9 68. 213.628 3631.7 % 239.939 4581.3 % 266.250 5641.2 H 292.561 6811.1 H 214.021 3645.0 yi 240.332 4596.3 % 266.643 5657.8 H 292.954 6829.5 % 214.414 3658.4 % 240.725 4611.4 85. 267.035 5674.5 Ys 293.346 6847.8 214.806 3671.8 H 241.117 4626.4 yi 267.428 5691.2 yi 293.739 6866.1 IZ 215.199 3685.3 % 241.510 4641.5 yi 267.821 5707.9 Ys 294.132 6884.5 \i 215.592 3698.7 77. 241.903 4656.6 % 268.213 5724.7 Yi 294.524 6902.9 H 215.984 3712.2 yi 242.295 4671.8 ^ 268.606 5741.5 Ys 294.917 6921.3 •u 216.377 3725.7 yi 242.688 4686.9 % 268.999 5758.3 94. 295.310 6939.8 69. 216.770 37.S9.3 •% 243.081 4702.1 H 269.392 5775.1 yi 295.702 6958.2 % 217.163 3752.8 1^1 243.473 4717.3 % 269.784 5791.9 Yi 296.095 6976.7 yi 217.555 3766.4 %\ 243.866 4732.5 86. 270.177 5808.8 Ys 296.488 6995.3 % 217.948 3780.0 «^l 244.259 4747.8 H 270.570 5825«7 Yl 296.881 7013.8 yi 218.341 3793.7 % 244.652 4763.1 yi 270.962 5842.6 Ys 297.273 7032.4 218.733 3807.3 78. 245.044 4778.4 Ys 271.355 5859.6 % 297.666 7051.0 H 219.126 3821.0 yi 245.437 4793.7 yi 271.748 5876.5 Ys 298.059 7069.6 % 219.519 3834.7 }4\ 245.830 4809.0 % .272.140 5893.5 95. 298.451 7088.2 70. 219.911 3848.5 % 246.222 4824.4 H 272.533 5910.6 yi 298.844 7106.9 yi 220.304 3862.2 y^ 246.615 4839.8 Y» 272.926 5927.6 Yi 299.237 7125.6 220.697 3876.0 % 247.008 4855.2 87. 273.319 5&44.7 ■ Ys 299.629 7144.? % 22-1.090 3888.8 247.400 4870.7 yi 273.711 5961.8 yi 300.022 7163.0 yi 221.482 3903.6 •u 247.793 4886.2 274.104 5978.9 Ys 300.415 7181.e % 221.875 3917.5 79. 248.186 4901.7 % 274.497 5996.0 Yi 300.807 7200.6 % 222.268 3931.4 H 248.579 4917.2 y^ 274.889 6013.2 Ys 301.200 7219.4 % 222.660 3945.3 yi 248.971 4932.7 H 275.282 6030.4 96. 301.593 7238.2 n. 223.053 3959.2 % 249.364 4948.3 r/ 275.675 6047.6 M 301.986 7257.1 H 223.446 3973.1 yi 249.757 4963.9 % 276.067 6064.9 Yi 302.378 7276 223.838 3987.1 % 250.149 4979.5 88. 276.460 6082.1 Ys 302.771 7294.9 y 224.231 4001.1 H 250.542 4995.2 yi 276.853 6099.4 ¥> 303.164 7313.8 yi 224.624 4015.2 y% 250.935 5010.9 yi 277.246 6116.7 % 303.556 7332.8 % 225.017 4029.2 80. 251.ar27 5026.5 Ys 277.638 6134.1 H 303.949 7351.8 % 225.409 4043.3 %\ 251.720 5042.3 yi 278.031 6151.4 Ys 304.342 7370.8 % 225.802 4057.4 .%\ 252.113 5058.0 Ys 278.424 6168.8 97. 304.734 7389.8 t2. 226.195 4071.5 Yi' 252.506 5073.8 H 278.816 6186.2 yi 305.127 7408.9 yi 226.587 4085.7 yi\ 252.898 5089.6 Ys 279.209 6203.7 Yi 305.520 7428. C 226.980 4099.8 %\ 253.291 5105.4 89. ■ 279.602 6221.1 Ys 305.913 7447.1 ^ 227.373 4114.0 %■ 253.684 5121.2 yi 279.994 6238.6 y^ 306.305 7466.2 227.765 4128.2 %! 254.076 5137.1 yi 280.387 6256.1 Ys 306.698 7485.3 ^ 228.158 4142.5 81. 254.469 5153.0 Ys 280.780 6273.7 Yi 307.091 7504.8 ^ 228.551 4156.8 1^1 254.862 5168.9 yi 281.173 6291.2 Ys 307.483 7523.7 % 228.944 4171.1 M: 255.204 5184.9 Ys 281.565 6308.8 98. 307.876 7543.C 78. 229.336 4185.4 Yi 255.647 5200.8 H 281.958 6326.4 yi 308.269 7562.2 yi 229.729 4199.7 y^\ 256.040 5216.8 Ys 282.351 6344.1 Yi 308.661 7581 .S 230.122 42141 %^ 256.438 5232.8 90. 282.743 6361.7 Ys 309.054 7600.8 % 230.514 4228.5 %. 256.825 5248.9 yi 283.136 6379.4 yi 309.447 7620.1 H 230.907 4242.9 % 257.218 5264.9 yi 283.529 6397.1 Ys 309.840 7639.5 % 231.300 4257.4 82. 257.611 5281.0 % 283.921 6414.9 Yi 310.232 7658.S % 231.692 4271.8 yi 258.003 5297.1 y^ 284.314 6432.6 Ys 310.625 7678.3 % 232.085 4286.3 H 258.396 5313.3 284.707 6450.4 99. 311.018 7697.7 74. 232.478 4300.8 J/g! 258.789 5329.4 %/ 285.100 6468.2 yi 311.410 7717.1 yi 232.871 4315.4 yi 259.181 5345.6 yi 285.492 6486.0 Yi 311.803 7736.6 23S263 4329.9 % 259.574 5361.8 91. 285.885 6503.9 % 312.196 7756.1 % 233.656 4344.5 Va 259.967 5378.1 yi 286.278 16521.8 yi 312.588 7775.6 H •«4.049 4359.2 Yt 260.359 5394.3 M 286.679 6539.7 Ys 312.981 7795.2 % 234.441 4373.8 83. 260.752 5410.6 Ys 287.063 6557.6 Yi 313.374 7814.8 % 234.834 4388.5 • H 261.145 5426.9 yi 287.456 6575.5 Ys 313.767 7834.4 X 235.227 4403.1 Yi 261.538 5443.3 % 287.848 6593.5 100. 314.159 785*.(i 76. 235.619 4417.9 yt 261.930 5459.6 H 288.241 6611.5 H 236.012 4432.6 yi 262.323 5476.0 }s 288.634 6629.6 166 CIRCLES. TABIii: 3 OF CIRCIiHS. Diameters in units and tenths. Dia. Circumf. Area. Dia. Circumf. Are*. Dia. Circumf. Area. 0.1 .314159 .007854 6.3 19.79203 31.17245 12.5 39.26991 122.7185 .2 .628319 .031416 .4 20.10619 32.16991 .6 39.58407 124.6898 .3 .942478 .070686 .5 20.42035 33.18307 .7 39.89823 126.6769 .4 1.256637 .125664 .6 20.73451 34.21194 .8 40.21239 128.6796 .5 1.570796 .196350 .7 21.04867 35.25652 .9 40.52655 130.6981 .6 1.884956 .282743 .8 21.36283 36.31681 13.0 40.84070 132.7323 .7 2.199115 .384845 .9 21.67699 37.39281 .1 41.15486 134.7822 .8 2.513274 .502655 7.0 21.99115 38.48451 o 41.46902 136.8478 .9 2.827433 .636173 .1 22.30531 39.59192 '.3 41.78318 138.9291 1.0 3.141593 .785398 .2 22.61947 40.71504 A 42.09734 141.0261 .1 3.455752 .950332 .3 22.93363 41.85387 .5 42.41150 143.1388 .2 3.769911 1.13097 .4 23.24779 43.00840 .6 42.72566 145.2672 ^ 4.084070 1.32732 .5 23.56194 44.17865 .7 43.03982 147.4114 .4 4.398230 1.53938 .6 23.87610 45.36460 .8 43.35398 149.5712 .5 4.712389 1.76715 .7 24.19026 46.56626 .9 43.66814 151.7468 .6 5.026548 2.01062 .8 24.50442 47.78362 14.0 43.98230 153.9380 .7 5.340708 2.26980 2.5^69 .9 24.81858 49.01670 .1 44.29646 156.1450 .8 5.654867 8.0 25.13274 50.26548 .2 44.61062 158.3677 .9 5.969026 2.83529 .1 25.44690 51.52997 .3 44.92477 160.6061 2.0 6.283185 3.14159 .2 25.76106 52.81017 .4 45.23893 162.8602 .1 6.597345 3.46361 .3 •26.07522 64.10608 .5 45.55309 165.1300 ^ 6.911504 3.80133 .4 26.38938 55.41769 .6 45.86725 167.4155 .3 7.225663 4.15476 .5 26.70354 56.74502 .7 46.18141 169.7167 .4 7.539822 4.52389 .6 27.01770 58.08805 .8 46.49557 172.0336 .5 7.853982 4.90874 .7 27.33186 59.44679 .9 46.80973 174.3662 .6 8.168141 5.30929 .8 27.64602 60.82123 15.0 47.12389 176.7146 .7 8.482300 5.72555 .9 27.96017 62.21139 .1 47.43805 179.0786 .8 8.796159 6.15752 9.0 28.27433 63.61725 .2 47.75221 181.4584 .9 9.110619 6.60520 .1 28.58849 65.03882 .3 48.06637 183.8539 3.0 9.424778 7.06858 .2 28.90265 66.47610 .4 48.38053 186.2650 .1 9.738937 7.54768 • .3 29.21681 .5 48.69469 188.6919 .2 10.05310 8.04248 .4 29.53097 69^39^78 .6 49.00885 191.1345 .3 10.36726 8.55299 .5 29.81513 70.88218 .7 49.32300 193.5928 .4 10.68142 9.07920 .6 30.15929 72.38229 .8 49.63716 196.0668 .5 10.99557 9.62113 .7 30.47345 73.89811 .9 49.95132 198.5565 ,6 11.30973 10.17876 .8 30.78761 75.42964 16.0 50.26548 201.0619 .7 11.62389 10.75210 .9 31.10177 76.97687 .1 50.57964 203.5831 .8 11.93805 11.34115 10.0 31.41593 78.53982 .2 50.89380 206.1199 .•9 12.25221 11.94591 .1 31.73009 80.11847 .3 51.20796 208.6724 4.0 12.56637 12.56637 .2 32.04425 81.71282 .4 51.52212 211.2407 .1 12.88053 13.20254 .3 32.35840 83.32289 .5 51.83628 213.8246 .2 13.19469 13.85442 .4 32.67256 84.94867 .6 52.15044 216.4243 .3 13.50885 14.52201 .5 32.98672 86.59015 .7 52.46460 219.0397 .4 13.82301 15.20531 .6 33.30088 88.24734 .8 52.77876 221.6708 .5 14.13717 15.90431 .7 33.61504 89.92024 .9 53.09292 224.3176 .6 14.45133 16.61903 .8 33.92920 91.60884 17.0 53.40708 226.9801 .7 14.76549 17.34945 .9 34.24336 93.31316 .1 53.72123 229.6583 .8 15.07964 18.09557 11.0 34.55752 95.03318 .2 54.03539 232.3522 .9 15.39380 18.85741 .1 34.87168 96.76891 .3 54.34955 235.0618 5.0 15.70796 19.63495 .2 35.18584 98.52035 .4 54.66371 237.7871 .1 16.02212 20.42821 .3 35.50000 100.2875 .5 54.97787 240.5282 .2 16.33628 21.23717 .4 35.81416 102.0703 .6 55.29203 243.2849 .3 16.65044 22.06183 .5 36.12832 103.8689 .7 55.60619 246.0574 .4 16.96460 22.90221 .6 36.44247 105.6832 .8 55.92035 248.8456 .5 17.27876 23.75829 .7 36.75663 107.5132 .9 56.23451 251.6494 .6 17.59292 24.63009 .8 37.07079 109.3588 18.0 56.54867 254.4690 .7 17.90708 25.51759 .9 37.38495 111.2202 .1 56.86283 257.3043 .8 18.22124 26.42079 12.0 37.69911 113.0973 .2 57.17699 260.1553 .9 18.53540 27.33971 .1 38.01327 114.9901 .3 57.49115 263.0220 6.0 18.84956 28.27433 .2 38.32743 116.8987 .4 57.80530 265.9044 .1 19.16372 29.22467 .3 38.64159 118.8229 .5 58.11946 268.8025 .2 19.47787 30.19071 .4 38.95575 120.7628 .6 58.43362 271.7163 CIRCLES. 167 TABIii: 3 OF CIRCIiES—CContinaed). ]>iaineters in units and tenths. Dia. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. 18.7 68.74778 274.6459 24.9 78.22566 486.9547 31.1 97.70353 759.6450 .8 59.06194 277.5911 25.0 78.53982 490.8739 2 98.01769 764.5380 .9 59.37610 280.5521 .1 78.85398 494.8087 !3 98.331&3 769.4467 19.0 59.69026 283.5287 .2 79.16813 498.7592 .4 98.64601 774.3712 .1 60.00442 286.5211 .3 79.48229 502.7255 .5 98.96017 779.3113 2 60.31858 289.5292 .4 79.79645 506.7075 .6 99.27433 784.2672 !3 60.^3274 292.5530 .5 80.11061 510.7052 .7 99.58849 789.2388 .4 60.94690 295.5925 .6 80.42477 514.7185 .8 99.90265 794.2260 .5 61.26106 298.6477 .7 80.73893 518.7476 .9 100.2168 799.2290 .6 61.57522 301.7186 .8 81.05309 522.7924- 32.0 100.5310 804.2477 .7 61.88938 304.8052 .9 81.36725 526.8529 .1 100.8451 809.2821 .8 62.20353 307.9075 26.0 81.68141 530.9292 .2 101.1593 814.3322 .9 62.51769 311.0255 .1 81.99557 535.0211 .3 101.4734 819.3980 20.0 62.83185 314.1593 .2 82.30973 539.1287 .4 101.7876 824.4796 .1 63.14601 317.3087 .3 82.62389 543.2521 .5 102.1018 829.5768 .2 63.46017 320.4739 .4 82.93805 547.3911 .6 102.4159 834.6898 .3 63.77433 323.6547 .5 83.25221 551.5459 .7 102.7301 839.8184 .4 64.08849 326.8513 .6 83.56636 555.7163 .8 103.0442 844.9628 .5 64.40265 330.0636 .7 83.88052 559.9025 .9 103.3584 850.1228 .6 64.71681 333.2916 .8 84.19468 564.1044 33.0 103.6726 855.2986 .7 65.03097 336.5353 .9 84.50884 568.3220 .1 103.9867 860.4901 .8 65.34513 339.7947 27.0 84.82300 572.5553 .2 104.3009 865.6973 .9 65.65929 343.0698 .1 85.13716 576.8043 .3 104.6150 870.9202 21.0 65.97345 346.3606 .2 85.45132 581.0690 .4 104.9292 876.1588 .1 66.28760 349.6671 .3 85.76548 585.3494 .5 105.2434 881.4131 .2 66.60176 352.9894 .4 86.07964 589.6455 .6 105.5575 886.6831 .3 66.91592 356.3273 .5 86.39380 593.9574 .7 105.8717 891.9688 .4 67.23008 359.6809 .6 86.70796 598.2849 .8 106.1858 897.2703 .5 67.54424 363.0503 .7 87.02212 602.6282 .9 106.5000 902.5874 .6 67.85810 366.4354 .8 87.33628 606.9871 34.0 106.8142 907.9203 .7 68.17256 369.8361 .9 87.65044 611.3618 .1 107.1283 913.2688 .8 68.48672 373.2526 28.0 87.96459 615.7522 .2 107.4425 918.6331 .9 68.80088 376.6848 .1 88.27875 620.1582 .3 107.7566 924.0131 22.0 69.11504 380.1327 .2 88.59291 624.5800 .4 108.0708 929.4088 .1 69.42920 383.5963 .3 88.90707 629.0175 .5 108.3849 934.8202 .2 69.74336 387.0756 .4 89.22123 633.4707 .6 108.6991 940.2473 .3 70.05752 390.5707 .5 89.53539 637.9397 .7 109.0133 945.6901 .4 70.37168 394.0814 .6 89.84955 642.4243 .8 109.3274 951.1486 .5 70.68583 397.6078 .7 90.16371 646.9246 .9 109.6416 956.6228 .6 70.99999 401.1500 .8 90.47787 651.4407 36.0 109.9557 962.1128 .7 71.31415 404.7078 .9 90.79203 655.9724 -.1 110.2699 967.6184 .8 71.62831 408.2814 29.0 91.10619 660.5199 .2 110.5841 973.1397 .9 71.94247 411.8707 .1 91.42035 665.0830 .3 110.8982 978.6768 28.0 72.25663 415.4756 .2 91.73451 669.6619 .4 111.2124 984.2296 .1 72.57079 419.0963 .3 92.04866 674.2565 .5 111.5265 989.7980 .2 72.8&495 73.19911 422.7327 .4 92.36282 678.8668 .6 111.8407 995.3822 .3 426..3848 .5 92.67698 683.4928 .7 112.1549 1000.9821 .4 73.51327 430.0526 .6 92.99114 688.1345 .8 112.4690 1006.5977 .5 73.82743 433.7361 .7 93.30530 692.7919 .9 112.7832 1012.2290 .6 74.14159 437.4354 .8 93.61946 697.4650 36.0 113.0973 1017.8760 .7 74.45575 441.1503 .9 93.93362 702.1538 .1 113.4115 1023.5387 .8 74.76991 444.8809 30.0 94.24778 706.8583 .2 113.7257 1029.2172 .9 75.08406 448.6273 .1 94.56194 711.5786 .3 114.0398 1034.9113 24.0 75.89822 452.3893 .2 94.87610 716.3145 .4 114.3540 1040.6212 .1 75.71238 456.1671 .3 95.19026 721.0662 .5 114.6681 1046.3467 .2 76.02654 459.9606 .4 95.50442 725.8336 .6 114.9823 1052.0880 .3 76.34070 463.7698 .5 95.81858 730.6166 .7 115.2965 1057.8449 .4 76.65486 467.5947 .6 96.13274 735.4154 .8 115.6106 1063.6176 .5 76.96902 471.4352 .7 96.44689 740.2299 .9 115.9248 1069.4060 .6 77.28318 475.2916 .8 96.76105 745.0601 37.0 116.2389 1075.2101 ,7 77.59734 479.1636 .9 97.07521 749.9060 .1 116.5531 1081.0299 .8 77.91150 483.0513 31.0 97.38937 754.7676 .2 116.8672 1086.8654 168 CIRCLES. TABIiE 3 OF CIRCI.es— (Continued). I>iaiiieters in units and tenths. Oia. Circunif. Area. Dia. Circumf. Area. Dia. Circumf. Area. 37.3 117.1814 1092.7166 43.5 136.6593 1486.1697 49.7 156.1372 1940.0041 .4 117.4956 1098.5835 .6 136.9734 1493.0105 .8 156.4513 1947.8189 .5 117.8097 1104.4662 .7 137.-2876 1499.8670 .9 156.7655 1955.6493 .6 118.1239 1110.3645 .8 137.6018 1506.7393 50.0 157.0796 1963.4954 .7 118.4380 1116.2786 .9 137.9159 1513.6272 .1 157.3938 1971.3572 .8 118.7522 1122.2083 44.0 138.2301 1520.5308 .2 157.7080 1979.2348 .9 119.0664 1128.1538 .1 138.5442 1527.4502 .3 158.0221 1987.1280 38.0 119.3805 1134.1149 .2 138.8584 1534.3853 .4 158.3363 1995.0370 .1 119.6947 1140.0918 .3 139.1726 1541.3360 .5 158.6504 2002.9617 .2 120.0088 1146.0844 .4 139.4867 1548.3025 .6 158.9646 2010.9020 .3 120.3230 1152.0927 .5 139.8009 1555.2847 .7 159.2787 2018.8581 .4 120.6372 1158.1167 .6 140.1150 1562.2826 .8 159.5929 2026.8299 .5 120.9513 1164.1564 .7 140.4292 1569.2962 .9 159.9071 2034.8174 .6 121.2655 1170.2118 .8 140.7434 1576.3255 61.0 160.2212 2042.8206 .7 121.5796 1176.2&30 .9 141.0575 1583.3706 .1 160.5354 2050.8395 .8 121.8938 1182.3698 45.0 141.3717 1590.4313 .2 160.8495 2058.8742 .9 122.2080 1188.4724 .1 141.6858 1597.5077 .3 161.1637 2066.9245 39.0 122.5221 1194.5906 .2 142.0000 1604.5999 A 161.4779 2074.9905 .1 122.8363 1200.7246 .3 142.3141 1611.7077 .5 161.7920 2083.0723 .2 123.1504 1206.8742 .4 142.6283 1618.8313 .6 162.1062 2091.1697 .3 123.4646 1213.0396 .5 142.9425 1625.9705 .7 162.4203 2099.2829 .4 123.7788 1219.2207 .6 143.2566 1633.1255 .8 162.7345 2107.4118 .5 124.0929 1225.4175 .7 143.5708 1640.2962 .9 163.0487 2115.5563 .6 124.4071 1231.6300 .8 143.8849 1647.4826 52.0 163.3628 2123.7166 .7 124.7212 1237.8582 .9 144.1991 1654.6847 .1 163.6770 2131.8926 .8 125.0354 1244.1021 46.0 144.5133 1661.9025 .2 163.9911 2140.0843 .9 125.3495 1250.3617 .1 144.8274 1669.1360 .3 164.3053 2148.2917 40.0 125.6637 1256.6371 .2 145.1416 1676.3853 .4 164.6195 2156.5149 .1 125.9779 1262.9281 .3 145.4557 1683.6502 .5 164.9336 2164.7537 .2 126.2920 1269.2348 .4 145.7699 1690.9308 .6 165.2478 2173.0082 .3 126.6062 1275.5573 .5 146.0841 1698.2272 .7 165.5619 2181.2785 .4 126.9203 1281.8955 .6 146.3982 1705.5392 .8 165.8761 2189.5644 .5 127.2345 1288.2493 .7 146.7124 1712.8670 .9 166.1903 2197.8661 .6 127.5487 1294.6189 .8 147.0265 1720.2105 53.0 166.5044 2206.1834 .7 127.8628 1301.0042 .9 147.3407 1727.5697 .1 166.8186 2214.5165 .8 128.1770 1307.4052 47.0 147.6549 1734.9445 .2 167.1327 2222.8653 .9 128.4911 1313.8219 .1 147.9690 1742.3351 .3 167.4469 2231.2298 41.0 128.8053 1320.2543 .2 148.2832 1749.7414 .4 167.7610 2239.6100 .1 129.1195 1326.7024 .3 148.5973 1757.1635 .5 168.0752 2248.0059 .2 129.4336 1333.1663 .4 148.9115 1764.6012 .6 168.3894 2256.4175 .3 129.7478 1339.6458 .5 149.2257 1772.0546 .7 168.7035 2264.8448 A 130.0619 1346.1410 .6 149.5398 1779.5237 .8 169.0177 2273.2873 .5 130.3761 1352.6520 .7 149.8540 1787.0086 .9 169.3318 2281.7466 „6 130.6903 13.59.1786 .8 150.1681 1794.5091 54.0 169.6460 2290.2210 .7 131.0044 1365.7210 • .9 150.4823 1802.0254 .1 169.9602 2298.7112 .8 131.3186 1372.2791 48.0 150.7964 1809.5574 .2 170.2743 2307.2171 .9 131.6327 1378.8529 .1 151.1106 1817.1050 .3 170.5885 2315.7386 42.0 131.9469 1385.4424 .2 151.4248 1824.6684 .4 170.9026 2324.2759 .1 132.2611 1392.0476 .3 151.7389 1832.2475 .5 171.2168 2332.8289 .2 132.5752 1398.6685 .4 152.0531 1839.8423 .6 171.5310 2341.3976 .3 132.8894 1405.3051 .5 152.3672 1847.4528 .7 171.8451 2349.9820 .4 133.2035 1411.9574 .6 152.6814 1855.0790 .8 172.1593 2358.5821 .5 133.5177 1418.6254 .7 152.9956 1862.7210 .9 172.4734 2367.1979 .6 133.8318 1425.3092 .8 153.3097 1870.3786 55.0 172.7876 2375.8294 .7 134.1460 1432.0086 .9 153.6239 1878.0519 .1 173.1018 2384.4767 .8 134.4602 1438.7238 49.0 153.9380 1885.7410 .2 173.4159 2393.1396 .9 134.7743 1445.4546 .1 154.2522 1893.4457 .3 173.7301 2401.8183 48.0 135.0885 1452.2012 .2 154.5664 1901.1662 .4 174.0442 2410.5126 .1 135.4026 1458.9635 .3 154.8805 1908.9024 .5 174.3584 2419,2227 .2 135.7168 1465.7415 .4 155.1947 1916.6543 .6 174.6726 2427.9485 .3 1:36.0310 1472.5352 .5 155.5088 1924.4218 .7 174.9867 24:^.6899 .4 136.3451 1479.3446 .6 155.8230 1932.2051 .8 175.3009 2445.4471 CIRCLES. 169 TABIiE 2 OF CIRCXES— (Continued). JDiameters in units and tentlis. Dia. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. o5.9 175.6150 2454.2200 62.1 195.0929 3028.8173 68.3 214.5708 3663.7960 66.0 175.9292 2463.0086 .2 195.4071 3038.5798 .4 214.8849 3674.5324 .1 176.2433 2471.8130 .3 195.7212 3048.3580 .5 215.1991 3685.2845 .2 176.5575 2180.6330 .4 196.0354 3058.1520 .6 215.5133 3696.0523 .3 176.8717 2489.4687 .5 196.13495 3067.9616 .7 215.8274 3706.8359 .4 177.1858 2498.3201 .6 196.6637 3077.7869 .8 216.1416 3717.6351 .5 177.5000 2507.1873 .7 196.9779 3087.6279 .9 216.4557 3728.4500 .6 177.8141 2516.0701 .8 197.2920 3097.4847 69.0 216.7699 3739.2807 .7 178.1283 2524.9687 .9 197.6062 3107.3571 .1 217.0841 3750.1270 .8 178.4425 2533.8830 63.0 197.9203 3117.2453 .2 217.3982 3760.9891 .9 178.7566 2542.8129 .1 198.2345 3127.1492 .3 217.7124 3771.8668 57.0 179.0708 2551.7586 .2 198.5487 3137.0688 .4 218.0265 3782.7603 .1 179.3849 2560.7200 .3 198.8628 3147.0040 .5 218.3407 3793.6695 .2 179.6991 2569.6971 .4 199.1770 3156.9550 .6 218.6548 3804.5944 .3 180.0133 2578.6899 .5 199.4911 3166.9217 .7 218.9690 3815.5350 .4 180.3274 2587.6985 .6 199.8053 3176.9042 .8 219.2832 3826.4913 .5 180.6416 2596.7227 • .7 200:1195 3186.9023 .9 219.5973 3837.4633 .6 180.9557 2605.7626 .8 200.4336 3196.9161 70.0 219.9115 3848.4510 .7 181.2699 2614.8183 .9 200.7478 3206.9456 .1 220.2256 3859.4544 .8 181.5841 2623.8896 64.0 201.0619 3216.9909 .2 220.5398 3870.4736 .9 181.8982 2«32.9767 .1 201.37(a 3227.0518 .3 220.8540 3881.5084 68.0 182.2124 2642.0794 .2 201.6902 3237.1285 .4 221.1681 3892.5590 .1 182.5265 2651.1979 .3 202.(X)44 3247.2209 • .5 221.4823 3903.6252 .2 182.8407 2660.3321 .4 202.3186 3257.3289 .6 221.7964 3914.7072 .3 183.1549 2669.4820 .5 202.6327 3267.4527 .7 222.1106 3925.8049 .4 183.4690 2678.6476 .6 2(J2.9469 3277.5922 .8 222.4248 3936.9182 .5 183.7832 2687.8289 .7 203.2610 3287.7474 .9 222.7389 3948.0473 .6 184.0973 2697.0259 .8 203.5752 3297.9183 71.0 223.0531 3959.1921 .7 181.4115 2706.2386 .9 203.8894 3308.1049 .1 223.3672 3970.3526 .8 184.7256 2715.4670 65.0 204.2035 3318.3072 .2 223.6814 3981.5289 .9 185.0398 2724.7112 .1 204.5177 3328.5253 .3 223.9956 3992.7208 59.0 185.3540 2733.9710 .2 204.8318 3338.7590 .4 224.3097 4003.9284 .1 185.6681 2743.2466 .3 205.1460 3349.0085 .5 224.6239 4015.1518 .2 185.9823 2752.5378 .4 205.4602 3359.2736 .6 224.9380 4026.3908 .3 186.2964 2761.8448 .5 205.7743 3369.5545 .7 225.2522 4037.6456 .4 186.6106 2771.1675 .6 206.0885 3379.8510 .8 225.5664 4048.9160 .5 186.9248 2780.5058 .7 206.4026 3390.1633 .9 225.8805 4060.2022 .6 187.2389 2789.8599 .8 206.7168 3400.4913 72.0 226.1947 4071.5U41 .7 187.5531 2799.2297 .9 207.0310 3410.8350 .1 226.5088 4082.8217 .8 187.8672 2808.6152 66.0. 207.3451 3421.1944 .2 226.8230 4094.1550 .9 188.1814 2818.0165 .1 207.659S 3431.5695 .3 227.1371 4105.5040 •0.0 188.4956 2827.4334 .2 207.9734 3441.9603 .4 227.4513 4116.8687 .1 188.8097 2836.8660 .3 208.2876 3452.3669 .5 227.7655 4128.2491 .2 189.1239 2846.3144' .4 208.6018 3462.7891 .6 228.0796 4139.6452 .3 189.4380 2855.7784 .5 208.9159 3473.2270 .7 228.3938 4151.0571 .4 189.7522 2865.2582 .6 209.2301 3483.6807 .8 228.7079 4162.4846 .5 190.0664 2874.7536 .7 209.5442 3494.1500 .9 229.0221 4173.9279 .6 190.3805 2884.2648 .8 209.8584 3504.6351 73.0 229.3363 4185.3868 .7 190.6947 2893.7917 .9 210.1725 3515.1359 .1 229.6504 4196.8615 .8 191.0088 2903.3343 67.0 210.4867 3525.6524 *.2 229.9646 4208.3519 .9 191.3230 2912.8926 .1 210.8009 3536.1845 .3 230.2787 4219.8579 61.0 191.6372 2922.4666 .2 211.1150 3546.7324 .4 230.5929 4231.3797 .1 191.9513 2932.0563 .3 211.4292 3557.2960 .5 230.9071 4242.9172 .2 192.2655 2941.6617 .4 211.7433 3567.8754 .6 231.2212 4254.4704 ,3 192.5796 2951.2828 .5 212.0575 3578.4704 .7 231.5354 4266.0394 .4 192.8938 2960.9197 .6 212..3717 3589.0811 .8 231.8495 4277.6240 .5 193.2079 2970.5722 .7 212.6858 3^599.7075 .9 232.1637 4289.2243 .6 193.5221 2980.2405 .8 213.0000 3610.3497 74.0 232.4779 4300.8403 .7 193.8363 2989.9244 .9 213.3141 3621.0075 .1 2.32.7920 4312.4721 .8 194.1504 2999.6241 68.0 213.6283 3631.6811 .2 233.1062 4324.1195 .9 194.4646 3009.3395 .1 213.9425 3642.3704 .3 233.4203 4335.7827 62.0 194.7787 3019.0705 .2 214.2566 3653.0754 .4 233.7345 4347.4616 170 CIRCLK8. TABIiE 2 OF ClRCIiES— (Continued). ]>iaineters in units and tenttis> Bia. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. 74.5 234.0487 4359.1562 80.7 253.5265 5114.8977 86.9 273.0044 5931.0206 .6 234.3628 4370.8664 .8 ■ 253.8407 5127.5819 87.0 273.3186 5944.6787 .7 234.6770 4382.5924 .9 254.1548 5140.2818 .1 273.6327 5958.3525 .8 2S4.9911 4394.3341 81.0 254.4690 5152.9974 -2 273.9469 5972.0420 .9 235.3053 4406.0916 .1 254.7832 5165.7287 .3 274.2610 5985.7472 75.0 235.6194 4417.8647 .2 255.0973 5178.4757 .4 274.5752 5999.4681 .1 235.9336 4429.6535 .3 255.4115 5191.2384 .5 274.8894 6013.2047 .2 236.2478 4441.4580 .4 255.7256 5204.0168 .6 275.2035 6026.9570 .3 236.5619 4453.2783 .5 256.0398 5216.8110 .7 275.5177 6040.7250 .4 236.8761 4465.1142 .6 256.3540 5229.6208 .8 275.8318 6054.5088 .5 237.1902 4476.9659 .7 256.6681 5242.4463 .9 276.1460 6068.3082 .6 237.5044 4488.8332 .8 256.9823 5255.2876 88.0 276.4602 6082.1234 .7 237.8186 4500.7163 .9 257.2964 5268.1446 .1 276.7743 6095.9542 .8 238.1327 4512.6151 82.0 257.6106 5281.0173 .2 277.0885 6109.8008 .9 238.4469 4524.5296 .1 257.9248 5293.9056 .3 277.4026 6123.6631 16.0 238.7610 4536.4598 .2 258.2389 5306.8097 .4 277.7168 6137.5411 .1 239.0752 4548.4057 .3 258.5531 5319.7295 .5 278.0309 6151.4348 .2 239.3894 4560.3673 .4 258.8672 5332.6650 .6 278.3451 6165.3442 .3 239.7035 4572.3446 .5 259.1814 5345.6162 .7 278.6593 6179.2693 .4 240.0177 4584.3377 .6 259.4956 5358.5832 .8 278.9734 6193.2101 .5 240.3318 4596.3464 .7 259.8097 5371.5658 .9 ^79.2876 6207.1666 .6 240.6460 4608.3708 .8 260.1239 5384.5641 89.0 279.6017 6221.1389 .7 240.9602 4620.4110 .9 260.4380 5397.5782 .1 279.9159 6235.1268 .8 241.2743 4632.4669 88.0 260.7522 5410.6079 .2 280.2301 6249.1304 .9 241.5885 4644.5384 .1 261.0663 5423.6534 .3 280.5442 6263.1498 77.0 241.9026 4656.6257 .2 261.3805 5436.7146 .4 280.8584 6277.1849 .1 242.2168 4668.7287 .3 261.6^7 5449.7915 .5 281.1725 6291.2356 .2 242.531C 4680.8474 .4 262.0088 5462.8840 .6 281.4867 6305.3021 .3 242.8451 4692.9818 .5 262.3230 5475.9923 .7 281.8009 6319.3843 .4 243.1593 4705.1319 .6 262.6371 5489.1163 .8 282.1150 6333.4822 .5 243.4734 4717.2977 .7 262.9513 5502.2561 .9 282.4292 6347.5958 .6 243.7876 4729.4792 .8 263.2655 5515.4115 90.0 282.7433 6361.7251 .7 244.1017 4741.6765 .9 263.5796 5528.5826 .1 283.0575 6375.8701 .8 244.4159 4753.8894 84.0 263.8938 5541.7694 .2 283.3717 6390.0309 .9 244.7301 4766.1181 .1 264.2079 5554.9720 .3 283.6858 6404.2073 18.0 245.0442 4778.3624 .2 264.5221 5568.1902 .4 284.0000 6418.3995 .1 245.3584 4790.6225 .8 264.8363 5581.4242 .5 284.3141 6432.6073 .2 245.6725 4802.8983 .4 265.1504 5594.6739 .6 284.6283 6446.8309 .3 245.9867 4815.1897 .5 265.4646 5607.9392 .7 284.9425 6461.0701 .4 246.3009 4827.4969 .6 265.7787 5621.2203 .8 285.2566 6475.3251 .5 246.6150 4839.8198 .7 266.0929 5634.5171 .9 285.5708 6489.5958 .6 246.9292 4852.1584 .8 266.4071 5647.8296 91.0 285.8849 6503.8822 .7 247.243? 4864.5128 .9 266.7212 5661.1578 .1 286.1991 6518.1843 .8 247.5575 4876.8828 85.0 267 0354 5674.5017 .2 286.5133 6532.5021 Jd 247.8717 4889.2685 .1 267 3495 5687.8614 .3 286.8274 6546.8356 79.0 248.1858 4901.6699 .2 267 6637 5701.2367 .4 287.1416 6561.1848 .1 248.5000 4914.0871 .3 267.9779 5714.6277 .5 287.4557 6575.5498 .2 248.8141 4926.5199 .4 268.2920 5728.0345 .6 287.7699 6589.9304 .3 249.1283 4938.9685 .5 268.6062 5741.4569 .7 288.0840 6604.3268 .4 249.4425 4951.43^8 .6 268.9203 5754.8951 .8 288.3982 6618.7388 .5 249.7566 4963.9127 .7 269.2345 5768.3490 .9 288.7124 6633.1666 .6 250.0708 4976.4084 .8 269.5486 5781.8185 92.0 289.0265 6647.6101 .7 250.3849 4988.9198 .9 269:8628 5795.3038 .1 289.3407 6662.0692 .8 250.6991 5001.4469 86.0 270.1770 5808.8048 .2 289.6548 6676.5441 .9 251.0133 5013.9897 .1 270.4911 5822.3215 .3 289.9690 6691.0347 8«.0 251.3274 5026.5482 .2 270.8053 5835.8539 .4 290.2832 6705.5410 .1 251.6416 5039 1??5 .3 271.1194 5849.4020 .5 290.5973 6720.0630 .2 251.9557 5051.7124 .4 271.4386 5862.9659 .6 290.9115 6734.6008 .3 252.2699 5064.3180 .5 271.7478 5876.5454 .7 291.2256 6749.1542 .4 252.5840 5076.9394 .6 272.0619 5890.1407 .8 291.5398 6763.7233 .5 252.8982 5089.5764 .7 272.3761 5903.7516 .9 291.8540 6778.3082 .6 253.2124 5102.2292 .8 272.6902 5917.3783 93.0 292.1681 6792.9087 CIRCLES. 171 TABIiE 2 OF CIRCIiES— (Continued). Diameters in units and tentbs. Ma. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. 9B.1 292.4828 6807.5250 95.5 300.0221 7163.0276 97.8 307.2478 7512.2078 .2 292.7964 6822.1569 .6 300.:^63 7178.0366 .9 307.5619 7527.5780 .3 293.1106 6836.8046 .7 300.6504 7193.0612 98.0 307.8761 7542.9640 .4 293.4248 6851.4680 .8 300.9646 7208.1016 .1 308.1902 7558.8656 .5 293.7389 6866.1471 .9 301.2787 7223.1577 .2 308.5044 7573.7830 .6 294.0531 6880.8419 9ft.0 301.5929 7238.2295 .3 308.8186 7589.2161 .7 294.3672 6895.5524 .1 301.9071 7253.3J70 .4 309.1327 7604.6648 .8 294.6814 6910.2786 .2 302.2212 7208.4202 .o 309.4469 7620.1293 .9 294.9956 6925.0205 .3 302.5354 7283.5391 .6 309.7610 7635.6095 94.0 295.3097 6939.7782 .4 302.8495 7298.6737 .7 310.0752 7651.1054 .1 295.6239 6954.5515 .0 303.1637 7313.8240 .8 310.3894 7666.6170 .2 295.9380 6969.3406 .6 303.4779 7328.9901 .9 310.7035 7682.1444 .3 296.2522 6984.1453 .7 303.7920 7344.1718 99.0 311.0177 7697.6874 .4 296.5663 6998.9658 .8 304.1062 7359.3693 .1 314.3318 7713.2461 .5 296.8805 7013.8019 .9 304.4203 7374.5824 .2 311.6460 7728.8206 .6 297.1947 7028.6538 97.0 304.7345 7389.8113 .3 311.9602 7744.4107 .7 297.5088 7043.5214 .1 305.0486 7405.0559 .4 312.2743 7760.0166 .8 297.8230 7058.4047 .2 305.3628 7420.3162 .5 312.5885 7775.6382 .9 298.1371 7073.3037 .3 305.6770 7435.5922 .6 312.9026 7791.2754 #6.0 298.4513 7088.2184 .4 305.9911 7450.8839 .7 313.2168 7806.9284 .1 298.7655 7103.1488 .5 306.3053 7466.1913 .8 313.5309 7822.5971 .2 299.0796 7118.0950 .6 306.6194 7481.5144 .9 313.8451 7838.2815 .3 299.3938 7133.0568 .7 306.9336 7496.a532 100.0 314.1593 7853.9816 A 299.7079 7148.0343 Circumferences when the diameter has more than one place of decimals. Diam. Circ. Diam. Circ. 1 Diam. Circ. Diam. Circ. Diam. Circ. .1 .314159 .01 .031416 .001 .003142 .0001 .000314 .00001 .000031 .2 .628319 .02 .062832 .002 .006283 .0002 .000628 .00U02 .000063 .3 .942478 .03 .094248 .003 .009425 .0003 .000942 .00003 .000094 .1 1.256637 .04 .125664 .004 .012566 .0004 .001257 .00004 .000126 .5 1.570796 .05 .157080 .005 .015708 .0005 .001571 .00005 .000157 .6 1.884956 .06 .188496 ..006 .018850 .0006 .001885 .00006 .000188 .7 2.199115 .07 .219911 .007 .021991 .0007 .002199 .00007 .000220 .8 2.513274 .08 .251327 .008 .025133 .0008 .002513 .00008 .000251 .9 2.827433 .09 .282743 .009 .028274 .0009 .002827 .00009 .000283 Diameter = 3.12699 Circumference = Circ for dia of 3.1 • •* .02 " .006 *** .0009 « .00009 Examples. Sum of ■■ 9.738937 : .062832 : .018850 : .002827 : .000283 9.823729 Circumfce =» Diameter = Dia for circ of 9.823729 9.738937 ■■ Sum oi 3.1 .062832 = .02 .021960" .018850 =, .006 .003110 .002827 = .0009 .000283 .000283 » .00009 172 CIRCLES. TABI.E 3 OF CIRCIiES. Diams in units and twelfths; as in feet and inebes. Dia. Circumf. Area, Dia. Circumf. 1 Area. Dia. Circumf. Area. Pt.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq.ft. 5 15.70796 19.63495 10 31.41593 78.53982 1 .261799 .005454 1 15.96976 20.29491 1 31.bV//3 79.85427 2 .523599 .021817 2 16.23156 20.96577 2 31.93953 81.17968 3 .785398 .049087 3 1 16.49336 21.64754 3 32.20132 82.51589 4 1.047198 .087266 4 16.75516 22.34021 4 32.46312 83.86307 5 1.308997 .136354 5 17.01696 23.04380 5 32.72492 85.22115 6 1.570796 .196350 6 17.27876 23.75829 6 32.98672 86.59015 7 1.832596 .267254 7 17.54056 24.48370 7 33.24852 87.97005 8 2.094395 .349066 8 17.80236 25.22001 8 33.51032 89.36086 9 2.356195 •441786 9 18.06416 25.96723 9 33.77212 90.76258 10 2.617994 .545415 10 18.32596 26.72535 10 34.03392 92.17520 11 2.879793 .659953 11 18.58776 27.49439 11 34.29572 93.59874 1 3.14159 .785398 6 18.84956 28.27433 11 34.55752 95.03318 1 3.40339 .921752 1 19.11136 29.06519 1 34.81932 96.47858 2 3.66519 1.06901 2 19.37315 29.86695 2 35.08112 97.93479 3 3.92699 1.22718 3 19.63495 30.67962 3 35.34292 99.40196 4 4.18879 1.39626 4 19.89675 31.50319 4 35.60472 100.8800 5 4.45059 1.57625 5 20.15855 32.33768 5 35.86652 102.3690 6 4.71239 1.76715 6 20.42035 33.18307 6 36.12832 103.8689 7 4.97419 1.96895 7 20.68215 34.03937 7 36.39011 105.3797 8 5.23599 2.18166 8 20.94395 34.90659 8 36.65191 106.9014 9 5.49779 2.40528 9 21.20575 35.78470 9 36.91371 108.4340 10 6.76959 2.63981 10 21.46755 36.67373 10 37.17551 109.9776 11 6.02139 2.88525 11 21.72935 37.57367 11 37.43731 111.5320 S 6.28319 3.14159 7 21.99115 38.48451 12 37.69911 113.0973 1 6.54496 3.40885 1 22.25295 39.40626 1 37.96091 114.6736 2 6.80678 3.68701 2 22.51475 * 40.33892 2 38.22271 116.2607 3 7.06858 3.97608 3 22.77655 41.28249 3 38.48451 117.8588 4 7.33038 4.27606 4 23.03835 42.23697 4 38.74631 119.4678 5 7.59218 4.58694 5 2S.30015 43.20235 5 39.00811 121.0877 6 7.85398 4.90874 6 23.56194 44.17865 6 39.26991 122.7185 7 8.11578 5.24144 7 23.82374 45.16585 7 39.53171 124.3602 8 8.37758 5.58505 8 24.08554 46.16396 8 39.79351 126.0128 9 8.63938 5.93957 9 24.34734 47.17298 9 40.05531 127.6763 10 8.90118 6.30500 10 24.60914 48.19290 10 40.31711 129.3507 11 9.16298 6.68134 11 24.87094 49.22374 11 40.57891 131.0360 8 9.42478 7.06858 8 25.13274 50.26548 13 40.84070 132.7323 1 9.68658 7.46674 1 25.39454 51.31813 1 41.10250 134.4394 2 9.94838 7.87580 2 25.65634 52.38169 2 41.36430 136.1575 3 10.21018 8.29577 3 25.91814 53.45616 3 41.62610 137.8865 4 10.47198 8.72665 4 26.17994 54.54154 4 41.88790 139.6263 5 10.73377 9.16843 5 26.44174 55.63782 5 42.14970 141.3771 6 10.99557 9.62113 6 26.70354 56.74502 6 42.41150 143.1388 7 11.25737 10.08473 7 26.96534 57.86312 7 42.67330 144.9114 8 11.51917 10.55924 8 27.22714 58.99213 8 42.93510 146.6949 9 11.78097 11.04466 9 27.48894 60.13205 9 43.19690 148.4893 10 12.04277 11.54099 10 27.75074 61.28287 10 43.45870 150.2947 11 12.30457 12.04823 11 28.01253 62.44461 11 43.72050 152.1109 4 12.56637 12.56637 9 28.27433 63.61725 14 43.98230 153.938a 1 12.82817 13.09542 1 28.53613 64.80080 1 44.24410 155.7761 2 13.08997 13.63538 2 28.79793 65.99526 2 44.50590 157.6250 3 13.35177 14.18625 3 29.05973 67.20063 3 44.76770 159.4849 4 13.61357 14.74803 4 29.32153 68.41691 4 45.02949 161.3557 5 13.87537 15.32072 5 29.58333 69.64409 5 45.29129 163.2374 6 14.13717 15.90431 6 29.84513 70.88218 6 45.55309 165.1301 7 14.39897 16.49882 7 30.10693 72.13119 7 45.81489 167.0335 8 14.66077 17.10423 8 30.36873 73.39110 8 46.07669 168.9479 9 14.92257 17.72055 9 30.63053 74.66191 9 46.33849 170.8732 10 15.18436 18.34777 10 30.89233 75.94364 10 46.60029 172.8094 U 15.44616 18.98591 11 31.15413 77.23627 11 46.86209 174.7565 CIRCLES. 173 Bia TABIii: 3 OF CIRCI.es— (Continued). \ in units and twelfths; as in feet and incbes. m. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. Pt.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.Iu. Feet. Sq. ft. [6 47.12389 176.7146 20 62.83185 314.1593 25 78.53982 490.8739 1 47.38569 178.6835 1 63.09365 316.7827 1 78.80162 494.1518 2 47.64749 180.6634 2 63.35545 319.4171 2 79.06342 497.4407 3 47.90929 182.6542 3 63.61725 322.0623 3 79.32521 500.7404 4 48.17109 184.6558 4 63.87905 324.7185 4 79.58701 504.0511 5 48.43289 186.6684 5 64.14085 327.3856 5 79.84881 507.3727 6 48.69469 188.6919 6 64.40265 330.0636 6 80.11061 510.7052 7 48.95649 190.7263 7 64.66445 332.7525 7 80.37241 514.0486 S 49.21828 192.7716 8 64.92625 335.4523 8 80.63421 517.4029 9 49.48008 194.8278 9 65.18805 338.1630 9 80.89601 520.7681 10 49.74188 196.8950 10 65.44985 340.8846 10 81.15781 524.1442 11 50.00368 198.9730 11 65.71165 343.6172 11 81.41961 527.5312 L« 50.26548 201.0619 21 65.97345 346.3606 26 81.68141 530.9292 1 50.52728 203.1618 1 66.23525 349.1149 1 81.94321 534.3380 2 50.78908 205.2725 2 66.49704 351.8802 2 82.20501 537.7578 3 51.05088 207.3942 3 66.75884 354.6564 3 82.46681 541.1884 4 51.31268 209.5268 4 67.02064 357.4434 4 1 82.72861 544.6300 5 51.57448 211.6703 5 67.28244 360.2414 5 82.99041 548.0825 6 51.83628 213.8246 6 67.54424 363.0503 6 83.25221 551.5459 7 52.09808 215.9899 7 67.80604 365.8701 7 83.51400 555.0202 8 52.35988 218.1662 8 68.06784 368.7008 8 83.77580 558.5054 9 52.62168 220.3533 9 68.32964 371.5424 9 84.03760 562.0015 10 52.88348 222.5513 10 68.59144 374.3949 10 84.29940 565.5085 11 53.14528 224.7602 11 68.85324 377.2584 11 84.56120 569.0264 17 53.40708 226.9801 22 69.11504 380.1327 27 84.82300 572.5553 1 53.66887 229.2108 1 69.37684 383.0180 1 85.08480 576.0950 2 53.93067 231.4525 2 69.63864 385.9141 2 85.34060 579.6457 3 54.19247 233.7050 3 69.90044 388.8212 3 85.60840 583.2072 4 54.45427 235.9085 4 70.16224 391.7392 4 85.87020 586.7797 5 54.71607 238.2429 5 70.42404 394.6680 5 86.13200 590.3631 6 54.97787 240.5282 6 70.68583 397.6078 6 86.39380 593.9574 7 55.23967 242.8244 7 70.94763 400.5585 7 86.65560 597.5626 8 55.50147 245.1315 8 71.20943 408.5201 8 86.91740 601.1787 9 55.76327 247.4495 9 71.47123 406.4926 9 87.17920 604.8057 10 56.02507 249.7784 10 71.73303 409.4761 10 87.44100 608.4436 11 56.28687 252.1183 11 71.99483 412.4704 11 87.70279 612.0924 L8 56.54867 254.4690 28 72.25663 415.4756 28 87.96459 615.7522 1 56.81047 256.8307 1 72.51843 418.4918 1 88.22639 619.4228 2 57.07227 '259.2032 2- 72.78023 421.5188 2 88.48819 623.1044 8 57.33407 261.5867 3 73.04203 424.5568 3 88.74999 626.7968 4 57.59587 263.9810 4 73.30383 427.6057 4 89.01179 630.5002 5 57.85766 266.3863 5 73.56563 430.6654 5 89.27359 634.2145 6 58.11946 268.8025 6 73.82743 433.7361 6 89.53539 637.9397 7 58.38126 271.2296 7 74.08923 436.8177 7 89.79719 641.6758 8 58.64306 273.6676 8 74.35103 439.9102 8 90.05899 645.4228 9 58.90486 276.1165 9 74.61283 443.0137 9 90.32079 649.1807 10 59.16666 278.5764 10 74.87462 446.1280 10 90.58259 652.9495 11 59.42846 281.0471 11 75.13642 449.2532 11 90.84439 656.7292 L» 59.69026 283.5287 24 75.39822 452.3893 29 91.10619 660.5199 1 59.95206 286.0213 1 75.66002 455.5364 1 91.36799 664.3214 2 60.21386 288.5247 2 75.92182 458.6943 2 91.62979 668.1339 3 60.47566 291.0391 3 76.18362 461.8632 3 91.89159 671.9572 4 60.73746 293.5644 4 76.44542 465.0430 4 92.15338 675.7915 5 60.99926 296.1006 5 76.70722 468.2337 5 92.41518 679.6367 6 61.26106 298.6477 6 76.96902 471.4352 6 92.67698 683.4928 7 61.52286 301.2056 7 77.23082 474.6477 7 92.93878 687.3597 8 61.78466 303.7746 8 77.49262 477.8711 8 93.20058 691.2377 9 62.04645 306.3544 9 77.75442 481.1055 9 93.46238 695.1265 10 62.30825 308.9451 10 78.01622 484.3507 10 93.72418 699.0262 11 62.57005 311.5467 11 78.27802 487.6068 11 93.98598 702.9368 174 CIRCLES. TABIiE 3 OF CIRCIiES— (Continued). Diams in units and twelfths; as in feet and incbes. Dia. Circumf. Area* Dia. Circumf. Area. Dia. Circumf. Area. ft. In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. SO 94.24778 706.8583 85 109.9557 962.1128 40 125.6637 1256.6371 1 94.50958 710.7908 1 110.2175 966.6997 1 125.9255 1261.8785 2 94.77138 714.7341 2 110.4793 971.2975 2 126.1873 1267.1309 3 95.03318 718.6884 8 110.7411 975.9063 3 126.4491 1272.3941 4 95.29498 722.6535 4 111.0029 980.5260 4 126.7109 1277.6683 5 95.55678 726.629';; 5 111.2647 985.1566 5 126.9727 1282.9534 6 95.81858 730.6166 6 111.5265 989.7980 6 127.2345 1288.2493 7 96.08088 734.6145 7 111.7883 994.4504 7 127.4963 1293.5562 8 96:34217 738.6233 8 112.0501 999.1137 8 127.7581 1298.8740 9 96.60397 742.6431 9 112.3119 1003.7879 9 128.0199 1304.2027 10 96.86577 746:6737 10 112.5737 1008.4731 10 128.2817 1309.5424 11 97.12757 750.7152 11 112.8355 1013.1691 11 128.5435 1314.8929 81 97.38937 754.7676 36 113.0973 1017.8760 41 128.8053 1320.2543 1 97.65117 758.8310 1 113.3591 1022.5939 1 129.0671 1325.6267 2 97.91297 762.9052 2 113.6209 1027.3226 2 129.3289 1331.0099 3 98.17477 766.9904 3 113.8827 1032.0623 3 129.5907 1336.4041 4 98.48657 771.0865 4 114.1445 1036.8128 4 129.8525 1341.8091 5 98.69837 775.1934 5 114.4063 1041.5743 5 130.1143 1347.225^ 6 98.96017 779.3113 6 114.6681 1046.3467 6 130.3761 1352.652ft 7 99.22197 783.4401 7 114.9299 1051.1300 7 130.6379 1358.0898 8 99.48377 787.5798 8 115.1917 1055.9242 8 130.8997 1363.5385 9 99.74557 791.7304 9 115.4535 1060.7293 9 131.1615 1368.9981 10 100.0074 795.8920 10 115.7153 1065.5453 10 131.4233 1374.4686 11 100.2692 800.0644 11 115.9771 1070.3723 11 131.6851 1379.9500 82 100.5310 804.2477 37 116.2389 1075.2101 42 131.9469 1385.4424 1 100.7928 808.4420 1 116.5007 1080.0588 1 132.2087 1390.9456 2 101.0546 812.6471 2 116.7625 1084.9185 2 132.4705 1396.4598 3 101.3164 816.8632 3 117.0243 1089.7890 3 132.7323 1401.9848 4 101.5782 821.0901 4 117.2861 1094.6705 4 132.9941 1407.5208 5 101.8400 825.3280 5 117.5479 1099.5629 5 133.2559 1413.0676 6 102.1018 829.5768 6 117.8097 1104.4662 6 133.5177 1418.6254 7 102.3636 833.8365 7 118.0715 1109.3804 7 133.7795 1424.1941 8 102.6254 838.1071 8 118.3333 1114.3055 8 134.0413 1429.7737 9 102.8872 842.3886 9 118.5951 1119.2415 9 134.3031 1435.3642 10 103.1490 846.6810 10 118.8569 1124.1884 10 134.5649 1440.9656 11 103.4108 850.9844 11 119.1187 1129.1462 11 134.8267 1446.5780 83 103.6726 855.2986 38 119.3805 1134.1149 43 135.0885 1452.2012 1 103.9344 859.6237 1 119.6423 1139.0946 1 135.3503 1457.8353 2 104.1962 863.9598 2 119.9041 1144.0851 2 135.6121 1463.4804 3 104.4580 868.3068 3 120.1659 1149.08C6 3 135.8739 1469.1364 4 104.7198 872.6646 4 120.4277 1154.0990 4 136.1357 1474.8032 5 104.9816 877.0334 5 120.6895 1159.1222 5 136.3975 1480.4810 6 105.2434 881.4131 6 120.9513 1164.1564 6 136.6593 1486.1697 7 105.5052 885.8037 7 121.213J 1169.2015 7 136.9211 1491.8693 8 105.7670 890.2052 8 121.4749 1174.2575 8 137.1829 1497.579S 9 106.0288 894.6176 9 121.7367 1179.3244 9 137.4447 1503.3012 10 106.2906 899.0409 10 121.9985 1184.4022 10 137.7065 1509.0335 11 106.5524 903.4751 11 122.2603 1189.4910 11 137.9683 1514.7767 84 106.8142 907.9203 39 122.5221 1194.5906 44 138.2301 1520.5308 1 107.0759 912.3763 1 122.7839 1199.7011 1 138.4919 1526.2959 2 107.3377 916.8433 2 123.0457 1204.8226 2 138.7537 1532.0718 3 107.5995 921.3211 3 123.3075 1209.9550 3 139.0155 1537.8587 4 107.8613 925.8099 4 123.5693 1215.0982 4 139.2773 1543.6565 5 108.1231 930.3096 5 123.8311 1220.2524 5 139.5391 1549.4651 6 108.3849 934.8202 6 124.0929 1225.4175 6 139.8009 1555.2847 7 108.6467 939.3417 7 124.3547 1230.5935 7 140.0627 1561.1152 8 108.9085 943.8741 8 124.6165 1235.7804 8 140.3245 1566.9566 9 103.1703 948.4174 9 124.8783 1240.9782 9 140.5863 1572.8089 10 109.4321 952.9716 10 125.1401 1246.1869 10 140.8481 1578.6721 11 109.6939 957.5367 11 125.4019 1251.4065 11 141.1099 1584.5462 CIRCLES. 175 TABIiE 3 OF CIRCIiES— (Continued). Diams in units and tivelfths; as in feet and inches. Di;i. Circamf. Area. Dia. Circumf. Area. Dia. Circumf. Area. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. 45 141.3717 1590.4313 50 157.0796 1963.4954 55 172.7876 2375.8294 1 141.6335 1596.3272 1 157.3414 1970.0458 1 173.0494 2383.0344 2 141.8953 1602.2341 2 157.6032 1976.6072 2 173.3112 2390.2502 3 142.1571 1608.1518 3 157.8650 1983.1794 3 173.5730 2397.4770 4 142.4189 1614.0805 4 158.1268 1989.7626 4 173.8348 2404.7146 5 142.6807" 1620.0201 5 158.3886 1996.3567 5 174.0966 2411.9632 6 142.9425 1625.9705 6 158.6504 2002.9617 6 174.3584 2419.2227 7 143.2043 1631.9319 7 158.9122 2009.5776 7 174.6202 2426.4931 8 143.4661 1637.9042 8 159.1740 2016.2044 8 174.8820 2433.7744 9 143.7279 1643.8874 9 159.4358 2022.8421 9 175.1438 2441.0666 10 143.9897 1649.8816 10 159.6976 2029.4907 10 175.4056 2448.3697 11 144.2515 1655.8866 11 159.9594 2036.1502 11 175.6674 2455.6837 46 144.5133 1661.9025 51 160.2212 2042.8206 56 175.9292 2463.0086 1 144.7751 1667.9294 1 160.4830 2049.5020 1 176.1910 2470.3445 2 145.0369 1673.9671 2 160.7448 2056.1942 2 176.4528 2477.6912 3 145.2987 1680.0158 3 161.0066 2062.8974 3 176.7146 2485.0489 4 145.5605 1686.0753 4 161.2684 2069.6114 4 176.9764 2492.4174 5 145.8223 1692.1458 5 161.5302 2076.3364 5 177.2382 2499.7969 6 146.0841 1698.2272 6 161.7920 2083.0723 6 177.5000 2507. 187» 7 146.3459 1704.3195 7 162.0538 2089.8191 7 177.7618 2514.5886 8 146.6077 1710.4227 8 162.3156 2096.5768 8 178.0236 2522.0008 9 146.8695 1716.5368 9 162.5774 2103.3454 9 178.2854 2529.4239 10 147.1313 1722.6618 10 162.8392 2110.1249 10 178.5472 2536.8579 11 147.3931 1728.7977 11 163.1010 2116.9153 11 178.8090 2544.3028 47 147.6549 1734.9445 52 163.3628 2123.7166 57 179.0708 2551.7586 1 147.9167 1741.1023 1 163.6246 2130.5289 1 179.3326 2559.2254 2 148.1785 1747.2709 2 163.8864 2137.3520 2 179.5944 2566.7030 8 148.4403 1753.4505 3 164.1482 2144.1861 3 179.8562 2574.1916 4 148.7021 1759.6410 4 164.4100 2151.0310 4 180.1180 2581.6910 5 148.9639 1765.8423 5 164.6718 2157.8869 5 180.3798 2589.2014 6 149.2257 1772.0546 6 164.9336 2164.7537 6 180.6416 2596.7227 7 149.4875 1778.2778 7 165.1954 2171.6314 7 180.9034 2604.2549 8 149.7492 1784.5119 8 165.4572 2178.5200 8 181.1652 2611.7980 9 150.0110 1790.7569 9 165.7190 2185.4195 9 181.4270 2619.3520 10 150.2728 1797.0128 10 165.9808 2192.3299 10 181.6888 2626.9169 11 150.5346 1803.2796 11 166.2426 2199.2512 11 181.9506 2634.4927 4S 150.7964 1809.5574 53 166.5044 2206.1834 58 182.2124 2642.0794 1 151.0582 1815.8460 1 166.7662 2213.1266 1 182.4742 2649.6771 2 151.3200 1822.1456 2 167.0280 2220.0806 2 182.7360 2657.2856 3 151.5818 1828.4560 3 167.2898 2227.0456 3 182.9978 2664.9051 4 151.8436 1834.7774 4 167.5516 2234.0214 4 183.2596 2672.5354 5 152.1054 1841.1096 5 167.8134 2241.0082 5 183.5214 2680.1767 6 152.3672 1847.4528 6 168.0752 2248.0059 6 183.7832 2687.8289 7 152.6290 1853.8069 7 168.3370 2255.0145 7 184.0450 2695.4920 8 152.8908 1860.1719 8 168.5988 2262.0340 8 184.3068 2703.1659 9 153.1526 1866.5478 9 168.8606 2269.0644 9 184.5686 2710.8508 10 153.4144 1872.9346 10 169.1224 2276.1057 10 184-8304 2718.5467 11 153.6762 1879.3324 11 169.3842 2283.1579 11 185.0922 2726.2534 49 153.9380 1885.7410 54 169.6460 2290.2210 59 185.3540 2733.9710 1 154.1998 1892.1605 1 169.9078 2297.2951 1 185.6158 2741.6995 2 154.4616 1898.5910 2 170.1696 2304.3800 2 185.8776 2749.4390 3 154.7234 1905.0323 3 170.4314 2311.4759 3 186.1394 2757.1893 4 154.9852 1911.4846 4 170.6932 2318.5826 4 186.4012 2764.9506 5 •155.2470 1917.9478 5 170.9550 2325.7003 5 186.6630 2772.7228 6 155.5088 1924.4218 6 171.2168 2332.8289 6 186.9248 2780.5058 7 155.7706 1930.9068 7 171.4786 2339.9684 7 187.1866 2788.2998 8 156.0324 1937.4027 8 171.7404 2347.1188 8 187.4484 2796.1047 9 156.2942 1943.9095 9 172.0022 2354.2801 9 187.7102 2803.9205 10 156.5560 1950.4273 10 172.2640 2361.4523 10 187.9720 2811.7472 11 156.8178 1956.9559 11 172.5258 2368.6354 11 188.2338 2819. 584i 176 CIRCLES. TABIiE 3 OF CIRCIiES— (Continued). Biams in units and twelftlis; as in feet and inebes. Dia. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. Ft.Iu. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. 60 188.4956 2827.4334 66 204.2035 3318.3072 70 219.9115 3848.4510 1 188.7574 2835.2928 1 204.4653 3326.8212 1 220.1733 3857.6194 2 189.0192 2843.1632 2 204.7271 3335.3460 2 220.4351 3866.7988 3 189.2810 2851.0444 3 204.9889 3343.8818 3 220.6969 3875.9890 4 189.5428 2858.9366 4 205.2507 3352.4284 4 220.9587 3885.1902 5 189.8046 2866.8397 5 205.5125 3360.9860 5 221.2205 3894.4022 6 190.0664 2874.7536 6 205.7743 3369.5545 6 221.4823 3903.6252 7 190.3282 2882.6785 7 206.0361 3378.1339 7 221.7441 3912.8591 8 190.5900 2890.6143 8 206.2979 3386.7241 8 222.0059 3922.1039 9 190.8518 2898.5610 9 206.5597 3395.3253 9 222 2677 3931.3596 10 191.1136 2906.5186 10 206.8215 3403.9375 10 222.5295 3940.6262 11 191.3754 2914.4871 11 207.0833 3412.5605 11 222.7913 3949.9037 61 191.6372 2922.4666 66 207.3451 3421.1944 71 223.0531 3959.1921 1 191.8990 2930.4569 1 207.6069 3429.8392 1 223.3149 3968.4915 2 192.1608 2938.4581 2 207.8687 3438.4950 2 223.5767 3977.8017 3 192.4226 2946.4703 3 208.1305 3447.1616 3 223.8385 3987.1229 4 192.6843 2954.4934 4 208.3923 3455.8392 4 224.1003 3996.4549 6 192.9461 2962.5273 5 208.6541 3464.5277 5 224.3621 4005.7979 C 193.2079 2970.5722 6 208.9159 3473.2270 6 224.6239 4015.1518 7 193.4697 2978.6280 7 209.1777 3481.9373 7 224.8867 4024.5165 8 193.7315 2986.6947 8 209.4395' 3490.6585 8 225.1475 4033.8922 9 193.9933 2994.7723 9 209.7013 3499.3906 9 225.4093 4043.2788 10 194.2551 3002.8608 10 209.9631 3508.1336 10 225.6711 4052.6763 11 ' 194.5169 3010.9602 11 210.2249 3516.8875 11 225.9329 4062.0848 62 194.7787 3019.0705 67 210.4867 3525.6524 72 226.1947 4071.5041 1 195.0405 3027.1918 1 210.7485 3534.4281 1 226.4565 4080.9343 2 195.3023 3035.3239 2 211.0103 3543.2147 2 226.7183 4090.3756 3 195.5641 3043.4670 3 211.2721 3552.0123 3 226.9801 4099.8275 4 195.8259 3051.6209 4 211.5339 3560.8207 4 227.2419 4109.2905 5 196.0877 3059.7858 5 211.7957 3569.6401 5 227.5037 4118.7643 6 196.3495 3067.9616 6 212.0575 3578.4704 6 227.7655 4128.2491 7 196.6113 3076.1483 7 212.3193 3587.3116 7 228.0273 4137.7448 8 196.8731 3084.3459 8 212.5811 3596.1637 8 228.2891 4147.2514 9 197.1349 8092.5544 9 212.8429 3605.0267 9 228.5509 4156.7689 10 197.3967 3100.7738 10 213.1047 3613.9006 10 228.8127 4166.2973 n 197.6585 3109.0041 11 213.3665 3622.7854 11 229.0745 4175.8866 «3 197.9203 3117.2453 68 213.6283 3631.6811 73 229.3363 4185.3868 1 198.1821 3125.4974 1 213.8901 3640.5877 1 229.5981 4194.9479 2 198.4439 3133.7605 2 214.1519 3649.505S 2 229.8599 4204.5200 3 198.7057 3142.0344 3 214.4137 3058.4337 3 230.1217 4214.1029 4 198.9675 3150.3193 4 214.6755 3667.3731 4 230.3835 4223.6968 5 199.2293 3158.6151 5 214.9373 3676.3234 5 230.6453 4233.3016 6 199.4911 3166.9217 6 215.1991 8685.2845 6 230.9071 4242.9172 7 199.7529 3175.2393 7 215.4609 3694.2566 7 231.1689 4252.5438 8 200.0147 3183.5678 8 215.7227 3703.2396 8 231.4307 4262.1813 9 200.2765 3191.9072 9 215.9845 3712.2335 9 231.6925 4271.8297 10 200.5383 3200.2575 10 216.2463 3721.2383 10 231.9543 4281.4890 11 200.8001 3208.6188 11 216.5081 3730.2540 11 232.2161 4291.1592 64 201.0619 3216.9909 69 216.7699 3739.2807 74 232.4779 4300.8408 1 201.3237 3225.3739 1 217.0317 3748.3182 1 232.7397 4310.5324 2 201.5855 3233.7679 2 217.2935 3757.3666 2 233.0015 4320.2353 3 201.8473 3242.1727 3 217.5553 3766.4260 8 233.2633 4329.9492 4 202.1091 3250.5885 4 217.8171 3775.4962 • 4 233.5251 4339.6739 6 202.3709 3259.0151 5 218.0789 3784.5774 5 233.7869 4349.4096 6 202.6327 3267.4527 6 218.3407 3793.6695 6 234.0487 4359.1562 7 202.8945 3275.9012 7 218.6025 3802.7725 7 234.3105 4368.9136 8 203.1563 3284.3606 8 218.8643 3811.8864 8 234.5723 4378.6820 9 203.4181 3292.8309 9 219.1261 3821.0112 9 234.8341 4388.4613 10 203.6799 3301.3121 10 219.3879 3830.1469 10 235.0959 4398.2515 11 203.9417 3309.8042 11 219.6497 3839.2935 U 235.3576 4408.0526 CIRCLES. 177 TABIiB 3 OF CIRCIiES— (Continued). Diams in units and twelfths; as in feet and Incbes. Dia. Circttinf. Area. Dia. Circumf. Area. Dia. Circumf. Area. rUa. Feet. Sq. ft. Ft.Ia. Feet. Sq. ft. Ft.In. Feet. Sq. ft. 75 235.6194 4417.8647 80 251.3274 5026.5482 85 267.0354 5674.5017 1 235.8812 4427.6S76 1 251.5892 5037.0257 1 267.2972 5685.6337 2 236.1430 4437.5214 2 251.8510 5047.5140 2 267.5590 5696.7765 8 236.4048 4447.3662 3 252.1128 5058.0133 3- 267.8208 5707.9302 4 236.6666 4457.2218 4 252.3746 5068.5234 4 268.0826 5719.0949 5 236.9284 4467.0884 6 252.6364 5079.0445 5 268.3444 5730.2705 6 237.1902 4476.9659 6 252.8982 5089.5764 6 268.6062 5741.4569 7 237.4520 4486.8543 7 253.1600 5100.1193 7 268.8680 5752.6543 8 237.7138 4496.7536 8 253.4218 5110.6731 8 269.1298 5763.8626 9 237.9756 4506.6637 9 253.6836 5121.2378 9 269.3916 5775.0818 10 238.2374 4:)16.5849 10 253.9454 5131.8134 10 269.6534 5786.3119 11 238.4992 4526.5169 11 254.2072 5142.3999 11 269.9152 5797.5529 76 238.7610 4536.4598 81 254.4690 5152.9974 86 270.1770 5808.8048 1 239.0228 4546.4136 1 254.7308 5163.6057 1 270.4388 5820.0676 2 239.2846 4556.3784 2 254.9926 5174.2249 . 2 270.7006 5831.3414 3 239.5464 4566.3540 3 255.2544 5184.8551 3 270.9624 5842.6260 4 239.8082 4576.3406 4 255.5162 5195.4961 4 271.2242 5853.9216 5 240.0700 4586.3380 5 255.7780 5206.1481 5 271.4860 5865.2280 6 240.3318 4596.3164 6 2»6.0398 5216.8110 6 271.7478 5876.5454 7 240.5936 4606.3657 7 256.3016 5227.4847 7 272.0096 5887.8737 8 240.8554 4616.3959 8 256.5634 5238.1694 8 272.2714 5899.2129 9 241.1172 4626.4370 9 256.8252 5248.8650 9 272.5332 5910.5630 10 241.3790 4636.4890 10 257.0870 5259.5715 10 272.7950 5921.9240 11 241.6408 4646.5519 11 257.3488 5270.2889 11 273.0568 5933.2959 t! 241.9026 4656.6257 82 257.6106 5281.0173 87 273.3186 5944.6787 1 242.1644 4666.7104 1 257.8724 5291.7565 1 273.5804 5956.0724 2 2i2.4262 4676.8061 2 258.1342 5302.5066 2 273.8422 5967.4771 8 242.6880 4686.9126 3 258.3960 5313.2677 3 274.1040 5978.8926 4 242.9i98 4697.0301 4 258.6578 5324.0396 4 274.3658 5990.3191 5 243.2116 4707.1584 5 258.9196 5334.8225 5 274.6276 6001.7564 6 243.4734 4717.2977 6 259.1814 5345.6162 6 274.8894 6013.2047 7 213.7352 4727.4479 7 259.4432 5356.4209 7 275.1512 6024.6639 8 243.9970 4737.6090 8 259.7050 5367.2365 8 275.4130 6036.1340 9 244.2588 4747.7810 9 259.9668 5378.0630 9 275.6748 6047.6149 10 244.5206 4757.9639 10 260.2286 5388.9004 10 275.9366 6059.1068 11 244.7824 4768.1577 11 260.4904 5399.7487 11 276.1984 6070.6097 SS 245.0442 4778.3624 83 260.7522 5410.6079 88 276.4602 6082.1234 1 245.3060 4788.5781 1 261.0140 5421.4781 1 276.7220 6093.6480 2 245.5678 4798.8046 2 261.2758 5432.3591 2 276.9838 6105.1835 3 ^5.8296 4809.0420 3 261.5376 5443.2511 3 277.2456 6116.7300 4 246.0914 4819.2904 4 261.7994 5454.1539 4 277.5074 6128.2873 5 246.3532 4829.5497 5 262.0612 5465.0677 5 277.7692 6139.8656 6 246.6150 4839.8198 6 262.3230 5475.9923 6 278.0309 6151.4348 7 246.8768 4850.1009 7 262.5848 5486.9279 7 278.2927 6163.0248 8 247.1386 4860.3929 8 262.8466 5497.8744 8 278.5545 6174.6258 9 247.4004 4870.6958 9 263.1084 5508.8318 9 278.8163 6186.2377 10 247.6622 4881.0096 10 263.3702 5519.8001 10 279.0781 6197.8605 11 247.9240 4891.3343 11 263.6320 5530.7793 11 279.3399 6209.4942 99 248.1858 4901.6699 84 263.8938 5541.7694 89 279.6017 6221.1389 1 248.4476 4912.0165 1 264.1556 5552.7705 1 279.8635 6232.7944 2 248.7094 4922.3739 2 264.4174 5563.7824 2 280.1253 6244.4608 3 248.9712 4932.7423 3 264.6792 5574.8053 3 280.3871 6256.1382 4 249.2330 4943.1215 4 264.9410 5585.8390 4 280.6489 6267.8264 5 249.4948 4953.5117 5 265.2028 5596.8837 6 280.9107 6279.5256 6 249.7566 4963.9127 6 265.4646 5607.9392 6 281.1725 6291.2356 7 250.0184 4974.3247 7 265.7264 5619.0057 7 281.4343 6302.9566 8 250.2802 4984.7476 8 265.9882 5630.0831 8 281.6961 6314.6885 9 250.5420 4995.1814 9 266.2500 5641.1714 9 281.9579 6326.4313 10 250.8038 5005.6261 10 266.5118 ' 5652.2706 10 282.2197 6338.1850 11 251.0656 5016.0817 11 266.7736 1 5663.3807 11 282.4815 6849.9496 12 178 CIRCLES. TABIiE 3 OF CIRCL.es— (Continued). Dlams in units and twelfths; as in feet and inebes. Dia. Circumf. Area. Dia. Circumf. Area. Dia. Circumf. Area. ' Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. 90 282.7433 6361.7251 93 5 293.4771 6853.9134 96 9 303.9491 7351.768ft 1 283.0051 6373.5116 6 293.7389 6866.1471 10 304.2109 7364.4386 2 283.2669 6385.3089 7 294.0007 6878.3917 11 304.4727 7377.1195 3 •283.5287 6397.1171 8 294.2625 6890.6472 97 301.7345 7389.8113 4 283.7905 6408.9363 9 294.5243 6902.9135 1 304.9963 7402.5140 5 284.0523 6420.7663 10 294.7861 6915.1908 2 305.2581 7415.2277 6 284.3141 6432.6073 11 295.0479 6927.4791 3 305.5199 7427.9522 7 284.5759 6444.4592 94 295.3097 6939.7782 4 305.7817 7440.6877 8 284.8377 6456.3220 1 295.5715 6952.0882 5 306.0435 7453.4340 9 285.0995 6468.1957 2 295.8333 6964.4091 6 306.3053 7466.1913 10 285.3613 6480.0803 3 296.0951 6976.7410 7 306.5671 7478.9595 11 285.6231 6491.9758 4 296.3569 6989.0837 8 306.8289 7491.7385 •1 285.8849 6503.8822 5 296.6187 7001.4374 9 307.0907 7504.5285 1 286.1467 6515.7995 6 296.8805 7013.8019 10 307.3525 7517.3294 2 286.4085 6527.7278 7 297.1423 7026.1774 11 307.6143 7530.1412 3 286.6703 6539.6669 8 297.4041 7038.5638 98 307.8761 7542.9640 4 286.9321 6551.6169 9 297.6659 7050.9611 1 308.1379 7555.7976 5 287.1939 6563.5779 10 297.9277 7063.3693 2 308.3997 7568.6421 6 287.4557 6575.5498 11 298.1895 7075.7884 3 308.6615 7581.4976 7 287.7175 6587.5325 95 298.4513 7088.2184 4 308.9233 7594.3639 8 287.9793 6599.5262 1 298.7131 7100.6593 6 309.1851 7607.2412 9 288.2411 6611.5308 2 298.9749 7113.1112 6 309.4469 7620.1293 10 288.5029 6623.5463 3 299.2367 7125.5739 7 309.7087 7633.0284 11 288.7647 6635.5727 4 299.4985 7138.0476 8 309.9705 7645.9384 •2 289.0265 6647.6101 5 299.7603 7150.5321 9 310.2323 7658.8593 1 289.2883 6659.6583 6 300.0221 7163.0276 10 310.4941 7671.7911 2 289.5501 6671.7174 7 300.2839 7175.5340 11 310.7559 7684.7338 3 289.8119 6683.7875 8 300.5457 7188.0513 99 311.0177 7697.6874 4 290.0737 6695.8684 9 300.8075 7200.5794 1 311.2795 7710.6510 5 290.3355 6707.9603 10 301.0693 7213.1185 2 311.5413 7723.6274 6 290.5973 6720.0630 11 301.3311 7225.6686 3 311.8031 7736.6137 7 290.8591 6732.1767 96 301.5929 7238.2295 4 312.0649 7749.6100 8 291.1209 6744.3013 1 i 301.8547 7250.8013 5 312.3267 7762.6191 9 291.3827 6756.4368 2 302.1165 7263.3840 6 312.5885 7775.6382 10 291.6445 6768.5832 3 302.3783 7275.9777 7 312.8503 7788.6681 11 291.9063 6780.7405 4 302.6401 7288.5822 8 313.1121 7801.7090 93 292.1681 6792.9087 5 302.9019 7301.1977 9 313.3739 7814.7608 1 292.4299 6805.0878 6 303.1637 7313.8240 10 313.6357 7827.8235 2 292.6917 6817.2779 7 303.4255 73L'6.4618 n 313.8975 7840.8971 S 292.9535 6829.4788 8 ! 303.6873 7339.1095 100 314.1593 7858.9816 4 293.2153 6&41.6907 1 Diam, 1-64" 1-32 3-64 1-16 5-64 3-32 7-64 ru 6-32 11-64 3-16 13^ Circumf, Dlam, Circumf, Diam, foot Inch. foot Inch .004091 7-32 .057269 27-64 .008181 15-64 .061359 7-16 .012272 H .065450 29-64 .016362 17-64 .069540 15-32 .020453 9-32 .0736ol 31-64 .024544 19-64 .077722 .^ .028634 5-16 .081812 .032725 21-64 .086903 17-32 .036816 11-32 .089994 36-64 .040906 23-64 .094084 9-16 .044997 % .098175 37-64 .049087 25-64 .102265 19-32 .063178 13-32 .106356 39-64 Circumf, Diam, Circumf, Diam, foot. Inch. 5-8 foot. Inch. .110447 .163625 53-64 .114537 41-64 .167716 27-32 .118628 21-32 .171806 56-64 .122718 43-64 .176896 7-8 .126809 11-16 .179987 57-64 .130900 46-64 .184078 29-32 .134990 23-32 .188168 69-64 .139081 47-64 .192259 15-16 .143172 % .196360 61-64 .147262 49-64 .200440 31-32 .151353 26-32 .204631 63-64 .165443 61-64 .208621 1 .169534 13-16 .212712 Circumf, foot. .216803 .220893 .224984 .229074 .233161 .237256 .241346 .245437 .249628 .263618 .257709 .261799 CIRCULAR ARCS. CIRCUIiAB ARCS. 179 e Fig:. 2. Kules for Fig. 1 apply to all arcs equal to, or less than, a semi-circle. Fig, 2 " " " or greater than, a semi-circle. Cliorcl, a bf of irliole arC) a db, -- 2 X \/radiu82 — (radius — rise)2. Fig. 1. = 2 X V radius^ — (rise — radiu8)2. Fig. 2. = 2 X N/rise X (2 X radius — rise). Figs. 1 and 2. ■■ 2 X radius X sine of }4 a cb. Figs. 1 and 2. rise 2 X Figs. 1 and 2. tangent of ahd.* - 2 X db^ X cosine of abd* Figs. 1 and 2. = 2 X \/d62 _ risez. Figs. 1 and 2.§ = approximately 8 X dbi^ — 3X length of arc a d 6 ^. Fig. 1. ,. ^ ^ »rc a d 6 in degrees _,. , ,_ «= 2 TT radius X ^^ — • ^igs- 1 and 2 ; •^ .01745 X radius X arc adb in degrees. Figs. 1 and 2. Ife-^^ _ = circumference of circle — length of small arc subtending angle ach. Fig. 2. S X^db^ — chord a&.** = approximately =^ ^ J^ig. 1. * a 6 d is = J^ of lihe angle a c 6, subtended by the arc. In Fig. 2 the latter angle exceeds 180". § d 6 = chord of d i 6, or of half adb = \/ri8e2 + (^ a 6)2. Figs. 1 and 2. f If rise = .5 chord, .4 « .333 « .3 multiply the result by 1.036 1.0196 1.0114 1.0083 If rise =- .25 chord, .2 " .125 « .1 « multiply the result by 1.0044 1.0021 1.00036 1.00016 ••If rise = .5 chord A .333 « .3 «* multiply the result by 1.012 1.0065 1.0038 1.0028 If rise = .25 chord .2 « .125 « .1 multiply the result ^w 1.0015 1.0007 1.00012 1.00006 180 CIRCULAR ARCS. E^ig.l Continued from p. 179. Fig. 2. Knits for Fig. 1 apply to all arcs equal to or less than a semi-circle. " " Fig. 2 " « • «♦ Qf greater than a semi-circle. Radius, c a, ed, ci or cb, ^ Q4ab)2 + riseg ^ j^igs. 1 and 2. 2 X rise ^ ^^^^ , Figs. 1 and 2. 2 X rise %_ab___ ^ YisB. 1 and 2. _ K ^'^ sine of y^acb rise d e 1 — cosine of ^acb , Figs. 1 and 2. , Fig. 1. Bine of }4bcdl rise de 1 4- cosine of ^ a c 6 f , Rgs. 1 and 2. , Fig. i. Rise, or middle ordinate, d9p =- radius — v radiusS — (J^ a fc)^, Fig. 1. "= radius + \/radiu82 — (i^ a 6)2, Fig. 2. = radius X (1 — cosine of bed ||), Fig. 1. = radius X (1 + cosine of 6cd||),f Fig. 2. db^^ . , Figs, 1 and 2. 2 X radius -^ }4ab X tangent of a 6 d * Figs. 1 and 2. ^ approximately /'^ ° ^ . Fig. 1. 2 X radius When radius = chord a 6, the result is 6.7 parts in 100 too shoft. « ** =- 3 X chord a b, the result is 0.7 parts in 100 too short Side ordinate, as n t, : ]/ radius* — e n* + rise — radius, Figs. 1 and 2. anX nb = approximately ^ ^ ^^^^^ . Fig. l.H * a 6 d is = 34 of the angle acb, subtended by the arc. f Strtetly, this should read 1 mimis cosine ; but the cosines of angles between 90* and 270° must then be regarded as miwxs or negative. Our rule, therefore, amounti to the same thing. § d 6 = chord of d i &, or of half a d 6, = v rise* (^a6)2. Figs. 1 and 2. \bcd =- half the angle acb subtended by* the arc. In Fig. 2, the latter angle exceeds 180°. f When radius = chord a b, this makes d e 6.7 parts in 100 too short. " " = 3 X chord a b, this makes d e 0.7 parts in 100 too short. The proportionate error is greater with the side ordinates. CIRCULAE ARCS. 181 Angle, ach, subtended by arc, adb. An angle and its supplement (as hce and bed, Fig. 2) have the same sine, the ime cosine and the same tangent. Caution. The following sines, etc., are those of only half acb. Sine of 1^ ac 6 = -Mj?L^ , Figs. 1 and 2. radius Cosine of 3^ a c 6 = Tangent of }^ acb radius — rise radius * _ _ Hah radius — rise , rise — radius „. ^ . 1; = • r-. , Fig. 2. . Fig.l; = radius rise — radius , Fig. 2 To describe tbe arc of a circle too large for tbe dividers. Ist Metliod. Let a c be the chord, and o b the height, of the required arc, as b laid down on the drawing. On a separate strip of paper, semn, draw a c. o b. and a& also b e, parallel to the chord a c. It is well to make 6 s, and b e, each a little longer than a b. Then cut off the paper carefully along the lines s b and b e, so as to leave remaining only the strip s ab emn. Now, if the straight sides s b and & e be applied to the drawing, so that any parts of them shall touch at the same time the points a and 6, or b and c, the point & on the strip will b© in the circumference of the arc, and may be pticked oflf. Thus, any number of points in the arc may be found, and afterward united to form the curve. /dd Metl&od* Draw the span ab; the rise re; and ac,b c From c with radius c r describe a circle. Make each of the arcs o t and 1 1 equal to ro or r i; and draw c t. cl. Divide ct, cl, cr, each into half as many equal parts as the curve is to be divided into. Draw the lines 61, 62, 63; and a4, a5, tt6, extended to meet the first ones at e, «, h. Then c, *. h, are points in one half the curve. Then for the other half, draw similar lines from a to 7, 8, 9; and others from b to meet them, as before. Trace the curve by hand. 182 CIRCULAR ARCS. Remark. —It may frequently be of use to remember, that ia any arc & o «, not exceeding 29°, or in o\ her words, whose chord b s is at least sixteen times its ris«, th« middle ordinate a o, will be one-halt of a c, quite n«ar enough tor many pur- poses; b c and s c being langents to the arc.f And vice ver«a, if in such an arc we make o c equal a o, then will c be, very nearly, the point at which tangents from th© ends of the arc will meet. Also the middle ordinate n, of tlie lialf arc o b, or o*, will be approximately 34 of a o, the middle ordinate of th© whole arc. Indeed, this last observation will apply near enough for many approximate uses even if the arc be as great as 45° ; for if in that case we take }^ofo a for the ordinate n, n will then be but 1 part in 103 too small; and therefore the principle may often be used in drawings, for finding points in a curve of too great radius to be drawn by the dividers; for in the same manner, % of n will be the middle ordinate for the arc n b or n o; and so on to any extent. Below will be found a table by 'wKlcli tlie rise or middle ordinate of a lialf arc can be obtained with greater accuracy when required for more exact drawings. CIRCUIiAR ARCS IN FREaUEMT USE. The fifth column is of use for finding points for drawing arcs too large for the beam-compass, on the principle given above. In even the largest oflBce drawings it will not be necessary to use more than the first three decimals of the fifth column; and after the arc is subdivided into parts smaller than about 35° each, the first two decimals .25 will generally suflBce. Original. Rise For For rise Rise 1 For For in Degrees For rad length of of half in Degrees For rad j length of rise of parts in whole mult rise aro mult arc parts in whole mult rise jarc mult halfare of arc. by chord mult rise of arc. by 1 chord mult chord. by by chord. by riseby 1-50 9 9.75 313. 1.00107 .2501 Xh o / 56 8.70 8.5 1.04116 .2538 1-45 10 10.75 253.625 1.00132 .2501 63 46.90 6.625 1.05356 .2549 1-40 11 26.98 200.5 1.00167 .2502 .155 68 53.63 6.70291 1.06288 .2557 1-35 13 4.92 153.625 1.00219 ,2502 1-6 73 44.39 6. 1.07250 .25«6 1-30 15 15.38 113. 1.00296 .2503 .18 79 11.73 4.35803 1.0842S .2576 1-25 18 17.74 78.625 1.00426 .2504 1-5 87 12.34 3.625 1.10347 .2593 1-20 22 50.54 50.5 1.00665 .2506 .207107 90 3.41422 1.11072 .2599 1-19 24 2.16 45.625 1.00737 .2507 .225 96 54.67 2.96913 1.12997 .2615 1-18 25 21.65 41. 1.00821 .2508 M 106 15.61 2.5 1.15912 .2639 1-17 26 50.36 36.625 1.00920 .2509 .275 115 14.59 2.15289 1.19082 .2665 1-16 28 30.00 32.5 1.01038 .2510 .3 123 5130 1.88889 1.22495 .2692 1-15 30 22.71 28.625 1.01181 .2511 yk 134 45.62 1.625 1.27401 .2729 1-14 32 31.22 25. 1.01355 .2513 .365 144 30.98 1.43827 1.32413 .2766 1-13 34 59.08 21.625 1.01571 .2515 .4 154 3835 1.28125 1.38322 .2808 1-12 37 50.96 18.5 L01842 .2517 .425 161 27.52 1.19204 1.42764 .2838 1-11 41 13.16 15.625 1.02189 .2520 .45 167 56.S3 1.11728 1.47377 .2868 1-10 45 14.38 13. 1.02646 .2525 .475 174 7.49 1.054U2 1.52152 .2899 1-9 50 6.91 10.625 1.03260 .2530 .5 180 1. 1.57080 .2929 ^▲t 29° o c thus found will be but about 3 parts too ihort in lOO. MENSURATION. liengttas of circular arcs. If arc exceeds a semicircle, seep 184 Knowing its chord and height, divide the height by the chord. Find in the column of heights the number equal to this quotient. Take out the corresponding number from the column of lengths, Multiplj this last number by the length of the given chord. TABI.E OF CIRCUI.AR ARCS. No error* H'ghts. Lengths. H'ghts. Lengths. H'ghts, Lengths. H'ghts. Lengths. H'ghts. Lengths. .00^ 1.00002 .076 1.01535 .151 1.05973 .226 1.13108 .301 1,22636 002 1.00002 ,077 1.01573 .152 1.06051 .227 1.13219 .302 1,22778 -003 1.00003 .078 1.01614 .153 1.06130 .228 1,13331 .303 1.22920 ,004 1.00004 .079 1.01656 .154 1.06209 .229 1,13444 .304 1.23063 .005 1.00007 .080 1.01698 .155 1.06288 .230 1.13557 .305 1.23206 ,006 1.00010 ,081 1.01741 .156 1.06368 .231 1.13671 .306 1,23349 .007 1.00013 .082 1.01784 .157 1.06449 .232 1,13785 .307 1.23492 ,008 1.00017 .083 1.01828 .158 1.06530 .233 1,13900 .308 1.23636 .009 1.00022 .084 1.01872 .159 1.06611 .234 1,14015 .309 1.23781 .010 1.00027 ,085 1.01916 .160 1.06693 .235 1.14131 .310 1.23926 .011 1.00032 ,086 1.01961 .161 1.06775 .236 1.14247 .311 1.24070 .012 1.00038 .087 1,02006 .162 1.06858 .2.37 1.14363 .312 1.24216 .013 1.00045 ,088 1,02052 .163 1.06941 .238 1.14480 ,313 1.24361 .OH 1.00053 ,089 1,02098 .164 1.07025 .239 1.14597 .314 1.24507 .01ft 1.00061 ,090 1.02146 .165 1.07109 .240 1.14714 .315 1.24654 .016 1,00069 .091 1.02192 .166 1.07194 .241 1.14832 .316 1.24801 017 1,00078 .092 1,02240 .167 1.07279 .242 1.14951 .317 1. '24948 .018 1.00087 ,093 1,02289 .168 1.07365 .243 1.15070 .318 1.25095 .019 1.00097 ,094 1,02339 ,169 1.07451 .244 1.15189 .319 1.25243 .020 1.00107 .095 1.02389 ,170 1.07537 .245 1.15308 .320 1.25391 .021 1.00117 ,096 1.02440 .171 1,07624 .246 1.15428 .321 L 25540 ,022 1.00128 087 1.02491 .172 1.07711 .247 1.15549 .322 1.2566* .023 1.00140 ,098 1,02542 ,173 1.07799 .248 1.15670 .323 1.25838 .024 1.00153 ,099 1.02593 ,174 1.07888 .249 1.15791 .324 1.25988 .025 1 00167 .100 1.02646 .175 1.07977 .250 1.15912 .325 1.26138 .026 1.C0182 .101 1.02698 .176 1.08066 .251 1.16034 .326 1.26288 .027 1.00196 .102 1.02752 .177 1.08156 .252 1.16156 .327 1.26487 .028 1,00210 .103 1,02806 .178 1.08246 .258 1.16279 .328 1.26568 .029 1.00225 ,104 1.02860 ,179 1.08337 .254 1.16402 .329 1.26740 ,030 ] .00240 ,105 1.02914 ,180 1.08428 .255 1,16526 .330 1,26892 .031 1.00256 ,106 1.02970 ,181 1.08519 .256 1.16650 ,331 1.27044 i)32 1.00272 ,107 1.03026 ,182 1.08611 .257 1.16774 ,332 1.271«« .033 1.00289 ,108 1.03082 .183 1.08704 .258 1,16899 .333 1.27.S4S .034 1.00307 ,109 1.03139 ,184 1.08797 .259 1,17024 .334 1. '27502 .035 1.00327 ,110 1.03196 ,185 1.08890 .260 1.17150 ,335 1.27656 .036 1.00345 .111 1.03254 ,186 1.08984 ,261 1.17276 .336 1.27810 .037 1,00364 .112 1.03312 ,187 1.09079 .262 1.17403 .337 1.27964 .038 1.00384 .113 1.03371 .188 1.09174 .263 1.17530 ,338 1.28118 .039 1.00405 .114 1.03430 .189 1.09269 .264 1.17657 .339 1.2827S .040 1.00426 .115 1.03490 .190 1.09365 .265 1.17784 .340 1.28428 .041 1.00447 .116 1.03551 .191 1.09461 .266 1.17912 .341 1.28583 .042 1.00469 .117 1,03611 ,192 1.09557 .267 1.18040 .342 1. '28739 .043 1.00492 .118 1.03672 .193 1.09654 .268 1.18169 .343 1.28895 .044 1.00515 .119 1.03734 .194 1.09752 .269 1.18299 .344 1.29052 .045 1.00539 .120 1,03797 .195 1.09850 .270 1.18429 .345 1.29209 .046 1.00563 .121 1.03860 .196 1.09949 .271 1.18559 .346 1. -29366 .047 1 .00587 .122 . 1.03923 .197 1.10048 .272 . 1.18689 .347 1.29523 .048 1.00612 .123 1.03987 ,198 1.10147 .273 1.18820 .348 1.29681 .049 1.00638 ,124 1.04051 .199 1.10247 .274 1.18951 ,349 1,29839 .050 1.00665 .125 104116 .200 1.10347 .275 1.19082 .350 1,29997 .051 1 .00692 .126 1.04181 .201 1.10447 .276 1.19214 ,351 1,30156 .052 1 00720 .127 1.04247 .202 1.10548 .277 1.19346 .352 1.30315 053 1 00748 .128 1.04313 .203 1.10650 .278 1.19479 .353 1.30474 .054 1.00776 ,129 1.04380 .204 1.10752 .279 1.19612 ,354 1.30634 .055 1.00805 .130 1.04447 .205 1,10855 .280 1.19746 .355 1.30794 ,056 1.00834 .131 1.04515 .206 1,10958 ,281 1.19880 .356 1.30954 ,OR7 I 1.00864 .132 1.04584 .207 1.11062 .282 1.20014 ,357 1.3111* ,058 1.00895 .133 1.04652 .208 1.11165 .283 1.20149 .358 1.31276 -059 1.00926 ,134 1.04722 .209 1.11269 .284 1.20284 ,359 1.31437 .060 1,00957 .135 1.04792 .210 1,11374 .285 1.20419 .360 1.31599 .061 1.00989 .136 1.04862 .211 1.11479 ,286 1.20555 .361 1.31761 .062 1.01021 .137 1 .04932 .212 1.11584 ,287 1 .20691 .362 1.3192S ,063 1.01054 .138 1 .05003 .213 1.11690 ,288 1.20827 .363 1.32086 ,064 1.01088 ,139 1.05075 .214 1.11796 ,289 1.20964 .364 1.32249 ,065 1.01123 .140 1.05147 .215 1.11904 ,290 L21102 ,365 1.32413 .066 1.01158 .141 1.05220 ,216 1.12011 .291 1.21239 ,366 1.32577 .067 1,01193 .142 1 .05293 .217 1.12118 ,292 1.21377 .367 1.32741 .068 1.01228 .143 1.05367 ,218 1.12225 ,293 1.21515 .368 1.32905 .069 1.01264 .144 1.05441 .219 1.12334 ,294 1.21654 ,369 1.33069 .070 1,01302 .145 1.05516 ,220 1.12444 ,295 1.21794 .370 1.. 33*234 .071 1,01338 .146 1.05591 ,221 1.12554 .296 1.21933 ,371 1.33399 .072 .073 .074 .075 1,01376 .147 1.05667 .222 1.12664 .297 1.22073 .372 1.33564 l!oi414 .148 1.05743 ,223 1.1'2774 ,-.i98 1.22213 373 1.33730 1 01453 .149 1,05819 .224 1.12885 .299 1.22354 ,374 1.33896 1.01493 .150 1,05896 .225 1 12997 .300 1.22495 .375 1.34063 184 MENSURATION. TABIi£ OF €IR€UIiAR ARCS — (CONTINUKD.) H'ghts. Lengths. H'ghts. Lengths. H'ghts. Lengths. H'ghts. Lengths. H'ghts. Lengths. .376 1.34229 .401 1.38496 .426 1.42945 .451 1.47565 .476 1.52346 .377 1.34396 .402 1.38671 .427 1.43127 .452 1.47753 .477 1.52541 .378 1.34563 .403 1.38846 .428 1.43309 .453 1.47942 .478 1.5273e .379 1.34731 .404 1.39021 .429 i. 43491 .454 1.48131 .479 1.52931 .380 1.34899 .405 1.39196 .430 1.43673 .455 1.48320 .480 1.53126 .381 1.35068 .406 1.39372 .431 1.43856 .456 1.48509 .481 1.53322 .382 1.35237 .407 1.39548 .432 1.44039 .457 1.48699 .482 1.53618 .383 1.35406 .408 1.39724 .433 1.44222 .458 1.48889 .483 1.53714 .384 1.35575 .409 1.39900 .434 1.44405 .459 1.49079 .484 1.53910 .385 1.35744 .410 1.40077 .435 1.44589 .460 1.49269 .485 1.54106 .386 1.35914 .411 1.40254 .436 1.44773 .461 1.49460 .486 1.5480J .387 1.36084 .412 1.40432 .437 1.44957 .462 1.49651 .487 1.5449» .388 1.36254 .413 1.40610 .438 1.45142 .463 1.49842 .488 1.54696 .389 1.36425 .414 1.40788 .439 1.45327 .464 1.50033 .489 1.54893 .390 1.36596 .415 1.40966 .440 1.45512 .465 1.50224 .490 1.55091 .391 1.36767 .416 1.41145 .441 1.45697 .466 1.50416 .491 1.55289 .392 1.36939 .417 1.41324 .442 1.45883 .467 1.50608 .492 1.55487 .393 1.37111 .418 1.41503 .443 1.46069 .468 1.50800 .493 1.55685 .394 1.37283 .419 1.41682 .444 1.46255 .469 1.50992 .494 1.55884 .395 1.37455 .420 1.41861 .445 1.46441 .470 1.51185 .495 1.56083 .396 1.37628 .421 1.42041 .446 1.46628 .471 1.51378 .496 1.56282 .397 1.37801 .422 1.42221 .447 1.46815- .472 1.51571 .497 1.56481 .398 1.37974 .423 1.42402 .448 1.47002 .473 1.51764 .498 1.56681 .399 1.38148 .424 1.425S3 .449 1.47189 .474 1.51958 .499 1.56881 .400 1.38322 .425 1.42764 .450 1.47377 .475 1.52152 .500 1.57080 If tbe arc is g^reater tlian a semicircle, then, as directed at top of p 187. find the diam of the circle. Then find its circumf. From diam take ht of arc. The rem will be ht of the smaller arc of the circle. By rule at top of p 183 find the length of this smaller aix>> Subtract it from circumf. The length of 1 degree of a circular arc is equal to .017453 292 520 X its radius. " " " 1 minute " " " " " .000290 888 209 X " " " " 1 second " " '< " " .000004 848 137 X " An arc of 1° of the earth's g-reat circle is but 4.6356 feet longer than its chord. Its length is 69.16 land or statute miles. Earth's equatorial rad = 3962.5705 miles. Polar 3949.67. An arc of 1°, rad 1 mile, is 92.1534 feet ; a minute is 1.5859 feet ; a second is .0266 of a foot: •r Tery nearly 5-8ixte«nths of an inch. Arc of 1°, rad 100 ft = 1.74633 fe«t. MENSURATION. 185 To find the leiig^tli of a circular arc by the following: table. Knowing the rad of the circle, and the measure of the arc in deg min &c fh?«^^ ^fni/^hfi^f^ the lengths in the table found respectively opposite to the deg, min, &c, of the arc. Mult the sum by the rad of the circle. •> rt- »> . , »»» Ex. In a circle of 12.43 feet rad, is an arc of 13 deg, 27 min, 8 sec. How long is the arc ? Here, opposite 13 deg in the table, we And, .2268928 27 min " " " .0078540 " 8 sec " » '< .0000388 And .2347856 X 12.43 or rad = 2.918385 feet, the reqd length of arc' I.ENOTHS OF CIRCUI.AR ARCS TO RAD 1. Deg. Length. Deg. Length. Deg. Length. Min. Length. Sec. Length. 1 .0174533 61 1.0646508 121 2.1118484 1 .0002909 1 ,0000048 2 .0349066 62 1.0821041 122 2.1293017 2 .0005818 2 .0000097 3 .0523599 63 1.0995574 123 2.1467550 3 .0008727 3 .0000145 4 .0698132 64 1.1170107 124 2.1642083 4 .0011636 4 .0000194 5 .0872665 65 1.1344640 125 2.1816616 5 .0014544 5 .0000242 6 .1047198 66 1.1519173 126 2.1991149 6 .0017453 6 .0000291 7 .1221730 67 1.1693706 127 2.2165682 7 .0020362 7 .0000339 8 .1396263 68 L 1868239 128 2.2340214 8 .0023271 8 .0000388 9 .1570796 69 1.2042772 129 2.2514747 9 .0026180 9 .0000436 10 .1745329 70 1.2217305 130 2.2689280 10 .0029089 10 .0000485 11 .1919862 71 1.2391838 131 2.2863813 11 .0031998 11 .0000533 12 .2094395 72 1.2566371 132 2.3038346 12 .0034907 12 .0000582 IS .2268928 73 1.2740904 133 2.3212879 13 .0037815 13 .0000630 u .2443461 74 1.2915436 134 2.3387412 14 .0040724 14 .0000679 15 .2617994 75 1.3089969 135 2.3561945 15 .0043633 15 .0000727 16 .2792527 76 1.3264502 136 2.3736478 16 .0046542 16 .0000776 17 .2967060 77 1.3439035 137 2.3911011 17 .0049451 17 .0000824 18 .3141593 78 1.3613568 138 2.4085544 18 .0052360 18 .0000873 19 .3316126 79 1.3788101 139 2.4260077 19 .0055269 19 .0000921 20 .3490659 80 1.3962634 140 2.4434610 20 .0058178 20 .0000970 21 .3665191 81 1.4137167 141 2.4609142 21 .0061087 21 .0001018 22 .3839724 82 1.4311700 142 2.4783675 22 .0063996 22 .0001067 23 .4014257 83 1.4486233 143 2.4958208 28 .0066904 23 .0001115 24 .4188790 84 1.4660766 144 2.5132741 24 .0069818 24 .0001164 25 .4363323 85 1.4835299 145 2.5307274 25 .0072722 25 .0001212 26 .4537856 86 1.5009832 146 2.5481807 26 .0075631 26 .0001261 2T .4712389 87 1.5184364 147 2.5656340 27 .0078540 27 .0001309 28 .4886922 88 1.5358897 148 2.5830873 28 .0081449 28 .0001357 29 .5061455 89 1.5533430 149 2.6005406 29 .0084358 29 .0001406 80 .5235988 90 1.5707963 150 2.6179939 30 .0087266 30 .0001454 SI .5410521 91 1.5882496 151 2.6354472 31 .0090175 31 .0001503 32 .5585054 92 1 .6057029 152 2.6529005 32 .0093084 32 .0001551 38 .5759587 93 1.6231562 153 2.6703538 33 .0095993 33 .0001600 34 .5934119 94 1.6406095 154 2.6878070 34 .0098902 34 .0001648 85 .6108652 95 1.6580628 155 2.7052603 35 .0101811 35 .0001697 86 .6283185 96 1.6755161 156 2.7227136 36 .0104720 36 .0001745 87 .6457718 97 1.6929694 157 2.7401669 37 .0107629 37 .0001794 38 .6632251 98 1.7104227 158 2.7576202 38 .0110538 38 .0001842 39 .6806784 99 1.7278760 159 2.7750735 39 .0113446 39 .0001891 40 .6981317 100 1.7453293 160 2.7925268 40 .0116355 40 .0001939 41 .7155850 101 1.7627825 161 2.8099801 41 .0119264 41 .0001988 42 .7330383 102 1.7802358 162 2.8274334 42 .0122173 42 .0002036 43 .7504916 103 1.7976891 163 2.8448867 43 .0125082 43 .0002085 44 .7679449 104 1.8151424 164 2.8623400 44 .0127991 44 .0002133 45 .7853982 106 1.8325957 165 2.8797933 45 .0130900 45 .0002182 46 .8028515 106 1.8500490 166 2.8972466 46 .0133809 46 .0002230 47 .8203047 107 1.8675023 167 2.9146999 47 .0136717 47 .0002279 48 .8377580 108 1.8849556 168 2.9321531 48 .0139626 48 .0002327 49 .8552113 109 1.9024089 169 2.9496064 49 .0142535 49 .0002876 50 .8726646 110 1.9198622 170 2.9670597 50 .0145444 80 .0002424 51 .8901179 ' 111 1.9373155 171 2.98451.30 51 .0148353 51 .0002473 62 .9075712 112 1.9547688 172 3.0019663 52 .0151262 52 .0002521 S3 .9250245 113 1.9722221 173 3.019419'' 53 .0154171 53 .0002570 »4 .9424778 114 1.9896753 174 8.0.368729 54 .0157080 54 .0002618 65 .9599311 115 2.0071286 175 3.0543262 65 .0159989 55 .0002666 66 .9773844 116 2.0245819 176 3.0717795 56 .0162897 56 .0002715 67 .9948377 117 2.0420352 177 3.0892328 57 .0165806 57 .0002768 68 1.0122910 118 2.0594885 178 3.1066861 58 .0168715 58 .0002812 69 1.0297443 119 2.0769418 179 3.1241394 59 .0171624 69 .0002860 •0 1.0471976 120 2.0943951 180 3.1415927 60 .0174533 60 .000290» 186 MENSURATION. CIRCUIiAR SECTORS, RINGS, SSGMKNTS, ETC. Area of a circular sector, a db c. Fig. A, arc adb , X radius c o. -=. area of entire circle X arc adb in degrees. 360 Area of a circular ring, Fig. B, -= area of larger circle, c d, — area of smaller one, a b. I , = .7854 X (sum of diams, cd -^ ah) X (diff- of diams. cd — ab.) = 1^708 X thickness ea \ sum of diameters c d and a b. To ftiid. tl&e radius ot a circle wlilcli sliall liave tlie same area as a given circular ring c a d ah, Fig. B, Draw any radius n r of the outer circle ; and from where said radius cuts th« Inner circle at t^ draw ( s at right angles to it. Then will t she the required radius. Breadtli, e a = b d, of a circular rlngy Fig. B, M> \^ difference of diameters c d and a 6, •a 34 (diameter cd— >/ 1.2732 area of circle a 6.) Area of a circular zone abed, «» area of circle m n — areas of segments amb and end, (for areas of segments, see below.) A circular lune is a crescent-shaped figure, comprised between two arcs abc and a o c of circles of different radii, a d and an. Area of a circular lune ab eo m= area of segment abc — area of segment aoe, i&tr areas of segments see below.) To And tlte area f%t a circular •Mf^ment, abed; Figs. iO, D. Ar«a of Segment adbn^ Fig. A (at top of page) — Area of Sector adbe — Area of Triangle abc, ■-J^(Arcad6 X radius a o — en X chord a 6). Hairing tlie area of a segment required to be out off from a given circle, to find its cliord and rise. <" Diride the area by the square of the diameter of the circle ; look for the quotient In tha column of areas in the table of areas, opposite ; take out from the table the corresponding number in the column of rises. Multiply this number by the diameter. The product will be the required rise. Then shord — 2 X %/ (diameter — rise) X riss^ MENSURATION. 187 TABIii: OF AREAS OF CIRCUIiAR I^EGMEXTS, Figs C, D. If the Se$^ineilt excc^eds a semicircle, its area is ^ area of circle- ar« of a segment whose rise is = (diam of circle — rise of given segment). Diam of circle = (squar of half choid -r rise) -\~ rise, whether the segment exceeds a semicircle or not. Rise Area = Rise Area = Rise Area = Rise Area= Rise Area =» div by (square div by (square div by (square div by (square div by (square diam of of diam) diam of of diam) diam of of diam) liam of of diam) diam of of diam circle. mult bj circle. mult by circle. mult by mult by circle. .253 mult by .001 .000042 .064 .021168 .127 .057991 .190 .103900 .156l4y .002 .000119 .065 .021660 .128 .058658 .191 .104686 .254 .157019 .003 .000219 .066 .022155 .129 .059328 .192 .105472 .255 ,157891 .004 .000337 .067 .022653 .130 .059999 .193 .106261 .256 .158763 .005 .000471 .068 .023155 .131 .060673 .194 .107051 .257 ,159636 .006 .U00619 .069 .023660 .132 .061349 .195 .107843 .258 .160511 .007 .000779 .070 .024168 .133 .062027 .196 .108636 .259 .161386 .008 .000952 .071 .024680 .134 ,062707 .197 .109431 .260 .162263 .009 . .00113". .072 .025196 .135 .063389 .198 .110227 .261 .163141 .010 .001329 .073 .025714 .136 .064074 .199 .111025 .262 ,164020 .011 .001533 .074 .026236 .137 .064761 .200 .111824 .263 ,164900 .012 .001746 .075 .026761 .138 .065449 .201 .112625 .264 ,165781 .013 .001969 .076 .027290 .139 .066140 .202 .113427 .265 .166663 .014 .002199 .077 .027821 .140 .066833 .203 .114231 .266 .167546 .015 =002438 .078 .028356 .141 .067528 .204 .115036 .267 ,168431 ,016 .002685 .079 .028894 .142 .068225 .205 .115842 .268 ,169316 .017 .002940 .080 .029435 .143 .068924 .206 .116651 .269 .170202 .bl8 .003202 .081 .029979 .144 .069626 .207 .117460 .270 .171090 .019 .003472 .082 .030526 .145 .070329 .208 ,118271 .271 .171978 020 .003749 .083 .031077 .146 .071034 .209 .119084 .272 .172868 .021 .004032 .084 .031630 .147 .071741 .210 .119898 .273 .173758 .022 .004322 .085 .032186 .148 .072450 .211 .120713 .274 . .174650 .023 .004619 .086 .032746 .149 .073162 .212 .121530 .275 ,175542 .024 .004922 .087 .033308 .150 .073875 .213 .122348 .276 ,176436 .025 .005231 .088 .033873 .151 .074590 .214 .123167 .277 .177330 .026 .005546 .089 .034441 .152 .075307 .215 .123988 .278 ,178226 .027 .005867 .090 .035012 .153 .076026 .216 ,124811 .279 .179122 .028 .006194 .091 .035586 .154 .076747 .217 ,125634 .280 .180020 .029 .006527 .092 .036162 .155 .077470 .218 .126459 .281 .180918 .030 .006866 .093 .036742 .156 .078194 .219 .127286 .282 .181818 X)31 .007209 .094 .037324 .157 .078921 .220 .128114 .283 .182718 .032 .007559 .095 .037909 .158 .079650 .221 .128943 ,284 .183619 ms .007913 .096 .038497 .159 .080380 .222 .129773 .285 .184522 .034 .008273 .097 .039087 .160 .081112 ,223 .130605 .286 .185425 .035 .008638 .098 .039681 .161 .081847 .224 .131438 ,287 .186329 .036 .009008 .099 .040277 .162 .082582 .225 .132273 .288 .187235 .037 .009383 .100 .040875 .163 .083320 .226 .133109 .289 .188141 .038 .009764 .101 .041477 .164 .084060 .227 .133946 .290 .189048 .039 .010148 .102 .042081 M65 .084801 .228 .134784 .291 .189956 .040 .010538 .103 .042687 .166 .085545 .229 .135624 ,292 .190865 041 .010932 .104 .043296 .167 .086290 .230 .136465 .293 .191774 .042 .011331 .105 .043908 .168 .087037 .231 .137307 .294 .192685 .043 .011734 .106 .044523 .169 .087785 .232 .138151 .295 .193597 .044 .012142 .107 .045140 .170 .088536 .233 .138996 .296 ,194509 .045 .012555 .108 .045759 .171 .089288 .234 .139842 .297 ,195423 .046 .012971 .109 .046381 .172 .090042 .235 140689 :298 .196337 .047 .013393 .110 .047006 .173 .090797 .236 .141538 .299 .197252 .048 .013818 .111 .047633 .174 .091555 .237 ,142388 .300 .198168 .049 .014248 .112 .048262 .175 .092314 .238 .143239 .301 .199085 .050 .014681 .113 .048894 .176 .093074 .2.39 .144091 .302 .20000S .051 .015119 .114 .049529 .177 .093837 .240 .144945 .303 .200922 .052 .015561 .115 .050165 .178 .094601 .241 .145800 .304 .201841 .053 .016008 .116 .050805 .179 .095367 .242 ,146656 .305 .202762 .054 .016458 .117 .051446 .180 .096135 .243 .147513 .306 ,203683 .055 .016912 .118 .052090 .181 .096904 .244 ,148371 .307 .204605 .066 .017369 .119 .052737 .182 .097675 .245 .149231 .308 .205528 .057 .017831 .120 .053385 .183 .098447 .246 .150091 .309 .206452 .058 .018297 .121 .0540^7 .184 .099221 .247 .150953 .310 .207376 .059 .018766 .122 .054690 .185 .099997 .248 .151816 .311 .208302 .060 .019239 ,123 .055346 .186 .100774 .249 .152681 .312 .209228 .061 .019716 .124 .056004 .187 .101553 .250 .153546 .313 .210155 .062 .020197 .125 .056664 .188 .102:^34 .251 .154413 .314 .211083 J063 .020681 .126 .067327 .18© .iu;iii6 .252 .15.'i281 315 iJ12011 188 MENSURATION. TABLE OF AREAS OF CIRCVIiAR JSEOMENTS— (Continued.) Rise Area = Rise Area = Rise Area = Riise Area = Rise Area =- divby (square divby (square divby (square divby (square divby (square diam of of diam) diam of of diam) diam of of diam) diam of of diam) diam of of diam oircle. mult by circle. mult by circle. mult by circle. .427 mult by circle. .464 mult by .316 .212941 .353 .247845 .390 ;283593 .319959 .356730 .317 .213871 .354 .248801 •391 .284569 .428 .320949 .465 .357728 ^18 .214802 .355 .249758 •392 .285545 .429 .321938 .466 .358726 .319 .215734 .356 .250715 •393 .286521 .430 .322928 .467 .359723 .320 .216666 .357 .251673 •394 .287499 .431 .323919 .468 .360721 .321 .217600 .358 .252632 •395 .288476 .432 .324909 .469 .361719 .322 .218534 .359 .253591 .396 .289454 .433 .325900 .470 .362717 .323 .219469 .360 .254551 •397 .290432 .434 .326891 .471 .363716 .324 .220404 .361 .255511 .398 .291411 .435 .327883 .472 .364714 .325 .221341 .362 .256472 •399 .292390 .436 .328874 .473 .365712 .326 .222278 .363 .257433 •400 .293370 .437 .329866 .474 .366711 .327 .223216 .364 .258395 •401 .294350 .438 .330858 .475 .367710 .328 .224154 .365 .259358 •402 .295330 .439 .331851 .476 .368708 .329 .225094 .366 .260321 •403 .296311 .440 .332843 .477 .369707 .330 .226034 .367 .261285 •404 .297292 .441 .333836 .478 .370706 .331 .226974 .368 .262249 •405 .298274 .442 .334829 .479 .371705 .332 .227916 .369 .263214 •406 .299256 .443 .335823 .480 .372704 .333 .228858 .370 .264179 •407 .300238 .444 .336816 .481 .373704 .334 .229801 .371 .265145 .408 .301221 .445 .337810 .482 .374703 .335 .230745 .372 .266111 •409 .302204 .446 .338804 .483 .375702 .336 .231689 .373 .267078 .410 .303187 .447 .339799 .484 .376702 .337 .232634 .374 .268046 •411 .304171 .448 .340793 .485 .377701 .338 .233580 .375 .269014 •412 .305156 .449 .341788 .486 .378701 .339 .234526 .376 .269982 •413 .306140 .450 .342783 .487 .379701 .340 .235473 .377 .270951 •414 .307125 .451 .343778 .488 .380700 .341 .236421 .378 .271921 .415 .308110 .452 .344773 .489 .381700 .342 .237369 .379 .272891 •416 .309096 .453 .345768 .490 .382700 .343 .238319 .380 .273861 •417 .310082 .454 .346764 .491 .383700 .344 .239268 .381 .274832 •41 P .311068 .455 .347760 .492 384699 .345 .240219 .382 .275804 •419 .312055 .456 .348756 .493 .385699 .346 .241170 .383 .276776 .420 .313042 .457 .349752 .494 .386699 .347 .242122 .384 .277748 .421 .314029 .458 .350749 .495 .387699 .348 .243074 .385 .278721 .422 .315017 .459 .351745 .496 .388699 .349 .244027 .386 .279695 .423 .316005 .460 .352742 .497 .389699 .350 .244980 .387 .280669 .424 .316993 .461 .353739 .498 . .390699 .361 .245935 .388 .281643 .425 .317981 .462 .354736 .499 .391699 .352 .246890 .389 .282618 .426 .318970 .463 .355733 .500 .392699 EI.L.IPSE (page 189). Focal distance ^fg = 2 yco'^—bo'^ ; * cf=gw = cw — fg c w ■ y/cO^ — bO^. * Because c o c w 2 fc+cg _ /b-{-bg 2 2 = /&. MEJJSURATION. THE ElililPSB. 189 ^><:^^^^Tn \ eCy:::^::!^ Fig. 1. Fig. 3. Jin ellipteU a curve, eeee, Pigl, formed by an oblique section of either a cone or a cylinder, pass, ing through its curved surface, without cutting the base. Its nature is such that if two lines, as »/ and n g, Fig. 2, be drawn from any point n in its periphery or circumf, to two certain points/ and g, in its long diam c w, (and called the foci of the ellipse.) their sum will be equal to that of any other two lines, as b /, and b g, drawn from any other point, a« b, in the cireumf, to the foci /and g; also the sum of any two such lines will be equal to the long diam c w. The line c t<; dividing the el lipae into two equal parts lengthwise, is called its transverse, or major axis, or long diam ; and a 6, which divides it equally at right-angles to c w, is called the conjugate, or minor axis, or short diam. To find the position of the foci of an ellipse, from either end, as 6, of the short diam, measure off the dists &/and b g, Fig 2, each equal to o c, or one-half the long diam. The parameter of an ellipse is a certain length obtained thus ; as the long diam i short diam : : short diam : parameter. Any line r v, or s d, Fig 3, drawn from the circumf, to, and at right angles to, either diam, is called an ordinate; and the parts c v and vw,bs and a a, of that diam, between the ord and the circumf, are called abacissce, or abscisaet. To find tbe leng^tb of any ordinate, rvorsd, drawn to eitbev diam 9 C to or b <*, Knowing the absciss, e o or « a, and the two diams, cw,bai e w^ i b cfi '.'. e V X V w, r ^. 6a*:cw*::6 exceeds 5 times il, then in* p «^^«^ stead of dividing (D — d)^ by 8.8, div it by - •^ • the number in this table. The following rule originated with Mr. M. Arnold Pears, of New South Wales, Australia, and was by him kindly communicated to the author, rate than our own, it is much neater. sssas I tto>o»oia>aiako>a>o»0>a>aioooc Q — — — - •e '8 13 -8 73 "« T3 •« -a M Circumf = 3.1416 d+ 2(D — d)- Although not more accu* d(D — d) ^(D +d) X (D + 2d) following table of semi-elliptic arcs was prepared by our rota. To use this table, div the height or rise of the are, by its span or chord. The quot will be the height of an arc whose span is 1. Find thia quot in the column of heights ; and take out the corresponding number from the^^ per cent ; ht= 10 X base, or more, about 15H per cent The following by the write* is correct within perhaps 1 part in 800, in ail cases ; and will therefore answer for many purposes. Let adb, Fig 3, or n o d, Fig 4, be the parabola, in which are given the base ab or n d; and the height c d or c a. Imagine the complete fig a d 6 «, or n a d 6, to be drawn ; and in et'tAer case, assume its long diam a 6 to be the chord or base ; and one- half the short diam, or c d, to be the height, of » circular arc. Find the length of this circular arc, by means of the rule and table given for that pur- pose. Then div tbe chord or base o 6, or n d of the parabola, by its height c d or c a. Look for the quot in the column of bases in the following table, and take from the table the correspondiag multiplier. Mult the length of the circular arc by this; the prod will be the length of arc adi,OT n a d, 3,3 the case may be. For bases of parabolas less than .05 of the height, or greater than 10 times the height, the multiplier is 1, and is very approx- imate; or in other words, the parabola will be of almost exactly the same length as the circular arc. To find tbe area of a parabola m anl>. Mult its base m n, Fig 5, by its height a b ; and take %ds of the prod. The area of any segment, as ub v, whose base « v is parallel to mn, is found in the same way, using u v and a b, instead of m n and a b. To find tbe area of a parabolic zone, or frus« tnm, as rn. ti t* v. RuLK 1. First find by the preceding rule the area of the whole pnrabola mbn\ then that of the segment ubv, and subtract the last from the first. Rule 2. From the cube of m n, take the cube of « « ; call the diff «.' From the square of m n, take the square of t« v; call the differ. Div c by «. Mult the quot by ^ds of the height a a. Fig. 5. MENSURATION, 193 Tabic lor Lengths of Parabolic Carves. See opp page. (Orvgiaal.) Base. Mult. Base. Mult. Base. Mult. Base. Mult. .05 1.000 1.10 .^99 2.15 .949 3.20 .983 .10 1.001 1.15 .997 2.20 .951 3.30 .984 .15 1.002 1.20 .995 2.25 .954 3.40 .985 .20 1.004 1.25 .993 2.30 .956 3.50 .986 .25 1.006 1.30 .990 2.35 .958 3.60 .987 .30 1.007 ' 1.35 .987 2.40 .960 3.70 .988 .35 1.007 1.40 .984 2.45 .962 3.80 .989 .40 1.008 1.45 .980 2.50 .963 3.90 .990 .45 1.009 1.50 .977 2.55 .965 4.00 .991 .50 1.010 1.55 .974 2.60 .967 4.25 .992 .55 1.010 1.60 .970 2.65 .969 4.50 .993 .60 1.010 1.65 .966 2.70 .970 4.75 .994 .65 1.011 1.70 .963 2.75 .972 5.00 .995 .70 1.011 1.75 .960 2.80 .973 5.25 .996 .75 1.010 1.80 .957 2.85 .975 • 6.50 .997 .80 1.009 1.85 .953 2.90 .976 5.75 .998 .85 1.008 1.90 .950 2.95 .978 6.00 .998 .90 1.006 1.95 .946 3.00 .979 7.00 .999 .95 1.004 2.00 .942 3.05 .980 8.00 ] .000 1.00 1.002 2.05 .944 3.10 .981 10.00 1.000 1.05 1.001 2.10 .946 3.15 .982 To draw a parabola, having base c s and height e o. e««, Fig6. Make o < equal to the height CO. Drawciand • t ; and divide ench of them into any number of equal parts ; numbering them as in the Pig. Join 1, 1 ; 2, 2 ; 3, 3, &c; then draw the curve by hand. It will be observed that the intersections of the lines 1, 1 ; 2, 2, &c, do not give points la the curve ; but a portion of each of those lines forms a tan- gent to the curve. By increasing the number of divisions on ct and s t, an almost perfect curve is formed, scarcely requiring to be touched up by hand. In practice it is best first to draw only the center portions of the two lines which cross each other just above o ; and from them to work down- ward; actually drawing only that small portion of each successive lower line, which is necessary to indicate the ourve. Or the parabola may be drawn thas: Let J c, Fig 7, be the base ; and a d the height. Draw the rectangle b nm c; div each half of the base into any num- ber of equal parts, and number them from the center each C' way. Div n h', and m c into the same number of equal parts ; and number them from the top, downward. From the points •n 6 c draw vert lines ; and from those at the sides draw lines to d. Then the intersections of lines 1,1; 2. 2, &c, will form points in the parabola. As in the pre- ceding case, it is not necessarj- to draw the entire lines ; but merely portions of'them, as shown be- tween d and c. Or a parabola may be drawn by first div the height a h. Fig 5, into any number of parts, either equal or unequal; and then calculating the ordi- sates us, &c; thus, as the height a b : square of half base am:: any absciss b s : square of its ord u 8. Take the sq rt for ms. Rem. —When the height of a parabola is not greater than 1-lOth part its base, the curve coin- cides so very closely with that of a circular arc, that in the preparation of drawings for suspen- sion bridges, &c., the circular arc may be em- ployed ; or if no great accuracy is reqd, the circle may be used even when the height is as great as •ne-eighth of the base. To draw a tangent tv v. Fig. 5, to a parabola, from any point v. Draw V s perp to axis a b ; prolong a b until b w equals a b. Join w v. 13 194 MENSURATION. Tlie Cycloid, ach is the curve described by a point a in the circumference of a circle, an during one complete revolution of the circle, rolled along a straight line ' ^ a6; which is called the base of the b cyclcfid. The vertex of the cycloid is at c. Base, a &, = circumference of generat- ing circle a n = diameter, cd^ of generat- ing circleXn" = 3.1416cci. Axis, or lieig^lit, cd= an. lien^tli, ac6, = 4cd Area, a c6 d = 3 X area of generating circle, an = 3^^ =cd^Xin = cd^X 2.3562. 4 Center of gravity of swface at g. c g = -^^ c d. Center of gravity of cycloid (curved line ac b) in axis c d at a point (as 5) distant ^ cd from c. To draw a tangent, eo, from any point e in a cycloid ; draw e s at right angles to the axis cd; on cd describe the generating circle dct; join t c ; frum e draw e parallel to t c. The cycloid is the curve of quickest descent ; so that a body would fall from b to c along the curve b m c, in less time than along the inclined plane b i c, or any other line. THE REGUIiAR BODIES. A regular body, or regular polyhedron, is one which has all its sides, and its solid angles, respectively similar and equal to each other. There are but five such bodies, as follows : Name. Bounded by Surface (=sum of surfaces of all the faces). Multiply the square of the length of one edge by Volume. Multiply the cube of the length of one edge by 4 equilateral triangles. 6 squares. 8 equilateral triangles. 12 " pentagons. 20 " triangles. 1.7320 6. 3.4641 20.6458 8.6602 .1178 Hexahedron or cube Octahedron 1. .4714 Dodecahedron 7.6631 Icosahedron 2.1817 Ouldinns- Fig. A. Theorem. Fig. B. To find the volume of any body Cas the irregularmassa 6 cm. Fig A, or the ring ab cm, Fig B), generated by a complete or partial revolution of any figure (as abca) around one of its sides (as IS. 195 Fig. 1. Fig. 2. Fig. 3. Fig. 4. A parallelopiped is any solid contained within six sides, all of which are parallelograms ; and those of each opposite pair, parallel to each other. We show but four of them ; corresponding to the four paralh lograras ; namely, th« cube. Fig 1, which has all its sides equal squares, aud all its angles right angles; the right rectangular prism, Fig 2, has all its angles right angles, each pair of opposite faces equal, but not all of its faces equal ; the Hhombohedron, Fig 3, which has all its sides equal rhombuses, aud which, like the rhombus, p 157, is sometimes called " rhomb " ; the Rhombic prism, Fig 4 ; its faces, rhombuses, or rhomboids, each pair of opposite faces equal, but not all its faces equal. All parallelepipeds are prisms. Volume of any ^ area of any face, y, perpendicular distance, p, as a, ^ to the opposite face. = cube of length of one edge, = 1.90985 X volume of inscribed sphere, = 1.27324 X " " cylinder, = 3.81972 X " " cone. ]>iag;oual of a cube = diameter of circumscribing sphere, = 1.7320508 X length of one edge of cub«. parallelopiped Volume of a cube h P § J PRISMS. A prism is any solid whose twoends are parallel, similar, and equal ; and whose sides are parallelograms, as Figs 5 i to 10. Consequently the fore- \p going parallelopipeds are i prisms. A right prism is one whose sides are perpendic- ular to its ends, as 5, 6, 7 ; when ^ot so, the prism is oblique, as 8, 9, 10. When all the sides of the 15 gures which form the ends are equal, and the angles included between those sides are alsv equal, the prism is said to be regular : otherwise, irregular. Volume of any prism (whether regular or irregular, right or oblique) = area of one end X perpendicular distance, jt), to the other end, «= area of cross section perpendicular to the sides X actual length, a b, Figs 5 to 10, =*= 3 X volume of pyramid whose base and height are — those of the prism. To find the volume of any frustum of any prism. Whose cross section, perpendicular to its sides, is either any triangle ; any parallelogram ; a square, (as in Fig loX^) or a regular polygon of any number of sides; no matter how the two ends of the frustum may be inclined with regard to each other; or whether one, or neither of them, is parallel to the base of the original prism. sum of lengths of parallel edges, Volume rr + 2^ + 3^ + T4 of frustum " number of such edges (4 in Fig 10^4) Figs. 101^. area of cross section X perpendicular to such edges. 196 MENSURATION. This rule may be used for ascertaining beforehand, the quantity of earth to be removed from a "borrow pit." The irregular surface of the ground is first staked oat iu squares; (the tape-line being stretched Aoriswito^, when meas- uring off their sides). These squares should be of such a size that without material error each of them may be considered to be a plane surface, either horizontal or in- clined. The depth of the horizontal bottom of the pit being determined on, and the levels being taken at every corner of the squares, we are thereby furnished with the lengths of the four parallel vertical edges of each of the resulting frustums of earth. In Figs 103^ y may be sup- posed to represent one of these frustums. If the frustum is that of an irregular 4-sided, or polyg- onal prism, first divide its cross section perpendicular to its sides, into tri- angles, by lines drawn from any one of its angles, as a. Fig 10^. Calculate the area of each of these triangles separately ; then consider the entire frustum to be made up of so many triangular ones; calculate the volume V,cr\ of each of these by the preceding rule for triangular frustums ; and add them together, for the volume of the entire frustum. Fig. 101^. Volume of any frnstum of any prism. Or of a cylinder. Consider either end to be the base ; and find its ^ area.' Also find the center of gravity c of the other end, and the n ^ — ^ perpendicular distance n c, from the base to said center of gravity. Fig. 10^. Then Tolume of frustum = area of base Xw c, Fig 10%;. The slant end, c, is an ellipse. Its area is greater than that of the circular end. Surface of any prism. Figs 5 to 10, whether right or oblique, regular or irregular _ ( circumference measured ^ , , ipntrfh ah\A. ^^^ ^^ *^^ ^^^^ - Vperpendicular to the sides ^ ^^^^^^ Jengin, a oj + ^^ ^^^ ^^^ ^^^^^ CYIilNDERS. A cylinder is any solid whose ends are parallel, similar, and eqxxi^i curved figures ; and whose sections parallel to the ends are everywhere the same as the ends. Hence there are circular cylindf rs, ellip- tic cylinders (or cylindroids) -^nd many others ; but when not otherwise expressed, the circular one is understood. A right cylinder is one whose ends are perpen- dicular to its sides, as Fig. 11 ; when other- wise, it is obliqve, as Fig 12. If the ends of a right circular cylinder be cut so as to make it oblique, it becomes an elliptic one ; because theni)oth its ends, and all sections parallel to them, are ellipses. An oblique circular cylinder seldom occurs ; it may be conceived of by imagining the two ends of Fig 12 to be circles, united by straight lines forming its curved sides. A cylinder is a prism having an infinite number of sides. Volume of any cylinder (whether circular or elliptic, &c, right or oblique) — area of one end X perpendicular distance, p, to the other end, _ f area of cross section ^ ^ j 1 ^^1^ ^ Pi j ^ ^ ^2, ( measured perp to the sides '^ b > > g » ^ 3 X volume of a cone whose base and height are — those of the cylinder. Surface of any cylinder (whether circular or elliptic, &c, right or oblioue) /circumference .... ... ., . \ , sum of the areas Fig. 11. Fig. 12. = ( measured perpendicularly X actual length, ah\ \to the sides, as at c o, Fig 12, / of the two ends. = diameter. Right circular cylinder whose height = Volume = 1^ X volume of inscribed sphere. Curved surface = surface of inscribed sphere. Area of one end = I snrface of inscribed sphere = a curved surface. Entire surface = 1^ X surface of inscribed sphere = 1^ X curved surface. CONTENTS OF CYLINDERS, OR PIPES. 197 Contents for one foot in length, in Cub Ft, and in U. S. Gallons of 231 cub ins, or 7.4805 Galls to a Cub Ft. A cub ft of water weighs about 623^ lbs ; and a gallon about 8M lbs. Diams 3* 8» or 10 times as great, give 4, 9, or 100 times the content. For 1ft. in For 1 ft in For 1 ft. in length. length. length. Diam. Diam. in deci- Diam. Diam. in deci- Diam. Diam. in deci- in ^.2 »- -i, Ts *- rn ■ • " Ins. mals of SS- °5 Ins. mals of 1'^^ a in mals of I'S . a a foot. '^t^ li a foot. ^l . i-2 Ins. a foot. ^i" Cx5 il^ ^^ ,0-6" s § " ■|5 il^ |5 0:5 ^i «5 ^S ^__ ^a Va. .0208 .0003 .0025 % .5625" .2485 1.859 19. 1.583 1.969 14.73 5-16 .0260 .0005 .0040 7. .5833 .2673 1.999 3^ 1.625 2.074 15.51 % .0313 .0008 .0057 1 74: .6042 .2867 2.145 20 1.667 2.182 16.32 7-16 .0365 .0010 .0078 .6250 .3068 2.295 K 1.708 2.292 17.15 9-f^ 0417 .0014 .0102 .6458 .3276 2.450 21.^ 1.750 2.405 17.99 .0469 .0017 .0129 8-. .6667 .3491 2.611 y2 1.792 2.521 18.86 % .0521 .0021 .0159 1^ .6875 .3712 2.777 22.^ 1.833 2.640 19.75 11-16 .0573 .0026 .0193 74 .7083 .3941 2.948 'A 1.875 2.761 20.66 13-?l .0625 .0031 .0230 .7292 .4176 3.125 23 1.917 2.885 21.58 .0677 .0036 .0269 9. .7500 .4418 3.305 M 1.958 3.012 22.53 % .0729 .0042 .0312 1 .7708 .4667 S.491 24:^ 2.000 3.142 23.50 15-16 .0781 .0048 .03r)9 .7917 .4922 3.682 25. 2.083 3.409 25.50 1. .0833 .0055 , .040S 74 .8125 .5185 3.879 26. 2.167 3.687 27.58 .1042 .0085 ' .0638 10. .8333 .5454 4.080 27. 2.250 3.976 29.74 .1250 .0123 .0918 K .8542 .5730 4.286 28. 2.333 4.276 31.99 74 .1458 .0167 .1249 U; .8750 .6013 4.498 29. 2.417 4.587 34.31 .1667 .0218 • .1632 % .8958 .6303 4.715 30. 2.500 4.909 36.72 "' Va. .1875 .0276 .2066 11. .9167 .6600 4.937 31. 2.583 5.241 39.21 I .2083 .0341 .2550 % .9375 .6903 5.164 32. 2.667 5 585 41.78 .2292 .0412 .3085 K! .9583 %\ .9792 .7213 5.S96 33. 2.750 5.940 44.43 3. .2500 .0491 .3672 .7530 5.633 34. 2.833 6.305 47.13 Va .2708 .0576 .4309 12. ilFoot. .7854 5.875 35. 2.917 6.681 49.98 y 74 .2917 .0668 .4998 1^; 1.042 .8.')22 6.375 36. 3.000 7.069 52.88 .3126 .0767 .5738 13. j 1.083 .9218 6.895 37. 3.083 7.467 55.86 4. .3333 .0873 .6528 M 1-125 .9940 7.436 38. 3.167 7.876 58.92 34 .3542 .0985 .7369 14. il.167 • 1.0R9 7.997 39. 3.250 8.296 62.06 .3750 .1104 .8263 M1I.2O8 1.147 8.578 40. 3.333 8.727 65.28 /a .3958 .1231 .9206 15. 11.250 1.227 9.180 41. 3.417 9.168 68.58 5. .4167 .1364 1.020 l^!l.292 1.310 9.801 42. 3.500 9.621 71.97 Y^ .4375 .1503 1.125 16. 11.333 1.396 10.44 43. 3.583 10.085 75.41 'i .4583 .1650 1.234 141.375 1.485 11.11 44. 3.667 10.559 78.99 .4792 .180.S 1.349 17. 11.417 1.576 11.79 45. 3.750 11.045 82.62 6. .5000 .1963 1.469 ■ 1^11.458 1.670 12.49 46. 3.833 !ll.541 86.33 'i .5208 .2131 1.594 18. 11.500 1.767 13.22 47. 3.917 12.048 90.13 .5417 .2304 1.724 )^jl.542 1.867 13.96 48. 4.000 12.566 94.00 Table continued, but witb tbe diams in feet. IMam. Cub. u. s. Diam. Cub. u. s. Dia. Cub. U.S. Dia. Cub. U.S. Feet. Feet. GaUa. Feet. Feet. Galls. Feet. Feet. Galls. Feet. Feet. Galls. 4 12.57 94.0 7 38.48 287.9 12 113.1 846.0 24 452.4 3384 '4 14.19 106.1 8 74 41.28 308.8 13 132.7 992.9 25 490.9 8672 15.90 119.0 44.18 330.5 14 153.9 1152. 26 530.9 3972 !^ 17.72 132.6 47.17 352.9 15 176.7 1322. 27 572.6 4283 5 19.63 146.9 8 50.27 376.0 16 201.1 1504. 28 615.8 4606 1^ 74 21.65 161.9 }4 56.75 424.5 17 227.0 1698. 29 660.5 4941 23.76 177.7 9 63.62 475.9 18 254.5 1904. 30 706.9 5288 25.97 194.2 U 70.88 530.2 19 283.5 2121. 31 754.8 5646 6 28.27 211.5 10 78.54 587.5 20 SI 4.2 2350. 32 804.2 6016 1^ 30.68 229.5 y2 86.59 647.7 21 346.4 2591. 33 855.3 6398 33.18 248.2 11 95.03 710.9 22 380.1 2844. 34 907.9 6792 35.78 267.7 y. 103.87 777.0 23 415.5 3108. 35 962.1 7197 198 CONTENTS AND LININGS OF WELLS. CONTENTS AWD I.ININOS OF WEI.I.S. For diams twice as great as those in the table, for the cub yds of digging, take out those opposlt* one half of the greater diam ; and mult them by 4. Thus, for the cub ?ds in each foot of depth of a well 31 feet in diam, farst take out from the table those opposite the diam of lo>6 feet • namelv 6 989 Then 6 989 X 4 := 27 956 cub yds reqd for the 31 ft diam-."^ But for the stone Uning or waH n|.'bHcks or plastering, mult the tabular quantity opposite half the greater diam, by 2. Thus, the perches of stone walling for each foot of depth of a well of 31 ft diam, will be 2.073 X 2 = 4 146 If the wall is more or less than one foot thick within usual moderate limits, it will generally be near enough for ^oSr ^^^"""^ number of perches, or of bricks, will increase or decrease inthe same pro- The size of the bricks is taken at 8>^ X 4 X 2 inches : and to be laid dry, or without mortar. Ic K bdcrthSk or'st^ fna°"' ^^'' °^"' ^^^"^"^ ^® ^^^^ ^*^^ ^''^*'^- ^^® ^^^^^ "'^^"^ ^^ supposed to CAUTIOX. — Be careful to observe that the diams to be used for the digeinff are greater than those for the walling, bricks, or plastering. No errors. For each foot of depth. For each foot of depth. For this For these three cols use the For this For these three cols nso tKo Diam. col use the Diameter of the diam in clear of the lining. Diam. in col use the Diameter of the diam in clear of the lining. in Feet. Digging. Stone No. of Square Yards of Plaster- Feet. Digging. Stone No. of Lining 1ft thick. Perches of Bricks in a Lining 1 Brick Lining 1 ft thick. Perches of Bricks in a Lining 1 Brick Square Cub Yds. of Digging. Cub Yds. Yards of Plas- 25 Cub Ft. thick. ing. of Digging. 25 Cub Ft. thick. tering. 1. .0291 .2513 57 .3491 H 5.107 1.791 750 4.625 Ya .0455 .2827 71 .4364 % 5.301 1.822 764 4.713 ]^ .0654 .3142 85 .5236 % 5.500 1>G54 778 4.800 H .0891 .3456 99 .6109 14. 5.701 1.885 792 4.887 2. .1164 .3770 114 .6982 Va. 5.907 1.916 806 4.974 M .1473 .4084 128 .7855 ^ 6.116 1.948 820 5.062 ^ .1818 • .4398 142 .8727 ?i 6.329 1.979 834 5.149 . ^ .2200 .4712 156 .9600 15. 6.545 2.011 849 5.236 3. .2618 .5027 170 1.047 34 6.765 2.042 S&^i 5.323 M .3073 .5341 184 1.135 3^ 6.989 2.073 877 5.411 ^ .3563 .5655 198 1.222 % 7.216 2.105 891 5.498 H .4091 .5869 212 1.309 16. 7.447 2.136 905 5.585 4. .4654 .6283 227 1.396 M 7.681 2.168 919 5.673 H .5254 .6597 241 1.484 % 7.919 2.199 933 5.760 Yl .5890 .6912 255 1.571 H 8.161 2.231 948 5.847 H .6563 .7226 269 1.658 17. 8.407 2.262 962 5.934 5. .7272 .7540 283 1.745 ]4. 8.656 2.293 976 6.022 34 .8018 .7854 297 1.833 8.908 2.325 990 6.109 }4 .8799 .8168 311 1.920 ' % 9.165 2.356 1004 6.196 H .9617 .8482 326 2.007 18. 9.425 2.388 1018 6.283 e. 1.047 .8796 340 2.095 34 9.688 2.419 1032 6.371 H 1.136 .9111 354 2.182 3^ 9.956 2.450 1046 6.458 1.229 .9425 368 2.269 % 10.23 2.482 1061 6.545 V 1.325 .9739 382 2.356 19. 10.50 2.513 1075 6.633 ». 1.425 1.005 396 2.444 34 10.78 2.545 1089 6.720 H 1.529 1.037 410 2.531 3^ 11.06 2.576 1103 6.807 H 1.636 1.068 425 2.618 H 11.35 2.608 1117 6.894 H 1.747 1.100 439 2.705 20. 11.64 2.639 1131 6.982 8. 1.862 1.131 455 2.793 3^ 11.93 2.670 IU5 7.069 34 1.980 1.162 467 2.880 12.22 2.702 1160 7.156 3^ 2.102 1.194 481 2.967 % 12.52 2.733 1174 7.243 % 2.227 1.225 495 3.054 21. 12.83 2.765 1188 7.331 9. 2:356 1.257 509 3.142 y* 13.14 2.796 1202 7.418 Va 2.489 1.288 523 3.229 H 13.45 2.827 1216 7.505 H 2.625 1.319 538 3.316 % 13.76 2.859 1230 7 593 % 2.765 1.351 552 3.404 22. 14.08 2.890 1244 7.HS0 M. 2.909 1.382 566 3.491 34 14.40 2.922, 1259 7.767 M 3.056 1.414 580 3.578 3^ 14.73 2.953 1278 7.854 M 3.207 1.445 594 3.665 % 15.06 2.985 1287 7.942 % 3.362 1.477 608 3.75S 23. 15.39 3.016 1301 8.029 11. 3.520 1.508 622 8.840 H 15.72 3.047 1315 8.116 Vi 3.682 1.539 637 8.927 H 16.06 3.079 1329 8.203 yi 3.847 1.571 651 4.014 % 16.41 3.110 1343 8.291 H 4.016 1.602 666 4.102 24. 16.76 3.142 1357 8.378 J2. 4.189 1.634 679 4.189 34 17.11 3.173 1372 8.465 1 4.365 1.665 693 4.276 H 17.46 3.204 1386 8.552 4.545 1.696 707 4.364 H 17.82 3.236 1400 8.640 H 4.729 1.728 721 4.451 25. 18.18 3.267 1414 8.72T 18. 4.916 1.759 736 4.538 A4 nbyd = = 202 1] r.s. gra Is. If perclies are named in a contract, it is necessary, in order to prevent fraud, to specify the number of cub feet contained in the perch ; for stone-quarriers have one perch, stone- masons another, &c. Engineers, on this account, contract by the cubic yard. The perch should be 4one away with entirely ; Perches of 25 cub ft X .926 = cub yds ; and cub yds-r .926— pers of 25 cub ft CYLINDRIC UNGULAS, ETC. 199 €IR€IJI.AR CYIilNDRIC UNOUIiAS. I. When the catting plane does not cnt the base. Figs 13, U area of base 5^ X 3^ sum of greatest & least perp heights, on, cm, area of cross sec measd w yi sum of greatest and least lengths, perp to sides, as a;, Volume of ungula Area of ciirved surface Add areas of ends if required. For areas of sections perpendicular to the sides, see Circles. For areas of sections oblique to the sides, see The Ellipse. _ r circumf measd perp ~ I to sides, as at x, gm,oty meabd along the sides. , half sum of greatest and least lengths, gm, 1, measd along the sides. II. When the catting plane toaches the base. Figs A to D. Tolame Fig A = FigB: FigC = = (?^a68 — a cX area adm& of base) • ^^^ am (whether right or oblique)v Figl> Carved „. ^ sarface * ig ^ = {right ^'^^ = ungula Fig C = "■"^^ Flg» = . (^ab^ + aeX&Te&admb of base) » l^ area of circle ym X^^ . }4 volume of cylinder xymn. 1 (ah X my ~~aeX length of arc dmb ) - ■ my X 'rnn. : {abXmy -\- aeX length of arc dm& ) - • 14 circumference of bate myXnin " "% curved surface of cylinder xymn. m n •meas'd perp to base JOO PYllAMIDS AND CONES, PYRAMIDS AlTD CONES. ^ d d ' ' ' T 5 A pyramid, Figs. 1, 2, 3, is any solid which has, for its base, a plane figure >f any number of sides, and, for its sides, plane triangles all terminating at one )oint d, called its apex^ or top. When the base is a regular figure, the pyramid s regular ; otherwise irregular. A cone, Figs. 4 and 5, is a solid, of which the base is a curved figure ; and rhich may be considered as made or generated by a line, of which one end is tationary at a certain point d, called the apex or top, while the line is being arried around the circumference of the base, which may be a circle, ellipse, )r other curve. A cone may also be regarded as a pyramid with an infinite lumber of sides. The axis of a pyramid or cone, is a straight line d o in Figs. 1, 2, 4 ; and d t ia rigs. 3 and 5, from the apex d, to the center of gravity of the base. When the .xis is perpendicular to the base, as in Figs. 1, 2, 4, the solid is said to be a right >ne; when otherwise, as Figs. 3, 5, an oblique one. When the word cone is used ilone, the right circular cone. Fig. 4, is understood. If such a cone be cut, as at i, obliquely to its base, the new base 1 1 will be an ellipse ; and the cone dtt )ecomes an oblique elliptic one. Fig. 5 will represent either an oblique circular tone, or an oblique elliptic one, according as its base is a circle or an ellipse. Volume of pyramid or cone, regular or irregular, right or oblique. Volume = 3^ area of base X perpendicular height d o. Figs. 1 to 5. = % volume of prism or cylinder having same area of base and same perpendicular height. *= ^ volume of hemisphere (\f same base and same height. Or, a cone, hemisphere and cylinder, of the same base and same height, have rolumes as 1, 2 and 3. Area o^ surface of sides of right regular pyramid or right circular cone. Area =. }^ circumference of base X slant height.* ^ In the cone, this becomes I Add area of base Area = area of base ^ ^j^^^ ^^^^ t if required, radius of base ^ "^ J Area of surface of oblique elliptic cone, d t ty Fig. 5i, cut from a right circular cone, d s s. From the point c where the axis d o of the right circular cone cuts the elliptic base tt, measure a perpendicular, r, in any direction, to the curved surface of the cone. Let v = the voriume of oblique elliptic cone, dtt; let a = the area of its elliptic base t i, and let h = the height d u measured perpendicularly to said base. Then , ^ a h S V Curved surface = = . r r Add. area of base if required No measurement has been devised for the surface of an oblique circular cone. *In the pyramid, this slant height must be measured along the middle of one )f the sides, and not along one of the edges. PYRAMIDS AND CONES. 201 To find tli« BTurface ot an irregular pyramid. Whether right or oblique, each side must be calculated as a separate triangle (see p. 148); and the several areas added together. Add the area of base if required. FRUSTUMS OF PYRAMIDS AND CONES. Fig. 6. Fig, 7. Frustum of pyramid (Fig. 6) or of eone (Fig. 7) with hase and tcp parallel. Volume (regular or irregular, right or oblique) — V V perpendicular ^ / area i area i / area v^ area \ ^ ^ height oo ^ ^of top "T" ©f base • V of top ^ of base/ — !• NX perpendicular ^ ( area t area , * ^, ^^ea of a section \ — >& X height X \ of top "T of base "T parallel to, and midway I ^ between, base and top / •*■ (for frustum of right or oblique circular cone only; See Fig. 7) H X "^KfgM^i" X S.U1« X (o«« + 0,* + ot . «,) Surface of frustum of right regular pyramid or cone, with top and base parallel^ figs. 6 and 7. — Xd^ /circumference i circumferenceX y. slant * ^V of top T of base ) ^ height «< Add areas of top and base if required. In tile frustum of a right circular cone, this becomes Surface- ^ 1''^]!^^ -L radius \ ^ , slant* V of top I of base^ ^ height tt (rr — 3.1416) . Add areas of top and base if required. Frustum of irregular or oblique pyraniid or eone. Surface •■ rom of surfaces of sides, each of which must be treated as a trapezoid. * In the frustum of the pvramid (Fig 6)» this slant height must be measured aloof th« middle of one of the ndet (as at i<), and not along one of the edges. 202 PRISMOIDS. PRISMOIl>gi. ¥is, 1. , A prlsmold is sometimes fteltnMl at a mlid haTing for Ita ends two parallel plane figures, connected by other plane figures on which, and through every point of which, a straight line may be drawn from one of the two parallel ends to the other. These connecting planes may be parallelograms or not, and parallel to each other or not. Tills definition vroiQd include the cube and all other parallelopipedi; the prism; the cylinder (considered as a prism having an infinite number of sides); the pyramid and cone (in which one of the two parallel ends, i e the one forming the apex, is considered to be infinitely small), and their frustums with top and base parallel ; and the wedge. But the use of the term prismoid is frequently restricted to six-sided eolida, in which the two parallel ends are unequal quadrangles; and the connecting planes, trapezoids ; as in Figs. 1 and 2 ; and, by some writers, to cases where the parallel quadrangular ends are rectangles. The following ^prismoidal fbrmnla" applies to all the foregoing solidi, and to others, as noted below. Let A =- the area of one of the two parallel ends. a = " '* the other of the two parallel ends. M = " "a cross section midway between, and parallel to^ th« two parallel ends. L — the perpendicular distance between the two parallel ends; Then Volume = L X ^ + ft^+'*M = L X mean area of cross section. The following six figures represent a few of the irregular solids which fall nnder tht above broad definition of " prismoid,'* and to which the prismoidal formula applies They may be regarded as one-chain lengths of railroad cuttings ; a o being the length, er perpendicular (horizontal) distance between the two parallel (vertical) ends. WEDGES. 203 Tine prlsmoidal formula applies also to the sphere, hemisphere, and other spherical segments; also to any sections such as a 6c d, and onidbe, of the cone, in which the sides ad, ac, or od, ic, are straight; as they are only when the •utting plane adc passes through the apex or top a. Also to tlie cylinder when a plane parallel to the sides passes through both ends; but not if the plane wz is oblique, as in the fig., though never erring more than 1 in 142. In tills last case we must imagine the plane to be extended until it cuts the side of the cylinder likewise extended; and then by page 199 find the solidity of the ungula thus formed Then find the solidity of the small ungula above w, also thus formed, and subtract it from the large one. This very extended applicability of the prismoidal formula was first discovered. and made known, by Ellwood Morris, C. E., of Philadelphia, in 1840. WEDGES. _b a b a A xvedjge Is usually defined to be a solid, Figs. 8 and 9, generated by a plane triangle, one, moving parallel to itself, in a straight line. This definition requires that the two triangular ends of the wedge should be parallel; but a wedge may be shaped as in Fig. 10 or 11. We would therefore propose the following definition, which embraces ftll the figs.; besides various modifications of them. A solid of five plane faces ; one of which is a parallelogram abed, two opposite sides of which, as a c and b d, are anited by means of two triangular faces a en, and bdm, to an edge or line nm, parallel to the other opposite sides ab and ed. The parallelogram abed may be either rectangular, or not ; the two triangular faces may be similar, or not ; and the same with regard to the other two faces. The following rule applies equally to all: Volume ^ ^ ^^'^^^IiZ^^}!f v Perphtpfrom w v "^H^V^^ QiwOze "">^ X of the 3 edges X edgetobiMsk X back (a 6 cd) ^* ub+cd + nm ougo w u«.». meas'd nerp to a & 202 PmSMOIDS. PRISMOID Fig, 1. . A prlsmold is sometimes clelin«cl bm a solid haTing for Ita ends two parallil plane figures, connected by other plane figures on which, and through every point of which, a straight line may be drawn from one of the two parallel ends to the other. These connecting planes may be parallelograms or not, and parallel to each other or not. Tills definition -woiild Include the cube and all other parallelopipedi; the prism; the cylinder (considered as a prism having an infinite number of sides); the pyramid and cone (in which one of the two parallel ends, i e the one forming the apex, is considered to be infinitely small), and their frustums with top and base parallel ; and the wedge. But the use of the term prisnioid is frequently restricted to six-sided solids, in which the two parallel ends are unequal quadrangles; and the connecting planes, trapezoids ; as in Figs. 1 and 2 ; and, by some writers, to cases where the parallel quadrangular ends are rectangles. The following ^prlsmoldal formitla" applies to all the foregoing solidi, and to others, as noted below. Let A =- the area of one of the two parallel ends. a = " '* the other of the two parallel ends. M -=- « "a cross section midway between, and parallel to, th« two parallel ends. L — the perpendicular distance between the two parallel eodflL Then Volume - L X ^ + '/ *" =» L X mean area of cross section. The following six figures represent a few of the irregular solids which fall nnder fht above broad definition of " prismoid," and to which the prismoidal formula applies They may be regarded as one-chain lengths of railroad cuttings ; a o being the length, er perpendicular (horizontal) distance between the two parallel (vertical) ends. WEDGES. 203 T9ie prismoidal formula applies also to the sphere, hemisphere, and other spherical segments; also to any sections such as abed, and onidbc, of the eone, in which the sides ad, ac, or od, ic, are straight; as they are only when the •utting plane adc passes through the apex or top a. Also to tlie cylinder when a plane parallel to the sides passes through both ends ; but not if the plane wz is oblique, as in the fig., though never erring more than 1 in 142. In this last case we must imagine the plane to be extended until it cuts the side of the cylinder likewise extended; and then by page 199 find the solidity of the ungula thus formed. Then find the solidity of the small ungula above w, also thus formed, and subtract it from the large one. This very extended applicability of the prismoidal formula was first discovered, •nd made known, by Ellwood Morris, C. E., of Philadelphia, in 1840. WEDGES. b a b a n rn ii m n m n m Flgr.8. Fig. 9. Fig.KX ¥ig.lL A -wedge Is usually defined to be a solid, Figs. 8 and 9, generated by a plane triangle, ane, moving parallel to itself, in a straight line. This definition requires that the two triangular ends of the wedge should be parallel ; but a wedge may be shaped as in Fig. 10 or 11. We would therefore propose the following definition, which embraces all the figs.; besides various modifications of them. A solid of five plane faces ; one of which is a parallelogram abed, two opposite sides of which, as a c and 6 d, are nnited by means of two triangular faces a en, and bdm, to an edge or line nm, parallel to the other opposite sides ab and cd. The parallelogram abed maybe either rectangular, or not ; the two triangular faces may be similar, or not ; and ths same with regard to the other two faces. The following rule applies equally to all: Volume of w«dse H X Snm of Isngths of the 3 edges ab +ed + nm perp ht p from edge to back width of back {abed) meas'd nerp to a & 204 MENSURATION. SPHERES OR GLOBES. A Spliere Ib a solid generated by the revolution of a semicircle around its diameter. Every point in the surface of a sphere is equidistant from a certain point called the center. Any line passing entirely through a sphere, and throagh its center, is called its oxt*, or diameter. Any circle described on the surface of a sphere, from the center oi the sphere as the center of the circle, is called a great circle of that sphere ; in other words any entire circumference of a sphere is a great circle. A sphere has a greater content or solidity than any other solid with the same amount of surface ; so that i| the shape of a sphere be any way changed, its content will be reduced. The inter- section of a sphere with any plane is a circle. Volume of sphere = I TT radius 3 ^ 4.1888 radius 3 = 3^ TT diameter* = 0.5236 diameter* , , circumference * „ „ , , , "^ % 5 "" 0.01689 circumference* — % diameter X area of surface "^ ^ diameter X area of great circle -= % volume of circumscribing cylinder «= 0.5236 volume of circumscribing cube. Axea of surface of sphere = 4 TT radius* -= 12.5664 radius* ==» TT diameter* = 3.1416 diameter* circumference* «„,oo x. • — == 0.3183 circumference* TT = diameter X circumference — 4 X area of great circle = area of circle whose diameter is equal to twice diameter of spheres = curved surface of circumscribing cylinder 6 X volume diameter. Radius of sphere = / Area of surface = \/ . 07958 X area of surface Circumference of sphere = 'v/e 7r2 volume =• '^^59.2176 volume =5 x/tt »rea of surface = v^34416 area of surfaee area of surface """ diameter. MENSURATION. SPHERES. (Original.) Some errors of 1 ia the last figure only. i 1 is i 1 1 a 1 2 a .5 1 ^ s a 3 o H a o 5 a s 3 o OQ OQ OQ QQ m OQ OQ OQ 1-64 .00077 13-82 18.190 7.2949 9i 170.87 210.03 H 921.33 2629.6 1-32 .00307 .00002 7-16 18.666 7.5829 M 176.71 220.89 Ya 934.83 2687.6 3-64 .00690 .00005 15-32 19.147 7.87«3 182.66 232.13 % 948.43 2746.5 1-16 .01227 .00013 H 19.635 8.1813 H 188.69 243.73 }4 962.12 2806.2 3-32 .02761 .00043 17-32 20.129 8.4919 Ve 194.83 255.72 % 975.91 2866.8 H .04909 .00102 9-16 20.629 8.8103 8. 201.06 268.08 Ya 989.80 2928.2 5-32 .07670 .00200 19-32 21.135 9.1366 3^ ■207.39 280.85 % 1003.8 2990.5 3-16 .11045 .00345 % 21.648 9.4708 H 213.82 294.01. 18. 1017.9 3053.6 7-32 .15033 .00548 21-32 22.166 9.8131 % 220.36 307.58 % 1032.1 3117.7 H .19635 .00818 11-16 22.691 10.164 ^ 226.98 321.56 % 1046.4 3182.6 9-32 .24851 .01165 23-32 23.222 10.522 H 233.71 335.95 % 1060.8 3248.5 5-16 .30680 .01598 H 23.758 10.889 H 240.53 350.77 H 1075.2 3315.3 11-32 .37123 .02127 25-32 24.302 11.265 K 247.45 366.02 % 1089.8 3382.9 H .44179 .02761 1316 24.850 11.649 9. 254.47 381.70 Ya 1104.5 3451.5 13-82 .51848 .03511 27-32 25.405 12.041 H 261.59 397.83 % 1119.3 3521.0 7-16 .60132 .04385 K 25.967 12.443 268.81 414.41 19. 1134.1 3591.4 15-32 .69028 .05393 29-32 26.535 12.853 % 276.12 431.44 ^ 1149.1 3662.8 34 .78540 •06545 15-16 27.109 13.272 ^ 283.53 448.92 M 1164.2 3735.0 17-32 .88664 .07850 31-32 27.688 13.700 % 291.04 466.87 % 1179.3 3808.2 9-16 .99403 .09319 3. 28.274 14.137 H 298.65 485.31 yi 1194.6 3882.5 19-32 1.1075 .10960 1-16 29.465 15.039 % 306.36 504.21 % 1210.0 3957.6 % 1.2272 .12783 H 30.680 15.979 10. 314.16 523.60 Ya 1225.4 4033.7 21-32 1.3530 .14798 3-16 31.919 16.957 H 322.06 543.48 % 1241.0 4110.8 11-16 1.4849 .17014 H 33.183 17.974 Ya 330.06 563.86 20. 1256.7 4188.8 23-32 1.6230 .19442 5-16 34.472 19.031 Vs 338.16 584.74 H 1272.4 4267.8 H 1.7671 .22089 Vs 35.784 20.129 H 346.36 606.13 % 1288.3 4347.8 25-32 1.9175 .24967 716 37.122 21.268 H 354.66 628.04 % 1304.2 4428.8 13-16 2.0739 .28084 J4 38.484 22.449 H 363.05 650.46 H 1320.3 4510.9 27-32 2.2365 .31451 916 39.872 23.674 K 371.54 673.42 H 1336.4 4593.9 M 2.4053 .35077 H 41.283 24.942 11. 380.13 696.91 Ya 1352.7 4677.9 29-32 2.5802 .38971 11-16 42.719 26.254 H 388.83 720.95 % 1369.0 4763.0 15-16 2.7611 .43143 H 44.179 27.611 H 397.61 745.5 1 21. 1885.5 4849.1 31-32 2.9483 .47603 13-16 45.664 29.016 % 406.49 770.64 H 1402.0 4936.2 1. 3.1416 .52360 % 47.173 30.466 34 415.48 796.33 Ya 1418.6 5024.S 1-32 3.3410 .57424 15-16 48.708 31.965 Vs 424.56 822.58 % 1435.4 5113.5 1-16 3.5466 .62804 4. 50.265 33.510 H 433.73 849.40 H 1452.2 5203.7 3-32 3.7583 .68511 1-16 51.848 35.106 % 443.01 876.79 H 1469.2 5295.1 H 3.9761 .74551 ^ 53.456 36.751 12. 452.39 904.78 Ya 1486.2 .5387.4 5-32 4.2000 .80939 316 55.089 38.448 M. 461.87 933.34 % 1503.3 5480.8 3-16 4.4301 .87681 H 56.745 40.195 34 471.44 962.52 22. 1520.5 5575.S 7-32 4.6664 .94786 5-16 58.427 41.994 % 481.11 992.28 H 15379 5670.8 H 4.9088 1.0227 % 60.133 43.847 H 490.87 1022.7 Ya 1555.3 5767.6 9-32 5.1573 1.1013 7-16 61.863 45.752 % 500.73 1053.6 % 1572.8 5865.1 5-16 5.4119 1.1839 H 63.617 47.713 H 510.71 1085.3 H 1590.4 5964.1 11-32 5.6728 1.2704 9-16 65.397 49.729 % 520.77 1117.5 % 1608.2 60«4.1 ^ 5.9396 1.3611 % 67.201 51.801 13. 530.93 1150.3 Ya 1626.0 6165.2 18-32 6.2126 1.4561 11-16 69.030 53.929 H 541.19 1183.8 K 1643.9 6267.3 7-16 6.4919 1.5553 H 70.888 56.116 Ya 551.55 1218.0 23. 1661.9 6370.6 15-32 67771 1.6590 13-16 72.759 58.359 % 562 00 1252.7 H 1680.0 6475.0 M 7.0686 1.7671 % 74.663 60.663 '%. 572.55 1288.3' 1698.2 6580.6 17-32 7.3663 1.8799 15-16 76.589 63.026 % 583.20 1324.4 % 1716.5 6687.3 9-16 7.6699 1.9974 6. 78.540 6.S.450 H 593.95 1361.2 }i 1735.0 6795.2 19-32 7.9798 2.1196 1-16 80.516 67.9.35 Ji 604.80 1398.6 % 1753.5 69042 H 8.2957 2.2468 H 82.516 70.482 14. 615.75 1436.8 Ya 1772.1 7014.3 21-32 8.6180 2.3789 3-16 84.541 73.092 H 626.80 1475.6 % 1790.8 7125.6 11-16 8.9461 2.5161 H 86.591 75.767 Ya 637.95 1515.1 24. 1809.6 7238.2 23-32 9.2805 2.6586 5-16 88.664 78.505 % 649.17 1555.3 H 1828.5 7351.9 H 9.6211 2.8062 H 90.763 81.308 H 660.52 ft96.3 Ya 1847.5 7466.7 25-32 9.9678 2.9592 7-16 92.887 84.178 % 671.95 1637.9 % 1866.6 7588.0 13-16 10.321 3.1177 34 95.033 87.113 Ya 683.49 1680.3 ^ 1885.8 7700.1 27-32 10.680 3.2818 9-16 97.205 90.118 % 695.13 1723.3 1905.1 7818.6 % 11.044 3.4514 % 99.401 93.189 15. 706.85 1767.2 ?i 1924.4 7938.3 29-32 11.416 3.6270 1116 101.62 96.331 M 718.69 1811.7 = ^ 1943.9 8059.2 15-16 11.793 3.8083 H 103.87 99.541 730.63 1857.0 25. 1963.5 8181.3 31-32 12.177 3.9956 13-16 106.14 102.82 yk 742.65 1903.0 ^ 1983.2 8304.7 2. 12.566 4.1888 Vb 108.44 106.18 H 754.77 1949.8 Ya 2002.9 8429.3 *l-32 12.962 4.3882 15-16 110.75 109.60 % 767.00 1997.4 % 2022.9 8554.9 1-16 13.364 4.5939 6. 113.10 113.10 Ya 779.32 2045.7 H 2042.8 8682.0 3-32 13.772 4.8060 H 117.87 120.31 K 791.73 2094.8 % 2062.9 8810.3 H 5-32 14.186 5.0243 M 122.72 127.83 16. 804.25 2144.7 Ya 2083.0 8939.9 14.607 5.2493 % 127.68 135.66 % 816.85 2195.3 % 2103.4 9070.6 3-16 15.033 5.4809 ^ 132.73 143.79 829.57 2246.8 26. 2123.7 9202.8 7-32 15.466 5.7190 % 137.89 152.25 % 842.40 2299.1 ^ 2144.2 9336.2 9.,1 15.904 5.9641 H 143.14 161.03 H 855.29 2352.1 Ya 2164.7 9470.8 16.349 6.2161 y» 148.49 170.14 ^ 868.81 2406.0 % 2185.5 9606.7 5-16 16.800 6.4751 7. 153.94 170.59 H 881.42 2460.6 H 2206.2 9744.0 11-82 17.258 6.7412 ' Ji 159.49 189.39 % 894.6S 2516.1 % 2227.1 9882.1 H 17.721 7.0144 3 166.18 199.53 17. 907.93 2572.4 H 2248.0 10022 206 MENSURATION. SPHERES — (Continued.) a" 1 is i 1 2 a .2 1 i a i 2 o S "o Q 3 o Q a o 3 cc 02 QQ 02 OS 02 a m Va 2269.1 10164 % 4214.1 25724 % 6756.5 52222 Y 9896.0 92570 27. 2290.2 10306 4243.0 25988 P 6792.9 52645 Y 9940.2 93190 % 2311.5 104:0 •u 4271.8 26254 6829.5 53071 % 9984.4 93812 M 233-2.8 10595 37. 4300.9 26522 6866.1 53499 Y 10029 94438 % 2354.3 10741 H 4330.0 26792 % 6902.9 53929 10073 96066 H 2375.8 10889 M 4359.2 27063 47. 6939.9 54362 % 10118 95697 % 2397.5 11038 % 4388.6 27337 Y. 6976.8 547S7 Va 10163 96330 Vi 2419.2 11189 3^ 4417.9 27612 Yi 7013.9 55234 57. 10207 96967 K 2441. 1 11341 % 4447.5 27889 % 7050.9 55674 Y 10252 97606 28. 2463.0 11494 H 4477.1 28168 H 7088.3 66115 10297 98248 14 2485.1 11649 % 4506.8 28449 7125.6 5655^ % 10342 98893 M 2507.2 11805 38. 4536.5 28731 % 7163.1 57006 Y 10387 99641 % 2529.5 11962 M 4566.5 29016 % 7200.7 67455 10432 100191 "A 2551.8 12121 Yi 4596.4 29302 48. 7238.3 57906 H 10478 100845 y% 2574.3 12281 % 4626.5 29590 % 7276.0 58360 Vb 10523 101501 H 2596.7 12443 ^ 4656.7 29880 M 7313.9 58815 58. 10568 102161 Ve 2619.4 12606 % 4686.9 30173 % 7351.9 69274 Y 10614 102825 29. 2642. 1 12770 % 4717.3 30466 Y. 7389.9 59734 Y 10660 103488 H 2665.0 12936 % 4747.9 30762 % 7428.0 60197 % 10706 104155 2687.8 13103 39. 4778.4 31059 % 7466.3 60663 Y 10751 104836 y 2710.9 13272 H 4809.0 31359 % 7504.6 61131 10798 105499 ■L/ 2734.0 13442 4839.9 31661 49. 7543.1 61601 % 10844 106175 % 2757.3 13614 % 4870.8 31964 Y 7581.6 62074 K 10890 106854 % 2780.5 137fc7 }i 4901.7 32270 Y 7620.1 62549 59. 10936 107536 % 2804.0 13961 % 4932.7 32577 H 7658.9 63026 Y 10983 108221 80. 2827.4 14137 H 4S64.0 32886 Y 7697.7 63506 11029 108909 % 2851.1 14315 % 4995.5 33197 % 7736.7 63969 % 11076 109600 Vi 2874.8 14494 40. 5026.6 33510 H 7775.7 64474 Y 11122 110294 % 2898.7 14674 ^ 5058.1 33826 H 7814.8 64861 Y 11169 110990 yi 2922.5 14856 M 5089.6 34143 50. 7854.0 65450 H 11216 111690 % 2946.6 15039 % 5121.3 3-1462 Y 7893.8 65941 ' % 11263 112892 , % 2970.6 15224 H 5153. 1 34783 7932.8 66436 60. 11310 113098 % 2994.9 15411 % 5184.9 351C6 % 7972.2 66234 Y 11357 113806 SI. 3019.1 15599 % 6216.8 35431 Y 8011.8 67433 11404 114618 ^ 3043.6 15788 % 5248.9 35758 % 8051.6 67935 % 11452 115232 yi 3068.0 15979 41. 6281.1 36087 % 8091.4 68439 Y 11499 115949 % 3092.7 16172 H 5313.3 36418 H 8131.8 68946 Y 11547 116669 ^ 3117.3 16366 M 5345.6 36751 51. 8171.2 69456 H 11596 117892 % 3142.1 16561 Ys 5378.1 37086 Y 8211.4 69967 Y 11642 118118 % 3166.9 16758 ^ 5410.7 37428 Y 8251.6 70482 61. 11690 118847 K 3192.0 16957 % 6443.3 37763 % 8292.0 70969 Y 11738 n957» «2. 3217.0 17157 H 5476.0 38104 Y 8332.3 71519 Y 11786 120315 H 3242.2 17359 % 6508.9 88448 Y 8372.8 72040 % 11884 121063 U 3267.4 17563 ^2. 5541.9 38792 H 8418.4 72565 Y 11882 121794 % 3292.9 17768 ^ 6574.9 39140 K 8454.1 73092 Y 11931 122588 ^. 3318.3 17974 5608.0 39490 52. 8494.8 73622 H 119F0 128286 3343.9 18182 % 5641.3 39841 Y 8535.8 74154 Y 12028 124036 ^ 3369.6 18392 y4 5674.5 40194 8576.8 74689 62. 12076 124789 K 3395.4 18604 % 5708.0 40551 % 8617.8 75226 Y 12126 125645 33. 3421.2 18817 % 6741.5 40908 Y 8658.9 75767 12174 126305 ^ 3447.3 19032 K 5775.2 41268 % 8700.4 76309 % 12223 127067 H 3478.3 19248 43. 5808.8 41630 % 8741.7 76854 Y 12272 12788J % 8499.5 19466 H 5842.7 41994 % 8783.2 77401 % 12322 128601 'A 3525.7 19685 5876.5 42360 53. 8824.8 77952 % 12371 129873 H 3552. 1 19907 % 5910.7 42729 Y 8866.4 78505 Y 12420 130147 H 3578.5 20129 r^ 5944.7 43099 Y 8908.2 79060 63. 12469 130925 Yh 3605.1 20354 % 5978.9 43472 % 8950.1 79617 Y 12519 131706 84. 3631.7 20580 % 6013.2 6047.7 43846 Y 8992.0 80178 12568 132490 % 3658.5 20808 % 44224 % 9034.1 80741 % 12618 133277 3685.3 21037 44. 6082.1 44602 H 9076.4 81308 Y 12668 134067 % 3712.3 21268 H 6116.8 449S4 Vs 9118.5 81876 Y 12718 134860 ^ 3739.3 21501 Yi 6151.5 45367 54. 9160.8 82448 % 12768 135657 % 3766.5 21736 % 6186.3 45753 Y 9203.8 83021 Y 12818 136466 % 3793.7 21972 Yl 6221.2 46141 Y 9246.0 83598 64. 12868 137259 K 3821.1 22210 % 62,56.1 46530 % 9288.5 84177 Y 12918 188065 86. 3848.5 22449 % 6291.2 46922 Y 9381.2 84760 Ya. 12969 138874 H 3876.1 22691 % 6326.5 47317 % 9374.1 85344 % 13019 139686 }4. 3903.7 22934 45. 6361.7 47713 % 9417.2 86931 Y 13070 140501 % 3931.5 23179 H 6397.2 48112 % 9460.2 86521 % 13121 141320 H 3959.2 23+25 M 6432.7 48513 55. 9503.2 87114 H 13172 142142 H 3987.2 23674 % 6168.3 48916 Y 9546.5 87709 K 13222 142966 H 4015.2 23924 A 6503.9 49321 9590.0 88307 65. 13273 143794 % 4043.8 24 J 76 % 6539.7 49729 % 9633.8 88908 Y 13324 144625 36. 4071.5 24429 H 6575.5 50139 Y 9676.8 89511 Y\ 13376 145460 H 4099.9 246R5 % 6611.6 .50551 H 9720.6 90117 %\ 13427 146297 4128.3 24942 46. 6647.6 60965 H 9764.4 90726 H' 13478 147138 % 4IS6.9 25201 H 6683.7 51382 % 9808.1 91338 % 13530 147983 H 4186.5 25461 Ya 6720.0 61801 56. 9852.0 91953 ?i' 1358? 1 148829 MENSURATION. 207 SPHERES— (Continued.) a 1 ^ S3 1 j3 i 1 1 1 s o 3 9 p m m s OQ CO 5 QQ m 5 CO CO Va 15633 149680 34 17437 216505 H 21708 300743 % 26446 40440S 66/* 13685 150533 y% 17496 217597 218693 M 21773 .302100 % 26518 406060 H 13737 151390 % 17554 % 21839 303463 92. 26590 407721 13789 152251 % 17613 219792 ^ 21904 304831 34 26663 409384 % 13841 153114 75. 17672 220894 % 21970 306201 Ya 26735 411054 3^ 13893 153980 H 17731 222001 % 22036 307576 % 26808 412736 % 13946 154850 M 17790 223111 K 22102 308957 34 26880 414405 % 13998 155724 % 17849 224224 84. 22167 310340 26953 416086 Va 14050 156600 ^ 17908 225341 ^ 22234 311728 % 27026 417774 67. 14103 157480 % 17968 226463 H 22300 318118 Ya 27099 419464 H 14156 158363 H 18027 227588 % 22366 314514 93. 27172 421161 34 14208 159250 % 18087 228716 H 22432 315915 ^ 27245 422862 % 14261 160139 76. 18146 229848 % 22499 317318 27318 424567 M 14314 161032 % 18206 230984 % 22565 318726 H 27391 426277 H 14367 161927 34 18266 232124 % 22632 320140 34 27464 427991 H 14420 162827 % 18326 233267 85. 22698 321556 % 27538 429710 H 14474 163731 34 18386 234414 H 22765 322977 H 27612 431433 68. 14527 164637 % 18446 235566 22832 324482 Ys 27686 433160 H 14580 165547 % 18506 236719 % 22899 325831 94. 27759 434894 H 14634 166460 Vh 18566 237879 34 22966 327264 % 27833 436630 % 14688 167376 77. 18626 239041 H 23034 328702 Ya 27907 438373 ^ 14741 168295 H 18687 24020C H 23101 330142 Ys 27981 440118 % 14795 169218 18748 241376 % 23168 331588 H 28055 441871 % 14849 170145 % 18809 242551 86. 23235 333039 % 28130 443625 % 14903 171074 J4 18869 243728 H 23303 33449* Yi 28204 445387 69. 14957 172007 % 18930 244908 Ya 23371 335951 X •28278 447151 H 15012 172944 H 18992 24609S % 23439 337414 95. 28353 448920 H 15066 173888 Vb 19053 247283 23506 338882 H 28428 45069J % 15120 174838 78. 19114 248475 % 23575 340352 Ya 28503 452475 14 15175 175774 ii 19175 249672 H 23643 341829 % 28577 454259 % 15230 176728 34 19237 250873 K 23711 343307 H 28652 456047 % 15284 17767T % 19298 252077 87. 23779 344792 % 28727 457839 K 15339 178635 34 19360 253284 H 23847 346281 Ya 28802 459638 70. 15394 179595 H 19422 254496 23916 347772 Ys 28878 461439 H 15449 180559 H 19483 255713 % 23984 349269 96. 28953 463248 H 15504 181526 % 19545 256932 34 24053 350771 H 29028 465059 15560 182497 79. 19607 258155 24122 352277 Ya 29104 466875 14 15615 183471 ^ 19669 259383 H 24191 353785 % 29180 466697 % 15670 184449 34 19732 260618 % 24260 355301 ^ 29255 470524 %/ 15726 185430 % 19794 261848 88. 24328 356819 Ya 29331 472354 h 15782 186414 34 19856 263088 34 24398 358342 Ya 29407 474189 71. 15837 187402 % 19919 264330 % 24467 359869 Yb 29483 476029 H 15893 188394 % 19981 265577 % 24536 361400 97. 29559 477874 15949 189389 % 20044 266829 34 24606 362935 ^ 29636 479725 % 16005 190387 80. 20106 268083 % 24676 364476 M 29712 481579 1^ 16061 191389 H 20170 269342 H 24745 366019 h 29788 483433 k£ 16117 192395 20232 270604 Vs 24815 367568 34 29865 485302 H 16174 193404 Yi 20296 271871 89. 24885 369122 % 29942 487171 % 16230 194417 H 20358 273141 H 24955 370678 Ya 30018 489046 n. 16286 195433 % 20422 274416 u 25025 372240 Ys 30095 490924 H 16343 196453 H 20485 275694 H 25095 373806 98. 30172 492808 H 16400 197476 % 20549 276977 }4 25165 375378 ^ 30249 494695 % 16456 198502 81. 20612 278263 % 25236 376954 Ya 30326 496588 ^ 16513 199532 34 20676 279553 % 25306 378531 Yb 30404 498486 % 16570 200566 34 20740 280847 Ys 25376 380115 ^ • 30481 500388 % 16628 201604 y% 20804 282145 90. 25447 381704 % 30558 502296 3 16685 202645 34 20867 283447 H 25518 383297 Ya 30636 504208 T3.^ 16742 203689 % 20932 284754 25589 384894 K .80713 506125 H 16799 204737 H 20996 286064 % 25660 386496 99. 30791 508047 16857 205789 % 21060 287378 H 25730 388102 H 30869 509975 az 16914 206844 82. 21124 188696 Yi 25802 389711 34 30947 511906 \Z 16972 207903 34 21189 290019 H 25873 391327 % 31025 51384S % 17030 208966 M 21253 291345 % 25944 392946 H 31103 515785 H 17088 210032 % 34 21318 292674 91. 26016 894570 % 31181 517730 % 17146 211102 21382 294010 H 26087 396197 % 31259 519682 U. 17204 212175 % 21448 295347 34 26159 397831 % 31338 521638 H 17262 213252 % 21512 296691 % 26230 399468 100. 81416 523596 34 17320 214333 % 21578 298036 Y, 26302 401109 % 17379 215417 83. 21642 299388 Ya 26374 402756 208 SEGMENTS, ETC., OF SPHERES. To find tlie solidity of a spberical seg^ment. RuLK 1.' Square the rad on, of its base; mult this square by 3 ; i the prod add the square of its height o s ; mult the sum by the heigl s ; and mult this last prod by .5236. Rule 2. Mult the diam a'b of the sphere by 3 ; from the pro take twice the height o s of the segment; mult the rem by the squai ef the height o s ; and mult this prod by .5236. The solidity of a sphere being %Aa that of its circumscribing cy lii der, if we add to any solidity in the table, its half, we obtain ths of a cylinder of the same diam aa the sphere, and whose heigi equals'its diam. To find tbe curved surface of a spherical segment. Rule 1. Mult the diam a 6 of the sphere from which the segment is cut, by 3.1416 malt the prod by the height o « of the seg. Add area of base if reqd. Rem. Having the diam n •f the seg, and its height o s, the diam a 6 of the sphere may be found thus: Div the square of hal the diam n r, by its height o s ; to the quot add the height o s. Rule 2. The curved surf of eithe • segment, lait Fig, or of a zone, (next Pig,) bears the same proportion to the surf of the whol ■phere, that the height of the seg or zone bears to the diam of the sphere. Therefore, first find th •urf of the whole sphere, either by rule or from the preceding table ; mult it by the height of the ae or zone ; div the prod bv dianl of sphere. Rule 3. Mult the circum.f of the sphere by the height o 9£ the seg. To find the solidity of a spherical zone Add together the square of the rad e d, the square of rad o I and 3^d of the square of the perp height eo; mult the sum b 1.5708; and mult this prod by the height 0. * To find the curved surface of a spher- ical zone. Bulk 1. Mult together the diam mn of the sphere ; the heigh e of the zone, and the number 3.1416. Or see preceding Rule for surf of segments. Rule 2. Mult the circumf of the sphere, b the height of the zone. * To find the solidity of a hollow spher ieal shell. Take from the foregoing table the solidities of two aphtres havin the diams a b, and c d. Subtract the least from th*sr«atest. fier aooebcliBthe thickac«s of th« akalL THE EI.I.IPSOID, OR SPHEROID, Is a solid generated by the revolution of an ellipse around either its long or its short diam. Win around the long (or transverse) diam, as at a. Fig 1, it is an oblongs or pre late spheroid; when around the short (or conjug'ate) one, as at w, in Fig it is oblate. Flg.l. Fig. 2. For the solidity in either case, mult the fixed diam or axis by the squar of the revolving one ; and mult the prod by .5236. * This rule applies, whether the zone includes the equator (as in our figure), o not, as in the earth's temperate zones. PARABOLOIDS, ETC. 20y THE PARABOI.OID, OR PARABOLIC CONOID, Qext Fig, is a solid generated by the revolution of a parabola a cb, around its axis, c r. For its solidity mult the area of its base, by half its height, r c. Or miilt togeilier the square of the rad o r of the base; the height r c; and the number 1.5708. For tlie solidity of a frnstmii, ui g h, the ends of which are perp to the axis r c ; add together tlM squares of the two diams a b and g h; mult the sum by the height r I; mult the prod by the deciiaal .3927. To find tbe surface of a paraboloid, Mult the rad a r of its base, by 6.2832; div the proi by 12 times the square of the height re, call the quotp. Then add together the square «f the rad a r, and 4 times the square of the height r c. Cube the aam ; take the sq rt of this cube ; from the sq rt subtract the cube of the rad a r. Mult the rem by ^. Either the solidity, or the surface of a frustum, ah g h, when ghis parallel to a h, may be found by calculating for the whole paraboloid, aad for the upper portion c g h, &a two separate paraboloids, and taking their diff. THE CIRCUIiAR SPINDI.E, |a « solid ab ny generated by the revolution of a circular segment mh ne a, around its chord a n as an axis. To find its solidity. RlTLS 1. First find the area of a 5 e, or half the generating circular i segment. Then to the square of a e, add the square of 6 e; div the sum by 6 y; from the quot take 6 e; mult the rem by the area of a e 6 ; call , the prod p. Cube a e ; div the cube by 3 ; from the quot take p. Mult the / rem by 12.5664. Rule 2. When th« dlst o e is known, from the center of the circle to \ the cen of the spindle, then mult that dist o e, by tbe area of a 6 e ; call \ the prod p ; cube a « ; div the cube by 3 ; from the quot take p ; mult the rem by 12.5664. To find its surface. RuLB 1. First find the length of the circular arc a 5 n ; and mult it by the dist o e from the center of the circle to the center of the spindle. Call the prod p. Next mult the length a n of the spindle, by the rad o 6 of the circle. Prom the prod take p ; mult the rem bv 6.2832. RuLK 2. First find the length of the arc abn. Square a e; also square b e; add these squares together; div their sum hjby; call the quot «; and mult it by a n; call the prod p. Next from stake 6 e ; malt the rem by the length of the arc abn. Subtract the prod from p ; mult the rem by 6.2832. To find tlie solidity of a middle zone of a circular spindle, AMhekp / (a e« — ^) X S «) - (o e X area of f ft i *) J X 6.2882. CIRCriiAR RIUTGS. ^^1^^^ area of cross section of bar ^ i sura of inner and outer ^ g j^^g^^ Volume => of which ring is made ^ diameters, « a and 6 & a o.i*io»* tUnrface = 14 circumference of bar ^ J sum of inner and outer ^ 3 ia^km of which rin« is made ^ diameters, a a and 66 ^ ' 210 SPECIFIC GRAVITY. SPEOIFIO GRAVITY. 1. The specific gravity, or relative density, D*, of a substance. is the ratio between the weight, W, of any given volume of that substance and the weight, A, of an equal volume of some substance adopted as a standard ol comparison. Or: D = — . 2. For gaseous substances, the standard substance is air, at a temper- ature of 0° Cent. = 32° Fahr., with barometer at 760 millimeters = 29.922 inches. 3. For solids and liquids, the standard substance is distilled water, at its temperature (4° Cent. = 39.2° Fahr.) of maximum density. 4. For all ordinary purposes of civil engineering, any clear fresh water, at any ordinary temperature, may be used. Even with water a1 30° Cent., ^ 86° Fahr., the result is only 4 parts in 1000 too great. 5. When a body is immersed in water, the upward force, or ** buoyancy," exerted upon it by the water, or the **loss of weight " of the body, due to its immersion, is equal to the weight of the water displaced by the immersion ol the body f ; or, if W = the weight of the body in air, w = its weight in water, D = its relative density or specific gravity, A = the weight of water displaced ; W W then A = W — w, and D = — = .=: . AW — w 6. Since the volume, V, of a body, of given weight, W, is inversely as ita density, or specific gravity, D ; the specific gravity is equal also to the ratio between the volume Vg of an equal weight of the standard substance, to the y volume, V, of the body in question ; or D =: ^^ 7. The specific gravities of substances heavier than water are ordi- narily determined by weighing a mass of the substance, first in air (obtain- ing its weight, W), and then when the mass is completely submerged in water (obtaining its diminished weight, w). Then D = , as in ^5. 8. If the body is lighter than water, it must be entirely immersed, and held down against its tendency to rise. Its weight, w, in water, or ita , upward tendency, is then a negative quantity, and means must be provided for measuring it, as by making it act upward against the scale pan. We then have, A = W — {—w)=W+w; or Loss due to immersion = weight of body in air, plus its buoyancy. 9. Or, first allow the body to float upon the water, and note the resulting dis- placement, V, of water, as by the rise of its surface level in a prismatic vessel. Then immerse the body completely, and again note the displacement, V.* Now r, the volume displaced by the body when floating, and V, the volume displaced by the body when completely immersed, are proportional respectively to the weight, W, of the body, and to the weight, W — w?, of a mass of water of equal W V volume with the body. Hence D = — - — = -,. W — w V 10. Or, attach to the light body, b, a heavier body, or sinker, S, of such den- sity and mass that both bodies together will sink in water. Let W be the weight of the light body, 6, in air ; Q the weight of both bodies in air, and q their combined weight in water. Then Q — g = the weight of a mass of water of equal volume with the two bodies, and Q — W = the weight, S, of the sinker in air. By immersing the sinker alone, find the weight, k, of water equal in volume to the sinker alone, = loss of weight in sinker, due to immersion. Then, for the weight. A, of water of equal volume with the light body, b, or for ♦Strictly speaking, " specific gravity " refers to weight, and " relative density " to mass (see Mechanics, Art. 14 a); but, as specific gravity and density are numerically equal, they are often treated as identical. t See Hydrostatics, Art. 18. SPECIFIC GRAVITY. 211 the loss of weight of 6, due to immersion, we have A = Q — q — k; and, for W W the specific gravity, D, of the light body, h. we have D = — -, = ~ where w — the (unknown) buoyancy of b. 11. A granular body, as a mass of saw-dust, gravel, sand, cement, etc., or a porous body, as a mass of wood, cinder, concrete, sandstone, etc., is a com- posite body, consisting partly of solid matter and partly of air. Thus, a cubic foot of quartz sand weighs about 100 lbs.; while a cubic foot of quartz weighs about 165 ft)s. 12. The specific gravity of porous substances is usually taken as that of the composite mass of solid and air. Thus,^ a wood, weighing (with its contained air) 62.5 lbs. per cubic foot, or the same as water, is said to have a specific gravity of 1. The absorption of water, when such bodies are immersed for the purpose of determining their specific gravities, may be prevented by a thin coat of varnish. 13. The specific gravity of granular substances is sometimes taken as that of tbe solid part alone. Thus, Portland cements ordinarily weigh (in air) from 75 to 90 lbs. per cubic foot, which would correspond to specific gravities of from 1.20 to 1.44; but the specific gravity of the solid portion ranges from 3.00 to 3.25 ; and the latter figures are usually taken as representing the specific gravities. 14. In determining the specific gravities of substances (such as cement) which are soluble in water or otherwise affected by it, the substances are weighed in some liquid (such as benzine, turpentine or alcohol) which will not affect them, instead of in water. The result, so obtained, must then be multi- plied by the ratio between the density of the liquid and that of water. 15. The specific gravity of a liquid is most directly determined by weighing equal volumes of the liquid and of water. 16. Or weigh, in the liquid, some body, whose weight, W, in air, and whose specific gravity, d, are known. Let u/ = its weight in the liquid. Then, for the specific gravity, D, of the liquid, we have W'.W — v/^d'.T); orD= ~^-y- ' . 17. Or, let the body, in ^ 16 (weighing W in air), weigh w in water, and (as before) w^ in the liquid in question. Then, since specific gravity of water = 1, we have W v/ W — w:W — w' = 1:1)', orD = ^^ —. W — w 18. The specific gravities of liquids are commonly obtained by observing the depth to which some standard instrument (called a hydremeter) sinks when allowed to float upon the surface of the liquid. The greater the depth, the less the specific gravity of the liquid. In Beaum^'s hydrometer the depth of immersion is shown by a scale upon the instrument. The graduations of the scale are arbitrary. For liquids heavier than water, 0° corresponds to a specific gravity of 1, and 76° to a specific gravity of 2. For liquids lighter than water, 10° correspond to a specific gravity of 1, and 60° to a specific gravity of 0.745. 19. In Twaddell's hydrometer, used for liquids heavier than water, 5 X No. of degrees 4- 1,000 specific gravity = ^m Thus, if the reading be 90° 5 X 90 + 1,000 1,450 specific gravity = ^^ = ITOOO = ^•^^• 30. In Nicholson's hydrometer, largely used also for solids, the specific gravity is deduced from the weights required to produce a standard depth of immersion. It consists of a hollow metal float, from which rises a thin but stiff wire carrying a shallow dish, which always remains above water. From the float is suspended a loaded dish, which, like the float, is always submerged. On the wire supporting the upper dish is a standard mark, which, in observations, is always brought to the surface of the water. The specific gravity is then deter- mined by means of the weights carried in the two dishes respectively. 31. The determination of the specific gravities of gaseous sub- stances requires the skill of expert chemists. 212 SPECIFIC GRAVITY. Table of specific gravities, and weig-tits. In this table, the sp gr of air, and gases also, are compared with that of water instead of that of air ; which last is usual. The specific gravity of any substance is = its wcigrlit in grams per cubic centimetre. Air, atmospheric ; at 60° Fah, and under the pressure of one atmosphere or 14.7 fts per sq inch, weighs -g-^-j- part as much as water at 60° Alcohol, pure ' ' of commerce '* proof spirit Ash, perfectly dry. , average.. 1000 ft board measure weighs 1.748 tons. Ash, American white, dry «' 1000 ft board measure weighs 1.414 tons. Alabaster, falsely so called; but really Marbles *' real; a compact white plaster of Paris average.. Aluminium Antimony, cast, 6.66 to 6.74 average .. " native " Anthracite. See Coal, below. Asphaltum, 1 to 1.8 " •• Basalt. See Limestones, quarried " •• Bath Stone, Oolite " .. Bismuth, cast. Also native " •• Bitumen, solid. See Asphaltum. Brass, (Copper and Zinc,) cast, 7.8 to 8.4 " .. " rolledi " .. Bronze. Copper 8 parts; Tin 1. (Gun metal.) 8.4 to 8.6 ** •• Brick, best pressed "■ common hard " soft, in ferior Brickwork. See Masonry. Boxwood, dry •' .. Calcite, transparent ** .. Carbonic Acid Gas, is 13.^ times as heavy as air " .. Cement. (See IT 13.) «* Portland, 3.00 to 3.25.- ~ " Natural, 2.75 to 3.00 Chalk, 2.2 to 2.8. See Limestones, quarried Charcoal, of pines and oaks Cheriy, perfectly dry Chestnut, perfectly dry Coal. See also page 215. Anthracite, 1.3 to 1.7 " piled loose Bituminous, 1.2 to 1.4 piled loose Coke .. piled loose In coking, coals swell from 25 to 50 per cent. Copper, cast 8.6 to 8.8 rolled, 8.8 to 9.0 Crystal, pure Quartz. See Quartz. Cork.. Diamond, 3.44 to 3.55 ; usually 3.51 to 3.55 Barth j common loam, perfectly dry, loose " " *' " " shaken " ** " '* " moderately rammed.. •* " " slightly moist, loose " " " more moist. shaken *• moderately packed as a soft flowing mud as a soft mud, well pressed into a box.. Ether Elm, perfectly dry. . . . . , 1000 ft board measure weighs 1 Ebony, dry Emerald, 2.63 to 2.76 Fat. Flint Feldspar, 2.5 to 2.8 Garnet, 3.5 to 4.3 ; Precious, 4.1 to 4.3 Glass, 2.5 to 3.45 " common window " Millville, New Jersey. Thick flooring glass . Granite, 2.56 to 2.88. See Limestone, 160 to 180 Average SpGr. .716 .56 2.6 2.65 4.2 2.98 2.52 2.53 2.72 SPECIFIC GRAVITY. 213 Table of specific gravities, and iFeiglits — (Continued.) The specific gravity of any substance is = its weigbt ill g^rams per cubic centimetre. Gneiss, common, 2.62 to 2.76 " in loose piles '• '• Hornblendic «♦ " " quarried, in loose piles " Gypsum, Plaster of Paris, 2.24 to 2.30 ♦• " in irregular lumps «'- .. " ground, loose, per struck bushel, 70 " " " well shaken, " " 80 " " " Calcined, loose, per struck bush, 65 to 76- " Greenstone, trap, 2.8 to 3. 2 ♦' " " quarried, in loose piles " Gravel, about the same as sand, which see. Gold, cast, pure, or 24 carat " " native, pure, 19.3 to 19.34 ' " " " frequently containing silver, 15.6 to 19.3.. " " pure, hammered, 19.4 to 19.6. " Gutta Percha " Hornblende, black, 3.1 to 3.4 " Hydrogen Gas, is 1434 times lighter than air; and 16 times lighter than oxygen average.. Hemlock, perfectly dry. " 1000 feet board measure weighs .930 ton. Hickory, perfectly dry. " 1000 feet board measure weighs 1.971 tons. Iron, and steel. «« Pig and cast iron and cast steel m..m.m.....*... «< Wrought iron and steel, and wire, 7.6 to 7.9 Ivory average . . Ice, .917 to .922 " .. India rubber '* Lignum vitae, dry " Lard .... " Lead, of commerce, 11.30 to 11.47 ; either rolled or cast " Limestones and Marbles, 2.4 to 2.86, 150 to 178.8 " '* •' ordinarily about " " " quarried "in irregular fragments, 1 cub yard solid, makes about 1.9 cub yds perfectly loose ; or about 1^ yds piled. In this last case, 571 of the pile is solid; and the remaining .429 part of it is voids piled . . Lime, quick, ground, loose, per struck bushel 62 to 70 Tbs " " " well shaken, " *• ....80 *' " " " thoroughly shaken, " ....93Ji " Mahogany, Spanish , dry * average . . " Honduras, dry " Maple, dry» , *' Marbles, see Limestones. Masonry, of granite or limestone?, well dressed throughout " •' " well-scabbled mortar rubble. About 4 of the mass will be mortar " " '* well-scabbled dry rubble ". " " " roughly scabbled mortar rubble. About %tf> % part will be mortar " *' " roughly scabbled dry rubble At 155 fts per cub ft, a cub yard weighs 1.868 tons j and 14.45 cub ft, 1 ton. Masonry of sandstone ; about % part less than the foregoing. " " brickwork, pressed brick, fine joints average.. " " " medium quality " " " " coarse ; inferior soft bricks " At 125 fts per cub ft, a cub yard weighs 1.507 tons; and 17.92 cub ft, 1 ton. Mercury, at 32° Fah 60O " 212° " Mica. 2.75 to 3.1 Mortar, hardened, 1.4 to 1.9 M ud , dry, close ' ' wet, moderately pressed " wet, fluid Average Sp Gr. 19.5 .98 3.25 7.2 7.75 1.82 * Green timbers usually weigh from one-fifth to nearly one-half more than dry ; and ordinary building timbers when tolerably seasoned about one-sixth more than perfectly dry. 214 SPECIFIC GRAVITY. Table of specific griravities, and weights — (Continued.) The specific gravity of any substance is = its iveig'lit ill g^rams per cubic centimetre. Naphtha Nitrogen Gas is about ^^ part lighter than air Oak, liTe, perfectly dry, .'88 to 1.02* average.. " white, " " .66 to .88 " " red, black, &c* " Oils, whale; olive " " ofturpentiae " Oolites, or Roestones, 1.9 to 2.5 " Oxygen Gas, a little more than J^ part heavier than air Petroleum Peat, dry, unpressed Pine, white, perfectly dry, .35 to .45* 1000 ft board measure weighs .930 ton.* " yellow, Northern, .48 to .62 1000 ft board measure weighs 1.276 tons.* " " Southern, .64 to .80 1000 ft board measure weighs 1.674 tons.* Pine, heart of long-leafed Southern yellow, unseas. ... 1000 ft board measure weighs 2.418 t*ns. Pitch Plaster of Paris ; see Gypsum. Powder, slightly shaken Porphyry, 2.66 to 2.8 .•. . Platinum 21 to 22 " native, in grains 16 to 19 Quartz, common, pure 2.64 to 2.67 •' " finely pulverized, loose " " " " well shaken " " " " well packed " quarried, loose. One measure solid, makes full 1^ broken and piled Ruby and Sapphire, 3.8 to 4.(^^ Rosin Salt.., Average SpQr. .95 .77 .92 .87 2.2 .00136 .878 .40 .55 .72 1.04 1.15 1. 2.73 21.5 17.5 2,65 3.41 *2.*6** .59 2.6 Sand, pure quartz, perfectly dry, loose *« " " '♦ " slightly shaken " '* •* rammed, dry Natural sand consists of grains of different sizes, and weighs more, per unit of volume, than a sand sifted from it and having grains of uniform size. Sharp sand with very large and very small grains may weigh as much as Sand is very retentive of moisture, and, when in large bulk, its natural moisture may diminish its weight from 5 to 10 per cent. " perfectly wet, voids full of water Sandstones, fit for building, dry, 2.1 to 2.73, 131 to 171. "■ quarried, and piled, 1 measure solid, makes about 1^ piled... Serpentines, good 2.5 to 2.66 Snow, fresh fallen = " moistened, and compacted by rain Sycamore, perfectly dry, 1000 ft board measure weighs 1.376 tons. Shales, red or black 2.4 to 2.8 average.. " quarried, in piles " Slate 2.7 to 2.9 " .. Silver " .. Soapstone, or Steatite 2.65 to 2.8 " .. Steel, 7. 7 to 7.9. The heaviest contains least carbon " Steel is not heavier than the iron from which It is made; unless the iron had impurities which were expelled during its conversion into steel. Bnlph ur average . . Spruce, perfectly dry. •* 1000 ft board measure weighs .930 ton. Spelter, or Zinc 6.8 to 7.2 " .. Sapphire; and Ruby, 3.8 to 4.... " Tallow " Tar '* Trap, compact, 2.8 to 3.2 " .. " quarried ; in piles ; " Topaz. 3.45to3.65 " .. ♦Green timbers usually weigh from one-fifth to nearly one-half more than dry ; and ordinary building timbers when tolerably seasoned about one-sixth more than perfectly dry. 2.8 10.5 2.73 7.85 7.00 3.9 .94 WEIGHT OF COAL. 215 Table of speeific g^ravities, and wei^bts — (Continued.) The specific gravity of any substance is = its iveig^llt in g^rams per cubic centimetre. Tin, cast, 7.2 to 7.5 average.. Turf, or Peat, dry, unpressed Water. See page 3'26. W.ix, bees average., Wiues, .993 to l.Oi <' .. Walnut, black, perfectly dry. " 1000 ft board measure weighs 1.414 tons. Zinc, or Spelter, 6.8 to 7.2 «« .. Zircon, 4.0 to 4.9 " .97 .998 .61 7.00 4.45 Average Wt of a Cub Ft. Lbs. 459. 20 to 30 62.417 60.5 62.3 38. Space occupied by coal. In cubic feet per ton of 2240 pounds. Pennsylvania Anthracite. Bro- ken. Egg. Stove. Nut. Pea. Buck- wheat. 38.6 39.2 39.8 40.5 41.1 39.4 39.6 39.6 39.6 39.8 39.8 39.0 39.6 40.2 40.8 41.5' 39.6 39.6 39.6 41.2 41.9 42.4 39.3 39.9 40.5 41.2 41.9 " 39.0 39.9 42.6 45.7 46.5 47.7 39.6 40.3 40.9 41.6 42.3 40.0 40.5 41.1 41.7 42.3 ( 44.8 45.2 45.7 46.2 46.7 144.2 44.3 44.3 45.0 46.1 46.5 40.0 39.8 39.4 39.4 38.8 38.5 38.4 42.1 41.4 38.5 38.8 40.1 40.3 40.3 40.5 .3: di St, 39.] Aver- age. Hard white ash* Free-burning white ash *.. Shamokiu * Schuylkill white ash *.. " red " *.. Lykens Valley* Wyoming free-burningf Lehigh t Lehigh ; Reading C. & I. Co. *. Lehigh:! Lump, 40.5; cupola, 40.3 39.6 40.2 40.7 40.6 43.6 40.9 41.1 45.7 45.1 39.7 40.0 39.7 Bituminous. From Coxe Bros. &Co.t From Jour. U. S. Ass'n Charcoal Iron Workers. Vol. Ill, 1882.§ Pittsburg Erie 48.2 46.6 Pittsburg 47.1 Cumberland, max 42.3 min 41.2 Blossburg, Pa 42.2 Clover Hill, Va 49.0 Ricbmond, Va. 45 4 (Midlothian) 41.0 Ohio Cannel Indiana Block Illinois 45.5 51.1 47.4 Caunelton, Ind 47.0 Pictou,N. S 45.0 Sydney, Cape Breton .47.0 Logarithm. 1 cubic foot per ton of 2240 pounds = _ 0.89286 cubic foot per ton of 2000 pounds 1.950 7820 2240 (exact) pounds per cubic foot 3.350 2480 1 cubic foot per ton of 2000 pounds = 1.12 (exact) cubic feet per ton of 2240 pounds 0.049 2180 2000 (exact) pounds per cubic foot 3.301 0300 1 pound per cubic foot = 2240 (exact) cubic feet per ton of 2240 pounds 3.350 2480 2000 " " " 2000 " 3.301 0300 *From Edwin F. Smith, Sup't & Eng'r, Canal Div., Phila. and Reading R. R. fFrom very careful weighings in the Chicago yards of Coxe Bros. & Co. Note the irregular variation with size of anthracite in Coxe Bros,' figures. g Quoted from The Mining Record. On the authority of " many years' experi- ence" of "a prominent retail dealer in Philadelphia," the Journal gives also figures requiring from 4 to 13 per cent, less volume per ton than those here quoted from the Journal and from other authorities. 216 WEIGHTS AND MEASURES, WEIGHTS AND MEASURES. United States and Britisb measures of leng-tli and weigbt, of the same denomination, may, /or all ordinary purposes ^ be considered as equal ; but the liquid and dry measures of the same denomination differ widely in the two countries. Tlie standard measure of leng-tli of both coun- tries is theoretically that of a pendulum vibrating seconds at the level of the sea, in the latitude of London, in a vacuum, with Faiirenheit's thermometer at 62°. The length of such a pendulum is supposed lo be divided into 39.1393 equal parts, called inches; and 36 of these inches were adopted as the standard yard of both countries. But the Parliamentary standard having been destroyed Dy fire, in 1834, it was found to be impossible. to restore it by measurement of a pendulum. The present British Imperial yard, as determined, at a temperature of 62° Fahrenheit, by the standard preserved in the Houses of Parliament, is the standard of the United States Coast and Geodetic Survey, and is recognized as standard throughout the country and by the Departments of the Govern- ment, although not so declared by Act of Congress. The yard between the 27th and 63d inches of a scale made for the U. S. Coast Survey by Troughton, of Lon- don, in 1814, is found to be of this standard length when at a temperature of 59°.62 Fahrenheit ; but at 62° is too long by 0.00083 inch, or about 1 part in 43373, or 1 46 inch per mile, or 0.0277 inch in 100 feet. The Coast Survey now uses, for purposes of comparison, two measures pre- sented by the British Government in 1855, as copies of the Imperial standard, namely : " Bronze standard. No. 11 ;" of standard length at 62°. 25 Fahr. " Malleable iron standard, No. 57 ;" " " " 62° 10 " See Appendix No. 12, Report of U. S. Coast and Geodetic Survey for 1877. The legral standard of iveiffht of the United States is the Troy pound of the Mint at Philadelphia. This standard, containing 5760 grains, is an exact copy of the Imperial Troy pound of Oreat Britain. The avoirdupois or commercial pound of the United States, con- taining 7000 grains, and derived from the standard Troy pound of the Mint, is found to agree within one thousandth of a grain with the British avoirdupois pound. The U. S. Coast Survey therefore declares the weights of the two coun- tries identical. The Ton. In Revised Statutes of the United States, 2d Edition, 1878, Title XXXIV, Collection of Duties upon Imports, Chapter Six, Appraisal, says : "Sec. 2951, Wherever the word *ton' is used in this chapter, in reference to weight, it shall be construed as meaning twenty-hundredweight, each hundred- weight being one hundred and twelve pounds avoirdupois." This appears to be the only U, S. Government regulation on the subject. The ton of 2240 ft)s (often called a gross ton or long ton) is commonly used in buying and selling iron ore, pig iron, steel rails and other manufactured iron and steel. Coke and many other articles are bought and sold by the net ton or short ton of 2000 lbs. The bloom ton had 2464 ft)s, = 22*40 lbs -4- 2 hundredweight of 112 ft)s each ; and the pig iron ton had 2268 lbs, = 2240 lbs + » "sandage" of 28 tbs, or one "quarter," to allow for sand adhering to the pigs, but some furnace men allowed only 14 ft)s. In electric traction woi'k the ton means 2000 fts. As a measure, the ton, or tun, is defined as 252 gallons, as 40 cubic feet of round or rough timber or in ship measurement, or as 50 feet of hewn timber. 252 U. S. gallons of water weigh about 2100 lbs ; 252 Imperial gallons about 2500 ft)s ; 50 cub ft yellow pine about 2500 lbs. The metric system * ivas legalized in the United States in * The metric system, as compared with the English, has much the same advantages and disadvantages that our American decimal coinage has in comparison with the English monetary system of pounds, shillings and pence. It will enormously facili- tate all calculations, but, like all other improvements, it will necessarily cause some inconvenience while the change is being made. The metric system has also tMs fur- ther and very great advantage, that it bids fair to become universal among c^ilizeo rations. WEIGHTS AND MEASURES. 217 1866, but has not been made obligatory. The government has since furnished very exact metric standards to the several States. The use of the metric system has been permitted in Great Britain, beginning with August 6, 1897, and in Russia, beginning with 1900. Its use is now at least permissive in most civil- ized nations. The metric unit of lengftli is tbe metre, or meter^ which was Intended to be one ten-millionth f J of the earth's quadrant, i.e., of that portion of a meridian embraced between either pole and the equator. This length was measured, and a set of metrical standards of weight and measure were prepared in accordance with the result, and deposited among the archives ^f France at Paris (Mfetre des Archives. Kilogramme des Archives, etc.). It has since been discovered that errors occurred in the calculations for ascertaining the length of the quadrant ; but the standards nevertheless remain as originally prepared. Tbe metric measures of surface and of capacity are the squares and cubes of the meter and of its (decimal) fractions and multiples. Tlie metric unit of weig-lit is tlie g-ramme or gram, which is the weight of a milliliter or cubic centimeter * of pure water at its tempera- ture of maximum density, about 4.5° Centigrade or 40° Fahrenheit. By the concurrent action of the principal governments of the world, an In- ternational Bureau of Weiglits and Measures has been estab- lished, with its seat near Paris. It has prepared two ingots of pure platinum- iridium, from one of which a number of standard kilograms (1000 grams) havf been made, and from the other a number of standard meter bars, both derived from the standards of the Archives of Frauce. Of these copies, certain ones were selected as international standards, and the others were distributed to the different governments. Those sent to the United States are in the keeping of the U. S. Coast Survey. The determination of the equivalent of the meter in Eng-lish measure is a very difficult matter. The standard meter is measured /row end to end of a platinum bar and at the freezing point ; whereas the standard yard is measured between two lines drawn on a silver scale inlaid in a bronze bar, and at 62° Fahrenheit. The United States Coast Survey f adopts, as the length of the meter at 62° Fahrenheit, the value determined by Capt. A. R. Clarke and Col. Sir Henry James, at the office of the British Ordnance Survey, in 1866, viz. : 39.370432 inches {= 3.2808666 + feet = 1.0936222 + yards) ; but the lawful equivalent, established by Congress, is 39.37 inches (= 3.28083 feet = 1.093611 yards). This value is as accurate as any that can be deduced from existing data. The gram weighs, by Prof. W. H. Miller's determination,^ 15.43234874 grains. An examination made at the International Bureau of Weights and Measures in 1884 makes it 15.43235639 grains. The legal value in the United States is 15.432 grains. * 1 centimeter = y^^j meter = 0.3937 inch. 1 milliliter {yi^xj liter) or cubic centi^ meter = 0.061 + cubic inches, t Appendix No. 22 to report of 1876, page 6. j Philosophical Transactions, 1866, pp. 893, etc. 218 FOREIGN COINS. Approximate Talnes of Foreign Coins, in U. S. Money. The references (i, 2, 3^ and *) are to foot-notes on next page. From Circular of U. S. Treasury Department, Bureau of the Mint, Jan. 1, 1887; from "Question Monetaire," by H. Costes, Paris, 1884; and from our 10th edition. Argentine Repub. — Peso = 100 Centavos, 96.5 ctg.2 3 Argentino = 5 Pesos, $4.82. Austria.— Florin = 100 Kreutzer, 47.7 cts.,2 35.9 cts.3 Ducat, $2.29. Maria Theresa Thaler, or Levantin, 1780, $1.00.2 Rix Thaler, 97 cts.4 Souverain, $3.57.'* Belgium. 1— Franc = 100 centimes, 17.9 cts.,2 19.3 cts.3 Bolivia.— Boliviano = 100 Centavos, 96.5 cts.,2 72.7 cts.s Once, $14.95. Dollar, 96 cts.4 Brazil.— Mil reis = 1000 Reis, 50.2 cts.,2 54.6 cts.s Canada. — English and U. S. coins. Also Pound, $4.4 Central America.'*— Doubloon, $14.50 to $15.65. Reale, average 5% cts. See Honduras. Ceylon. — Rupee, same as India. Chili.— Peso = 10 Dineros or Decimos = 100 Centavos, 96.5 cts.,2 91,2 cts.* Con- dor = 2 Doubloons = 5 Escudos = 10 Pesos. Dollar, 93 cts.^ Cuba.— Peso, 93.2 cts.3 Doubloon, $5.02. Denmark.— Crown = 100 Ore, 25.7 ct8.,2 26.8 cts.3 Ducat, $1.81.4 Skilling, % ct.* Ecuador.— Sucre, 72.7 cts.^ Doubloon, $3.86. Condor, $9.65. Dollar, 93 cts.* Reale, 9 cts.* Egypt.— Pound -= 100 Piastres -= 4000 Paras, $4.94,3.3 Finland.— Markka = 100 Penni, 19.1 cts.2 10 Markkaa, $1.93. France.i— Franc =100 Centimes, 17.9 cts.,2 19.3 cts.3 Napoleon, $3.84.4 Livre, 18.5 cts.4 Sous, 1 ct.4 Germany.— Mark = 100 Pfennigs, 21.4 ct8.,2 23.8 cts.s Augustus (Saxony), $3.98.* Carolin (Bavaria), $4.93.4 Crown (Baden, Br.varia, N. Germany), $1.06.'^ Ducat (Hamburg, Hanover), $2.28.4 Florin (Prussia, Hanover), 55 cts.4 Groschen, 2.4 cts.4 Kreutzer (Prussia), .7 ct. Maximilian (Bavaria), $3.30.4 Rix Thaler (Hamburg, Hanover), $1,104 (Baden, Brunswick), $1,004 (Prussia, N. Germany, Bremen, Saxony, Hanover), 69 ct8.4 Great Britain.— Pound Sterling or Sovereign (£) = 20 Shillings = 240 Pence, $4.86.65.3 Guinea = 21 Shillings Crown = 5 Shillings. Shilling (s), 22.4 cts.,2 24.3 cts. (^ pound sterling). Penny (d), 2 cts. Greece.i— Drachma = 100 Lepta, 17 cts.,2 19.3 cts.3 Hay ti.— Gourde of 100 cents, 96.5 cts.2 3 Honduras.— Dollar or Piastre of 100 cents, $1.01. See Central America. India.— Rupee = 16 Annas, 45.9 cts.,2 345 cts.^ Mohur = 15 Rupees, $7.10. Star Pagoda (Madras), $1.81.4 Italy, etc.i— Lira = 100 Centesimi, 17.9 cts.,2 19.3 cts.s Carlin (Sardinia), $8.21.* Crown (Sicily), 96 cts.4 Livre (Sardinia), 18.5 cts.4 (Tuscany, Venice), 16 5ts.4 Ounce (Sicily), $2.50.4 Paolo (Rome), 10 cts.4 Pistola (Rome), $3.37.* Scudo* (Piedmont), $1.36 (Genoa), $1.28 (Rome), $1.00 (Naples, Sicily), 95 cts. (Sardinia), 92 cts. Teston (Rome), 30 cts.4 Zecchino (Rome), $2.27.4 Japan.— Yen = 100 Sen (gold), 99.7 cts.3 (silver), $1,042, 78.4 cts.3 Liberia.— Dollar, $1.00.3 4 Mexico.— Dollar, Peso, or Piastre = 100 Centavos (gold), 98.3 cts. (silver), $1.05,2 79 ct8.3 Once or Doubloon = 16 Pesos, $15.74. Netherlands.— Florin of 100 cents, 40.5 cts.,2 40.2 cts.a Ducatoon, $1.32.4 Guilder, " 40 cts.4 Rix Dollar, $1.05.4 Stiver, 2 cts.4 New Granada.— Doubloon, $15.34.4 Norway.— Crown = 100 Ore = 30 Skillings, 25.7 cts.,* 26.8 cts.* Paraguay. — Piastre = 8 Reals, 90 cts. Persia.— Thoman = 5 Sachib-Kerans = 10 Banabats = 25 Abassis = 100 Scahls, $2.29. Peru.— Sor= 10 Dineros = 100 Centavos, 96.5 cts.,2 72.7 cts.3 Dollar, 93 ct8.4 Portugal.— Milreis = 10 Testoons = 1000 Reis, $1.08.3 Crown = 10 Milreis. Moidore, $6.50.4 Russia.— Rouble = 2 Poltinniks = 4 Tchetvertaks = 5 Abassis = 10 Griviniks = 20 Pietaks = 100 Kopecks, 77 cts.,2 58.2 cts.3 Imperial =- 10 Roubles, $7.72. Ducat = 3 Roubles, $2.39. Sandwich Islands.— Dollar, $1.00.4 Sicily.— See Italy. Spain.— Peseta or Pistareen = 100 Centimes, 17.9 cts.,* 19.3 cts.s Doubloon (new) = 10 Escudos = 100 Reals, $5.02. Duro = 2 Escudos,4 $1.00.2 Doubloon (old), $15.65.4 Pistole = 2 Crowns, $3.90.* Piastre, $1.04.4 Reale Plate, 10 ct«.4 Beale vellon, 5 cts.* 1, 2, 3, 4. See ibot-aotes, next page. FOREIGN COINS. 219 (Foreigrn Coins Untinued. Small figures 0, 2, 3^ <) refer to foot notes.) Sweden.— Crown ^ 100 Ore, 25.7 cts.,2 26.8 cts.3 Ducat, $2.20.* Rix Dollar, 81.05 * Switzerland.!— Franc = 100 Centimes, 17.9 cts.,- 19.3 cts.3 Tripoli.— Mahbub = 20 Piastre s, 65.6 cts.3 Tunis.— Piastre = 16 Karobs, 12 cts.2 10 Piastres, Si. 16.6. Turkey.— Piastre = 40 Paras, 4.4 cts.3 Zecchin, ^1.40.4 United States of Colombia.— Peso = 10 Dineros or Decimos = 100 Centayos, 96.5 cts.,2 72.7 cts.3 Condor = 10 Pesos, S9.65. Dollar, 93.5 cts.* Uruguay.— Peso = 100 Centavos or Centesiraos (gold), $1.03 (silver), 96.5 cts.2 Venezuela. — Bolivar = 2 Decimos, 17.9 cts.,2 19.3 cts.3 Venezolano = 5 Bolivars. Sizes and Weights of United States Coins.* Gold, 10 per cent, alloy : Double eagle , . . $20 Eagle 10 Half eagle 5 Three dollars 3 Quarter eagle 2.50 Dollar ........ . . 1.00 Silver, 10 per cent, alloy : Trade dollar 100 cts. Standard dollar 100 " Half dollar 50 " Quarter dollar 25 " Twenty cents 20 " Dime 10 " Half dime 5 " Three cents 3 " Minor. 5 cents, 75^ copper, 25% nickel . . 2 *' 95% " 5 % tin and zinc. 1 " Diameter. Thickness. Legal weight of coin. Inch. Inch. Grains. Grams. 1.35 .077 516 33.436 1.05 .060 258 16.718 .85 .046 129 8.359 .8 .034 77.4 5.015 .75 .034 64.5 4.179 .55 .018 25.8 1.672 1.5 .082 420. 27.215 1.5 .080 412.5 26.729 1.2 .057 192.9 12.6 .95 .045 96.45 6.25 .875 .047 77.16 5. .7 .032 38.58 2.5 .6 .023 19.2 1.244 .55 .018 11.52 .746 .8 .062 77.16 5.0 .725 .034 30. 1.944 .9 .060 96. 6.22 .75 .043 48. 3.11 Perfectly pnre g-old is worth SI per 23.22 grs = $20.67183 per troy oz = S18.84151 per avoir oz. Standard (U. S. coin) is worth $18.60465 per troy oz = $16.95736 per avoir oz. It consists of 9 parts by weight of pure gold, to 1 part alloy. Its value is that of the pure gold only ; the cost of the alloy and of the coinage being borne by Government. A cubic foot of pure g'oid ireighs about 1204 avoir 5)s; and is- worth $362963. A cubic inch weighs about 11.148 avoir oz ; and is worth $210.04. Pure gold is called fine, or 24 carat gold ; and when alloyed, the alloy is sup- posed to be divided into 24 parts by weight, and according as 10, 15, or 20, &c, of these parts are pure gold, the alloy is said to be 10, 15, or 20, &c, carat. The averag-e fineness of California native grold, by some thou- sands of assays at the U. S. Mint in Philada., is 88.5 parts gold, 11.5 silver. Some from Georgia, 99 per cent. gold. Pure silver fluctuates in value: thus, during 1878-1879 it ranged between $1.05 and $1.18 per troy oz., or $.957 and $1,076 per avoir, oz. A cubic inch weighs about 5.528 troy, or 6.065 avoir, ounces. 1 France Belgium, Italy, Switzerland, and Greece form the Latin Union. Their coins are alike in diameter, weight, and fineness. 2 = 19 3 times the value of a single coin in francs as given by Costes. 3 Par of exchange, or equivalent value iu terms of U. S. gold dollar.— Treasury Circular. * From our 10th edition. * Thirteenth Anuual Report of the Director of the Mint. 1885 ; pp. 106 and 148. 220 WEIGHTS AND MEASURES. Troy Weight. U. S. and British. 24 grains 1 pennyweight, dwt. 20 pennyweights 1 ounce ~ 480 grains. 12 ounces 1 pound = 240 dwts. = 5760 grains. Troy weigrlit Is used for g-old and silver. A carat of the jewellers, for precious stones is, in the U. S. = 3.2 grs. ; in London, 3.17 grs. ; in Paris, 3.18 grains., divided into 4 jewellers' grs. In troy, apothecaries' and avoirdupois, the grain is the same. Apothecaries' liVeight. U. S. and British. 20 grains 1 scruple. 3 scruples 1 dram = 60 grains. 8 drams 1 ounce = 24 scruples = 480 grains. 12 ounces 1 pound = 96 drams = 288 scruples = 5760 grains. In troy and apothecaries' weights, the grain, ounce and pound are the same. Avoirdupois or Commercial liVeight. U. S. and British. 27.34375 grains - 1 dram. 16 drams 1 ounce = 437^^ grains. 16 ounces 1 pound = 256 drams = 7000 grains. 28 pounds 1 quarter = 448 ounces. 4 quarters 1 hundredweight = 112 lbs. 20 hundredweights 1 ton = 80 quarters = 2240 lbs. A stone = 14 pounds. A quintal = 100 pounds avoir. The standard of the avoirdupois pound, which is the one in common commercial use, is the weight of 27.7015 cub ins of pure distilled water» at its maximum density at about 39°.2 Fahr, in latitude of London, at the level of the sea ; barometer at 30 ins. But this involves an error of about 1 part in 1362, for the lib of water = 27.68122 cub ins. A troy lb = .82286 avoir lb. An avoir ft) = 1.21528 troy ft), or apoth. A troy oz. = 1.09714 avoir, oz. An avoir, oz. = .911458 troy oz., or apoth. liOng Measure. U. S. and British. 12 inches 1 foot = .3047973 metre. 3 feet 1 yard = 36 ins = .9143919 metre. 514 vards 1 rod, pole, or perch = 16^ feet = 198 ins. 40 rods 1 furlong = 220 yards = 660 feet." 8 furlongs 1 statute, or land mile = 320 rods = 1760 yds =[5280 ft = 63360 ins. 3 miles 1 league = 24 furlongs = 960 rods = 5280 yds = 15840 ft. A point = 7^5 inch. A line = 6 points = yV ii^ch. A palm = 3 ins. A hand = 4 ins. A span = 9 ins. A fathom = 6 feet. A cable's length = 120 fathoms = 720 feet. A Ounter's surveying chain is 66 feet, or 4 rods long. It has 100 links, 7.92 inches long. 80 Gunter's chains = 1 mile. A nautical mile, geographical mile, sea mile, or knot, is variously defined as being = the length of 1 min of longitude at the equator 1 " latitude " " 1 " " " pole 1 ♦' " atlat45° 1 "a great circle of a true') sphere whose surface area is > equal to that of the earth J British Admiralty knot The above lengths of minutes, in metres and feet, are those published by the U. 8. Soast and Geodetic Survey in Appendix No 12, Keport for 1881, and are calculated from Clarke's spheroid, which is now the standard of that Survey. At the equator 1° of lat = 68.70 land miles ; at lat 20° = 68.78 ; at 40° = 69.00 ; at 60° - 69.23 ; »t 80° = 69.89 ; at 90° = 69.41. metres feet statute miles = 1855.345 6087.15 1.15287 = 1842.787 6045.95 1.14607 = 1861.665 6107.85 1.15679 = 1852.181 6076.76 1.15090 ( value adopted by U. . S. Coast =^ and Oeodetic Survey 1 1853.248 6080.27 1.15157 = 1853.169 6080.00 1.15152 WEIGHTS AND MEASURES. liengrths of a Begree of L.on^itiic1e in different liatitndefi, And at the level of tlie Siea. These lengths are iu common land or statute miles, of 5280 ft. Since the figure of the earth has never been precisely ascertained, these are but close ap proximations. Intermediate ones may be found correctly by simple proportion. 1° of longitude ••rresponds to i mins of civil or clock time ; 1 min of longitude to 4 sees of lime. Degof Lat. Miles. Degof Lat. Miles. Degof Lat. MileB. Degof Lat. Miles. Degof Lat. Miles. Degof Lat. Miles. 69.16 14 67.1-2 28 61.11 42 51.47 56 38.76 70 23.72 2 69.12 16 66.50 30 59.94 44 49.83 58 36.74 72 21.43 4 68.99 18 65.80 32 58.70 46 48.12 60 34.67 74 19.12 6 68.78 20 65.02 34 57.39 48 46.36 62 32.55 76 16.78 8 68.49 •22 64.15 36 56.01 50 44.54 64 30.40 78 14.42 10 68.12 24 63.21 38 54.56 52 42.67 66 28.21 80 12.05 12 «7.66 26 62.20 40 53.05 54 40.74 68 26.98 82 9.66 Indies reduced to Decimals of a Foot. No errors. Ins. Foot. Ins. Foot. Ins. Foot. Ins. Foot. Ins. Foot. Ins. Foot. o .0000 2 .1667 4 .3333 6 .5000 8 .6667 10 ,8333 1-32 .00-26 .1693 .3359 .5026 .6693 ,8359 1-16 .0052 .1719 .3385 .5052 .6719 ,8385 3-32 .0078 .1745 .3411 .5078 .6745 ,8411 c,^ .0104 H .1771 H .3438 \i .5104 H .6771 ^ .8438 5-32 .0130 .1797 .3464 .5130 .6797 ,8464 3-16 .0156 .1823 .3490 .5156 .6823 .8490 7-32 .0182 .1849 .3516 .5182 .6849 .8516 H .0208 H .1875 34 .3542 34 .5208 H .6875 y*. .8542 9-32 .0234 .1901 .3568 .5234 .6901 .8568 5-16 .0260 .1927 .3594 .5260 .6927 ,8594 11-32 .0286 .1953 .3620 .5286 .6953 ,8«20 H .0313 % .1979 % .3646 % .5313 % .6979 Va .8646 13-32 .0339 .'2005 .3672 .5339 .7005 - ,8672 7-16 .0365 .2031 .3698 .5365 .7031 .8698 16-32 .0391 .2057 .3724 .5391 .7057 ,8724 H .0417 % .2083 }4 .3750 14 .5417 H .7083 H .8750 17-32 .0443 .2109 .3776 .5443 .7109 ,8776 9-16 .0469 ,2135 .3802 .5469 .7135 .8802 19-32 .0495 .2161 .3828 .5495 .7161 .8828 % .0521 % .2188 % .3854 H .5521 % .7188 H .8854 21-32 .0547 .2214 .3880 .5547 .7214 ,8880 11-16 .0573 .2240 .3906 .5573 .7240 ,8906 23-32 .0599 .2-266 .3932 ,5599 .7266 ,8932 H .0625 H .2292 H .3958 H ,5625 h .7292 H .8958 25-32 .0651 .'2318 ,3984 ,6651 .7318 ,8984 13-16 .0677 .2344 .4010 .5677 .7344 .9010 27-32 .0703 .'2370 .4036 .5703 .7370 ,9036 H .0729 Ji .'2396 }i .4063 H .57'29 % .7396 % .9063 29-32 .0755 .2422 .4089 .5755 .74-22 ,9089 15-16 .0781 .2448 .4115 .5781 .7448 .9116 31-32 .0807 .2474 .4141 .5807 .7474 .9141 1 .0833 S .-2500 5 .4167 7 .5833 9 .7500 11 .9167 1-32 .0859 .2526 .4193 ,5859 .7526 .9193 1-16 .0885 .2552 .4219 .5885 .7552 .9219 3-32 .0911 .2578 " .4245 .5911 .7578 .9246 5-M .0938 H .2604 H .4-271 H .5938 H .7604 H .9271 .0964 .2630 .4297 .5964 .7630 .9297 3-16 .0990 .'2656 .43-23 .6990 .7656 .9823 7-32 .1016 .2682 .4349 .6016 .7682 ,9349 }4 .1042 % .2708 H ,4375 H .6043 H ,7708 yi ,9376 9-32 .1068 .2734 .4401 .6068 .7734 .9401 5-16 .1094 .2760 .4427 .6094 .7760 .9427 11-32 .1120 .2786 .4453 .6120 .7786 ,9463 % .1146 % .2813 % ,4479 % .6146 % .7813 % ,9479 13-.(2 .1172 .2839 .4505 .6172 .7839 .9605 7-16 .1198 .2865 .4531 .6198 .7865 .9531 15-32 .1224 .2891 .4557 .6224 ,7891 ,9557 3^ .1250 % .2917 H .4583 H .6250 ^ .7917 M .9583 17-32 .1276 .2943 .4609 .6276 .7943 .9609 9-16 .1302 .2969 .4635 .6302 .7969 .9635 19-32 .1328 .2995 .4661 .6328 .7995 .9661 % .1354 H .3021 % .4688 H ,6354 % .8021 % .9688 21-3-2 .1380 .3047 .4714 .6380 .8047 .9714 11-16 .1406 .3073 .4740 .6406 .8073 .9740 23-32 .1432 .3099 .4766 .6432 .8099 .9766 H .1458 H .3125 H .4792 H .6458 H ,8125 . % .9792 25-3-2 .1484 .3151 .4818 .6484 ,8151 .9818 13-16 .1510 .3177 .4844 .6510 ,8177 .9844 27-32 .1536 .3203 .4870 ,6536 .8203 .9870 29-32 .1563 % .3229 % .4896 % ,6563 K .8229 K .9896 .1589 .3255 .4922 .6589 .8255 ,9922 15-16 .1615 .3-281 .4948 .6615 .8281 .9948 81-32 .1641 .3307 .4974 .6641 .8307 .9974 222 WEIGHTS AND MEASURES. Square, or liaiid Pleasure. U. S. and British. 144 square inches 1 sq foot. 100 sq ft = 1 square. 9 sq feet 1 sq yard = 1296 aq ins. 30J4 sq yards 1 sq rod = 272}^ sq feet. 40 sq rods 1 rood = 1210 sq yds =r 10890 sq feet. . 4 roods 1 acre ~ 160 rods = 4840 sq yds = 43560 sq feet. A section of land is 1 mile sq, or 27878400 sq ft ; or 3097600 sq yds ; or 640 acres. An acr« contains 10 sq Gunter's chains. A sq acre is 208.710 feet; a sq half acre, 147.581 ft; and a sq quarter acre, 104.355 ft on each side. A circular acre is 235.504 feet: a circular half acre = 166.527 ft; and a circular quarter acre = 117.752 ft diaui. A circular inch is a circle of 1 inch diani; a sq ft = 183.346 cir ins. Also 1 sq inch = 1.27324 cir ins ; and 1 cir inch — .7854 of a sq inch. Cubic, or $$olicl Measure. U. S. and British. 1728 cubic inches 1 cubic, or solid foot. 27 cubic feet 1 cubic, or solid yard. A cord of wood = 128 cub ft ; being 4 ft X 4 ft X 8 ft. A perch of masonry actually con- taiDS 24?^ cub ft; being 16}^ ft X 134 ft X 1 ft. It is generally taken at 25 cub ft; but by some at 22, &c ; and there is every probability that a payer will be cheated unless the number of cubic ft be dis- tinctly agreed upon in his contract. It is gradually falling into disuse among engineers ; and the cub yd is very properly taking its place. To reduce cub yds to perches of 25 cub ft, mult by 1.080; and to reduce perches to cub yds, mult by .926. The Brit rod of brickwork, of house-builders, is 16K feet square, by 14 inches'ci)^ English bricks; thick r: 272>4 sq ft of 14 inch wall. It is conven- tionally taken at 272 sq ft j which gives 317}^ cub ft. In Brit engineering works the rod is 306 cub ft, or UH cub yds. The Montreal, (Canada,) toise = 261>^ cub ft; or 9.6852 cub yds, or 10.46 perches of 25 cub ft. The Canadian chaldron = 58.64 cub ft. A ton (2240 fts) of Pennsylvania anthracite, when broken for domestic use, occupies from 11 to 43 cub ft of space; the mean of which is equal to 1.556 cub yds: or a cube of 3.476 ft on each edge. Bituminous coal 44 to 43 cub ft; meaa equal to 1.704 cub yd ; or a cube of 3.583 ft on each edge. Coke 80 cub ft. A cubic foot is equal to 1728 cub ioB, or 3300.23 spherical ins. .037037 cub yard, or 1.90985 spherical fU .002832 myriolitre, or decastere. r Htere. .028316 kilolitre, or cubic metre, o .283161 hectolitre, or decistere. 2.83161 decalitres, or centisteres. 28.3161 litres, or cub decimetres. 283.161 decilitres. 2831.61 centilitres. 28316.1 millilitres, or cub centimetres. .803564 U.S. struck bushel of 2150.42 cub ins, or 1.24445 cub ft. .779013 Brit bushel of 2218.191 cub ins, or 1.28368 cub ft. 3.21426 U. S. peeks. A cubic iuch is equal to 1^38«63 millilitres; or 1.638663 centilitres; or .1638663 decilitre; or .01638663 litre; or to .0005783 f«b ti; or to .138528 U. S. gill; or 1.90985 spherical ins. A cubic yard is equal to 76.4534 decalitres. 764.534 litre.s, or cub decimetres. 7645.34 decilitres. 3.11605 Brit pecks. 7.48052 U. S. liquid galls of 231 cub ins. 6.42851 U. S. dry galls. 6.23210 Brit galls of 277.274 cub ins. 29.92208 U. S. liquid quarts. 25.71405 U. S. drv quarts. 24.92842 Brit quarts. 59.84416 U. S. liquid pints. 51.42809 U. S. dry pints. 49.85684 Brit pints. 239.37662 U. S. gills. 199.42737 Brit gills. .26667 flour barrel of 3 struck bushels. .23748 U. S. liquid barrel of 313^ galls. n eub feet, or to 201.974 U. S. galls. 46656 cub ins. .0764534 myriolitre. .764534 kilolitre, or cub metre. 7.64534 hectolitres. f.2 flour barrels of 3 struck bushels. 21.69623 U. S. bushels (struck). 21.03336 Brit bushels. A spliere 1 foot in diameter, contains .01939 cub yard. .5236 cub foot. 904.781 cub inches. .42075 U. S. bushel. 1.6830 U. S. pecks. 13.4639 U. S. drv quarts. 26.9278 U. S. drV pints. 3.9168 U. S. liquid gallons, 11.6672 U. S. liquid quarts. A sphere 1 incb .000303 cub foot. .5236 cub inch. .07263 U. S. gill. 31.3344 U. S. liquid pints. 125.3376 U. S. liquid gills. 3.2631 Brit imp gallons. 13.0625 Brit imp quarts. 26.1050 Brit imp pints. 104.4201 Brit imp gills. 14.826;5 litres. 1.48263 decalitres. .148263 hectolitres. in diameter, contains .06043 Brit gill. 8.580 mlllilitre. .8580 centilitre. .08580 deciUtr*. WEIGHTS AND MEASUBES. 223 A cylinder 1 foot in diameter, and 1 foot bigrli, contains .02909 cub yard. .7854 cub foot. i357. 1712 cub inches. .63112 U. S. dry bushels. 2.5245 U. S. dry pecks. 20.1958 U. S. dry quarts. 40.3916 U. S. dry pints. 5.8752 U. S. liquid gallons. 23.5008 U. S. liquid quarts. 47.0016 U. S. liquid pints. 188.0064 U. S. liquid gills. 4.8947 Brit imp gallons. 19.5788 Brit imp quarts. 39.1575 Brit imp pints. 156.6302 Brit imp gills. 222.395 decilitres. 22.2395 litres. 2.22395 decalitres. .222395 hectolitre. A cylinder 1 inch in diameter, and 1 foot higb, contains .005454 cub foot. 9.4248 cub inches. .2805 U. S. dry pint. .3264 liquid pint. 1.3056 U. S. gill. .2719 Brit imp pint. 1.0877 Brit imp gill. 15.4441 centilitres. 1.54441 decilitres. .154441 litres. liiqnid Measure, v. S. only. The 1>asi8 of this measure in the U. S. is the old Brit wine gallon of 231 cub ins ; or 8.33888 B>« avoir of pure water, at its max density of about 39°. 2 Fahr ; the barom at 30 ins. A cylinder 7 ins diam, and 6 ins high, contains 230.904' cub ins, or almost precisely a gallon ; as does also a cube of 6.1358 ins on fn edge. Also a gallon — .13.308 of a cnh ft ; and a cub ft contains 7.48052 galls ; nearly 7 Ji galls. This basis however involves an error of about 1 part in 1362, for the water actu- ally weighs 8.34o008 lbs. cub ins. 4 gills 1 pint =28.875. 63 gallons 1 hogshead. 2pints 1 quart = 57.750 = 8 gills. 2 hogsheads 1 pipe, or butt. 4 quarts 1 gallon =: 231. =8 pints ~ 32 gills. 2 pipes 1 tun. In the U. S. and Great Brit. 1 barrel of wine or brandy = 31}^ galls : in Pennsylvania, a half barrel, 16 galls; a double barrel, 64 galls; a puncheon, 84 galls; a tierce, 42 galls. A liquid measure barrel of 31^^ galls contains 4.211 cub ft = a cube of 1.615 ft on an edge; or 3.38i U. S. struck bushels. A still = 7.21H75 cub ins. The following cylinders contain some o." these measures rery approximately. Diam. cub ins. Ins. Gill v7. 21875) 1% J^pint 2% Pint SVjj' Quart 33^ Height. Ins, Diam. Ins. 7 .. Gallon 3^ 2 gallons 8 gallons 14 10 gallons 14 Apothecaries' or "Wine Measure. Height. Ins. Symbol. Pints. Fluid ounces. Fluid drachms. Minims. Cubic inches. Weight of water.J Pounds, av. Grains. 1 Gallon 1 Pint Cong* ot m 8 128 16 1024 128 8 61440 7680 480 60 1 231 28.875 1.8047 0.2256 0.0038 8.345 1.043 Ounces, av. 1.043 68415 7301.9 1 Fluid ounce ... 1 Fluid drachm.. IMinira 456.4 67.05 0.91 To reduce U. S. liquid measures to Brit ones of the same denomina- tion, divide by 1.20032; or near enough for common use, by 1.2; or to reduce Brit to U. S. multiply by 1.2. I>ry Measure, U. 8. only. The basis of this is the old British Winchester struck bushel of 2150.42 cub ins; or 77.627413 pounds avoir of pure water at its max density. Its dimensions by law are 181.^ ins inner diam ; 19}^ ins outer diam ; and 8 ins deep ; and when heaped, the cone is not to be less than < ins high ; which makes a heaped bushel equal to 1J4 struck ones : or to 1.55556 cub tt. Edge of a cube of • equal capacity. 2 pints 1 quart, = 67.2006 cub ins := 1.16365 liquid qt 4.066 ms. 4 quarts 1 gallon, = 8 pints, = 268.8025 cub ins, =r 1.16365 liq gal 6.454 " 2 gallons 1 peck, rr 16 pints, = 8 quarts. = 537.6050 cub ins 8.131 " 4 pecks 1 struck bushel, = 64 pmts, = 32 quarts. = 8 gals, =r 2150.4200 cub ins. 12.908 •« * Abbreviation of Latin, Congius. t Abbreviation of Latin, Octariu^. t At its maximum density, 62,425 pounds per cubic foot, corresponding to a temperature of 4° Centigrade = 39.2° p^ahrenheit. 224 WEIGHTS AND MEASURES. A struck bushel = 1.24445 cub ft. A cub ft = .80356 of a struck bushel The dry flour barrel = 3.75 cub ft; = 3 struck bushels. The dry barrel h not, however, a legalized measure; and no great attention is given to its capacity; consequently barrels varj considerably. A barrel of flour contains by law, 196 H>s. In ordering by the barrel, th< amount of its contents should be specified in pounds or galls. To reduce U. S. dry measures to Brit imp ones of the same name, dii by 1.031516; and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposes use 1.0S2. British Imperial Measure, both liquid and dry. This system is established throughout Great Britain, to the exclusion of the old ones. Its basis ii the imperial gallon of 277.274 cub ins, or 10 fts avoir of pure water at the temp of 62° Fahr, whei the barom is at 30 ins. This basis involves an error of about 1 part in 1836, for 10 as of the water = only 277.123 cub ins. igills 1 pint 1.25 2 pints 1 quart 2.50 3 quarts 1 pottle 5. 2 pottles 1 gallon 2 gallons 1 peck ♦ peeks 1 bushel 80. 1 Dry 4 bushelsl coomb 320. j meas. 2 coombs 1 quarter 1640. Avoir lbs. of water. Cub. ins. 34.6592 69.3185 138.637 277.274 554.548 2218.192 8872.768 17745.586 Cub. ft. 1.2837 5.1347 10.2694 Edge of a cube ol equal capacity. Inches. 3.2605 4.1079 5.1756 6.5208 8.2157 13.0417 The imp gall = .16046 cub ft; and 1 cub (1=6.23310 galls. Measure. Symbol. Pints. Fluid ounces. Fluid drachms. Minims. Cubic inches. Weight of water .J Pounds, av. Oraios. 1 Gallon 1 Pint c* or fl. oz. fl.dr. 8 1 160 20 1 1280 160 8 1 76800 9600 480 60 1 277.274 36.659 1.733 0.217 0036 10 1.25 Ounces, av. 1 70000 8750 1 Fluid ounce ... I Fluid drachm.. 1 Minim 437.5 54.6875 9114 The weight of water affords an easy way to find the cubic contents of a vessel. To obtain the size of commercial measures by means of the weig^ht of water. At the common temperature of from 70° to 75° Fah, a cub foot of fresh water weighs very approxl mately 623^ lbs avoir. A cubic half foot, (6 ins on each edge,) 7.78125 H)s. A cub quarter foot, (3 ini on each edge,) .97266 lb. A cub yard, 1680.75 B)s ; or .75034 ton. A cub half yd, (18 ins on each edge,] 210.094 tts; or .0938 ton. A cub inch, .036024 lb ; or .576384 ounce ; or 9. 2222 drams ; or 252.170 grains, An inch square, and one foot long, .432292 S). Also 1 9> = 27.75903 cub ins, or a cube of 3.028 ins on an edge. An ounce, 1.735 cub ins ; a ton, 35.984 cub ft, all near enough for common use. Original. liiquid Measures. Lbs Avoir, of Water. U. S. Gill 26005* U. S. Pint 1.0402 U. S. Quart 2.0804 U. S. Gallon 8 fts 5^ oz 8.3216 U. S. Wme Barrel, 31 Ji Gall 262.1310 JDry Measures. tJ. S. Pint 1.2104 tJ. S. Quart 2.4208 U. S. Gallon 9.6834 U. S. Peck 19.3668 U. S. Bushel, struck 77.4670 * Or 4 ounces ; 2 drams ; 15.6625 grs. liiquid and Dry. Lbs Avoir, •'of Water. British Imp Gill 31214» " Pint 1.24858 " " Quart 2.49715 " " Gallon 9.9886 " " Peck 19.9772 " " Bushel 79.9088 * 4.9942 ; or very nearly 5 ounces. Metric Measures. Centilitre 021981 Decilitre .2198J Litre 2.1981 Decalitre, or Centistere 21.9808 Metre, or Stere 2198.0786 t Or 5.6271 drams ; or 153.866 gr». } 3.5169 ounces. ♦ Abbreviation of Latin, Congius. t Abbreviation of Latin, Octarius. t At the standard temperature, 62° Fahrenheit = about 16.7° Centigrade. WEIGHTS AND MEASURES. 225 ]^Ietric Measures of lieng-tb. By U. 8. and British Standard. Ins. Ft. Yds. Miles. Millimetre* .039370 .39370428 3 9370428 .003281 .032809 .3280869 3.280869 32.80869 328.0869 3280.869 32808.69 Centimetref Decimetre .1093623 1.093623 10.93623 109.3623 1093.623 10936.23 MetreJ 39.370428 393.70428 Road measures. Decametre "I Hectometre .0621375 .6213760 6.213750 Kilometre [ Myriametre J * Nearly the -^-^ part of an inch. t Full % inch. X YtTj nearly 3 ft, 3% ins, which is too long by only 1 part in 8616. Metric Square Measure. By U. S. and British Standard. Sq. Ins. Sq. Feet. Sq. Yds. Acres. Sq Millimetre .001550 .155003 15.5003 1550.03 155003 .00001076 .00107641 .10764101 10.764101 1076.4101 10764.101 107641.01 10764101 .0000012 .0001196 .0119601 1.19601 119.6011 1196.011 11960.11 1196011. Sq Centimetre Sq Decimetre Sq Metre, or Centiare Sq Decametre, or Are Decare (not used) .000247 .024711 .247110 2 47110 Hectare Sq Kilometre .3861090 sq miles. 38.61090 " 247 110 8q Myriametre 24711.0 Metric Cubic or Solid MLeasure. Aocordinc to U. S. Standard. Only those marked " Brit" are British. Millilitre, or cub Centimetre- Centilitre. Decilitre .. Litre, or cubic Decimetre .. Decalitre, Centistere.. Hectolitre, Decistere.. Kilolitre, or Cubic Metre, or Stere Myriolitre, or Decastere Cub Ins. .0610254 .610254 61.0254 610.254 Cub Ft. .353156 . 3.53156 35.3156 {Liquid. Dry. ( Liquid. iDry. ■ ( Liquid. (Dry. {Liquid. Dry. r Liquid. iDry. {Liquid. Dry. I Liquid. (Dry. ( Liquid. (Dry- .0084537 gill. .0070428 Brit gill. .0018162 dry pint. .084537 gill. .070428 Brit gill. .018162 dry pint. .84537 gill = .21134 pint. .70428 Brit gill = .17607 Brit pint. .18162 dry pint. 1.05671 quart = 2.1134 pints. .88036 Brit quart = 1.7607 Brit pints. .11351 peck = .9081 dry qt = 1.8162 dry pt. 2.64179 U. S. liquid gal. 2.20090 Brit gal. .283783 bush = 1.1351 peck ^ 9.081 dry qts. 26.4179 U. S. liquid gal. 22.0090 Brit gal. 2.83783 bush. 264.179U.S. liquid gal.) 220.090 Brit gal. yf»h yds, 1.3080. 28.3783 bush. j 2641.79 U. S. liquid gal. 283.783 bush. } Cub yds, 13.060. 15 226 WEIGHTS AND 301ASUBES. Metric Weisrhts, reduced to eommoii Commercial or ATOii Wel^bt, of 1 pound = 16 ounces, or 7000 g^rains. Milligramme Centigramme Decigramme Gramme By law a 5-cent nickel = 5 grammes- Decagramme Hectogramme Kilogramme Myriogramme Quintal* Tonneau; Millier; or Tonne Grains. .015432 .15432 1.5432 15.432 Pounds av. .022046 .22046 2.2046 22.046 220.46 2204.6 The gramme !■ the basis of French weights ; and la the weight of a cub centimetre of distillal Vater at Its max density, at sea level, in lat of Paris ; barom 29.922 ins. Frencli Measures of the " Systeme Usuel." This system was in use from about 1812 to 1840, when it was forbidden by law to use even its names, This was done in order to expedite the general use of the tables which we'have before given. But a£ the Systeme Usuel appears in books published during the above interval, we add a table of some of ita Measures of liCng^th. Yards. Feet. Inches. .09113 .09113 1.09362 3.93708 6.56181 1.09362 .36454 1.31236 2.18727 13.12344 Aune nanel, or ell <, Toi8«nBuel,=6pieds 47.245 78.74172 Weights, TJeuel. Cubic, or Solid, TTsuel. .8375 grains. 60.297 " 1.10258 avoir OS. .66129 avoir lb. 1.10258 avoir lb. Litron usuel, or 1 litre Bolsseaa usuel = 1.7608 British pint. 2.7512 British gals. Livre usuel, > or pound, J " Before 1812, or before the "Systeme usuel," the Old System, •' Systeme Ancien," was in use. Frencb Measures of tbe *' Systeme Ancien.'' LlneaU SquaM. Cubic. Point ancien, .0148 ins Ligne ancien, .0888 ins Sq. ins. .00789 1.1359 Sq.ft. 'i."i359 40.8908 Sq. yds. 4.5434 C. ins. .0007 1.2106 C. ft. 1.2106 261.482 C.yds. Pouce ancien, 1.06577 ins — .0888 ft Pied ancien 12 7892 ins — 1 06577 ft Aune ancien, 46.8939 ins =3.90782 ft= 1.30261 yds Toise ancien "~ 6 3946 ft — 2 1315 yds 9.684S There is, however, much confusion about these old measures. Different measures had the j lame in difereat provinces. • Tke av9ir4ujfoi$ quintal is 100 avoirdupois pwamds. WEIGHTS AND MEASURES. Rassian. 227 Foot ; same as U. S. or British foot. Sactiine = 7 feet. Terst- — 50f sachine = 3500 feet = 1166^ yards = .6629 mile. Pood = 36.114 lbs avoirdupois, Spanish. Tlie castellano of Spain and New Granada, for weighing gold, is variously estimated, from 71.07 to 71.04 grains. At 71.055 grains, (the mean between the two,) an avoirdupois, or common commercial ounce contains 6.1572 castellano; and a ft) avoirdupois contains 98.515. Also a troy ounce = 6.7553 castellano ; ana a troy ft) = 81.064 castellano. Three U. S. gold dollars weigh about 1.1 castellano. Tbe ISpanisli mark, or marco, for precious metals, in South America, may be taKen in practice, as .5065 of a ft) avoirdupois. In Spain, .5076 ft). In other parts of Europe, it has a great number of values; most of them, however, being between .5 and .54 of a pound avoirdupois. The .5065 of a ft) = 3545^ grains ; and .5076 ft) = 3553.2 grains. 1 marco = 50 castellanos = 400 tomine =« 4800 Spanish gold-grsAns. The arroba has various values in different parts of Spain. That of Cas- tile, or Madrid, is 25.4025 ft)s avoirdupois; the tonelada of Castile = 2032.2 ft)s avoirdupois; the quintal = 101.61 ft)s avoirdupois; the libra = 1.0161 ft)s avoirdupois; the cantara of wine, &c, of Castile = 4.263 U. S. gallons; that of Havana = 4.1 gallons. The vara of Castile = 32.8748 inches, or almost preciselv 32'^ inches; or 2 feet %% inches. The faneg^ada of land since 1801 = 1.5871 acres = 69134.08 square feet. The fane^a of corn, &c = 1.59914 U. S. struck bushels. In California, the vara by law = 33.372 U. S. inches ; and (he le§^aa = 5000 taras; or 2.6335 U. S. mile*. 228 CONVERSION TABLES. S S ^ D 1^ O ?2>^S®l of? ^ o g p. O o >. = *-• o o-C bObO o o ^'^ rf Z, cS'-ip: '= 0) o) ^Op, ^-5 X co5^ii^-.£:o S 00 cole's ^ S '^^D ^ S CO 2 U)^ bij^ a 5 S C v^ 9^ E J 19 ^ o 'So o O c ;-H t* in a Jo H o |i a 'J c8 2 as ,0 ^ ' ':3 i ^ • cS ft?^ a ^^ u *^S fl .•^ SCN 0) P fa cS — II aR o S tl}^ ^ s? a o 5 > (M -1— b^" «8 > gj; «2 2 -*« d*5 a .. :S o a» CO oj 3 CI -r C ^ O 'So C ^ -§5 . cJ-O -r; +^ 03 t: C n O C O 03 .2 >^ te- 'O o ■-=< |§a2§a^> CI to ITS g d I c3 a E 13 S eS 4:^ CO O^ 03 C3 ^^^'O - 3-5 ^^03bcaa 03 "5b~ "03 o .ti 3 -J -Cl 03 ?,'^^ JS^-^^ t_( ir C3 o 'w S.t_i -^ ^ 2* CONVEESION TABLES. 229 H-^U5«Ot^l>QOQOOSOJOT-li-l^(^^(Nc^cs^c^<^^(^^(N(^^(^^c^c5(^^c^^(^^N(^^<^^(^^(^^<^^ B S 8 c I I CD S r- -- -T? O -i- ) 1 = s S s ^^-3^ ^ o o s a a a 2 |'g,^||'§o'g a a <^ Q $ ^ >_' *-Qj s«2tS«2 : 3 a> 2 o) S <^ O^fC o^3 QJ « a; o) , , , -So ^ ►>. ^^ ^^ ^ ^>^ t- fc- t_ _, S o S 5? ^2 L^^ a. « 0) ^ ^?)f|||| ; q3 q; o 'o "o 'S "'C!« aoS,= -^^^ o p„H^ OS o S — ^ cTea >^ 3 ^r >> -d ^ g _ r =2 2 3 o 5 _, I c = 2 ^ ■♦^ -^ ^ « « o rt S) S H g -"^ ^ -S ^ f '0 1^ ST.^ a-^ ^^^^ ft«! g bc 11 I, ©ft -0"^ 0't:?<2 cJC' Ijtl is ill Jli^ .2g-|5.|ft?|| fa © Oi-i ^ (M CO rH CO CO CO CO sc :o ^ C;>a-g I ft.^ bCbciiO- Ot-4 ftCl- •rt^-eSoP'XJOPfl a^ -tJ O OQ O OJ rt ° - ■■a W),^ -^ -■ a> c t, I c» CONVERSION TABLES. —. bOC s n: .§ .2 o arts « o o Ef .- .S .S ce K c; c - c B S'- CIS a> oi a> o) oj oj a; a> r- o 1-1 T-H c» QO H O* O th (m' (m' 5 s !!i^-i .Sb^i:? ^ ;- ^^^Z ^-S 9 w ^3 ^^ ^5^^ = o tc cc o o> ?o t~ t^ o CO «r5 1^ ^ «o IC lO T-< to U5 «0 O 00 GO t^ O^ iO (N CC o d i-J th " fl ? 8 8 En : «^ : ^ S '■ -'^ r a fc- s 2 r? i3 abi)§§.2t8ftll ^'i ^ tH ^ T-l tH rH tH bO f+tr P.?; Srrj ©CO I- St_3 -*^ = '^ o O w ^^r S'^ t^fe ^ ^ S^i^ oO- a> jH c3 o .5c«o §«c« II £ I ^ ^ -•^^H-^S'C II § a> PQ • &Q S p II £ '*^ a _a < '2 +rcOiCCOiM'*'-HCC«OOCO bCCO eOt>;QOr-i,-H « - d $ rr ^- = = 9) B^ - a eS- ej- fa bO bO «> ^ :: - - " 5 sg §s ■thOOo S OtH •« II II II 1 II 1 II II a> d !=^ P 7 -s s®^^??*^*?' ^i^ ►. 4* meter, gra meter, grj 1 Myria- 1 Kilo- 1 Hecto- 1 Deka- illl s dS ? c^ lo o o iM o,o as(Ncoio-^coooic SlTifN OJ O ^ (N O ^^ T-i CC C^ »C »-< CO QO O ?0 IC -f ■«- «> O CO ^ 1-^ (J> O r- IM <£! C '-^ Tfi r- Tf ^cqco ^cc-^t^a>ai oe^icotc^g a '^ II li 11^ * II i- " ^ s- c O S o t- . r-» 5 S iw t^'3 £ Hi: coll :i[!-;|[i|-gii:ii^i|- a c5 a a 232 CONVERSION TABLES. O o looeoc OOOCOOrHOOOCO i-HC0(NiO^C0oo DOOO 0O(N I>CO (N •-J-CO •-»-iOO locodd II "* ^ooooot^oo O p 1> i-H COOCOOOJ Tt< CO Tt^ T-H lO »CO>0 0i>0 O5; tH (N -"^^ tH 1-J C0 8 ^ 73 s .s>.as 234 CONVERSION TABLES. Eh05 t^ t> lo (N «i: »o CO .-I Tt CO l> t^ Tt< CO CO CO "* ■* lOOiCDcOOO lO 00 OS CO i>cou:)t-i O5C0(M -"^ oooocooooo 0'-Hi-H cocoooooco CDOt>. CO 05 05 i-i lO »0 OO cicooo gxoo foiopo CQ ,-; d (m" o ooo 82 eoTj4»oooo COOOOOO CDXOOOO OlOOt^OOO i-fMCOOOO CO 05 00000 q CO »o q o p c2^|aa-2:3 o 6 3 S W TjHCO OOO T-ICO Tt^O CONVERSION TABLES. 235 (N cOOOO dOO u -t^ooi-HOO 4i '*.'-; p CO cop -H- "Sco-^d»0'-Hdo *^(NCD{NCO O gpr-t C0f0 05I>'-l rHcoeOOOS lO rt| CO Csj p »0 »0 ^' CO CO CQ ft w ^3 ;3 3 3 bObOo o o CO CO •coos • ■»-* "^ OrH(NOcoeo «H lO t^ CD rH CO • OC^^iCCO^fN tilOi-HCO'T-TrH T-lO'-< oooco 05 00^0 OiOiN C01>Tt< QOOCOCO 05(MC0O Oi-Hcoco lo-^coco "^OkOiO 00 Ol p (£> CO T-i • CO H-COTt* 05IO n*^ II »0 O CO II -KH-iO ^, 1 Oo t>do6 ©CO (M _ o CO CO d 'dO'-i'o C3 flp ^ H S tf o Ph «CDO Oo^ph' t-3 «a CO ^t^05 o 2§ OSO Tj«0 III 2^ poo f-Jr-i lOO ooo t^p lOCO oooo t>:00 (n'coco i Is I|Ss X nCOrfH CO 00 I b-l003t^ ^.•a d Q (fc.^ : is oo oo oo oo oo oo oo C »0 CI O r-< OS CO iO 3025 1513 7888 4959 COI^iO t-iOCO §8 COiOOOiOCOCO C5 1-H CO T-1 Tt rjl lOt^i-HlO OOi-Hi-l t^; '^ooco p,J>co '^ c«od •CO"* OCOCO oor^t^ > S O^ •? O c3 O o3 e«oo SoO'^'* SocOcO $ 1 II M Ton poun tons ton ( kilogi 1 '« • ! '. '. »H • V .iO . a ^ ,-iO • 0) 4) . ^ : (NTH . « ^<]( '"'O -"^i-iOO CJO O fl'"' C(M p fiCSJ r-« s o c H CONVERSION TABLES. 237 Koooco Sosooo ^00 05 CO Oeco^' d rH CO -^ ■^" (MCOOOrH tOCOOCOO 0500^ COO (N Tf ■<* '^ 00 dcocbiocsi swsg 3 G (U bc'iH faC^l' S^ "x ■ V si 00 O CO o ^ . . .(NO • -oo-* "^ • ■^ C5 t^ TtllCCOCOO XidTt^fNt^ C^(MXCOCO OX'-HOO lOOOOOO lOCOr-lOOO ., -- .-£0 Wa-^ M S t: o ■a O O --ti lO oi o O 05 1— • CO 05 >— ' 00005 TfiCO. 005COCO coco OO'^^ t^x OOiCOO OO o 00 p -*_ 00 p a II 0) cc c)*^ s^§aa '^^ s ?? s W p '*' CO O rH *> — 'O ''f ''^ '^ ■'^ O"0TfO P^o'fN'iOCO Kx xt^ (NlOO x»ox r-HUOCO COO) CO Ot^T-i»0 05 X C-l o ocococo ooii>co Oi-HTf 05»0 DOiiMOCO ftpT-HCO-^ 6d t-, bc^L tA'i^-*^ -•-•■ HH'd'^t-o . o o xr^ OCO coos X OJ '0505 ^ -oo • (NiO aiO»COi -COrt^W SOriC^ -cocot^ C0COO5 •* (MC^ coo fi CO .-I O '-I r-" O O l^rH S(NOOXOOO dd '^t>'d>d>d>dd>d 238 CONVERSION TABLES. mOOOO .' Sos«oeo '-' tJcOOSrH » » H O 1-9 « II- Pd -l-QOCO coco OOS t^tJ* i-HiO 0505 ^CO coGOoeo " $ cr c o* t>l>C0CO cooo»oo O5O5 5O00 C0COr-t(N OC0 05r-J m ^^ S ^. ft m P, CO CO CO o> irjlCr-HCO «OCO'^J<0 O 0)0 ^ ddcoiH w o ft *> -T J.a| Sees co^os l>«Or-l wscoeo cowt> II « o ^ —I CO (N«0 »CCO li O'er 1^*0 00 o 00 t^"^ TP ^COCO CO 00«CO 00 O'-HTt^ Tj< IN'^OO 0)000''^ WC0c4»H Cflfl O O - ftft«- «- ■g-gftft «2»2 0) g WOS coos ^^ lOCO coco ooo 0 4) -OJ O iOCC-^ ,-icOcO O M O 03 ;h a fe fto- o OJCO qGOtHCO •^OCOb- ri H'-i'-HCO S«>^2 O O a, .. i^+-(N CO ^-CONOO D.5 cooco l^COrH H • MOO "^00 J3 'd-»- CO" 00© »H©COI> CJiOOO ftOCOlC O* I> ^dodd Co ocso cooo Tj*OOTt< 00-^05 wooo o»ooi C^OirH T-tTj^OO © 05 00 -^ Tt< << « i>eo CO 1-t T-i O X "^. lO '^ p o o c^' »H r-1 .-^ d d S o ^ O O) o o3 a obc2 pCj .0505 &odfo wcocsjec 005 di-5 P C JH a> o C coo t^eoeo OiCOCO COOO 1 §.o ooooo ,-((NCO^ 00l>.05i0 (NC^COOO cocoTt^o CO OC 00 CO dcM'cid QJ Qj B rj Vs m Vi cc II 5 lis O* -r-MCO COOCOGO ^*COCX)0»0 OJOcot^co p,coooo _^(Nddct>" OfO'^'OTH o^oX'-ico O) O S ^ £ i^ S c ■ ■ ' " 4) P( -Tt^KCOO •coi^oot^ S- (M CO O »0 • (N 05 00 05 g 1-H TtJ d (m" 00 gj rH CO(N ooooo r-ioa>io i>t>00 CO'* CO 00 rt<©cO-<# C^r-ii-i,-; his ® ^ C ?! 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HO + s s ® Js fljpl— ( o fl fl o s I> 'OfOTtt H -t^OOi-i ?;^T-ioco coTt^oio aoicocqco o p O - o w M 00 00 »-(OS »Or-t ""^co 1-1 CO(N coo 00 »o CC05 coos 6(N bo ^.-i I ' -=5 : ^ ^ ' «o a M . y, . . . 95 . . ^ . • • ^ : • : ^ : : be : : : bi) .^ . be . . "S .(N iO fT -t^TjH jTiO • ^OO OS S'-iO^O So5»o oco 00 '^COX'-' !>'-• ^h"50> i-i^aOpOT^i-iCO 0)00 Soo: »0 «c Utlr-H> (N M (N rt^O^ lO ffit-cc 00 «3 cc t^ OOCO-H <^^^^ ■^ 1-Hrt^CC ®S5 M^^ COiC oc t-ooo T-irt* ^TtH 1^05 "5 d t H i IC oot-io lOCC Tt^ rH05(M 05CC50 II CO-* (M o»oq l>rH ec 05-^ in COt-hCO 0(N(N c C CC '^ (NO c "^ C '. ^ ^1 3 m H -d 13 H c ^ ^1 o O 3 5 .s 6 T O C II '^ II 1 5 w m M II g'i ^ 1 per Second = 826446. . .acre per ho 0309917. .sq mile per Ft2 sn mftfprs n per Second = sq feet per 137741. , .acre per da 2259 sq meters p( •3 H P 11 .11 o 1^ o o S o 2^ 1 ^?c ^-co(N . M'- t4 ;h s H 1 s -si < CO H 244 CONVERSION TABLES. n HOCO OQOOS Ococo ^3 O) r|Dg Ah ' o • • iU :t^ U •lO ^ S :§ P A :g H^ ^^o>o o Sqq 2^^ ^^. o CO w S U 12 9§ GO to CO oocdtj^ iOI>iO (MlO rH ec6 5 O M O l i^fM' =^CC w 'S^ bi'^ bcS; ■!3 2 St- (£3 3 eo CO CO CO »HGOOOI> t>-C0O3l> »0 00 O C.iOt> 00 00O •H— »— 05I>0'-' q o o «^ 6 1-i 6 (MOiOTt^ T-< O 10 t^ Pi*'*' co- os CO I ^ 2!; d CO CC P^ P o g ^ ^ w Sooco Wt^oso t^OSCOGO SgoiM (NO5C0 TjHt^lO OTjHTt^ 05C0C0 , dcoco 00 1-1 CO t^03^ co-*-^ (NCO^ CX)(M CO ^( ft 2 1^ fH h ^ ft II a» a> " aa-g «+3-+e o -M 0) 0«4H S3 « o ^ M O O o3 ft05 ftCO 10 lJ © © © © © OJ .-^ O) • 'i-t 4> "^ ftCOO ft -0005 ft -lO coo '^T}H(M.-I "^ -1-1 -* 00 r-OON^ -. lOCO i-w., Sooo Soc^i'* S^-* i> TtH— f ©005 0(NOO OiOCOfN tqoq S bC Weo-<*i-^ 00 cooco '* 05 1-5 d OM ee o bfi d -00 o Qo &- t^CO">if« COCOIO coo-* 1-HCOCO eo'sJ^io cico(N IS _^ o ^ ft • ; ; h '•<£> • « -OSOJ Soooo o *>l8 M M SCO »-j t^ OS o W n:co»c 7303 3795 0613 0168 OOiCO «ocoi>05 oot^ooos 00 "i* 1-1 (N Wi3t>. -00 «COTt. Ph O .2 a 73 ; ; gcoco OlOr-t OOOJ aQcoco T— ( T-l CO ftddd CO « CO o 1^ ft OiOCO S.-ooo CO 00 (NCO C3 O bCg I o > Ph 0-73 a ^§a o o d w ^D.-i. ft-"-! • CO t^'-' ^^ Sooo gooo §,°^ I >** o o ;3W) S : ft ; I •♦a o o 248 CONVERSION TABLES. .2 ^ o COCC(N coooo OOJ eo(N ooo r^Oi-H l-HO--^ i-HO t-HrH oo ^;S;S OOP, ?3 3 O « O fH ^, . a> « QJ ft CJ O 01 rj B 5? -ft ft . I'd :3 3 o « o p. f^ . a» ^ftbo ti b HiH 0^r-;66 4)00 •^ C OQ M 'SS:ag§ too .HrHTHr-l . QJ ftft 03 s ft§ M a w 2i n. S o. CO ^ Tjioooo >>Tt<00 flOJoq 00 Tj* _ COIN 005 ftfOl^CO •■ ■ i-HOO M)d66 (^ coco P,-K-_ 1-1 1^ "^ co66»o ooTt<»oco ftO-^CDCO ^(6<6d>d) A o CONVERSION TABLES. 249 S0i050 KOiOO ^ < a o T e8 >>>> QJ u. f-i »h05(NO ^ ^ _; _r 'O O CI 5 S^ ?? O 0) o 5 M 0) 4> L. W M 5 fc, II no A >>-0 o3 (S " -00 ^1— l'«T 1— I ooo ^ dc>>>o3 ^ o3T3 S <» ft I ^ >>o3 I, t^H^D M »O»Ot>t>G0 lO cO d th (n csi 3 :3 Jj 3 o o 3 o 11 cj ft«i3 B t-CO00<©iO P^T-H O (N rH . ^'(Ndd o "co 0)00 CO lo p^O-^co . lO CD o»ooo C050 (HrH,H C ti a O 3 O ftoj ft Soft m • • M _rt fH tH 3 ^ 3 3 O i « g '^ ^ o ft bl HoOfM . CO'^t^ Sroooi * ooo CO OJOOS-^ ftddd 250 CONVERSION TABLES. ""a o • CCOiOCO ooo5i>oo o © W H ;?; cS c3 ^ O) $ 0) a» 03 teg bCfl P op o CO "5 • ceo •coo O5(M00 OOI>i-l o^o CC(NO0 COOOTfH ""^rH CO 2^-s ' ?^ CO H-COO flot^oo ^o o Oco" co" 0r-^ 0005 0^05 OiOCX) OrHCO S So* o* « fe ^ »H P.O. i^ -CO^ SooJd .-KM OrHOS 0 OiOOO OrHCO CJ . o ft Q •"oco-^ ViOt-KM S005 '3 ^:coi>t^«o Scot^t>oo «0>0(N«0 l>COCCt^ »ocOTt P. ^ (h P< m 5? 9^ P^ TO ' 5,0.0. ^ „ fl CD TO H fl - r o P « OJO S . C O. • • fa ^ Pi h! 3 O 2 5 2-^ !h ;h (U 0. <1^ <1^ O. r« ^^toS „ TO TO t, rt cc ft ■ -O CD 3f"<^«Do6id (H^tCO CO -^ CO OOi 1^00 0000 lO'^ oot^ OSOO «00 t^TH »OCO >«^ 'd bO :J'2 ^.. „. 2 a ft "S ft ^ .§ 2 C3 C S C S H gj O) C CJ 0) re o p. c3 ^ ft 3 : : S t^OCOO g^CDCO Sl> t- O-^J* ooo 00 00 i>o kOl^ c^t> U5C0 05 10 (Nl>. l>00 Kloo CO 00 Tt^O 00^ Soo 1-1 CO rJHlO OSrH , ^0 CO C<>(N COCO Tt<(N H ^TtHlO (NO lO-H COOS 00 (M 00 CO CO Tt*O0 Tj^OO (NCO coc^ c^o OICO Oi-l -^Oi p '^"l dc^ (Ni-i g fl &^ ^ ^ '^ !h tH £l^ ^ t- ^ 02 S ^^-2 ft ; •^00 1>. ft «J^ S05 • ^^^ ^^^ *2^ o^of: *ss CCo «^^ QC 00 1 p5§S p§^ Pq2. -COO" P«oo p^ P§^ P+^00 . co>o K- ^m6 fl^?^' fi^g rt-^'^ c^ fl^^ ■cSi fe o> © O 0 lOrH O"^ 1> (NtH CO^ iOI> 05^ lo 1>^ 05'^ o»o CO 05 OCC (M COl> 10 02 Tt^OC OC£ oc< Tf e^ CDiO w goa co^ rt*(N (NO l> 1-HCS »o": O5 05 So^ tHCO tHCC IOC Oi^ 1 tH coic 1-Ht- t-oc t^OO d H ^jTt^TjH rt^Tt^ I>b- Oi-i 005 CO (M«M a>c 00 in Tfl-I g C^QC 00 (M 00 cs (NCC o^ ■^ ,-iC- i>^ COCC TjHTfJ II -:: COC^ CO(M 6c 6^ ft c . ft^^ £=°S 1 Xtl .^ c r^*" II ri-C g O > SS S^£ S^s ot^i »»s 1 feiMCC »oa ^H* C^ II g| 00 (N Sio a S : «< : ^ : CO las ;^i-lOS 73 •43 S(MCC Soscc ) «oc M 02Tt^OC ftt-OC P8^ c^ -c Si ojoocT 0,05 o- (Mb' ^COCC a)(M ft22c ^CDOI sii ^oc 4) 10 CO ^(N(N * §^^ 1^= g^-^ ft'-c %^ Or-;.- o ©"-"it ©dc © ;>4 h o fe ^ ^ W ^ o^ "M ;iH i s ^ 1 02 M ^ !3 S s f-l o H 3 H 252 CONVERSION TABLES. Hi io B C0<£) El 0(N (5 II T-iO> 0) ^ ^ ooco oo»o 05 05.0 05 05X CO C0 1-1 OJ5 «J ft ^^ ft o o c3 c) 00 II '"' «0 050 T-llO lOfO II '^'^ (M(MO l-HTt^05 11 o>eo 11 xo .-HC^ «o II »oeo " --t > ^ : ^ ft ft ftft Jh Ol> (5COI>. ^fO --hOO TOi— I ^li ^11 ^05 : CJiOCO aj 0) '-I ^x : ^00 05 Rl0»0 111! S 5f? »^ ^ fH U O 0) « CO fl p.-H(NO ,0606 © o o^ p !! 5^ o s S(NCO C8!>0> OS M CO OS coo COtH rH(N COTt ooco M "^ CO 10 rfiiO 00 Tt Oiir 10 1-1 RO . ir. COTJ^ II 10 00 H ^0 OCCN csir- s 3' (M "^ 6 II COr-i U5T-i Tjieo c6»- 6^ c e II ! f4 • • ^ : • : : a • P$ fl P A C fl II : &; • • Ah II : U '. A^ 11 [° II ts • 0^ K-^ • b : • • W : i 73 1 ^5 IS 1' 3 o-c 1^ fl : . ftfc- |5 a' 1! 0) ft ^ l!S Wi oj CJ ft tH ^Sft P^ H 3 c8 1| ^ *-• ft l!S T ga «i «ia «ia Ah 111 |li ft (U H ^ vi-* Vi K : 0* ;-i H g H fecoos ISoo ^ ! ! . . • CO 1 CJ1> V) . 0) . « V ^,t^o >^2 ^^05 ft : Vi • ft ft OS CO g ^11 COCO w§2 ^000 0)0 CJiC OJOS aji> a;rH«0 -rH ^osco ."So "Sc ^2- |§s ^^^ g-" ? 060 ©60 w d !3 s so s 3^" =^1 d 0' S u U ;;> ^ U qW z § « ^ o CONVERSION TABLES. 253 ,:; OS ^ t>. 00 00 K T-i 00 o CO -^ fHi-Ht>OS»00 <.T^Ot^I^00 53 OS QO '-H 00 1> o Tj^' csi T-I o CO Q ^ a H^P g » QC O CO On OT3 t, C m « o 3 3 :Dt)i oo a • o : « . So "O Tt^" I> CO »!NTfOS ^oso CCOSrHTjteO »O(NI>iOC0 CO'* --too —I OS <-< t>. i> t>. 00'<^CO(Nt^ OOiOioXfN oowosoos CO (N* r-i r-i rH O-rt ' s 01 « " ^1 « u, C+5 0) O (U .« "^ ft K Q _^gbM^ -Oft OSI> «oc^ l>rH ooo OSt^ »OI> -+3 u ly o ©t>r o „§p t^ ft SOO* ^CJO ^ fc,eoo< t^ t^ 1-* 3 » tH c 4 (N Tt ) O : c . II 73 O -d II • -d * O) V o . As 0) Q,tH ^ a O CO po ft-a O " ■ei d i-i(M lOCO io«o coos COfH II Pc -d • fl • o : cc -o o 6 at. _ o CIS ft 254 WEIGHTS AND MEASURES. TABIiE OF ACRES REQUIRED per mile, and per 100 feet, for different widtlis. Width. Feet. Acres Mile. Acres per 100 Ft. Width. Feet. Acres per Mile. Acres per 100 Ft. Width. Feet. Acres per Mile. Acres KxTpt. Width. Feet. Acres Acres lOO^FU 1 .121 .002 26 3.15 .060 52 6.30 .119 78 9.45 .179 2 .242 .005 27 3.27 .062 53 6.42 .122 79 9.58 .181 3 .364 .007 28 3.39 .064 54 6.55 .124 80 9.70 .184 4 .485 .009 29 3.52 .067 55 6.67 .126 81 9.82 .186 5 .606 .011 30 3.64 .069 56 6.79 .129 82 9.94 .188 6 .727 .014 31 3.76 .071 67 6.91 .131 K 10. .189 7 .848 .016 32 3.88 .073 M 7. .133 83 10.1 .190 8 .970 .018 33 4.00 .076 58 7.03 .133 84 10.2 .193 H 1. .019 34 4.12 .078 59 7.15 .135 85 10.3 .195 9* 1.09 .021 35 4.24 .080 60 7.27 .138 86 10.4 .197 10 1.21 .023 36 4.36 .083 61 7.39 .140 87 10.5 .200 11 1.33 .025 37 4.48 .085 62 7.52 .142 88 10.7 .202 12 1.46 .028 38 4.61 .087 63 7.64 .145 89 10.8 .204 13 1.58 .030 39 4.73 .090 64 7.76 .147 90 10.9 .207 14 1.70 .032 40 4.85 .092 65 7.88 .149 % 11. .209 15 1.82 .034 41 4.97 .094 66 8. .151 91 11.0 .209 16 1.94 .037 % 5. .094 67 8.12 .154 92 11.2 .211 IT^ 2. .038 42^ 5.09 .096 68 8.24 .156 93 11.3 .213 2.06 .039 43 5.21 .099 69 8.36 .158 94 11.4 .216 18 2.18 .041 44 5.33 .101 70 8.48 .161 95 11.5 .218 19 2.30 .044 45 5.45 .103 71 8.61 .163 96 11.6 .220 20 2.42 .046 46 5.58 .106 72 8.73 .165 97 11.8 .223 21 2.55 .048 47 5.70 .108 73 8.85 .168 98 11.9 .225 22 2.67 .051 48 5.82 .110 74 8.97 .170 99 12. .227 23 2.79 .053 49 5.94 .112 H 9. .170 100 12.1 .230 24 2.91 .055 H 6. .114 75 9.09 .172 % 3. .057 50^^ 6.06 .115 76 9.21 .174 36* 3.03 .057 51 6.18 .117 77 9.33 .177 GRADES. 255 Table of g>radeN per mile, and per 100 feet measured hori- zontally, ana corresponding^ to different ang^les of incli* nation. ^ .9 Feet per Feet per ti d F eet per Feet per « a Feet per Feet per lA Feet per Feet per o S mile. 100 ft. Q S mile. 100 ft. Q a mile. 100 ft. mile. 100 ft. 1 1.536 .0291 45 69.11 1.3090 1 58 181.3 3.4341 3 26 316.8 5.9994 2 3.072 .0582 46 70.64 1.3381 2 184.4 3.4924 28 319.8 6.0579 3 4.608 .0873 47 72.18 1.3672 2 187.5 3.5506 30 322.9 6.1163 4 6.144 .1164 48 73.72 1.3963 4 190.6 8.6087 32 326.0 6.1747 5 7.680 .1455 49 75.26 1.4254 6 193.6 S.6669 84 329.1 6.2330 6 9.216 .1746 60 76.80 1.4545 8 196.7 3.7250 36 332.2 6.2914 7 10.75 .2037 51 78.33 1.4837 10 199.8 3.7833 38 335.3 6.3498 8 12.29 .2328 52 79.87 1.5128 12 202.8 3.8416 40 338.4 6.4083 9 13.82 .2619 53 81.40 1.5419 14 205.9 8.8999 42 341.4 6.4664 30 15.36 .2909 54 82.94 1.5710 16 208.9 3.9581 44 344.5 6.5246 11 16.90 .3200 55 84.47 1.6000 18 212.0 4.0163 46 347.6 6.5832 12 18.43 .3491 56 86.01 1.6291 20 215.1 4.0746 48 350.7 6.6418 13 19.96 .3782 57 87.54 1.6583 22 218.1 4.1329 50 353.8 6.7004 14 21.50 .4073 58 89.08 1.6873 24 221.2 4.1911 62 356.8 6.7583 15 23.04 .4364 59 90.62 1.7164 26 224.3 4.2494 54 359.9 6.8163 16 24.58 .4655 1 92.16 1.7455 28 227.4 4.3076 66 363.0 6.8751 17 26.11 .4946 2 95.23 1.8038 30 230.5 4.3659 68 366.1 6.9339 18 27.64 .5237 4 98.30 1.8620 32 233.5 4.4242 4 369.2 6.9926 19 29.17 .5528 6 ] 01,4 1.9202 34 236.6 4.4826 6 376.9 7.1384 20 30.72 .5818 8 ] 04.5 1.9784 36 239.7 4.5409 10 ?84.6 7.2842 21 32.26 .6109 10 ] 07.5 2.0366 38 242.8 4.5993 15 392.3 7.4300 22 33.80 .6400 12 1 10.6 2.0948 40 245.9 4.6576 20 400.1 7.5767 23 35.33 .6691 14 1 13.6 2.1530 42 248.9 4.7159 25 407.8 7.7234 24 86.86 .6982 16 1 16.7 2.2112 44 252.0 4.7742 80 415.5 7.8701 25 38.40 .7273 18 1 19.8 2.2694 46 255.1 4.8325 35 423.2 8.0163 26 39.94 .7564 20 1 22.9 2.3277 48 258.2 4.8908 40 431.0 8.1625 27 41.47 .7855 22 1 26.0 2.3859 50 261.3 4.9492 45 438.7 8.3087 28 43.01 .8146 24 1 29.1 2.4441 52 264.3 5.0075 60 446.5 8.4554 29 44.54 .8436 26 1 32.1 2.5023 64 267.4 5.0658 55 454.2 8.6021 30 46.08 .8727 28 1 35.2 2.5604 56 270.5 5.1241 5 461.9 8.7489 31 47.62 .9018 30 1 38.3 2.6186 68 273.6 6.1824 6 469.6 8.8951 32 49.16 .9309 32 1 41.3 2.6768 3 276.7 6.2407 10 477.4 9.0413 33 50.69 .9600 34 I 44.4 2.7350 a 279.7 6.2990 15 485.1 9.1875 34 52.23 .9891 36 1 47.4 2.7932 4 282.8 6.3573 20 492.9 9.3347 35 53 76 1.0182 38 1 50.5 2.8514 6 285.9 6.4158 25 500.6 9.4819 36 55.30 1.0472 40 1 53.6 2.9097 8 289.0 6.4742 80 508.4 9.6292 37 56 83 1.0763 42 1 56.6 2.9679 10 292.1 6.5326 35 516.1 9.7755 38 58.37 1.1054 44 1 59.7 3.0262 12 295.1 6.5909 40 523.9 9.9218 39 59.90 1.1345 46 1 62.8 3.0844 14 298.2 6.6493 45 531.6 10.068 40 61.44 1.1636 48 1 65.9 3.1427 16 301.3 6.7077 50 539.4 10.215 41 62.97 1.1927 50 1 69.0 3.2010 18 304.4 J. 7660 55 547.2 10.362 42 6*.51 1.2218 52 1 72.0 3.2592 20 307.5 5.8244 6 555. 10.510 4S 66.04 1.2509 54 1 75.1 3.3175 22 310.5 5.8827 44 67.57 1.2800 56 1 78.2 3.3758 24 313.6 5.9410 On a turnpike road 1° 38', or about 1 in 35, or 151 feet per mile, is the greatest slope that will allow horses to trot down rapidly with safety. In crossing mountains, this is often increased to 3°, or eren to 6°. It should never exceed 2^^°, except when abaolutely necessary. Any iior dist is = sloping dist X cosine ang of slope. ** sloping* dist is = hor dist -*- cosine " " " " vert beigbt is = hor dist X tangent " " " or = sloping dist X sine " " " A grade of n feet rise per 100 feet is usually called a grade of n per cent. 256 WEIGHTS AND MEASURES. SLOPES IN FEKT PER 100 FT. HORIZONTAI<. The fractions of minutes are given onlj to 34 feet in 100. A clinometer graduated by the 3d columr, and numbered by the first one, will give at sight the slopes in feet per 100 feet. No errors. Original. (6 Length of slope per 100 ft hor. Angle of slope. Length of slope per 100 ft hor. Angle of slope. 1§g Length of 100 ft hor. Angle of slope. Feet. Deg. Min. Feet. Deg. Min. Feet. Deg. Min. 1 100.005 34.4 35 105.948 19 17 69 121.495 34 36 2 100.020 1 8.7 36 106.283 19 48 70 122.066 35 3 100.045 1 43.1 37 106.626 20 18 71 122.642 35 23 4 100.080 2 17.5 38 106.977 20 48 72 123.223 35 45 5 100.125 2 51.8 39 107.336 21 18 73 123.810 36 8 6 100.180 3 26.0 40 107.703 21 48 74 124.403 36 30 7 100.245 4 0.3 41 108.079 22 18 75 125.000 36 52 8 100.319 4 34.4 42 108.462 22 47 76 125.603 37 14 9 100.404 5 8.6 43 108.853 23 16 77 126.210 37 36 10 100.499 5 42.6 44 109.252 23 45 78 126.823 37 57 11 100.603 6 16.6 45 109.659 24 14 79 127.440 38 19 12 100.717 6 50.6 46 110.073 24 42 60 128.062 38 40 IS 100.841 7 24.4 47 110.494 25 10 81 128.690 39 1 14 100.975 7 58.2 48 110.923 25 38 82 129.321 39 21 15 101.119 8 31.9 49 111.359 26 6 83 129.958 39 42 16 101.272 9 5.4 50 111.803 26 34 84 130.599 40 2 17 101.435 9 38.9 51 . 112.254 •27 1 85 131.244 40 22 18 101.607 10 12.2 52 112.712 27 28 86 131.894 40 42 19 101.789 10 45.5 53 113.177 27 55 87 132.548 41 1 20 101.980 11 18.6 54 113.649 28 22 88 133.207 41 21 21 102.181 11 51.6 55 114.127 28 49 89 133.869 41 40 22 102.391 12 24.5 56 114.612 29 15 90 134.536 41 59 23 102.611 12 57.2 57 115.104 29 41 91 135.207 42 18 24 102.840 13 29.8 58 115.603 30 7 92 135.882 42 3T 25 103.078 14 2.2 59 116.108 30 32 93 136.561 42 55 26 103.325 14 34.5 CO 116.619 30 58 94 137.244 43 14 27 103.581 15 6.6 61 117.137 31 23 95 137.931 43 32 28 103.846 15 38.5 62 117.661 31 48 96 138.622 43 50 29 104.120 16 10.3 63 118.191 32 13 97 139.316 44 8 30 104.403 16 42.0 64 118.727 32 37 98 140.014 44 25 31 104.695 17 13.4 65 119.269 33 1 99 140.716 44 43 32 104.995 17 44.7 66 119.817 33 25 100 141.421 45 00 33 105.304 18 15.8 67 1J0.370 33 49 101 142.180 45 17 34 105.622 18 46.7 69 120.930 34 13 102 142.843 45 34 In describing railroad g'rades, it is usual, as in our tables, to refer the rise, A, to the corresponding horizontal length, B. We then have , r = ^ = length B the tangent of the angle, a, between the plane and the horizontal. If the rise, A, be referred to the sloping length, C, we have . ^ ^ r ^ "**^ -^ ' ^^^ *^^^ fraction is proportional to the component, S, of the weight, W, in the direction of the slope. Thus, on a grade where rise, A, = 0.1 X sloping length, C, we have sin a = 0.1, and S = 0.1 W. The tangent of a is only approximately proportional to S ; but the steepest grades, surmounted by traction only, even on electric railways, rarely, if ever, exceed from 13 to 15 per cent ; and, on these, the error, due to using tan a in- stead of sin a, is less than a difference of 0.2 per cent in the grade, and about = 1 per (Tent of the true vakie of S. For steeper grades, such as those of rack railways, it should always be specified whether the rise refers to the horizontal or to the sloping measurement. Transvense slopes, such as those of earthwork, are sometimes, like railroad ^ i. X J . ft vertical A , , „ . ft horizontal B „ grades, stated in -^-r — -. —;-, =,, but usually in -^^ — , as -r- Here ft horizontal ' B' •'ft vertical ' A - is the cotangent of the angle, a, with the horizontal, or the tangent of the angle (90°-^) with the vertical. Thus stated, a slope of 2 to 1 means a slope of 2 horizontal to 1 vertical. GRADES. 257 Table of srrades per mile; or per 100 feet measured liorl* zontally. Grade Grade Grade Grade Grade Grader Grade Grade in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. per mile. per 100 ft. per mile. per 100 ft. per mile. per 100 "ft. per mile. per 100 ft. 1 .01894 39 .73S64 77 1.45833 115 2.17803 2 .03788 40 .75758 78 1.47727 116 2.19697 3 .05682 41 .77652 79 1.49621 117 2.21591 4 .07576 42 .79545 80 1.51515 118 2.23485 5 .09470 43 .81439 81 1.53409 119 2.25379 6 -11364 44 .83333 82 . 1.55303 120 2.27273 7 .13258 45 .85227 83 1.57197 121 2.29167 8 .15152 46 .87121 84 1.59091 122 2.31061 9 .17045 47 .89015 85 1.60985 123 2.32955 10 .18939 48 .90909 86 1.62879 124 2.34848 11 .20833 49 .92803 87 1.64773 125 2.36742 12 .22727 50 .94697 88 1.66666 126 2.38636 13 .24621 51 .96591 89 1.68561 127 2.40530 14 .26515 52 .98485 90 1.70455 128 2.42424 15 .28409 63 1.00379 91 1.72348 129 2.44318 16 .30303 54 1.02273 92 1.74242 130 2.46212 17 .32197 65 1.04167 93 1.76136 131 2.48106 18 .34091 66 1.06061 94 1.78030 132 2.50000 19 .35985 67 1.07955 95 1.79924 133 2.51894 20 .37879 68 1.09848 96 1.81818 134 2.53788 21 .39773 59 1.11742 97 1.83712 135 2.55682 22 .41667 60 1.13636 98 1.85606 1.36 2.57576 23 .43561 61 1.15530 99 1.87500 137 2.59470 24 .45455 62 1.17424 100 1.89394 138 2.61364 25 .47348 63 1.19318 101 1.91288 139 2.63258 26 .49242 64 1.21212 102 1.93182 140 ^65152 27 .51136 65 1.23106 103 1.95076 141 2.67045 28 .53030 66 1.25000 104 1.96970 142 2.68939 29 .54924 67 1.26894 105 1.98864 143 270833 30 .56818 68 1.28788 106 2.00758 144 2.72727 31 .58712 69 1.30682 107 2.02652 145 2.74621 32 .60606 70 1.32576 108 2.04545 146 2.76515 33 .62500 71 1.34470 109 2.06439 147 2.78409 34 .64394 72 1.36364 110 2.08333 148 2.80303 35 .66288 73 1.38258 111 2.10227 149 2.82197 36 .68182 74 1.40152 112 2.12121 150 2.84091 37 .70076 75 1.42045 113 2.14015 151 2.85986 38 .71970 76 1.43939 114 2.15909 152 2.87879 If the grade per mile should consist of feet and tenths, add to the grade per 100 feet in the foregoing table, that corresponding to the number of tenths taken from the table below ; thus, for a grade of 43.7 feet per mile, we have .81439 + .01326 = .82765 feet per 100 feet. Ft. per Mile. Per 100 Feet. Ft., per Mile. Per 100 Feet. Ft. per Mile. Per 100 Feet. .05 .00094 .4 .00758 .7 .01326 .1 .00189 .45 .00852 .75 .01420 .16 .00283 .6 .00947 .8 .01515 .2 .00379 .65 .01041 .85 .01609 .26 .00473 .6 .01136 .9 .01705 A .00568 .65 .01230 .95 .0179i .35 .00662 17 258 WEIGHTS AND MEASURES. TABIiJG OF HEABiS OF WATFR CORRESiPOlTDINO TO OITEN PRESSURES. Water at maximum density, 62.425 lbs. per cubic foot = 1 gram per cubia centimeter ; correspon^iing to a temperature of 4° Centigrade = 39.2"^ Fahrenheit. Head in feet = 2.306768 X pressure in lbs. per square inch. " " = 0.0160192 X pressure in lbs. per square foot. Heads corresponding to pressures not given in the table can be found by these formulae, or taken from the table by simple proportion. Pregsure. Head. Pressure, Head. Pressure. Head. lb«. per lbs. per Feet. lbs. per lbs. per Feet. lbs. pel lbs. per Feet. aq. in. sq. ft. sq. in. sq. ft. sq. in. sq. ft. 1 144 2.3068 51 7344 117.645 101 14544 232.984 2 288 4.6135 52 7488 119.952 102 14688 235.290 3 432 6.9203 53 7632 122.259 103 14832 237.597 4 576 9.2271 54 7776 124.565 104 14976 239.904 5 720 11.5338 55 7920 126.872 105 15120 242.211 6 864 13.8406 56 8064 129.179 106 15264 244.517 7 1008 16.1474 57 8208 131.486 107 15408 246.824 8 1152 18.4541 58 8352 133.793 108 15552 249.131 9 1296 20.7609 59 8496 136.099 109 15696 251.438 10 1440 23.0677 60 8640 138.406 110 15840 253.744 11 1584 25.3744 61 8784 140.713 111 15984 256.061 12 1728 27.6812 62 8928 143.020 112 16128 258.358 13 1872 29.9880 63 9072 145.326 113 16272 260.665 14 2016 32.2948 64 9216 147.633 114 16416 262.972 15 2160 34.6015 65 9360 149.940 115 16560 265.278 16 2304 36.9083 66 9504 152.247 116 16704 267.585 17 2448 39.2151 67 9648 154.553 117 16848 269.892 18 2592 41.5218 68 9792 156.860 118 16992 272.199 19 2736 43.8286 69 9936 159.167 119 17136 274.505 20 2880 46.1354 70 10080 161.474 120 17280 276.812 21 3024 48.4421 71 10224 163.781 121 17424 279.119 22 3168 50.7489 72 10368 166.087 122 17568 281.426 23 3312 53.0557 73 10512 168.394 123 17712 283.732 24 3456 55.3624 74 1065S 170.701 124 17856 286.039 25 3600 57.6692 75 10800 173.008 125 18000 288.346 26 3744 59.9760 76 10944 175.314 126 18144 290.653 27 3888 62.2827 77 11088 177.621 127 18288 292.960 28 4032 64.5895 78 11232 179.928 128 18432 295.266 29 4176 66.8963 79 11376 182.235 129 18576 297.573 30 4320 69.2030 80 11520 184.541 130 18720 299.880 31 4464 71.5098 81 11664 186.848 131 18864 302.187 32 4608 73.8166 82 11808 189.155 132 19008 304.493 33 4752 76.1233 83 11952 191.462 133 19152 306.800 34 4896 78.4301 84 12096 193.769 134 19296 309.107 35 5040 80.7369 85 12240 t 196.075 135 19440 311.414 36 5184 83.0436 86 12384 198.382 136 19584 313.720 87 5328 85.3504 87 12528 200.689 137 19728 316.027 38 5472 87.6572 88 12672 202.996 138 19872 318.334 39 5616 89.9640 89 12816 205.302 139 20016 320.641 40 5760 92.2707 90 12960 207.609 140 20160 322.948 41 5904 94.5775 91 13104 209.916 141 20304 325.254 42 6048 96.8843 92 13248 212.223 142 20448 327.561 43 6192 99.1910 93 13392 214.529 143 20592 329.868 44 6336 101.4978 94 13536 216.836 144 20736 332.175 45 6480 103.8046 95 13680 219.143 145 20880 334.481 46 6624 106.1113 96 13824 221.450 146 21024 336.788 47 6768 108.4181 97 13968 223.756 147 21168 339.095 48 6912 110.7249 98 14112 226.063 148 21312 341.402 49 7056 113.0316 99 14256 228.370 149 21456 343.708 50 7200 115.3384 100 14400 230.677 150 21600 346.015 WEIGHTS AND MEASURES. 259 TABIi£ OF PRESSURES CORRESPONDINO TO OIVEIT HEADS OF WATER. Water at maximum density, 62.425 lbs. per cubic foot = 1 gram per cubic •entimeter ; corresponding to a temperature of 4° Centigrade = 39.2^ Fahrenheit. Pressure in lbs. per square inch = 0.433507 X head in feet. Pressure in lbs. per square foot = 62.425 X head in feet. Pressures carresponding to heads not given in the table can be found by thes« formulae, or taken from the table by simple proportion. Head. Inches. Pressure. lbs. per sq. in. lbs. per sq. ft. 0.036126 0.072251 0.108377 0.144502 0.180628 0.216753 5.202083 10.404167 15.606250 20.808333 26.010417 31.212500 Head. Inches. 9 10 11 12 Pressure. lbs. per sq. in. lbs. per sq. ft. 0.252879 0.289005 0.325130 0.361256 0.397381 0.433507 36.414583 41.616667 46.818750 52.020833 57.222917 62.425000 Pressure. Pressure. Pressure. Head. Feet. Head. Feet. Head. lbs. per lbs. per lbs. pel lbs. per Feet. lbs. per IbB. per sq. in. sq. ft. sq. in. sq. ft. sq. in. sq. ft. 1 0.4335 62.425 38 16.4733 2372.150 75 32.5130 4681.875 2 0.8670 124.850 39 16.9068 2434.575 76 32.9465 4744.300 3 1.3005 187.275 40 17.3403 2497.000 77 33.3800 4806.725 4 1.7340 249.700 41 17.7738 2559.425 78 33.8135 4869.150 5 2.1675 312.125 42 18.2073 2621.850 79 34.2471 4931.575 6 2.6010 374.550 43 18.6408 2684.275 80 34.6806 4994.000 7 3.0345 436.975 44 19.0743 2746.700 81 35.1141 5056.425 8 3.4681 499.400 45 19.5078 2809.125 82 35.5476 5118.850 9 3.9016 561.825 46 19.9413 2871.550 83 35.9811 5181.275 10 4.3351 624.250 47 20.3748 2933.975 84 36.4146 5243.700 11 4.7686 686.675 48 20.8083 2996.400 85 36.8481 5306.125 12 5.2021 749.100 49 21.2418 3058.825 86 37.2816 5368.550 13 5.6356 811.525 50 ■ 21.6753 3121.250 87 37.7151 5430.975 14 6.0691 873.950 51 22.1089 3183.675 88 38.1486 5493.400 15 6.5026 936.375 52 22.5424 3246.100 89 38.5821 5555.825 16 6.9361 998.800 53 22.9759 3308.525 90 39.0156 5618.250 17 7.3696 1061.225 54 23.4094 3370.950 91 39.4491 5680.675 18 7.8031 1123.650 55 23.8429 3433.375 92 39.8826 5743.100 19 8.2366 1186.075 56 24.2764 3495.800 93 40.3162 5805.525 20 8.6701 1248.500 57 24.7099 1 3558.225 94 40.7497 5867.950 21 9.1036 1310.925 58 25.1434 3620.650 95 41.1832 5930.375 22 9.5372 1373.350 59 25.5769 3683.075 96 41.6167 5992.800 23 9.9707 1435.775 60 26.0104 3745.500 97 42.0502 6055.225 24 10.4042 1498.200 61 26.4439 3807.925 98 42.4837 6117.650 25 10.8377 1560.625 62 26.8774 3870.350 99 42.9172 6180.075 26 11.2712 1623.050 63 27.3109 3932.775 100 43.3507 6242.500 27 11.7047 1685.475 64 27.7444 3995.200 101 43.7842 6304925 28 12.1382 1747.900 65 28.1780 4057.625 102 44.2177 6367.350 29 12.5717 1810.325 66 28.6115 4120.050 103 ^4.6512 6429.775 30 13.0052 1872.750 67 29.0450 4182.475 104 45.0847 6492.200 31 13.4387 1935.175 68 29.4785 4244.900 105 45.5182 6554.625 32 13.8722 1997.600 69 29.9120 4307.325 106 45.9517 6617.050 33 14.3057 2060.025 70 30.3455 4369.750 107 46.3852 6679.475 34 14.7392 2122.450 71 30.7790 4432.175 108 46.8188 6741.900 36 16.1727 2184.875 72 31.2125 4494.600 109 47.2523 6804.325 36 15.6063 2247.300 73 31.6460 4557.025 110 47.6868 6866.750 87 16.0398 2309.725 74 32.0795 4619.450 111 48.1193 6929.175 260 WEIGHTS AND MEASIJRES. TABI.I: OF PRESISURES (Continued). Pressure. Pressure. Pressure. Head. Feet. Head. Feet. Head. Feet. lbs. per lbs. per lbs. per lbs. per lbs. per lbs. per sq. in. sq. ft. sq. in. sq. ft. sq. in. sq. ft. 112 48.5528 6991.600 144 62.4250 8989.200 176 76,2972 10986.800 113 48.9863 7054.025 145 62.8585 9051.625 177 76.7307 11049.225 114 49.4198 7116.450 146 63.2920 9114.050 178 77.1642 11111.650 115 49.8533 7178.875 147 63.7255 9176.475 179 77.5978 11174.075 116 50.2868 7241.300 148 64.1590 9238.900 180 78.0313 11236.500 117 50.7203 7303.725 149 64.5925 9301.325 181 78.4648 11298.925 118 51.1538 7366.150 150 65.0260 9363.750 182 78.8983 11361.350 119 51.5873 7428.575 151 65.4596 9426.175 183 79.3318 11423.775 120 52.0208 7491.000 152 65.8931 9488.600 184 79.7653 11486.200 121 52.4543 7553.425 153 66.3266 • 9551.025 185 80.1988 11548.625 122 52.8879 7615.850 154 66.7601 9613.450 186 80.6323 11611.050 123 53.3214 7678.275 155 67.1936 9675.875 187 81.0658 11673.475 124 53.7549 7740.700 156 67.6271 9738.300 188 81.4993 11735.900 125 54.1884 7803.125 157 68.0606 9800.725 189 81.9328 11798.325 126 54.6219 7865.550 158 68.4941 9863.150 190 82.3663 11860.750 127 55.0554 7927.975 159 68.9276 9925.575 191 82.7998 11923.175 128 55.4889 7990.400 160 69.3611 9988.000' 192 83.2333 11985.600 129 55.9224 8052.825 161 69.7946 10050.425 193 83.6669 12048.025 130 56.3559 8115.250 162 70.2281 10112.850 194 84.1004 12110.450 131 56.7894 8177.675 163 70.6616 10175.275 195 84.5339 12172.875 132 57.2229 8240.100 164 71.0951 10237.700 196 84.9674 12235.300 133 57.6564 8302.525 165 71.5287 10300.125 197 85.4009 12297.725 134 58.0899 8364.950 166 71.9622 10362.550 198 85.8344 12360.150 135 58.5234 8427.375 167 72.3957 10424.975 199 86.2679 12422.575 136 58.9570 8489.800 168 72.8292 10487.400 200 86.7014 12485.000 137 59.3905 8552.225 169 73.2627 10549.825 201 87.1349 12547.425 138 59.8240 8614.650 170 73.6962 10612.250 202 87.5684 12609.850 139 60.2575 8677.075 171 74.1297 10674.675 203 88.0019 12672.275 140 60.6910 8739.500 172 74.5632 10737.100 204 88.4354 12734.700 141 61.1245 8801.925 173 74.9967 10799.525 205 88.8689 12797.125 142 61.5580 8864.350 174 75.4302 10861.950 206 89.3024 12859.550 143 61.9915 8926.775 175 75.8637 10924.375 207 89.7359 12921.975 Table showing the total pressure against a vertical plane one foot wide, extending from the surface of the water to the depth named in the first column. Water at its maximum density. 62.425 lbs per cubic foot = 1 gram per cv^M centimeter, corresponding to a temperature of 4° Cent. -= 39.2° Fahr. Total pressure in poun^-^ = 31.2125 X square of depth in feet. Depth. Total pressure. Depth. Total pressure. Depth. Total pressure. Depth. Total pi-essurt Feet. Pounds. Feet. Pounds. * Feet. Pounds. Feet. Pounds. 1 31.21 17 9020 33 33990 49 74941 2 124.85 18 10113 34 36082 50 78031 3 280.9 19 11268 35 38235 51 81184 ' 4 499.4 20 12485 36 40451 52 84399 5 780.3 21 13765 37 42730 53 87676 6 1124 22 15107 38 45071 54 9ioie 7 1529 23 16511 39 47474 55 94418 8 1998 24 17978 40 49940 60 112365 9 2528 25 19508 41 52468 65 131873 10 3121 26 21100 42 55059 70 152941 11 3777 27 22754 43 57712 75 175570. 12 4495 28 24471 44 60427 80 1997G0 13 5275 29 26250 45 63205 85 225510 14 6118 30 28091 46 66046 90 252821 15 7023 31 29995 47 68948 95 281(593 16 7990 32 31962 48 71914 100 312125 WEIGHTS AND MEASURES. 261 TABIiE OF I>ISCHARO£S IN CUBIC FEET PER SECOND CORRESPONOINO TO GIVEX OUSCHAROESi IN IJ. S. OAEEONS PER 24 HOURS. U. S. gallon = 231 cubic inches. Discharge in cubic feet per second = 1.54728 X discharge in millions of U. S. gal' Ions per 24 hours. Millions Millions Millions Millions of U.S. Cubic feet of U. S. Cubic feet of U. S. Cubic feet of U. S. Cubic feet gals, per ' per second. gals, per per second. gals, per per second. gals, per per second. 24 hrs. 24 hrs. 24 hrs. 24 hrs. .010 .0154723 13 20.1140 43 66.5308 72 111.400 .020 .0309446 14 21.6612 44 68.0781 73 112.948 .030 .0464169 15 23.2084 45 69.6253 74 114.496 .040 .0618891 16 24.7557 46 71.1725 75 116.042 .050 .0773614 17 26.3029 47 72.7197 76 117.589 .060 .0928337 18 27.8501 48 74.2670 77 119.137 .070 .108306 19 29.3973 49 75.8142 78 120.684 .080 .123778 20 30.9446 50 77.3614 79 122.231 .090 .139251 21 32.4918 51 78.9087 80 123.778 .100 .154723 22 34.0390 52 80.4559 81 125.326 .200 .309446 23 35.5863 53 82.0031 82 126.873 ; .300 .464169 24 37.1335 54 83.5503 83 128.420 .400 .618891 25 38.6807 56 85.0976 84 129.967 .500 .773614 26 40.2279 56 86.6448 85 131.514 .600 .928337 27 41.7752 57 88.1920 86 133.062 .700 1.08306 28 43.3224 58 89.7393 87 134.609 .800 1.23778 29 44.8696 59 91.2865 88 136.156 .900 1.39251 30 46.4169 60 92.8337 89 137.703 1 1.54723 31 47.9641 61 94.3809 90 139.251 2 3.09446 32 49.5113 62 95.9282 91 140.798 3 4.64169 33 51.0685 63 97.4754 92 142.345 4 6.18891 34 52.6058 64 99.0226 93 143.892 5 7.73614 35 54.1530 .65 100.570 94 145.439 6 9.28337 36 55.7002 66 102.117 95 146.987 7 10.8306 37 57.2475 67 103.664 96 148.534 8 12.3778 38 58.7947 68 105.212 97 150.081 9 13.9251 39 60.3419 69 106.759 98 151.628 19 15.4723 40 • 61.8891 70 108.306 99 153.176 11 17.0195 41 63.4364 71 109.853 100 154.723 12 18.5667 42 64.9836 262 WEIGHTS AND MEASURES. TABIiE OF I>I8CHAROES i:sr CUBIC FEET PER SECOND CORRESPOIVDINO TO GIVEN DISCHARGES IN IM- PERIAI. GAI.EONS PER 24 HOURS. Imperial gallon = 277.274 cubic inches. Discharge in cubic feet per second = 1.85717 X discharge in Imperial gallons per 24 hours. Millions Millions Millions Millions of Imp. Cubic feet of Imp. Cubic feet of Imp. Cubic feet of Imp. Cubic feet •gals, per per second. gals, per per second. gals, per per second. gals, per per second. 24hrs. 24 hrs. 24 hrs. 24 hrs. .010 .0185717 13 24.1432 43 79.8583 72 133.7162 .020 .0371434 14 26.0004 44 81.7155 73 135.5734 .030 .0557151 15 27.8576 45 83.5727 74 137.4306 .040 .0742868 16 29.7147 46 85.4298 75 139.2878 .050 .0928585 17 31.5719 47 87.28/0 76 141.1449 .060 .111430 18 33.4291 48 89.1442 77 143.0021 .070 .130002 19 35.2862 49 91.0013 78 144.8593 .080 .148574 20 37.1434 50 92.8585 79 146.7164 .090 .167145 21 39.0006 51 94.7157 80 148.5736 .100 .185717 22 40.8577 52 96.5728 81 150.4308 .200 .371434 23 42.7149 53 98.4300 82 152.2879 .300 .557151 24 44.5721 54 100.2872 83 154.1451 .400 .742868 25 46.4293 55 102.1444 84 156.0023 .500 .928585 26 48.2864 56 104.0015 85 157.8595 .600 1.11430 27 50.1436 57 105.8587 86 159.7166 .700 1.30002 28 52.0008 58 107.7159 87 161.5738 .800 1.48574 29 53.8579 59 109.5730 88 163.4310 .900 1.67145 30 55.7151 60 111.4302 89 165.2881 1 1.85717 31 57.5723 61 113.2874 90 167.1453 2 3.71434 32 59.4294 62 115.1445 91 169.0025 3 5.57151 33 61.2866 63 117.0017 92 170.8596 4 7.42868 34 63.1438 64 118.8589 93 172.7168 6 9.28585 35 65.0010 65 120.7160 94 174.5740 6 11.1430 36 66.8581 66 122.5732 95 176.4312 7 13.0002 37 68.715"3 67 124.4804 96 178.2883 8 14.8574 38 70.5725 68 126.2876 97 180.1455 9 16.7145 39 72.4296 69 128.1447 98 182.0027 10 18.5717 40 74.2868 70 130.0019 99 183.8598 11 20.4289 41 76.1440 71 131.8591 1»0 185.7170 12 22.2860 42 78.0011 WEIGHTS AND MEASURES. 263 TABL.I: OF DlfSCHAROES IN OAT.I.ONS PER 34 HOURS €ORR£SPONI>INO TO GIVEN DISCHARGES IN CUBIC FEET PER SECOND. U. S, gallon = 231 cubic inches. Imperial gallon = 277.274 cubic inches. Discharge in U. S. gallons per 24 hours = 646317 X discharge in cubic feet per second. Discharge in Imperial gallons per 24 hours = 538454 X discharge in cubic feet per second. Cub. ft. per sec. Millions of Millions of Cub. ft. Millions of Millions of U. S. gallons Imperial gallons per sec. U. S. gallons Imperial galloni per 24 hours. per 24 hours. per 24 hours. per 24 hours. 1 0.646317 0.538454 53 34.254795 28.538W4 2 1.292634 1.076907 54 34.901112 29.076498 3 1.938951 1.615361 55 35.547428 29.614951 4 2.585268 2.153815 56 36.193745 30.153405 5 3.231584 2.692268 57 36.840062 30.691859 6 3.877901 3.230722 58 37.486379 31.230312 7 4.524218 3.769176 59 38.132696 31.768766 8 5.170535 4.307629 60 38.779013 32.307220 9 5.816852 4.846083 61 39.425330 32.845673 10 6.463169 5.384537 62 40.071647 33.384127 11 7.109486 5.922990 63 40.717963 33.922581 12 7.755803 6.461444 64 41.364280 34.461034 13 8.402119 6.999898 65 42.010597 34.999488 14 9.0484:^6 7.538351 66 42.656914 35.537942 15 9.694753 8.076805 67 43.303231 36.076395 16 10.341070 8.615259 68 43.949548 36.614849 17 10.987387 9.153712 69 44.595865 37.153303 18 11.633704 9.692166 70 45.242182 37.691756 19 12.280021 10.230620 71 45.888498 38.230210 20 12.926338 10.769073 72 46.534815 38.768664 21 13.572654 11.307527 73 47.181132 39.307117 22 14.218971 11.845981 74 47.827449 39.845571 23 14.865288 12.384434 75 48.473766 40.384025 24 15.511605 12.922888 76 • 49.120083 40.922478 25 16.157922 13.461342 77 49.766400 41.460932 26 16.804239 13.999795 78 50.412717 41.999385 27 17.450556 14.538249 79 51.059034 42.5378.39 28 18.096873 15.076702 80 51.705350 43.076293 29 18.743190 15.615156 81 52.351667 43.614746 30 19.389506 16.153610 82 52.997984 44.153200 31 20.035823 16.692063 83 53.644301 44.691654 32 20.682140 17.230517 84 54.290618 45.230107 33 21.328457 17.768971 85 54.936935 45.768561 34 21.974774 18.307424 86 55.583252 46.307015 35 22.621091 18.845878 87 66.229569 46.845468 36 23.267408 19.384332 88 66.875885 47.383922 37 23.913725 19.922785 89 57.522202 47.922376 38 24.560041 20.461239 90 58.168519 48.460829 39 25.206^58 20.999693 91 58.814836 48.999283 40 25.852675 21.538146 92 69.461153 49.537737 41 26.498992 22.076600 93 60.107470 50.076190 42 27.145309 22.615054 94 60.753787 50.614644 43 27.791626 23.153507 95 61.400104 51.153098 44 28.437943 23.691961 96 62.046420 51.691551 46 29.084260 24.230415 97 62.692737 52.23000« 46 29.7.30576 24.768868 98 63.339054 52.7684.59 47 30.376893 25.307322 99 63.985371 53.306912 48 31.023210 25.845776 100 64.631688 53.845366 49 31.669527 26.384229 101 65.278005 54.383820 60 32.^15844 26.922683 102 65.924322 54.922273 51 32.962161 27.461137 103 66.570639 55.460727 52 33.608478 27.999590 104 67.216956 55.999181 264 WEIGHTS AND MEASURES. TABIiE OF I>IS€HARO£S (Continaed). Cub. ft. per sec. Millions of Millions of Cub. ft. Millions of Millions of U. S. gallons Imperial gallons per sec. U. S. gallons Imperial gallon! per 24 hours. per 24 hours. per 24 hours. per 24 hours. i05 67.863272 56.537634 167 107.934919 89.921761 106 68.509589 57.076088 168 108.581236 90.460215 107 69.155906 57.614542 169 109.227553 90.998669 108 69.802223 58.152995 170 109.873870 91.537122 109 70.448540 58.691449 171 110.520186 92.075576 110 71.094857 59.229903 172 111.166503 92.614030 111 71.741174 59.768356 173 111.812820 93.152483 112 72.387491 60.306810 174 112.459137 93.690937 113 73.033807 60.845264 175 113.105454 94.229391 114. 73.680124 61.383717 176 113.751771 94.767844 115 74.326441 61.922171 177 114.398088 95.306298 116 74.972758 62.460625 178 115.044405 95.844751 117 75.619075 62.999078 179 115.690722 96.383205 118 76.265392 63.537532 180 116.337038 96.921659 119 76.911709 64.075986 181 116.983355 97.460112 120 77.558026 64.614439 182 117.629672 97.998566 121 78.204342 65.152893 183 118.275989 98.537020 122 78.850659 65.691347 184 118.922306 99.075473 123 79.496976 66.229800 185 119.568623 99.613927 124 80.143293 66.768254 186 120.214940 • 100.152381 125 80,789610 67.306708 187 120.861257 100.690834 126 81.435927 67.845161 188 121.507573 101.229288 127 82.082244 68.383615 189 122.153890 101.767742 128 82.728561 68.922068 190 122.800207 102.306195 129 83.374878 69.460522 191 123.446524 102.844649 130 84.021194 69.998976 192 124.092841 103.383103 131 84.667511 70.537429 193 124.739158 103.921556 132 85.313828 71.075883 194 125.885475 104.460010 133 85.960145 71.614337 195 126.031792 105.098464 134 86.606462 72.152790 196 126.678108 105.536917 135 • 87.252779 72.691244 197 127.324425 106.075371 136 87.899096 73.229698 198 127.970742 106.613825 137 88.545413 73.768151 199 128.617059 107.152278 138 89.191729 74.306605 200 129.263376 107.690732 139 89.838046 74.845059 201 129.909693 108.229186 140 90.484363 75.383512 202 130.556010 108.767639 141 91.130680 75.921966 203 131.202327 109.306093 142 91.776997 76.460420 204 131.848644 109.844547 143 92.423314 76.998873 205 132.494960 110.383000 144 93.069631 77.537327 206 133.141277 110.921454 145 93.715948 78.075781 207 133.787594 111.459908 146 94.362264 78.614234 208 134.433911 111.998361 147 95.008581 79.152688 209 135.080228 112.536815 148 95.654898 79.691142 210 135.726545 113.075269 149 96.301215 80.229595 211 136.372862 113.613722 150 96.947532 80.768049 212 137.019179 114.152176 151 97.593849 81.306503 213 137.665495 114.690630 1521 98.240166 81.844956 214 138.311812 115.229083 153 98.886483 82.383410 215 138.958129 115.767537 154 99.532800 82.921864 216 139.604446 116.305991 155 100.179116 83.460317 217 140.250763 116.844444 156 100.825433 83.998771 218 140.897080 117.382898 157 101.471750 84.537225 219 141.543397 117.921352 158 102.118067 85.075678 220 142.189714 118.459805 159 102.764384 85.614132 221 142.836030 118.998259 160 103.410701 86.152586 222 143.482347 119.536713 361 104.057018 86.691039 223 144.128664 120.075166 162 104.703335 87.229493 224- 144.774981 120.613620 163 105.349651 87.767947 225 145.421298 121.152074 164 105.995968 88.306400 226 146.067615 121.690527 165 106.642285 88.844854 227 146.713932 122 ?'?8981 166 107.288602 89.383308 228 147.360249 122.767434 TIME. 265 TABI.C OF ]>IS€IIARO£S (Continued). Cub. ft Millions of Millions of Cub. ft. Millions of Millions of per sec. U. S. gallons Ijpperial gallons per sec. U. S. gallons Imperial galloni per 24 hours. per 24 hours. per 24 hours. per 24 hours. 229 148.006566 123.305888 240 155.116051 129.228878 230 148.652882 123.844342 241 155.762368 129.767332 231 149.299199 124.382795 242 . 156.408685 130.305786 232 149.945516 124.921249 243 157.055002 130.844239 233 150.591833 125.459703 244 157.701519 131.382693 234 151.238150 125.998156 245 158.347636 131.921147 235 151.884467 126.536610 246 158.993952 132.459600 236 152.530784 127.075064 247 159.640269 132.998054 237 153.177101 127.613517 248 160.286586 133.536508 238 153.823417 128.151971 249 160.932903 134.074961 239 154.469734 128.690425 250 161.579220 134.613415 .TIME. 60 seconds,*! marked s, = 1 minute 60 minutes,! '* m, = 1 hour = 3600 seconds 24 hours, " h, = 1 day = 1440 minutes = 86400 secondB 7 days, " d, = 1 week = 168 hours = 10080 minutes Arc Time 1° = 4 minutes 1' = 4 seconds I'l = 0.066... second Time Arc 24 hours = 360° 1 hour = 15° 1 minute = 0° 15' 1 second = 0° 0' 15" Methods of reckoning' time. Astronomers distinguish between mean solar time, true or apparent solar time, and sidereal time. At a standard meridian (see page 267) mean solar time is the same as ordinary clock time. At any point not on a standard meridian, standard time is the local mean solar time of the meridian adopted as standard for such point ; and local time is = time at a standard meridian phis correction for longitude from that meridian if the place is east of the meridian, and vice versa. For the amount of such correction, see second table above. A true or apparent solar day is the interval of time between two successive culminations of the sun, i.e., between two successive transits or passages of the sun across the meridian of the same point on the earth ; but, since these intervals are unequal, they do not correspond with the uniform movement of clock time. A fictitious or imaginary sun, called the "mean sun," is therefore supposed to move along the equator in such a way that the interval between its culminations is con- stant. This interval is called. a day, or mean solar day, and is the average of the lengths of all the apparent solar days in a year. Apparent and mean time agree at four points in the year, viz., about the middle of April and of June, September 1 and December 24. The sun is sometimes behind and sometimes in advance of the mean sun, and is called " slow " or " fast " accordingly. The sun is " slow " in winter, the maximum being about February 11, when it passes any standard meridian, or " souths " (making apparent noon), about 14m, 288, after noon by a correct clock. The sun is " fast," or in advance of the clock, in May and in the fall, with a maximum, about November 2, of about 16m, 20s. The difference between apparent and mean time is called the equation of time. It can be obtained from the Nautical Almanac, or, approximately, by taking the mean between the times of sunrise and sunset, as given in ordinary almanacs. As solar time is measured by the apparent daily motion of the sun, so sidereal time is measured by that of the fixed stars, or, more strictly speaking, by the motion of the vernal equinox which is the point where the sun crosses the equator in the spring. * The second was formerly divided into 60 equal parts called thirds (marked '") ; but it is now divided decimally. t The old and confusing practice of designating minutes, seconds and thirds of time (see footnote *) as ', " and '" , is no longer in vogue. Days, hours, min- utes and seconds are now designated by d, h, m, and s, respectively, thus: 2d, 20h, 48m, 55.43 s., and the symbols ' and " designate minutes and seconds of arc. 266 TIME. A sidereal day is the interval of time between two successive passages of the vernal equinox (or, practically, of any star) past the meridian of a given f>oint on the earth. It is, practically, the time required for one complete revo- ution of the earth on its axis, relatively to the stars. The length of the sideral day is 23 h, 56 m, 4.09 s, of mean solar time, or 3 m, 55.91 s of mean solar time less than the mean solar day of 24 hours. lu other words, a star will, on any night, appear to set 3 m, 55.91 s earlier by a correct clock than it did on the preceding night. Hence, substantially, the number of sidereal days in a year is greater by 1 than the number of solar days. The sidereal day, like the solar day, is divided into 24 hours. These hours are, of course, shorter than those of the solar day in the same proportion as the sidereal day is shorter than the solar day. They are counted from to 24, com- mencing with sidereal noon, or the instant when the vernal equinox passes the upper meridian. Tlie civil day (= 24 hours of clock or mean solar time) commences at mid- night; and tlie astronomical solar day at noon on the civil day of the same date. Thus, on a standard meridian, Thursday, May 9, 2 a. m. civil time, is Wednesday, May 8, 14 h, astronomical time ; but Thursday, May 9, 2 P. M., civil time, is Thursday, May 9, 2 h, astronomical time. The civil month is the ordinary and. arbitrary month of the calendar, varying in length from 28 to 81 mean solar days. A sidereal month is the time required for the moon to perform an entire revolution with reference to the stars. Its mean length, in mean solar time, is about 27 d, 7 h, 43 m, 12 s. A lunation, or synodic month is the time from new moon to new moon. Its mean length is about 29 d, 12 h, 44 m, 3 s. The tropical or natural year is the time during which the earth describes the circuit from either equinox to the same again. Its mean length, in mean solar time, is now about 365 d, 5 h, 48 m, 49 s. The sidereal year is the time during which the earth describes its orbit with reference to the stars." Its mean length, in mean solar time, is about 365 d, 6 h, 9 m, 10 s. The civil year is that arbitrary or conventional and variable division of time comprised between the 1st of .January and the 31st of the following Decem- ber, both inclusive. It contains ordinarily 365 mean solar days of 24 hours, but each year whose number is divisible by 4 contains 366 days, and is called a leap year, except that those years whose numbers end in 00 and are not multiples of 400 are not leap years. To regulate a watch by the stars. The author, after having regu- lated his chronometer for a year by this method only, differed but a few seconds from the actual time as deduced from careful solar observations. Select a window, facing west if possihle, and commanding a view of a roof-crest or other fixed horizontal line, preferably about 40° above the horizon, in order to avoid disturbance due to refraction, and distant say 50 feet or more. Note the time when any bright fixed star (not a planet) passes the Vange formed between the roof, etc., and any fixed horizontal line about the window frame, as a pin fixed in either jamb. The sight in the window, and the watch, must be illumi- nated. The star will pass the range 3 m. 55.91 s. earlier on each succeeding evening. Those stars which are nearest the equator appear to move the fastest* and are therefore best suited to the purpose. If the first observation of a given star be made as late as midnight, that same star will answer for about three months, until at last it will begin to pass the range in daylight. Before this happens, transfer the time to another star which sets later. By thus tabulating, throughout the year, about half a dozen stars which follow each other at nearly equal intervals of time, we may provide a standard by means of which correct clock time may be ascertained on any clear night. Experimenting in this way with two of the best chronometers, the author found that their i^aies varied, at times, as much as from three to eight seconds per day. An average man takes two steps (one right, one left) per second. Hence, march music usually takes one second per measure (or " bar"). Modern cratches usually tick five times, and clocks either one, two, or four times, per second. STANDARD RAILWAY TIME. 267 STAKDARn RAH.WAY TIME, ADOPTED 1883. The following arrangement of standard time was recommended by the General and Southern Time Conventions of the railroads of the United States and Canada, held respectively in St. Louis, Mo., and New York city, April, 1883, and in Chicago, 111., and New York city, in October, 1883, and went into effect on most of the rail- roads of the United States and Canada, November 18th, 1883. Most of the principal cities of the United States have made their respective local times to correspond with it. This system was proposed by Mr. W. F. Allen, Secretary of the Time Conven- tions, and its adoption was largely due to his efforts. We are indebted to Mr. Allen for documents from which the following has been condensed. Five standards of timet or five " times," have been adopted for the United States and Canada. These are, respectively, the mean times of the 60th, 75th, 90th, 105th, and 120th meridians west of Greenwich, England. As each of these meridians, in the above order, is 15° west of its predecessor, its time is one hour slower. Thus, when it is noon on the 90th meridian, it is 1 p. m. on the 75th, and 11 a. m. on the 106th. The following gives the name adopted for the standai'd time of each meridian, and the conventional color adopted, and uniformly adhered to, by Mr. Allen, for the purpose of designat- ing it and its time, &c, on the maps published under his auspices: Longitude west from Greenwich. Name of Standard Time. Conventional color. 60° 900 105° 120° Intercolonial. Eastern. Central. Mountain. Pacific. Brown. Red. Blue. Green. Yellow. Theoretically, each meridian may be said to give the time for a strip of country 15° wide, running north and south, and having the meridian for its center. Thus the meridian on which the change of time between two standard meridians is sup- posed to take place, lies half-way between them. But it would, of course, not bo practicable for the railroads to use an imaginary line in passing from one time standard to another. The changes are made at prominent stations forming the ter- mini of two or more lines; or, as in the case' of the long Pacific roads, at the ends of divisions. As far as practicable, points at which changes of time had previously been made, were selected as the changing points under the new system. Detroit, Mich., Pittsburgh, Pa., Wheeling and Parkersburg, W. Ya., and Augusta, Ga., al- though not situated upon the same meridian, are points of change between eastern and central standard times. A train arriving at Pittsburgh from the east at noon, and leaving for the west 10 minutes after its arrival, leaves (by the figures shown upon its time-table, and by the watches of its train. hands) not at 10 minutes after 12, but at 10 minutes after 11. • The necessity for making the changes of time at principal points, instead of on a true meridian line, necessitates also so^ie "overlapping" of the times, or of their colors on the map. Thus, most of the roads between Buffalo and Detroit, on the north aide of Lake Erie, run by "eastern," or " red," time; while those on the south side of the Lake, between Buffalo and Toledo, immediately opposite to and directly south of them, run by " central " or " blue " time. If the changes of time were made at the meridians midway between the standard ones, it would not be necessary for any town to change its time more than 30 min- utes. As it is, somewhat greater changes had to be made at a few points. Thus, standard time at Detroit is 32 minutes ahead, and at Savannah 36 minutes back, of mean local time. In most cases the necessary change was made npon the railways by simply setting clocks and watches ahead or back the necessary number of minutes, and without making any change in time-tables. Halifax, and a few adjacent cities, use the time of the 60th meridian, that being the nearest one to them ; but the railroads in the same district have adopted the 75th meridian, or eastern, time; so that, for railroad purposes, intercolonial time has never come into force. In 1873 there were 71 time standards in use on the railroads of the United States and Canada. At the time of the adoption of the present system this number had been reduced, by consolidation of roads, &c, to 53. By its adoption, the number be- came 5, or, practically, 4, owing to the adoption of eastern time by the intercolonial roads, as already explained. 268 DIALS. DIALLING. To make a borizontal Sun-dial, Draw a line a h ; and at right angles to it, draw 66. From any convenient point, aa c, in a by draw the perp c o. Make the angle c a o equal to the lat of the place ; also the angle c o e equal to the same ; join o e. Make e n equal to o e; and from n as a center, with the rad en, describe a quadrant e 5; and div it into 6 equal parts. Draw « y, parallel to 6, 6; and from n, through the 5 ^ DIAL points on the quadrant, draw lines n t, n i, &c, terminating in e y. From a draw lines a 5, a 4, &c, passing through t, i, &c. From any convenient point, as c, describe an arc rmh,a,9& kind of fin- ish or border to half the dial. All the lines may now be effaced, except the hour lines a 6, a 5, a 4, Ac, to a 12, or a ^ ; unless, as is generally " the case, the dial is to be divided to quarters of an hour at least. In this case each of the divisions on the quad- rant e s, must be subdivided into 4 equal parts; and lines drawn from n, through the points of subdivision, terminating in ey. The quarter-hour lines must be drawn from a, as were the hour lines. Subdivisions of 5 min may be made in the same way ; but these, as well as single min, may usually be laid off around the border, by eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi- nary dial. To draw the other half of the Fig, make a d equal to the intended thick- ness of the gnomon, or style, of the dial ; and draw d 12, parallel, and equal to a 12 ; and drawthearc^j^rM;, precisely similar to the arc rm^. Between a; and w, on thearcx^w, space off divisions equal to those on the arc rmh', and number them for the hours, as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be triangular ; its thickness must throughout be equal to ad or hw; its base must cover the space adhw; its point will be at ad; and its perp height h m, over h w, must be such that lines vd,ua, drawn from its top, down to a and d, will make the angles uah^vdw. each equal to the lat of the place. Its thickness, if of metal, may conveniently be from 3^ to 1^ inch ; or if of stone, an inch or two, or more, according to the size of the dial. Usually, for neatness of appearance, the back huvw of the style is hollowed inward. The upper edges, ua, vd, which cast the shadows, must be sharp and straight. The dial must be fixed in place hor, or perfectly level ; ah and dw must be placed truly north and south ; ad being south, and hw north. The dial gives only sun or solar time ; but clock time can be found by means of the " fast or slow of the sun," as given by all almanacs. If by the almanac the sun is 5 min, Ac, fast, the dial will be the same ; and the clock or watch, to be correct, must be S uin slower than it ; and vice versa. To make a Tertical Sun-Dial. Proceed as directed above, except that the angles cao and coe on the drawing, and the angle uah or vdw of the style, must be equal to the co-latitude (= dif- ference between the latitude and 90°) of the place, and the hours must be num- bered the opposite way from those in the above figure ; i e, from htoy number 12, 11, 10, 9, 8, 7 ; and from w to g number 12, 1, 2, 3, 4, 5. The dial plate must be placed vertically, in the position shown in the figure, facing exactly south, and with ah and dvt vertical. BOARD MEASURE. 269 BOARD MEASUEE. Remarlc on following' table. The table extends to 12 ins by 24 ins, but it Is easy to find for greater sizes ; thus, for example, the board measure in a piece of 19 by 2'2, -n-ill be twice that of a piece of 19 by 11, or 17.42 X 2 = 34.84 ft board meas ; or that of 19)4 ^7 22, will b« that of 1034 by 22 added to that of 9 by 22, or 18.79 -j- 16.50 = 35.29. A foot of board meas is equal t« I foot square and 1 inch thick, or to 144 cub ins. Hence 1 cub ft = 12 ft board meas. pj Feet of Board Measure contained in one running foot of Scantlings _d . "^a of different dimensioas . (Original.) |5 1000 ft board measure = = 83Kcubft. il •d o THICKNESS IN INCHES. 13 O ^H 1 IK 1^ IH 2 2H 2^ 2% 3 Ft. Bd.M. FtBd-M. Ft. Bd.M. FtBd.M. Ft.Bd.M. Ft. Bd.M. Ft. Bd.M. Ft Bd.M. FtBd.M. H .0208 .0260 .0313 .0365 .0417 .0469 .0521 .0573 .0625 1. ^ .0417 .0521 .0625 .0729 .0833 .0938 .1042 .1146 .1250 .0625 .0781 .0938 .1094 .1250 .1406 .1563 .1719 .1875 1. .0833 .1042 .1250 .1458 .1667 .1875 .2083 .2292 .2500 .1042 .1302 .1563 .1823 .2083 .2344 .2604 .2865 .3125 2^ iz .1250 .1563 .1875 .2188 .2500 .2813 .3125 .3438 .3750 a^ .1458 .1823 .2187 .2552 .2917 .3281 .3646 .4010 .4375 3. .1667 .2083 .2500 .29ir .3333 .3750 .4166 .4583 .5000 .1875 .2344 .2813 .3281 .3750 .4219 .4688 .5156 .5625 1 .2083 .2604 .3125 .3646 .4167 .4688 .5208 .5729 .6250 a^ .2292 .2865 .3438 .4010 .4583 .5156 .5729 .6302 .6875 8. .2500 .3125 .3750 .4375 .5000 .5625 .6250 .6875 .7500 .2708 .3385 .4063 .4739 .5416 .6094 .6771 .7448 .8125 ■ 4. M .2917 .3646 .4375 .5104 .5833 .6563 .7292 .8021 .8750 r/ .3125 .3906 .4689 .5469 .6250 .7031 .7813 .8594 .9375 4. .3333 .4167 .5000 .5833 .6667 .7500 .8333 .9167 1.000 .3542 .4427 .5312 .6198 .7083 .7969 .8854 .9740 1.063 5" IZ .3750 .4688 .5625 .6563 .7500 .8438 .9375 1.031 1.125 H .3958 .4948 .5938 .6927 .7917 .8906 .9896 1.086 1.188 5. .4167 .5208 .6250 .7292 .8333 .9375 1.042 1.146 1.250 .4375 .5469 .6563 .7656 .8750 .9844 1.094 1.203 1.313 g ^ .4583 .5729 .6875 .8020 .9167 1.031 1.146 1.260 1.375 ^ .1792 .5990 .7188 ,8385 .9583 1.078 1.198 1.318 1.438 H t. .5000 .6250 ,7500 .8750 1.000 1.125 1.250 1.375 1.500 6 y< .5208 .6510 .7813 .9115 1.042 1.172 1.302 1.432 1.563 li .5417 .6771 .8125 .9479 1.083 1.219 1.354 1.490 1.625 H .5625 .7031 .8438 .9844 1.125 1.266 1.406 1.547 1.688 7. .5833 .7292 .8750 1.021 1.167 1.312 1.458 1.604 1.750 7.* .6042 .7552 .9063 1.057 1.208 1.359 1.510 1.661 1.813 }4 ^ .6250 .7813 .9375 1.094 1.250 1.406 1.563 1.719 1.875 ^ .6458 .8073 .9688 1.130 1.292 1.453 1.615 1.776 1.938 H 8. .6667 1.000 1.167 1.333 1.500 1.667 1.833 2.000 8. .6875 .8594 1.031 1.203 1.375 1.547 1.719 1.891 2.063 9.* \/ .7083 .8854 1.063 1.240 1.417 1.594 1.771 1.948 2.125 x/ .7292 .9114 1.094 1.276 1.458 1.641 1.823 2.005 2.188 9. .7500 .9375 1.125 - 1.313 1.500 1.688 1.875 2.062 2.250 .7708 .9635 1.156 1.349 1.542 1.734 1.927 2.120 2.313 14 .7917 .9895 1.188 1.385 1.583 1.781 1.979 2.177 2.375 s^ .8125 1.016 1.219 1.422 1.625 1.828 2.031 2.234 2.438 10. 10. .8333 1.042 1.250 1.458 1.667 1.875 2.083 2.292 2.500 .8542 1.068 1.281 1.495 1.708 1.922 2.135 2.349 2.563 iz .8750 1.094 1.313 1.531 1.750 1.969 2.188 2.406 2.625 y} &£ .8958 1.120 1.344 1.568 1.792 2.016 2.240 2.463 2.688 x/ 11. .9167 1.146 1.375 1.604 1.833 2.063 2.292 2.521 2.750 11. .9375 1.172 1.406 1.641 1.875 2.109 2.344 2.578 2.813 g IZ .9583 1.198 1.438 1.67T 1.917 2.156 2.396 2.635 2.875 az .9792 1.224 1.469 1.714 1958 2.203 2.448 2.693 2.938 % 12. 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 12. H 1.042 1.302 1.563 1.823 2.083 2.344 2.604 2.865 3.125 H 13 1.083 1.354 1.625 1.896 2.167 2.438 2.708 2.979 3.250 13 H 1.125 1.406 1.688 1.969 2.250 2.531 2.813 3.094 3.375 }4 u. 1.167 1.458 1.750 2.042 2.333 2.625 2.917 3.208 3.500 14. H 1.208 1.510 1.813 2.115 2.417 2.719 3.021 3.322 3.625 % 15. 1.250 1.563 1.875 2.188 2.500 2.813 3.125 3.438 3.750 15 ^ 1.292 1.615 1.938 2.260 2.583 2.906 3.229 3.552 3.875 H 16. 1.333 1.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000 16. H 1.375 1.719 2.063 2.406 2.750 3.094 3.438 3.781 4.125 U 17. 1.417 1.771 2.125 2.479 2.833 3.188 3.542 3.896 4.250 17 14 1.458 1.823 2.187 2.552 2.917 3.281 3.646 4.010 4.375 \i 18 1.500 1.875 2.250 2.625 S.OOO 3.375 3.750 4.125 4.500 18. 19. 1.583 1.979 2.375 2.771 3.167 3.563 3.958 4.354 4.750 19. 20. 1.667 2.083 2.500 2.917 3.333 3.750 4.167 4.583 5.000 20. 21. 1.750 2.188 2.625 3.063 3.500 3.936 4.375 4.812 5.250 21. 22. 1.833 2.292 2.750 3.208 3.667 4.125 4.583 5.042 5.500 22. 23. 1.917 2.396 2.875 3.354 3.833 4.313 4.792 5.270 5.750 23. 2i. 2.000 2.500 3.000 3.500 4.000 4.500 6.000 5.500 6.000 24. 270 BOARD MEASURE. Table of Board Measure — (Continued.) . Feet of Board Mea. ure contained in one runnine foot of Scantlines 5^ il of different dimensions. (Original.) ^1 ■ ga THICKNESS IN INCHES. %- 3Ji 3K 8« 4 4M 4>^ 4% 5 5>i FtBd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M Ft.Bd.M. "■ g .0677 .0729 .0781 .0833 .0885 .0938 .0990 .1042 .1094 34 .1354 .1457 .1562 .1667 .1770 .1875 .1979 .2083 .2188 3^ H 1. .2031 .2187 .2344 .2500 .2656 .2813 .2969 .3125 .3281 % .2708 .2917 .3125 .3333 .3542 .3750 .3958 .4167 .4375 1. g .3385 .3646 .3906 .4167 .4427 .4688 .4948 .5208 .5469 M .4063 .4375 .4688 .5000 .5313 .5825 .5938 .6250 .6563 >i H .4740 .5104 .5469 .5833 .6198 .6563 .6927 .7292 .7656 H 2. .5417 .5833 .6250 .6667 .7083 .7500 .7917 •8333 .8750 a. .6094 .6563 .7031 .7500 .7969 .8438 .8906 .9375 .9844 Ya, .6771 .7292 .7813 .8333 .8854 .9375 .9896 1.042 1.094 yi •/ .7448 .8021 ,8594 .9167 .9740 1.031 1.089 1.146 1.203 H sf* .8125 .8750 .9375 1.000 1.062 1.125 1.188 1.250 1.313 3. H .8802 .9479 1.016 1.083 1.151 1.219 1.286 1.354 1.422 34 H .9479 1.021 1.094 1.167 1.240 1.313 1.385 1.458 1.531 3i H 1.016 1.094 1.172 1.250 1.327 1.406 1.484 1.563 1.641 j^ 4. 1.083 1.167 1.250 1.333 1.416 1.500 1.588 1.667 1.750 4. 1.151 1.240 1.328 1.417 1.504 1.594 1.682 1.771 1.859 ^ 14 1.219 1.313 1.406 1.500 1.593 1.688 1.781 1.875 1.969 yi % 1.286 1.384 1.484 1.583 1.681 1.781 1.880 1.979 2.078 % i. 1.354 1.457 1.566 1.666 1.770 1.875 1.979 2.083 2.188 5. Yi. •1.422 1.530 1.644 1.750 1.858 1.969 2.078 2.188 2.297 % ^ 1.490 1.603 1.722 1.833 1.947 2.063 2.177 2.292 2.406 % ' 1.557 1.676 1.800 1.917 2.035 2.156 2.276 2.396 2.516 %. 1.625 1.750 1.875 2.000 2.125 2.250 2.375 2.500 2.625 6.* -u 1.69S 1.823 1.953 2.083 2.214 2..344 2.474 2.604 2.7.^54 ^ rz 1.760 1.896 2.031 2.167 2.302 2.438 2.573 2.708 2.843 y*. 1.828 1.969 2.109 2.250 2.391 2.531 2.672 2.813 2.953 % 7. 1.896 2.042 2.188 2.333 2.479 2.625 2.771 2.917 3.063 7. 1.964 2.115 2.266 2.416 2.568 2 719 2.870 3.021 3.172 M ^ 2.031 2.187 2.344 2.500 2 656 2 813 2.969 3.125 3.281 3^ 2.099 2.260 2.422 2.583 2.745 2.906 3.068 8.229 3.891 8. 2.167 2.333 2.500 2.667 2.833 3.000 3.167 3.333 3.500 8. M 2.234 2.406 2.578 2.750 2.922 3.094 3.266 8.438 3.609 ^ >^ 2.302 2.479 2 656 2.833 3.010 3.188 3.365 8.542 3.718 ^ 2.370 2.552 2.734 2.916 3.099 3.281 3.464 3.646 3.828 5i 9. 2. t38 2.625 2.813 3.000 3.187 3.375 3.563 3.750 3.938 9. M 2.505 2.698 2.891 3.083 3.276 3.469 3.661 3.854 4.047 ^ ^ 2.573 2.771 2.909 3.167 3.365 3.563 3.760 3.958 4.156 ?i 2.641 2.844 3.047 3.250 3.453 3.656 3.859 4.063 4.266 ^ 10. 2.708 2.917 3.125 3.333 3.542 3.750 3.958 t.l67 4.375 10. ^ 2.776 2.990 3.203 3.416 3.630 3.844 4.057 . 4.271 4.484 34 2.844 3.063 3.281 3.500 3.719 3.938 4.156 4.375 4.594 H 9i 2.911 3.135 3.359 3.583 3.807 4.031 4.255 4.479 4.703 H 11. 2.979 3.208 3.488 3.666 3.896 4.125 4.354 4.588 4.813 11. S 3.047 8.281 3.516 3.750 3.984 4.219 4.453 4.688 4.922 ^ 3.115 8.354 3.594 3.833 4.073 4.313 4.552 4.792 5.031 J< ?i 3.182 3.427 3.672 3.916 4.161 4.406 4.651 4.896 5.141 y* IJ. 3.250 3.500 3.750 4.000 4.250 4.500 4.750 5.000 5.250 12. >^ 3.385 3.646 3.906 4.167 4.427 4.688 4.948 5.208 5.469 Ji 18. 3..521 3.792 4.063 4.333 4.604 4.875 5.146 5.417 5.688 13. ^ 3.656 3.938 4.219 4.500 4.781 5.063 5.344 5.625 5.906 yi 14. 3.792 4.083 4.375 4.667 4.958 5.250 5.542 5.833 6.125 14. H 3.927 4.229 4.531 4.833 5.135 5.438 5.740 6.042 6.344 « 15. 4.063 4.375 4.688 5.000 5.313 5.625 5.938 6.250 6.563 15. ^ 4.198 4.521 4.844 5.166 5.490 5.813 6.135 6.458 6.781 Ji 16. 4.. 333 4.667 5.000 5.333 5.667 6.000 6.333 6.667 7.000 1«. % 4.469 4.813 5.156 5.500 5.844 6.188 6.531 6.875 7.219 X IT. 4.604 4.958 5.313 5.667 6.021 6.375 6.729 7.083 7.488 IT. )^ 4.740 5.104 5.469 5.833 6.198 6.563 6.927 7.292 7.656 J« 18. 4.875 5.250 5.625 6.000 6.375 6.750 7.125 7.500 7.875 18. 19. 5.146 5.542 5.938 6.333 6.729 7.125 7.521 7.917 8.318 19. 20. 5.417 5.838 6.250 6.667 7.083 7.500 7.917 8.333 8.750 20. 21. 5.688 6.125 6.563 7.000 7.438 7.875 8.313 8.750 9.188 21. 22. 5.958 6.417 6.875 7.333 7.792 8.250 8.708 9.167 9.625 22. 23. 6.229 6.708 7.188 7.667 8.145 8.626 9.104 9.583 10.06 23. Si. C.500 7.000 7.500 8.000 8.500 9.000 9.600 10.00 10.60 ai. BOARD MEASURE. 271 Table of Board HCeasnre— -(Contintied.) .s« Feet of Board Measure contained in one running foot of Scantlings | H - * of different dimensions (Original.) j A^ P 2rt THICKNESS IN INCHES. 53^ ^H 6 63i 63i ^Vi T 1H 7K Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. FtBd.M. Ft.Bd.M. FkBd.M. Ft.Bd.M. FtBd.M. 34 .1146 .1198 .1250 .1302 .1354 .1406 .1458 .1510 .1563 H 1^ .2292 .2396 .2500 .2604 .2708 .2813 .2917 .3021 ,3125 H i^ .3438 . .3594 .3750 .3906 .4063 .4219 .4375 .4531 .4688 h 1. .4583 .4792 .5000 .5208 .5417 .5625 .5«33 .6042 .6250 1. M .5729 .5990 .6250 .6510 .6771 .7031 .7292 .7552 .7813 g 8 .6875 .7188 .7500 .7812 .8125 .8438 .8750 .9062 .9375 ^ .8021 .8385 .8750 .9115 .9479 .9844 1.020 1.057 1.094 H X. .9167 .9583 1.000 1.042 1.083 1.125 1.167 1.208 1.250 2. 1.031 1.078 1.125 1.172 1.219 1.266 1.313 1.359 1.406 1^ 1.146 1.198 1.250 1.302 1.354 1.406 1.458 1.510 1.563 }i a^ 1.260 1.318 1.375 1.432 1.490 1.547 1.604 1.661 1.719 H 8. 1.375 1.438 1.500 1.562 1.625 1.688 1.750 1.813 1.875 8. ^ 1.490 1.557 1.625 1.693 1.760 1.828 1.896 1.964 2.031 H ^ 1.604 1.677 1.750 1.823 1.896 1.969 2.042 2.115 2.188 H H 1.719 1 1.797 1.875 1.953 2.031 2.109 2.188 2.266 2.344 H 4. 1.833 , 1.917 2.000 2.083 2.167 2.250 2.333 2.417 2 500 4. 34 1.948 1 2.036 2.125 2.214 2.302 2.391 2.479 2.568 2.656 ^ 3^ 2.063 2.156 2.250 2.344 2.438 2.531 2.625 2.719 2.813 H 2.177 2.276 2.375 2.474 2.573 2.672 2.771 2.870 2.969 H 5. 2.292 2.396 2.500 2.604 2.708 2.813 2.917 3.021 3.125 5. 34 2.406 2.516 2.625 2.734 2 844 2.953 3.063 3.172 3.281 H 3^ 2.521 2.635 2.750 2.865 2.979 3.094 3.208 3.323 3.438 « ^ 2.635 2.755 2.875 2.995 3.115 3.234 3.354 3.474 3.594 H 6 2.750 2.875 3.000 3.125 3.250 3.375 , 3.500 3.625 3.750 6. 34 2.865 2.995 3.125 3.255 3.385 3.516 3.646 3.776 3.906 14 2.979 3.115 3.250 3.385 3.521 3.656 3.792 3.927 4.063 H 3.094 3.234 3.375 3.516 3.656 3.797 3.938 4.078 4.219 ^ 7. 3.208 3.354 3.500 3.646 3.792 3.938 4.083 4.229 4.375 7. 34 3.323 3.474 3.625 3.776 3.927 4.078 4.229 4.380 4.531 g 3.438 3.594 3.750 3.906 4.063 4.219 4.375 4.531 4.688 M 3.552 3.714 3.875 4.03G 4.198 4.359 4.521 4.682 4.844 ^ 8. 3.667 3.833 4.000 4.167 4.333 4.500 4.667 4.833 5.000 8. 3.781 3.953 4.125 4.297 4.469 4.641 4.813 4.984 5.156 1^ 34 3.896 4.073 4.250 . 4.427 4.604 4.781 4.957 5.135 5.313 ^ a^ 4.010 4.193 4.375 4.557 4.740 4.922 5.103 5.286 5.469 ^ 9. 4.125 4.313 4.500 4.687 4.875 5.063 5.249 5.438 5.625 9. 34 4.240 4.432 4.625 4.818 5.010 5.203 5.395 5.589 5.781 ^ 3I 4.354 4.552 4.750 4.948 5.146 5.344 5.541 5.740 5.938 h S 4.469 4.672 4.875 5.078 5.281 5.484 5.687 5.891 6.094 H 10. 4.583 4.792 5.000 5.208 5.417 5.625 5.833 6.042 6.250 10. H 4.698 4.911 5.125 5.339 5.552 5.766 5.979 6.193 6.406 ^ H 4.813 5.031 5.250 5.469 5.688 5.906 6.125 6.344 6.5C3 ^ H 4.927 5.151 5.375 5.599 5.823 6.047 6.271 6.495 6.719 ^ 11. 5.042 5.271 5 500 5.729 5.958 6.188 6.417 6.646 6.876 11. 34 5.156 5.391 5.625 5.859 6.094 6.328 6.563 6.797 7.031 g S 5.271 5.510 5.750 5 990 6.229 6.469 6.708 6.948 7.188 H 5.385 5.630 5.875 6.120 6.365 6.609 6.854 7.099 7.344 H 12. 5.500 5.750 6.000 6.250 6.500 6.750 7.000 7.250 7.500 12. ^ 5.729 5.990 6.250 6.510 6.771 7.031 7.292 7.552 7.813 H 13 5.958 6.229 6.500 6.771 7.042 7.313 7.583 7.854 8.125 13. 3>i 6.188 6.469 6.750 7.031 7.313 7.594 7.875 8.156 8.-; 08 H U 6.417 6.708 7.000 7.292 7.583 7.875 8.167 8.458 8.750 14. 3i 6.646 6.948 7.250 7.552 7.854 8.156 8.458 8.760 9.063 H 15 6.875 7.188 7.500 7.812 8.125 8.438 8.750 9.063 9.375 15. 3^ 7.104 7.427 7.750 8.073 8.396 8.719 9.042 9.365 9.688 3i 16. 7.333 7.667 8.000 8.333 8.667 9.000 9..S33 9.667 10.00 16. }i 7.563 7.906 8.250 8.594 8.938 9.281 9.625 9.969 io.;'.i H 17. 7.792 8.146 8.500 8.854 9.208 9.563 9.917 10.27 10.6;} 17. yi 8.021 8.385 8.750 9.115 9.479 9.844 10.21 10.57 10.94 H 18. 8.250 8.625 9.000 9.375 9.750 10.13 10.50 10.88 11.25 18. 19. 8.708 9.104 9.500 9.896 10.29 10.69 11.08 11.48 11.88 19. 20. 9.167 9.583 10.00 10.42 10.83 11.25 11.67 12.08 12.50 20. 21. 9.625 10.06 10.50 10.94 11.38 1K81 12.25 12.69 13.13 21. 32. 10.08 10.54 11.00 11.46 11.92 12.38 12.83 13.29 13.75 22. 23. 10.54 11.02 11.50 11.98 12.46 12.94 13.42 13.90 14.38 2». 34. 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 24. 272 BOARD MEASURE. Table of Board Measure — (Continued.) a- Feet of Board Measure contained in one running foot of Scantlings fi of dififerent dimensions . (Original.) •^ • 5^ TI IICKN] ESS IN INCHES. 1 1 m 8 8M 8>^ m 9 9M 9J^ m Ft.Bd.M. Ft.Bd.M. FtBd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. FtBd.M. FtBd.M. Ft.Bd.M. M .1615 .1667 .1719 .1771 .1823 .1875 .1927 .1979 .2031 Va. K .3229 .3333 .3438 .3542 .3646 .3750 .3854 .3958 .4063 M ?i .4844 .5000 .5156 .5313 .5467 .5625 .5781 .5938 .6094 H 1. .6458 .6667 .6875 .7083 .7292 .7500 .7708 .7917 .8125 1. K .8073 .8333 .8594 .8854 .9115 .9375 .9633 .9896 1.016 M .9688 1.000 1.031 1.063 1.094 1.125 1.156 1.188 1.219 >i ?i 1.130 1.167 1.203 1.240 r.276 1.313 1.349 1.385 1.422 y^ 2. 1.292 1.333 1.375 1.417 1.458 1.500 1.542 1.583 1.625 2. H 1.453 1.500 1.547 1.594 1.641 1.688 1.734 1.781 1.828 ^ ^ 1.615 1.667 1.719 1.771 1.822 1.875 1.927 1.979 2.031 M ?i 1.776 1.833 1.891 1.948 2.005 2.063 2.120 2.177 2.234 ^ 8. 1.938 2.000 2.063 2.125 2.188 2.250 2.313 2.375 2.438 8. K 2.099 2.167 2.234 2.302 2.370 2.438 2.505 2.573 2.641 /4 }4 2.260 2.333 2.406 2.479 2.552 2.625 2.698 2.771 2.844 ^ % 2.422 2.500 2.578 2.656 2.734 2.813 2.891 2.969 3.047 ^ 4. 2.583 2.667 2.750 2.833 2.917 3.000 3.083 3.167 3.250 4. V 2.745 2.833 2.922 3.010 3.099 3.188 3.276 3.365 8.453 yi j^ 2.906 3.000 3.094 3.188 3.281 3.375 3.469 3.563 3.656 H ?i 3.068 3.167 3.266 3.365 3.464 3.563 3.661 3.760 3.859 H 5. 3.229 3.333 3.438 3.542 3.646 3.750 3.854 3.958 4.063 5. yi 3.391 3.500 3.609 3.719 3.836 3.938 4.047 4.156 4.266 Ji /^ 3.552 3.667 3.781 3.896 4.010 4.125 4.240 4.354 4.469 H % 3.714 3.833 3.953 4.073 4.193 4.313 4.432 4.552 4.672 H B. 3.875 4.000 4.125 4.250 4.375 4.500 4.625 4.750 4.875 6. M 4.036 4.167 4.297 4.427 4.557 4.688 4.818 4.948 5.078 34 J^ 4.198 4.333 4.469 4.604 4.740 4.875 5.010 5.146 5.281 }4 ?^ 4.359 4.500 4.641 4.781 4.922 5.063 5.203 5.344 5.484 H 7. 4.521 4.667 4.813 4.958 5.104 5.250 5.360 5.542 5.688 7. 1/ 4.682 4.8.33 4.984 5.135 5.286 5.438 5.590 5.740 5.891 yi iz 4.844 5.000 5.156 5.313 5.469 6.625 5.782 5.938 6.094 1^ fi 5.005 5.167 5.328 5.490 5.651 5.813 5.975 6.135 6.297 t£ 8. 5.167 5.333 5.500 5.667 5.833 6.000 6.167 6.333 6.500 8. >i 5.328 5.500 5.672 5.844 6.016 6.188 6.359 6.531 6.703 Ji M 5.490 5.667 5.844 6.021 6.198 6.375 6.552 6.729 6.906 ^ ?i 5.^)51 5.833 6.016 6.198 6.380 6.563 6.745 6.927 7.109 ^ 9 5.813 6.000 6.188 6.375 6 563 6.750 6.938 7.125 7.318 9. M 5.974 6.167 6.359 6.552 6.745 6.938 7.130 7.323 7.516 1^ s 6.135 6.3.33 6.531 6.729 6.927 7.125 7.323 7.521 7.719 ^ H 6.297 6.500 6.703 6.906 7.109 7.313 7.516 7.719 7.922 % 10. 6.458 6.667 6.875 7.083 7.292 7.500 7.708 7.917 8.125 10. H 6.620 6.833 7.047 7.260 7.474 7.688 7.901 8.115 8.328 y^ ^ 6.781 7.000 7.219 7.438 7.656 7.675 8.094 8.313 8.531 ^ ^ 6.943 7.167 ]1& 7.615 7.839 8.063 8.286 8.510 8.734 ^ 11. 7.104 . 7..S33 7.792 8.021 8.250 8.479 8.708 8.938 11. M 7.266 7.500 7.735 7.969 8.203 8.438 8.672 8.906 9.141 ^ ^ 7.427 7.667 7.906 8.146 8.386 8.625 8.865 9.104 9.344 ^ ^ 7.589 7.833 8.078 8.323 8.568 8.813 9.057 9.302 9.547 ^ 12. 7.750 8.000 8.250 8.500 8.750 9.000 9.250 9.500 9.750 12. H 8.073 8.333 8.594 8.854 9.115 9.375 9.635 9.896 10.16 H 13. 8.396 8.666 8.938 9.208 9.479 9.750 10.02 10.29 10.56 IS. ^ 8.719 9.000 9.281 9.563 9.844 10.13 10.41 10.69 10.97 H U. 9.042 9..S33 9.625 9.917 10.21 10.50 10.79 11.08 11.38 14. ^ 9.365 9.666 9.969 10.27 10.57 10.88 11.18 11.48 11.78 H 15. 9.688 10.000 10.31 10.63 10.94 11.25 11.56 11.88 12.19 15. ^ 10.01 10.33 10.66 10.98 11.30 11.63 11.95 12.27 12.59 H 1«. 10.33 10.67 11.00 11.33 11.67 12.00 12.33 12.67 13.00 16. 3^ 10.66 11.00 11.34 11.69 12.03 12.38 12.72 13.06 13.41 H 17. 10.98 11.33 11.69 12.04 12.40 12.75 13.10 13.46 13.81 IT. % 11.30 11.66 12.03 12.40 12.76 13.13 13.49 13.85 14.22 H 18. 11.63 12.00 12.38 12.75 13.13 13.50 13.88 14.25 14.63 18. 19. 12.27 12.67 13.06 13.46 13.85 14.25 14.65 15.04 15.44 19. 20. 12.92 13.33 13.75 14.17 14.58 15.00 15.42 15.83 16.25 20. 21. 13.56 14.00 14.44 14.88 15.31 15.75 16.19 16.63 17.06 21. 22. 14.21 14.66 15.13 15.58 16.04 16.50 16.96 17.42 17.88 22. 23. 14.85 15.33 15.81 16.29 16.77 17.25 17.73 18.21 18.69 28. 24. 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 14. BOARD MEASURE. 273 Table of Board Measure— (Continued.) •s. Feet of Board Measure contained in one running foot of Scantlings 3i of dififerent dimensioas. (Original.) •So 55 TE [icKNEss m INCH ES. 10 103^ 10>^ 10?^ 11 nn 11^ UH 12 Ft Bd.M. Ft. Bd.M. Ft. Bd.M. Ft.Bd.M. Ft. Bd.M. Ft.Bd.M. FtB^ 4.375 4.484 4.594 4.703 4.813 4.922 5.031 5.141 5.250 M 4.583 4.698 4.813 4.927 5.042 5.156 5.270 5.385 5.500 34 H 4.792 4.911 5.031 5.151 5.271 5.391 5.510 5.6.30 5.750 H 6. 5.000 5.125 5.250 5.375 5.500 5.625 5.750 5.875 6.000 6. ^ 5.208 5.339 5.469 5.599 5.729 5.859 5.990 6.120 6.250 H 5.417 5.552 5.688 5.823 5.958 6.094 6.229 6.365 6.500 \y ?^ 5.625 5.766 5.906 6.047 6.188 6.328 6.469 6.609 6.750 H. 7. 5.833 5.979 6.125 6.271 6.417 • 6.563 6.708 6.854 7.000 7. 6.042 6.193 6J544 6.495 6.646 6.797 6.948 7.099 7.250 ^ J^ 6.250 6.406 8.563 6.719 6.875 7.031 7.188 7.344 7.500 % p. 458 6 620 6.781 6.943 7.104 7.266 7.427 7.589 7.750 K 8. 6.667 6.833 7.000 7.167 7.333 7.500 7.667 7.833 8.000 8. ^ 6.875 1.047 7.219 7.391 7.563 7.734 7.906 8.078 8.250 H, ^ 7.083 7.260 7.438 7.615 7.792 7.969 8.146 8-323 8.500 H H 7.292 7.474 7.656 7.839 8.021 8.203 8.385 8.568 8.750 % 9. 7.500 7.688 7.876 8.06:i 8.250 8.438 8.625 8.813 9.000 9.* >4 7.708 7.901 8.094 8.286 8.479 8.672 8.865 9.057 9.250 >^ 7.917 8.115 8.313 8.510 8.709 8.906 9.104 9.302 9.500 ^ H 8.125 8.323 8.531 8.734 8.839 9.141 9.344 9.547 9.750 ^ 10. 8.333 8.542 8.750 8.958 9.167 9.375 9.583 9.792 10.00 10. 34 8.542 8.755 8.969 9.182 9.396 9.609 9.823 10.04 10.25 3i 3^ 8.750 8.969 9.188 9.*06 9.625 9.844 10.06 10.28 10.50 ^ ?i 8.958 9.182 9.406 9.630 9.854 10.08 10.30 10.53 10.75 H 11. 9.167 9.396 9.625 9.854 10.08 10.31 10.54 10.77 11.00 11. ^ 9.375 9.609 9.844 10 08 10.31 10.55 10.78 11.02 11.25 H M 9.583 9.823 10.06 10.30 10.54 10.78 11.02 11.26 11.50 H 9i 9.792 10.04 10.28 10.53 10.77 11.02 11.26 11.51 11.75 H 12. 10.00 10.25 10.50 10.75 11.00 11.25 11.50 11.75 12.00 12. 14 10.42 10.68 10.94 11.20 11.46 11.72 11.98 12.24 12.50 H 13. 10.83 11.10 11. .38 11.65 11.92 12.19 12.46 12.73 13.00 13 3^ 11.25 11.53 11.81 12.09 12.38 12.66 12.94 13.22 13.50 H U. 11.67 11.96 12.25 12.54 12.83 13.13 13.42 13.71 14.00 14 J^ 12.08 12.39 12.69 12.99 13.29 13.59 13.90 14.20 14.50 ^ 15. 12.50 12.81 13.13 13.44 13.75 14.06 14.38 14.69 15.00 15 ^ 12.92 13.24 13.56 13.89 14.21 14.53 14.85 15.18 15.50 }i 16. 13.33 13.67 14.00 14.33 14.67 15.00 15.33 15.67 16.00 16. 3^ 13.75 14.09 14.44 14.78 15.13 15.47 15.81 16.16 16.50 H 17. 14.17 14.52 14.88 15.23 15.58 15.94 16.29 16.65 17.00 17. 34 14.58 14.95 15.31 15.77 16.04 16.41 16.77 17.14 17.50 H re. 15.00 15.38 15.75 16.13 16.50 16.88 17.25 17.63 18.00 18. 19. 15.83 16.23 16.63 17.02 17.42 17.81 18.21 18.60 19.00 19. 20. 16.67 17.08 17.50 17.92 18.33 18.75 19.17 19.58 20.00 20. 21. 17.50 17.94 18.38 18.81 19.25 19.69 20.13 20.56 21.00 21. 22. 18.33 18.79 19.25 19.71 20.17 20.63 21.08 21.54 22.00 22. 23. 19.17 19.65 20.13 20.60 21.08 21.56 22.04 22.52 23.00 28. a. 30.00 20.50. ai.oo 21.50 22.00 22.50 23.00 23.50 24.00 M- 18 274 LAND SURVEYING. LAND SURVEYING. In surveying a tract of ground, the sides which compose its outline are desig» nated by numbers in the order in which they occur. That end of each side whicn first presents itself in the course of the survey, may be called its w«z?- end; and the other its /ar end. The number of each side is placed at its far end. Thus, in'Fig. 1, the survey being supposed to commence at the corner 6, and to follow the'direo- tion of the arrows, the first side is 6, 1 ; and its number is placed at its far end at 1; and so of the rest. Let NS be a meridian line, that, is, a north and south line; and EW an east and west line. Then in any side which runs northwardly. wliether northeast, as side 2; or northwest, as sides 5 and 1 ; or due north; the distance in a due north direction between its near end and its far end, is called its northing; thus, a 1 is the northing of side 1; 16 the northing of side 2; 4c of side 5. In like manner, if any side runs in a southwardly direction, whether southeastwardly, as side 3; or southwestwardly, as sides 4 and 6 ; or due south ; the corresponding distance in a due south direction between its near end and its far end, is called its southing; thus, d3 is the southing of side 3; 3e of side 4; /6 of side 6. Both northings and southings are included in the general term Difference of Latitude of a side ; or, more commonly but erroneously, its latitude. The distance due east, or due west, between the near and the far end of any side, is in like manner called the easting, or westing, of that side, as the case may be; thus, 6 a is the westing of side 1 ; 5/ of side 6; c5 of side 5; «4 of side 4; and 6 2 is the easting of side 2 ; 2d of side 3. Both eastings and westings are included in the general term Departure of a side; implying that the side departs so far from a north or south direction. We may say that a side norths, wests, southeast*, &c. We shall call the northings, southings, &c. the Ns, Ss, £s, and Ws ; the lati- tudes, lats ; and the departures, deps. Perfect accuracy is unattainable in any operation involving the measure- ments of angles and distances.* That work is accurate enough, which cannot be made more so without an expenditure more than commensurate with the object to be gained. There is no great difficulty in confining the uncertainty within about one-half per cent, of the content, and this probably never pre- vents a transfer in farm transactions. But errors always become apparent when we come to work out the field notes ; and since the map or plot of the survey, and the calculations for ascertaining the content, should be consistent within them-* selves, we do what is usually called correcting the errors, but what in fact is simply humoring them in, no matter how scientific the process may appear. We distrib- ute them all around the survey. Two methods are used for this purpose, both based upon precisely the same principle; one by means of drawing; the other, more exact but much more troublesome, by calculation. The graphic method, in the hands of a correct draftsman, is sufficiently exact for all ordinary purposes. Add all the sides in feet together; and divide the sum by their number, for the average length. Divide this average by 8; the quotient will be the proper scale in feet per inch. In other words, take about 8 ins. to represent an average side. We shall take it for granted that an engineer does not consider it accurate work to * A 100 ft. chain may vary its length 5 feet per mile, between winter and summer, by mere change of temperature ; and this alone will make a difference of about 1 acie in 533. The atu. dent should practise plotting from perfectUr accurate data ; as from the example in table, p. 281, or lAND SURVEYING. 275 ntaanre his angles to the nearest quarter of a degree, •which is the usual praetice among land-survey ors. They can, by means of the engineer's transit, now in universal use on our public works, be readily measured within a minute or two ; and being thus much more accurate than the compass courses, (which cannot be read off so closely, and which are moreover subject to many sources of error,) they serve to correct the latter in the office. The noting of the courses, however, should not be confined t« the nearest, quarters of a degree, but should be read as closely as the observer can guess at the minutes. The back courses also should be taken at every corner, as an additional check, and for the detection of local attraction. It is well in taking the com- pass bearings, to adopt as a rule, always to point the north of the compass- box toward the object whose bearing is to be taken, and to read ofiF from the north end of the needle. A person who uses indifferently the N and the S of the box, and of the needle, will be very liable to make mistakes. It is best to measure the least angle (shown by dotted arcs, Fig 2,) at the corners ; whether it be exterior, as that at corner 5; or interior, as all the others ; because it is al- ways less than 180°; so that there is less danger of reading it off incor- rectly, than if it exceeded 180° ; taking it for grant- ed that the transit instrument is graduated from the same zero to 180° each way ; If it is graduated from zero to 360° the precaution is useless. When the small angle is exterior, subtract it from 360* for the interior one. Supposing the field work to be finished, and that we require a plot from which the contents may be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be measured by scale, and their areas calculated one by one,) a protraction of it may be made at once from the field notes, either by using the angles, or by first correcting the bearings by means of the angles, and then using them. The last is tlie best, because in the first the protractor must be moved to each angle ; whereas in the last it will remain stationary while all the bearings are being pricked off. Every movement of it increases the liability to errors'. The manner of correcting the be»ring3 is explained on the next page. In either case the protracted plot will certainly not close precisely ; not only in consequence of errors in the field work, but also in the protracting itself. Thus the last side, No 6, Fig 2, instead of closing in at corner 6, will end somewhere else, say, for instance, at t; the dist t 6 being the closing error, which, however, as represented in Fig 2, is more than ten times as great, proportionally to the size of the survey, as would be allowable in practice. Now to humor-in this error, rule through every corner a short line parallel to «6; and, in all cases, in the direction from « (wherever it may be) to the •tarting point 6. Add all the sides together ; and measure « 6 by the scale of the plot. Then befiM* mine at corner 1, at the far end of side 1, say, as the 6-^.'^ Fig. 2. Sum of all the sides Total closing error t 6 Bidel Error for side 1. Lay «£f this error from 1 to a. Sum of all the sides Then at corner 2, say, as the Total closing , , Sum of , Error error < 6 • • sides 1 and 2 • for side 2. "Which error lay off from 2 to & ; and so at each of the corners ; always using, as the third term, th« «um of the sides between the starting point and the given corner. Finally, join the points o, b, c, d, e, 6; and the plot is finished. The correction has evidently changed the length of every side; lengthening some and shortening others. It has also changed the angles. The new lengths and angles may with tolerable accuracy ba found by means of the scale and protractor ; and be marked on the plot instead of the old ones. from those to be found in books on surveying. This is the only way in which he can learn what i« meant by accurate work. His semicircularprotractor should be about 9 to 12 ins in diam and gradu- ated to 10 min. His straight edge and triangle should be of metal; we prefer German silver, which does not rust as steel does ; and they should be made with scrupulous accuracy by a skilful instru- ment-maker. A very fine needle, with a sealing-wax head, should be used for pricking off dists and angles ; it must be held vertically ; and the eye of the draftsman must be directly over it. The lead pencil should be hard (Faber's No. 4 is good for protracting), and must be kept to a sharp point by rubbing on a fine file, after using a knife for removing th« wood. The scale should be at least as long as the longest side of the plot, and should be made at the edge of a strip of the same paper as the plot is drawn on. This will obviate to a considerable extent, errors arising from contraction and expan- sion. Unfortunately, a sheet of paper does not contract and expand in the same proportion length- «rise and crosswise, thus preventing the paper scale from being a perfect corrective. In plots of com- mon farm surveys, &c, however, the errors from this source may be neglected. For such plots as majf *e protracted, divided, and computed within a time too shorttoadmit of appreciable change, theordi- iary scales of wood, ivory or metal may be used ; but satisfactory accuracy cannot be obtained with ;hem on plots requiring several davs. if the air be meanwhile alternately moist and dry, or subject to considerable variations in temperature. What is called parchment paper is worse in this respect thiu» good ordinary drawing-paper. With the foregoing precautions we may worli from a drawing, with as much accuracy as is uauaUy attained in the field work. 276 LAND SURVEYING. When the plot baa many sides, tbis calculating the error for each ol them becomes tedious ; aa4 since, in a well-performed survey and protraction, the entire error will be but a very small quantity, fit should not exceed about x^tt P**"' °^ '^^ periphery,) it may usually be divided among the sides by merely placing about J4, ^, and 5i of it at corners about J4» /^» "■'id % ^^7 around the plot ; and at intermediate corners propor- tion it by eye. Or calculatioa may be avoided entirely by drawing a line a 6 of a length equal to the united lengths of all the sides ; dividing it Into distances a, 1 ; 1, 2 ; &c, equal to the respective sides. Make 6 c equal to the entire closing error ; join a c ; and draw 1 , 1' ; 2, 2' , &c, which will give the error at each corner. When the plot is thus completed, it may be divided by fine pencil lines into triangles, whose bases and heights may be measured by the scale, in order to compute the contents. With care in both the survey and the drawing, the error should not exceed about ^4^7 P^^' °^ *'^® ^'""^ *''^'=^' ^* least two distinct sets of triangles should be drawn and computed, as a guard against mistakes ; and if the two sets differ in calculated contents more than about j^-^ part, they have not been as carefully prepared as they should have been. The closing error due to imperfect field-work, may be accurately •alculated, as we shall show, and laid down on the paper before beginning the plot ; thus furnishing a perfect test of the accuracy of the protraction work, which, if correctly done, will not close at the point of beginning, but at the point which indicates the error. But this caiculation of the error, by a little additional trouble, furnishes data also for dividing it by calculation among the difiT sides ; besides the means of drawing the plot correctly at once, without the use of a protractor ; thus ena- bling us to make the subsequent measurements and computations of the triangles with more cer- tainty. We shall now describe this process, but would recommend that even when it is employed, and especially in complicated surveys, a rough plot should first be made and corrected, by the first of the two mechanical methods already alluded to. It will prove to be of great service in using the method by calculation, inasmuch as it furnishes an eye check to vexatious mistakes which are otherwise apt to occur ; for, although the principles involved are extremely simple, and easily remembered when once understood, yet the continual changes in the directions of the sides will, without great care, cause us to use Ns instead of Ss ; Es instead of Ws, &c. We suppose, then, that such a rough plot has been prepared, and that the angles, bearings, and distances, as taken from the field book, are figured upon it in lead pencil. Add together the interior angles formed at all the corners : call their sum a. Mult the number of sides by 180° ; from the prod subtract 360° : if the remainder is equal to the sum a, it is a proof that the angles have been correctly measured.* This, however, will rarely if ever occur; there will always be some discrepancy ; but if the field work has been performed with moderate care, this will not exceed about two min for each angle. In this case div it in equal parts among all the angles, adding or subtracting, as the case may be, unless it amounts to less than a min to each angle, when it may be entirely disregarded in common farm surveys. The corrected angles may then be marked on the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be a« shown in Fig 3. Next, by means of these corrected angles, correct the bearings also, thus. Fig 3 ; Select some side (the longer the better) from the two ends of which the bearing and the reverse bearing agreed ; thus showing that that bearing was probably not influenced by local attraction. Let '?ide 2"be the one so selected ; as- sume its bearing, N 75° 32' E, as taken on the ground, to be correct; through either end of it, as at its far end 2, draw the short meridian line ; par- allel to which draw others through every corner. Now, having the bearing of side 2, N 75° .32' E, and requiring that of side 3, it is plain that the reverse bearing from cor- ner 2 is S 75° 32' W; and that therefore the angle 1, 2, m, is 75° 32'. Therefore, if we take 75° 32' from the entire corrected angle 1, 2, 3, or 144° 57', the rem 69° 25' will be the angle m 23 ; consequently the bearing of side 3 must be 8 69° 25' E. For finding the bearing of side 4, we now have the angle 23 a of the reverse bearing of side 3 also equal to 69° 25' ; and if we add this to the entire corrected angle 234, or to69° 32', we have the angle a 34 =69° 25' +69° 32' = 138° 57'; which taken from 180°, leaves the angle 6 34 — 410 3'; consequently the bearing of side 4 must be S 41° 3' W. For the bearing of side 5 we now have the angle 34 c = 41° 3', which taken from the corrected angle 345, or 120° 43', leaves the angle c 45 =-"9° 40' • consequently the bearing of side 5 must be N 79° 40' W. At corner 5, for the bearing of side 8, we have the angle 45 d = 79° 40', which taken from 133° 10', leaves the angle d 56 = 53° 30' ; consi*- quently the bearing of side 6 must be S 53° 30' W. And so with each of the sides, nothing but Fig. 3. « Because in every straight-lined figure the sum of all its interior angles is equal to twice as vanj fight angles as the figure has sides, minus 4 right angles, or 360°., LAND SUBVEYING. 277 earcfnl observation la necessary to see hoir the several angles are to be employed at each oomer. Eules are sometimes given for this purpose, but unless frequently used, they are soon forgotten. The plot mechauically prepared obviates the necessity for such rules, inasmuch as the principle of proceeding thereby becomes merely a matter of sight, and tends greatly to prevent error from using the wrong bearings ; while the protractor will at once detect any serious mistakes as to the angles, and thus prevent their being carried farther along. After having obtained all the corrected bearings, they may be figured on the plot instead of those taken in the field. They will, however, require a still further correction after a while, since they will beafiFected by the adjustment of the closing error. We now proceed to calculate the closing error < 6 of Fig i, which is done on the principle that in a correct survey the northings will be equal to the southings, and the eastings to the westings. Pre- pare a table of 7 columns, as below, and in the first 3 cols place the numbers of the sides, and their oor. xected courses ; also the dists or lengths of the sides, as measured on the rough plot, ifsuchaooQ has been prepared; but if not, then as measured on the ground. Let them be as follows : Side. Bearing. Dist. Ft. Latitudes. Departures. N. S. E. W. 1 2 3 4 6 « N 16° 40' W N 75° 32' E S 69° 25' E S 41° 3' W N 790 40' W g 530 30' W 1060 1202 1110 850 802 705 1015.5 300.3 143.9 390.2 641. 419.3 1163.9 1039.2 304. 558.2 789. 566.7 1459.7 1450.5 1450.5 Error in Lat. 2203.1 Error in Dep. 2217.9 2203.1 9.2 14.8 Kow, by means of tne Table of Sines, etc., find the N, S, E, W, of the several sides, and place them in the corresponding four columns. Thus, for side 1, which is 1060 feet long, with bearing N 16° 40' W ; cos 16° 40' = O.itoSO ; sin le-^ 40' = 0.2868. Here N = 1060 x 0.9580 = 1015.5; and W = 1060 X 0.2868 = 304. Proceed thus with all. Add up the four cols ; find the diff between the N and S cols ; and also between the E and W ones. In this instance we find that the Ns are 9.2 feet greater than the Ss; and that the Ws are 14.8 ft greater than the Es; in other words, there is a closing error which would cause a eorrect protraction of our first three cols, to terminate 9.2 feet too far north of the starting point ; and 14.8 feet too far west of it. So that by placing this error upon the paper before beginning to protract, We should have a test for the accuracy of the protracting work ; but, as before remarked, a little more trouble will now enable us to di v the error proportionally among all the Ns, Ss, Es, and Ws, and thereby give ns data for drawing the plot correctly at once; without using a protractor at all. To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaces are farther apart ; and that the addings-up of the old N, S, E, W columns are omitted. The additioai here noticed are made subsequently. The new table is on the next page. Remark. The bearing: and the reverse bearing: from the two ends of a line will not read preciat-ly the same angle ; and the difference varies with the latitude and with the length of the line, but not in the same proportion with either. It is, however, generally too small to be detected by the needle, being, according to Gummere, only three quarters of a minute in a line one mile long in lat 40°. In higher lats it is more, and in lower ones less. It is caused by the fact that meridians or north and south lines are not truly parallel to each other ; but would if extended meet at the poles. Sence the only bearing: that can be run in a straig:ht line, with strict accuracy, is a true N and S one ; except on the very equator, where alone a due E and W one will also be straight. But a true curved E and W line may be found anywhere with sufficient accuracy for the surveyor's purposes thus. Having first by means of the N staV p 284, or otherwise got a true N and S bearing at the starting point, lay off from it 90o, for a true E and W bearing at that point. This E and W bearing will be tangent to the true E aud W curve. Run this tangent carefully ; and at intervals (say at the end of each mile) lay off from it (towards the N if in N lat, or vice versa) an offset whose length in feet is equal to the proper one from the following table, multiplied by the square of the distance in mUes from the starting point. These offsets will mark points in the true E and W curve. l,atitude :N^ or IS. 350 40° 450 50° 550 60= 65» 6° 10° 150 20° 250 30° Offsets in ft one mile from starting^ point. .058 .118 .179 .243 .311 .385 .467 .559 .667 .795 .952 1.15 1.43 Or, any offset in ft = .6666 X Total Dist in miles2 x Nat Tang of Lat. A rhumb line is any one that crosses a meridian obliquely, that is, i« neither due N and S, nor E and W. 278 LAND SUKVEYING. Side. Bearing. Dist. Ft. Latitudes. Departures. N. S. E. W. 1 N 16° 40' W N 75° 32' E S 69° 25' E S 41° 3' W N 79° 40' W S 53° 30' W 1060 1202 1110 850 802 705 1015.5 1.7 399.2 1.8 1163.9 3.1 304.0 2.7 2 1013.8... 300.3 1.9 ... 301.3 3 298.4 143.9 1.3 ... 1167.0 1039.2 2.9 4 392 ... 641.0 1.3 ... 1042.1 558.2 2.2 6 642.3... 419.3 1.1 ... 556.0^ 789.0 2.1 6 142.6... ... 786.9 566.7 1.8 420.4... 564.9 5729 Sum of Sides. 1454.8 Cor'd Ns. 1454.7 Cor'd Ss. 2209.1 Cor'd Es. 2209.1 Cor'd Ws. Now -we have already found by the old table that the Ns and the Ws are too long ; consequently they must be shortened ; while the Ss, and Ea, must be lengthened ; all in the following proportioas : As the Sum of all , Any given ,, Total err of , Err of lat, or dep, the sides • side • • lat or dep • of given side. Thus, commencing with the lat of side 1, we have, as Sum of all the sides. . Side 1. , . Total lat err. . Lat err of side 1. 5729 • 1060 • • 9.2 • 1.7 Now as the lat of side 1 is north, it must be shortened ; hence it become8 = 1015.5 — 1.7^10ia.8, M figured out in the new table. Again we have for the departure of side 1, Sum of all the sides. , Side 1. , , Total dep err. , Dep err of side 1. 5729 • 1060 • • 14.8 • 2.7 Now as the dep of side 1 is west, it must be shortened ; hence it becomes 304 — 2.7 = 301 .3 as figured out in the new table. ' Proceedinft thus with each side, we obtain all the corrected lats and deps as shown in the new table ; where they are con- nected with their respective Bides by dotted lines; but ia practice it is better to cross out the original ones when the cal- culation is finished and proved. If we now add up the 4 cols of corrected N, S, E,W, we find that the Ns =: the Ss ; and the E8= the Ws ; thus proving that the work is right. There is, it is 5/ \ / true, a discrepancy of .1 of a ft I -Y jl5_- — y between the Ns, and the Ss ; but this is owing to our carrying out the corrections to only one decimal place ; and is too small to be regarded. Discrepancies of 3 or 4 tenths of a foot will sometimes occur from this cause; but may be neglected. The corrected lats and deps must evidently change the bearing and distance of every •ide; but without knowing either of these, we ean now plot the survey by means of the corrected Fig. 4. LAND SURVEYING. 279 lats and deps alone. The principal is self-evident, explaining itself. First draw a meridian line N 8, Fig. 4; and upon it fix on a point 1, to represent the extreme west* comer of the survey. Then from the point 1, prick off by scale, northward, the dist 1, 2'=the corrected northing 298.4 of side 2, taken from the last table ; from 2' southward prick off the dist 2', 3', the corrected south- ing 392 of side 3 ; from 3' southward prick otf 3', 4',=southing 642.3 of side 4 ; from 4' northward prick off 4', 5'=northing 142.6 of side 5; from 5' prick off southward 5', G'^southing of side 6t. Then from the points 2^, 3', 4/, i/, &, draw indefinite lines due eastward, or at right angles to the neridian line. Make by scale, 2', 2 = corrected departure of side 2; and join 1, 2. Make 3', 3 = dep of side 24-depof side 3; and join 2, 3; make 4', 4=3', 3 — depof side 4; and join 3, 4; make 5', 5 =4', 4 — aep of side 5; andjoin4,5; make 6', 6=5', 5 — dep of side 6; andjoin5,64 Finally join S, 1 ; and the plot is complete. If scrupulous accuracy is not requited, the contents may be found by the mechanical method of triangles; the bearings, by the protractor; and the lengths of the sides by the scale ; all with an approximation sufficient for ordinary purposes ; and perhaps quite as close M by the method by calculation, when, as is customary, the bearings are taken only to the nearest faarter of a degree. We have already said that with a scale of feet per inch= ' 1 o 'the error of area need not exceed the '2'7)"5"th part. But if it is required to calculate the area of the corrected survey with rigorous exactness, it may be done on the following principle, (see Fig 5.) If a ai meridian line N S be sup- IN posed to be drawn through a g the extreme west corner 1 of ^ — a survey; and lines (called niddle dittancea) drawn (as the dotted ones in the Fig) •it right angles to said me- ridian, from the center of each side of the survey; ♦,hen if each of the middle lists of such sides as have aorthings, be mult by the jorrected northing of its cor- responding side ; and if each 9f the middle dists of such tides as have southings, be mult by the corrected south- ing of its corresponding side ; if we add all the north prods into one sum ; and all the south prods into another lum ; and subtract the least •f these sums from the great- Mt( the rem will be the area Fifir.5. * The extreme east corner would answer as well, with a slight change in the sabsequeat «p«r* atious, as will become evident. t Instead of pricking off these northings and southings in succession, /Vom each other, It will be more correct in practice to prepare first a table showing how far each of the points 2', 3', &o, is north or south from 1. This being done, the points can be pricked off north or south from 1, without mov- ing the scale each time; and of course with greater accuracy. Such a table is readily formed. Rule it as below ; and in the first three columns place the numbers of the sides (starting with side 2 from point 1 ;) and their respective corrected northings and southings. The formation of the 4th and 5th colt by means of the 3d and 4th ones, explains itself. Its accuracy is proved by the final result being 0. Dist Nor Sft-om Point 1.1 Side. N. lat. 8. lat. N. S. 298.4 298.4 392. 9e.s 642.3 735.9 142.6 693.3 420.4 1013.7 1013.8 000.0 000.0 J A similar table should be prepared beforehand for the dists of the points 2, 8, 4, Ac, east from the xheridian line. It is done in the same manner, but requires one col less, as all the dists are on th« •am* Bide of the mer line. Thus, starting from point 1, with side 2: Side. E. dep. W. dep. Dist east from meridian line. * 1167.0 1042.1 656.0 786.9 564.9 301.3 1167.0 2209.1 1653.1 866.S 301.8 000.0 This wwrk likewise proves itself by the final result being 0. 280 LAND SURVEYING. •r the BQrrey.* The corrected northings and southings we have already found ; as also the eastings and westings. The middle dists are found by means of the latter, by employing their halves ; adding half eastings, and subtracting half westings. Thus it is evident that the middle dist 2' of side 2, is equal to half the easting of side 2. To this add the other half easting of side 2, and a half easting of side 3 ; and the sum is plainly equal to the middle dist 3' of side 3. To this add the other half easting of side 3, and subtract a half westing of side 4. for the middle dist 4' of side 4. From this subtract the other half westing of side 4, and a half westing of side 5, for the middle dist 5' of sidt 5; and so on. The actual calculation may be made thus : Half easting of side 2 = = 583.5 B = mid dist of iide 2. 2 583.5 E 1042.1 1167.0 E Half easting of side 3 = = 521.0 E 1688.0 E = mid dist of side 8. 521.0 £ 556 2209.0 E Half westing of side 4 = — = 278.0 W 1931.0 E = mid dist of side 4. 278.0 W 786.9 1653.0 E Half westing of side 5 = = 393.5 W 1259.5 E = mid dist of «ide 6. 393.5 W Half westing of side 6 = ■ 301.3 Half westing of side 1 = = 583.6 E = 282.4 W 301.2 E 150.6 W mid dist of aide 6. 150.6 E = mid dist of side 1. The work always proves itself by the last two results being equal. Next make a table like the following, in the first 4 cols of which place the numbers of the sldee, the middle dists, the northings, and southings. Mult each middle dist by its corresponding northing or southing, and place the products in their proper col. Add up each col ; subtract the least from the Side. Middle dist. 150.6 583.5 1688 1931 1259.5 583.6 Northing. 1013.8 298.4 142.6 Southing. 392 642.3 North prod. South prod. 152678 174116 1240281 S45345 2147322 506399 43560)1640923(37.67 Acret. * Proof. To illustrate the principle upon which this rule is based, let ab, be, and c a, Fig 6, represent in order the 3 sides of the triangular plot of a survey, with a meridian line d/ drawn through the extreme west cor- ner, a. Let lines b d and cf be drawn from each corner, perp to the meridian line ; also from the middle of each side draw lines we, mn, so, also perp to meridian ; and representing the middle dists of the sides. Then since the sides are regarded in the order ab, be, ca, it is plain that a d represents the northing of the side a b ; fa the northing of ca; and df the southing of 6c. Now if we mult the northing ad oT the side ab, by its mid dist ew, the prod is the area of the triangle abd. In like manner the northing fa of the side c a, mult by its mid dist s o, gives the area of the triangle a cf. Again, the southing df of the side 6 c, mult by its mid dist w»n, gives the area of the entire Gg db cfd. If from this area we subtract the areas of the two triangles abd, and a cf, the rem is evidently the area of the plot a 6 e. So with any other plot, however oomplicata«U liAND SURVEYING. 281 greatest. The rem will be the area of the survey in sq ft ; which, div by 43560, (the number of sq ft u an acre,) will be the area in acres ; in this instance, 37.67 ac. It now remains only to calculate the corrected bearings and lengths of the sides of the survey, all of which are necessarily changed by the adoption of the corrected lats and deps. To find the bearing of any side, div its departure (E or W) by its lat (N or S) ; in the table of nat tang, find the quot ; the angle opposite it is the reqd angle of bearing. Thus, for the course of side 1, we have • '- ' 1013.8 N = .2972— nat tang ; opposite which in the table is the reqd angle, 16° 33' ; the bearing, therefore, ie K 163 33' w. Again : for the dist or length of any side, from the table of nat cosines take the cos opposite to the angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the cos. Taos for the dist of side 1, we find opposite 16° 33', the cos .9586. And Lat. Cos. 1013.8 -i- .9586 = 1057.6 the reqd dist. The following table contains all the corrections of the foregoing survey ; consequeutly, if the bear. Side. Bearing. Dist. Ft. 1 2 3 4 5 6 N 16° 33' W N 75° 39' E S 69° 23' E S 40° 53' W N 79044' W S 530 21' W 1057.6 1204.0 1113.3 849.6 800.1 704.3 lags and dists are correctly plotted, they will close perfectly. The young assistant is advised ta practise doing this, as well as dividing the plot into triangles, and computing the content. In this manner he will soon learn what degree of care is necessary to insure accurate results. The following hints may often be of serrloe. Ist. Avoid taking bearings and 4Ust3 along a circuitous bound- a ary line like a 6 c, Fig 7; but run *. —----._-_. •* i* .....*X the straight line a c; and at • ^ ' ' right angles to it, measure oflf- eets to the crooked line. 3d. Vishing to survey a straight Jine from a to c, but being una- ble to direct the instrument precisely toward c, on account •f intervening woods, or other •bstacles ; first run a trial line, as a m, as nearly in the proper direction as can be guessed at. Measure m c, and say, as o m is to t» c, so Is 100 ft to ? Lay off o o equal to 100 ft, and o a equal to ? ; and run the final line aa c. Or. if m c is quite small, calculate offsets like o « for every 100 ft along a w», and thus avoid the necessity for running a second line. 8d. When c is visible from a, but the intervening ground difficult to measure along, on account of marshes, &c. extend the side y a to good ground at t: then, making the angle y t d equal to y a c, run the line t n to that point d at which the angle n s \ v \ • N^. ^ f" , 1 ^ \ N *s N s N ^ >h < "J^ h •x 1*^ *v sv. ^ ^e 1 *" ;:; b; ^ ^ ^ ^ > L, 1; =n =5 ^ p^ ^ ^ "^ r-\ "*■ "^ ■is =55 aas ..^ *■ --- "-0.5 "" — — 5 10 15 20 23 Pull in LJts. on Steel Tape Weighing 0.75 Lh. per ±00 Ft. The diagram shows that a span of 100 ft of tape weighing 0.75 lb per 100 ft, requires a pull of 11.2 lbs to reduce the correction to zero ; a span of 50 ft, 7.3 lbs, etc. At these tensions (which are called normal pulls), the opposite effects 01 sag and of stretch are equal. At higher tensions, the lengthening due to stretch exceeds the shortening due to sag, and vice versa. Tapes of otber weights require pulls proportional to their weights or to their areas of cross-section. Thus, a tape, of any length, weighing 1 lb p6r 100 ft, would require, for any given correction, a pull of a; = -=r- 2/ = 13^ y, where y = the pull for the same correction on the standard tape, weighing 0.75 ib per 100 ft. Conversely ; given a pull of 10 lbs on a 50 ft span of a tape weighing 0.6 lb per 100 ft ; required the correction. To produce the same error in the tape weighing 0.75 lb per 100 ft would require a pull of y = 10 X 7;:^ = 12.5 lbs. Referring to 60 the diagram at 12.5 lbs on the curve for a 50 ft span, we find correction = — 0.16. This is the proper correction for either the heavier tape with 12.5 lbs or for the lighter tape with 10 lbs pull. Corrections for temperature. Tapes are usually graduated so as to be of standard length at 62° Fahr. For ordinary steel tape, the correction for temperature is about 0.0000065 ft per ft per degree Fahr. Corrections for temperature are uncertain, since the temperature of the tape cannot be determined with any accuracy. Measurements requiring great accuracy should therefore be made in cloudy weather, or at night, and the tape and the' thermometer should be kept off the ground. When measuring over sloping* ground, in ordinary work, the chain or tape should be held as nearly horizontal as possible, transferring the position of the raised end to the ground by means of a plumb line. Where the ground is steep, it becomes necessary to use a short length of tape, as the down-hill chain- man could not otherwise hold his end high enough ; or the tape may be held parallel with the slope, and the distance corrected by the following formulas : H TT -5- = cos A ; H = S. cos A ; S = r = H. sec A ; S ' ' cos A R =5: = tan A ; R = H. tan A ; o^ = sin A ; R = S. sin A. 284 LOCATION OF THE MERIDIAN. liOCATION OF THE MERIDIAN. By means of circnmpolar stars. (1) Seen from a point O (Figs. 1 and 2) on the earth, a circumpolar star e (star near the pole P) appears to describe daily* and counterclockwise a small circle, euwl, about the pole. The angle P O e, P O w, etc., subtended by the radius Pe, P w, etc., of this circle, or the apparent distance of the star from the pole, is called Its polar distance. The polar distances of stars vary slightly from year to year. See Table 3. They vary slightly also during each year. In the case of Polaris this latter variation amounts to about 50 seconds of arc. (2) The altitude of the pole is the angle N O P of the pole's elevation above the horizon N E S W, and is = the latitude of the point of obser- Fig. 1. Fig. 2. vation. Declination = angular distance north or south from the celestial equator. Thus, declination of pole = 90°. Declination of any star = 90°— its polar distance. (3) Let Z e H be an arc of a vertical circlet passing through a circumpolar Btar, e, and let H be the point where this arc meets the horizon N E S W. Then the angle N Z H at the zenith Z, or N O H at the point of observa- tion, between the plane N Z O of the meridian and the plane H Z O of the star's vertical circle (or the arc N H), is called the azimutlit of the star. If this angle N O H be laid oflf from O H, on the ground, the line O N will be in the plane of the meridian N Z S, or will be a nortli-and-soutli line.ll (4) When a star is on the meridian Z N of the observer, above or below the pole P, as at u or I, it is said to be at its upper or loi¥er culmina- tion, respectively. Its azimuth is then = 0, the line O H coinciding with the meridian line O N. (5) When the star has reached its greatest distance east or west from the pole, as at e or w, it is said to be at its eastern or western elonga- tion. J * In 23 h. 56.1 m. t A great circle is that section of the surface of a sphere which is formed by a plane passing through the center of the sphere. A vertical circle is a great circle passing through the zenith Z. X Astronomers usually reckon azimuth from the south point around through the west, north, and east points, to south again ; but for our pur- pose it is evidently much more convenient to reckon it from the north point, and either to the east or to the west, as the case may be. II The point N,on the horizon, is called the north point, and must not be confounded with the north pole P. § As seen from the equator, a star, at either elongation, is, like the pole itself, on the horizon; and the two lines Pe, Viv, joining it with the pole, form a single straight line perpendicular to the meridian, and lying in the LOCATION OF THE MERIDIAN. 285 (6) The honr ang-le of any star, at any given moment, is the time which has elapsed since it was in upper culmination.* (7) Evidently the azimuth of a star is continually changing. In cir- cumpolar stars it varies from 0° to maximum (at elongation) and back to 0° twice daily, as the star appears to revolve about the pole ; but when the star is near either elongation the change in azimuth takes place so slowly that, for some minutes, it is scarcely perceptible, the star appearing to travel vertically. (8) Given the polar distance of a star and the latitude of the point of observation, the azimutb of the star, at elong-ation, may be found by the formula.f c,. ^ . .1, ^ ^ sine of polar distance of star Sine of azimuth of star = ; . ..f , ^ t-^ — -^—r- r^ — cosine of latitude of point of observation or see (11) and Table 3. (9) The following circumpolar stars are of service in connection with observations for determining the meridian. See Fig. 3. Constellation Letter Called Ursa minor (Little bear) a (alpha) Polaris Ursa major (Great bear) e (epsilon) Alioth " " ( " " ) ^(zeta) Mizar Cassiopeia 6 (delta) Delta % poi*^^' (10) Polaris, or the north star, is fortunately placed for the determi- nation of the meridian, its polar distance being only about lVi°. See Table 3. Fig. 3 shows the circumpolar stars as they appear about midnight in July ; inverted, as in January ; with the left side uppermost, as in April ; and, with the right side uppermost, as in October. || horizon. The azimuth of the star is then = its polar distance. But in other latitudes P c and P w form acute angles with the meridian, as shown, and these angles decrease, and the azimuth of the star at elongation in- creases, as the latitude increases. * In lat. 40° N., the hour angle, ZPc = ZPty, of Polaris, at elongation, is = 5 h. 55 m. of solar time. Caution. It will be noticed that, except for an observer at the equator, the elongations do not occur at 90° from the meridian. t In the spherical triangle Z P e, we have : sin e Z P ^ sin P e sin Z e P " sin P Z But, since Z e P = 90°, sin Z e P = 1. Also, sin P Z = cos (90° — P Z), and c Z P ^ azimuth of c. sin P e _ sin polar distance P O e cos latitude Hence, sin azimuth of e -- sin P Z t 5 Cassiopeia is here called Delta, for brevity. |l Polaris is easily found by means of the two well-known stars called the " pointers " in " the dipper," Fig. 3, which forms the hinder 286 LOCATION OF THE MERIDIAN. (11) Table 3 gives the polar distances of Polaris and their log sines for January 1 in each third year from 1900 to 1930 inclusive, the log cosines of each fifth degree of latitude from 20° to 50°, and the corresponding azimuths of Polaris at elongation. Intermediate values may be taken by interpolation.* (12) By observation of Polaris at elongation. This method has the convenience, that at and near elongation the star appears to travel vertically for some minutes, its azimuth, during that time, remaining practically constant ; but during certain parts of the year (see Table 1), the elongations of Polaris take place in daylight; so that this method cannot then be used. | See (18), (19), (22). Nor can it be used at any time in places south of about 4° N. lat., because there Polaris is not visible. (13) The approximate times of elongation of Polaris for certain dates, in 1900, are given in Table 1, with instructions for finding the times for other dates. Or, watch Polaris in connection with any of those stars which are nearly in line with it and the pole, as Delta, Mizar, and Alioth. See Fig. 3. The time of elongation is approximated, with sufficient closeness for the determination of the azimuth, by the cessation of apparent hori- zontal motion during the observation. (14) From fifteen to thirty minutes before the time of elongation, have the transit, see (21), set up and carefully centered over a stake previously driven and marked with a center point. The transit must be in adjust- ment, especially in regard to the second adjustment, p. 294, or that of the horizontal axis, by which the line of collimation is made to describe a ver- tical plane when the transit is leveled and the telescope is swung upward or downward. (15) Means must be provided for illuminating the cross-hairs of the tran- sit. J This maybe done by means of a bull's eye, or a dark lantern, so held as not to throw its light into the eye of the observer ; or, better, by means of a piece of tin plate, cut and per- forated as in Fig. 4, bent at an angle of 45°, as in Fig. 5, and painted white on the surface next to the telescope. The ring, formed by bending the long strip, is placed around tlie object end of the telescope. A light, screened from the view of the observer, is then held, at one side of the instrument, in such a way that its' rays, falling upon the oblique and whitened surface of the tin plate, are reflected directly into the telescope. (16) Bring the vertical hair to cut Polaris, and, by means of the tangent screw, follow the star as it appears to move, to the right if approaching east- ern elongation, and vice versa, keeping the hair upon the star, as nearly as* may be. As elongation is approached, the star will appear to move more and more slowly. AVhen it appears to travel vertically along the hair, it has practically reached elongation, and the vertical plane of the transit, with the vertical hair cutting the star, is in the plane of the star's vertical circle. Depress the telescope, and fix a point in the line of sight, preferably 300 feet or more distant from the transit.f Immediately reverse the transit, (swinging it horizontally through an arc of 180°), sight to the star again, Fig. 4. Fig. 5. portion of the " great bear " (Ursa major), a line drawn through these two stars passing near Polaris. As the stars in the handle of the dipper form the tail of the great bear, as shown on celestial maps, so Polaris and the stars near it form the tail of the little bear (Ursa minor.) Polaris is also nearly midway and in line between Delta and Mizar, Polaris forms, with three other and less brilliant stars, a quite symmetrical cross, with Polaris at the end of the right arm. In Fig. 3 this cross is inverted. Its height is about 5°, or = the distance between the pointers. * Part of a table computed by the Surveying Class of 1882-3, School of Engineering, Vanderbilt University, Nashville, Tenn., and published by Prof Olin H. Landreth. t The stake must be illuminated. This may be done by throwing light upon that side of the stake which faces the transit, or, better, by holding a sheet of white paper behind the stake, with a lantern behind the paper. In the latter case, the cross-hairs of the transit, as well as the stake, and the knife-blade or pencil-point with which the assistant marks it, show out dark against the illuminated surface of the paper. :j: See Note, pnge 290. LOCATION OF THE MERIDIAN. 287 again depress, and, if the line of sight then coincides perfectly with the mark first set, both are in the plane of the star's vertical circle. If not, note where the line of sight does strike, and make a third mark, midway between the two. The line of sight, when directed to this third mark, is in the req^uired plane, from which the azimuth, found as in (8), has yet to be laid ofl to the meridian, to the left from eastern elongation, and vice versa. (17) To avoid driving the distant stake and marking it during the night, a fixed target at any convenient point may be used, and the horizontal angle formed between the line of sight to the star and that to the target merely noted, for use in ascertaining and laying off the azimuth of the target. (18) By observation of Polaris at ciilminatioii. Owing to, its greater difficulty, this method will generally be used only when that by elongation is impracticable. It consists in watching Polaris in connec- tion with another circumpolar star (such asMizar*or Delta) until Polaris is seen in the same vertical plane with such star, and then waiting a short and known time T, as follows,! until Polaris reaches culmination, where- upon Polaris is sighted and the line of collimation is in the meridian. At their upper culminations, Mizar and Delta are too near the zenith to be conveniently observed at latitudes north of about 25° and 30° respectively. At their lower culminations they are too near the horizon to be used to advantage at places much below about 38° of N. latitude. In general, Delta is conveniently observed at lower culmination from February to August, and Mizar during the rest of the year. Mizar Delta T= T = In 1900 2.6 mins 3.4 mins In 1910 6.5 mins 7.2 mins Mean annual increase, 1900-1910 . 0.39 min 0.38 min (19) By observation of Polaris at any point in its patli. Table 1 gives the mean solar times of upper culmination of Polaris on the 1st of each month in 1900, and directions for ascertaining the times on other dates ; and Table 2 gives the azimuths of Polaris corresponding to different values of its hour angle in civil or. mean solar time, for different latitudes from 30° to 50°, and for the years 1901 and 1906. For hour angles and lati- tudes intermediate of those in the table, the azimuths may be taken by interpolation. See Caution and formula, p. 290. (20) The local time % of observation must be accurately known, and the time of the preceding upper culmination (as obtained from*Table 1) deducted from it. The difference is the hour angle. If the hour angle, thus found, is 11 h. 58 m. or less, the star is west of the meridian. If it is greater than 11 h. 58 m., the star is east of the meridian. In that case deduct the hour angle from 23 h. 56 m. and enter the table with the remainder as the hour angle. See Fig. 1. (31) Where great accuracy is not required, Polaris may be observed by means of a plumb-line and sight. A brick, stone, or other heavy object will answer perfectly as a plumb-bob. It should hang in a pall of water. A compass sight, or any other device with an accurately straight slit about 1//16 inch wide, may be used. The sight must remain always perfectly verti- cal, but must be adiustable horizontally for a few feet east and west. The plumb-line and sight should be at least 15 feet apart, and so placed that the star and plumb-line can be seen together through the sights throughout the observation. The plumb-line must be illuminated. It is well to arrange all these matters on an evening preceding that of the observation. When the star reaches elongation, the sight must be fastened in range with the plumb-line and the star. From the line thus obtained, lay off the azimuth ; to the west for eastern elongation, and vice versa. (33) By any star at eqnal altitudes. This method, applicable to south as well as to north latitudes, consists in observing a star when it is at any two equal altitudes, E. and W. of the meridian, thus locating, on the horizon, two points of equal and opposite azimuth. The meridian will be midway between the two points. * Mizar will be recognized by the small star Alcor, close to it. t Deduced from values calculated in astronomical time (p. 266) by the U. S. Coast and Geodetic Survey. X liOcal time agrees with standard time (p. 267) on the standard meridians only. For other points add to standard time 4 minutes for each degree of longitude east of a standard meridian, and vice versa. 288 LOCATION OF THE MERIDIAN. (33) By equal shadows from the son. Fig. 6. Approximate. At the solstices (about June 21 and December 21) the path abed traversed before and after noon, by the end of the solar shadow O a, etc., of a verti- cal object O, or by the shadow oi a knot tied in a plumb-line suspended over O, will intersect a circular arc a N d, described about O, at equal dis- tances, am, md, from the meridian O N. The observations should be made within two hours before and after noon. At the vernal equinox (March 21) the line thus located will then be west, and at the autumnal equinox (Sept. 21) east, of the merid- ,. . . .^ ian, by less than 2U minutes of arc. For intermediate dates the error is nearly proportional to the time elapsed. It is well to draw several arcs of difierent radii, O a, O 6, etc., note two points where the path of the shadow intersects each arc, and take the mean of all the results. A small piece of tin plate, with a hole pierced through it, may be placed with the hole vertically over O ; and the bright spot, formed by the light shining through the hole, used in place of the end of the shadow. Table 1. Approximate civil times of elongation and cnlmination of Polaris in lat. 40° N., long. 90° W. from Greenwich, on the first of each month, in 1900. The times given in this table are mean solar or local times. P. M. times (between noon and midnight) are printed in bold-face. In latitude 25^, W. elongations occur later and E. earlier ) , nearlv o mins In latitude 50°, W. " " earlier and E. later P^ ^^^^^^^ ^'^^• The correction for longitude amounts to scarcely a minute of time in any part of the United States. For other days of the month, deduct 5.94 min. for each succeeding day. In general, the times are a little later each year. In 1904 ihey will beabout 5^ minutes later, but in 1905, only about 3 minutes later, than in 1900. This discrepancy is due to the occurrence of leap-year in 1904. Inasmuch as this table serves chiefly to put the observer on guard, and as he should be at his post from 15 to 30 minutes in advance of these times, the gradual increase in the times is of little consequence. The position of the star at elongation is determined by observation. At culmination, where the change in azimuth is most rapid, an error in time of 2 minutes in observing Polaris involves an error of about 1 minute of azimuth. At elongation, an error in time of' 20 minutes 10 minutes 5 minutes 1 minute will make an error in azimuth of less than 30 seconds less than 6 " less than 2 " about 0.06 second In all latitudes, each culmination follows the preceding one bv 11 h. 58 m. mean solar time. The times between successive elongations vary with the latitudes. . (E, eastern : W, western.) 1900. May 1. E. Jun. 1. E 4.50 A. M. 2.49 A. M. Nov.l.W. Dec.l.W. 4.33 A. M. 2.35 A. M. Elong>ations. Jan. 1. W. 12.31 A. M. July 1. E. Feb. 1. W. 10..30 P. M. Aug. 1. E. 12.51 A. M. 10.46 P. M Mar. 1. W. 8.40 P. M. Sept. 1. E. 8.45 P. M. Apr. 1. W. 6.38 P. M. Oct. 1. E . 6.47 P. M. Culminations. (U, upper ; Jan. 1. U. Feb. 1. L. Mar. 1. L. Apr. 1. L. 6.38 P. M. 4.38 A. M. 2.47 A. M. 12.45 A. M. L. Aug. 1. U. Sept. 1. U. Oct. 1. U. . M. 4.45 A. M. 2.43 A. M. 12.46 A. M. July 1. 6.44 P. L, lower.) 1900. May 1. L. 10.43 P. M Nov. 1. U. 10.40 P. M. June 1. L. 8.43 P. M. Dec. 1. U. 8 42 P. M. LOCATION OF THE MERIDIAN. 289 Table 3. AZIMUTHS OF POLARIS. Hour A.ugle Azimuth for latitude Hour Angle Azimuth for latitude Mean solar time Meau solar time 1901 1906 30° 35° 40° / 450 / 50° 1901 1906 30° f 35° / 40° / 45° / 50° h m h m o / / / h m h m f 4 4 2 20 210 2 2 6 57 6 27 1 21 1 26 1 32 1 39 1 49 8 9 .0 3 30 410 4 4 7 12 6 52 1 20 1 24 1 80 1 37 1 47 13 13 5 5,0 50 6 6 7 25 7 9 1 18 1 22 1 28 1 35 1 44 17 17 6 7[0 70 8 8 7 38 7 25 1 16 1 20 1 26 1 83 1 42 21 21 8 8 9 10 11 7 4^ 7 38 1 14 1 19 1 24 1 31 1 39 25 26 9 10 11|0 12 13 7 57 7 47 1 13 1 17 1 22 1 29 1 37 29 30 11 12 12!0 14 15 8 5 7 57 1 11 1 15 1 20 1 27 1 35 33. 34 12 13 14 16 17 8 13 8 6 1 10 1 14 1 18 1 25 1 33 38 38 14 15 16 18 19 8 20 8 13 1 8 1 12 1 17 1 23 1 31 42 43 15 17,0 18 20 21 8 27 . 8 21 1 7 1 11 1 15 1 21 1 29 46 47 17 no 19 21 23 8 34 8 28 1 5 1 9 1 13 1 19 1 27 50 51 19 20,0 21 23 25 8 40 8 35 1 3 1 7 1 12 1 18 1 25 54 55 20 22 23 25 27 8 47 8 42 1 2 1 6 1 10 1 16 1 23 59 1 22 23 25 27 29 8 53 8 48 1 1 4 1 8 1 14 1 21 1 3 1 4 23 25 26 29 32 8 58 8 53 59 1 3 1 7 1 12 1 19 1 8 1 9 2.-, 26 28 31 34 9 4 8 69 58 1 1 1 5 1 10 1 17 1 13 1 15 27 29 31 34 37 9 9 9 5 56 1 1 3 1 8 I 15 1 19 1 20 29 31 ;0 33 36 39 9 15 9 11 55 58 1 2 i 7 1 13 1 24 1 25 31 33' 35 38 42 9 20 9 16 53 56 1 1 5 1 11 1 29" 1 31 32 35 37 40 44 9 25 9 22 51 54 58 1 3 1 9 1 34 1 37 34 37 39 43 47 9 31 9 27 50 53 56 1 1 1 7 1 40 1 42 36 39 41 45 49 9 36 9 33 49 52 55 59 1 5 1 45 1 47 38 41 43 47 51 9 41 9 38 47 50 53 57 1 2 1 50 1 53 39 42 45 49 54 9 47 9 44 45 48 51 55 1 1 56 1 58 41 44 47 51 56 9 52 9 49 44 46 49 63 58 2 1 2 3 43 46 49 53 59 9 57 9 55 42 44 47 51 56 2 6 2 9 45 48 51 55 1 1 10 2 10 40 42 45 49 54 2 11 2 14 46 50 53 57 1 3 10 8 10 5 39 41 43 47 51 2 17 2 20 48 51 54 59 1 5 10 13 10 11 37 39 41 45 49 2 22 2 25 50 53 56 1 1 1 8 10 18 10 16 35 37 40 43 47 2 27 2 31 51 54 58 1 3 1 10 10 24 10 21 33 35 38 41 44 2 33 2 36 53 56 1 15 1 12 10 29 10 27 32 34 36 39 42 2 38 2 42 54 5811 2 17 1 14 10 34 10 33 30 32 34 37 40 2 43 2 47 56,0 59 1 3 1 9 1 16 10 39 10 38 28 29 31 34 37 2 49 2 53 57 1 ill 5 1 11 1 18 10 45 10 43 26 27 '19 32 35 2 54 2 59 59,1 3;i 7 1 13 1 20 10 50 10 49 24 25 27 29 32 3 3 5 1 Oil 4'1 8 1 15 1 22 10 55 10 54 23 24 25 27 30 3 5 3 10 I 2I1 6 1 10 1 16 1 24 10 59 10 58 21 22 24 26 28 3 11 3 16 1 3 1 7;i 12^1 18 1 26 11 4 11 3 20 21 22 24 26 3 18 3 23 1 5|l 91 13 1 20 1 28 11 8 11 7 18 19 20 22 25 3 24 3 30 1 6 1 10 1 15 1 22 1 30 11 12 11 11 17 18 19 20 22 3 31 3 37 1 8^1 12:1 17|1 24 1 32 11 16 11 15 15 9 16 17 18 20 3 38 3 45 1 9 1 14|l 19'l 26 1 34 11 20 11 20 14 14 15 16 18 3 45 3 52- 1 11 1 15 1 2011 27 1 36 11 25 11 24 12 13 14 15 16 3 53 4 1 1 12 1 17 1 22jl 29 1 38 11 29 11 28 11 11 12 13 14 4 1 4 11 1 14 1 1811 241 31 1 40 11 33 11 32 9 9 10 11 12 4 10 4 20 1 15 1 20 jl 25 1 33 1 42 11 37 11 37 8 80 80 9 10 4 20 4 33 1 17 1 22 1 27,1 35 1 44 11 41 11 41 6 60 70 7 8 4 33 4 49 1 19 1 24 1 29:1 37 1 47 11 45 11 45 5 5 50 5 6 4 46 5 6 1 20 1 25|l 3111 39 1 49 11 50 11 49 3 30 30 4 4 5 1 5 31 I 22 1 27! 1 33 1 41 1 51 11 54 11 54 1 1,0 20 2 2 When the star is near elongation (hour angles hetween 5 h and 7 h). a con- siderable change in hour angle corresponds to but a small change in azimuth. At such times it will usually be better to use the method employing the elongation. 19 290 LOCATION OF THE MERIDIAN. Table 3. POLARIS. POLAR DISTANCES, AND AZIMUTH AT ELONGATION. Azimuth at Elongation, in Latitude »H Polar Dist. of Polaris Log sin pol dist. $ ao° 35° 30° 35° 40° 45° 50° o / // O f o / o / o r O f O f o / 1900 1 13 33 8.33 027 1 18.3 1 21.1 1 24.9 1 29.8 1 36.1 1 44.1 1 54.4 1903 1 12 37 8.32 472 1 17.3 1 20.1 1 23.8 1 28.7 1 34.8 1 42.7 1 53.0 1906 1 11 41 8.31 910 1 16.3 1 19.1 1 22.8 1 27.6 1 33.6 1 41.4 1 51.5 1909 1 10 45 8.31 341 1 15.3 1 18.1 1 21.7 1 26.4 1. 32.3 1 40.1 1 50.1 1912 1 9 49 8.30 765 1 14.3 1 17.0 1 20.6 1 25.2 1 81.1 1 38.7 1 48.6 1915 1 8 53 8.30181 1 13.3 1 16.0 1 19.5 1 24.1 1 29.9 1 37.5 1 47.2 1918 1 7 58 8.29 594 1 12.3 1 15.0 1 18.5 1 23.0 1 28.7 1 36.1 1 45.7 1921 1 7 2 8.28 999 1 11.4 1 14.0 1 17.4 1 21.9 1 27.5 1 34.8 1 44.3 1924 1 6 7 8.28 401 1 10.4 1 13.0 1 16.3 1 20.7 1 26.3 1 33.5 1 42.9 1927 1 5 12 8.27 794 1 9.4 1 11.9 1 15.3 1 19.6 1 25.1 1 32.2 1 41.4 1930 1 4 16 8.27 169 1 8.4 1 10.9 1 14.2 1 18.5 1 23.9 1 30.9 1 40.0 Log cos lat 9.97 299 9.95 728 9.93 753 9:91 337 9.88 425 9.84 949 9.80 807 Owing to changes in the position of Polaris during the year, the positioni given in the table may at times be in error by as much as a minute. Thi error is greater in the higher latitudes. Having the north polar distance, /), of a star, and the latitude, L, of th( point of observation, we have, declination of star = 5 = 90° — p ; and th( aKiiiiiith, a, of the star, corresponding to any hour angle, h, may b< found by the following formulas : rr, i.r c ot p t&uS COS M . tau h Tan M = r: = r. Then Tan a = — -z—. . cosh cosh cos(L — M) The declinations, 5, of Polaris are given in the U. S. Ephemeris or Nautica! Almanac. From these the polar distances may be obtained more accuratelj than from our Table 3. Caution. When it is desired to determine the meridian within out minute of arc, it is well to use more than one method and compare th€ results. For example, observe Polaris both E. and W. of the meridian, anc a star at equal altitudes south of the zenith. Note. — If Polari.s be found during twilight, in the morning or evening, obser vations of it may be made without artificial illumiuation of the cross-hairs For times of elongation, see Table L Conversion of Arc into Time, and vice versa. Arc Time 1° = 4 minutes 1' = 4 seconds V = 0.066... second Time Arc 24 hours = 360° Ihour = 15° 1 minute = 0° 15' 1 second = 0° O' 15" THE engineer's TRANSIT. 291 THE ENGINEER'S TRANSIT. 292 THE ENGINEER S TRANSIT. The details of the transit, like those of the level, are differently arranged by diff makers, and to suit particular purposes. We describe it in its modern form, as made by Heller and Brightly, of Philada. Without the long- bubble-tube F F, Fig 1, under the telescope, and the graduated arc g, it is their plain transit. With these appendages, or rather with a graduated circle in place of the arc, it becomes virtually a Complete Theodolite. B D D, Fig 2, is the tripod-head. The screw-threads at v receive the screw of a wooden tripod-head-cover when the instrument is out of use. S S A is the lower parallel plate. After the transit has been set very nearly over the center of a stake, the shifting^-plate^ c^c^ cc, enables us, by slightly loosening the levelling'-screwis K, to shift the upper parts horizontally a trijfle, and thus bring the plumb-bob exactly over the center with less trouble than by the elder method of pushing one or two of the legs further into the ground, or spread- ing them more or less. The screws, K, are then tightened, thereby pushing up- ward the upi>er parallel plate mmmxx, and with it the half-ball &, thus pressing c b tightly up against the under side of S. The plumb-line through tlie vert hole in b. Screw-caps. /, g, protect the levelling-screws from dust &c. The feet, % of the screws, work in loose sockets,./, made flat at bottom, to preserve S from being indented. The parts thus far described are generally left attached to the legs at all times. Fig 1 shows the method of attachment. To set the upper parts upon the parallel plates. Place the lower end of U U in y x, holding the instrument so that the three blocks on m wi (of which the one shown at F is movable) may enter the three corresponding THE engineer's TRANSIT. 293 recesses in a, thus allowing a to bear fully on m, upon which the upper paite then rest. (The inner end of the spring-catch, I, in the meantime enters a groove around U, just below a, and prevents the upper parts from falling off, if the in- strument is now carried over the shoulder.) Revolve the upper parts horizontallj a trifle, in either direction, until they are stopped by the striking of a small lug on a against one of the blocks F. The recesses in a are now clear of the blocks. Tighten q, thereby pushing inward the movable block F, which clamps the bevelled flange a between it and the two fixed blocks on m m, and confines the spindle U to the fixed parallel plates. It remains so clamped while the instrument is being used. To remove the upper parts from tlie parallel plates. Loosen g, bring the recesses in a opposite the blocks F, Hold back I, and lift the upper parts, which are then held together by the broad head of the screw inserted into the foot of the spindle w. T T is the outer revolving spindle, east in one with the support- ing-plate Z Z, to which is fastened the graduated limb O O. The limb extends beyond the compass-box, and thus admits of larger graduations than would otherwise be obtainable, w wi% the inner revolving spindle. At it« top it has a broad flange, to which is fastened the vernier pl-ate P. To the latter are fastened the compass-box C, the two bubble-tubes M M, the standards V V, supporting the telescope, &c. Each bubble-tube is supported and adjusted by four capstan-head nuts, two at each end. The bent strip, curving over the tube, protects the glass from accidental blows in swinging the telescope. Control of motions of graduated limb O O and vernier plate P. — The tangent-screw G and a spiral spring (not shown) opposite to it are fixed to the graduated limb 0, and hold between them a projection y from the loose collar t, which is thus confined to the limb and made to travel with it. The clamp-screw H passes through the collar t and presses against the small lug shown at its inner end. When H is tiglitened, this lug is pressed against the fixed spindle U U, to whicii the graduated limb is thus made fast. A slow mo- tion may, liowever, still be given to the limb by means of the tangent-screw G. The motion of the vernier plate P over the graduated limb O O is similarly governed by the tangent-screw b and its spiral spring (not shown), fixed to the vernier plate P, and the clamp-screw e, which passes through the collar z, and presses against the small lug shown at its inner end. In Heller and Brightly's instruments, the screw b is provided with means for taking up its "wear," or " lost-motion." There are two verniers. One is shown at p. Fig 1. Both may be read, and their mean taken, when great accuracy is required. Ivory reflectors, c, facilitate their reading. Before the instrument is moved from one place to another, the eompass-needle, k, Fig 2, should always be pressed up against the glass cover of the compass-box by means of the upright milled-head screw seen on the ver- nier-plate in Fig 1, just to the right of the nearest standard. The pivot-point is thus protected from injury.. R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope in any required position. It is best to let the e2/e-end,E, of the telescope revolve downward, as otherwise the shade on O, if in use, may fall otf. The tangent-screw, d, moves a vert arm attached to R, and is thus used for slightly changing the elevation of the telescope. In the arm is a slit like that seen in the vernier-arm I. By means of the screw D, the movable vernier-arm Y may be clamped at any desired point on the vertical limb g. When 0° of the vernier is placed at 30° on the arc g, and the index of the opposite arm is placed over a small notch on the horizontal brace (not seen in our figs) of the standards, the two slit^ will be opposite each other, and may be used for laying off offsets, &c, at right-angles to the line of sight. One end, R, of the telescope axis rests in a movable box, under which is a screw. By means of the screw, the box may be raised or lowered, and the axis thus ad- justed for very slight derangements of the standards. For E, B, 0, and A, see Level, p 306. « is a dAist-guard for the object-slide. Stadia Hairs. Immediately behind the capstan-screw, p, Fig 1, is seen a smaller one. This and a similar o'ne on the opposite side of the telescope, work in a ring inside the telescope, and hold the ring in position. Across the ring are stretched two additional horizontal hairs, called stadia hairs, placed at such a distance apart, vertically, that they will subtend say 10 divisions of a graduated rod placed 100 ft from the instrumeut, 16 divisions at 150 ft, &c. They are thus used for measuring hor and sloping distances. The long bubble- tube. B' F, Fig 1, enables us to use the transit as a level, aJthough it is not so well adanVid as the latter to this purpose. 294 THE engineer's transit. To adjust a plain Transit* When either a level or a transit is purcliased, it is a good precaution (but on* which the writer has never seen alluded to) to first screw the object-glass firmly horn* to its place ; and then make a short continuous scratch upon the ring of the glass, and upon its slide; so as to be able to see at any time when at work, that the glass is always in the same position with regard to the slide. For if, after all the adjustments are completed, the position of the glass should become changed, (as it is apt to be if unscrewed, and afterward not screwed up to the same precise spot,) the adjustments may thereby become materially derauged ; especially if the object-glass is eccentric, or not truly ground, which is often the case. Such scratches should be prepared bj the maker. In making adjustments, as well as when using a transit or level, be careful that the eye-glass and object-glass are so drawn out that there shall be no parallax. The eye-glass must first be drawn out so as to obtain perfect distinctnegs of the cross-hairs ; it must not be disturbed afterward ; but the object-glass must be moved for different distances. First, to ascertain tliat tlie bnbble-tubes, M m, are placed parallel to the vernier-plate, and that therefore when both bubbles are in the centers of their tubes the axis of the inst is vert. By means of the four levelling- screws, K, bring both bubbles to the centers of their tubes in one position of the inst ; then turn the upper parts of the inst half-way round. If the bubbles do not remain in the center, correct half the error by means of the two capstan-nuts r r ; and the other half by the levelling-screws K. Repeat the trial until both bubbles remain in the center while the inst is being turned entirely around on its spindle. Second, to see that the standards have suffered no derang^e- ment ; that is, that they are of equal height and perpendicular to the vernier- plate, as they always are when they leave the maker's hands. Level the inst perfectly; then direct the intersection of the hairs to some point of a high object (as the top of a steeple) near by; clamp the inst by means of screws H and e, and lower the telescope until the intersection strikes some point of a low object. (If there is none such drive a stake or chain-pin, &c, in the line.) Then un- clamp either H or e, and turn the upper parts of the inst half-way round ; fix the intersection again upon the high point; clamp; lower the telescope to the low point. If the intersection still strikes the low point, the standards are in order. If notj correct one-half of the difference by means of the adjusting-block and screw at the end, R, of the telescope axis, Fig. 1, and repeat the trial de novo, resetting the stake or chain-pin at each trial. If the inst has no adjusting- block for the axis, it should be returned to the maker for correction of any derange- ment of the standards. A transit may be used for running straight lines, even if the standards become Slightly bent, by the process described at the end of the fourth adjustment. Third, to see that the cross-hairs are truly vert and hor M^hen the inst is level. When the telescope inverts, the cross-hairs are nearer the eye-end than when it shows objects erect. The maker takes care to place the cross-hairs at right-angles to each other in their ring, or diaphragm ; and gene- rally he so places the ring in the telescope, that when levelled, they shall be vert and hor. Sometimes, however, this is neglected ; or the ring may by accident be- come turned a little. To be certain that one hair is vert, (in which case the other must, by construction, be hor,) after having adjusted the bubble-tubes, level the in- strument carefully, and take sight with the telescope at a plumb-line, or other vert- straight edge. If the vert hair coincides with this object, it is, so far, in adjustment; but if not, then loosen slightlf only tioo adjacent screws of the four, p p i i. Fig 1 ; and with a knife, key, or other small instrument, tap very gently against the screw-heads, so as to turn the ring a little in the telescope; i)ersevering until the hair be* comes truly vertical. When this is done, tighten th« screws. In the absence of a plumb-line, or vert straight edge, sight the cross-hair at a very small distinct point; and see if the hair still cuts that point, when the telescope is raised or lowered by revolving it on its axis. The mode of performing the foregoing will be readily nnderstood from this Fig, which represents a section across the top part of the tele- iscope, and at the cross-hairs. The hair-ring, or diaphragm, a; vert hair, v; tele- scope tube, g ; ring outside of telescope tube, d; & is one of the four capstan-headed screws which hold the hair-ring, a, in its place, and also serve to adjust it. The lower. ends of these screws work in the tnickness of the hair-ring; so that wheci they are loosened somewhat, they do not lose their hold on the ring. Small loose THE engineer's TRANSIT. 295 ^m washers, c, are placed under the heads h cf the screws. A space y y is left around each screw where it passes through the telescope tube, to allow the screws and ring together to be moved a little sideways when the screws h are slightly loosened. Fourth, to see tliat ttie vertical hair is in the line of colli- mation. Plant the tripod firmly upon the ground, as at a. Level the inst ; clamp it; and direct the vert hair by means of tangent-screw G (figs. 1 and 2) upon some convenient object b\ or if there is none such, drive a thin stake, or a chain-pin. Then revolving the telescope vert on its axis, ^q, observe some object, as c, where the vert hair now strikes; t. a ^^""^ or if there is none, place a second pin. Unclamp the instru- , •^rrrl ,o ment by the clamp-screw H; and turn the whole upper • ^^-^ part of it around until the vert hair again strikes ^- -.^Fifi". 4, Clamp again; and again revolve the telescope vert on its axis. If the vert hair now strikes c, as it did before, it shows that c is really at ; and that b, a, c, are in the .same straight lute ; and therefore this adjustment is in order. If not, observe where it does strike, say at m, (the dist a m being taken equal to a c,) and place a pin there also. Measure m c ; and place a pin at t', in the line m. c, making m v = one-fourth of m c. Also put a pin at o, half- way between m and c, or in range with a and b. By means of the two hor screws that move the ring carrying the cross-hairs, adjust the vert hair until it cuts V. Now repeat the entire operation ; and persevere until the telescope, after being directed to 6, shall strike the same object o, both times, when revolved on its axis. See whether the movement of the ring in this 4th adjustment has dis- turbed the verticality of the hair. If it has, repeat the 3d adjustment. Then re- peat the 4th, if necessary ; and so on until both adjustments are found to be riyht at the same time. 'I'hus a straight line may be run, even if the hairs are out of adjustment ; but with somewhat more trouble. For at each station, as at a, two back-sights, and two foresights, a c and a m, may be taken, as when making the adjustment ; and the point o, half-way between c and ni, will be in the straight line. The inst may then be moved to o, and the two back-sights be taken to a ; and so on. Angles measured by the transit, whether vert or hor, will evidently not be affected by the hairs being out of adjustment, provided either that the vert hair is truly vert, or that we use the intersection of the hairs when measuring. The foreg-oing are all the adjustments needed, unless the tran- sit is requirea for levelling, in which case the folio wiug one must be attended to : Big. 5 To adjust the long bubble* tnbe, F F, Fig. 1, we first place the line of sight of the telescope hor, and then make the bubble-tube hor, so that the two are parallel. Drive two pegs, a and b Fig. 5, with their tops at precisely the same level (see Rem. p. 296) and at least about 100 ft. apart ; 300 or more will be better. Plant the inst firmly, in range with them, as at c, making h c an aliquot part of a b. and as short as will permit focusing on a rod at b. The inst need not be leveled. Suppose the line of sight, to cut e and d. Take the readings b e and a d. Their diff is be — ad = an — ad = dn; and a h : a c : : dn: d s] s being the height of the target at a when the readings (a s, b o) on the two stakes are equal, a s = a d-\-d s=a d -\ — • If the reading on a exceeds that on 6 (as when the line of sight is vfg) the diff of readings is = a ^r — b /= ag — ai = gi; and as = ag — gs = ag — giXac ab Sight to s, bring the bubble to the cen of its tube by means of the two small nuts n n at one end of the tube. Fig. 1, and assume that the telescope and tube are parallel.* The zeros of * This neglects a small error due to the curvature of the earth ; for a hor line at t; is t; h, tan- gential to the curved (or " level") surface of still water at v, whereas u s is tangential to water surf at a point midway between a and h. Hence if the telescope at v points to a it will not be parallel to the level bubble-tube. To allow for this, and for the refraction by the air, which diminishes the error, raise the target on o to a point A above ». h s = .0000000205 X square of a c in ft ; but whea • c is 660 ft, ha\s only aboufc one tenth of an inch and barely covers the apparent thickness of the w«M-hair in the tele.scoi>» j:> .rrii 296 THE engineer's transit. the vert circle, and of its vernier, may now be adjusted, if they require it, by loosening the vernier screws and then moving the vernier until the two coin- cide. , ^ . , , Rem. If no level is at hand for levelling the two pegs a and 6, it may be done by the transit itself, thus : Carefully level the two short bubbU-s, by means of the levelling-screws K. Drive a peg m, from 100 to 300 feet froni the instrument o. Then placing a target-rod on m, clamp the target tight at whatever height, as ««, the hor hair happens to cut it ; it being of no im- portance whether the telescoi>e is level or not; although it might as well be as nearly so as can conveniently be guessed at. Clamp the telescope -Q- ■ H in its position by the clamp-ring R, Fig. 1. Re- ^ volve the Inst a considerable way round ; say ^S- 0« nearly or quite half way. Place another peg 71, at precisely the same dist from the instrument that m is ; and continue to drive it un- til the hor hair cuts the target placed on it, and still kept clamped to the rod, at the same height as when it was on m. When this is done, the tops of the two pegs are on a level with each other, and are ready to be used as before directed. When a transit is intended to be used for surveying farms, &c, or for retracing lines of old surveys, it is very useful to set the compass so as to allow for the "va- riation " during the interval between the two surveys. For this purpose a " variation- vernier " is added to such transits; and also to the compass. When the graduations of a transit are figured, or numbered, so as to read both 10 10 ways from zero, thus, 1 1 t 1 1 i i i 1 ii i i I i i i 1 1 i i 1 1 I i i i the vernier also is made double; that is, it also is graduated and numbered from its zero both ways. In this case, if the angle is measured from zero toward the right hand, the reading must be made from the right hand half of the vernier ; and vice versa. If the figuring is single, or only in one direction, from zero to 360°, then only the single vernier is necessary, as the angles are then measured only in the direction that the figuring counts. Engineers differ in their preferences for various manners of figuring the graduations. The writer prefers from zero each way to 180°, with two double ver- niers. To replace cross-hairs in a level, or transit. Take out the tube from the eye end of the telescope. Looking in, notice which side of the cross- hair diaphragm is turned toward the eye end. Then loosen the four screws which hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new hairs with beeswax, varnish, glue, or gum-arabic water, &c. This requires care. Then, to return the diaphragm to its place, press firmly into one of the screw-holes on the circiimf of the diaphragm itself, the end of a piece of stick, long enough to reach easily into the telescope as far as to where the diaphragm belongs. By this stick, as a handle, insert the diaphragm edgewise to its place in the telescope, and hold it there until two opposite screws are put in place and screwed. Then draw the stick out of the hole in the diaphragm ; and with it turn the diaphragm until the same side presents itself toward the eye end as before ; then put in the other two screws. The so-called cross hairs are actually spider-web, so fine as to be barely visible to the naked eye. Holler & Brightly use very fine platina wire, which is much better. Human hair is entirely too coarse. To replace a spirit-level, or bubble-g-lass. Detach the level from the instrument; draw off its sliding ends; push out the broken glass vial, and the cement which held it ; insert the new one, with the proper side up (the upper side is always marked with a file by the maker); wrapping some paper around it? ends, if it fits loosely. Finally, put a little putty, or )nelted beeswax over the ends of tlio vial, to secure it against moving in its tube. In purchasing instruments, especially when they are to be used far from a maker, it is advisable to provide extras of such parts as may be easily broken or lost; such as glass compass-covers, and needles; adjusting pins; level vials; magnifiers, Ac. Tbeodolite adjustments are performed like those of the level and transit. let. That of the cross-hairs ; the same as in the level. 2d. The long bubble-tube of the telescope ; also as in the level. 3d. The two short bubble-tubes ; as in the transit. 4th The vernier of the vert limb ; as in the transit with a vert circle. 5th. To see that the vert hair travels vertically ; as in the fourth adjustment of the transit. In some theodolites, no adjustment is provided for this; but in large ones it is provided for by screws under the feet of the standards. 6om«time§ a second telescope is added ; it is p'aced below the hor lin^b, and is THE BOX OR POCKET SEXTANT. 297 called a watcher. It has its own clamp, and tangent-screw. Its use is to ascertain whether the zero of that limb has moved during the measurement of hor angles. When, previously to beginning the measurement, the zero and upper telescope are directed toward the first object, point the lower telescope to any small distant object, and then clamp it. During the subsequent measurement, look through it, from time to time, to be sure that it still strikes that object ; thus proving that Tom slipping has occurred. THE BOX OE POCKET SEXTANT. The portability of the pocket sextant, and the fact that it reads to single minutes, render it at times very useful to the engineer. By it, angles can be measured while in a boat, or on horseback: and in many situations which preclude the use of a transit. It is useful for obtaining latitudes, by aid of an artificial horizon. When closed, it resembles a cylindrical brass box, about 3 inches in diameter, and 1]/^ inches deep. This box is in two parts; by unscrewing which, then inverting one part, and then screwing them to- gether again, the lower part becomes a handle for holding the instrument. Looking down upon its top when thus arranged, we see, as in this figure, a movable arm I C, called the iiKlex, which turns on a center at C, and car- ries the vernier V at its other end. Gr Gr is the graduated arc or limb. It actually subtends about 73°, but is di- vided into about 146°. Its zero is at one end. Its graduations are not shown in the Fig. Attached to the index is a small mov- able lens, (not shown in the figure,) likewise revolving around C, for read- ing the fine divisions of the limb. When measuring an angle, the index is moved by turning the milled-head P of a pinion, which works in a rack placed within the box. The eye is applied to a cir- cular hole at the side of the box, near A. A small telescope, about 3 inches long, accompanies the instrument", but may generally be dispensed with. When so, the eyehole at A should be partially closed by a slide which has a very small eye-hole in it; and which is moved by the pin h, moving in the curved slot. Another slide, at the side of the box, carries a dark glass for covering the eye-hole when observing the sun. When the telescope is used, it is fastened on by the milled-head screw T. The top part shown in our figure, can be separated from the cylindrical part, by removing 3 or 4 small screws around its edge ; and the interior can then be exam- ined, and cleaned if necessary. Like nautical, and other sextants, this one has two principal glasses, both of them mirrors. One, the liidex-g-lass, is attached to the underside of the index, at C ; its upper edge being indicated by tlie two dotted lines. The other, the liorlzoii-g-lass, (because, when meas- uring the vert angles of celestial bodies, it is directed toward the horizon,) is also within the box; the position of its upper edge being shown by the dotted lines at R. The horizon-glass is silvered only half-way down ; so that one of the observed objects may be seen directly through its lower half, while the image of the other object is seen in the upper half, reflected from the index-glass. That the instrument may be in adjustment, ready for use, these two glasses must be at right angles to th« plane of the instrument ; that is, to the under side of the top of the, box, to which they are attached; and must also be parallel to each other, when the zeros of the vernier and of the limb coincide. The index-glass is already permanently fixed by the maker, and requires no other adjustment. But the horizon-glass has two adjust- ments, which are made by a key like that of a watch, and having a milled-head K. It is screwed into the top of the box, so as to be always at hand for use. When needed, it is unscrewed. This key fits upon two small square-heads, (like that for 298 THE COMPASS. winding a watch;) one of which is shown at S; while the other is near it, but on the SIDE of the box. These squares are the heads of two small screws. If the horizon glass H should, a^; in this sketch, (where it is shown endwise,) not be at right angles to the top U ij of the box, it is brought right by turning the square- bead S of the screw S T ; and if, after being so far rectified, it still is not parallel to the index-glass when the zeros coincide, it is moved a little backward or forward by the square head at the side. To adjust a box sextant, bring the two zeros to coincide precisely ; then look through the eye-hole, and the lower or unsilvered part of the horizon-glass, at some distant object. If the instru- ment is in adjustment, the object thus seen directly, will coincide precisely with its reflected image, seen at the same time, at the same spot. But if it is not in adjustment, the two will appear separated either hor or vert, or both, thus, * * ; in which case apply the key K to the square-head S ; and by turning it slightly in whichever direc- tion may be necessary, still looking at the object and its image, bring the two into a hor position, or on a level with each other, thus, * *. Then applj the key to the square- head in the side of tlie box; and by turning it slightly, bring the two to coincide perfectly. The instrument is then adjusted. In some instruments, the hor glass has a hinge at v, to allow it play while being adjusted by the single screw ST; but others dispense with this hinge, and use ttoo pcrews like S on top of the box, in addition to the one in the side. If a sextant is used for measuring vert angles by means of an artificial horizon, the actual altitude will be but one-half of that read off on the limb ; because we then read at once both the actual and the reflected angle. The great objection to the sextant for engineering purposes, is that it does not measure angles horizontally, as the transit does; unless when the observer, and the two ob- jects happen to be in the same hor plane. Thus an observer with a sextant at A, if measuring the angle subtended by the mountain-peaks B and C, must hold the graduated plane of the sextant in the plane of A B C ; and must actually meas- ure the angle BAG; whereas what he wants is the hor angle n Am. This is greater than BAG, because the dists A n and A m are shorter than A B and A C. The transit gives the hor angle nAm, be- <;ause its graduated plane is first fixed hor by the le veiling-screws : and the subse- quent measurement of the angle is not affected by his directing merely the line of sight upward, to any extent, in order to fix it upon B and G. For more on this sub- ject ; and for a method of partially obviating this objection to the sextant, see the note to Example 2, Gase 4, of " Trigonometry." The nautical sextant, used on ships, is constructed on the same principle as the box sextant ; and its adjustments are very similar. In it, also, the index- glass is permanently fixed by the maker; and the horizon-glass has the two adjust- ments of the box sextant. It also has its dark glasses for looking at the sun ; and * small sight-hole, to be used when the telescope is dispensed with. THE COMPASS. To adjust a Compass. The first adjustment is that of the bubbles. Plant firmly ; and level the Instrument, in any position ; that is, bring the bubbles to the centers of their tubes. Then turn the instrument half-way round. If the bubbles then remain at the cen- ters, they are in adjustment; but if not, correct one-half the difif in each bubble, by means of the adjusting-screws of the tubes. Level the instrumemt again; turn it half round ; and if the bubbles still do not remain at the center, the adjusting- •crews must be again mored a little, so as to rectify half the remaining diflf. Gener- THE COMPASS. 299 ally several trials must bo thus made, until the bubbles will remain at the cente while the compass is being turned entirely around. Second adjust/iiient. Level the compass, and then see that the needle ii hor ; and if not, make it so by means of the small piece of wire which is wrapped around it ; sliding the wire toward the high end. A needle thus horizontally ad- Justed at one place, will not remain so if removed far north or south from that place. If carried to the north, tlie north end will dip down ; and if to the south, the south end will do so. The sliding wire is intended to counteract this. Third adjustment. This is always fixed right at first by the maker; that is, the sights, or slits for sighting through, are placed at right angles to the compass plate ; so that when the latter is levelled by the bubbles, the sights are vert. To test whether they are so, hang up a plumb-line ; and having levelled the compass, take sight at the line, and see if the slits coincide with it. If one or both slits should prove to be out of plumb, as shown to an exaggerated extent in this sketch, it should be unscrewed from the compass, and a portion of its foot on the high side be filed or ground off, as per the dotted line ; or ^^^_^ aa a temporary expedient, a small wedge may be placed under the ^ j low side, so as to raise it. Fourth adjustment, to straighten the needle, if it should become bent. The compass being levelled, and the needle hor, and loose on its pivot, see whether its two ends continue to point to exactly opposite graduations, (that is, graduations 180° apart;) while the compass is turned completely around. If it does, the needle is straight ; and its pin is in the center of the graduated circle ; but if it does not, then one or both of these require adjusting. First level the compass. Then turn it until some graduation (say 90°) comes precisely to the north end of the needle. If the south end dues not then point precisely to the opposite 90° division, lilt off the needle, and bend the pivot-point until it does; remembering that every time said point is bent, the compass must be turned a hairsbreadth so as to keep the north end of the needle at its 90° mark. Then turn the compass half-way round, or until the opposite 90° mark comes precisely to the north end of the needle. Make a fine pen- cil mark where the soiith end of the needle now points. Then take off the needle, and bend it until its south end points half-way between its 90° mark and the pencil mark, while its north end is kept at 90° by moving the compass round a hairsbreadth. The needle will then be straight, and must not be altered in making the following, adjustment, although it will not yet cut opposite degrees. Fifth adjustment, of the pivot-pin. After being certain that the needle is straight, turn the compass around until a part is arrived at where the two ends of the needle happen to cut opposite degrees. Then turn the compass quarter way around, or through 90°. If the needle then cuts opposite degrees, the pivot-point is already in adjustment ; but if the needle does not so cut, bend the pivot-point until it does. Repeat, if necessary, until the needle cuts opposite degrees while being turned entirely around. Oftre and nicety of observation are necessary in making these adjustments properly ; because the entire error to be rectified is, in itself, a minute quantity ; and the novice is very apt to increase his trouble by not knowing how to use his mag'nifier, when looking at the end of the needle and the corresponding graduations. The mag- nilier must always be held with its center directly over the point to be examined ; antl it must be held parallel to the graduated circle. Otherwise annoying errors of several minutes wTLl be made in a single observation ; and the accumulation of two or three such errors, arising from a cause unknown to him, may compel him to abandon the adjustments in despair. This suggestion applies also to the reading of angles taken by the transit, &c ; although the errors are not then likely to be so great as in the case of the compass. In purchasing a magnifier for a compass, see that no part of it, as hinges, or rivets, are made of iron ; for such would change the direction of the needle. If the sight-slits of a compass are not fixed by the maker in line with the two opposite zeros, the engineer cannot remedy the defect. This can be ascertained by passing a piece of fine thread through the slits, and observing whether it stands precisely over the zeros. 300 THE COMPASS. 100 200 300 400 500 GOO 6 100 200 300 400 500 ' ^ THE COMPASS. .2-? ^ 3-1 ;> <» o :3 ■2 -" "" fi r , a oj o a a> fl jj^o^ c3 a f> ;; r^ - « sa S a =« 5 = =^ fl .2't5 f rt M a a S 3? 5 O 'r ^"^ ^ ^ » rt 2'd '^■e 2-13 3 '-I 13 -. -tJ (« hr 3> c5 f>;3 T a; ^ _, >»;= = 58 oj-ca J3 3) > ^-- O a3 t3 3 - O = a .3'^ a is»c2a>-5 = = - ^H 3 •- -^^ ■= H r'.'^ ^ 2 2 2 -13 r^S flt;:a^%- ^ T3 >4 q 5 O*^ CJ = -.2lS =^ ^ ^^ • - -2 -2i:;t; =5 S a rt r S -S a rt ^ e .5 aj t-.iJ es =^ o G rt a>. a» o aJ 2i^2.2a'' ^ -Jj S fl "* C 5b ^^a^^-^lg r^ ^ S « =« P 73 o o -sga 2Q 1-2 rS .2.2 rt^oi i:^ , :3 r .'^o cs .^ oaa •pK^'-^'^^s .2 S.S'^ t. « o* .fa ^ a 2-° 3 N ^ ^^ CO CO -g i, ^ OOOi-hXJ o ^«8 o •^ o: ctvh C o t 3 a; o .^2 « 3J '^ a^ ^ aii ■ O D.S ? a © a • * r^ a A « -tf •" a aa o .- O) •* 2 ^ ■.tSB fi -"^ 3 a O A i^ *J p *"* bct^ n B u ^ 3 «8 iJ ? o 2=«a'h--.r •2.2^S5-;|f| Slo^a'-^^o^ .t5'« I §.2 g.g a o o © S bc -— ' 3 o-a -S^^ja-g J^^o (u a en'^ a g*- g^bowfl.Ef2««a ^ 2^^ O o 3> a.;:; o rt £ 2Mi2o a >.2c3 S-^' 2 : «^^ S ^ ^r^ o c3 S o i^ 2 © 2 2 -H a •^^'^ «o Ig-o^^a^lo, l-=?^» :s^ w -^"S '' OJ :8 © ;i.2»|a^.2 S ^=^2-2 S ^-3 -eg ' a^ ^ 03 ® rt'^ o'^ 2 3 -a S a .S V a si ^ g 03 -^ g ''3 "^ '^ 2 af S 61) a> ^^ o rtj.ii^ s a ^ ^0 ^^ -^ :!t2Ha2| = 8 §^^§a.2 2 .a: 2 a a ^-^ S - = a ^ 2-^ S « «« a w "^ dj 'N ^ o — ^^^-^ 'l^a^b t;.2 2" :2a?53ail .5.2 q!" 3 ' a a ^^.'ij x^o 5aar^.2a ^z: a^-g=! ^«.aSSa .|;r^ii o a^^ cr-.S P^C3-».s-M £]»IAOXi:TIZATIO]!ir. The needle, if of soft metal, sometimes loses part of its magnetism, and consequently does not work well. It may be restored by simply drawing the north pole of a common magnet (either straight or horseshoe) about a dozen times, from the center to the end of the south half of tlie needle ; and the south pole, in the same way, along the north half; pressing the magnet gently upon the needle. After each stroke, remove the magnet several inches from the needle, while bringing it back to the center for making another stroke. Each half of the needle in turn, while being thus operated on, should be held flat upon a smooth hard surface. Sluggish action of the needle is, however, more generally produced by the dulling or other injury of the point of the pivot. Remagnetizing will throw the needle out of balance ; which must be counteracted by the sliding wire. In order to prevent mistakes by reading* sometimes from one end, B,nd sometimes from the other end of the needle, it is best to always point the N of the compass- box toward the object whose bearing is to be taken ; and to read oflf from the noith end of the needle. This is also more accurate. CONTOUE LINES. A CONTOUR LINE is a curved hor one, every point in which represents the same level; thus each of the contour lines 88c, 9ic, 94c, «S;c, Fig 1, indicates that every point in the ground through which it is traced is at the same level ; and that that level or height is everywhere 88, 91 , or 94 ft above a certain other level or height called datum: to which all others are referred. Frequently the level of the starting point of a survey is taken as being 0, or zero, or datum ; and if we are sure of meeting with no points lower than it, this answers every purpose. But if there is a probability of many lower points, it is better to assume the starting point to be so far above a certain supposed datum, that none of these lower points shall become minus quantities, or below said supposed datum or zero. The only object in this is to avoid the liability to error which arises when some of the levels are +, or plus ; and some — , or minus. Hence we may assume the level of the starting point to be 10, 100, 1000, &c, ft above datum, according to circumstances. The vert dists between each two contour lines are supposed to be equal ; and in railroad surveys through well-known districts, where the engineer knows that his actual line of survey will not require to be much changed, the dist may be 1 or 2 ft only ; and the lines need not be laid down for widths greater than 100 or 200 ft on each side of his center-stakes. But in regions of which the topography is compara- tively unknown ; and where consequently unexpected obstacles may occur which require the line to be materially changed for a considerable dist back, the observa- tions should extend to greater widths ; and for expedition the vertical dists apart may be increased to 3, 5, or even 10 ft, depending on the character of the country, &c. Also, when a survey is made for a topographical map of a State, or of a county, vert dists of 5 or 10 ft will generally suffice. Let the line A B, Fig 1, starting from O, represent three stations (S 1, S 2, S 3,) of the center line of a railroad survey ; and let the numbers 100, 103, 101, 104, along that line denote the heights at the stakes above datum, as determined by levelling. Then the use of the contour lines is to show in the office what would be the effect of changing the surveyed center line A B, by moving any part of it to the right oi CONTOUR LINES. 303 left hand.* Thus, if it should be moved 100 ft to the left, the starting point would be on ground about 6 ft higher than at present ; inasmuch as its level would then be about 106 ft above datum, instead of 100. Station 1 would be about 7 ft higher, or 110 ft instead of 103. Station 2 would be about 7 ft higher, or 108 ft instead oi 101. If the line b?i thrown to the right, it will plainly be on lower ground. The field observations for contour lines are sometimes made with the spirit-level; but more frequently oy a slope-man, with a straight 12-ft graduated rod, and a slope Instrument, or clinometer. At each station he lays his rod upon the ground, as Fig. 1. nearly Sii right angles to the center line A B as he can judge by eye; and placing the slope instrument upon it, he takes the angle of the slope of the ground to the nearest %ofsi degree. He also observes how far beyond the rod the slope continues the same ; and with the rod he measures the dist. Then laying down the rod at that point also, he takes the next slope, and measures its length ; and so on as far as may be judged necessary. His notes are entered in his field-book as shown in Fig 2 ; the angles of the slopes being written above the lines, and their lengths below ; and should be accompanied by such remarks as the locality suggests ; such as woods, rocks,> mari»h. sand, field, garden, across small run, j&c, &c. * In thus using the words right and left we are supposed to have our backs turned to the starting point of the survey. In a river, the ri^bt bank or shore is that which IS on the right hand as we descend it, that is, in speaking of its right or lefl bank, we are supposed to have our backs turned towards its head, or origin ; and so with a survey 304 CONTOUR LINES. It is not absolutely necessary to represent the slopes roughly in the field-book, ai in Fig 2 ; for by using the sign + to signify the slopes may be writ- ten in a straight line, as in Fig 2i^. The notes having been taken, the preparation of the contour lines by means of them, is of course office-work ; and is usually done at the *!ame time as the draw- ing of the map, &c. The field observations at each station are then sepa- rately drawn by protrac- tor and scale, as shown in Fig 3 for the starting up;" — "down;" and cz "level," point 0. The scale should not be less than about j^ inch to a ft, if anything lik« accuracy is aimed at. Suppose that at said station the slopes to the right, taken iu their order, are, as in Fig 2, 15°, 4°. and 26° ; aiid those to the left, 20°, 10°, and 16° ; and their lengths as in the same Fig. Draw a hor line ho, Fig 3; and consider the center of it to be the station-stake. From this point as a center, lay off these angles with a protractor, as shown on the arcs in Fig 3. Then beginning say on the right hand, with a parallel ruler draw the first dist a c, at its proper slope of 15° ; and of its proper length, 45 ft, by scale. Then the same with c y and y t. Do the same with those on the left hand. We then have a cross-section of the ground at Sta 0. Then on the map, as in Fig 1, draw a line as in n, or h w, at right angles to the line of road, and passing through ths station-stake. On this line lay down the hor dists ad,ds,sVf ae,eg, g k, marking them with a small star, as is done and lettered in Fig 1, at Sta 0. When extreme accuracy is pretended to, these hor dists must be found by measure on Fig 3; but as a general rule it will be near enough, when the slopes do not ex- ceed 10° to assume them to be the same as the sloping dists measured in the field. Next ascertaiQ how high each of the points cytlniis above datum. Thus, measure by scale the vert dist d c. Suppose it is found to be 5 ft ; or in other words, that c is 5 ft below station-stake 0. Then since the level at stake is 100 ft above datum, that at c must be 5 ft less, or 100 — 5 = 95 ft above datum ; which may be marked in . light lead-pencil figures on the map, as at d, Fig 1. Next for the point y, suppose we find s 2/ to be 11 ft, or y to be 11 ft below stake ; then its height above datum must be 100 — 11 = 89; Avliich also write in pencil, as at s. Proceed in the same way with t. Next going to the left hand of the station-stake, we find eZ to be say 2 ft; but I is above the level of the station-stake, therefore its height abov» datum it h-^ Fig. 8. 100 -f- 2 == 102 ft, as figured at e on the map. Let ng be 5 ft; then is n, 100 -f- 5 =. 105 ft above datum, as marked at g ; and so on at each station. When this has been done at several stations, we may draw iu the contour lines of that portion by hand thus : Suppose they are to represent vert heights of 3 ft. Beginning at Station O (of which the height above datum is 100 ft) to lay dow^n a contour line 103 ft above datum, we see at once that the height of 103 ft must be at t, or at 3/^ the dist from e to g. Make a light lead-pencil dot at t; and then go to the next Station 1. Here we see that the heiglit of 103 ft coincides with the station-stake itself; place a dot there, and go to Sta 2. The level at this stake is 101 ; therefore the contour for 103 CONTOUR LIKES. 305 ft must evidently be 2 ft higher, or at i,.% of the dist from Sta 2 to +104; thereior* make a dot at i. Then go to Sta 3. Here the level being lOi above datum, the con- tour of 103 must be at y, or -5- of the dist from Sta 3 to +99 ; put a dot at y. Finally draw by hand a curving line through t, SI, i, and y 5 and the contour line of 103 ft ig done. All the others are prepared in the same way, one by one. The level of each must be figured upon it at short intervals along the map, as at 103 c, 106 c, &c. Or, instead of first placing the + points on the map, to denote the slope dists actu- ally measured upon the ground, we may at once, and with less trouble, find and show those only which represent the points t, S 1, t, y, &c, of the contours themselves. Thus, say that at any given station-stake. Fig 4, the level is 104 ; that the cross-sec- tion c s of the ground has been prepared as before ; and that we want the hor diiii from the stake, to contour lines for 94, 97, 100 ft, &c, 3 ft apart vert. Draw a vert line v I, through the station-stake, and on it by scale mark levels of 94, 97, 100, &c. ft. This is readily done, inasmuch as we have the level 104 of the stake already given. Through these levels draw the hor lines «, ft, m, n, Ac. to the ground-slopes. Then these lines, measured by the scale, plainly give the required dists. When the ground is very irregular transversely, the cross-sections must be taken in the field nearer together than 100 ft. The preparation of contour lines will be greatly facilitated by the use of paper ruled into small squares of not less than about ^V inch to a side, for drawing the cross-sections upon. When the ground is very steep, it is usual to shade such portions of the map to represent hill-side. The closer together the contours come, the steeper of course is the ground between thera ; and the shading should be proportionally darker at such portions. But for ivorking maps it is best to omit the shading. In surveys of wide districts, the transit instrument with a graduated vertical circle or arc, g, p. 291, is used for measuring the angles of slope, instead of. the common slope-instrument. In many cases, notes similar to the following will serve the purpose of contour lines on railroad surveys. Sta 60 — 3. 1 R. + 2. 1 L. 61 + 2. 2 R. —1.3 L. 62 = 1. R. + 4. 2 L. 63.. "Which means that at station 60, the slope of the ground on the rieht, as nearly as he can judge by eye, or by his hand-level, is about 3 ft downward, for I chain, or 100 ft ; and on the left, about 2 ft upward in 1 chain. A/ 61 , 2 ft up, in 2 chains to the right; and 1 ft down in 3 chains to the left. At 62, level for 1 chain to the right; and ascending 4 ft in 2 chains to the left. At 68, the same as at 62. At some spots it will be well to add a sketch of a cross-section, like Fig 2; only, instead of the angles, use ft of rise or fall, to indicate the slopes, as judged by eye, or by a hand-level. By this method, the result at every stntion will be somewhat in error; but these small errors will balance each other so nearly that the total may be regarded as sufficiently correct for all the purposes of a preliminary estimate of the cost of a road. When the final stakes for guiding the workmen are placed, the slopes should be carefully taken, \n order to calculate the quauiity of excavation accu- rately for payment. 20 306 TH£ LJBV£IL. THE LEVEL. Although the levels of different makers vary somewhat in their details, still theil principal parts will be understood from the following figure. The telescope T T rests upon two supports YY, called Ys ; out of which it can be lifted, first removing the pins s s which confine the semicircular clips e e, and then opening the clips. The pins should be tied to the Ys, by pieces of string, to prevent their being lost. The slide of the object-glass O, is moved backward or forward by a rack and pinion, by means of the milled head A. The slide of the eye-glass E, is moved in the same way by the milled head e. A cylindrical tube of brass, called a shade, is usually furnished with each level. It is intended to be slid on to the object-end O of the telescope, to prevent the glare of the sun upon the object-glass, when the sun is low. At B is an outer ring encircling the telescope, and carrying 4 small capstan- headed screws ; two of which, »j9, are at top and bottom; while the other two, of which i is one, are at the sides, and at right angles top p. Inside of this outer ring is another, inside of the telescope, and which has stretched across it two spider-webs, usually called the cross-hairs. These are much finer than they ap- pear to be, being considerably magnified. They are at right angles to each other ; and, in levelling, one is kept vert, and the other hor. They are liable at times to b« thrown out of this position by a partial revolution of the telescope, when carrying the level, or when setting the tripod down suddenly upon the ground ; but since, in levelling, the intersection of the hairs is directed to the target-rod, this derangement does not affect the accuracy of the work. Still it is well to keep them nearly vert and hor, by keeping the bubble-tube D D as nearly directly over the bar V F as can be judged by eye. This euables the leveller to see that the rod-man holds his rod nearly vert, which is absolutely essential for correct levelling. If perfect verticality is desired, as is sometimes the case, when staking out work, it may be obtained (if the instrument is in perfect adjustment, and levelled) by sighting at a plumb-line, or other vert object, and then turning the telescope a little in its Ys,so as to bring the hair to correspond. When this is done, a short continuous scratch may be made on the telescope and Y, to save that troncle in future. Heller & Brightly, however, provide their levels with a small projection inside of the Ys, and a corresponding stop on the telescope, the contact of which insures the verticality of the hair. Should the hairs be broken by accident, they may be replaced as directed here- after. The small holes around the heads of the 4 small capstan-screwsp, i, just referred to, are for admitting the end of a small steel pin. or lerer, for turning them. If first the upper screw p be loosened, and then the lower one tightened, the interior ring will be lowered, and the horizontal hair with it. But on looking through the tele- THE LEVEL. 307 scope they will appear to be raised. If first the lower one be loosened, and the upper one tightened, the hor hair will be actually raised, but apparently lowered. This is because the glasses in the eye-piece E reverse the apparent position of objects inside cf the telescope ; which effect is obviated, as regards exterior objects, by means of the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a hair appears to strike too high, it must be raised still higher; if it appears to be already too far to the right or left, it must be actually moved still more in the same direction. This remark, however, does not apply to telescopes which make objects appear inverted. There is no danger of injuring the hairs by these motions, inasmuch as the four screws act against the ring only, and do not come in contact with the hairs them- selves. Under the telescope is the bubble-tube D D. One end of this tube can be raised or lowered slightly by means of the two capstan-headed nuts n n, one of which must be loosened before the other is tightened. On top of the bubble-tube are scratchei for showing when the bubble is central in the tube. Frequently these scratches, or marks, are made on a strip of brass placed above the tube, as in our fig. There are several of them, to allow for the lengthening or shortening of the bubble by changes of temperatuie. At the other end of the bubble-tube are two small capstan-screws, placed on opposite sides horizontally. The circular head of one of them is shown near t. By means of these two screws, that end of the tube can be slightly moved hor, or to right or left. Under the bubble-tube is the bar V ¥ ; at one end of which, as at V,are two large capstan-nuts w w, which operate upon a stout interior screw which forms a prolongation of the Y. The holes in these nuts are larger than the others, as they require a larger lever for turning them. If the lower nut is loosened and the upper one tightened, the Y above is raised ; and that end of the telescope becomes farther removed from the bar; and vice versa. Some makers place a similar screw and nuts under both Ys ; while others dispense with the nuts entirely, and substitute beneath one end of the bar a large circular milled head, to be turned by the fingers. This, however, is exposed to accidental alteration, which should be Avoided. When the portions above m are put upon m, and fastened by the screw Y, all the upper part may be swung round hor, in either direction, Dy loosening the clamp-screi¥ H ; or such motion may be prevented by tightening thatserew. It frequently happens, after the telescope has been sighted very nearly upon an object, and then clamped by H, that we wish to bring the cross-hairs to coincide more precisely with the object than we can readily do by turning the telescope bp hand; and in this case we use the tan^eiit-stcrenr b, by means of which a Blight but steady motion may be given after the instrument is clamped. For fuller remarks on the clamp and tangent-screws, see "Transit." The parallel plates m and S are operated by four levellin^-screws; three of which are seen in the figure, at K K. The screws work in sockets R ; ■which, as well as the screws, extend above the upper plate. When the instrument is placed on the ground for levelling, it is well to set it so that the lower parallel plate S shall be as nearly horizontal as can be roughly judged by eye ; in order to avoid much turning of the levelling screws K K in making the upper plate m hor. The lower plate S, and the brass parts below it, are together called the tripod-bead ; and, in connection with three wooden legs Q Q Q, constitute the tripod. In the figure are seen the heads of wing-nuts J which confine the legs to the tripod-head. Under the center of the tripod-head should always be placed a small ring, from which a plumb-bob may be suspended. This is not needed in ordinary levelling, but becomes useful when ranging center-stakes, &c. To adjust a liCvel. This is a quite simple operation, but requires a little patience. Be careful to avoid ttraining any of the screws. The large Y nuts ww sometimes require some force to start them ; but it should be applied by pressure, and not by blows. Before begin- ning to adjust, attend to the object-glass, as directed in the first sentence under "To adjust a plain transit." Three adjustments are necessary; and must he made in the following order: First, that of tlie cross-hairs ; to secure that their intersection shall continue to strike the same point of a distant object, while the telescope is being turned round a complete revolution in its Ys. This is called adjusting the line of collimatton, or sometimes, the line of sight; but it is not strictly the line of sight until all the adjustments are finished; for until then, the line of collimation will not serve for taking levelling sights. If cross-hairs break, see t» 296. Second, that of the bubble-tube D D, to place it parallel to th^ line 308 THE LEVEL. of eollimation. previously adjusted ; so that when the bubble stands at the centre of Its tube, indicating that it is level, we know that our sight through the telescope ii hor. To replace broken bubC>le tube, see p 296. Tbird, that ©f the Ys, by which the telescope and bubble-tube are supported; so that the bubble-tube, and line of sight, shall be perp to the vert axis of the instru- ment; so fis to remain hor while the telescope is pointed to objects in difif directions, as when taking baiCk and fore sights. To make tlie first adjustment, or that of the cross-hairs, plant the ivi^od. firmly upon the ground. In this adjustment it is not necessary to level the instrument. Open the clips of the Ys ; unclamp ; draw out the eye-glass E, until the cro88-ha;irs are seen perfectly clear ; sight the telescope toward some clear dis- tant point of an object ; or still better, toward some straight line, whether vert or Dot. Move the object-glass 0, by means of the milled head A, so that the object shall be clearly seen, i¥ithout parallax, that is, without any apparent dancing about of the cross-hairs, if the eye is moved a little up or down or sideways. To secure this, the object-glass alone is moved to suit different distances ; the eye-giass is not to be changed after it is once properly fixed upon the cross-hairs. The neglect of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the tangent-screw b, bring either one of the cross-hairs to coincide precisely with the object. • Then gently, and without jarring, revolve the telescope half-way round in its Ys. When this is done, if the hair still coincides precisely with the object, it is in adjustment ; and we proceed to try the other hair. But if it does not coincide, then by means of the 4 screws jo, i, move the ring which carries the hairs, so as to rectify, as nearly as can be judged by eye, only one-half of the error; remembering that the ring must be moved in the direction opposite to what appears to be the right one ; unless the telescope is an inverting one. Then turn the telescope back again to its former position ; and again by the tangent-screw bring the cross-hair to coincide with the object. Then again turn the telescope half-way round as before. The hair Mill now be found to be more nearly in its right place, but, in all probabil- ity, not precisely so ; inasmuch as it is diliicult to estimate one-half the error accu- rately by eye. Therefore a little more alteration of the ring must be made ; and it may be necessary to repeat the operation several times, before the adjustment is perfect. Afterward treat the other hair in precisely the same manner. When both are adjusted, their intersection will strike the same precise spot while the telescope is being turned entirely round in its Ys. This must be tried before the adjustment can be pronounced perfect; because at times the adjustment of the second hair, slightly deranges that of the ;first one ; especially if both were much out in the be- ginning. To make the second adjustment, or to place the bubble-tube parallel to the line of eollimation. This consists of two dis- tinct adjustments, one vert, and one hor. The first of these is effected by means of the two nuts n n on the vert screw at one end of the tube ; and the second by the two hor screws at the other end, t, of the tube. Looking at the bubble-tube endwise, from t in the foregoing Fig, its two hor adjusting-screws 1 1 are seen as in this sketch. The larger capstan-headed nut below, has nothing to do with the adjustments; it merely holds the end of the tube in its place. To make the vert adjustment of the bubble-tiibe, by means of the two nuts nn. Place the telescope over a diagonal pair of the levelling-screws K K ; and clamp it there. Open the clips of the Ys ; and by means of the levelling-screws bring the bubble to the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and put it back again in its reversed position. This being done, if the bubble still remains at the center of its tube, this adjustment is in order ; but if it moves toward one end, that end is too high, and must be lowered; or else the other end must be raised. First, correct half the error by means of the levelling-screws K K, and then the re- maining half by means of the tM'o small capstan-headed nuts n n. To raise the end w, first loosen the upper nut and then tighten the lower one ; to do which, turn each nut so that the near side moves toward your right. To lower it, first loosen the lowet nut, then tighten the upper one, moving the near side of each nut toward your left Having thus brought the bubble to the middle again, again lift the telescope out of its Ys ; turn it end for end, and replace it. The bubble will now settle nearer th« center than it did before, but will probably require still further adjustment. If so, correct half the remaining error by the levelling-screws, and half by the nuts, as be- fore ; and so continue to repeat the operation until the bubble remains at the centei in both positions. For another method, see " To adjust the long bubble-tube," p 295. Horizontal adjustment of bubble-tube ; to see that its axis is in the same plane with that of the telescope, as it usually is in new instruments. It is not easily de^ THE LEVEL. 309 ranged, except by blows. Have the bubble-tube, as nearly as may be, directly under the telescope, or over the center of the bar V F. Bring the telescope over two of the levelling-screws K K ; clamp it there ; center the bubble with said screws ;' turn the telescope in its Ys, say about ^^ inch, bringing the bubble-tube out from over the center of the bar, first on one side, then on the other. If the bubble stays centered while so swung out, this adjustment is correct. If it runs toward opposite ends of its tube when swung out on opposite sides of the center, move the end t of the tube by the two horizontal screws 1 1 until the bubble stays centered when the tube is swung out on either side. If the bubble runs toward the same end of its tube on both sideSy the tube is not truly cylindrical, but slightly conical,* so that if the telescope is turned in its Ys the bubble will leave the center, even when the horizontal adjust- ment is correct. It is known to be correct, in such tubes, if the bubble runs the same distance from the center when swung out the same distance on each side. Having made the horizontal adjustment, turn the telescope back in its Ys until the bubble-tube is over the bar. Repeat the vertical adjustment (p 308), which may have become deranged in making this horizontal one. Persevere until both adjustments are found to be correct at the same time. To make the third ad justinent, or to adjust the heights of the Ys, m fts to make the line of collimation parallel to the bar V F, or perp to the vert axis of the instrument. The other adjustments being made, fasten down the clips of the Ys. Make the instrument nearly level by means of all four of the levelling-screws K. Place the telescope over two of the levelling-screws which stand diagonally; and leave it there undamped. Then bring the bubble to the center of its tube, by the two levelling-screws. Swing the upper part of the instrument half-way around, BO that the telescope shall again stand over the same two screws; but end for end. This done, if the bubble leaves the center, bring it half-way back by the large cap- stan nuts w, w; and the other half by the two levelling-screws. Remember that to raise the Y, and the end of the bubble over w, w, the lower w must be loosened ; and the upper one tightened ; and vice versa. Now place the telescope over the other diagonal pair of levelling-screws: and repeat the whole operation with them. Hav- ing completed it, again try with the first pair; and so keep on until the bubble re- mains at the center of its tube, in every position of the telescope. Correct levelling may be performed even if all the foregoing adjustments are out of order ; provided each fore-sight be taken at precisely the same distance from the instrument as the back-sight, is. But a good leveller will keep his instrument always in adjustment; and will test the adjustments at least once a day when at work. As much, however, depends upon the rodman, or target-man, as upon the leveller. A rod- man who is careless about holding the rod vert, or about reading the sights correctly, ■hould be discharged without mercy. The levelling-screws in many instruments become very hard to turn if dirty. Clean with water and a tooth-brush. Use no oil on field instruments. Forms for level note-books. When the distance is short, so as not to require two sets of books, the following is perhaps as good as any. UlU'I^tl's.IsSitJ ""f- heve..|Grade.| Cut. | Fill. | But on public works generally the original field-books have only the first five cols. After the grades have been determined by means of the profile drawn from these, the results are placed in another book, which has only the first col and the last four. In both cases, the right-hand page is reserved for memoranda. The writer considers it best, both with the level and with the transit, to consider the term " Station " to a^ply to the whole dist between two consecutive stakes ; and that its number shall be that written on the last stake. Thus, with the transit. Station 6 means the dist from stake 5 to stake 6 ; that it has a bearing or course of so and so ; and its length is so and so. And with the level. Station 6 also means the dist from stake 5 to stake t) ; the back-sight for that dist being taken at stake 5, and the fore-sight on stake 6 ; and that the level, grade, cut, or fill is that at stake 6. The starting-point of the survey, whether a stake, or any thing else, we call and mark simply 0. * This defect can be remedied only by removing the tube and inserting a correctly- shaped one, and this is best done by an instrument -maker ; but coiTect work can be done in^ spite of it, thus: Make all the adjustments as nearly correct as possible. Level the instrument. By turning the telescope in its Ys, make the vertical hair coincide with a plumb-line or other vertical line, and make a short continuous knife- scratch on the collar nearest the object-glass, and on the adjoining Y. Lift the tele- scope out of its Ys, turn it end for end, replace it in its Ys ; again bring the upright hair vertical, and make on the other Y a scratch coinciding with that on the collar. Then, in levelling or in adjusting, always see that the scratch on the collar coincides with that on the adjoining Y when the bubble-tube is under the telescope. 310 THE HAND-LEVEL. THE HAJ^I>-I.EVEI.. si This very useful little instrument, as arranj2;ed by Professor Locke, of Cincinnati, is but about five or six inches long. Simply holding it in one hand, and looking through It in any direction, we can ascertain at once, approximately, what objects are at the same level with the eye. E is the eye end : and the object end. L is a small level, enclosed in a kind of brass boxing t g, the bottom of which is open, with a cor- responding opening under it, through the top of the main tube E 0. Immediately at the bottom of the small level L, is a cross-wire, stretched across said opening, and carried by a small plate, which, for adjusting the wire, can be pushed backward a trifle by tightening the screw t, or pushed forward by a small spring within the box- ing, near g, when the screw t is loosened. At m is a small semicircular mirror a a, silvered on the back m. This is placed at an angle of 45°, and occupies one-half the width of the tube E 0. Through the forementioned openings, the images of the cross-wire and of the level-bubble are reflected down on the unsilvered face a a of the mirror, and thence to the eye, as shown by the single dotted lines c and w; and when the instrument is adjusted, and held level, the wire will appear to be at the center of the bubble. At 7e is one-half of a plano-convex lens, at the inner end of a short tube k p, which may be moved backward or forward by a pin n, projecting through a short slit in the main tube. By this means the image of the cross-wire is rendered distinct ; and the half lens must be moved until, when viewing an object, tbe wire shall show no parallax; but appear steady against the object when the ey* is slightly moved up or down. At each end of the tube E is a circular piece of plain glass for excluding dust. To adj ust tlie band-level, first fix two precisely level marks, say from 50 feet to 100 yards apart. This being done, rest the instrument against one of the level marks, and take sight at the other. If, then, the wire does not appear to be precisely at the center of the bubble, move it slightly backward or forwai d, as the case may be, by the screw t, until it does so appear. The two level marks may be fixed by means of the ^,^ hand-level itself, even if it is entirely out of adjust- ^ rtrr-^ ment, thus : First, by the pin n arrange the half lens "^ •' JSZIZ____ TTl k, so as to show the wire distinctly and without paral- ^ ~ /. lax. Then holding the level steadily, at any selected object, as a, so that the wire appears to cut the center of the bubble, see where it cuts any other convenient object, as b. Then go to 5, and from it, in like manner, sight back toward a. If the instrument is in adjust- ment, the wire will cut a ; but if not, it will strike either above it or below it, as at c. In either case, make a mark m, half-way between c and a. Then b and m will be the two level marks required. With care, these adjustments, when once made, will remain in order for years. The instrument generally lias a small ring r, for hanging it around the neck : it is not adapted to very accurate work, but admirably so for exploring a route. The height of a bare hill can be found by begii^ning at the foot, and sighting ahead at any little chance object which the cross-wire may strike, as a pebble, twig, &c; then going forward, stand at that object, and fix the wire on another one still farther on, and so to the top. At each observation we plainly rise a height equal to that of the eye, say 5% feet, or whatever it may be. Whether going up or down it, if the hill is covered with grass, bushes, &c, a target rod must be used for the fore-sights ; and the constant height of the eye may be regarded as the back-sight at each station. An attachment may be made for screwing the level to a small ball and socket on top of a cane, or of a longer stick, for occasional u»e, when rather more accuracy is desired. LEVELS. 311 To adjust a builder's plnmb- level, t'b d% stand it upon any two sup- ports m and n, and mark where the plumb- line cuts at o. Then reverse it, placing the foot t upon n, and d upon w, and mark where the line now cuts at c. Half-way between o and c maKe the permanent mark. Wlienever the line cuts this, the feet t and d are on a level. To adjust a slope-instrament, or clinometer. As usually made, the bubble-tube is attached to the movable bar by a screw near each end, and the head of one of the screws conceals a small slot in the bar, which allows a slight vert motion to the screw when loose, and with it to that end of the tube. Therefore, in order to adjust the bubble, this screw is first loosened a little, and then moved up «r down a trifle, as may be reqd. It is then tightened again. 312 liE YELLING BY THE BAROMETER. liETEIililHrO BY THE BAROMETER. 1. Many circumstances combine to render the results of this kind ofierellin? un- reliable where great accuracy is required. This fact was most conclusively proved by the observations made by Captain T, J. Cram, of the U. S. Coast Survey. See Report of U. S. C. S., vol. for 1854. It is difficult to read off from an aneroid (the kind of barom generally employed for engineering purposes) to within from two to five or six ft, depending on its size. The moisture or dryness of the air affects the results; also winds, the vicinity of mountains, and the daily atmospheric tides, which cause incessant and irregular fluctuations in the barom. A barom hanging quietly in a room will often vary jq of an inch within a few hours, corresponding to a diff of elevation of nearly 100 ft. No formula can possibly be devised that shall embrace these sources of error. The variations dependent upon temperature, lati- tude, Ac, are in some measure provided for; so that with very delicate instruments, a skilful observer may measure the diff of altitude of two points close together, such as the bottom and top of a steeple, with a tolerable confidence that he is within two or three feet of the truth. But if as short an interval as even a few hours elapses between his two observations, such changes may occur in the condition of the atmo- sphere that he may make the top of the steeple to be lower than its bottom ; or at least, cannot feel by any means certain that he is not ten or twenty ft in error; and this may occur without any perceptible change in the atmosphere. Whenever prac- ticable, therefore, there should be a person at each station, to observe at both points at the same time. Single observations at points many miles apart, and made on dif- ferent days, and in different states of the atmosphere, are of little value. In such cases the mean of many observations, extending over several days, weeks, or months, and made when the air is apparently undisturbed, will give tolerable approximations to the ttuth. In the tropics the range of the atmospheric pres is much less than in other regions, seldom exceeding ^ inch at any one spot ; also more regular in time, and, therefore, less productive of error. Still, the barometer, especially either the aneroid, or Bourdon's mietallic, may be rendered highly useful to the civil engi- neer, in cases where great accuracy is not demanded. By hurrying from point to point, and especially by repeating, he can form a judgment as to which of two sum- mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying party, may materially lessen their labors, especially in a rough country, by select- ing the general route for them in advance. The accounts of the agreement within a few inches, in the measurements of high mountains, by diff observers, at diff periods ; and those of ascertaining accurately the grades of a railroad, by means of an aneroid, while riding in a car, will be believed by those only who are ignorant of the subject. Such results can happen only by chance. When possible, the observations at different places should be taken at the same time of day, as some check upon the effects of the daily atmospheric tides ; and in very important cases, a memorandum should be made of the year, month, day, and hour, as well as of the state of the weather, direction of the wind, latitude of the place, &c, to be referred to an expert, if necessary. Ttie effects of latitude are not included in any of our formulas. When reqd they may be found in the table page 314. Several other corrections must be made when great accuracy is aimed at ; but they require extensive tables. In rapid railroad exploring, however, such refinements may be neglected, inas- much as no approach to such accuracy is to be expected ; but on the contrary, errors oi from 1 to 10 or more feet in 100 of height, will frequently occur. As a very rong-li average we may assume that the barometer falls y^^ . inch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But in fact its rate of fall decreases continually as we rise ; so that at one mile high it falls ^Q inch for about 106 ft rise. Table 2 shows the true rate. LEVELLING BY THE BAROMETER. 313 To ascertain the difT of heigrht between two points. KuLE 1, Take readings of the barom and therm (Fah) in the shade at both Btations. Add together the two readings of the barom, and div their sum by 2, for their mean ; which call b. Do the same with the two readings of the thermom, and call the mean t. Subtract the least reading of the barom from the greatest ; and call the diff d. Then mult together this diff d; the number from the next Table No. 1, opposite t; and the constant number 30. Div the prod by b. Or Height _ DiflF( Tabular number opposite ^ p „^far.<- -^n in feet "" barom ^ mean (Q of thermom ^ i^oDstant du . mean (b) of barom. Example. Reading of the barom at lower station, 26.64 ins ; and at the upper 6ta 20.82 ins. Thermom at lowest sta, 70°; at upper sta, 40°. What is the ditf io height of the two stations ? Here, Barom, 26.64 Therm, 70^ 20.82 " 40° Also, 2)47.46 2)110 23.73 mean of bar, or b. 55° mean of therm, or t. The tabular number opposite 55°, is 917.2. Bar. Bar. Again, 26.64 — 20.82 = 6.82, diff of bar ; or d. Hence, d, Tab No. Con. Height _ 5.82X917.2X30 _ 160143.12 _ 6748.5 ft; answer, in feet 23.73 (or 6) 23.73 Then correct for latitude, if more accuracy is reqd, by rule on next page. The screw^ at the bach of an aneroid is for adjusting the index by a stand- ard barom. After this has been done it must by no means be meddled with. In Bome instruments specially made to order with that intention, this screw may be lised also for turning the index back, after having risen to an elevation so great that the index has reached the extreme limit of the graduated arc. After thus turning it back, the indications of the index at greater heights must be added to that at- tained when it was turned back. TABIi£ 1. For Rale 1. Mean Meao Mean Mean of No. of No. of No. of No. Ther. Ther. Ther. Ther. 0° 801.1- 30° 864.4 60° 927.7 90° 991.0 1 803.2 31 866.5 61 929.8 91 993.1 2 805.3 32 868.6 62 931.9 92 995.2 3 807.4 33 870.7 63 934.0 93 997.3 i 809.5 34 872.8 64 9.36.1 94 999.4 5 8H.7 35 874.9 65 938.2 95 1001.6 6 813.8 36 877.0 66 940.3 96 1003.7 7 815.9 37 879.2 67 942.4 97 1005.8 8 818.0 38 881.3 68 944.5 98 1007.9 9 820.1 39 883.4 69 946.7 99 1010.0 10 822.2 40 885.4 70 948.8 100 1012.1 11 824.3 41 887.5 71 950.9 101 1014.2 12 826.4 42 889.6 72 953.0 102 1016.3 13 828.5 43 891.7 73 955.1 103 1018.4 U 830.6 44 893.8 74 957.2 104 1020.5 15 832.8 45 896.0 75 959.3 105 1022.7 16 834.9 46 898.1 76 961.4 106 1024.8 17 837.0 47 900.2 77 963.5 107 1026.9 18 839.1 48 902.3 78 965.6 108 1029.0 19 841.2 49 904.5 79 967.7 109 1031.1 20 843.3 50 906.6 80 969.9 110 1033.2 21 845.4 51 908.7 81 972.0 111 1035.3 22 847.5 52 910.8 82 974. J 112 1037.4 23 849.6 53 913.0 83 976.2 113 1039.5 24 851.8 54 915.1 84 97^.3 114 1041.6 25 853.9 55 917.2 86 980.4 115 1043.8 26 856.0 56 919.3 86 982.6 116 1045.9 27 858.1 57 921.4 87 984.7 117 1048.0 28 860.2 58 923.5 88 986.8 118 1050.1 29 862.3 59 925.6 89 988.9 119 1052.2 314 LEVELLING BY THE BAROMETER. Rule 2. Belville's short approx rule is the one best adapted to rapid field use, namely, add together the two readings of the barom only. Also find the diff between said two readings ; then, as the Slim of the two reading's is to their diff, so is 55000 feet to the reqd altitude. Correction for latitude is usually omitted where great accui-acy is not required. To apply it, first find the altitude by the rule, aa before. Then divide it by the number in the following table opposite the latitude of the place. (If the two . places are in different latitudes, use their mean.) Add the quotient to the altitude if the latitude is less than 45°. Subtract it if the latitude is more than 45°. No cor- rection required for latitude 45°. Table of corrections for latitude. Lat. Lat. Lat. Lat. Lat. Lat. 0° 352 14° 399 280 630 420 3367 540 1140 680 490 2 354 16 416 30 705 44 10101 56 941 70 460 i 356 18 436 32 804 45 00 58 804 72 436 6 360 20 460 34 941 46 10101 60 705 74 416 8 367 22 490 36 1140 48 3367 62 630 76 399 10 375 24 527 38 1458 50 2028 64 572 78 386 12 386 26 572 40 2028 52 1458 66 527 80 375 LiCTelling^ by Barometer; or by the boiling point. Rule 3. The following table. No. 2, enables us to measure heights either by means of boiling water, or by the barom. The third column shows the approximate alti- tude above sea-level corresponding to diff heights, or readings of the barom ; and to the diff degrees of Fahrenheit's thermom,at which water boils in the open air. Thus when the barom, under undisturbed conditions of the atmosphere, stands at 24.08 inches, or when pure rain or distilled water boils at the temp of 201° Fah ; the place is about 5764 ft above the level of the sea, as shown by the table. It is therefore very eeisy to find the diff of altitude of two places. Thus : take out from table No 2, the altitudes opposite to the two boiling temperatures ; or to the two barom readings. Subtract the one opposite the lower reading, from that opposite the upper reading. The rem will be tlie reqd height, as a rough approximation. To correct this, add together the two therm readings ; and div the sum by 2, for their mean. From table for temperature, p 816, take out the number opposite this mean. Mult the ap- proximate height just found, by this tabular number. Then correct for lat if reqd. Ex. The same as preceding ; namely, barom at lower sta, 26.64 ; and at upper sta, 20.82. Thermom at lower sta, 70° Fah ; and at the upper one, 40°. What is the dilf of height of the two stations ? Alt. Here the tabular altitudes are, for 20.82 9579 and for 26.64 3115 6464 ft, approx height. 70° -f 40° 110° To correct this, we have ^ = — ^ = 55° mean ; and in table p 316, opp to 55°, we find 1.048. Therefore 6464 X 1.048 = 6774 ft, the reqd height. This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft in each 100 ft. At 70° Fah, pure water will boil at 1° less of temp, for an average of about 550 ft of elevation above sea-level, up to a height of l^ a mile. At the height of 1 mile, 1° of boiling temp will correspond to about 560 ft of elevation. In table p 315 the mean of the temps at the two stations is assumed to be 32° Fah ; at which no correc- tion for temp is necessary in using the table; hence the tabular number opposite 32°, in table p 316, is 1. This diff produced in the temp of the boiling point, by change of elevation, must not be confounded with that of the atmosphere, due to the same cause. The air be- comes cooler as we ascend above sea-level, at the rate (very roughly) of about 1° Fah for every 200 ft near sea-level, to 350 ft at the height of 1 mile. The follonring' table, Jfo. 2, (so far as it relates to the barom.) was de- duced by the writer from the standard work on the barom by Lieut.-Col. R. S. Wil- liamson, U. S. army.* • Published bj permissioo of Ooverament la 1868 by Van Nostraud, N. Y. LEVELLING BY THE BAROMETER, ETC. 315 TABLE 3. Levelling: by Barometer; or by the boiling point. Assumed temp in the shade 32° Fah. If not 32° mult barom alt as per Table, p 316 Boil Altitude Boil Altitude Boil Altitude Boil Altitud* point Barom. above point Barom. above point Barom. above point Barom. above indeg aea level in deg sea level in deg sea level indeg sea level Fah. Ins. Feet. Fah. Ins. Feet. Fah. Ins. Feet. Fah. Ins. Feet. 1840 16.79 15221 .3 19.66 11083 .6 22.93 7048 .9 2659 3164 .1 16.83 15159 .4 19.70 11029 .7 22.98 6991 206 26.64 3115 .2 16.86 15112 .5 19.74 10976 .8 23.02 6945 .1 26-69 3066 .3 16.90 15050 .6 19.78 10923 .9 23.07 6888 26.75 3007 .4 16.93 15003 .7 19.82 10870 199 23.11 6843 26.80 2958 .5 16.97 14941 .8 19.87. 10804 ,1 23.16 6786 26.86 2899 .6 17.00 14895 .9 19.92 10738 .2 23.21 6729 26.91 2850 .7 1704 14833 192 19.96 10685 ,3 23.26 6673 26.97 2792 .8 17.08 14772 .1 20.00 10633 .4 23.31 6617 27.02 2743 .9 17.12 14710 .2 20.05 10567 .5 23.36 6560 27.08 2685 186 17.16 14649 .3 20.10 10502 .6 23.40 6516 27.13 2637 .1 17.20 14588 .4 20.14 10450 .7 23 45 6460 207 27.18 2589 .2 17.23 14543 .5 20.18 10398 .8 23.49 6415 27.23 2540 .3 17.27 14482 .6 20.22 10346 .9 23.54 6359 27.29 2483 .4 17.31 14421 .7 20.27 10281 200 23.59 6304 27.34 2435 .5 17.35 14361 .8 20.31 10230 .1 23 64 6248 27.40 2377 .6 17.38 14315 .9 20.35 10178 .2 23.69 6193 27.45 2329 .7 17.42 14255 193 20.39 10127 .3 23.74 6137 27.51 2272 .8 17.46 14195 .1 20.43 10075 .4 23.79 6082 27.56 2224 .9 17.50 14135 ,2 20.48 10011 .5 23 84 6027 27.62 2167 186 17.54 14075 .3 20.53 9947 .6 23.89 5972 27.67 2120 .1 17.58 14015 .4 20.57 9896 .7 23.94 5917 208 27.73 2063 .2 17.62 13956 .5 20.61 9845 .8 23.98 5874 27.78 2016 .8 17.66 13886 .6 20.65 9794 .9 24.03 5819 27.84 1959 .4 17.70 13837 .7 20.69 9743 201 24.08 5764 27.89 1912 .5 17.74 13778 .8 20.73 9693 .1 24.13 5710 27.95 1856 .6 17.78 13718 .9 20.77 9642 .2 24.18 5656 28.00 1809 .7 17.82 13660 194 20.82 9579 .3 24.23 5602 28.06 1753 17.86 13601 .1 20.87 9516 .4 24.28 5547 28.11 1706 17.90 13542 .2 20.91 9466 .5 24.33 5494 28.17 1650 187 17.93 13498 .3 20.96 9403 .6 24.38 5440 28.23 1595 17.97 13440 .4 21.00 9353 .7 24.43 5386 209 28.29 1539 18.00 13396 .5 21.05 9291 .8 24.48 5332 28.35 1483 18.04 13338 .6 21.09 9241 .9 24.53 5279 28.40 1437 18.08 13280 .7 21.14 9179 202 24.58 5225 28.45 1391 18.12 13222 .8 21.18 9130 .1 24.63 5172 28.51 1336 18.16 13164 .9 21.22 9080 .2 24.68 5119 28.56 1290 18.20 13106 195 21.26 9031 .3 24.73 5066 28.62 1235 18.24 13049 .1 21.31 8969 .4 24,78 5013 28.67 1189 18.28 12991 .2 21.35 8920 .5 24.83 4960 28.73 1134 188 18.32 12934 .3 21.40 8859 .6 24.88 4907 28.79 1079 18.36 12877 .4 . 21.44 8810 .7 24.93 4855 210 28.85 1025 18.40 12820 .5 21.49 8749 .8 24.98 4802 28.91 970 18.44 12763 .6 21.53 8700 .9 25.03 4750 28.97 916 18.48 12706 .7 21.58 8639 203 25.08 4697 29.03 862 18.52 12649 .8 21.62 8590 .1 25.13 4645 29.09 808 18.56 12593 .9 21.67 8530 ,2 25.18 4593 29.15 754 18.60 12536 196 21.71 8481 .3 25.23 4541 29.20 709 18.64 12480 .1 21.76 8421 .4 25.28 4489 29.25 664 18.68 12424 .2 21.81 8361 .5 25.33 4437 29.31 610 189 18.72 12367 .3 21.86 8301 .6 25.38 4:^86 29.36 565 18.76 12311 .4 21.90 8253 .7 25.43 4334 211 29.42 512 18.80 12256 .5 21.95 8193 .8 25.49 4272 29.48 458 18.84 12200 .6 21.99 8145 .9 25.54 4221 29,54 405 18.88 12144 .7 22.04 8086 204 25.59 4169 29.60 352 18.92 12089 .8 22.08 8038 .1 25.64 4118 29.65 308 18.96 12033 .9 22.13 7979 .2 25.70 4057 29.71 255 19.00 11978 197 22.17 7932 .3 25.76 3996 29.77 202 19.04 11923 .1 22.22 7873 .4 25.81 3945 29 83 149 19.08 11868 .2 22.27 7814 .5 25.86 3894 29,88 105 190 19.13 11799 .8 22.32 7755 .6 25.91 3844 29.94 52 19.17 11745 .4 22.36 7708 .7 25.96 3793 212 30.00 sea leT=0 19.21 11690 .5 22.41 7649 .8 26.01 3742 B elow sea level. 19.25 11635 .6 22.45 7602 .9 26.06 3692 30.06 — 52 19.29 11581 .7 22.50 7544 205 26.11 3642 2 30.12 —104 19.33 11527 .8 22.54 7498 .1 26.17 3582 3 30.18 —156 19.37 11472 .9 22.59 7439 .2 26.22 3532 4 30.24 —209 19.41 11418 198 22.64 7381 .3 26.28 3472 5 30.30 —261 19.45 1 1364 .1 22.69 732,4 .4 26.33 3422 6 30.35 -304 19.49 11310 .2 22.74 7266 .5 26.38 3372 7 30.41 -356 Itl 19.54 11243 .3 22.79 7208 .6 26.43 3322 8 30.47 —408 19.58 11190 .4 22.84 7151 .7 26.48 3273 9 30.53 —459 '.2 1 19.62 11136 .6 22.89 7001 .8 26.54 3213 213 30.59 —611 316 SOUND. Corrections for temperature; to be used in connection with Rule 3, when $?reater accuracy is necessary. Also in con^ nection with Table 2 when the temp is not »2°. Mean Mean Mean Mean temp Mult temp Mult temp Mult temp Mult in the by in the by in the by in the by shade. shade. shade. shade. Zero. .933 28° .992 56° 1.050 84° 1.108 2° .937 30 .996 58 1.054 86 1.112 4 .942 32 1.000 60 1.058 88 1.117 6 .946 34 1.004 62 1.062 90 1.121 8 .950 36 1.008 64 1.066 92 1.125 10 .954 38 1.012 66 1.071 94 1.129 12 .958 40 1.016 • 68 1.075 96 1.133 14 .962 42 1.020 70 1.079 98 1.138 16 .967 44 1.024 72 1.083 100 1.142 18 .971 46 1.028 74 1.087 102 1.146 20 .975 48 1.032 76 1.091 104 1.150 22 .979 50 1.036 78 1.096 106 1.164 24 .983 52 1.041 80 1.100 108 1.158 26 .987 54 1.046 82 1.104 110 1.163 SOUND. ■JOke velocity of sound in quiet open air, has been experimentally deter- xnlned to be very approximately 1U90 feet per second, when the temperature is at freezing point, or 3'^° Fahaenheit. For every degree Fahrenheit of increase of temperature, the velocity increases by from i^ foot to 1}^ feet per second, according to different authorities. Takiug the increase at 1 foot per second for each degree (which agrees closely with theoretical calculations), we have at _ 30° Fahr 1030 feet per sec = 0.1951 mile per sec == 1 mile in 5.13 seconda. 20° " 1040 " <.' = 0.1970 " " = 1 " 5.08 10° " 1050 <' (( = 0.1989 « " —=1 " 5.03 " 1060 '• " = 0.2008 (I « = 1 " 4.98 10° « 1070 <' '< = 0.2027 " « = 1 " 4.93 20° " 1080 « « = 0.2045 " " = 1 " 4.88 32° " 1092 (; '• = 0.2068 " « = 1 " 4.83 40° " 1100 " « = 0.2083 " « ==1 " 4.80 50° " 1110 " « = 0.2102 " <« = 1 " 4.78 60° '' 1120 " « = 0.2121 " " = 1 " 4.73 70° «' 1130 " « = 0.2140 « «• = 1 " 4.68 80° « 1140 " « = 0.2159 " " = 1 " 4.63 90° " 1150 « « = 0.2178 « <* = 1 " 4.59 100° " 1160 " i( = 0.2197 " « = 1 " 4.55 110° " 1170 « «« = 0.2216 " « = 1 " 4.51 120° " 1180 " « = 0.2235 u « = 1 " 4.47 If the air is calm, fog or rain does not appreciably affect the result ; but winds do. Very loud sounds appear to travel somewhat faster than low ones. The watchword of sentinels has been heard across still water, on a calm night, 10}^ miles; and a cannon 20 miles. Separate sounds, at intervals of ^^ of a second, cannot be distin- guished, but appear to be connected. The distances at which a speaker can be understood, in front, on one side, and behind him. are about as 4, 3, and 1. Dr. Charles M. Cresson informs the writer that, by repeated trials, he found that in a Philadelphia gas main 20 inches diameter and 16000 feet long, laid and covered in the earth, but empty of gas, and having one horizontal bend of 90°, and of 40 feet radius, the sound of a pistol-shot travelled 16000 feet in precisely 16 seconds, or 1000 feet per second. The arrival of the sound was barely audible; but was rendered very apparent to the eye by its blowing off a diaphragm of tissue-paper placed over the end of the main. Two boats anchored some distance apart may serve as a base line for triangulating objects along the coast; the distance between them being first found by firing guns on board one of them. In -water the velocity is about 4708 feet per second, or about 4 times that in air. In '«voods, it is from 10 to 16 times ; and in metals, from 4 to 16 times greater than in air, according to some authorities. HEAT. 317 Approximate expansion of solids by heat: and tlieir melt* ing^ points by Fahrenheit's thermometer. | Fire-brick.. Granite.... ; from J fn i to Glass rod. Glass tube " crown " plate Platina Marble, granular, white, dry.. " " *' moist. " * black, compact Antimony Obst iron Slate Steel " blistered " untempered " tempered yellow " hardened " annealed Iron, rolled " soft, forged •* wire Bismuth .... Gold, annealed Copper average Sandstone^*. Brass average '• wire Silver .... Tin average Lead average Pewter Zinc (most of all metalf) "White pine For 1 degree. 1 part in 365220 187560 221400 214200 211500 209700 173000 128000 405000 166500 16200011 173000 151200 159840 167400 131400 147600 14994011 147420 146340 129600 123120 104400 103320 97740 94140 95040 87840 63180 78840 61920 440530 ^ inch in 3804 ft. 1954 2375 2306 2231 2203 2184 2175 1802 1333 4219 1722 1688 1802 1575 1665 1744 1369 1530 1537 1562 1536 1524 1350 1282 1088 1076 1018 981 990 915 658 821 645 4588 For 180 degrees.* 1 part in % inch in 2029 21.14 ft. 1042 10.85 1267 13.20 1230 12.81 1190 12.40 1175 12.24 1165 12.13 1160 12.08 961 10.00 711 7.41 2250 23.44 925 9.63 900 9.38 961 10.00 840 8.75 888 9.25 930 9.69 730 7.60 816 8.50 820 8.54 833 8.68 819 8.53 8K 8.55 720 7.50 684 7.12 580 6.04 574 5.98 543 5.66 523 5.45 528 11 488 351 3.66 438 4.56 344 3.58 2447 25.49 Melting point in Deg.t 955 1920 to 2800 2370 to 2550 3000 t* 3500 506 2016 2000 1873 1861 444 612 Heat of a common wood-fire* variously estimated from 800 to 1140 deg. That of a ohareoal eue about 2200 ; coal about 2400°. Each 12° to 15° of heat produce in wrot iron an expansion equal to that produced by a tension of about 1 ton per sq inch of section ; varying with the quality of the iron. * By adding y^^f P'*^'"' *^ f^- lengths in the two cola under 180°, ire get the lengths corresponding t« a number of degrees -Jh less than 180° ; or to 1630.63 deg; which may be taken as about the extremet of temp in the colder portions of the United States. In the Middle States the extremes rarely reach 135°, or U part less than 180°. No dependence whatever is to be placed on results obtained by Wedgewood's pyrometer. t The table shows that the contraction and expansion of stone will cause open joints in winter ; and crushing of the mortar in summer, at the ends of long coping-stones. I The melting: points are quite uncertain. We give the mean of the best authorities. Assuming that with a change of temp of about 163°, wrought iron will alter its length 1 part in 916 ; this in a mile amounts to 5.764 ft, or about 5 ft 9J^ ins ; and in 100 ft to .109 of a foot; or lys ins ; so that a ditf of 5 ft, or more, can readily result from measuring a mile in winter and in summer with the same chain ; and a 25 ft rail will change its length full % of an inch. II Hence fvrought escpands by heat about one eleventh part more than cast; whereas under tension within elastic limits cast stretches twice as much as wrt. 318 THERMOMETERS. THEEMOMETERS. Below about 35° below zero of Fah. the mercurial thermometer and barometer become too irregular It be depended on. Mercury begins to freeze at abou t— 40° Fp.h. Below — 4(Ppure alcohol is used. To cbaiige degrees of Fahrenheit to the corresponding* de* g'rees of Centigrade ; take a Fah reading 32° lower than the given one; mult this lower reading by 5 ; divide the prod by 9. Thus : -|-UO Fah = (14 —32) X 5 -r 9 = —18 X 5 -r 9= — IQoCent. Again, —13° Fah = (— 13 — 32) X 5-r9 = — 45 X 5-r9 = — '25° Cent. To chang-e Fah toReau; take a Fah reading 32° Zower than the given one; mult this lower reading by 4 ; div the prod by 9. Thus: +14° Fah = (14— 32) X 4-r9 = — 18X 4-r9=— SOReau. Again, —13° Fah = (—13 —32) X 4-r9 = — 45 X 44-9 = — 20° Reau. To change €ent to Fah; mult the Cent reading by 9; divide by 5. Take a Fah reading 320 Uqher than the quot. Thus : -fioo Cent = (10 X 9 -j- 5) +32 = 18 +32 = -j-SQO Fah. Again, —20° Cent = (—20 X 9 -j- 5) -f 32 = —4° Fah. To change Cent to Iteau; mult by 4; div by 5. Thus: -t-10°Cent =10X 4-r5-=+80Reau. Again,— 10oCent=— lOX 4t-5=:— 8° R^au. To change Reau to Fah; mult by 9; div by 4. Take a Fah reading 32° higher. Thus : +16° Reau = fie X9 + 4) 4- 82= 36 + 32 = +68° Fah. Again. —8° Reau = (-8 X 9 •r 4) 4-32 = —18 +32 = +14" Fah. To change Reau to €ent; mult by 5 ; div by 4. Thus : +8° Reau = + 8° X5 +4= +10° Cent. Again,— 8° Reau =—8X5+4= —10° Cent. TABIiF 1. Fahrenheit compared with Centigrade and R^an- mnr. In this table the Cent and Reau readings are given to the nearest decimal. F. c. R. F. c. R. F. c. R. F. c. R. F. c. R. o o o o o o o o o o o o o o 212 100 80.0 158 TO.O 56.0 104 40 32.0 50 10.0 8.0 —3 -19.4 -15.6 211 99.4 79.6 157 69.4 55.6 103 39.4 31.6 49 9.4 7.6 —4 -20.0 —16.0 210 98.9 79.1 156 68.9 55.1 102 38.9 31.1 48 8.9 7.1 —5 -20.6 —16.4 209 98.3 78.7 1.55 68.3 54.7 101 38 3 30.7 47 8.3 6.7 -6 -21.1 —16.9 208 97.8 78.2 154 67.8 54.2 100 37.8 30.2 46 7.8 6.2 —7 -21.7 -17.3 207 97.2 77.a 153 67.2 53.8 99 37.2 29.8 45 7.2 5.8 —8 —22.2 -17.8 206 96.7 77.3 152 66.7 53.3 98 36.7 29.3 44 6.7 5.3 —9 -22.8 —18.2 205 96.1 76.9 151 66.1 52.9 97 36.1 28.9 43 . 6.1 4.9 -10 -23.3 -18.7 204 95.6 76.4 i.y) 65.6 52.4 96 35.6 28.4 42 5.6 4.4 —11 —23.9 —19.1 203 95.0 76.0 U9 65.0 52.0 95 35.0 28.0 41 . 5.0 4.0 —12 -24.4 -19.6 202 94.4 75.6 148 64.4 51.6 94 84.4 27.6 40 4.4 3.6 —13 -250 -20.0 201 93.9 75.1 147 63.9 51.1 93 33.9 27.1 39 3.9 3,1 —14 -25.6 -20.4 200 93.3 74.7 146 63.3 50.7 92 33.3 26.7 38 8.3 2.7 -15 —26.1 -20.9 199 92.8 74.2 145 62.8 50.2 91 32.8 26.2 37 2.8 2.2 —16 -26.7 -21.3 198 92.2 73.8 144 62.2 49.8 90 32.2 25.8 36 2.'/ 1.8 —17 -27.2 —21.8 197 91.7 73.3 143 61.7 49.3 89 31.7 25.3 85 1.7 1.8 —18 —27.8 -22.2 196 91.1 72.9 142 61.1 48.9 88 31.1 24.9 34 1.1 0.9 —19 —28.3 -22.7 195 90.6 72.4 141 60.6 48.4 87 30.6 24.4 33 0.6 0.4 —20 -28.9 —23.1 194 90.0 72.0 140 60.0 48.0 86 30.0 24.0 32 0.0 0.0 —21 —29.4 —23.6 193 89.4 71.6 139 59.4 47.6 85 29.4 23.6 31 —0.6 -0.4 —22 —30.0 —24.0 192 88.9 71.1 138 58.9 47.1 84 28.9 23.1 .SO — l.l —0.9 —23 -30.6 —24.4 191 88.3 70.7 137 58.3 46.7 83 28.3 22.7 29 —1.7 —1.3 —24 -31.1 -24.9 190 87.8 70.2 136 57.8 46.2 82 27.8 22,2 28 —2.2 —1.8 —25 -31.7 —25.3 189 87.2 69.8 135 57.2 45.8 81 27.2 21.8 27 —2.8 —2.2 —26 —32.2 -25.8 188 86.7 69.3 134 56.7 45.3 80 26.7 21.3 26 —3.3 —2.7 —27 —32.8 —26.2 187 86.1 68.9 133 56.1 44.9 79 26.1 20.9 25 —8.9 —3.1 —28 -33.3 -26.7 186 85.6 68.4 132 55.6 44.4 78 25.6 20.4 24 —4,4 —3.6 —29 —33.9 -27.1 185 85.0 68.0 131 55.0 44.0 77 25.0 20.0 33 -6,0 —4.0 —80 -34.4 -27.« 184 81.4 67.6 130 54.4 43.6 76 24.4 19.6 22 -5.6 —4.4 —31 —35.0 —28.0 18:^ 83.9 67.1 129 53.9 43,1 75 23.9 19.1 21 —6.1 —4.9 —32 —85.6 —28.4 182 83.3 66.7 128 53.3 42.7 74 23.3 18.7 20 -6.7 -5.3 —33 —36.1 —28.9 181 82.8 66.2 127 52.8 42.2 73 22.8 18.2 19 -7.2 —5.8 —34 —36.7 -29.3 180 82.2 65.8 126 52.2 41.8 72 22.2 17.8 18 -7.8 -6.2 -35 —87.2 -29.8 179 81.7 65.3 125 51.7 41.3 71 21.7 17.3 17 —8.3 -6.7 -36. -37.8 -30.2 178 81.1 64.9 124 51.1 40.9 70 21.1 16.9 16 —8.9 —7.1 -87 -38.8 —30.7 177 80.6 64.4 123 50.6 40.4 69 20.6 16.4 15 —9.4 -7.6 -38 —38.9 —31.1 176 175 80.0 79.4 64.0 63.6 122 121 50.0 49.4 40.0 39.6 16.0 15.6 14 13 —10-0 —10.6 1 -8.0 —8.4 —39 —40 -39.4 -40.0 —81.6 -32.0 67 \%A 174 78.9 63.1 120 48.9 39.1 66 18.9 15.1 12 —11.1 —8.9 —41 -40.6 —32.4 173 78.3 62.7 119 48.3 38.7 65 18.3 14.7 11 —11.7 \ —9.3 -42 —41.1 -.S2.9 172 77.8 62.2 118 47.8 38.2 64 17.8 14.2 10 —12.2 —9.8 -43 —41,7 — 33.S 171 77.2 61.8 117 47.2 37.8 63 17.2 13,8 9 —12.8 —10.2 —44 —42,2 -33.8 170 76.7 61.3 116 46.7 37.3 62 16.7 13.3 8 —13.3 —10.7 —45 -42,8 -34.2 169 76.1 60.9 115 46.1 36.9 61 16.1 12.9 7 —13.9 -11.1 -46 -43.3 -34.7 168 75.6 60.4 114 456 36.4 60 15.6 12.4 6 -14.4 -11.6 —47 —43.9 -35.1 167 75.0 60.0 113 45.0 36.0 59 15.0 12,0 5 —15.0 —12.0 —48 —44.4 -36.6 166 74.4 59.6 112 44.4 35.6 58 14.4 11.6 4 —15.6 —12.4 —49 —45.0 -36.0 165 73 9 59.1 111 43.9 35.1 57 13.9 11.1 3 -16.1 —12.9 -50 -45.6 i— 36.4 164 73.3 58.7 110 43.3 34.7 56 13.3 10.7 2 -16.7 —18.3 —51 —46.1 1—36.9 163 72.8 58.2 109 42.8 34.2 55 12.8 10.2 1 —17.2 -13.8 —52 —46.7 -87.1 162 72.2 57.8 108 42.2 .33 8 54 12.2 9.8 —17.8 -14.2 —53 —47.2 — ST.8 '161 71.7 57.3 107 41.7 33.3 53 11.7 9.3 —1 -18.3 —14.7 —54 -47.8 -S8.I 160 71.1 56.9 106 41.1 32.9 52 11.1 8.9 —2 —18.9 j— 15.1 -55 — 4«.3 — S«.1 1S9 70.6 56.4 105 40.6 32.4 51 10.6 8.4 1 THERMOMETERS. 319 TAB1.E 2, Centigrade compared with FatireaiBieit and Reaumur. c. F. R. c. F. R. c. F. R. €. F. R. o o o o c o o o o o o Exact. Exact. Exact. Exact. Exact. Exact. Exact. Exact. (00 212.0 80.0 62 143.6 49.6 24 75.2 19.2 —14 6.8 -11.2 99 210.2 79.2 61 141.8 48.8 23 73.4 18.4 -15 5.0 —12.0 98 208.4 78.4 60 140.0 48.0 22 71.6 17.6 —16 3.2 -12.8 97 206.6 77.6 59 138.2 47.2 21 69.8 16.8 -17 1.4 — 1S.6 96 204.8 76.8 58 136.4 46.4 20 68.0 16.0 -18 -0.4 -14.4 95 203.0 76.0 57 134.6 45.6 19 66.2 15.2 -19 -2.2 —15.2 94 201.2 75.2 56 132.8 44.8 18 64.4 14-4 —20 -4.0 —16.0 93 199.4 74.4 55 131.0 44.0 17 62.6 13.6 -21 —5.8 —16.8 92 197.6 73.6 54 129.2 43 2 16 60.8 12.8 —22 —7.6 -17.6 91 195.8 72.8 53 127.4 42.4 15 59.0 12.0 —23 -9.4 -18.4 90 194.0 72.0 52 125.6 41.6 14 57.2 11.2 —24 -11.2 —19.2 89 192.2 71.2 51 123.8 40.8 13 55.4 10.4 -25 —13.0 —20.0 88 190.4 70.4 50 122.0 40.0 12 53.6 9.6 —26 —14.8 -20. g 87 188.6 69.6 49 120.2 39.2 11 51.8 8.8 -27 -16.6 —21.6 86 186.8 68.8 48 118.4 38.4 10 50.0 8.0 —28 -18.4 —22.4 85 185.0 68.0 47 116.6 37.6 9 48.2 7.2 —29 —20.2 —23.2 84 183.2 67.2 46 114.8 36.8 8 46.4 6.4 -30 —22.0 -24.0 83 181.4 66.4 45 113.0 .S6.0 7 44.6 5.6 —31 —23.8 -24.8 82 179.6 65.6 44 111.2 35.2 6 42.8 4.8 —32 -25.6 -25.6 81 177.8 64.8 43 109.4 34.4 5 41.0 4.0 -33 —27.4 —26.4 80 176.0 64.0 42 107.6 33.6 4 39.2 3.2 —34 —29.2 —27.2 79 174.2 63. 2 41 105.8 32.8 3 37.4 2.4 —35 —31.0 —28.0 78 172.4 62.4 40 104.0 32.0 2 35.6 1.6 —36 —32.8 —28.8 77 170.6 61.6 39 102.2 31.2 1 33.8 0.8 —37 -34.6 -29.6 76 168.8 60.8 38 100.4 30.4 32.0 0.0 -38 —36.4 -30.4 75 167,0 60.0 37 98.6 29.6 -1 30.2 -0.8 -39 —38.2 —31.2 74 165.2 59.2 36 96.8 28.8 -2 28.4 -1.6 —40 ^40.0 —32.0 7S 163.4 58.4 35 95.0 28.0 —3 2«.6 —2.4 —41 —41.8 -32.8 72 161.6 57.6 34 93.2 27.2 —4 24.8 —3.2 —42 -43.6 -33.6 71 159.8 56.8 33 91.4 26.4 —5 23.0 —4.0 — 43 -45.4 —34.4 70 158.0 56.0 32 89.6 25.6 -6 21.2 -4.8 —44 —47.2 — S5.2 69 156.2 55.2 31 87.8 24.8 —7 19.4 -5.6 —45 —49.0 -36.0 68 154.4 54.4 30 86.0 24.0 —8 17.6 -6.4 —46 -50.8 -36.8 67 152.6 53.6 29 84.2 23.2 -9 15.8 —7.2 —47 -52.6 -37.6 66 150.8 52 8 28 82.4 22.4 —10 14.0 -8.0 —48 -54.4 —38.4 65 149.0 52.0 27 80.6 21.6 -11 12.2 —8.8 —49 -56.2 —39,2 64 147.2 51.2 26 78.8 20.8 -12 10.4 —9.6 -50 -.78.0 — 40.# 68 145.4 50.4 25 77.0 20.0 —13 8.6 —10.4 TABL.F 3. Reaumur compared witli Falirenbeit and Centig-rade. R. F. c. R. F. €. R. F. c. K. F. c. o o o o o o o o o O o o Exact. Exact. Exact. Exact. Exact. Exact. Exact. Exact. 80 212.00 100.00 49 142.25 61.25 19 74.75 23.75 — li 7.25 —13.75 79 209.75 98.75 48 HO.OO 60.00 18 72.50 22.50 —12 5.00 —15,00 78 207.50 97.50 47 137.75 58.75 17 70.25 21.25 —13 2.75 —16.25 77 205.25 96.25 46 135 50 57.50 16 68.00 20.00 —14 0,50 —17,50 76 203.00 95.00 45 133.25 56.25 15 65.75 18,75 —15 -1,75 -18,75 75 200.75 93.75 44 131.00 55 00 14 63.50 17,50 —16 -4.00 —20.00 74 198.50 92,50 43 128.75 53.75 13 61.25 16.25 —17 —6.25 —21.25 73 196.25 91,25 42 126.50 52.50 12 59.00 15.00 -18 -8.50 -22,50 72 194.00 90.00 41 124.25 51.25 11 56.75 13.75 -19 —10.75 -!«}.75 71 191.75 88.75 40 122.00 50.00 10 54.50 12,50 -20 —13.00 -25.00 70 189.50 87.50 39 119.75 48.75 9 52.25 11.25 —21 —15.25 —26.25 69 187.25 86.25 38 117.50 47.50 8 50.00 10,00 —22 —17.50 —27.50 68 185.00 85 00 37 115.25 46.25 7 47.75 8.75 -23 -19.75 —28.75 67 182.75 83.75 36 113.00 45.00 6 45,50 7.50 —24 —22.00 —30.00 66 180.50 82.50 35 110.75 43.75 5 43.25 6.25 -25 —24,25 -31.25 65 178 25 81.25 34 108.50 42.50 4 41.00 5.00 -26 —26,50 -32.50 64 176.00 8(t.0O 33 106.25 41.25 3 38.75 3.75 -27 —28.75 —33.75 63 173.75 78.75 32 104.00 40.00 2 36,50 2.50 -^28 —31.00 —35.00 62 171.50 77,50 31 101.75 38.75 1 34,25 1.25 -29 -33.25 -36.25 61 169.25 76.25 30 99.50 37 50 32.00 0.00 -30 —35.50 —37.50 60 167.00 75.00 29 97.25 36.25 —1 •29.75 -1.25 -31 —37.75 -38.75 59 164.75 73.75 28 95.00 35.00 —2 27,50 -2.50 —32 -40.00 -40.00 58 162.50 72.50 27 92.75 33 75 —3 25,25 —3.75 -33 —42.25 -41.25 57 160.25 71.25 26 90.50 32 50 —4 23.00 -5.00 -34 -44.50 -42.50 56 158.00 70.00 25 88.25 31.25 -5 20.75 -6.25 -35 -46.75 -43.75 55 155.75 68.75 24 86.00 30.00 —6 18.50 -7.50 -36 -49.00 -45.00 54 153.50 67.50 23 83.75 28.75 —7 16.25 —8.75 --37 -51.25 —46.25 53 151.25 66.25 22 81,50 27.50 -8 14.00 —10.00 -38 —53.50 -47.50 52 149.00 65.00 21 79,25 26.25 —9 11.75 —11,25 -39 -.55.75 -48.75 61 146,75 63.75 20 77,00 25.00 —10 9.50 —12.50 -40 —58.00 -50.08 50 144,50 62.50 320 AIK. AIR-ATMOSPHERE. The atmosphere is known to extend to at least 45 miles above the earth. It is a mixture of about 79 measures of nitrogen gas and 21 of oxygen gas; or about 77 nitrogen, 23 oxygen, by weight. It generally con- tains, however, a trace of water, and of carbonic acid and carburetted hydrogen gases, and still less ammonia. Density of air. Under *' normal" or "standard" conditions (sea level, lat45°, barometer 760 mm = 29.922 ins, temperature 0°C = 32°F) dry air weig'hs 1.292673 kilograms per cubic meter * = 2.17888 flbs avoir per cubic yard. For other lats and elevations — R Density, in kg per cu m, = 1.292673 X r^ , ^ ^ X (1 —0.002837 cos 2 lat) * Is, -f- z a where R = earth's mean radius = 6,366,198 meters; h == elevation above sea level, in meters. For other temperatures, see below. Under normal conditions, but with 0.04 parts carbonic acid (C Oj) in 100 parts of air, density = 1.293052 kg per cu m.f = 2.17952 lbs avoir per cu yd. J The atmospheric pressure, at any given, place, may vary 2 inches or more from day to day. Tne average pressure, at sea level | varies from about 745 to 770 millimeters of mercury according to the latitude and locality. 760 millimeters* is generally accepted as the mean atmospheric pressure, and called an atmosphere. The '^ metric atmosphere,'' taken arbitrarily at 1 kilogram per square centimeter, is in general use in Continental Europe. The pressure diminishes as the altitude increases,! Therefore, a pump in a high region will not lift water to as great a height as in a low one. The pressure of air, like that of water, is, at any given point, equal in all directions. It is often stated that the temperature of the atmosphere lowers at the rate of 1° Fah for each 300 feet of ascent above the earth's surface: but this is liable to many exceptions, and varies much with local causes. Actual observation in balloons 'seems to show that, up to the first 1000 feet, 1° in about 200 feet is nearer the truth ; at 2000 feet, 1° in 250 feet ; at 4000 feet, 1° in 300 feet ; and, at a mile, 1° in 350 feet. In breathing*, a grown person at rest requires from 0.25 to 0.35 of a cubic foot of air per minute; which, when breathed, vitiates from 3.5 to 5 cubic feet. When walking, or hard at work, he breathes and vitiates two or three times as much. About 5 cubic feet of fresh air per person per minute are required for the perfect ventilation of rooms in winter ; 8 in summer. Hospitals 40 to 80. Beneath the general level of the surface of the earth, in temperate regions, a tolerably uniform temperature of about 50° to 60° Fah exists at the depth of about 50 to 60 feet ; and increases about 1° for each additional 50 to 60 feet; all subject, however, to considerable deviations owing to many local causes. In the Rose Bridge Colliery, England, at the depth of 2424 feet, the temperature of the coal is 93.5° Fah ; and at the bottom of a boring 4169 feet deep, near Berlin, the temperature is 119°. The air is a very sloiv conductor of heat^ hence hollow walls serve to retain the heat in dwellings; besides keeping them dry. It rushes into a vacuum near sea level with a velocity of about 1157 feet per second ; or 13.8 miles per minute ; or about as fast as sound ordinarily travels through quiet air. See Sound. liike all other elastic fluids, air expands equally with equal increases of temperature. Every increase of 5° Fah, expands the bulk of any of them slightly more than 1 per cent of that which it has at 0® Fah ; or 500° about doubles its bulk at zero. The bulk of any of them diminishes inversely in proportion to the total pressure to which it is subjected. This holds good with air at least up to pressures of about 750 lbs per square inch, or 50 times its natural pressure ; the air in this case occupying one-fiftieth of its natural bulk. In like manner the bulk will increase as the total pressure is diminished. Substances which follow these laws, are said to be perfectly *H. V. Regnault, Memoires de 1' Academic Rovale des Sciences de I'lnstitut de France, Tome XXI, 1847. Translation in abstract, Journal of Franklin Insti- tute, Phila., June, 1848. fTravaux et Memoires du Bureau International desPoidset Mesures, Tomel, page A 54. Smithsonian Meteorological Tables, 1893, published in Smithsoniam Miscellaneous Collections, Vol. XXXV, 1897. t See Conversion Tables. I See Leveling by the Barometer. WIND. 321 elastiCo Under a pressure of about 5}^ tons per square inch, air would become as dense as water. Since the air at the surface of the earth is pressed 14% Bbs per square inch by the atmosphere above it, and since this is equal to the weight of a column of water 1 inch square and 34 feet high, it follows that at the depths of 34, 68, 102 feet, &c, below water, air will be compressed into 3^, %, %, &c, of its bulk at the surface. In a diviiig'-bell, men, after some experience, can readily work for several hours at a depth of 51 feet, or under a pressure of 2^ atmospheres ; or 373^ lbs per square inch. But at 90 feet deep, or under 3.64 atmospheres, or nearly 55 fcs per square inch, they can work for but about an hour, without serious suffer- ing from paralysis, or even danger of death. Still, at the St. Louis bridge, work was done at a depth of 1103^ feet ; pressure 63.7 lbs per square inch. Tlie dew^ point is that temp (varying) at which the air deposits its vapor. Tlie greatest tieat of tlie air in tlie sun probably never exeeeds 145° Fah ; nor the greatest cold — 74° at niglit. About 130° above, and 40° below zero, are the extremes in the U. S. east of the Mississippi; and 65° below in the N. W. ; all at common ground level. It is stated, however, that — 81° has t.een observed in N. E. Siberia;' and +101° Fah in the shade in Paris; and +153° in the sun at Greenwich Observatory, both in July, 1881. It has frequently ex- ceeded +100° Fah in the shade in Philadelphia during recent years. WIND. The relation between tlie velocity of wind, and its press- ure against an obstacle placed either at right angles to its course, or inclined to it, has not been weM determined ; and still less so its pressure against curved surfaces. The pressure against a large surface is probably proportionally greater than against a small one. It is generally supposed to vary nearly as the squares of the velocities; and when the obstacle is at right angles to its direction, the pressure in Bbs per square foot of exposed surface is considered to be equal to the square of the velocity in miles per hour, divided by 200. On this basis, which is probably quite defective, the following table, as given by Smeaton, is prepared. Vel. in Miles Vel. in Ft. Pres. in Lbs. Remarks. per Hour. per Sec. per Sq. Ft. 1 1.467 .005 Hardly perceptible. Pleasant. ^^^ff 2 2.933 .020 3 4.400 .045 c 4 5.867 .080 5 10 7.33 14.67 .125 .5 rk^/jh 12}4 18.33 .781 Fresh breeze. O 15 22. 1.125 20 29.33 2. ^ . , . ^ The pres against 25 36.67 3.125 Brisk wind. a semicylindrical 30 44. 4.5 Strong wind. surface acbnom 40 58.67 8. High wind. is about half that 50 73.33 12.5 Storm. against the flat 60 88. ' 18. Violent storm. surf a 6 nm. 80 117.3 32. Hurricane. 100 146.7 50. Violent hurricane, uprooting large trees. Tredg-old recommends to allow 40 lbs per sq ft of roof for the pres of wind against it ; but as roofs are constructed with a slope, and consequently do not receive the full force of the wind, this is plainly too much.* Moreover, only one-half of a roof is usually ex- posed, even thus partially, to the wind. Probably the force in suoh cases varies approximately as the sines of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per hour, produced a pres of 14 lbs per sq ft against an object perp to it: and one of 70 miles per hour, (the severest gale on record at that city,) 42 lbs per sq foot. These would make the pres per sq ft, more nearly equal to the square of the vel in miles per hour, div by 100; or nearly twice as great as given in Smeaton's table. We should ourselves give the preference to the Liverpool observations. A very violent gale in Scotland, registered by an excellent anemometer, or wind-gauge, 45 R)s per sq ft. It is stated that as high as 55 B)s has been observed at Glasgow. High winds often lift roofs. The gauge at Girard College, Philada, broke under a strain of 42 lbs per sq ft; a tornado passing at the moment, within 34 mile. By inversion of Smeaton's rule, if the force in lbs per sq ft, be mult by 200, the sq rt of the prod will give the vel in miles per hour. Smeaton's rule is used by the U. S. Signal Service. * The writer thinks 8 lbs per sq foot of ordinary double-sloping rooft, or 16 lbs for shed-roofs, sufia- cient allowance for pres of wind. 21 322 RAIN AND SNOW. RAIN AND SNOW. Tlie annnal precipitation "i" at any given place varies greatly from year to year, the ratio between maximum and minimum being frequently greater than 2:1. Beware of averag'es. In estimating ^00^5, take the maximum falls, and in estimating water supply, the minimum, not only per annum, but for short periods. In estimating water supply, make deductions for evaporation and leakage. Maxima and minima deduced from observations covering only 4 or 5 years are apt to be misleading. Data covering even 10 or more years may just miss includ- ing a very severe flood or drought. Records of from 15 to 20 years may usually be accepted as sufficient. Table 1. Average Precipitation* in the United States, in ins. (From Bulletin C of U. S. Department of Agriculture, compiled to end of 1891.) State. Spr. Alabama 14.9 Arizona 1.3 Arkansas 14.3 California 6.2 Colorado 4.2 Connecticut 11.1 Delaware 10.2 Dist. Columbia.11.0 Florida 10.2 Georgia 12.4 Idaho 4.4 Illinois 10.2 Indiana 11. Indian T'y 10.6 Iowa 8.3 Kansas 8.9 Kentucky 12.4 Louisiana 13.7 Maine 11.1 Maryland 11.4 Massachusetts. ..11.6 Michigan 7.9 Minnesota 6.5 Mississippi 14.9 Missouri 10.0 Sum. Aut. Win. Ann'l 13.8 10.0 14.9 53.6 4.3 2.2 3.1 10.9 12.5 11.0 12.8 50.6 0.3 3.5 11.9 21.9 5.5 2.8 2.3 14.8 12.5 11.7 11.5 46.8 11.0 10.0 9.6 40.8 12.4 9.4 9.0 41.8 21.4 14.2 9.1 54.9 15.6 10.7 12.7 61.4 2.1 3.6 7.0 17.1 11.2 9.0 7.7 38.1 11.7 9.7 10.3 42.7 11.0 8.9 5.7 36.2 12.4 8.1 4.1 32.9 11.9 67 3.5 31.0 12.5 9.7 11.8 46.4 15.0 10.8 14.4 53.9 10.5 12.3 11.1 45.0 12.4 10.7 9.5 44.0 11.4 11.9 11.7 46.6 9.7 9.2 7.0 33.8 10.8 5.8 8.1 26.2 12.6 10.1 15.4 53.0 12.4 9.1 6.5 38.0 State. . . Spr. Montana 4.2 Nebraska 8.9 Nevada 2.3 N. Hampshire. 9.8 New Jersey 11.7 New Mexico 1.4 New York 8.5 N. Carolina 12.9 N. Dakota 4.6 Ohio 10.0 Oregon 9.8 Pennsylvania. ..10.3 Rhode Island ...11.9 S. Carolina 9.8 S. Dakota 7.2 Tennessee 13.5 Texas 8.1 Utah 3.4 Vermont 9.2 Virginia 10.9 Washington 8.6 W. Virginia 10.9 Wisconsin 7.8 Wyoming 4.3 United States... 9.2 Sum. 4.9 10.9 0.8 12.2 13.3 5.8 10.4 16.6 8.0 11.9 2.7 12.7 10.7 16.2 9.7 12.5 8.6 1.5 12.2 12.5 3.9 12.9 11.6 3.5 10.3 Aut. Win. Ann'l 2.6 2.3 14.0 4.9 2.2 26.9 1.3 3.2 7.6 11.4 10.7 44.1 11.2 11.1 47.3 3.5 2.0 12.7 9.7 7.9 36.5 12.0 12.2 53.7 2.8 1.7 17.1 9.0 9.1 40.0 10.5 21.0 44.0 10.0 9.5 42.5 11.7 12.4 46.7 e.7 9.7 45.4 3.5 2.5 22.9 10.2 14.5 50.7 7.6 6.0 30.3 3.5 10.6 9.3 42.1 9.7 42.6 16.8 39.8 9.0 10.0 42.8 7.8 5.2 32.5 2.2 1.6 11.6 8.3 8.6 36.3 2.2 11.4 9.5 10.5 At Philadelphia, in 1869, during which occurred the greatest drought known there for at least 50 years, 43.21 inches fell ; August 13, 1873, 7.3 inches in 1 day; August, 1867, 15.8 inches in 1 month; July, 1842, 6 inches in 2 hours; 9 inches per month not more than 7 or 8 times in 25 years. From 1825 to 1893, greatest in one year, 61 inches, in 1867 ; least, 30 inches, in 1825 and 1880. At Norristown, Pennsylvania, in 1865, the writer saw evidence that at least 9 inches fell within 5 hours. At Genoa, Italy, on one occasion, 32 inches fell in 24 hours ; at Geneva, Switzerland, 6 inches in 3 hours ; at Marseilles, France, 13 inches in 14 hours; in Chicago, Sept., 1878, .97 inch in 7 minutes. Near la to 3630 cnMc feet; or 27155 U. S. gallons; or 101.3 tons per acre; or to 2323200 cubic feet; or 17378743 U. S. gal- lons ; or 64821 tons per square mile at 623^ ft)s per cubic foot. The most destructive rains are usually those which fall upon snow, under which the ground is frozen, so as not to absorb water. Table 2. Maximum intensity of rainfall for periods of 5. 10, and 60 minutes at Weather Bureau stations equipped with self-registering gauges, compiled from all available records to the end of 1896. (From Bulletin D of U. S. Department of Agriculture.) Stations. Rate per hour for — Stations. Rate per hour for — 5min. lOmins. 60mins. 5min. lOmins. 60mins. Ins. 9.00 8.40 8.16 7.80 7.80 7.50 7.44 7.20 7.20 6.72 6.60 6.60 6.60 Inches. 6.00 6.00 4.86 4.20 6.60 5.10 7.08 6.00 4.92 4.98 6.00 3.90 4.80 Inches. 2.00 1.30 2.18 1.25 2.40 1.78 2.20 2.15 1.60 1.68 2.21 1.60 1.86 Chicago Ins. 6.60 6.48 6.00 6.00 5.76 5.64 5.46 5.40 5.40 4.80 4.56 3.60 3.60 Inches. 5.92 5.58 4.80 4.20 5.46 3.66 5.46 4.80 4.02 3.84 4.20 3.30 2.40 Inches. 1.60 St. Paul .. .. Galveston Omaha 2.55 New Orleans 1.55 Milwaukee Kansas City Washington Jacksonville .. Dodge City Norfolk 1.34 1 55 Cleveland Atlanta 1.12 1 50 Detroit Key West Philadelphia... St. Louis Cincinnati Denver 2.25 New York City.. Boston 1.50 2.25 1.70 Indianapolis 1.18 Duluth 1.35 The w^ei^ht of freshly fallen snow^, as measured by the author, Taries from about 5 to 12 ft)s per cubic foot ; apparently depending chiefly upon the degree of humidity of the air through which it had passed. On one occasion, when mingled snow and hail had fallen to the depth of 6 inches, he found its weight to be 31 ibs per cubic foot. It was very dry and incoherent. A cubic foot of heavy snow may, by a gentle sprinkling of water, be converted into about half a cubic foot of slush, weigbing 20 lbs.; which will not slide or run off from a shingled roof sloping 30°, if the weather is cold. A cubic block of snow saturated with water until it weighed 45 lbs per cul)ic foot, just slid on a rough board inclined at 45°; on a smoothly planed one at 30°; and on slate at 18°; all approximate. A prism of snow, saturated to 52 Tbs per cubic foot, one inch square, and 4 inches high, bore a weigrht of 7 fibs ; which at first compressed it about one-quarter part of its length. European engineers consider 6 lbs per square foot of roof to be sufficient allowance for the Aveig^ht of snow^ ; 324 RAIN AND SNOW. and 8 !bs for the pressure of wind ; total, 14 lbs. The writer thinks that in the U. S. the allowance for snow should not be taken at less than 12 lbs ; or the total for snow and wind, at 20 lbs. There is no danger that snow on a roof will become saturated to the extent just alluded to ; because a rain that would supply the necessary quantity of water would also by its violence wash away the snow ; but we entertain no doubt whatever that the united pressures from snow and wind, in our Northern States, do actually at times reach, and even surpass, 20 fibs per square foot of roof. Tbe limit of perpetual $iiiow at the equator is at the height of about 16000 feet, or say 3 miles above sea-level; in lat 45° north or south, it is about half that height; while near the poles it is about at sea-level. Rain Oau^es. Plain cylindrical vessels are ill adapted to service as rain gauges; because moderate rains, even though sufficient to yield a large run-off from a moderate area, are not of sufficient depth to be satisfactorily measured unless the depth be exaggerated. The inaccuracy of measurement, always con- siderable, is too great relatively to the depth. In its simplest and most usual form, the gauge (see Fig.) consists essentially of a funnel, A, which receives the rain and leads it into a measuring tube, B, of smaller cross-section. The funnel should have a vertical and fairly sharp edge, and, in order to minimize the loss through \A/^ evaporation, it should fit closely over the tube, and its lower end C^ should be of small diameter. The depth of water in the tube is ascertained by inserting, to the bottom of the tube, a measuring stick of some unpolished wood which will readily show to what depth it has been wet. The stick may be permanently graduated, or it may be compared with an ordi- nary scale at each observation. The tube is usually of such diameter that the area of its cross-section, minus that of the'stick, is one-tenth of the area of the funnel mouth. The depth of rainfall is then one- tenth of the depth as measured by the stick. Dimensions of Standard U. S. Weather Bureau Rain Gauge. Ins. A. Receiver or funnel. Diameter 8 B. Measuring tube. Height 20 ins. " 2.53 C C. Overflow attachment and snow gauge. " 8 Such gauges, with the tubes carefully made from seamless drawn brass tubing, cost about $5.00 each ; but an intelligent and careful tinsmith, given the dimen- sions accurately, can construct, of galvanized iron, for about $1.00 a gauge that will answer every purpose of the engineer. The exposure has a very marked effect upon the results obtained. The funnel should be elevated about 3 ft, in order to prevent rain from splashing back into it from the ground or roof. If on a roof, the latter should be fiat, and pref- erably 50 ft wide or wider, and the gauge should be placed as far as possible from the edges ; else the air currents, produced by the wind striking the side of the building, will carry some of the rain over the gauge. No objects much higher than the gauge should' be near it, as they produce variable air currents which may seriously affect its indications. An overflow tank, C, should be provided, for cases of overfilling the tube. Water, freezing in the gauge, may burst it, or force the bottom off", or at least so deform the gauge as to destroy its accuracy. To measure snow, the funnel is removed, and the snow is collected in the overflow attachment or other cylindrical vessel deep enough to prevent the snow from being blown out, and the cross-sectional area of which is accurately known. The snow is then melted, either by allowing it to stand in a warm place, or, with less loss through evaporation, by adding an accurately known quantity of luke-warm water. In the latter case, the volume of the added water must of course be deducted from the measurement. Sainfall equivalent of snow. Ten inches of snow are usually taken as equivalent to 1 in of rain ; but, according to various authorities, the equiva- lent may vary between 2>^ and 34; i. e., between 25 and 1.84 ft)s. per cubic foot. Self-recording: gauges, of which several forms are on the market, are quite expensive, and, even when purchased from regular makers, seldom per- fectly reliable. Gauges using a small tipping bucket register inaccurately in heavy rains ; those using a float are limited as to the total depth which they can register ; while those which weigh th« rain, if exposed, are affected by wind. EAIN AND 8NOW. 825 Table 3. Details of Precipitation, in the United States. Records from 1871 or earlier, to 1891 inclusive. (From Bulletin C of U. S. Department of Agriculture, 1894.) State. City. ' o ^ Trace to 0.25 inch. Average percentage days with rain. * 0.26 to 0.50 in. 0.51 to 1.00 in. 1.012.01 to to 2.00 3.00 ins. ins. Max. No. of con- sec, days -a 5 Alabama Arizona Arizona California Colorado Connecticut Dis of Columbia, Florida Florida Georgia Georgia Illinois Illinois Indiana Iowa Iowa i. Kansas Louisiana Louisiana Maine Maryland Massachusetts... Michigan Michigan Minnesota Minnesota Mississippi Missouri Nebraska New Jersey New Mexico New York New York New York North Carolina.. Ohio Ohio Ohio Pennsylvania ... Pennsylvania ... South Carolina.. Tennessee Tennessee Tennessee Texas Virginia Virginia Wisconsin Wyoming Mobile Yumaf Whipple Barr'ks San Francisco.... Denver New London Washington Jacksonville Key West Augusta Savannah Cairo Chicago Indianapolis Davenport Keokuk Leavenworth New Orleans Shreveport Portland Baltimore Boston Detroit Marquette Duluth St. Paul Vicksburg St. Louis Omaha Atlantic City J... Fort Wingate Buffalo New York Oswego Wihuington Cincinnati Cleveland Toledo Philadelphia §.... Pittsburg , Charleston , Knoxville , Memphis , Nashville , Galveston , Lynchburg , Norkfolk Milwaukee Cheyenne , 40.1 6.6 15.7 21.8 28.7 43.0 40.8 44.7 38.8 37.2 36.4 38.5 44.0 44.2 40.8 35.4 31.9 40.6 35.9 43.4 41.2 39.8 46.7 49.0 44.5 38.9 36.4 38.4 34.1 41.0 14.6 54.2 39.6 58.3 36.2 46.4 52.2 44.3 43.6 51.3 34.7 43.4 39.8 42.0 34.6 40.3 42.0 44.7 30.5 24.3 5.6 10.2 14.2 24.2 28.1 27.2 29.8 27.9 22.9 22.3 25.0 32.5 30.3 29.7 24.5 20.7 24.9 22.6 29.8 27.5 25.8 36.0 39.0 34.7 30.3 21.1 26.4 24.4 27.5 10.1 41.3 25.7 46.5 20.7 32.5 40.2 33.9 30.6 38.3 20.6 26.8 24.4 26.4 22.8 26.2 26.2 34.0 26.7 7.3 1.7 4.2 4.7 6.5 11.8 4.2 6.2 8.2 4.9 8.6 4.2 5.6 4.3 5.2 4.8 3.6 8.9 6.9 3 4.5 4.4 3.9 5.2 3.1 3.7 5.4 4.5 5.0 9.0 4.5 3.2 6.2 3.6 8.0 3.0 3.6 3.2 5.2 3.6 8.3 5.6 8.9 5.2 7.9 4.7 4.6 3.7 1.9 * For instance, Alabama, Mobile, trace to 0.25 inch, 24.3, means that on 24.3 per cent, of the days embraced within the 20 years, rain fell to a depth of from a trace to 0.25 inch. t From October 1875 only. J From January 1874 only. § From May 1872 only. 326 WATER. WATER. Pure water, as boiled and distilled, is composed of the two gases, hydro, gen and oxygen; in the proportions of 2 measures hydrogen to 1 of oxygen; or 1 weight of hydrogen to 8 of oxygen. Ordinarily, however, it contains sev- eral foreign ingredients, as carbonic and other acids; and soluble mineral, or organic substances. When it contains much lime, it is said to be hard; and will not make a good lather with soap. Tlie air in its ordinary state contains about 4 grains of water per cubic foot. The average pressure of tlie air at sea level, wiW balance a column of ivater 34 feet high ; or about 30 inches of mercury. At its boil- ing point of 212° Fah, its bulk is about one twenty-third greater than at 70°. Its wei^lit per cubic foot is taken at 623^ lbs, or 1000 ounces avoir; but 623^ ft)s would be nearer the truth, as per table below. It is about 815 times heavier than air, wlien both are at the temperature of 62°; and the barometer at 30 inches. With barometer at 30 inches the weight of perfectly pure water is as follows. At about 39° it has its maximum density of 62.425 lbs per cubic foot. Temp, Fah. Lbs per Cub Ft. 32° 62.417 40° 62.423 50° 62.409 60° 62.367 Temp, Fah. Lbs per Cub Ft. 70° 62.302 80° 62.218 90°.. 62.119 212°- .59.7 Weight of sea iprater 64.00 to 64.27 Rs per cubic foot, or say 1.6 to 1.9 fts per cubic foot more than fresh water. See also p 328. Water has its maximum density when its temperature is a little above 39° Fah ; or about 7° above the freezing point. By best authorities 39.2°. From about 39° it expands either by cold, or by heat. When the temperature of 32° reduces it to ice, its weight is but about 57.2 ft)s. per cubic foot ; and its specific gravity about .9175, according to the investigations of L. Dufour. Hence, as ice, it has expanded one-twelfth of its original bulk as water ; and the sudden expansive force exerted at the moment of freezing, is sufficiently great to split iron water-pipes; being probably not less than 30000 lbs per square inch. Instances have occurred of its splitting cast tubular posts of iron bridges, and of ordinary buildings, when full of rain water from exposure. It also loosens and throws down masses of rock, through the joints of which rain or spring water has found its way. Retaining- walls also are sometimes overthrown, or at least bulged, by the freezing of water which has settled between their backs and the earth filling which they sustain ; and walls which are not founded at a sufficient depth, are often lifted upward by the same process. It is said that in a grlass tube }^ inch in diameter, water will not freeze until the temperature is reduced to 23°; and in tubes of less than -^ inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in rapid running streams. Anchor ice, sometimes found at depths as great as 25 feet, consists of an aggregation of small crystals or needles of ice frozen a*' the surface of rapid open water; and probably carried below by the force of tho stream. It does not form under frozen water. Since ice floats in water; and a floating body displaces a weight of the liquid equal to its own weight, it follows that a cubic foot of floating ice weighing 57.2 ibs, must displace 57.2 lbs of water. But 57.2 Bbs of water, one foot square, is 11 inches deep: therefore, floating ice of a cubical or parallelopipedal shape. \v\\\ have \}i of its volume under water; and only ^^ above ; and a square foot of ice of any thickness, will require a weight equal to ^ of its own weight to sink it to the surface of the water. In practice, however, this must be regarded merely as a close approximation, since the weight of ice is somewhat affected by en- closed air-bubbles. Pure water is usually assumed to boil at 212° Fah in the open air, at the level of the sea ; the barometer being at 30 inches ; and at about 1° less for every 520 feet above sea level, for heights within 1 mile. In fact, its boiling point varies like its freezing point, with its purity, the density of the air, the material of the vessel, &c. In a metallic vessel, it may boil at 210°; and in a glass one, at from 212° to 220° ; and it is stated that if all air be previously extracted, it requires 275°. It evaporates at all temperatures; dissolves more substances than any other agent: and has a greater capacity for heat than any other known substance. It is compressed at the rate of about one-21740th, (or about y^ of an inch in 18^^ feet,) by each atmosphere or pressure of 15 ibs per square inch. When the pressure is removed. U« pinaHoUy restores its original bulk. WATER. 327 Effect on metals. The lime contained in many waters, forms deposits in metallic water-pipes^ and in channels of earthenware, or of masonry ; especially if the current be slow. Some other substances do the same; obstructing the flow of the water to such an extent, that it is always expedient to use pipes of diameters larger than would otherwise be necessary. The lime also forms very hard incrustations at tlie bottoms of boilers; very much impair- ing their efficiency; and rendering them more liable to burst. Such water is unfit for locomc^ives. We have seen it stated that the Southwestern R R Co, England, prevent this lime deposit, along their limestone sections, by dissolving 1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms similar deposits in boilers; as also does mud, and other impurities. Water, either when very pure, as rain water ; or when it contains carbonic acid, (as most water, does,) produces carbonate of lead in lead pipes; and as this is an active poison, such pipes should not be used for such wat^ers. Tinned lead pipes may be substituted for them. If, however, sulphate of lime also be present, as is very frequently the case, this effect is not always produced; and several other substances usually found in spring and river water, also diminish it to a greater or less degree. Fresb i¥ater corrodes ^¥roug-bt iron more rapidly than cast; but the reverse appears to be the case with sea ivater; although it also affects wrought iron very quickly; so that thick flakes may be detached from it with ease. The corrosion of iron or steel by sea water increases with the carbon. Cast-iron cannons from a vessel which had been sunk in the fresh water of the Delaware River for more than 40 years, were perfectly free from rust. Gen. Pasley, who had examined the metals found in the ships Royal George, and Edgar, the first of which had remained sunk in the sea for 62 years, and the last for 133 years, "stated that the cast iron had generally become quite soft; and in some cases resembled plumbago. Some of the shot when exposed to the air became hot; and burst into many pieces. The wrought iron was not so much injured, except when in cmiiact with copper, or brass gun-metal. Neither of these last was much afiected, except when in contact with iron. Some of the wrought iron was reworked by a blacksmith, and pronounced superior to modern iron." "Mr. Cottam stated that some of the guns had been carefully removed in their soft state, to the Tower of London ; and in time (within 4 years) resumed their orig- inal hardness. Brass cannons from the Mary Rose, which had been sunk in the sea for 292 years, were considerably honeycombed in spots only; (perhaps where iron had been in contact with them.) The old cannons, of wrought-iron bars hooped together, were corroded about i^ inch deep; but had probably been pro- tected by mud. The cast-iron shot became redhot on exposure to the air; and fell to pieces like dry clay!" "Unprotected parts of cast-iron sluice-valves, on the sea gates of the Cale- donian canal, were converted into a soft plumbaginous substance, to a depth of % of an inch, within 4 years; but where they had been coated with common Swedish tar, they were entirely uninjured. This softening effect on cast iron appears to be as rapid even when the water is but slightly brackish ; and that only at intervals. It also takes place on cast iron imbedded in salt earth. Some water pipes thus laid near the Liverpool docks, at the expiration of 20 years were soft enough to be cut by a knife; while the same kind, on higher ground beyond the influence of the sea water, were as good as new at the end of 50 years." Observation has, however, shown that tbe rapidity of this action «lepends mucb on tbe quality of tbe iron ; that which is dark- colored, and contains much carbon mechanically combined with it, corrodes most rapidly : while hard white, or light-gray castings remain secure for a long time. Some cast-iron sea-piles of this character, showed no deterioration in 40 years. Contact witb brass or copper is said to induce a galvanic action which greatly hastens decay in either fresh or salt water. Some muskets were recovered from a wreck which had been submerged in sea water for 70 years near New York. The brass parts were in perfect condition ; but the iron parts had entirely disappeared. Oalvanizing' (coating with zinc) acts as a pre- servative to the iron, but at the expense of the zinc, which soon disappears. The iron then corrodes. If iron be well heated, and then coated with bot coal-tar, it will resist the action of either salt or freshwater for many years. It is very important that the tar be perfectly purified. Such a coat- ing, or one of paint, will not prevent barnacles and otber sbells from attaching themselves to the iron. Asphaltum, if pure, answers as well as coal-tar. Copper and bronze are very little aflfected by sea water. No galvanic action has been detected where brass feriiles are inserted intt the water-pipes in Philadelphia. 328 TIDES. Tbe most prejudicial exposure for iron, as well as for wood, is that to alternate wet and dry. At some dangerous spots in Long Island Sound, it has been the practice to drive round bars of rolled iron about 4 inches diam- eter, for supporting signals. These wear away most rapidly between high and low water; at the rate of about an inch in depth in 20 years; in which time the 4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under fresh water especially, or under ground, a thin coating of coal-pitch varnish, carefully applied, will protect iron, such as water-pipes, Ac, for a long time. See page 655. The sulphuric acid contained in the water from coal minei corrodes iron pipes rapidly. In the fresb water of canals, iron boate have continued in service from 20 to 40 years. l¥oocl remains sound for centuries under either fresh or salt water, if not exposed to be worn away by the action of currents ; or to be destroyed by marine insects. Sea water weig^ns from 64 to 64.27 lbs per cubic foot, or say from 1-6 to 1.9 R)s per cubic foot more than fresh water, varying with the locality, and jiot appreciably with the depth. Theexcess, over the weight of fresh water, is chiefly common salt. At 64 ft)s per cubic foot, 35 cubic feet weigh 2240 lbs. Sea ivater freeases at about 27° Fahr. The ice is fresh; but (especially at low tempera- tures) brine may be entrapped in the ice. A teaspoonful of powdered alum, well stirred into a bucket of dirty w^ater, will generally purify it sufficiently within a few hours to be drinkable. If a bole 3 or 4 feet deep be dug in the' sand of the sea-shore, the infiltrating water will usually be sufficiently fresh for washing vi'ith soap; or even for drinking. It is also stated that water may be preserved sweet for many years by placing in the containing vessel 1 ounce of black oxide of manganese' for each gallon of water. It is said that water kept in zinc tanks; or flowing through iron tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc formed thereby; and it is recommended to coat zinc surfaces with asphalt varnish to prevent this. Yet, in the city of Hartford, Conn, service pipes of iron, galvanized inside and out, were adopted in 1855, at the recommendation of the water commissioners; and have been in use ever since. They'are like- wise used in Philadelphia and other cities to a considerable extent. In many hotels and other buildings in Boston, the "Seamless Drawn Brass Tube" of the American Tube Works at Boston, has for many years been in use for service pipe; and has given great satisfaction. II is stated that the softest water may be kept in brass vessels for years without any deleterious result. The action of lead upon some waters (even pure ones) is highly poison- ous. The subject, however, is a complicated one. An injurious ingredient may be attended by another which neutralizes its action. Organic matter, whether vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm- less. Ice may be so impure that its water is dangerous to drink. The popular notion that hot water freezes more quickly than cold, with air at the same temperature, is erroneous. TIDES. . ^ The tides are those well-known rises and falls of the surface of the sei. and of some rivers, caused by the attraction of the sun and moon. There are two rises, floods, or high tides; and two falls, ebbs, or low tides, every 24 hours and 50 minutes (a lunar day) ; making the average of 5 hours 123x^ minutes between high and low water. These intervals are,, however, subject to g-reat variations; as are also the heights of the tides; and this not only at different places, but at the same place. These irregularities are owing to the shape of the coast line, the depth of water, winds, and other causes. Usually at new and full moon, or rather a day or two after, (or twice in each lunar month, at intervals of two weeks,) the tides rise higher, and fall lower than at other times; and these are called spring tides. Also, one or two days after the moon is in her quarters, twice in a lunar month, they both rise and fall less than at other times ; and are then called neap tides. From neap to spring they rise and fall more daily ; and vice versa. The time of hig-h w^ater at any place, is generally two or three hours after the moon has passed over either the upper or lower meridian; and is called the establishment of that place; because, when this time is established, the time of high water on any other day may be found from it in most cases. The total height of spring tides is genei'ally from 1}/^ to 2 times as great as that of neaps. The great tidal wave is merely an undulation, unattended by any current, or progressive motion of the particles of water. Each successive high tide occurs about 24 minute* later than the preceding one; aod so with the low tides EVAPORATION AND LEAKAGE. 329 EVAPOEATION, FILTRATION, AND LEAKAGE. The amount of evaporation from surfaces of water exposed to the natural effects of the open air, is of course greater iu summer than in winter; although it is quit© perceptible in eren the coldest weather. It is greater in shallow water than iu deep, inasmuch as the bottom also becomes heated by the sun. It is greater in runuing, than in standing water; on much the same principle that it is greater during winds than calms. It is probable that the average daily I088 from a reservoir of moderate depth, from evaporation alone, throughout the 3 warmer month* of the year, (June, July, August,) rarely exceeds about y^-g- inch, in any part of the United States. Or JLj. inch during the 9 colder months ; except in the Southern States. These two averages would give a daily one of .15 inch ; or a total annual loss of 55 ins, or 4 ft 7 ins. It probably is 3.5 to 4 ft. By some trials by the writer, in the tropics, ponds of pure water 8 ft deep, in a stiff retentive clay, and fully exposed to a very hot sun ail day, lost during the dry sea- son, precisely 2 ins in 16 days ; or ^ inch per day ; while the evaporation from a glass tumbler wa« yi inch per day. The air in that region is highly charged with moisture; and the dews are heavy. Every day during the trial the thermometer reached from 115° to 125° in the sun. The total annual evaporation in several parts of England and Scotland is stated to average from 22 to 38 ins ; at Paris, 34 ; Boston, Mass, 32 ; many places in the U. S., 30 to 36 ins. This last would give a daily average of -^t^ inch for the whole year. Such statements, however, are of very little value, unless accompanied by memoranda of the circumstances of the case ; such as the depth, exposure, size and nature of the vessel, pond. &c, which contains the water. branches, namely: Kinematics; or the study of the motions of bodies, without reference to the causes of niutiou; and Dynamics, or the study of force and its effects. The latter is sub-divided into Kinetics; which treats of the relations between force and motion; and iiilatics; which considers those special, but very numerous, cases, where equal and opposite forces counteract each other and thus destroy each other's motions. Art. 3 (a). Matter, or substance, may be defined as whatever occupies space; as meta', stone, wood, water, air, steam, gas, etc. (to) A toody is any portion of matter which is either more or less completely separated in fact from all other matter, or which we take into consideration by itself and as if it were so separated. Thus, a stone is a body, whether it be fallinsr through the air or lying detached upon the ground, or built up into a wall. Also, the wall is a body ; or, if we wish, we may consider any portion of the wall, as any particular cubic foot or inch in it, as a body. The earth and the other planets are bodies, and their smallest atoms are bodies. A tram of cars may b© regarded as a body ; as may also each car, each wheel or axle or other part of the car, each passenger, etc., etc. Similarly, the ocean is a body, or we may take as a body any portion of it at pleas- ure, such as a cubic foot, a certain bay, a drop, etc. (c) But in what follows we shall (as already stated) consider chiefly rigid bodies; i. e., bodies which undergo no change in shape, such as by being crushed or stretched or pulled apart, or penetrated by another body. All actual bodies are of course more or less subject to some such changes of shape ; t. e., no body i« in fact absolutely rigid; but we may properly, for convenience, suppose such bodies to exist, because many bodies are so nearly rigid that under oniinary circumstances they undergo little or no change of shape, and because such change as does occur may be con- sidered under the distinct head of Strength of Materials. (d) But while bodies are thus to be regarded as incapable of change of form, it is squally important that we regard them as susceptible to change of position as wholes. Thus, they may be upset or turned around horiKontally or in any other direction, or moved along in any straight or curved line, with or without turning around a point within themselves. In short they are capable of motion, as wholes. • FORCE IN RIGID BODIES. 331 Art. 3 (a). Motion of a body is change of its position in relation to another body or to some real or imaginary point, which (for conyenience) we regard as fixed, or at rest. Thus, while a stone falls fron. a roof to the ground, its position, relatively to the roof, is constantly changing, as is also that relatively to the ground and that relatively to any given point in the wall : and we say that the stone is in motion rela- tively to either of tfiose bodies, or to any point in them. But if two stones, A and B, fall from the roof at the same instant and reach the ground at the same (subsequent) instant, we say that although each moves, relatively to roof and ground, yet they have no motion relatively to each other ; or, they are at rest relatively to each other; for their position in regard to each other does not change ; i. e., in whatever direction and at whatever distance stone A may be from stone B at the time of starting, it remains in that same direction, and at that same distance from B during the whole time of the fall. Similarly, the roof, the wall and the ground are at rest relatively to each other, yet th«y are in motion relatively to a falling stone. They are also in motion relatively to the tun. owing to the earth's daily rotation about its axis, and its annual movement around the sun. (1>) If a train-man walks toward the rear along the top of a freight train ju«t ae fast as the train moves forward, he is in motion relatively to the train; but, as a whole, he is at rest relatively to buildings, etc. near by ; for a spectator, standing at a little distance from the track, sees him continually opposite the same part of such building, etc. If the man on the train now stops walking, he comes to rest relatively to the train, but at the same time comes into motion relatively to the surrounding buildings, etc., for the spectator sees him begin to move along with the train. (c) Since we know of no absolutely fixed point in space, we cannot say, of any body, what its absolute motion is. Consequently, we do not know of such a thing as absolute red, and are safe in saying that all bodies are in motion. Art. 4 (a). The velocity of a moving body is its rate of motion. A body (as a railroad train) is said to move with uniform velocity , or constant velocity, when the distances moved over in equal times are equal to each other, no matter how small those times may be taken. (to) The velocity Is expressed by stating the distance passed over during some given time, or which would be passed over during that time if the uniform motion continued so long Thus, if a railroad train, moving with constant velocity, passes over 10 miles in half an hour, we may say that its velocity, during that time, is (t. e., that it moves at the rate of) 20 miles per hour, or 105,600 feet per hour, or 1760 feet per minute, or 29% feet per second. Or, we may, if desirable, say that it moves at the rate of 10 miles in half an hour, or 8*^ feet in three seconds, etc.; but it is generally more convenient to state the distance passed over in a unit of time, as in one day. one hour, one second, etc. (c) If, of two trains, A and K, moving with constant velocity, A moves 10 miles in half an hour, B moves 10 miles in quarter of an hour, then the velocities are, A, 20 niilps per hour, B, 40 iniles per hour. In other words, the velocity of a body (which may be defined as the distance passed over in a given time) is inversely as the time required to pans over a given distance. (d) By unit velocity is meant that velocity whi«h, by common consent, is taken as equal to uniti/ or one. Where English measures are used, the unit velocity gen- erally adopted in the study of Mechanics is 1 foot per second. (e) When we say that a body has a velocity of 20 miles per hour, or 10 feet per second, etc.. we do not imply that it will necessarily travel 20 miles, or 10 feet, etc. ; for it may not have suflScient time for that. We mean merely that it is traveling at the rate of 20 miles per hour, or 10 feet per second, etc. ; so that if it continued to move at that same rate for an hour, or a second, etc., it would travel 20 miles, or 10 feet. etc. (f) When velocity increases, it is said to be accelerated. When it decreases. it is said to be retarded. If the acceleration or retardation is in exact proportion to the time ; that is, when during any and every equal interval of time, the same degree of change takes place, it is uniformly accelerated, or retarded. When otherwise, the words variable and variably are UHed. (g) A body may have, at the same time, t-wo or more Independent veloci- ties requiring to be considered. For instance, a ball fired vertically upward from a 332 FORCE IN RIGID BODIES. gun, and then falling again to the earth, has, during the whole time of its rise and fell, (Ist) the uniform upward velocity with which it leaves the muzzle, and (2nd) the continually accelerated downward velocity given to it by gravity, which acts upon it during the whole time. Its resultant (or apparent) velocity at any moment is the difftrence between these two. Thus, immediately after leaving the gun, the downward velocity given by gravity is very small, and the resultant velocity is therefore upward and very nearly equal to the whole upward velocity due to the powder. But after awhile the downward velocity (by constantly increasing) becomes equal to the upward velocity; i. e., their difference, or the resultant velocity, becomes nothing ; the ball at that instant stands still ; but its downward velocity continues to increase, and immediately becomes a little greater than the upward velocity ; then greater and greater, until the ball strikes the ground. At that instant its resultant velocity is {the downward velocity which it would "j ( the uniform upward have acquired by falling during the > — -< velocity given by the whole time of its rise and fall. ) [ powder. We have here neglected the resistance of the air, which of course retards both the Bficent and the descent of the ball. (h) As a further illustration, regard a 6 n c as a raft drifting in the directioQ c a or n b. A man on the raft walks with uniform velocity from corner n to corner c while the raft drifts (with a uniform velocity a little greater than that of the man) through the distance n b. /^^^ Therefore, when the man reaches corner c, that corner has \^i^^*\ moved to the point which, when he started, was occupied by /"Iff^^^^ a. The man's resultant motion, relatively to the bed of the / ; river or to a point on shore, has therefore been w tion is called negative. Art. 8 (a). The rate of acceleration f is the acceleration which takes place in a given time, as one second. (to) The unit rate of acceleration is that which adds unit of velocity in a unit of time ; or, where English measures are used, one foot per second, per second. (c) For a given rate of acceleration, the total accelerations are of course propor- tional to the times during which the velocity increases at that rate. Art. 9 (a). Lavrs of acceleration. Suppose two blocks of iron, one (which we will call A) twice as large as the other (a), placed each upon a perfectly frictionless and horizontal plane, so that in moving them horizontally we are opposed by no force tending to hold them still. Now apply to each block, through a spring balance, a pull such as will keep the pointer of each balance always at the same mark, as, for instance, constantly at 2 in both balances. We thus have equal forces acting upon unequal masses.^ Here the rate of acceleration of a is double that of A ; for vrlien tlie forces are equal tlie rates of accelera- ration are inversely as the masses. In other words, in one second (or in any other given time) the small block of iron, a, will acquire twice the increase of velocity that A (twice as large) will acquire; so that if both blocks start at the same time from a state of rest, the smaller one, a, will have, at the end of any given time, twice the velocity of A, which has twice its mass. (to) Again, let the two masses, A and a, be equal, but let the force exerted upon a be twice that exerted upon A. Then the rate of acceleration of a will (as before) be twice that of A ; for, "wlien tlie masses are equal, the rates of accelera- ration are directly as tlie forces. (c) We thus arrive at the principle that, in any case, tlie rate of acceleration is directly proportional to tlie force and inversely proportional to ttoe mass. * We here speak of the force of gravity, exerted in a given place, as constant, because it is so for all practical purposes. Strictly speaking, it increases a very little as the stone approaches the earth. t Since the rate of acceleration is generally of greater consequence, in Mechanics, than the total acceleration, or the "acceleration" proper, scientific writers (for the sake of brevity) use the term "acceleration" to denote that rate, and the term ^^ total acceleration" to denote the total increase or decrease of velocity occurring during any given time. Thus, the rate of acceleration of gravity (about' 32.2 ft. per second per second) is called, simply, the " acreleration of gravity." As we shall not have to use either expression very frequently, we shall, generally, to avoid misappre- hension, give to each idea its full name; thus, "total acceleration" for the whole change of velocity in a given case, and " rate of acceleration " for the rate of that change. t The mass of a body Is the quantity of matter that it contains. FORCE IN RIGID BODIES. 335 (d) Hence, if we make the two forces proportional to the two masses, tne rates of acceluratioa will be equal ; or, for a given rate of acceleration) tlie forces must l)e directly as tlie masses. (e) Hence, also, a greater force is required to impart a given velocity to a given body in a short time than to impart the same velocity in a longer time. For instance, the forward coupling links of a long train of cars would snap instantly under a pull sufficient to give to the train in two seconds a velocity of twenty miles per hour, sup- posing a sufficiently powerful locomotive to exist. In many such cases, therefore, we have to be contented with a slow, instead of a rapid acceleration. A string may safely sustain a weight of one pound suspended from our hand. If we wish to impart a great upward velocity to the weight in a very short time, we evi- dently can do so only by exerting upon it a great force; in other words, by jerking the string violently upward. But if the string has not tensile strength sufficient to transmit this force from our hand to the weight, it will break. We might safely give to the weight the desired velocity by applying a less force during a longer time. (f ) When a stone falls, the force pulling the earth upward is (as remarked above) equal to that which pulls the stone downward, but the mass of the earth is so vastly greater than that of the stone that its motion is totally imperceptible to ns, and would still be so, even if it were not counteracted by motions in other directiona in other parts of the earth. Hence we are practically, though not absolutely, right when we say that the earth remains at rest while the stone falls. (g) But in the case of the two billiard balls (Art. 5 c p. 333), we can clearly see the result of the action of the force upon each of the two bodies; for the second ball, B, which was at rest, now moves forward, while the forward velocity of the first one, A, is diminished or destroyed, its backward motion thus appearing as a retardation of ita forward motion. And, (since the same force acts upon both balls) mass mass rate of acceleration rate of negative acceleration of A • of B ■ • of B • of A or (since the force acts for the same time upon both balls) mass mass forward velocity . loss of forward velocity of A • of B • • of B * of A (b.) Bemark. A man cannot lift a. weight of 20 tons; but if it be placed upon proper friction rollers, he can move it horizontally, as we see in some drawbridges, turntables, &c. ; and if friction and the resistance of the air could be entirely removed, he could move it by a single breath; and it would continue to move forever after the force of the breath had ceased to act upon it. It would, however, move very slowly, because the force of the single breath would have to diffuse.itself among 20 tons of matter. He can move it, if it be placed in a suitable vessel in water, or if suspended from a long rope. Apowerful locomotive that may move 2000 tons, cannot lift 10 tons vertically. If we imagine two bodies, each as large and heavy as the earth, to be precisely balanced in a pair of scales without friction, a single grain of sand added to either Bcale-pan, would give motion to both bodies. Art. 10 (a). The constant force of gravity is a uniformly accelerating force when it acts upon a body faUing freely ; for it then increases the velocity at the uni- form rate of .322 of a foot per second during every hundredth part of a second, or 32.2 feet per second in every second. Also when it acts upon a body moving down an in- clined plane; although in this CRse the increase is not so rapid, because it is caused by only a part of the gravity, while another part presses the body to the plane, and a third part overcomes the friction. It is a uniformly retarding force, upon a body thrown vertically upward; for no matter what may be the velocity of the body when projected upward, it will be diminished .322 of a f< ot per second in each hundredth part of a second during Its rise, or 32.2 feet per second during each entire second. At least, such would be the case were it not for the varying resistance of the air at different velocities. It is a uniformly straining force when it causes a body at rest, to press upon another bodv ; or to pull upon a striner by which it is suspended. The foregoing expressions, like those of momentum, strain, push, pull, lift, work, &c., do not indicate different Mnds of force ; but merely different kinds of effects produced by the one grand principle, force. (1>) The above 32.2 feet per second is called the acceleration of gravity ; and by scientific writers is conventionally denoted by a small g ; or, more correctly speak- 836 FORCE IN RIGID BODIES. fng, Biuce the acceleration is not precisely the same at all parts of th« earth, f^ denotes the acceleration per second, whatever it may be, at nay particular place. Art. 11 (a). Relation betTreen force and mass. The mass of a body is the quantity of matter which it contains. One cubic foot of water has twic9 us great a mass as half a cubic foot of water, but a less mass than one cubic foot of iron. Thus, the size of a body is a measure of mass between bodies of the same material, but not between bodies of different materials. (b) When bodies are allowed to fall freely in a vacuum at a given place, they are found to acquire equal velocities in any given time, of whatever different materials they may be composed. From this we know (Art. 9(d), p. 335), that the forces moving them downward, viz. : their respective weights at that place, must be proportional to their masses. Thus, in any given place, the weight of a body is a perfect measure of its mass. But the loeight of a given body changes when the body is moved from one level above the sea to another, or from one latitude to another; while the mass of the body of course remains the same in all places. Thus, a piece of iron which weighs a pound at the level of the sea, will weigh less than a pound by a spring balance, upon the top of a mountain close by, because the attraction between the earth and a given mass diminishes when the latter recedes from the earth's center. Or if the piece of iron weighs one pound near the North or South Pole, it will, for the same reason, weigh less than a pound by a spring balance if weighed nearer to the equator and at the same level above the sea. The difference in the weight of a body in different localities is so slight as to be of no account in questions of ordinary practical Mechanics;* but scientific exactness requires a measure of mass which will give the same expression for the quantity of matter in a given body, wherever it may be; and, since weighing is a very convenient way of arriving at the quantity of matter in a body, it is desirable that we should still be able to express the mass in terms of tlie weight. Now, when a given body is carried to a higher level, or to a lower latitude, its loss of weight is simply a decrease in the force with which gravity draws it downward, and this same decrease also causes a decrease of the velocity which the body acquires in falling during any given time. The change in velocity, by Art. 9 (6), p. 334, is necessarily propor- tional to the change in weight. Therefore, if the weight of a body at any place be divided by the velocity which gravity imparts in one second at the same place ^and called s, or the acceleration of gravity for that place), the quotientwUl be tne same at aU places, and therefore serves as an invariable measure of the mass. (c) By common consent, the nnlt of mass, in scientific Mechanics, is said to be that quantity of matter to which a unit of force can give unit rate of acceleration. This unit rate, in countries where English measures are used, is one foot per second, per second. It remains then to adjust the units of forc4 and of mass. Two methods (an old and a new one) are in use for doing this. We shall refer' to them here as methods A and B respectively. (d) In method A, still generally used in questions of statics, the unit of force is fixed as that -force which is equal to the weight of one pound in a certain place; t. c, the force with which the earth at that place attracts a certain standard piece of platinum called a pound; and the unit of mass is not this standard piece of metal, but, as stated in (c), that mass to which this unit force of one pound gives, in one second, a velocity of one foot per second. Now the one pound attraction of the earth upon a mass of one pound will (Art. 1, p. 330) in one second give to that mass a velocity = gr or about 32 feet Ser second ; and (Art. 9 (a), p.334), for a given force the masses are inversely as le velocities imparted in a given time. Therefore, to give in one second a Telocity of only one foot per second (instead of g or about 32) the one pound nnit of force would have to act upon a mass g times (or about 32 times) that which weighs one pound. This could be accomplished, with an Attwood's machine. Art. 16 (c), p. 339, by making the two equal weights each — 15 3^ lbs. and the third weight =» 1 lb. * The greatest discrepancy that can occur at various heights and latitudes, by adopting weight as the measure of quantity, would not be likely to exctea 1 in 300 ; or, under ordinary circumstances, 1 in 1000. FORCE IN RIGID BODIES. 337 By method A, therefore, the unit of mass is g times (or about 32 times) the mass of the standard piece of metal called a pound; i. e., a body containing one such unit of mass weighs g lbs. or about 32 lbs. ; or, l>y metliod A, Or, the weight of any given body _ in lbs. = a V *^® inass of the body, "y -^ in units of mass. the mass of a body, in units of For instance: mass =3 — 9 in a body weighing J^ pound the mass is about ^1 unit of mass 1 " 2 « 32- •♦ A " " 1 " " 64 « . 2 « M It has been suggested to call this unit of mass a " Matt." (e) In metliod B, the mass of the standard pound piece of platinum is taken as the unit of mass and is called a pound; and the force which will give to it in one second a velocity of one foot per second is taken as the unit offeree. This small unit of force is called a poundal. In order that it may in one second give to the mass of one pound a velocity of only one foot per second, it must (by Art. 9 6), be (ov about i j of the weight of said pound mass. Hence, l>y nietliod B, the mass of any given body, in pounds = the weight of the body in pouvdals and the weight of a body, in poundals == gr X the mass of the body in pounds. the mass of the body Is about "or instai'ce : in a body weighing 14 pouDdal -= ^^ pound 1 " 2 •' 32 - 1 « 64 " -.2 « JL pound 1 (f ) For conventence, we sometimes disregard the scientific require- ment that the unit of force must be that which will give unit rate of accele- ration to unit mass, and take a pound of matter as our unit of mass^ and a pound weight as our unit of force. Our unit of force will then in one second give a velocity of g (or about 32.2 feet per second) to our unit of mass. In Statics^ we are not concerned with the masses of bodies, but only with the forces acting upon them, including their weights. Art. 13 (a). Impulse. By taking, as the unit of force, that force which, in one second, will give to unit mass a velocity of one foot per second, we. have (by Art. 9, p. 334), in any case of unbalanced /orce acting upon a mass during a fiven time: force X time mass velocity X mass time force X- time Velocity =• Force = Mass =- also 22 velocity mass X velocity force Force X time -= mass X velocity. Time =- (1) (2) (3) (4) (5) 338 FORCE IN RIGID BODIES. To ihe product, force X time, in equation (5), writers now give the name Impi&lse, which was formerly given to collision (now called impaot). See Art. 24 (ft). The term impulse^ as now used, conveys merely the idea of force acting through a certain length of time. Equation (5) tells us that aa impulse (the product of a force by the time of its action) is numerically equal to the momentum* which it produces. Equation (2) tells us that any force is numerically equal to the momentum which it can produce in one second. In other words, ttie momentum of a body moving with a given velocity is numerically equal to the force which in one second can produce or destroy that velocity in that body; or, a force is numerically equal to the rate per eeoond at which it can produce momentum. Thus, forces are proportional to the momentums which they can produce in a given time; or, in a given time, equal forces produce equal momentums. Therefore a force must always give equal and opposite momentums to the two bodies between which it acts. Art. 13 (a). Tl&e usual way of measuring a force is by ascertaining the amount of some other force which it can counteract. Thus we may meas- ure the weight of a body by hanging it to a spring balance. The scale of the balance then indicates the amount of tension in the spring; and we know that the weight of the body is equal to the tension, because the weight just pre vents the tension from drawing the hook upward. Thus, forces are conveniently expressed in weighiMf as in pounds, tons, &c., and they are generally so measured in Statics, and in our following articles. (b) A force may toe constant or variatole. When a stone rests upon the ground, the pull of gravity upon it (i. e., its weight) remains constant, neither increasing nor decreasing. But when a stone is thrown upward its weight decreases very slightiy as it recedes from the earth, and again increases as it approaches it during its fall. In this case, the force of gravity, acting upon the stone, decreases or increases steadily. But a force may change suddenly f or irregularly, or may be intermittent; as when a series of unequiU blows are struck by a hammer. In what follows we shall have to do only with forces supposed to be constant. Art. 14: (a). Density. The densities of materials are proportional to the masses contained in a given volume, as a cubic inch ; or inversely as the volume required to contain a given mass. Or, since the weights at a given place are proportional to the masses, the densities are proportional to the weights per unit of volume (or " specific gravities ") of the materials. Thus, a body weigh- ing 100 lbs. per cubic foot is twice as dense as one weighing only 60 lbs. per cubic foot at the same place. Art. 15 (a). Inertia. The inability of matter to set itself in motion, or to change the rate or direction of its motion, is called its inertia, or inertness. When we say that a certain body has twice the inertia (inertness) of a smaller one, we mean that twice tYiQ force is required to give it an equal rate of accele- ration ; and that, since all force (Art. 5/), acts equally in both direo tions, we experience twice as great a reaction (or so-called " resistance") from the larger body a? from the smaller one. The " inertia" of a body is therefore ft measure of the force required to produce in it a given rate of jacceleration; or, which is the same thing, it is a measure of the mass of the body. We may therefore consider "inertia" and "mass" as identical. (to) What is called the "resistance of inertia" of a body, is simply the reaction, (*. e., one of the two equal and opposite actions) of whatever force we apply to the body. Hence, its amount depends not only upon the mass of the Dody, but also upon the rate of acceleration which we choose to *The momentum of a body (sometimes called its "quantity of motion'*) is equal to the product obtained by multiplying its mass by its velocity. If we adopt the pound as the unit of mass, as in "method B," Art. 11 (e), the •proawci, weight in pounds X velocity, is numerically either exactly or nearly the same as the product, mass in pounds X velocity, depending upon whether or not the body is in that latitude and at that level where a mass of one pound is said to weigh one pound. But the product, weight in poundals X velocity, is exactly a times (about 32.2 times) the product, mass in pounds X velocity ; alsc^ by "meftiod A," weight in pounds X velocity = gX ««ws in " matts*' X velocity. T.2.4 FORCE IN RIGID BODIES. 339 gire to it. Therefore we cannot tell, from the mass or weight of a body alone, what its " resistance of inertia" in any given case will be. Art. 16 (a). Forces in opposite directions. When two equal and opposite forces act upon a body at the same time, and in the same straight line, we say that they destroy each other's tendencies to move the body, and it remains at rest. If two unequal forces thus act in opposition, the smaller force and an equal portion of the greater one are said to counteract each other in the same way, but the remainder of the greater force, acting as an unbalanced or unresisted force, moves the body in its own direction, as it would do if it were the only force acting upon it. Thus, when we move bodies, in practice, we encounter not only the "resist- ance of inertia" {i. e., we not only have to exert force in order to move inert matter), but we are also opposed by other /orces, acting against us, as friction, the resistance of the air, and, often, all or a part of the iveight of the body. By "resistances," in the following, we mean such resisting /orce^, and do not include in the term the " resistance of ineriia." (b) If separated, the two bodies, A and B, of 3 Rs and 2 lbs respectively, would fall with equal accelerations = g ; each unit, — , of mass being acted upon by its own weight, W. But, connected as they are, A will move downward, and B upward, with an acceler- T='2.4 ation = only f : for now an unbalanced force of 5 only 3 — 2 = 1 ft) must give acceleration to a mass m ^ a 3 + 25 2 * -i5.4 of = ~. But, to give to a mass, B, of -, an g g 2 2 accel of f , requires a force of - . f = - ft) = 0.4 ft). , — ™ 5 ^ g 5 6 III ill B This, plus 2 ft)s (required to balance the weight of B) is the tension, 2.4 ft)s existing throughout the cord. Exerted at A, this tension balances 2.4 of the 3 ft)s weight of A. The remainder (3 — 2.4 = 0.6 ft)) of the weight, actinj? downward upon the mass, -, of A, gives to it the required acceleration of ^; f„-, here '^^ = 0.6 ^ *=% = 0.2 g = ?. mass g 3 5 Or we may regard the total tension, 2.4 ft)s, in the cord at A, as acting upon A g and giving to it a negative or upward acceleration of 2.4 -r- - = 0.8 g, which, deducted from g (the acceleration which A would otherwise have) leaves Acceleration = g — 0.8 g = 0.2 g = f . 5 Let W = weight of A w = weight of B F = net force available for acceleration = W — w M = combined mass of both bodies = g ra = mass of B = - g a = acceleration T = tension in cord. Then: a=™=(W — w)- M _ w T = w+ma = wH a = g (c) An " Atwood's Macliine" consists essentially of a pulley, a flexible cord passing over the pulley, two equal weights (one suspended at each end of the cord), and a third weight, generally much lighter than either of the other two. The two equal weights balance each other by means of the pulley and cord. The third weight is laid upon one of the other two weights. The force of gravity, acting upon the third weight, then sets the masses of the three weights in motion at a small but constantly increasing velocity. In order to do this it must also overcome the friction of the pulley and cord, and the rigidity 340 FORCE IN RIGID BODIES. of the latter; but, as these are made as slight as possible, they are, for con- veuience, neglected. The machine is used for illustrating the acceleration given to inert matter by unbalanced force, and forms an excellent example of the two distinct duties which a moving force generally has to perform, viz: (1st) the balancing of resistance, and (2nd) acceleration, (d) In the case of a locomotive, drawing- a train on a level, fric- tion and the resistance of the air are the only resistances to be balanced ; for the weight of the train here opposes no resistance. Unless the force of the steam is moie than sufficient to balance the resistances, it cannot move the train. If it exceeds the resistances, the excess, however slight, gives motion to the inert matter of the train. If, at any moment while the train is moving, the force of the steam becomes just equal to the resistances (whether by an increase of the latter or by diminishing the force) the train will move on at a unifoi-m velocity equal to that which it had at the moment when the force and resistance were equalized; and, if these could always be kept equal, it would so move on forever. But so long as the excess of steam pressure over the resistances continues to act, the velocity is increased at each instant ; for during each such instant the excess of force gives a small velocity in addition to that already existing. On a level railroad, let P = the total tractive force of the locomotive = say 13 tons W = weight of locomotive = 50 tons- "W = weight of train = 336 tons R = resistance of locomotive (including internal friction, etc.) = 3 tons r = resistance of train = 1 ton F = net force available for acceleration = P — R — r = 9'toM8 - . ,, . W + \7 50+336 ,. M == mass of engine and train = ■ « — = 12 * g 32.2 1. X • ^ 336 .„ ,. m = mass of train = — — —— = 10.44 g 32.2 a = acceleration T = tension on draw-bar. F 9 Then : Acceleration = a =» ;rj: — r^ = 0.75 ft per second per second. The tension T on the draw-bar = resistance of train + force causing accel- eration a, or T = r + m a =- 1 + 10.44 X 0.75 = 1 -f 7.83 = 8.83 tons. This tension, T, pulling backward against the locomotive, causes there a T 8 83 ff retardation, or negative acceleration, of ^-j -^- = ' ^ ^ = 5.69 ft ' ^ mass of locomotive 50 per sec per sec, and thus reduces, by that amount, the acceleration which the (P — r) 2 10 X. 32 2 locomotive would otherwise have, and which would be — i — ■ = — "j^ 50 50 = 6.44. This, less 5.69, = 0.75 ft per sec per sec = acceleration of train. (e) If the tractive force of a locomotive exceeds the resistances, due to friction, grades, and air, the velocity will be accelerated ; but it then becomes more diffi- cult to maintain the excess of force, for the pistons must travel faster through the cylinders, and the boiler can no longer supply steam fast enough to maintain the original cylinder pressure. Besides, some of the resistances increase with increase of velocity. We thus reach a speed at which the engine, although exerting i,ts utmost force, can do no more than balance the resistances. The train then moves with a uniform velocity equal to that which it had when this condition was reached. When it becomes necessary to stop at a station some distance ahead, steam is shut off, so that the steanj force of the engine shall no longer counterbalance or destroy the resisting forces; and the number of the resistances themselves is in- creased by adding to them the friction of the brakes. The resistances, thus increased, are now the only forces acting upon the train, and their acceleration is negative, or a retardation. Hence, the train moves more and more slowly, and must eventually stop. (f) Caution. When two opposite forces are in equilibrium, an addition to one of the forces does not always form an unbalanced force ; for in many cases the other force increases equally, up to a certain point. For instance, when we attempt to lift a weight, W, its downward resistance, R, remains constantly just equal to our upward pull, P, however P may vary, until P exceeds W. Thus, R can never exceed. W, but may be much less than it. Indeed, when we stop pull- ing, R ceases, although W (the attraction between the earth and J,he weighj) of FORCE IN RIGID BODIES. 341 course remains unchanged throughout. Such variation of resisting force, to meet varying demands, occurs in all those innumerable cases where structures sustain varying loads within their ultimate strength. Art. 17 (a). Work. Force, when it moves a body,* is said to do *' work " upon it. The whole work done by the force in moving the body through any dis- tance is measured by multiplying the/orce by the distance; or: Work = Force X distance. If the/orre is taken in pounds, and the distance infect, the product (or the work done) will be in foot-pounds ; if the force is in tons and the distance in inches, the product will be in inch-tons ; and so on.f Thus, if a force of moves a body through we have work = 1 pound 10,000 feet 10,000 foot-pounds 100 pounds 100 " 10,000 " 10,000 " 1 foot 10,000 " or, in any case, if the force be F pounds, the "whole work done by it in moving a body through s feet, is F s foot-pounds. (b) The foot-pound, the foot-ton, the inch-pound, the inch-ton, etc., etc., are called nnits of ivorli.t For practical purposes, in this country, forces are most frequently stated in pounds, and the distances (through which they act) in feet. Hence the ordi- nary unit of worlt is the foot-pound. Tlie metric unit of worK is the kilog'rani-nieter, i. e. 1 kilogram raised 1 meter = 2.2046 pounds raised 3.2809 feet, = 7.2331 foot-pounds. 1 foot-pound == 0.13825 kilogram-meter. (c) In most cases, a portion at least of the \¥ork done by a force is ex- pended in overcoming; resistances. Thus, when a locomotive begins to move a train, a portion of its force works against, and balances, the resist- ances of friction or of an up-grade, while the remainder, acting as unbalanced force upon the inert mass of the train, increases its velocity. An upward pull of exactly one pound will not raise a one pound weight, but will merely balance the downward force of gravity. If we increase the upward pull from one pound (= 16 ounces) to 17 ounces, the ounce so added, being unbalanced force, will give motion to the mass, and Mill accelerate its upward velocity as long as it continues to act. If we now reduce the upward pull to 1 pound, thus making it just equal to the downward pull of gravity, the body will move on upward with a uniform velocity ; btit if we reduce the upward force to 15 ounces (= ^| pound), then there will be an unbalanced downward force of 1 ounce acting upon the body, and this downward force will generate in the body a downward or negative acceleration or retardation, and will destroy the upward velocity in the same time as the upward excess of 1 ounce required to produce it. During any time, while the 17 ounces upward " force" were acting against the 16 ounces downward •' resistance," the product of total upward force X distance must be greater than that of resistance X distance. The excess is the work done in accelerating the velocity, by virtue of which the body has acquired kinetic energy or capacity for doing work in coming to rest. On the other hand, while the upward velocity was being retarded, the product of total upward force X dist was less than that of resistance X dist, the difference being the work done by the kinetic energy against the resistance of gravity. In practice, the term " work" is usually restricted to that po7-tion of the work which a force performs in balancing the resistances which act against it ; in other words, to the work done by so much of the force as is equal to the resistance. With this restriction, we have work = force X dist, = resistance X dist. Thus, if the resistance be a friction of 4 ft)s., overcome at every point along a distance of 3 feet; or if it be a weight of 4 lbs., lifted 3 feet high, then the work done amounts to 4 X 3 = 12 foot-fi)s, provided the initial and the final velocities are equal. (d) In cases nrhere the velocity is uniform, as in a steadily running machine, the force is necessarily equal to the resistance ; and where the velocities at the beginning and end of any work are equal (as where the machine starts from rest and comes to rest again) the mean force is equal to the mean resistance. In such cases, therefore, the two products, mean force X distance, and mean resistance X distance, are equal; and we have, as before. Work = force X dist = resistance X dist. * A man who is standing still is not considered to be working, any more than is a post or a rope when sustaining a heavy load; although he may be support- ing an oppressive burden, or holding a car-brake with all his strength ; for his force moves nothing in either case. fThese products must not be confounded with moments, = force X leverage. 342 FORCE IN RIGID BODIES. (f ) In calculating the work done by macliiner^^ etc., allowance must be made for this expenditure of a portion of the work in overcoming resistances. Thus, in pump- ing water, part of the applied force is required to balance the friction of the different parts of the pump; so that a steam or water "power," exerting a force of lUO lbs., and moving 6 feet per second, cannot raise 100 lbs. of water to a height of 6 feet per second. Therefore machines, so far from gaining power, according to the popular idea, actually lose it in one sense of the word. In starting a piece of machinery, the forces employed have (Ist) to balance, react against, or destroy the resisting force of friction and the cohesive forces of the material which is to be operated on ; and (2d) to give motion to the unresisting matter of the machine and of the material operated on, after the resisting forces which had acted upon them have thus been rendered ineffective. But after the desired velocity has been established, the forces have merely to balance the resistances in order that the velocity may continue uniform. (g) That portion of the work of a machine, etc., which is expended against fric- tion is ^metimes called ** lost work" or *' prejudicial work,*' while only that portion is called " useful -work " which renders visible and tangible service in the ^hape of output, etc. Thus, in pumping water, the work done in overcoming the fri9^on of the i)ump and of the water is said to be lost or prejudicial, while the useful ^ork would be represented by the product, weight of water delivered X height to whi(5i it is lifted. The distinction, although artificial, and somewhat arbitrary, is often a very con- venient one; but the work is of course not actually " lost," and still less is it "pre- judicial ;" for the water could not be delivered without first overcoming the resist- ances. A merchant might as well call that portion of his money lost which he expends for clerk-hire, etc. (li) For a given force and distance^ ttie^work done is independent of the time ; for the product, force X distance, then remains the same, whatever the time may be. But the distance through which a given force will work at a given velocity is of course proportional to the time during which it is allowed to work. Thus, in order to lift 50 pounds 100 feet, a man must do the same work, (= 5000 foot-pounds) whether he do it in one hour or in ten ; but, if he exerts constantly the same force, he will lift 50 lbs. ten times as high in ten hours as in one, and thus will do ten times the work. Thus, for a given force, tl»e work is proportional to tlie time* Art. 18 (a), Power. The quantity of any work may evidently be considered without regard to the time required to perform it ; but we often require to know the rate at which work can be done ; that is, how much can be done within a certain time. The rate at which a machine, etc. can work is called its power. Thus, in selecting a steam-engine, it is important to know how much it can do pei- minute, hour, or day. We therefore stipulate that it shall be of so many horse-powers; which means nothing more than that it shall be capable of overcoming resisting forces at the rate of so many times 33,000 foot-pounds per minute when running at a uniform velocity, i. «., when force X distance = resistance X distance. (to) The liorse-po-wer, 33,000 foot-pounds per minute, or 550 foot-pounds per second, is the unit of power, or of rate of Avork, commonly used in connec- tion with engines. The metric liorse-po-wer, called "force di cheval," " cheval-vapeur," or (German) " Pferdekraft," is 75 kilogram-meters pel second = 542.48 ft.-lbs. per sec. =32,549 ft.-Jbs. per minute = 0.9863 horse-power. 1 horse-power = 1.0138 " force de cheval." In theoretical Mechanics the foot-pound per second is used in English measure; and the kilogram-meter per sec- ond in metric measure, 1 foot-pound per second = 0.13825 kilogram-meter per second. 1 kilogram-meter per second = 7.2331 foot-pounds per second. (c) Up to the time when the velocity becomes uniform, the po^wer, or rate ot • -work, of the train, in Art. 16 (d), is variable, being gradually accelerated. For in each second it overcomes its resistances (and moves its point of application) through a greater distance than during the preceding second. Also, after the steam is shut oft", the rate of work is variable, being gradually retarded. When the foi'ce of the steam just balances the resistances, the rate of work is uniform. (d) Power = force X velocity. Since the rate of work is equal to the work done in a given time, as so many foot-pounds per second, we may find it by dividing the work in foot-pounds done during any given time by the number of seconds in that lame. Thus _ , , force in pounds X distance in feet Power = rate of work = :- — ; — • time in seconds FORCE IN RIGID BODIES. 343 But this is equivalent to . Power = rate of work - force in pound. X t^tTn^Ltla = Corce in lbs. X velocity in feet per second. Or if we treat only of the work of that force which overcomes resistances : or m * where the velocity is either uniform throughout or the same at the beginning and end of the work; Power rate of work resistance, v, velocity, in ft-lbs. per sec. "" in fi^lbs. per sec. ™ in lbs. ^ in ft per sec. Thus if the resistance is 3300 lbs. and is overcome through a distance of 10' feet in every minute; or if the resistance is 33 lbs. and is overcome through a distance of 1000 feet per minute, the rate of the work is in each case the same, namely, 33,000 foot-pounds per minute, or one horse-power; for lbs. vel. lbs. vel. . 3300 X 10 = 33 X 1000 -= 33,000 foot-pounds per mmute. (e) The same "power" which will overcome a given resistance through a given distance, in a given time, will also overcome any other resistance through any other distance, in that same time, provided the resistance and distance when multiplied together give the same amount as in the first case. Thus, the power that will lift 50 pounds through 10 feet in a second, will m a second lift 500 pounds, 1 foot; or 25 pounds, 20 feet; or 5000 pounds ^ of a foot. In practice, the adjustment of the speed to suit different resistances, is usually effected by the medium of cog-wlieels, toelts, or levers. By means of these the engine, water-wheel, horse, or other motive power, exerting a given force and running at a given velocity, may be made to overcome small resist- ances rapidly, or great ones slowly, as desired. Art. 19 (a). Tlie -work, vrlitcli a bodjr can do "hy virtue of its motion; or (which is the same thing) tlie work required to bring tlie body to rest. Kinetic energy, vis viva, or •'living force." As already remarked, a force equal to the weight of any body, at any place, will, in one second, give to the mass or matter of the body a velocity = g, or (on the earth's surface) about 32.2 feet per second. Or if a body be thrown Mpward with a velocity = gr, its weight will stop it in one second. Since, in the latter case, the velocity at the beginning and at the end of the second are, respectively, = g feet per second, and == 0, the mean velocity of th8 fcody is -1— feet per second. Therefore, during the second it will rise ^— feet, or about 16 feet. In other words, the work which any body can do, by virtue of being thrown vertically upward with an initial velocity (velocity at the Start) of g feet per second, is equal to the product of its weight multiplied bf -|-feet. Or, work in foot-pounds =. weight X ^ Notice that in this case (since the initial velocity v is equal to gf), ^ ». 1. g Suppose now that the same body be thrown upward with double the former velocity; i. 6., with an initial velocity equal to 2 g (or about CA feet per second). Since gravity requires (Art. 8 c), two seconds to impart or destroy this velocity, the body will now move upward during two seconds, or twice as long a time as before. But its mean velocity now is (7, or twice as great as before. Therefore, moving for double the time and with double the velocity, it will travel /oMr times as far, overcoming the same resistance as before (viz.: iia own weight) through /owr times the distance. Thus, by making its initial velocity t? = 2 ^, i. e., by doubling its JL., making it = 2, we have enabled the body to do four times the work which it could do when its -^ was 1; so that the work in the second case is equal to the 344 FORCE IN RIGID BODIES. tofWbs. - -«-'>' ^ ^ product of that in the first case multiplied by the square of ^,; or, » weight X -SL X (-^)' 2 ^ g ^ - weight X -f- X -?^ 2 ff ^ = weight X -^ 2g And it is plain that this would be ithe case for any oi^er velocity. Now the total amount of the work which the body can do, is independent of the amount of the resistance against which it is done; for if we increase the resistance we diminish the distance in the same proportion, so that their product, or the amount of worlc, remains the same. The above formula, therefore, applies to ail cases; i. e., the total amount of work, in foot pounds, which any body will do, against any resistance, by virtue of its motion alone, in coming to rest, is Work = weight of moving body, in lbs. X square of its velocity in ft per sec^d = weight of moving body, in lbs. X fall in ft required to give the velocity ^ weightof moving body, in lbs, s^ square of its velocity in ft per second 9 2 In these equations, the weight is that which the body has in any given place, and g is the acceleration of gravity at that same place. (to) Since the weight of a body .^ .^^ ^^^^ (Art. 11, p. 336) , the last formula 9 becomes, by "method A," Art. 11 (d), Mrork ^ mass of moving body w s quare of its velocity in ft per second in foot-pownds in "waiii" '^ 2 and by "method B," Art. 11 («), vvorls: _ mass of moving body y, square of its velocity in ft per second in foot-poundals ~ in pounds ^ 2 (c) In the above equations the ie/i hand side represents the work (or resis- tance overcome through a distance) in any given case, while the right hand side represents the Iclnetic energy of the body, by which it is enabled to do that work. Some writers call this energy "vis viva," or " living force" a name formerly given (for convenience) to a quantity just double the energy, or = mass X velocity^. (d) As an illustration of the foregoing, take a train weighing 1,120,000 pounds, and moving at the rate of 22 feet per second. The kinetic energy •f such a train is energy » weight X — r — ~ ; or, 1,120,000 lbs. X -^ = 8,400,000 ft.-lbs. 64.4 That is, if steam be shut off, the train will perform a work of 8,400,000 ft.-lbe. in coming to rest. Thus, if the sum of all the resistances (of friction, air, grades, curves, etc.) remained constantly == 5000 lbs.,* the train would travel 8,400,000 ft.-lbs. ^ jggQ f^^ 5000 lbs. (e) We thus see that the total quantity of work which a body can do by virtue of its motion alone, and without assistance from extraneous forces, is in pro- portion to the weight of the body and to the square of its velocity when it begins to do the work. For example, suppose that a tram, at the moment when steam is shut off, has a velocity of 10 miles an hour and that the kinetio energy, which that velocity gives it, will by itself carry the train agamst th« •In practice, this would not be the case. FORCE IN RIGID BODIES. 345 resistances of the road, etc., for a distance of one quarter of a mile before it stops. Then, if steam be shut oflf while the train is moving at 5, 20, 30 or 40 miles per hour (i. e. with J^, 2, 3 or 4. times 10 miles per hour) the train will travel JL.» 1» 2 J^ or 4 miles (or J^, 4, 9 or IG times ^ mile) before coming to rest* But the rate of work done is proportional simply to the resistance and the v^ocity (Art. 18d, p. 342). Therefore, the locomotive whose steam is shut ofl at 20, 30 or 40 miles pelr hour, will require, for running its 4, 9 or 16 quarters «f amile, but 2, 3 or 4 times as many seconds as itrequired at 10 miles per hour. The same principle applies to all cases of acceleration or of retardation.f For instance, in the case of a falling body, the distance through which it must fall in order to acquire any given velocity is as the square of that velocity, but the time required is simply as the velocity. Also, if a body is thrown vertically upward with any given velocity, the height to which it will rise by the time gravity destroys that velocity, will be as the square of the velocity, l5ut the time will be simply as the velocity. Art. 20 (a). The momentuni of a moving body (or the product of its mass by its velocity) is the rate, in foot-pounds per second, at which it works against a resisting force equal to its own weight, as in the case of a body thrown vertically upward. At the instant when it comes to rest, its momentum, or rate of work, is of course = nothing. Therefore its mean rate of work, or mean momentum, is one-half of that which it has at the moment of starting. Thus, suppose such a body to weigh 5 lbs. Then, whatever its velocity niay be, 5 pounds is the resisting force, against which it must work while coming lo rest. Let the initial velocity be 96 feet per second. Then its momentum = mass X velocity — 5 X 96 — 480 foot-pounds per second* and, while coming to rest, its mean momentum =» mass X T^ Qc^ty ^ 240 foot-pounds per second. Now, in falling, the weight of the body (5 lbs.), would give it a velocity of 9d feet per second in about three seconds. Consequently, in rising, it will destroy its velocity in the same time. In other words, the time — — \e oci y _ ^ ve ocity acceleration g BB £6 «= 3. Three seconds, therefore, is the time during which it can work. Now, if the mean rate of work in foot-pounds per second (at which a body can work against a resistance) be multiplied by the time during which it can continue so to work, the product must be the total work done. Or, in this case, work mean rate of work ^ time, _ 940 v ^ L 720 foat.nnnnrl. in ft-lbs. ■" in ft-lbs. per sec. >< or No. of sees. - 240 X ^ =« 7^0 loot-pounds. - weight X^^^^^^X^^^^ 2 g — weight X ^^^^^^^^ , as in Art. 19 (a), = 5 X -^ -= 720 ft-pound«. (b) We may notice also that since, in the case of a falling body, or. of one thrown upward, 7^ ^^^ ^ is the time during which it must fall in order to g acquire a given velocity, or during which it must rise in order to lose it, therefore, velocity ^ yelocity ^ ^^^^ velocity X time = distance traversed; 2 g so that weight X I^l^in^' = weight X I2l!!2i*L x l^^^S}^ » 2^ 2. g weight X distance traversed «=» the work. * This supposes, for convenience, that the resistances remain uniform through- out, and are the same in all the cases, which, however, would not hold good in practice. t Retardation is merely acceleration in la direction opposite to tlaat of the motion which we happen to be considering. 346 FORCE IN RIGID BODIES. Art, ai (a). Energy is In.lestmctible. Energy, expended in work, is not destroyed. It is either transterred to other bodies, or else stored up in the body itself; or part may be thus transferred, and the rest thus stored. But, altliough energy cannot be destroyed, it may oe rendered useless to us. Thus, a moving train, in coming to rest on a level track, transfers its kinetic energy into other kinetic energy; namely, the useless heat due to friction at the rails, brakes and journals ; and this heat, although none of it ia destroy ed^ is dissipated in the earth and air so as to be practically beyond our recovery. Art. Ji'4 (a). Potential energy, or possible energy, may be defined as stored-up energy. We lift a one-pound body one-foot oy expending upon it one foot-pound of energy. But this foot-pound is stored up in the " system " (composed of the earth and the body) as an addition to its stock of potential energy. For, while the stone falls through one foot, the system will acquire a kinetic energy of one foot-pound, and will part with one foot-pound of its potential energy. (b) The potential energy of a "system" of bodies (such as the earth and a weight raised above it, or the atoms of a mass of powder, or those of a bent spring) depends upon the relative positions or those bodies, and upon their tendencies to change those positions. The kinetic energy of a system (such as the earth and a moving train of cars) depends upon the masses of its bodies and upon their motion relatively to each other. Familiar instances of potential energy are— the weight or spring of a clock when fully or partly wound up, and whether moving or not; the pent-up water in a reservoir; the steam pressure in a boiler; and the explosive energy of powder. We have mechanical energy in the case of the weight or springs or water; heat energy in the case of the steam, and chemical energy in that of the powder. (c) In many cases we may conveniently estimate the total potential energy of a system. Thus (neglecting the resistance of the air) the explosive energy of a pound of powder is =- the weight of any given cannon ball X the height to which the force of that powder could throw it, =» tlie weight of the ball X (the square of the initial velocity given to it by the explosion) -i- 2o. But in other cases we care to find only a certain definite portion of the total potential energy. Thus, the total potential energy of a clock-weight* would not be exhausted until the weight reached the center of the earth; but we generally deal only with that portion which was stored in it by winding-up, and which it will give out again as kinetic energy in running down. This portion is = the weight X the height which it has to run down >= the weight X (the square of the velocity which it would acquire in falling/reeZ^/ through that height) -f- 2^g. (d) There are many cases of energy in which we may hesitate as to whether the term "kinetic" or "potential" is the more appropriate. Thus, the pres- sure of steam in a boiler is believed to be due to the violent motion of the particles of steam, which bombard the inner surface of the boiler-shell; so that, from this point of view, we should call the energy of steam kinetic. But, on the other hand, the shell itself remains stationary; and, until the steam is permitted to escape from the boiler, there is no outward evidence of energy in the shape of work. The energy remains stored up in the boiler ready for use. From this point of view, we may call the energy of steam potential energy. (e) It seen\s reasonable to suppose that further knowledge as to the nature of other forms of energy, apparently potential (as is that of steam), mighl reveal the fact that all energy is ultimately kinetic. Art. 23 (a). There is much confusion of ideas in regard to those actions to which, in Mechanics, we give the names, ** force," *• energy," " power," etc. This arises from ihe fact that in every-day language these terms are used indiscriminately to express the same ideas. Thus, we commonly speak of the " force " of a cannon-ball flying through the air, meaning, however, the repulsive force which ivoidd be exerted between the ball and a building, etc. with which it might come into contact. This force would tend to move a part of the building along in the direction of the flight of the ball, and would move the ball backward ; (i.e., would retard its forward motion). But this great repulsive "/o^ce" does not exist until the ball strikes the building. Indeed, we cannot even tell, from the velocity and weight of the ball, what the.amount of the force will be, for this depends upon the strength, etc., of the building. If the building is of glass, the force may be so slight as scarcely to retard the motion of the* ball perceptibly, while, if the building is an * For convenience we may thus speak of the energy of a system of bodies (the earth and tlie clock-weight) as residing in only one of the bodies. FORCE IN RIGID BODIES. 347 earth embankment, the force will be much greater, and may retard the motion of the ball so rapidly as to entirely stop it before it has gone a foot farther. The moving ball has great (kinetic) energy; but the only force that it exerts during its flight is the comparatively very slight one required to push aside the particles of air. The energy of the ball, and therefore the total work which it can do, are inde- pendent of the nature of the obstruction which it meets ; but since the work is the product of the resistance ottered and the distance through which it can be overcome, the distance must be inversely as the resistance ottered ; or (which is the same thing) inversely as the force required of, and exerted by, the ball in balancing that resistance. Since work, in ft.-tbs. = force, in fibs., X distance traversed, in feet, we have force in lbs = wo rk, in ft.-ft>s. ^ rate of work, ' ' distance traversed, in feet in ft.-lbs. per foot. Art. 24 (a). An impact, blow, stroke or collision takes place when a •loving body encounters another body. The peculiarity of such cases is that the time of action of the repulsive force due to the collision is so skvH that gen- erally it is impossible to measure it, and we therefore cannot calculate the force from the momentum produced by it in either of the two bodies : but since both bodies undergo a great change of velocity (i, e., a great acceleration) during this short time, we know that the repulsive force acting between them must be very great. We shall consider only cases of direct impact, or impact where the centers of gravity of the two bodies approach each other in one straight line, and where the nature of the surfaces of contact is such that the repulsive force caused by the impact also acts through those centers and in their line of approach. (b) This force, acting equally upon the two bodies (Art. Sjf), for the same length of time (namely, the time during which they are in contact), neces- sarily produces equal and opposite changes in their momentums (Art. 12, p. 338). Hence, the total momentum (or product, mass X velocity) of the two bodies is always the same after impact as it was before. (c) But the relative behavior of the two bodies, after collision, depends upon their elasticity. If they could be perfectly inelastic, their velocities, after im- pact, would be equal. In other words, they would move on together. If they could be perfectly elastic, they would separate from each other, after collision, with the same velocity with which they approached each other before collision. (cl) Between these two extremes, neither of which is ever perfectly realized in practice, there are all possible degrees of elasticity, with corresponding differences in the behavior of the bodies. The subject, especially that of indirect impact, is a very complex one, but seldom comes up in practical civil engineering. (e) "In some careful experiments made at Portsmouth dock-yard, England, a man of medium strength, and striking with a maul weighing 18 fibs., the handle ©f which was 44 inches long, barely atarted a bolt about Yq of an inch at each blow ; and it required a quiet pressure of 107 tons to press the bolt down the same quantity ; but a small additional weight pressed it completely home." 348 GRAVITY — FALLING BODIES. GRAVITY, FAI.I.IXO BODIES. Bodies falliiig;> vertically. A body, falling freely in racuo from a state of rest, acquires, by tlie end of tlie fir^t second, a velocity of about 32.2 feet per second; and, in each succeeding second, an addition of velocity, or acceleration, of about 32.2 feet per second. In other words, tlie velocity receives in each second an acceleration of about a2.2 feet per second, or is accelerated at th« rate of about 32.2 feet per second, per second. This rate is generally called (for brevity, see foot-note,f p. 334) , simply the acceleration of gravity (but see * belowj, and is denoted by g. It increases from about 32.1 f>.et per second, per second, at the equator, to about 32.5 at the poles. In the latitude of London it is 32.19. These are its values at sea-level ; but at a height of 5 miles above that level k is diminished by only abuut i part in 400. For most practical purposes it may be taken at 32.2. Caution. Owing to tbe resistance of the air none of the follow- ing rules give perfectly accurate results in practice, especially at great vels. Thp greater the specific gravity of the body the better will be the result. The air resists botfi rlsiii;; and fiiHin^ bodies. If a body be tliroiFn vertically upwards with a given vel, it will rise to the same height from wliich it inust have fallen in order to acquire said vel; and its vel will be retarded in each second 32.2 ft per sec. Its average ascend' ing velocity will be half of that with which it started ; as in all other cases of uniformly retarded vel.. In falling it will acquire the same vel that it started up with, and in the same time. See above Caution. Acceleration acquired* in a given time = 9 X time in a given fall from rest = y' 2 g xTall. in a given fall frotn rest ) _ twice the fall and given time j ~~ time Time required acceleration for a given acceleration = for a given fall from rest = /fall fall fall ^ final velocity fall for a given fall from rest ■> _ or otherwise J ~~ mean vel ^^ (initial vel + final vel) Fall in a given time (starting from rest) = time X H ^'^^^ ^^^ == ^^°^^ ^ X j^Sf in a given time (starting) ,. . , i ^. ^ . initial vel -f final vel ^ ^ ^ ^, . ° [= time X mean vel = time X tt from rest or otherwise) ) 2 reqd for a given acceleration ") _ acceleration^ (starting from rest) j "~ 2 g during any one given second (counting from rest) = g X (number of the second (1st, 2d, &c) — ^ j during any equal consecutive times (starting from rest) oc 1, 3, 5, 7, 9, &c. At tlie end of tlie 4th. 5th. 6th. 7th. 8th. 9th. 10th. seconds ''^"'wfLTf-'} »^'- 2.1. Velocity ; ft per sec. Dist fallen since end of preceding sec ; ft. Total dist fallen ; ft. 32.2 64.4 96.6 128.8 161.0 193.2 225.4 257.6 289.8 16.1 48.3 80.5 112.7 144.9 177.1 209.3 241.5 273.7 16.1 64.4 144.9 257.6 402.5 579.6 788.9 1030.4 1304.1 322.0 305.9 1610.0 * By " acceleration," in this article, we mean the totat acceleration ; t. e., the whol« change of velocity occurring in tbe givea time or Mi. J^'or tbe rate of acceleration we use simply tbe letter g. DESCENT ON INCLINED PLANES. 349 Descent on inclined planes. When a body, U, is placed upon an inclined plane, AC, its whole weight W is not employed in giving it velocity (as in ihe case of bodies falling vertically) but a portion, P, of it (= W X cosine of o — W X cosine of a*) is expended in perpendicular pressure against the plane; while only S, (= W X sine of o = W X sine of a,*) acts upon U in a direction parallel to the surface A C of the plane, and tends to slide it down that surf. The acceleration, generated in a given body in a given time, is proportional to the force acting upon the body in the direction of the acceleration Hence if we make W to represent by scale the acceleration g (say 32.2 ft per sec) which grav would give to U in a sec if falling freely, then S -will give, by the same scale, the acceleration in ft pei- sec which the actual sliding force S would give to U in one sec if there were no friction between U and the plane. • We have therefore theoretical acceleration down the plane = g x sine of a. Therefore we have only to substitute "(7. sin a" in place of "5';" and the sloping distance or "slide" A C in place of the corresponding vertical distance or " fall " A E in the equations, in order to obtain the accelerations etc as follows: on an inclined plane wifliont friction. In the following, the slides A € are in feet, the times in seconds, and the velocities and accelerations in feet per second. f Accelerationfof sliding velocity i« Q o-itroTi tir^no - ^^^^ ^^^^cl acQulred in falling) ^ ^. „ in a given time - ^^^^ ^^^.^"^ ^^^^ ^^^^^ ^.^| j X sin a- = g. sin a X time in a given slide, as A C, "> ^ slide from rest i 3^ time ("vert accel acquired in falling"| _ =< freely thro the corresponding >= j/ 2 ^r AB t vert ht A E J «a y 2 g. sin a X slide Time required ,.,. , .. sliding acceleration for a given sliding acceleration = ^ ^ ^ g.sma for a given slide, as A C, from slide _ / slide rest ' ^ final sliding velocity ^ /^9- sin a time reqd to fall freely thro the correspond- ^ ing ve rt ht A E sin a for a given slide, from> slide _ slide rest or otherwise J mean sliding vel 3^ (initial + final sliding vels) horizontal stretch, as EC, ^_ . _ bas e E C _ of any leng th, as A C \/ AC2 — A E2 "~ length A C ~ that length ~^ Sine a = ^^^^S^IA? — fall, A E, in any giv en length, AC length A C ~ that length * Because and a are equal. fBy acceleration, in this article, we m.ati the lotai acceleration, t. e., the wholo chana^e in velocity occurring in the given time or slide. For the rate of accelwation ipe use simply the letter g. 350 GRAVITY — PENDULUMS. Slide, as A C in a given time, starting from rest = time X K final sliding vel = time 2 X K ^. sin a. in a given time, s<;arting from rest ^. or otherwise = "me X mean sliding vel = time X li (initial -f final, sliding vels) required for a given sliding aceel- _ sliding acceleration g eration (starting from rest) 2 g. sin a - g X [sin a - (cos a. coefF fric)] " in place of " g. sin a " in the above equations. Because Friction = Perpendicular pressure P X coefficient of friction = weight W X cosine a X coefficient of friction retardation of friction = gX cosine a X coefficient of friction. Resultant slidinjs^ acceleration = theoretical sliding; accel (due to the sliding force, S) — retardation of fric = {g. sin a) — {g. cosine a. coeff fric) = ^ X ^sin a — (cosine a. coeff fric)~| If The retardation of friction (= g. cos a X coeff fric) is not less than the total or theoretical accel ("p. sin a") the body cannot slide down the plane. PENDULUMS. The numbers of vibrations which diff pendulums will make in any given place* in It given time, are inversely as the square roots of their lengths; thus, if one of them is 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; but its number of vibrations will be but ]/^, %, or % as great. The times in which diff pendulums will mak« a vibration, are directly as the sq rts of their lengths. Thus, if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great; and so also will be the time occupied in one of its vibrations. The length of a pendulum vibrating seconds at the level of the sea, in a vacuum, in the lat of London (513^° North) is 39.1393 ins ; and in the lat of N. York (40%° North) 39.1013 ins. ^I'j the ecpiator about -^-q inch shorter ; and at the poles, about -^-^ Inch longer. Approximately enough for experiments which occupy but a few sec, we may at any place call the length of a seconds pendulum in the open air, 39 ins ; half sec, 9^ ins ; and may assume that long and short vibrations of the same pen- dulum are made in the same time ; which they actually are, very nearly. For meas- uring depths, or dists by sound, a sulficiently good sec pendulum may be made of a pebble (a small piece of metal is better) and a piece of thread, suspended from 8 common pin. The length of 39 ins should be measured from the cpntre of the pebble. PKJ^TDULTJMS, ETC, 351 In starting the vibrations, the pebble, or bob, mxist not be thrown into motion, but merely let drop, after extending the string at the proper height. To find the leiig>t;ti of a pendulum reqd to make a given number of vibrations in a min, divide 375 by said reqd number. The square of the quot will be the length in ins, near enough for such temporary purposes as the foregoing. Thus, for a pendulum to make 100 vibrations per min, we have |^^4 = 3.75 ; and the square of 3.75 = 14.06 ins, the reqd length. To find tbe number of vibrations per min for a pendulum of given length, in ins, take the sq rt of said length, and div 375 by said sq rt. Thus, for a pendulum 14.06 ins long, the sq rt is 3.75 ; and ^ = 100, the reqd number. Rem. 1. By practising before the sec pendulum of a clock, or one prepared as just stated, a person will soon learn to count 5 in a sec, for a few sec in succession ; and will thus be able to divide a sec into 5 equal parts ; and this may at times be useful for very rough estimating when he has no pendulum. Centre of Oscillation and Percussion. Rem. 2. When a pendulum, or any other suspended body, is vibrating or oscillating backward and forward, it is plain that those particles of it which are far from the point of suspension move faster than those which are near it. But there is always a certain point in the body, such that if nil the particles were concentrated at it, so that all should move with the same actual vel, neither the number of oscillations, nor their angular vel, would be changed. This point is called the center of osciUa- tion. It is not the same as the cen of grav, and is always farther than it from the point of suspension. It is also the centre of percussion of the suspended vibrating body. The dist of this point from the point of susp is found thus : Suppose the body to be divided into many (the more the better) small parts; the smaller the better. Find the weight of each part. Also find the cen of grav of each part ; also the dist from each such con of grav to the poiat of susp. Square each of these dists, and mult each square by the wt of the corresponding small part of the body. Add the products together, and call their sump. Next mult the weight of the entire body by the dist of its cen of grav from the point of susp. Call the prod g. Divide p hyg» This jo is the moment of inertia of the body, and if divided by the wt of the k)ody the sq rt of the quotient will be the Radius of Oy ration. Ang-ular Velocity. When a body revolves around any axis, the parts which are farther from that axis move faster than those nearer to it. Tlierefore we cannot assign a stated linear velocity in feet per second, or miles per hour etc, that shall apply to every part of it. But every part of the body revolves around an entire circle, or throutrh an angle of 360°, in the same time. Hence, all the parts have the same velocity in degrees- per second, or in revolutions per second. This is called the angular velocity. Scientific writers measure it by the length of the arc de- scribed by any point in the body in a given time, as a second, the length of the arc being measured by the number of times the length of its oion radius is con- tained in it. When so measured, Angular velocity _ li:aear velocity (in feet etc) per sec in radii per second - length of radius (in feet etc) Here, as before, the angular velocity is the same for all the points in the body, because the velocities of the several points are directly as their radii or dis- tances from the axis of revolution. Ih each revolution, each point describes the circumference of the circle in which it revolves = 2 7rr (tt = 3.1416 etc; r = radius of said circle). Conse- quently, if tlie body makes n revolutions per second, the length of the arc de- scribed by each point in one second is 2 7rr?i; and the angular velocity of the body, or linear velocity of any point measured in its own radii, is a = = 2 TT w = say 6.2832 X revs per second = say .1047 X revs pp' i^inute. Moment of Inertia. Suppose a body revolving around an axis, as a grindstone; or oscillating, like a pendulum. Suppose that the distance from the axis of revolution (which, in the pendulum, is the point of suspension) to each individual particle of the body, has been measured; and that the square of each such distance has been multiplied by the weight of that particle to which said distance was measured. 352 MOMENT OF INERTIA. The sum of all these products is the moment of inertia of the body. Or Moment of inertia -{ ■.%d^w. thpsnm • ) (weight square of dist forallthen^rticlesr ^^ 1 ''^' X «t Partielefrom lor all the particles j (particle axis of revolution Scientific writers frequently use the mass of each particle ; its weii^ht . , „ . . . , . t e, -, --: 7-7 — -^ ~. r — r^^FT^ instead of its weight, m calculatiug acceleration {g) of gravity, or about 32.2 ° the moment of inertia. In practice we may suppose the body to be divided into portions measuring a cubic inch (or some other small size) each ; and use these insteaO of the theo- retical infinitely small particles. The smaller these portions are taken, the more nearly correct will be the result. When tl)e moment of inertia of a mere surface is wanted (instead of that of a body), we suppose the surface to be divided into a number of small areas^ and use ihem instead of the weights of the small portions of the body. Moment of inertia = weight of body. square of area of surface '^^^"« «^' gyration Table of Radii of Oyration. Body Revolving; around Any body or figure Solid cylin- der ditto ditto, infinitely short (circular surface; Hollow cyl- inder > ditto, infinitely thin ditto, of any thickness ditto, infinitely thin ditto, infinitely thin and infinitely short (circumfer- ence of a circle) Solid spliere any given axis its longitudinal axis a diam, midway between its ends a diameter its longitudinal axis ditto a diam midway between its ends ditto a diameter a diameter Radius of Gyration V moment of inertia around the given axis weight of body, or area of surface radius of cylinder X -\/-|- — radius of cylinder X about .7071 4 'length 2 4 radius^ of cylinder .2 ' 4 radius of cylinder 2 [inner rad2 + outer rad^ 4 radius of cylinder inne r rad^ + outer rad^ length* 4 + ~l2~" V radius^ of cylinder lengths 2 + 12 radius of cylinder X \ '= radius of cylinder X about .7071 / radius^ of sphere - radius of sphere X 1/^4" = radius of sphere X about .63246 RADII OF GYRATION. 353 Table of Radii of Oyration.— Continued. Body Revolving^ around Radius of Oyration Hollow sphere of any thickness ditto, thin ditto, infinitely thin (spherical surface) Straight lin^, ab Solid cone Circular plate, of rect- angular cross sec- tion Circular ring, of rectan- jular cross section Square, rect- ang^le and other sur- faces a diameter ditto ditto any point, «, in its length either end, a or 6 its center, e its axis 3 Solid cylin- der e Hollow cylin- der 4 2 (outer rad^ — inner rad^) 5 (outer rad^ — inner rad^) approx (outer rad + inner rad) X 4085 radius of sphere X 'v-v" «= radius of sphere X about .8165 4 /axs ^ xb» 3 ab length abX -yj-Y- — length abX abont .5775 ac X »= length abX about .2887 radius of base of cone X l/^3~ -= radius of base of cone X -5477 For the thickness of plate or ring, . measured perpendicularly to the plane of the circumference, take the length of the cylinder. For least radius of gyration, or that around the longett axii, see p 496 and 497. 23 354 CENTRIFUGAL FORCE. CENTRIFUGAI. FORCE. When a body a, Fig. 1, moves in a circular path abd, it tends, at each point; as a or b, to move in a tangent at or bi' to the circle at that point. But at each point, as a, etc., in the path, it is deflected from the tangent by a force acting toward Llie center, c, of the circle. This force may be the tension of a string, ca, or the attraction between a planet at c and its moon a, or the inward pressure of the rails, a 6, on a curve, etc., etc. Like all force, it is an action between two bodies, tending either to separate them or to draw them closer together, and act- ing equally upon both. (See Art. 5 (fc), p. 332). In the case of the string, it pulls the body a, toward the center, c, and the nail or hand, etc., at c, toward the body at a or 6, etc. ; i. e.,from the center. In the case of a car on a curve it pushes the car toward the center, and the rails frojn the center. The pull or push on the revolving body toward the center is called the centripetal force; while the pull or push tending to move the deflecting body from the center is called the ceiitrifug-al force. These two " forces," being merely the two "sides " (as it were) of the same stress, are necessarily equal and opposite, and can only exist together. The moment the stress or tension exceeds the strength (or inherent cohesive force) of the string, etc., the latter breaks. The centripetal and centrif- ugal forces therefore instantly cease ; and the body, no longer disturbed by a deflecting force, moves on, at a uniform velocity,* in a tangent, at or bt', etc., to its circular path ; i. e., at right angles to the direction which the centrifugal force had at the moment it ceased. {a). A sing>1e revolving body, a, Fig. 1. Let / = the centrifugal or centripetal force, in pounds. W = the weight of the body a, in pounds, K = the radius ca of the path of the center of gravity of the body a, in feet, V = the uniform velocity of the body a in its circular path abd, in feet per second, n = the number of revolutions per minute, g = the acceleration of gravity = say 32.2 feet per second per second, 900 g = about 28980. n = circumference -f- diameter = say 3.1416. ir^ = about 9.8696. Then, for the centrifug^al force, /: If we have the velocity v in feet per second : f = W ;^— t • • • (1) R g If we have the number ?i of revolutionsper minute :/= W-^ J ... (2) / = about .0003406 WR7i« § . . . (3). * Neglecting friction, gravity, the resistance of the air, etc. t For let at, Fig. 1, represent the amount and direction of the velocity v of the body at a in feet per second. Then at the end of one second the body will have reached the point b (the arc ab being made = at), and the amount and direction of its velocity at b will then be represented by the line bt' = at in length, but dilTeriMg in direction. Drawing cu and cu' at the center, equal and parallel respectively to ai and bt', we find that the change in the direction of the motion {i.e., the acceleration toward the center) during the second is represented by the arc uu'; and, since angle acb = angle ucu', we have the proportion, radius R or ac : ab or at :: cu^ or ai : arc uu\ In other words, the acceleration mm' in one second, or rate of acceleration, is = at^ v^ — = — ; ana, for the force causing that acceleration, we have A ii / = mass of body X rate of acceleration = mass of body X tT X By formula (1),/=. W ^^ ^. But . = —^ ; and v'^ = — ^go^— ^ Txr 'T^ I^"" W'' XTr'T'^Bn* Hence, /=^W^---^^ = W^^. § Formula (3) is obtained from (2) by substituting the va^^e8 9.8696 and 28980 for w* and 900 g respectively. CENTRIFUGAL FORCE. 355 (6> "Wheels and discs. Suppose the rim of a wheel to be cut into very ishort slices, as shown (much exaggerated) at a, Fig. 2. Then for each slice, as a, by formula (1): /= weight W of slice X :^~ ;* and if each slice were connected n m ' c n Fig. 7 with the center by a separate string, the sum of the stresses in all the strings (neglecting friction between adjacent slices) would be : F = sum of centrifugal forces of all the slices t = weight of rim X ^g' . (4). But the stress with which we are usually concerned in such cases (viz. : flie tension in tlie rim itself in the direction of a tangent to its own cir- cumference) is much less than the theoretical quantity F ol»tained from formula (4), being in fact only ■ of it. For suppose first that the same thin rim is cut only at two opposite points m and n, Fig. 3, and that its two halves are held together only by tne string S. * If the rim is very thin in proportion to its diameter mn, we may take the center ef gravity of each slice as being in a circle mn midway between the inner and outer , r i.1, • J.^ ^ -^ inner radius -f outer radius ^ . » . , , edges of the nm, so that R = . In a rjm of appreciable thickness, this is not the case, because each slice is a little thicker at its outer than at Its inner end. See Fig. 6. Hence its center of gravity is a little outside of the curved line win, Fig. 2. t In a perfectly balanced rim (i. e., a rim whose center of gravity coincides with its center of rotation, as in Fig. 3) the centrifugal forces of the particles on one side of c counterbalance those on the opposite side. Here, too, R = Hence, as a whole, such a rim has no centrifugal force ; i. e., no tendency to leave the center in any one direction by virtue of its rotation. But if the two centers do not coincide (Fig. 4), then the rim is a single revolving body, and its centrifugal force is : / = weight 1)2 of entire rim X :;;" ; where R is the distance between the two centers, and v the Rj7' velocity of the center of gravity a. The force / acts in the line joining the two tenters. 356 CENTRIFUGAL FORCE. Then : * F semi-circumference mzn : diameter mn - '• '^ ' pull on the string S ; so that pull on _ half weight J^ 2 _ weight v^ F_ F strings S " of rim R^ ^ tt ~ of rim ^ Ugn" it~ 3.1416 • • • ^ )' and if the rim is now made complete by joining the ends at m and n, and if the string S is removed, then the pull on the string by formula (5) will be equally- divided between m and n. Hence each cross-section, as m or w, of the rim, will sustain a tensile stress equal to half the pull on the string ; or ^ . . . F F weight of rim X v^ tension m rim = - = -- = 6.2832 Rg • <«'• The centripetal force,/, Fig. 2, holding any part o of the rim to its circular path, is the resultant of the two equal tensions at the ends of that part. For the stress per square inch of cross-section of rim, we have : .^ ^ tension in rim unit stress = r ^ 1 . jr-^ ; ; j— - area A of cross-section of rim, in square inches _ F _ weight of rim X v^ ,-. ~ 6.2832 A ~ 6.2832 KB.g ^ '' We shall arrive at the same result if we reflect that the pull in the string S or the sum of the two tensions at m and w, is equal to the centrifugal force/ of either half of the rim, revolving, as a single body, about the center c. Find the center of gravity G of the half rim, and then, in formula (1), use the velocity of that point, and the radius cG instead of velocity at z and radius cz respectively ; thus: p«U in string = /= -'„f,f^r,/r^= weight of half-rim X -^'-f^^i and half of this is the tension in each cross-section of the rim.f If the rim were infinitely thin, cG, Fig. 3, would be 0.6366 cz. If its thickness must be taken into consideration, and if it is of rectangular cross-section, find the centers of gravity g and g\ Fig. 6, of the whole semicircular segment cz and of the small segment ch respectively {eg = 0.4244 cz, and eg' = 0.4244 cb. Then , ,, .^ area of entire segment cz g'^ = gg>y^ -^—-J^^^^~ . For rims of other than rectangular cross-section, use formulse (4), (5) and (6). In a disc, sucli as a ;s:rindstonc, the tension in each full cross-section mn. Fig. 7, is equal to the centrifugal force / of half the disc. Let W = weight of half disc. The distance cG from the center c to the center of gravity G of the half disc, is cG = 0.4244 cz ; and the * In Fig. 2, let the centrifugal force of any slice, o, be represented by the diagonal, /, of a rectangle, whose sides, H and V, are respectively parallel and perpendicular to the given diameter mn. Then H and V represent the components of / in those two directions. The equal and opposite horizontal components H, of o and of the corresponding slice o\ being parallel to mn, have no tendency to pull the rim apart at m or n. Hence, the pull on a string S, Fig. 3, perpendicular to mn, is the sum of the components V of all the slices. For each very thin slice, Fig. 5 (greatly exaggerated) we have (since angle A = angle A') : Length I . its horizontal . . centrifugal force . its vertical of slice • projection, J) ** /, of slice * component V. Hence, for the entire half-rim mn. Fig. 3 (made up of such slices), we have : ♦v.« o,iT« r.f *>,o ^oT,fr4f t^® s"°^ of the Length mzn its horizontal , f^ill^f oUfT.! . vertical compo- of half-rim • projections,™ = ^^t^rthelf/iit ' t^tsTicS?" which is identical with the proportion at top of page. t The rims of revolving wheels are usually made strong enough to resist the tension dtie to the centrifugal force, without aid from the spokes, which thus have merely to support the weight of the wheel. But if the rim breaks, the centrifugal forces of its fragments come entirely upon the spokes ; and, since the breakage is always irregu- lar, some of the spokes will always receive more than their share. CENTRIFUGAL FORCE. 357 tension In mn = f = W ,' ^ ' = rad. cGXg ■ W 0.4244 (vel. at g)a czXg 0.4244 ir^ n^ cz 900 g 0.42448 (vel. at g)" 0.4244 czXg . . . (8). = W (9). The stress per square inch in any full section mn is tension in mn unit stress = area of cross-section in square inches = W 0.4244 (velocity at z)^ diam. mn, ins. X thickness, ins. X cz X g ' ^ 0.4244 7r» n^ cz diam. m»i, ins. X thickness, ins. X 900 g ' . (10). • (11). Fig. 5 ^n nv c n Fig. 7 /= the centripetal force, in pounds, acting upon a single revolving body, a, Figs. 1, 2, 4 and 5, or upon the half-rim or half-disc, Figs. 3, 6 and 7 = the centrifugal force exerted by such body. F = the sum of the centrifugal forces /, of all the particles of a rim, Fig. 3. W = the weight of the body, in pounds. R = the radius ca, Figs. 1, 4 and 5, of the path of the center of gravity of the body. V = the uniform velocity of the body in its circular path, in feet per second. M = tlie number of revolutions per minute. g = the acceleration of gravity = say 32.2 feet per second. 900 g = about ■■ say 3.1416. n^ = about 9.8696. circumference diameter Ilia rolling' "wlieel, each point in the rim, during the moment when it touches the ground, is stationary witk respect to the earth; but each particle has the same velocity abovi the center as if the latter were stationary, and hence the centrifugal force has no effect upon the weight. 358 STATICS. STATICS. FORCES. 1. Statics Defined. The science of statics, or of equilibrium of forces; takes account of those very numerous cases where the forces under con* sideration are in equilibrium, or balanced. It embraces, therefore, all cases of bodies which are said to be "at rest."* 2. In the problems usually presented in civil engineering, a certain given force, or certain given forces, applied to a stationary * body (as a bridge or building) tend to produce motion, either in the structure as a whole or in one or more of its members ; and it is required to find and to apply another force or other forces which will balance the tendency to motion, and thus permit the structure and its members to remain at rest. See 1[ 33, below. 3. Equilibrium. Suppose a body to be acted upon by certain forces. Then those forces are said to be in equilibrium, when, as a whole, they pro- duce no change in the body's state of rest or of motion, either as regards its motion as a whole along any particular line (motion of translation), or as regards its rotation about any point, either within or without the body. In such cases the body also is said to be in equilibrium. See 1 84, below. 4. A body may be in equilibrium as regards the forces under consideration, even though not in equilibrium as regards other forces. Thus, a stone, held between the thumb and finger, is in equilibrium as regards their two equal pressures, even though it may be lifted upward by the excess of the muscular force of the arm over the attraction between the earth and the stone. Simi- larly, on a level railroad, a car is in equilibrium as regards gravity and the upward resistance of the rails, although the horizontal pull of the locomotive may exceed the resistance to traction. 5. Molecular Action. Any force, applied to a body, is in fact made up of a system of forces, often parallel or nearly so, applied to the several particles of the body. Thus, the attraction exerted by the earth upon a grain of sand or upon the moon is, strictly speaking, a cluster of nearly par- allel forces exerted upon the several particles of those bodies ; but, for con- venience, and so far only as concerns their tendency to move the body as a whole, we conceive of such forces as replaced by a single force, equal to their sum and acting in one line. In thus considering the forces, we as- sume that the bodies are absolutely rigid, so that each of them acts as a single " material particle" or ** material point.'* 6. Transmission of Force. Theupwardpressureof the ground, upon a stone resting upon it, acts directly only upon those particles which are nearest to the ground. These, in turn, exert a (practically) equal upward force upon those immediately above them, and so on ; and the force is thus transmitted throughout the stone. 7. Rigid Bodies. In treating of bodies as- rigid, we assume that the intermolecular forces hold the several particles absolutely in their original relative positions. It is not the material that resists being broken, but the forces which hold its particles in their places. Thus, a cake of ice may sustain a great pressure ; but its particles yield readily when its cohesive forces are destroyed by a melting temperature. 8. Force Units, The force units generally used in statics are those of weight, as the pound and the kilogram. See Conversion Tables, p. 235. In statics we have no occasion to consider the masses of bodies (except * Strictly speaking, absolute rest is scarcely conceivable, since all bodies are actually in motion (see Art. 3, p. 331), so that unbalanced forces produce merely changes in the states of motion oi bodies. Yet, for a body to be at rest, relative to other bodies, is a very common condition, and, in practical statics, we usually regard the body under consideration as being at rest relatively to the earth or to some other large body, so that the change of state of motion, due to the action of unbalanced force upon it, consists in a change from relative rest to relative motion. See ^ 33, below. FORCES. 359 in so far as these determine their weights, or the force of gravity exerted upon them), bodies being regarded merely as the media upon and through which the forces under consideration are exerted. Hence we require, in statics, no units of mass; and, as the bodies are regarded as being "at rest," no units of time, velocity, acceleration, momentum, or energy. 9. Forces, how Determined. A force is fully determined when we know (1) its amount (as in pounds, or in some other weight unit), (2) its direction, (3) its sense (see ^ 10), and (4) its position or its point of applica- tion. 10. When a force is represented by a line, the length of the line may be made to represent by scale the amount of the force, and its direction and position may often be made to indicate those of the force, while the sense of the force may be shown by arrows or letters affixed to the lines, or by the signs, + and — . Thus, the directions of the forces represented by lines a and h, Fig. 1, are vertical, and those of c and d are horizontal. The sense of a is upward, of h downward, of c right-handed, of d left-handed. Thus, a and h are of like direction, but of opposite sense ; and so with c and d. In treating of vertical or horizontal forces, we usually call upward or right-handed forces posi- tive, and downward or left-handed forces negative, as indicated by the signs, + and — , in Fig. 1, When a force is designated by two letters, at- tached to the line representing it, one at each end of the line, the sense of the force may be indicated by the order in which the letters are taken. Thus, in Fig. 1, having regard to the directions of the arrows, we have forces, e f, hg, k I, and n m. 11. Line of Action, etc. The point (see ^ 5) at which a force P, Fig. 2, is supposed to be applied, as a, is called its point of application. But the force is transmitted, by the particles, throughout the body (see H 6), and 'A :i '-' fa the effect of the force, as regards the body as a whole, is not changed if it be regarded as acting at any other point, as b, in its line of action. We may therefore regard any point in that line as a point of application of the force. For instance, the tendency to move the stone, Fig. 2, as a whole, will not be changed if, instead of pushing it, at a, we apply a pull (in the same direction and in the same sense) at b; and if a weight, P, be laid upon the top of the hook, at b, Fig. 3, it will have the same tendency, to move the hook as a whole, as it has when suspended from the hook as in the Fig. A force cannot actually be applied to a body at a point outside of the sub- stance of the body, as between the upper and lower portions of the hook in Fig. 3, yet this portion also of the line a 6 is a part of the line of action of the force. The vertical force, exerted by the weight, P, is transmitted to b by means of bending moments in the bent portion of the hook. 12. Stress. (See Art. 1, Strength of Materials, p. 454.) Opposing forces, applied to a body by contact (see Art. 5 c, p. 332), cause stress, or the exertion of intermolecular force, within it, or between its particles, tending to pull them apart (tension) or to press them closer together (compression). The stress, due to two equal opposing forces, is equal to one of them. Tension and Compression. Ties, Struts, etc. If the action of the forces tends to pull farther apart the particles of the body upon which they act, the stress is called a tension or pull, or a tensile stress. If it tends to press them closer together, the stress is called a pressure, com- pression or push, or a compressive stress. A long slender piece sustaining tension is called a tie. One sustaining compression is called a strut or post. One capable of sustaining either tension or compression is called a tie-strut or strut-tie. 360 STATICS. MOMENTS. 13. Moments. If, from any point, o, or o', Fig. 4, a line, o c or o' s, be drawn normally to the line of action, n m, of a force, Pi, whether the point, o or o', be within or outside of the body upon which the force. Pi, is acting, said line, o c or o' s, is called the arm or leverage of the force about such point; and if the amount of the force, in lbs., etc., be multiplied by the length of the arm, in ft., etc., then the product, in ft.-lbs., etc., is called the moment of the force about that point.* The moment represents the total tendency of the force to produce rotation about the given point. A force has evidently no moment about any point in its line of action. 14. Sense of Moments. Since the moment of Pi about o. Fig. 4, tends to cause rotation (about that point) in the direction of the motion of the hands of a clock, as we look at the clock and at the figure, or from left to right, as indicated by the arrow on the circle around o, it is called a clock- wise or right-hand moment; but the moment of the same force about o' tends to produce rotation from right to left. Hence it is called a counter- clockwise or left-hand moment, as is also that of P2 about o. Right- hand or clockwise moments are conventionally considered as positive, or +, and left-hand or counter-clockwise moments as negative, or — . 15.^ The plane of a moment is that plane in which lie both the line of action and the arm of the force. 16. The resultant or combined tendency of two or more moments in the same plane is equal to the algebraic sum of the several moments. Thus, Fig. 4, if the forces. Pi, P2, and Pg, are respectively 6, 5, and 3 lbs., and if the arms, o c, oy, and o e, of their moments about o are respectively 7, 6, and 8 ft., we have Pi . o c — P2 . o 2/ -f P3 . o e = 6X7— 5X 6 + 3X3 = 42 — 30 + 9 =21 ft.-lbs. k — -m ■» I K— w— ^ «* w ? Fig. 4. Tig. 6. 17. If the algebraic sum of the moments is zero, they are in equilibrium and tend to cause no rotation of the body about the given point. Thus, in Fig. 5, where W is the weight, and G the center of gravity of the body, and R the upward reaction of the left support, a, taking moments about the right support, b, we have R Z — W x = zero; or R Z = W x. Hence, W X having W, x and I, to find R, we have R = — , . Similarly, in Fig. 6, where W = weight of beam alone, and g, the center of gravity of W, is at the center of the span I, so that the leverage b g of the weight of the beam about 6, is = -^f we take moments about b, thus: R I + O o R = W^— M m — M m + N n + W • Oo I ♦Note that a very small force may have a great moment about a point, while a much greater force, passing nearer to the same point, may have a smaller moment about it ; or, passing through the point, no moment at all. MOMENTS. 361 In Fig. 7, where W is the weight of the beam itself, and w its leverage, tak- ing moments about b, we have + RZ + Oo — Nn — Ww; + Mm = 0; Hence, Reaction at a = R = Ww + Nn — -Mm — Oo I ^ In any case, if W be the combined weight and G the common center of gravity, of the beam and its several loads, and x the horizontal distance of that center from the right support, b ; and if I be the span, R the reaction of the left support, a, and R' that of the right support, b, we have R = Ifx = Ras = - Wrc , I ' R'. and R' = W R. Fig. 7. Note that the moments of two or more forces, about a given point, may be in equilibrium, while the forces themselves are not in equilibrium. See ^ 84, below. 18. Center of Moments. So far as concerns equilibrium of moments, it is immaterial what point is selected as a center of moments ; but it is gen- erally convenient to take the center of moments in the line of action of one (or more, if there be concurrent forces, see ^ 19) of the unknown forces, for we thus eliminate that force or those forces from the equation. CLASSIFICATION OF FORCES. 19. Classification of Forces. Concurrent, Colinear, Coplanar, and Parallel Forces. Forces are called concurrent when their lines of Fig. 8. Tig. 9. action meet at one point, as a, b, e, d, e and /, or / and g, Fig. 8 ; non-concur- rent when they do not so meet, as c and g ; colinear when their lines of action coincide, as a and b, or c and d; non-colinear when they do not coincide, as b and /; coplanar when their lines of action lie in one plane,* as a, b, c, d and c, or 6, / and g, etc. ; non-coplanar, as c and g, or 6, / and d, when they do not lie in one plane; parallel when their lines of action are parallel, as b and g\ non-parallel when those lines are not parallel, as b and /. ♦Acting upon a plane, as in Fig. 9, must not be confounded with acting in that plane, as in Figs. 70, etc. 362 STATICS. Any two parallel forces must be coplanar. Three or more parallel forces may or may not be coplanar. Any two concurrent forces must be coplanar. Three or more concurrent forces may or may not be coplanar. Any two coplanar forces must be either parallel or concurrent. COMPOSITION AND RESOLUTION OF FORCES. ^ 30. Resultant. A single force, which can produce, upon a body con- sidered as a whole, the same effect as two or more given forces combined, is called the resultant of those forces. Thus, in Fig. 10 (6), a downward pres- sure, G, = w + W, is the resultant of the downward pressures w and W; and, in Fig. 11 (6), a downward pressure, = W — w, is the resultant of the downward pressure W and the upward pull w of the left-hand string.* 31. Component. Any two or more forces which, together, produce, upon a body considered as a whole, the same effect as one given force, are called the components of that force, which thus becomes their resultant. Thus, in Fig. 10 (b), w and W are the components of the total force, G, = w + W. In Fig. 11 (6), + W (= 5) and ty (== — 3) are the components of G.* 33. If we take into account the resultant of any given forces, those forces (components) themselves must of course be left out of account, as regards their action upon the body as a whole; although we may still have to con- sider their effect upon its particles. Vice versa, if the forces (components) are considered, their resultant must be neglected. (a) Fig. 10. =3 >J=3 (C) Fig. 11. 33. Anti-resultant. The anti-resultant of one or more forces is a single force which, acting upon any body or system of bodies considered as a whole, produces an effect equal, but opposite, to that of their resultant. In other words, the anti-resultant is the force required to hold the given force or forces in equilibrium. Thus, in Fig. 10 (5), the upward reaction, G, of the ground, is the anti-resultant of the two downward forces, w and W ; and the downward resultant, W -j- w, oi W and w, is the anti-resultant of G. In Fig. 11 (6), G (upward) is the anti-resultant of W (downward) and w (acting upward through the left-hand string). Similarly, this upward pull of w is the anti-resultant of W and G. 34. In any system of balanced forces (forces in equilibrium), any one of the forces is the anti-resultant of all the rest ; and any two or more of them have, for their resultant, the anti-resultant of all the rest. In such a system, the resultant (and the anti-resultant) of all the (balanced) forces is zero. 35. Anti-component. The anti-components of a given force, or of a given system of forces, are any two or more forces whose resultant is the anti- resultant of the given force or of the given system of forces. 36. Composition and Resolution of Forces. The operation of finding the resultant of any given system of forces is called the composition of forces ; while that of finding any desired components of a given force is called the resolution of the force. * For convenience, we here reverse the convention of H 10. COLINEAR FORCES. 863 Colinear Forces. 27. Let the vertical line, w. Fig, 10 (6), represent, by any convenient scale, the weight of the upper stone in Fig. 10 (a), and W that of the lower stone. Then, w + W, = G, = the combined length of the two lines, gives, by the same scale, the combined weight of the two stones, and a vertical line G, coincident with them, equal to their sum, and pointing upward, would represent their anti-resultant, or the reaction of the ground. (a) (b) (€) c \ \ V / / a \ \ A / / & J m m Z=l€ Zive 10 10 10 Dead_2^ _2 2 Total 12 12 12 10 10 10 _2 2 _ 12 12 12 2 n= X+2> >--12 n==s6< 12 ^24 Tig. 12. 28. Similarly, if, at each panel point of the lower chord in the bridge truss in Fig. 12 (o), we have 2 tons dead load (weight of bridge and floor, etc,*) and 10 tons live load (train, vehicles, cattle, passengers, etc.), the com- bined length of the two lines in Fig. 12 (6), L = 10, and D = 2, gives the total panel load of 12 tons. 29. In Fig. 11 the pressure, 5 lbs., of W upon the ground, is diminished by the 3 lbs. upward pull of the cord, transmitted from the smaller weight to, leaving 2 lbs. upward pressure to be exerted by the ground in order to main- tain equilibrium. The upward reaction, R, of the pulley is = ly + W — G = 3+5 — 2 =«6. This is represented graphically in Fig. 11 (c). 30. In the truss shown in Fig. 12 (o), the total dead and live load is =• 6 X 12 = 72 tons, and half this total load, or 36 tons, rests upon each abut- ment. Hence, to preserve equilibrium, each abutment must exert an up- ward reaction of 36 tons; but, in order to ascertain how much of these 36 tons is transmitted through the end-post, a c, we must deduct from it the 12 tons which we assume to be originally concentrated, as dead and live load, at the panel point a; for this portion is evidently not transmitted through a c. Accordingly, in Fig. 12 (c), we draw R upward, and equal by scale to 36 tons; and, from its upper end, draw p downward and = 12 tons. The - remainder of R, «= R — p == 36 — 12 = 24 tons, is then the pressure trans- mitted through a c. 31. Colinear forces are called similar when they are of like sense, and opposite when of opposite sense. The same distinction applies to result- ants. -^-h Fig. 13. d 32. For equilibrium, under the action of colinear forces, it is, of course, necessary that the sum of the forces acting in one sense be equal to the sum of those acting in the opposite sense, or, in other words, that the algebraic sum of all the forces be zero. Thus, in Fig. 13, if the forces are in equilibrium, the sum, 6 a + a o, of the two right-handed forces must be equal to the sum, ed-^dc + co, oi the three left-handed forces. Or, con- sidering the right-handed forces, b a and a o, as positive, and the left-handed forces, e d, d c and c o, as negative, as in ^ 10, we have, as the condition of equilibrium of colinear forces: ha-\-ao — oc — cd — de — 0. * The dead load is, of course, never actually concentrated upon one chord, as here indicated; but it is often assumed, for convenience, that it is so concentrated. 364 STATICS. In other words, the algebraic sum of all the forces must be zero ; or, more briefly, 2 forces = 0, where the Greek letter S (sigma), or sign of summation, is to be read "The sum of — ." 33. Two equal and opposite forces, acting upon a body, are com- monly said to keep it at rest ; but, strictly speaking, they merely prevent each other from moving the body, and thus permit it to remain at rest, so far as they are concerned ; for they cannot keep it at rest against the action of any third force, however slight and in whatever direction it may act; and the body itself has no tendency to move. 34. Unequal Opposite Forces. If two opposite forces, acting upon a body, are unequal, the smaller one, and an equal portion of the greater one, act against each other, producing no effect ujjon the body as a whole ; while the remainder, the resultant, moves the body in its own direction. Concurrent Coplanar Forces. The Force Parallelogram. 35. Composition. Let the two lines, a o, h o, in any of the diagrams of Fig. 14, represent, in magnitude, direction and sense, concurrent forces whose lines of action meet at the point o. Then, in the parallelogram, acbo, formed upon the lines a o, b o, the resultant of those two forces is repre- sented, in magnitude and in direction, by that diagonal, R, which passes through the point, o, ex concurrence. The parallelogram, a c b o, is called a force parallelogram. \«' ""W \^ ^ \\ c< \ V V \b / '.&' Fig. 14. 36. Resolution. Conversely, to find the components of a given force, o c, Fig. 14, when it is resolved in any two given directions, o a, ob, draw the lines, o a\ o b', in those directions and of indefinite length, and upon these lines, with the diagonal R = o c, construct the force parallelogram a c b o. The sides, o a, o b, of the parallelogram then represent the required compo- nents in amount and in direction. ■ 37. Caution. The two forces, a o and b o, Fig. 14, may act either toward or from the point o; or, in other words, they may act either as pulls or as pushes ; but the lines representing them in the parallelogram, and meeting at the point, o, must be drawn, either both as pushes or both as pulls ; and the resultant, R, as represented by the diagonal of the parallelogram, will be a pull or a push, according as the two forces are represented as pulls or as pushes. 38. Thus, in Fig. 15 (a), the inclined end-post of the truss pushes obliquely downward toward o, with a force represented by a' o, while the lower chord pulls away from o, toward the right, with a force represented by o b'. If, now, we were to construct, in Fig. 15 (a), the parallelogram o a' c' b\ we should obtain the diagonal o c' or c' o, which does not represent the true re- sultant. In fact, as one of the two forces acts toward, and the other from, the point, o, we could not tell (even if R' were the direction of the resultant) in which sense its arrow should point. We must first either suppose the push, a' o, in the end-post, toward o, to be carried on beyond o, so as to act as a pull, o a, Fig. 15 (6) (of course, in the same direction and sense a& before), thus treating both forces as pulls; or FORCE PARALLELOGRAM. 365 else we must similarly suppose the pull, o h', in the chord, to be transformed into the push, h o, of Fig. 15 (c), thus treating both forces as pushes. In either case we obtain the true resultant, R (= a' b', Fig. 15 a), which, in this case, represents the vertical downward pressure of the end of the truss upon the abutment. Figr. 15. Caution. The tensile force, exerted at the end of a flexible tie, neces- sarily acts in the line of the tie; but, in general, the pressure, exerted at the end of a strut, acts in the line of the axis of the strut only when all the forces producing it are applied at the other end of the strut. Thus, in Fig. 15 (d), the components, R and H, of the weight, W, do not coin- cide with the axis of the beam which supports the load; but in Fig. 15 (e), where the weight acts at the intersection of the two struts, its com- ponents, R and H, do coincide with the axes of the struts. See also Figs. 143 and 145 (6). 39. Demonstration. The rational demonstration of the principle of the force parallelogram is given in treatises on Mechanics. (See Bibliog- raphy.) It may be established experimentally as indicated in Fig. 16, where c o represents by scale the pull shown by the spring balance C, while » a and o b represent those shown by A and B respectively. Tig, 16. 40. Equations for Components and Resultant. Given the amounts of the forces, a and c, or of the resultant, R, and the angles formed between them, Fig. 17 (a), we have*: * See dotted lines, Fig. 17 (a), noting that c' X, and a. sin (x + y) = R. sin y. c; c. sin (x -\- y) = R. sin 366 STATICS. .^ sin (x K = c ^ c = R y) sin X sin X sin (X -\- y)' a = R , sin (3; + y) sin y sin y sin (x +3/)* If the angle between the two forces is 90°, Fig. 17 (6), these formulas be- come: R = ^ = "^ cos y cos X c = R cos y, a = R cos x. Fig. 17. 41. Position and Sense of Resultant. Figs. 18. If the lines representing the components be drawn in accordance with 1[1[ 37 and 38, and if a straight line, mnor m' n\ be drawn through the point, o, of concur- rence, in such a way that both forces are on one side of that line, then the line representing the resultant will be found upon the same side of that line with the components, and between them; and it will act toward the line, mnor m' n', if the components act toward it, and vice versa. The resultant is necessarily in the same plane with its two components. Fig. 18. Figr. 19. 42. If one of the components is colinear with the force, it is the force itself, and the other component is zero. In other words, a force cannot be resolved into two non-colinear components, one of which is in the line of action of the force. Thus the rope, o c, Fig. 19, may receive assistance from two ad- ditional ropes, pulling in the directions a c, and c 6; for the resultant of their pulls may coincide with o c; but, so long as o c remains vertical, no single force, as c a or c 6, can relieve it, unless acting in its own direction c o. 43. In Fig. 20, the load, P, placed at C, is suspended entirely by the verti- cal member B C, and exerts directly no pull along the horizontal member, C E. Neither does a pull in the latter exert any effect upon the force acting in B C, so long as B C remains vertical. But the tension in B C, acting at B, does exert a thrust o a along B D, although that member is at right angles to B C ; for B C meets there also the inclined member A B ; and the tension o d is thus resolved into o a and o b, along B D and B A respectively. The horizontal thrust, o a, in B D, is really the anti-resultant of the horizontal component, d b, of the oblique thrust in the end-post B A, at its head, B, which thrust is==the pull in A E, due to P. FORCE TRIANGLE. 367 44. In Fig. 21, the tension, o c, in the inclined tie, D G, is resolved, at D, into o a and o b, acting at right angles to each other along D F and D E re- spectively. 45. A resultant may be either greater or less than either one of its two oblique components, but it is always less than their sum. If the components are equal, and if the angle between them = 120°, the resultant is equal to one of them. Therefore the same weight which would break a single vertical rope or post, would break two such ropes or posts, each inclined 60° to the vertical. Fig. 20. The Force Triangle. 46. The Force Triangle. Inasmuch as the two triangles, into which a parallelogram is divided by its diagonal, are similar and equal, it is suffi- cient to draw either one of these triangles, a o c or b o c, Figs. 14, 16, 18, in- stead of the entire parallelogram. 47. If three concurrent coplanar forces are in equilibrium, the lines rep- resenting them form a triangle; and the arrows, indicating their senses, follow each other around the triangle. Thus, in Fig. 22 (a), we have, acting at o and balancing each other there, three forces: viz., (1) the vertical down- ward force o c of the weight, acting as a pull through the rope o c, (2) the horizontal thrust a o through the beam a o, and (3) the upward inclined thrust 6 o of the strut o b, all acting in the senses (o c, a o,b o) in which the letters are taken, and as indicated by the arrows. 48. ICach of the forces in Fig. 22 (Jb) and (c) is the anti-resultant of the other two in the same triangle; and, if its sense be reversed, it becomes their resultant. Thus, o c, Fig. 22 (6), is the anti-resultant, and c o the resultant, of c a and a o; and o c. Fig. 22 (c), is the anti-resultant, and c o the resultant of c 6 and b o, c b being parallel to a o, Fig (6), and representing the thrust exerted by the horizontal beam against the joint o, Fig. (a).* (b) (c) Tig. 22. *Fig. 22 (d) and (e), representing the same two forces, a o, b o, of Fig. 22 (a), show the erroneous resultant (a b) obtained if the lines are drawn with their arrows pointing both toward or both from the meeting-point of the lines. See 1^ 37, 38. A comparison of any force parallelogram, as that in Fig. 18, with either of the two force triangles composing it, will show that this, while apparently contradicting Ifl 37 and 38, is merely another statement of the same fact. The apparent contradiction is due to the fact that, in the force triangle, the lines representing the forces do not meet at the point, o, of concurrence of the forces. 368 STATICS. 49. Conversely, if the thtee sides of a triangle be taken as representing, in direction and in amount, three concurrent forces whose senses are such that arrows, representing them and affixed to their respective sides in the triangle, follow each other around it, then those forces are in equilibrium. 50. The three forces, Fig. 23, are proportional, respectively, to the sines of their opposite angles. Thus : Force a : force h : force c = Sin A : sin B : sin C. Fig. 23. 51. Example. In Fig. 24, the half arch and its spandrel, acting as a single rigid body, are assumed to be held in equilibrium by their combined weight, W, the horizontal pressure h at the crown, and the reaction R of the skewback, which is assumed to act through the center of the skewback. In the force triangle c s t, c s, acting through the center of gravity of the half arch and spandrel, represents the known weight W, and s t is drawn hori- zontal, or parallel to h . From c, where h, produced, meets the line of ac- tion of W, draw c t through the center of the skewback. Then a t and c t give "lis the amounts of h and R respectively. Fig. 24. Fig. 25. 52. Example. Let Fig. 25 represent a roof truss, resting upon its abut- ments and carrying three loads, as shown by the arrows. Draw a R ver- tically, to represent the proportion of the loads carried by the left abut- ment, a, or, which is the same thing, the vertical upward reaction of that abutment. Then, drawing R c, parallel to the chord member, a d, to inter- sect a e in c, we have, for the stresses in a e and a d, due to the three loads: Stress in a e = a c *' ** o d = R c Fig. 26. 53. While any two or more given forces, as o 6 and h c, Fig. 26 (a) (arrows reversed), or o h' and h' c, or o a and a c, or o a' and a' c, can have but one re- sultant o c ; a single force, as o c, may be resolved into two or more concur- rent components in any desired directions. In other words, there is an infinite number of possible systems of concurrent forces which have o c for their resultant. EECTANGULAE COMPONENTS. 369 Rectangular Componentf. 54. Resolutes, or Rectangular Components. A very common case of resolution of forces is that where a force, as the pressure, c n, of the post, Fig. 27, is to be resolved into components at right angles to each other, as are the vertical and horizontal components c t and t n in Fig. 27 (a). Two such components, taken together, are called the resolutes or rectangular compo- nents of the force. The joint, o d, in Fig. 27 (a), is properly placed at right angles to c n; but the joint ci b, Fig. 27 (6), provides also against accidental changes in the direction of c n. In Fig. 27 (6), the surfaces, c i and i b, are preferably proportioned as the components, c t and t n. Fig. 27 (a), respec- tively, by similarity of triangles, cib, ctn. Fig. 27. Fig. 28. 55. Example. In bridge and roof trusses it is often required to find the vertical and horizontal resolutes of the stress in an inclined member, or to find the stress brought upon an inclined member by a given vertical or hori- zontal stress applied at one of its ends, in conjunction with another stress (whose amount may or may not be given) at right angles to it. Thus, in Fig. 28, the tension C p in the diagonal C rf is resolved into a com- pression e p along the upper chord member C D* and a compression C e in the post C c* Adding to C e the load at c, and representing their sum by / c, we have tension f g in chord member c d, and tension c g in the diagonal B c. Making B h = c ^,we have j h, compression in B C, and B ;, compression in the end-post or batter post B A. But the load at b also sends to B, through the hip vertical B 6, a load (tension) equal to itself. Representing this by B k, we have I k as its component along the chord member B C, and B i as its component along the end-post B A. Now, making A m = the sum of B; and B I, we find the vertical resolute A n = so much of the vertical reaction of the abutment as is due to the three loads only, and the horizontal resolute m n = the corresponding stress in the chord member, A c. Fig. 29. Fig. 30. 56. Example. Inclined Plane. Again, in Fig. 29, let it be required to find the two resolutes of P (the weight of the ball) respectively parallel and perpendicular to the inclined plane. The former is the tendency of the ball to move down the plane, and is called the tangential component. The *The stress thus found is not necessarily the total stress in the member. The compression in C c (neglecting its own weight and that of the top chord) is due entirely to the tension C p in C d, acting at its top, and hence C e rep- resents the total compression in C c; but e p is only a portion of the com- pression sustained by C D ; for B C also contributes its share toward this. 24 370 STATICS. latter is the pressure of the ball against the plane, and is called the normal component. Here we have only to draw the triangle of forces o a c,* drawing o c = P to represent the weight of the ball, and o a and a c in the required directions. Then o a and a c give respectively the normal and the tangential components of the force, P.f 57. If we suppose the inclined plane g m, Fig. 29, to be frictionless, and if the body o is to be prevented from sliding down the plane, by means of a force applied in a direction parallel to the plane, that force must be = c a. Thus, in Fig. 30, supposing the plane o m to be frictionless, we have a c = pressure against the stop, s. 58. Table of normal and tangential components for different angles of inclination: Pres on Tendency Inclination or Slope of the Plane. vertical height The sloping length is _ ^.^^^ ^^^ ^ Plane, in parts of the wt. Or, nat. 00a. of angle of Plane. Pres. on Plane, in fits per ton. down the Plane, in parts of the wt. Or. nat. sine of angle of Plane. Tendency down the Plane, in lbs per ton. Tert. Hor. Ft. per mile. Deg. Min. 1 in 3. 1760.00 18 26 .9487 2125 .3162 708. 1 in 4. 1320.00 14 2 .9702 2173 .2425 543. 1 in 5. 1056.00 11 19 .9806 2196 .1962 439. 1 in 6. 880.00 9 28 .9864 2210 .1645 368. 1 in 8. 660.00 7 8 .9923 2223 .1242 278. 1 in 9. 588.66 6 20 .9939 2226 .1103 247. 1 in 10. 528.00 5 43 .9950 2229 .0996 223. 1 in 11.4 461.94 5 00 .9962 2231 .0872 195. 1 in 12. 440.00 4 46 .9965 2232 .0831 186. 1 in 14.8 369.23 4 00 .9976 2232 .0698 156. 1 in 15. 352.00 3 49 .9978 2233 .0666 149. 1 in 19.1 276.73 3 00 .9986 2237 .0523 117. 1 in 20. 264.00 2 52 .9987 .0500 112. 1 in 23.1 229.04 2 30 .9990 »' .0436 97.7 1 in 25. 211.20 2 17 .9992 2238 .0398 89.2 1 in 28.6 184.36 2 00 .9994 .0349 78.2 1 in 30. 176.00 1 55 " .0334 74.8 1 in 32.7 161.47 1 45 .9995 2239 .0305 68.4 1 in 35. 150.86 1 38 .9996 '< .0285 63.8 1 in 38.2 138.22 1 30 .9997 2240 .0262 58.6 1 in 40. 132.00 1 26 " '< .0250 56.0 1 in 45.8 115.29 1 15 «< »« .0218 48.8 1 in 50. 105.60 1 9 .9998 '< .0201 45.0 1 in 57.3 92.16 1 « .0175 39.1 1 in 60. 88.00 571^ .9999 «« .0167 37.4 1 in 70. 75.43 49 " »< .0143 32.0 1 in 76.4 69.12 45 '< «< .0131 29.3 1 in 80. 66.00 43 " " .0125 28.0 1 in 90. 58.67 38 " <( .0111 24.9 1 in 100. 52.80 34 1.0000 .0100 22.4 1 in 114.6 46.07 30 " .0087 19.6 1 in 125. 42.24 27^ " «* .0080 17.9 1 in 150. 35.20 23 »< " .0067 15.0 1 in 175. 30.17 19% " «' .0057 12.8 1 in 200. 26.40 17 " «« .0050 11.2 1 in 229.2 23.04 15 '< " .0044 9.77 1 in 250. 21.12 14 «' «« .0041 9.18 1 in 300. 17.60 UH <« «« .0033 7.39 I in 343.9 15.35 10 « «« .0029 6.52 1 in 400. 13.20 s% << • « .0025 5.60 1 in 500. 10.56 7 <« «< .0020 4.48 1 in 600. 8.80 6 << <« .0017 3.81 1 in 800. 6.60 4% " " .0013 2.91 1 in 1000. 5.28 SH " •• .0010 2.24 1 in 3437. 1.54 1 . .0003 0.66 Level. 0.00 " " .0000 0.00 * Or o 6 c. If both triangles are drawn, we have the force parallelogram, o a c b. fThe line a c (or c a) is called the projection of o c upon the inclined plane; and o a (or a o) is the projection of o c upon a normal to the inclined plane. STRESS COMPONENTS. 371 59. Equations. In Fig. 29, o a = P . cos c o a a c = P . sin c o a and, since the angle c o a between the vertical o c and the normal component o a is equal to the angle A of inclination between the plane g m and the hori- zontal g n, we have : Normal component, o a = P . cos A. Tangential compo^ient, a c = P . sin A. 60. When a force is resolved into rectangular components, as in Figs. 29 and 30, each of these components represents the total effort or tendency which that force alone can exert in that direction. Tig. 31. Thus, in Fig. 31, the utmost force which the weight a c alone can exert perpendicularly against the plane is that represented by the component o a. True, if, in order to prevent the body from sliding down the plane, we apply a force in some other direction, such as the horizontal one, h o, instead of the tangential one 5 o, and find the components of o c in the directions h o and o a, we shall find the normal component o d greater than before ; but the increase a d is due entirely to the normal component, h b, of the horizontal force h o. Thus, the only effect upon the body o, and upon the plane, of substituting h o for b o, is to add the normal component, h b, of the former, to that (o a) of o c. Stress Components. 61. Stress Components. In Fig. 32, let o o and 6 o be any two forces, and c o their resultant. From a and b draw a a' and b b' at right angles to the diagonal o c of the force parallelogram a o b c, and construct the sub- parallelograms (rectangles), o a' a a'' and ob' b b" . Each of the original com- ponents, o a, o b, is thus resolved into two sub-components, perpendicular to each other, one of which is perpendicular also to the resultant, o c, while the other coincides with o c in position and in sense. Now, perpendiculars, let fall from the opposite angles of a parallelogram upon its diagonal, are equal. fc --^ ^ a 1 / > h'/ / Figr. 32. Hence the two colinear forces, o a" and o b", acting upon the body at o, are equal and opposite (although the lines, a' a and b' b, representing them, are not opposite). Hence also they are in equilibrium, and their only effect upon the body is a stress of compression in Fig. 32 (a), and of tension in Fig. 32 (6). They may therefore be called the stress coniponents. The other two sub-components (o a' of o a, and o 6' of o b) combine to form the resultant o c, which is equal to their sum, and which tends to move the body o in its own direction. 372 STATICS. 62. The two great forces, o a, oh, in Fig. 33 {h) have the same resultant, oc, = oc', as the two small forces, o a' oh', in Fig. 33 (a), although their stress components, a" a, = h" h, are much greater. 63. It often happens that one of the components is itself normal to the resultant. Thus, in Fig. 22, where o c is vertical, its component, o a, is hori- zontal, and the perpendicular, let fall from a upon o c, represents its hori- zontal anti-component, a o. Here the horizontal and the inclined l5eam sustain equal horizontal pressures; but the vertical pressure, o c, = the weight, W, is borne entirely by the inclined beam. ^^~--^_£ ■ — -— ^._^ b Tig. 33. Fig. 34. 64. When, as in Fig. 34, the resultant, o c, forms, with one of the original components, o a and o h, an angle, a o c, greater than 90°, the perpendiculars, a a', h b', from a and b, must be let fall upon the line of the resultant produced. Here, however, as before, the two equal and opposite sub-components, o a" and o h", are in equilibrium at o, while the other two sub-components, o h' and o a\ go to make up the resultant o c; which, however (since a h' and o a' here act in opposite senses) is equal to their difference, and not to their sum, as in Fig. 32. Fig. 34 shows that a downward force, o c, may be so resolved that one of its components is an upward f<^e, o a, greater than the original downward force, and that the pressure, o b, nas a component, o b' or V h, parallel to o c, and greater than o c itself; for b" h = a b' = o c -\- c h'. Applied and Imparted Forces. 65. Applied and Imparted Forces. In Fig. 29, the ball is free to roll down the inclined plane. Hence, although the entire weight P of the ball is applied to the body g m n, only the normal component o a is imparted to it or exerts any pressure upon it, and this pressure is in the direction o a. But in Fig. 30, the body g m n receives and resists not only the normal component o a, but also (by means of the stop s) the tangential component o h; and the entire force P, or o c, is thus imparted to the body g m n, press- ing it in the direction o c. Composition and Resolution of Concurrent Forces by Means of Co-ordinates. 66. In Fig. 35 (a) let the three coplanar forces E, F and G act through the point x. Draw two lines, H H, and V V, Fig. 35 (6), crossing each other at right angles, as at o* These lines are called rectangular co-ordin- ates. From o, draw lines E o, F o, G o, parallel to E x, F x, Gx, Fig. 35 (a), and equal respectively to the forces E, F, and G by any convenient scale. Re- solve each of these forces, Fig. 35 (6), into two components, parallel to H H and V V respectively. Thus, E o is resolved into t o and n o, F o into u o and e o, G o into i o and m o. Then, summing up the resolutes, we have: Sum of horizontal resolutes = u o — io — to = — so, and Sum of vertical resolutes = n o -\- e o — m o = a o; *It is only for convenience that the co-ordinates are usually drawn (as in Fig. 35) at right angles. They may be drawn at any other angle (see Fig. 36) ; but, in any case, the forces must of course be resolved into components 'parallel to the co-ordinates, whatev^ the directions of those co-ordinates may COMPOSITION AND RESOLUTION. 373 and — 8 and a o are the resolutes of the resultant, R, of the three forces, E, F and G. 67. When a system of (concurrent) forces is in equilibrium, the algebraic* sum of the components of all the forces, along either of the two co-ordinates, is zero. Thus, in Fig. 35 (6) or 36, if the sense of R be such that it shall act as the anti-reeultant of the other three forces E, F and G, its component, o s or o a, along either co-ordinate, will be found to balance those of the other forces along the same co-ordinate. /« F •^ w u o / i 8 t H^ 1 n — ^^^ (a) Fig. 35. (&) Hence we have the very important proposition that: When a system of concurrent coplanar forces is in equilibrium, the algebraic sums of their com- ponents, in any two directions, are each equal to zero. Figr. 36. 68. Conversely, in a system of concurrent forces, if the algebraic sums of the components in any two directions are each equal to zero, the forces are in equilibrium. If the sum of the components in one of any two directions is not equal to zero, the forces cannot be in equilibrium. Thus, in Fig. 35 (6) or 36 (6), the sum of the components, along either one (as VV) of the two co-ordinates, may be zero; and yet, if the sum of those along the other co-ordinate is not zero, their resultant, or algebraic sum, will move the body, on which they act, in the direction of that resultant. *The components being taken as + or — , according to the sense of each. 374 STATICS. 69. With vertical and horizontal co-ordinates, the condition of equilibrium* becomes: The sum of the horizontal resolutes must be equal to zero; The sum of the vertical resolutes must be equal to zero; or, more briefly: 2 horizontal resolutes = 2 vertical resolutes = Conversely, if these conditions are fulfilled, the forces are in equilibrium. Figr. 37. Fig. 38. Tig, 39. 70. Resultant of More than Two Coplanar Forces. Where it is required to find the resultant of more than two concurrent and coplanar forces, as in Fig. 37, we may first find the resultant Ri of any two of them, as of P] and P2 ; then the resultant, Rq, of Ri and a third force, as P3 ; and so on, until we finally obtain the resultant R of all the forces. This resultant is evidently concurrent and coplanar with the given forces. 71. It is quite imniaterial in what order the forces are taken. Thus, we may, as in Fig. 38, first combine P] and P3; then their resultant Ri with P2, obtaining R2 ; and, finally, R2 with P4, obtaining R ; or, as in Fig. 39, we may first combine any two of the forces, as Pi and P2, obtaining their resultant Ri ; then proceed to any other two forces, as P3 and P4, and obtain their resultant R2; and finally combine the two resultants, Ri and R2, ob- taining the resultant R. The Force Polygon. 72. The Force Polygon. Comparing Figs. 37 and 38 with Figs. 40 and 41, respectively, we see that we may arrive at the same resultant R by simply drawing, as in Fig. 41, lines representing the several forces in any order, but following each other according to their senses. It will be noticed that this is merely an abbreviation of the process of drawing the several force parallelograms. 73. Resultant and Anti-resultant. The line, — R, required to com- plete the polygon, represents the an^i-resultant of the other forces if its sense is such that it follows them around the polygon, as in Fig. 40. If its sense is opposed to theirs, as in Fig. 41, it is their resultant, R. 74. In other words, if any number of concurrent forces, as Pj, P2, P3, P4 and R, Figs. 37 and 38, f are in equilibrium, the lines representing them, if drawn in any order, but so that their senses follow each other, will form a closed polygon, as in Fig. 40 (or in Fig. 41 if the sense of R be reversed). 75. Conversely, if the lines representing any system of concurrent coplanar forces, when drawn with their senses following each other, form a closed polygon, as in Fig. 40, those forces are in equilibrium. *With non-concurrent forces, another condition must be satisfied. See ^ 83. fR is here regarded as tending upward, so as to form the an^t-resultant of the other forces. FORCE POLYGON. 375 It will be noticed that the force triangle, and the straight line representing a system of colinear forces, Figs. 10 and 11, If^H 20, etc., or a system of parallel forces, Figs. 55, etc., 1[^ 111, etc., are merely special cases of the force polygon. 76. In a force polygon. Fig. 42, any one of the forces is the anti-resultant of all the rest. Any two or more of the forces balance all the rest; or, their resultant is the anti»resultant of all the rest. If a line acorh d. Fig. 42, be drawn, connecting any two corners of a force polygon, that line represents the resultant, or the anti-resultant (according as its arrow is drawn) of all the forces on either side of it. Thus: a c is the resultant of Pi P2 and the anti-resultant of P3 P4 P5 ca - " " P3 P4 P5 " " " Pi P2 ^ 6 d " " •• P2 P3 " " " P4 P5 Pi d 6 " •• •' P4 P5 Pi " " " P2 P3 77. Knowing the directions of all the forces of a system, as Pi P5, Fig. 42, and the amounts of all hut two of them, as Po and P3, we may find the amounts of those two by first drawing the others, P4, P5 and Pi, as in the figure. Then two lines h c and c d, drawn in the directions of the other two and closing the polygon, will necessarily give their amounts. Fig. 43. Fig. 44. 78. If any two points, as o and c, Fig. 43, be taken, then the force or forces represented by any line or system of lines joining those two points will be equivalent to o c. Thus : oc = oabc = odc = onpc = o h k m c = o h m c = o f c = o g c, etc., etc. Similarly, in Fig. 42, the force polygon ah c d e ais equivalent to the force polygon ah f d e a, and to the force triangle, ah c a, each being = zero. Non-concurrent Coplanar Forces. 79. Non-concurrent Coplanar Forces. Fig. 44. The process of finding the resultant of three or more coplanar but non-concurrent forces is the same as if they were concurrent. Thus, let Pi, P2 and P3 represent three such forces.* We may first find the resultant Ri of any two of them, as P2 *Any two coplanar non-parallel forces, as Pi and P2, or P2 and P3 are necessarily concurrent (see *[f 19) ; but there is no single point in which the three forces meet. 376 STATICS. and P3; and then, by combining Ri with the remaining force Pi, we find the resultant R of the three forces. Here the line R represents the resultant, not only in amount and in direction, but also in position. That is, the line of action of the resultant coincides with R. 80. The resultant R is the same, in amount and in direction, as if the forces were concurrent, and its position is the same as it would have been if their point of concurrence were in the line of R. If there are more than three forces, we proceed in th^ same way. 81. Conversely, the resultant R, or any other force, may be resolved into a system of any number of concurrent or nonconcurrent coplanar forces, in any directions, at pleasure. Thus, we may first resolve R into Pi and Ri; then either of these into two other forces, as Ri into Pa and P3, and so on. 83. If a system of non-concurrent coplanar forces is in equilibrium, the forces will still be in equilibrium if they are so placed as to be concurrent; provided, of course, that their directions, senses and amounts remain un- . changed ; but it does not follow that a system of forces, which is in equilib- rium when concurrent, will remain in equilibrium when so placed as to be non-concurrent. Thus, the five forces, Pi P-„ Fig. 45 (a), may be so placed, as in Fig. 45 (6), that the resultant a c, of Pi and P2, does not coincide with the re- sultant c a of P3, P4 and P5, but is parallel to it. These two resultants then form a couple. (See 1[Tf 155, etc.) Fig. 45. 83. Third Condition of Equilibrium. Hence, the conditions of equilibrium for concurrent forces, stated in H 69, 2 vertical components = 2 horizontal components = do not suffice for non-iconcurrent forces, and a third condition must be added, viz.: — 2 moments = 0; i. e., the moments of the forces, taken about any point, must be in equilib- rium. A system of forces in equilibrium has no resultant ; hence it has no moment about any point. In other words, the moments of the forces, as well as the forces themselves, are in equilibrium. 84. The resultant of a system of unbalanced non-concurrent forces, acting upon a body, may be either (1) a single force, acting through the center of gravity of the body; or (2) a couple; i. e., two equal and parallel forces of opposite sense (see It 155, etc.) ; or (3) either (a) a single force, acting through the center of gravity of the body, and a couple ; or (b) a single force, acting elsewhere than through the center of gravity of the body. In Case (3), the two alternative resultants are interchangeable; i. e., a single force, acting elsewhere than through the center of gravity of the body, may always be replaced by an equivalent combination consisting of an equal CORD POLYGON. 377 parallel force, acting through the center of gravity of the body, and a couple, and vice versa. See 11[ 161, etc. The resultant gives to the body, in Case (1), motion of translation in a straight line, without rotation; in Case (2), rotation without translation; and in Case (3), both translation and rotation. See foot-note (*), ^ 1. 85. The force polygon, H 72, Figs. 40, etc., and the method by co- ordinates, Tf 66, Fig. 35, therefore, give us only the amount, direction and sense of the resultant of non-concurrent forces, and not its position. To find the position of the resultant of non-concurrent forces, we may have recourse to a figure, like Fig. 44, where the forces are represented in their actual posi- tions, or to the cord polygon, 1[^ 86, etc., Fig. 46. The Cord Polygon. 86. In the force triangle any two of the three lines may be regarded as representing, by their directions, the positions of two members (two struts or two ties, or one strut and one tie) of indefinite length, resisting the third force ; while their lengths give the amounts of the forces which those mem- bers must exert in order to maintain equilibrium. %4 XA4 V\ Fi^. 86 (repeated). 87. Thus, in Fig. 26 (6), are shown four different systems, of two mem- bers each, inclined respectively like the forces c b and 6 o in Fig. 26 (a) and balancing the third force o c. The stresses in these two members are given by the lengths of the lines c b and 6 o in Fig. 26 (o) . The members acting as struts are represented, in Fig. 26 (6), as abutting against flat surfaces, while, those acting as ties are represented as attached to hooks, against which they pull. In Fig. 26 (c) and (d) are indicated systems of members, inclined like the forces c a' and a' a, ca and a o, respectively, of Fig. 26 (a), by which the third force o c might be supported. 88. In the force polygon ahcd ea, Fig. 46 (6), representing the four forces, Pi, Ps, P3, P4, of Fig. 46 (a), if w-e select, at pleasure, any point o (called the pole) and draw from it a series of straight lines o a, o b, etc. (called rays), radiating to the ends, a, b, c, etc., of the lines P], P2, etc., representing the forces, we shall form a series of force triangles, a o b, b o c, etc. Thus, in the triangle ab o we have the force Pi, or a b, balanced by the two forces o a and b o; in the triangle b c o, the force Pg, or b c, balanced by the two forces o b and c o; and so on. 89. The Cord Polygon. If, now, in Fig. 46 (a), we draw the lines a and b, parallel respectively to the rays o a and o 6 of Fig. 46 (6) and meeting in the line representing the force Pi, they will represent the positions of two tension members of indefinite length, which will balance the force Pi by ex-, erting forces represented, in amount as well as in direction, by the rays o a and b o. Fig. 46 (6). Again, taking pole o^ Fig. 46 (b), instead of o, we have a' and b\ Fig. 46 (a'), parallel respectively to the rays, o' a and o' b, and rep- resenting a pair of struts performing the same duty. 90. Similarly, the lines b and c. Fig. 46 (a), parallel respectively to rays o b and c, represent two tension members, which, with stresses equal respec- tively to o 6 and c o, Fig. 46 (6), balance the force Pg. 378 STATICS. 91. We thus obtain, finally, a system of five tension members, a b c d «, Fig. 46 (a), which, if properly fastened at the ends a and e respectively, will, by exerting forces represented respectively by the rays, o a, o b, o c, etc.. Fig. 46 (6), balance the four given forces Pi, P2, P3 and P4. 93. The figure a b c d e, Fig, 46 (a), is called a cord polygon, funicular polygon, or equilibrium polygon. 93. Resultant, Anti-resultant. Amount and Direction. In the force polygon, Fig. 46 (6) or (d), the line e a, joining the end of the last force- line de with the beginning of the first one a b, represents the anti-resultant of the given system of four forces, and a e their resultant. Evidently, there- fore, the rays a o and o e, which represent two components of a e, represent also, in direction and in amount, two forces which would balance e a, or which would be equivalent to the given system of (four) forces. Fig^s. 46 (a), {a') and (6). 94. Position of Resultant. Hence, in the cord polygon. Fig. 46 (a), the intersection, i, of the cords a and e, parallel respectively to the rays o a and c o, is a point in the line of action of the resultant R ; and, if we imagine a % and e i to be rigid rods, and apply, at i, a force, — R, equal and parallel to o e, but of opposite sense, that force will be the anti-resultant of the (four) given forces, and we shall have a frame-work b c dioi cords and rods, kept in equilibrium by the action of the five forces. Pi, P2, P3, P4 and — R. 95. By choosing other positions of the pole, as o', Fig. 46 (6), or by differ- ently arranging the given forces, as in Fig. 46 (c), we merely change the shape of the cord polygon, and (in some cases) reverse the sense of the stresses in the members. Thus, in Fig, 46 (a), all the stresses are tensions, or pulls ; while in Fig. 46 (c) a, b, d and e are tensions or pulls, and c is a com- pression or push. 96. In constructing the cord polygon. Fig. 46 (a), (a'), (c), and (e), care must be taken to draw the cords in their proper places ; and for this it is nec- essary to remember, simply, that the two rays pertaining to any particular force line in the force polygon. Fig. 46 (&), represent those members which, in the cord polygon, Fig. 46 (a), take the components of that force. COED POLYGON. 379 Thus, o a and h o, Fig. 46 (Jb), pertain to the force Pi; o 6 and c o to the force Po. Hence, in Fig. 46 (a) or (c) we draw a and h (parallel respectively to o a and h o) meeting in the line of action of Pi ; b and c (parallel respect- ively to o 6 and c o) meeting in the line of action of Po, etc., etc. 97. Each ray in the force polygon, Fig. 46 (6), including the outside ones, is thus seen to pertain to two force?, and each force has two rays. The two cords, parallel respectively to the two rays of any force, must be drawn to meet in the line of action of that force; and each cord must join the lines of action of the two forces to which its parallel ray pertains. The lines, a, b, c. etc., in the cord polygon. Fig. 46 (a) and (c), give merely the inclinations of members which, as there arranged, would sustain the given forces. The lengths of these lines have nothing to do with the amounts of the stresses. These are given by the lengths of the corresponding rays in the force polygon. Fig. 46 (6). Figs. 46 (c), (d) and (c). 98. Tf the anti-resultant force, — R, is not applied, the cords a and e may be supposed fastened to firm supports, against which they exert stresses rep- resented, in amount and in direction, by the rays a o and o e respectively. But the resistances of those two supports are plainly equal and opposite to those stresses, or equal to o a and e o respectively. Hence, their resultant is the anti-resultant, — R, of the four original forces. 99. If, Fig. 46 (e), the two end members a and e were attached merely to two ties, V and V, parallel to the anti-resultant, — R, they would evidently draw the ends of those ties inward toward each other. To prevent this, let the strut k be inserted, making it of such length that the ties V and V may remain parallel to — R, and draw o k. Fig. 46 (6), parallel to k. Then a k and k e give the stresses in V and V respectively. 100. If the anti-resultant, — R, found by means of the force polygon, be applied in a line passing through the intersection of the outer (initial and final) members in the cord polygon, all the forces, includinc: of course the anti-resultant, will be in equilibrium. In other words, coplanar forces are in equilibrium if they may be so drawn as to form a closed force polygon, and if a closed cord polygon may be drawn between them. But if the anti-re- sultant be applied elsewhere, we shall have a couple, composed of the anti- resultant, — R, and the resultant R of the forces. 380 STATICS. Concurrent Non-coplanar Forces. 101. Any two of the concurrent forces, as o a and o c. Fig. 47 (a) or (6), are necessarily coplanar. Find their resultant, o r, which must be coplanar with them and with a third force o b. Then the resultant, R, of o r and o 6 is the resultant of the three forces. If there are other forces, proceed in the same way. 103. No three non-coplanar forces, whether concurrent or not, can be in equilibrium. 103. Force Parallelopiped. The resultant of any three concurrent non-coplanar forces, o a, o b, o c. Figs. 47, will be represented by the diagonal o R, of a parallelopiped, of which three converging edges represent the three forces. 104. Methods by Models. (a) For three forces. Construct a box. Fig. 47 (a) or (b), with three convergent edges representing the three forces in position and amount. Then a string o R, joining the proper corners, will represent the resultant. Fig. 47. Or, let ao, bo, c o, Fig. 48 (a), be three forces, meeting at o. Draw on pasteboard the three forces a o, b o, c o, as in Fig. 48 (6), with their actual angles a o b, b o c, c o a, and find the resultant w o of the middle pair, 6 o and c o. Cut out neatly the whole figure, a o a c lo b a. Make deep knife- scratches along o b, o c, so that the two outer triangles may be more readily turned at angles to the middle one. Turn them until the two edges o a, o a meet, and then paste a piece of thin paper along the meeting joint to keep («) (ft) Fig. 48. (c) them in place. Stand the model upon its side o 6 ty c as a base, and we shall have the slipper shape a o b w, P'ig. 48 {c); o w being the sole, and a ob the hollow foot. In the model, the force a o and the resultant w o oi the other two forces, are now in their actual relative positions. To find their resultant, cut out a separate piece of pasteboard, ^ a o w, with R a and R w parallel respectively to w o and a o. Draw upon each side of it the diagonal R o. Paste this piece inside the model, with its lower edge w o on the line w o, Fig. 48 (6), and its edge a o in the corner a o. This done, R o represents the re- sultant oi a o,b o, c o, Fig. 48 (a), in its actual position relative to them. 105. (b) For four forces, as a o, 6 o, c o, rfo, in Fig. 49. Draw them as in Fig. 49 (a), with their angles a o b, b o c, etc. Draw also the resultants v o, of a o and b o; and w o, of c o and d o. Then cut out the entire figure, as before, and paste together the two edges a o, a o. Hold the model in such a way that two of its planes (as a o 6 and b o c) form the same angle with each other NON-COPLANAE FORCES. 881 as do the two corresponding planes between the forces. Then we have the two resultants v o, w o. Fig. 49 (6), in their actual relative positions. Cut out a separate piece of pasteboard K v o w, Fig. 49 (6), draw the diagonal R o on each side of it, and paste it inside the model, with o v and o w on the corre- sponding lines of the model. Then R a will represent the resultant of the four forces, a o, bo, c o, do, in its actual position relative to them. The model may be made of wood, the triangles a o b, b o c, etc., being cut out separately, the joining edges bevelled, and then glued together. Non-concurrent Non-coplanar Forces. 106. Non-concurrent Non-coplanar Forces. Fig. 50 (a). (For par- allel non-coplanar forces, see ^1 110, etc.) Resolve each force into two rec- tangular components, one normal to an assumed plane, the other coin- ciding with the plane.* Find the resultant of the (coplanar) components coinciding with the plane, by methods already given, and that of the normal (parallel) components, by ^1f 110, etc. If these two resultants are coplanar, they are also concurrent, and their resultant (which is the resultant of the system) is readily found. 107. If not, let V, Fig. 50 (6), be the resultant normal to the plane, and H the resultant lying in the plane. By ^ 162, substitute, for H, the equal and parallel force H', meeting V at O, and the couple H . O a, and find the result- ant, R', of V and H'. The system of forces is thus reduced to the single force R' and the couple H . O a. For Couples, see 1 155. 108. Moments of Non-coplanar Forces. The action of the weight W of the wall. Fig. 51 (a), and of the non-coplanar forces Pi and Pg, may be represented as in Fig. 51 (6), where the axle o' c' represents the edge a c about which the wall tends to turn, while the bars or levers represent the leverages of the forces. So far as regards the overturning stability of the wall, regarded as a rigid body and as capable of turning only about the edge a c, it is immaterial whether an extraneous force, as P], is applied at p or at q; but it is plainly not immaterial as regards a tendency to swing the wall around horizontally, or to fracture it; or as regards pressures (and conse- quent friction) between the axle a' c' and its bearings. For equilibrium. Pi m = Pg ^ -i- W. -r-. Here a torsional or twisting stress is exerted in the axle, ♦Wires, stuck in a board representing the plane, will facilitate this. 382 STATICS. and the pressures of its ends in the bearings are more or less modified ; but, so far as merely the equilibrium of the moments is concerned, we may sup- pose all of the forces and their moments to be shifted into one and the same plane, as in Fig. 51 (c). 109. In cases like that represented in Fig. 51, it is usual, for convenience, to restrict ourselves to a supposed vertical slice, s, 1 foot thick, and to the forces acting upon such slice ; supposing the weight of the slice to be concen- trated at its center of gravity, and the extraneous forces to be applied in the same vertical plane with gravity. In effect, we are then dealing with a slice indefinitely thin, but having the weight of the 1-ft. slice. m\ '^''' 3.^ f' srj^^^X^ \ — - 1^ --1 — r- ' [" 1 / i\ ! ( 1 <' m^ ^ ^n' ^p -^ ' e (a) Figr- 56. 118. Resolution. Let Fig. 57 (a) represent a beam bearing a single concentrated load, a, elsewhere than at its center; and let it be required to find the pressure on each of the two supports, w and x. Fig. 57. Draw X a, Fig. 57 (6), to represent the load a by scale, and rays X O, o O, to any point O not in the line X a. In Fig. (a), from any point, ^, in the vertical through the point, a, where the load is applied, draw i s and i r, parallel respectively to O X and O a. Join r s, and in Fig. (6) draw O w par- allel to rs. Then the two segments, w a and X w, of X a, give by scale the pressures upon the two supports, w and x respectively. The greater pres- sure will of course be upon the support nearest to the load ; but we may be guided also by remembering that the segment X w, adjoining the radial line O X in Fig. (b) represents the pressure on that support, x, Fig. (a), which pertains to the line i s parallel to O X ; and vice versa. 119. Fig. 58 represents a case where there are several loads on the beam. Here tHe intersection, i, of the lines h s and k r, Fig. (a), drawn parallel respectively to O X and c O, Fig. (6) shows the position of the resultant of the three loads. Here, as in Fig. 57, we join r a. Fig. (a), PARALLEL FORCES. 385 and draw O w, Fig. (6), parallel to r s. Then X w, Fig. (6), gives the pressure upon x, and w c that upon w. (a) Fig. 58. Non-coplanar Parallel Forces. 120. Non-coplanar Parallel Forces. Fig. 59 (a). Between the lines of action of any two of the forces, as a and b, draw any straight line, u v, and make u I = u V X r ; or V I = u V X T ' a -t b a + 6 Through i draw R', parallel to a and b, and equal to their sum. Then is R' the resultant of a and b. Then, from any point, i, in the line of action of R', draw i z to any point, 2, in the line of action of c, and make c R' ik = i z X — -^^, ; or zk = i z X — -^^, . Through k draw R parallel to a, b and c, and equal by scale to their sum. Then is R the resultant of the three forces, a, b and c. If there are other forces, proceed in the same way with them. c=6 Fig. 59. 121. In Fig. 59 (a) we have shown the forces, a and c, acting upon surfaces raised above the general plane, merely in order to illustrate the fact that it is not at all necessary that the forces be supposed to act upon or against a plane surface. 122. Although Fig. 59 (a) illustrates the method of finding the resultant of non-coplanar parallel forces, yet it plainly does not give the actual relative positions of the forces and their resultant ; because it is necessarily drawn in a kind of perspective, and therefore all the parts cannot be measured by a scale. The true relative positions may of course be represented in plan, as by the five stars, a, b, c, i and k. Fig. 59 (6) , corresponding to the points where 25 386 STATICS. the forces and resultants intersect some one chosen plane. But it is now impossible to represent the forces themselves by lines. They must there- fore be stated in figures, as is here done. It is then easy to find the positions of the resultants, as before. 133. If there are also forces acting in the opposite direction, as d and e. Fig. 59 (a), find their resultant separately. We thus obtain, finally, two resultants of opposite sense. These resultants may be equal or unequal, and colinear or non-colinear. If they are non-colinear, see ^ 84, and Couples, tt 155, etc. 134. Method by projections. Fig. 60. First find the projections, a', b' and c' of the forces, a, b and c, upon any plane, as x y, parallel to them ; and then their projections, a" , b", and c", upon a second plane, x v, parallel to them and normal to the first. Find the position, R', of the re- sultant of a', b' and c', in plane x y, and that R", of a", b" and c", in plane X V. Now, as the lines, a', b\ c', and a", b", d', are projections of the forces, a, b and c, so R', R", are projections of the resultant, R, of the forces. The position of R is therefore at the intersection of two planes, R R' and R R", perpendicular to the planes, x y and x v, and standing upon the projections R' and R", of the resultant, R. R = a + 6 + c. CENTER OF GRAVITY. 125. If a body. Fig. 1,* or a system of bodies, Fig. 2, be held successively in different positions, (a), b), etc., the resultant of the parallel forces of grav- ity, acting upon its particles and indicated by the arrows in the figures, will occupy different positions, relatively to the figure of the body or system. That point, where all these positions, or lines of gravity, meet, is called the center of gravity of the body or system. Thus, if a homogeneous cylinder be stood vertically upon either end, the line of gravity will coincide with the axis of the cylinder; but if the cylinder be then laid upon its side, the line of gravity will intersect the axis at right angles and will bisect it. Hence, in the cylinder, the center of gravity is at the center of the axis. 126. About the center of gravity the moments of all the forces of gravity are in esquilibrium, in whatever position the body or system may be. Hence, the body, or system, if suspended by this point, and acted upon by gravity alone, will balance itself; i. e., if at rest it will remain at rest; or, if set ii* motion revolving about its center of gravity, and then left to itself, it will continue to revolve about that center indefinitely and with uniform angulaf velocity. Or, if suspended freely from any point, it will oscillate until the center of gravity comes to rest vertically under such point. * Figs. 1 to 45, relating to Center of Gravity, are numbered independently of the rest of the series of figures relating to Statics. CENTER OF GRAVITY. 387 127. In some bodies, such as the cube, or other parallelopiped, the sphere, etc., the center of gravity is also the center of the weight of the body; but very frequently this is not the case. Thus, in a body a b, Fig, 2, with its center of gravity at G, there is more weight on the side a G, than on the side G6. Fig. 1. Stable, Unstable, and Indifferent Equilibrium. 128. A body is said to be in stable equilibrium when, as in the pendulum, it is so suspended that, if swung a little to either side, it tends to oscillate until it comes to rest again, with its center of gravity vertically under the point of suspension. 129. It is said to be in unstable equilibrium when, as in the case of an egg, stood upon its point, it is so supported that, if swung a little to either side, and left to itself, it swings farther out from the vertical and eventually falls. 130. It is said to be in indifferent equilibrium when, as in the case of a grindstone, supported by its horizontal axis, or of a sphere resting upon a horizontal table, it is so suspended or supported that, if made to rotate about its center of gravity and then left to itself, it will continue in that state of rest or of angular motion in which it is left. mmmm . (a) (b) Fig. 2. General Rules. 131. The following general rules (1) to (6), form the basis of the special rules, (7) to (39). In speaking of the center of gravity of one or more bodies, we shall assume, for simplicity, that they are homogeneous (i. e., of uniform density through- out) and of the same density with each other. The center of gravity is then the same as the center of volume, and we may use the volumes of the bodies (as in cubic feet, etc.) in the rules, instead of their weights (as in pounds, etc.). In applying these general rules to surfaces, use the areas of the surfaces, and in applying them to lines, use the lengths of the lines, in place of the weights or volumes of the bodies. In all of the rules and figures, pp. 388 to 398, G represents the center of gravity, except where otherwise stated. 388 FORCE IN RIGID BODIES. (1). Any tw^o bodies^ Fig. 3. if each body/ g and g' ; and , , . , „ Having found the center of gravity, g, gr', of each body, by means of the rules given below : then G- is in the line joining weight of g' sum of weights of g and gf g'G-gg' X weight of g sum of weights of g and / Fig. 3 (3). Any nnmlier of bodies, as a, b and c, Fig. 4, whether their center* of gravity are in the same plane or not. First, by means of rule (1> find the center of gravity, g, of any two of the bodies, as a and b. Then the center of gravity, G, of the three bodies, a, b and c, is in the line grgr' joining g with thts center of gravity, g' of c; and gG = gg' X weight of c g'G^gg' X sum of weights of a, b and c ' sum of weights of a and b . sum of weights of a, b and c and so on, if there are other bodies. (3). In many cases, a single complex body may be supposed to be divided into parts whose several centers of gravity can be readily found. Then the center of gravity of the whole may be found by the foregoing and following rules. Thus, in Fig. 5, we may find separately the centers of gravity of the two parallelopipeds and of the cylinder between them (each in the center of its respective portion of the whole solid) ; and in Fig. 6 the centers of gravity of the square prism and the square pyramid (the latter by rule (36), and then, knowing in either case the weights of the several parts, find their common center of gravity as directed in rules (1) and (2). CENTER OF GRAVITY. 389 (4). Any liolloiv body, or body containing one or more openings. Fig. 7, Find the common center of gravity, g\ of the openings by rule (1) or (2), anc jnig. r the center of gravity, gr, of the entire figure, as though it had no opentoga Then G is in the line fir y', extended, and qG ^ oQf sy sum of volumes of openings volume of entire body — volumes of openings • Q^ ^v^ volume of entire body volume of entire body — volumes of openings Bemark. For convenience, we have shown the several centers of gravity. g, g,' G, upon the surface of the figure. In the real solid (suwosed to be ol uniform thickness) they would of course be in the middle of its thickness- and immediately under the positions shown in the figure. (5). In any line, figure or body, or in any system of lines, figures or bodies, any plane passing through the center of gravity is called a** plane of gravity" for said line, etc., or system of lines, etc. The intersection of two such planes of gravity is called a " line of gfravlty.'* The center of gravity is (1st) the intersection of two lines of gravity; (2nd) the intersection of three planes of gravity, or (3rd) the intersection of a plane of gravity with a line of gravity not lying in said plane. If a figure or body has an axis or plane of symmetry (i. e., 8 'ine or plane dividing it into two equal and similar portions) said axis or plane is a line or plane of gravity. If a figure or body has a central point, said point is the center of gravity. In Fig. 1, the string represents a line of gravity ; and any plane with which the string coincides is a plane of gravity. Thus O may often be con- veniently found, especially in the case of a flat body, by allowing it to hang freely from a string attached alternately at different comers of it, or by bal- ancing it in two or more positions over a knife-edge, etc., and finding G ia either case by the intersection of the lines or planes of gravity thus found. (6). Tbe grapbic method of finding the resultant of parallel forces may often be advantageously used for finding the center of gravity of a com- pound body or figure, or of a system of bodies or figures, when the centers of gravity of the several parts are known. Thus, in Fig. 8, let o, b and c represent three figures or bodies whose centers of gravity are in one plane. Draw vertical lines through said centers, and construct the polygon of forces, xabc, Fig. 9, making the lin<-s a? a, a 6, etc., Sroportional to the weights of a, b and c; and from any convenient point O raw radial lines Ox, Oa, etc. In Fig. 8, draw inh,inn, npy and p k, parallel Tespectively to O a;, O a, O 6, O c. Then a vertical line, i G, drawn through the "intersection, %oi mh and p Ar, is a line of gravity of the system or figure. If the body or figure is symmetrical^ as in the cross section of- a T rail, I oeam or deck beam, etc., the axis of symmetry, dividing the figure, etc. into two simi- lar and equal parts, is also a line of gravity, and its intersection with the line tG- already found is the required center of gravity G. In such cases it is generally most convenient to draw the lines through the several centers of gravity perpendicular to the axis of symmetry, so that the line of gravity found will also be perpendicular to it. But if, as in Fig. 8, the body or figure, etc., is not symmetrical, we must find a second line of gravity, the intersection of which with the first will give the center of gravity, G. To do this, repeat the process, drawing another set of parallel lines through the several centers of gravity, Fig. 8. It will be most convenient to draw them horizontally, or at right angles to those already drawn, and in the following instructions we suppose this to be dona. 390 FORCE Iiq- RIGID BODIES. Then draw a second funicular polygon, m^n'p^i', Fie. 8, making the lines, m'n/ etc., perpendicular (instead of parallel) to the radial lines O x, etc., Fig. 9; and draw the second line of gravity, i' G, through i', perpendicular to the first* Then G is at the intersection of the two lines of gravity. y-Qa // \\m, \ 4F-rf -\1~1 / N(i ^ ^^^^ I^XS. 8 fi^iSi.9 The drawing of the second funicular polygon is often less simple than tha* of the first, because in the second the parallel lines through the several centers of gravity do not necessarily follow each other in the same order as in the first. Bear in mind that the two lines (as n' p' ^ n' m!) meeting in the parallel line (as 6nO pertaining to any given pait, 6, of the tigure, must be perpendicular respectively to those radial lines (O a, O 6) which meet the ends of the line, a 6, that represents that same part. Figs. 10 and 11 show the application of the same process to an irregular fig- ure composea of three rectangles, o, 6 and c The lettering is the same as in Figs. 8 and 9; but in Fig. 10 it happens that t' and p' of the second funicular polygon fall upon the game point. B^igr. lO Kig. 11 If the centers of gravity of the several bodies, or of the several parts of the body, etc., are in more than one plane, we must find their projections upon certain planes, and apply the process to those projections. CENTER OP GRAVITY. 391 Special Rules. 132. Special Rules, derived from the general rules, (1) to (6), Liines. (7). Straiglit line. G is in the line, and at the middle of its length. (8). Circular arc,* aob, Figs. 12 and 13 (center of circle at c). G is in th« line CO joining the center of the circle with the middle of the arc, and c G =■ radius ac X chord a b length of arc aob ' Fig. IS <9a). If the arc is a semi-olrcle,* cG radius ac X ^ias. 13 radius ac X 0.6366. = .15 « t< . '« =.6ft3 so ==.20 « " > « =.660 so = .25 «t u . « =.657 «o (8 6). Approximate rules for distance 8 G, Fig. 12, from chord to center of gravity. If rise so =. .01 chord a b; sG= .6fi6 so If riae so =- .30 chord ab; sQ=- .653 s o •■ .10 " " ; « = .665 so " " " = .35 " " ; " =- .649 8 O «< « =.40 " ♦*; *• =-.645 8 « « =,45 « «; «• =..64ls a « « =.50 •' '*; « =- .637*0 (9). Triangle, a 6 c, Fig. 14. The center of gravity, G, of its three sides* is the center of the circle inscribed by a triangle, d e/, wl;>ose corners are in the centers of the sides of the given triangle. (lOj. Paralleloi^ram (square, rectangle, rhombus of rhomboid). The center of gravity of the four sides* is at the intersection of the liiagonals. (11). Circle, ellipse, or regular polygon. The center of gravity of the outline or circumfer- ence* IS the center of the figure. (13). Reisular prism, right or oblique, and ri{^lit regular pyramid, or frustum. The center of gravity of the edges* is thf^ center of the axis. in the »rtsw, the position of G is not affected by either including or excludinc the sides of bolh of the polygons forming the ends. (12 a). Cycloid.* tiee p. 194. Surfaces. A. Plane surfitoes. We now treat of the centers of gravity of plane surfaces, which may b© regarded as infinitely thin flat bodies. The rules for surfaces may be used also for actual flat bodies, in which, however, the center of gravity (s in the middle of the thickness, immediately under the points found by the rules. (13) Parallelogram (square, rectangle, rhombus or rhomboid), circle, ellipse or regular polygon. G is the center of the figure; or the inter- section of any two diameters, or the middle of any diameter. In a Parallelo- gram, G is the intersection of the two diagonals. (14). Triangle, Fii^. 15. G is at the intersection of lines (as a c and cd) drawn from any two angles, a and c, to the centers, e and d, of the sides, 6« * We are now treating of lines only; not of the surfaces bounded by them, For surfaces, see rules (13), etc., eta. 392 FORCE IN RIGID BODIES. and a 6, respectively opposite to said angles. Such lines are called " medial linea." •C* "• /^ o«; rf<* =» J^ cd; /a '^libf (J being the middle of ae), b Figi VIS a' h'G' C (14o), Fig. 15. Or, on either one of the sides (as a ft), meeting at any angle, a, make ao ^ }4 ab. Draw op parallel to the other side, a c. Then o G «= i^ op, and G is at the intersection of op with any medial line, as ae, etc. (14:6). Fig. 16. If aa\ bb\ cc^ and GG' are the distances of the three cor- ners and of G from any straight line or plane a' c' ; then CtG' — K {aa' + 66' + cc'). This gives us the position of the line of gravity G G''. In the same wa3r we find the distance G G'' of G from any second line or plane, 6" c". This gives us the position of a second line of gravity G G'. G is at the intersecton of G G' and G G''. (14: c). Fig. 17. The distance Gn of Gin any direction from any side, as a « (extended if necessary) is =- ^ the distance w' 6 measured in a parallel direo* tion from the same side to the opposite angle, 6. It follows firom this that the shortest distance, G o, of G from any side (as oc) is — 3^ the shortest distance, o' 6, from the same side to its opposite angle b. It foHows also that p G = 3^ p 6, as in Rule (14). (15). Trapeziiun or trapexold, Fig. 18. For trapezoids, see also Rule (16). Draw the two diagonals, o c and bd. Divide either of them, as a c, into two equal parts, am a.nd cm. From 6, on b d, lay off 6 » — d s (or from d lay ofl dn — 8). Join mn. G is in m n, and mG =^%mn, (G is the center of gravity of the triangle a e n). ^ig- 18 Fig. 19 (15 a). Or, Fig. 19, find first the centers of gravity, m and n, of the two tri- angles, cb d and a 6 d, into which the trapezium is divjded by one of its diago- nals, b d. Join m n. Then find the centers of gravity, o and p, of the two triangles, d a c and b a c, into which the trapezium is divided by its other diagonal, a c. Join op. Then G is the intersection of m w and op. CENTER OF GRAVITY. 393 (16), Trapezoid only 9 Fig. 20. O is in the line ef joining the centers, «and/, of the two parallel sides, a 6 and cd. To find its position in said line, prolong either parallel si(Je, as a 6, in either direction, say toward i; and make hi equal to the opposite side, cd. Then prolong said opposite side, cd, in the oi)posite direction, making dh'=' ah. Join h i. Then Gr is the intersection of hi&ndef. Or fO ef 2ab + cd 3 ^ 06 -f cd ^ en ^^ 2ab + ed or oG __x___. h e a n.'=.^^h (ID. Regnlar polygon. G is the center of the figure. (11' a). Irregular polygon. If the polygon be divided into any two portions, as by any diagonal, G must be in the line (of gravity) joining the centers of gravity of those two portions. If we again divide the whole polygon into two other parts by another diagonal, and join the centers of gravity of those two parts, Cr is the intersection of the two lines of gravity. (176). Or we may divide the polygon into triangles, find the center of gravity of each triangle, by Rules (14), etc., and then find O by general Rul« 5). (2) or (6). inig..si (IS). Cirenlar sector, aobCy Fig. 21. (Center of circle at c). ^^ 2 chord a 5 radius^ x chord c w — ■— radius ac X r- — » ^^ — 3 arc a o 6 3 X area For length of arc, see p. 141. (18a). If the sector is a sextant^ eO — radius X — -» radius X 0.6386, ft (18&). If the sector is a qnadrant. Fig. 22, cO — -^ radius X ^ -~ — radius X 0.6002. 4 •vX 2 — radius X 3 IT c« — a: G — -i radius X — 3 IT (18 e). If the sector is a semi-circley ^ 4 ,. 1 c » "" -;r radius X 3 T — radius X 0.4244 ^ (approximately) radius X -^ • 394 FORCE IN RIGID BODIES. (19). ClrculMr se^m«iit, a o 6 s, Fig. 23. (Center of bircle at c). cube of chord a b cG — 12 X area of segment (19 a). If the segment is a semi-circle, 4 I «C> — -r radius X =- radius X 0.4244 o jr 14 • (approximately) radius X -—• Fig. S3 1»0). Cycloid, Fig. 24. (Vertex at v). (ai). Parabola, a 6 c, Fig. 25. «c IPig. S5 is the base; ax and ca;, ordinates; and the height or axis, 6 a:, an ab- scissa. Center of gravity at G, in the axis X 6, and 2 X G — -r a? 5. 5 (/SI a). Semi-parabola, a 6 a; or ebx. Center of gravity at G^, and xQ xb; GCy -^-^aa (aa). £:ilipse, mnop. Fig. 26. The center of gravity, c, of the whote •Uipse is at the center of the figure. JFig. S6 G is the center of gravity of the quarter ellipse, one. G' « ** " " « half " nop, Q// « « a « M u p. mTio, 4 1 14 C G' a. — o c X "= 0.4244 oe '-^ (approximately) -—- o c. 3 IT <*5 C&'-^G/G-^^en X -^^ 0.4244 cw — (approximately) -^ 0«k 3 IT o3 CENTER OF GRAVITY, 395 (83). Any plane figure. Draw the figure to scale on stout card-board. Cut it out and balance it in two or more positions over the edge of a table or on a knife-edge; and mark on it the several positions of the supporting edge. Where these intersect is the center of gravity. Considerable care is of course necessary to obtain very close results by this method. Before balancing the card, its upper edges should be marked off into small equal spaces. Otherwise it will be difficult to locate the positions of the supporting edge. The paper on which the figure is prepared must of course be so stiff that the figure will not bend when balanced on the knife-edge. See Rule (5). B. SSarfaees of ISolids.* (34). Curved surface * of spliere or splieroid (ellipsoid). G is the center of the figure. (S5). Curved surface* of any splierical zone, as a splierical segmemtf lienilspliere . etc., Figs. 27. G is the center of the axis or height, ao.f In the liemispliere, oO = }i radius.f Fig. sr (86). Right or oblique prism, whose ends are either regular figures or parallelograms (this includes the cnbe and other parallelopipeds) ; and right or oblique cylinder (circular or elliptic). Surface* (either including both or excluding both of the two parallel ends). G is the center of the axis, or line joining the centers of the two parallel ends. (87). Curved surface*! of right cone, Fig. 28 (circular or elliptic), or slanting surfaces*! of right regular pyramid. Fig. 29. G is in the axis oa (the line joining the apex and the center of the base); and oOt -= % oa. In an oblique cone or pyramid, the perpendicular distance of G* from the base is one-third of the perpendicular height, as in the right cone and pyramid; but does not lie in the axis. (88). Frustums with top and base parallel, Figs. 30 and 31 Curved sur face*t of frustum of right cone (circular or elliptic); or slanting surfaces *t of- frustum of right regular pyramid. G is in the axis'o a (the line joining the centers of the two parallel ends) ; and . j b o G = -L a X ^^^^""^^Q ^Q^^Q of o -f 2 circumference of a. 3 circumference of o -f circumference of a. * We treat now of the surfaces of solids, not of their contents or volumes or weights. For these, see Rules (29), etc. t If the top or 1>ase is to be included, see Rules (1) and (2), 396 FORCE IN RIGID BODIES. In the conic frustum, Fig. 30, we may use the radii of the two ends; and in the frustum of a regular pyramid. Fig. 31, any side of each end (as b c and d e) instead of the circumferences. T^is.30 ITig. 31 Solids. In the following rules for center of gravity of solids, the solid is supposed to be homogeneous; i. e., of uniform density throughout; so that the center of gravity is the center of magnitude or of volume. (39). Spl&ere and spheroid (ellipsoid). G is the center of the body. (30). Hemisphere, Fig. 32. (Center of sphere at c). Height c T = radius cb. G is in the axis, c T, and cG =■ cT ■■ — - radius eb. (31> Splierical sector, Fig. 33. (Center of sphere at c). c G = — (radius ch ^). (33). Spherical segment, a m 6 T, Fig. 34. Center of sphere at c. of base aim. Rise or height of segment = wT — ^. G is in the axis wi 3 (2 radius c b of sphere — height h)^ ^ '^^ 4 3 radius c 5 of sphere — height h _ height, h 2 (radius m 6 of base)^ + (height, h)^ Center T; and height, h ■" 4 3 (radius m 6 of base)^ + (height, h)^ 4 X radius c 6 of sphere — height, h 3 X radius c 6 of sphere ~ height, k ' (33). Spherical zoue. Fig. 35. ot 2 (radius o 6 of base)^ -f 4 (radius t c of top)2 + (height o <)» 2^3 (radius o b of base)2 + 3 (radius t c of top)2 -|- (height o t)^ ' oG = (34). Prism, regular or irregular, right or oblique (including the cube and other parallelopipeds), and cylinder, circular or elliptic, etc., regular or irregular, right or oblique. G is the center of the axis joining the centers of gravity of the two ends. CENTER OF GRAVITY. 397 (34 a). A flat body, such as an iron plate, etc, may be treated as a very- short cylinder or prism. See (34) (35). Ungula of a cylinder, circular, or elliptic (provided one of the axes ©f the ellipse coincides with the oblique cutting plane); right or oblique. Figs. 36 and 37. C I«et O T be the axis (joining the centers of gravity of the ends), and X N • line drawn parallel to the axis, in the plane, A B C D, passing through the •xis and through the uppermost and lowermost points C and D of the oblique " cutting pUne. Then the position of G in the plane A B C D, is found thua: OX OB XG 2 •^ 2h + a * = l(2h + a+i_J^). 4 ^ 2h +»/ (35 a). Figs 38 and 39. If the oblique plane G D meets the base, A 6, at A, so that *»»0, while C D remains a complete ellipse or circle, this becomes OX - OB XG XN 2 -•A* (36). Cone, Figs. 40 and 41, circular, elliptic, etc., right or oblique; or pyra- mid, regular or irregular, right or oblique. The center of gravity G is in the axis O T, drawn from the apex, or top, T, to the center of gravity O of the base; and OT OG =^. T^ig. 40 (37). Frustum of a 'cone. Figs. 42 and 43, circular or elliptic, right or oblique ; or of a pyramid, regular or irregular, right or oblique ; provided the two ends A B and C D are parallel. Call the area of the large end A, and that of the small end a ; and let h be the height O Z of th'3 frustum, measured along its axis. Then 398 FORCE IN RIGID BODIES, G is in the axis O Z, which joins the centers of gravity O and Z of the t •nds; and its distance from the base, A B, measured along the axis, is OG - iL 4 A +_2 V^Ai + 3 a A + /a« (37 a). In a frustum of a circular cone, right or oblique, with parallel ends, this becomes 2L X R^ + 2 R r + 3 r8 OG « R2 + Rr + r2 ■ where R and r are the radii of the large and small ends of the frustum respectively. (38). Figs. 44 and 45. Frustum, A B C D, of a cone, circular, elliptic, etc., right or oblique; or of a pyramid, regular or irregular, right or oblique; whether the ends are parallel or not. By rule (36) find the center of gravity N of the entire pyramid (or cone, as the case may be) A B T, of which the frustum forms the lower part; and the center of gravity S of the smaller pyramid or cone D C T (== entire pyramid or cone, minus the frustum). Also find the volume of each: thus, Volume of pyramid or cone =. area of base X perpendicular height ^ •ad Volume of volume of volume ^f the frustum = entire pyramid — smaller A B C D or cone, A B T one, DOT Then the center of gravity G of the frustum A B C D is in the extension oC the line S N ; and ... ., ^«« vol ume of smaller pyramid or cone, DOT NG =. SN X volume of frustum, A B C D ' (39). Paraboloid. G is in the axis, and at one-third of its length £ro«i the base. CENTER OF PRESSURE. 399 LINE OF PRESSURE. CENTER OF FORCE OR OF PRESSURE. Position of Resultant. 133. In Tf ^ 133 to 154 we discuss the position of the resultant, or line of pressure, of a system of parallel forces acting against a surface. For the changes in that position within a structure, due to the action of non-parallel forces, see Arches, Dams, etc., 1[t 251, etc. 134. In a system of parallel forces, acting against a surface, the line of pressure, or pressure line, is the position of the resultant of the forces; and the center of force or center of pressure is the point where the pressure line meets that surface against which the forces act. 135. If the lengths of the lines which represent the forces be taken as rep- resenting weights, to scale, theii the position of the pressure line is the line of gravity (see (5), ^ 131) corresponding to those weights. 136. Thus, in Fig. 55 (a), 1 117, if the three forces, a, b and c, be taken as weights, represented to scale by the arrows, a, b and c, respectively, then the resultant R of the three forces occupies the position of the line of gravity of the three weights. 137. Again, in a mass of sand, Fig. 61,* with an irregular surface, we may Fig. 61. suppose the mass to consist of innumerable vertical columns of sand, of different heights, and exerting pressures proportional to those heights. Here, also, the pressure line is the vertical line of gravity of the mass, and the cen- ter of pressure against the base of the containing box is the point where said pressure line meets that base. 138. Although we are usually concerned with forces acting against sur- faces, so that the lines representing the forces form a solid and not merely a surface, yet, ii. a majority of the cases which occur in civil engineering, we may, for con/cnience, regard the forces as concentrated in a single plane, and therefore as acting against a mere line. 139. Thus, in the case of an arch, pressing against its skewback, the pres- sure is ordinarily distributed over all or a considerable part of the bearing surface of the skewback; but we may, for convenience, regard it as concen- trated in a single plane, midway between, and parallel to, the two faces of the arch. 140. Similarly, in the case of the water pressure against the back of a dam (or against a small strip of the back, extending from the water surface to the bottom, or to any other depth), the water, of course, presses upon the entire surface of such strip; but we may, for convenience, regard the pressure as concentrated in a vertical plane normal to the back of the dam and meeting it in the vertical axis of the assumed strip. 141. We have just seen (tH 138 to 140) that, when a system of parallel pressures acts against a surface, they may often be assumed to act, in one plane, against a single line — viz., the intersection of that plane with the sur- face. It also frequently happens that such forces are so distributed along that line that the lines representing the forces are either ojf equal length or of lengths increasing uniformly from one end of the line to the other. ♦Following Fig. 60, of Parallel Forces, ^ 124. Figs. 1 to 45, illustrating Center of Gravity, are numbered independently of the rest of the series ol figures relating to Statics. 400 STATICS. 143. Thus, in the case of water resting upon a horizontal surface, Fig. 62, the pressure is uniformly distributed, and the diagram, Fig. (6), representing the pressures, is a rectangle bounded by a horizontal line, and its center of gravity, G, is at the center of the figure. Hence, the center of pressure, c, is at the center of the line a b, or I. Here the unit pressure, p, is uniform, and R = p Z. JB (a) mi; ]p(b) I Figr. 62. Fi^. 63. 143. But when the water presses horizontally against a vertical or in- clined surface, a b, Fig. 63, the unit pressure increases uniformly from zero, at the water surface, b, to a max at the bottom, a ; and the hor pressures are represented, in Fig. (6), by the ordinates of the triangle b' a' d. Since the resultant passes through the center of gravity, G, of the triangle, the center of pressure, c, is at such a depth that c a = i a fe, and c' a' = ^ h. See Rule (14 c) under Center of Gravity . Here the mean horizontal unit pressure, p, is half the maximum horizontal pressure at a, and the total horizontal pressure ia = p h. (a) Figr. 64. 144. Again, if we consider only the water pressures against a certain part, a b, Fig. 64, of the depth of the back of a dam, the diagram. Fig. (b), repre- senting the horizontal unit pressures, becomes a trapezoid, composed of a rectangle b' d, and a triangle e d f, with their centers of gravity at g and g' respectively ; and the center of pressure, c, on a b, is opposite their common center of gravity (center of gravity of trapezoid), G. If h be the vertical depth of the portion considered, then - = 4 See Rule (16) under Center of Gravity, under Hydrostatics. 2¥ e -\- a' f b' e -\- a' f See also Center of Pressure, Distribution of Pre'ssmre. 145. Conversely, if two surfaces, as those of a masonry joint, are in such contact that the pressure is, or may be regarded as, regularly distributed, and if the position of the resultant is known, the rectilinear figure, represent- ing the distribution of pressure, may be drawn by means of the principles just stated. DISTRIBUTION OF PRESSURE. 401 I 146. In Figs. 65 to 68 inclusive, let o = the center of the joint a b between the two surfaces; R = the total pressure = resultant of all the pressures ; c = point of application of resultant, R ; ab = the length of the joint ; ■ o c = the distance of the center of pressure from the center of the joint; ! — — X = a c = distance of center of pressure from nearest end of joint; R the mean unit pressure = r pa = the maximum unit pressure ; pb = the minimum unit pressure. 1^ 147 to 154 apply equally whether the surface is horizontal, vertical or inclined, and whether the forces are normal or inclined to it. 147. If X is not greater than —» or, in any case, if the joint is capable of o ^ ■ Buetaining tension, as well as compression, we have : 6 X Maximum unit pressure = Pa = P (1 + ~z~^' 6 X, Minimum unit pressure = Pb = P (1 • I ). If a; exceeds — , and if the joint is incapable of resisting tension, see tH 151, 152, 154. Fi^. 65. Tig. 66. 148. Demonstration. In Fig. 66, where the parallelogram a' d repre- sents the total pressure R as it would be if uniformly distributed along I, we see that the moment of R, about o, which changes the parallelogram a' d into the trapezoid a' ¥ n m, is equivalent to a couple (see Couples, ^ 155, etc.) composed of two forces — viz., a pressure, / (not shown) distributed over o a and represented by the shaded triangle on the left, and a tension, — /, or diminution of pressure, distributed over o b and represented by the triangle on the right. The forces, / and — /, act through the centers of gravity of these two triangles respectively; and the distance of each of these centers of gravity from the-center, o, of the joint, measured parallel to the joint, 2 I is = :^ . ^. Hence the distance between the two centers of gravity, meas- 26 402 STATICS. 2 I ured parallel to the joint, is = — . Let x be the eccentricity, c o, of R, measured along the joint, and let Ar and Ac (not shown) be the lever arms of R and of the couple, respectively, about the center, o, of the joint. Then, since R is parallel to / and — /, Ae to Ac, and x to I, we have: Ar : Ac = X : -5-. 2 I If R is normal to the joint, we have: Ab = x; and Ac = — . Figr* 65 (repeated). moment of R Now Hence, / = arm of couple R^ 3 Fig. 66 (repeated). R.Ar Ac * R 3a; Zx = 7--2- = PT-- The mean additional pressure on o a (or mean tension on o 6) is = ^-r 7 • and the corresponding maximum additional pressure is ^ f 4 , 4 3x 6a: ■KT t J , 6 X ,, , 6 a:. Now Pa =P + /m = p4-p-y-=P(l+ ^) and Pb -/m V ■ 6 X M 6 X, p(i-7-). 149. If, as in Fig. 65, the center of pressure, c, is at the center, o, of the surface, we have x = o c = zero, and the pressure, R, is uniformly distrib- uted over the surface, 150. " The Middle Third." If, as in Fig. 67, x = i;—i. e., if the re- o sultant, R, of all the forces, meets the surface at the edge of the middle third of that surface, then pa = 2 p; and Pb = 0. See ^^ 143 and 148. 151. When, as in Fig. 68 (a), x exceeds — , — i. e., when the center of pres- sure, c, falls beyond the middle third of the surface of pressure, a portion, s b, of the surface, is in tension, the maximum tension, Pb, Fig. 68 (6), 6x, ^ . ^^ , 6 X. being = p (1 —) as above; maximum pressure = p (1 - - ) , and total pressure on a s = — " — = R plus the tension in s b ; but if, as usually DISTRIBUTION OP PRESSURE. 403 happens in masonry, the surfaces are incapable of resisting tension, the total pressure, R, is simply concentrated upon a portion a v. Fig. 68 (c), of the sur- face, o V being = 3 y. Then, mean unit pressure on a v == ^ — = ^-~ \ p» = 23^ = 2 r^- Fig. B7. Figr. 68. 153. Hence, in a joint incapable of sustaining tension, Fig. 68 (c), if pa =" 2 — - is the maximum permissible unit stress, the distance a c, from the cen* ter of pressure, c, to the nearest end of the joint, must not be less than y "^ 2R 3 Pa' 153. If the joint can j-esist tension. Fig. 68 (6), we substitute, in the equa* 6 X I tion, Pa = P (1 + -y), the value of a? = -^ — V, and, solving for y, we have 2 , Pa Z ^==■3^-- 6^- 154. The influence diagrams, Fig. 69 (see %^ 339, etc., and Trusses, ^1^ 79, etc.), show the changes in the maximum and minimum unit pres- sures, Pa and Pb, as the center of pressure, c, recedes from the center, o, of the joint. The diaecrams are constructed for a mean unit pressurp, p, of 1. If the surfaces of the joint are capable of sustaining tension, every part of the joint always sustains either pressure or tension; and (see dotted lines. 404 STATICS. 6 X Fig. 69) the maximum unit pressure, Pa = P (1 + -7 ). see I 146, increases 4R proportionally with x; becoming = 4 p = — ,— when c reaches the end, a, of the joint, and y = — . The maximum tension, 2>b. is then = 2 p = 2R I But if the surfaces are incapable of sustaining tension (see solid lines, Fig. 69), the increase of Pa is proportional to x only so long as a; < — ; — i. e., so long as the resultant of all the pressures falls within the middle third of the base a b. When that limit is exceeded, the maximum unit pressure, Paf begins to increase more rapidly than does the distance, x, of c, from the center, o, of the joint, the diagram becoming a rectangular hyperbola; so that, if the resultant could be actually applied at the very edge of the joint, the unit pressure there would become infinite. Values ofije = oc =distance from center of joint-to center of pressure Fig. 69. COUPLES. 155. Couples. Two equal parallel forces, p and q, or p' and q\ Fig. 70,* of opposite sense, are called a couple. A couple has no tendency to move the body t as a whole in any straight line. In other words, the two forces, forming a couple, can have no resultant. Their only tendency is to make the body revolve about its center of gravity, G, and in the plane of the couple — i. €., the plane in which the two forces lie. A body with a fixed axis can revolve only in a plane normal to that axis. The actual plane of rotation of a free body depends upon the distribution of mass in the body, and is not necessarily the plane of the couple. * Figs. 70 to 75 are supposed to be seen in perspective, and the forces are supposed to act in the planes shown. t See foot-note (*), K 1. COUPLES. 405 156. The moment of a couple is equal to the product of one of the two forces, v or q, into the perpendicular distance, d, between the two forces. Or, in our figures, moment of couple = p . d = q . d. 157. Graphic Representation of Couples. A couple, M or N, Fig. 70, is indicated, in amount, in direction and in sense, by a line, L or L'. normal to the plane of the couple, so placed that, looking along it toward that plane, the couple appears positive or right-handed, and of such length as to represent, by scale, the moment of the couple. In Fig. 70, the two couples M and N are of opposite sense. Hence the lines L and W, repre- senting them, project in opposite directions from their respective planes. Fig. 70. Fig. 71. 158. Composition of Couples. If the lines, L and L', Fig. 71, repre- sent two couples, in accordance with ^ 157, then the line R, completing the triangle, will, in the same way, represent their resultant or anti-resultant. As drawn, with its arrow following those of the other two sides, it represents their anft-resultant. _ For their resultant, the arrow on R, and that indicating the direction of rotation, must be reversed. 159. Equality of Couples. Two couples, M and N, in the same plane. Fig. 72 or Fig. 73, or in parallel planes. Fig. 70, are equal if their moments are equal, whether or not the forces of one of the couples be equal or parallel to those of the other. In Fig. 73, the two couples, M and N, are of like sense; in Figs. 70 and 72, of opposite sense. Fig. 72. Fig. 73. 160. Since a couple has no resultant (^ 155), it can have no anti-resultant; i. c, no single force can balance a couple and thus preserve equilibrium. (But see ^ 168.) To do this requires an equal and opposite couple. Thus, in Fig^ 72 the couple M is balanced by the equal and opposite couple N. If, as in Fig. 72, the two couples are in the same plane, and if we find first the resultant of either pair of non-parallel forces, as p and p', and then those of the other pair, q and g', we shall find these resultants equal and opposite, maintaining equilibrium. 161. Any couple, as M, Fig. 73, may be replaced by any other equal couple, N, in the same plane or in a parallel plane, and of like sense. 163. If, to a force, P, Fig. 74 (a), we add a couple, M, Fig. 74 (6), in the same plane with the force, we may replace the couple, M, by an equal and like couple, N, Fig. (c), composed of the forces, — P and P', each = P, placing — P opposite P, as shown. Then P and — P counteract each other, and we have left only P', equal and parallel to P ; and, since Pd = M, we have 406 STATICS. d = — . In other words, the effect of the addition of the couple, M, Fig. (b), to the forcfe, P, is simply to shift the line of action of P, parallel with itself, through the distance, d. If the couple M is left-handed, as in the figure, P will be shifted to the right (looking in its own direction), and vice versa. 163. Conversely, the force, P', Fig. (c), is equivalent to the combination of force P and couple M, Fig. (6) . 164. Again, having only the force P', Fig. (c), if we apply, at a distance, d, from P', the two opposite forces, P and — P, each equal and parallel to P', we shall thus substitute, for P', the equal and parallel force, P, and a couple = Pd = M. 165. Hence, also, the combination of the force P and the couple M, Fig. (6), is equivalent to the combination of the force P and the couple N, Fig. (c). 166. If the moment of the couple, M, Fig. (6), or N, Fig. (c), be equal and opposite to the moment of the force P about the center of gravity, G, of a body, we have d = p = distance from P to G. In other words, the effect of such a couple is to shift the force, P, parallel with itself, to a line passing through the center of gravity, G. 167. Hence, the effect of a force, P, Fig. (a), applied to a body at a dis- tance, d, from its center of gravity, G, is equivalent to the combined effect of an equal and parallel force, P', Fig. (c), applied at the center of gravity, and a couple (as M, Fig. b) = Fd, and of like sense, applied to any part of the body in a plane parallel to P and P'. 168. It will be seen that, although (^ 160) no single force can balance a couple and establish equilibrium, yet, if a force, P, be so applied that its moment, Fd, about the center of gravity, G,. of the body, is equal and oppo- site to the moment of the couple, it will counteract the tendency to rotatioQf due to the couple, and substitute for it a motion of translation only. Fig. 75. 169. Thus, in Fig. 75, where the force, p, acts through the center of gravity, G, of the body, let a force, — q, equal and opposite to q, be applied in the same line with it. Then rotation will be prevented, and the body will move * under the action of p (= the resultant of the three forces), which acts through the center of gravity, G, of the body. The rotation will similarly be prevented if a force less than q be applied farther from G than g is ; or if a force greater than q be applied nearer G than q is ; provided always that the moment of said third force, about G, be equal and opposite to that of the couple p q. But in the first case the resultant of the three forces (being always equal to the third force) will be less, and in the second case greater, than p. 170. If, to a couple, be added a third force, colinear with one of the forces of the couple, we have the case of two unequal parallel forces of opposite sense. See H 112, under Parallel Forces. * See foot-note (*), If 1. FKICTION. ^ 407 FRICTION.* 171. When one rough body rests upon another, the projections and de- pressions, forming the roughnesses of their surfaces of contact, interlock to a greater or less extent ; and, in order to slide one over the other, we must expend a portion of the sliding force, either in separating the bodies (as by lift- ing the upper one) suflSciently to clear the projections, or in breaking off some of the projections and clearing the others. 173. Even the most highly polished flat surface, as xy. Fig. 76, is not (as it appears to the eye) a plane, but is, in fact, a more or less jagged surface, as would appear under a sufficiently powerful microscope; so that the force, a b, instead of forming the apparent angle, ah x, with one smooth surface, x y, of application, really becomes a series of parallel forces, as c, d and e, which form other angles with a number of surfaces, m m, nn, etc., of application, inclined (often in different directions) to the general surface, x y, as shown. Among these surfaces may be some, as m m, at right angles to the applied force; and, the force c will be imparted to them in its original direction, although applied obliquely to the apparent surface, x y. In the case of the two forces, d and e, applied to the surfaces, n n and s s, if the sliding tendencies along the two surfaces are equal and act in opposition to each other, the combined resistance of the two surfaces, n n and s s, is directly opposite to the forces, as would be that of a single surface at right angles to those forces. 173. It is of course entirely out of the question to ascertain the exact resistance of each such microscopic projection in any given case. Instead of this, we find by experiment the combined resistance which all of the projec- tions, in a given case, offer to the sliding force, and give to this resistance the name of friction. 174. Friction always tends to prevent relative motion of the two bodies between which it acts; i. e., motion of one of the bodies relatively to the other. In doing so, however, it tends equally to cause relative motion f between each of those two and a third, or outside body. Thus, the fric between a belt and the pulley driven by it tends to prevent slipping between them; but t-hus tends to make the belt slip on the driving pulley, and sets the driven pulley and its shaft in motion relatively to the bearing in which the shaft revolves. This motion is resisted by the fric between journal and bearing; and this fric, in turn, tends equally to make the bearing revolve with the journal, and to make the belt slip on the driven pulley. 175. The fric between two bodies at rest relatively to each other is called static friction, or fric of rest. That between two bodies in relative motion is called liinetic friction or fric of motion. 176. The ultimate or maximum static fric between two bodies, as U and L, Fig. 77 (or the greatest fric resistance which they are capable of opposing to any sliding force when at rest), is equal to a force (as that of * ** Friction" (meaning rubbing) is a misnomer in so far as it implies that rubbing must take place in order to produce the resistance. For we meet this resistance, not only during rubbing, but also before motion (or rubbing) takes place. * ' Resistance of roughness " would better express its nature. t See foot-note (*), t 1. 408 ' STATICS. the wt F) which is just upon the point of making TJ begin to slide upon L.* Thus fric, like other forces, may be expressed in weights, as in lbs. 177. A resistce cannot exceed the force which it resists. t Therefore if F is less than the ult static fric between U and L, the frictional resistce actually exerted by them is also less. When F is = the ult fric (and U is therefore on the point of sliding) the actual resistce is = the ult stat fric. If F exceeds the ult stat fric, the excess gives motion to U. 178. If, when a body is in motion, all extraneous forces and resistces are removed or kept in equilib, it moves at a uniform vel. Hence, if the force, F, Fig, 77, is just = the ultimate kinetic fric between U and L, their vel is uni- form. If F exceeds this, the excess accelerates the vel. If the ult kinetic fric exceeds F, the excess retards the vel. Thus the actual frictional resistce exerted by two bodies in relative motion is = their ult kinetic fric = that force (as F) which can just maintain their relative vel uniform. 179. Hence, if the hor surf S upon which L rests, could be made perfectly frictionless, the pres of L against the lug m (which would then always be = the actual fric resistce between U and L) would also be = their ult fric so long as U continued in motion over L, and might therefore be greater or less than or = F; but when U was at rest the pres against m would be = F, and less than (or at most just = ) the ult fric. Coefficient of Friction. 180. Since no surface can be made absolutely smooth, some separation of the two bodies must in all cases take place in order to clear such projections as exist. Hence the fric is always more or less affected by the amount of the perp pres which tends to keep them together. 181. The ratio of the ult fric, in a given case, to the perp pres, is called the coefficient of friction for that case. Or, Coefl&cient of friction = ultimate friction and Ultimate friction = perpendicular pressure perp pres X coeffoffric. Thus, if a force F, Fig. 77, of 10 lbs, just balances the ult fric between U and L, and if the wt of U (the perp pres in this case since the surf between XJ and L is hor) is 50 lbs, then the coeff of fric between U and L is =» ^ = 0.2. ^\ Fig. 77. Fig. 78. 182. The coeff Is usually expressed decimally, or by a common fraction; but sometimes, as in the case of railroad cars and engines, in lbs (of fric) per ton (of perp pres). Or by the "angle of fric" in degs and mins. * We here neglect the fric of the string and pulley, and assume that all the force of the wt F is transmitted by the string to U. t If a resisting force exceeds the force resisted, the excess is not resistce, but motive force. ANGLE OF FRICTION. 409 183. Angle of Friction. In Fig. 78, let W = the weight of the body, P = its pressure normal to the plane, and S = the component tending to slide the body down the plane. When the angle a is such that the body is just on the point of sliding down the plane, it is called the angle of friction, or angle of repose. The friction F and the sliding force S are then equal. But p == p" == r» ^ coefficient of friction = tan a. Hence F = P tan a = W cosin a . tan a. 184. Frictional Stability. Let R, Fig. 79, be the resultant of all the forces pressing a body against a plane, and N a normal to the plane. If the angle i between R and N exceeds the angle of friction (a. Fig. 78) between the two surfaces in contact, the body will slide on the plane, but not otherwise. If i does not exceed the angle of friction, the entire resultant R will be im- parted to the plane and in its own direction, and not merely its normal com- ponent V, as would be the case if the surfaces were frictionless. Fig. 79. Fig. 80. 185. To find the coeff of kinetic fric, allow one of the bodies, U, Fig. 80, to slide down an inclined plane A C formed of the other one and hav- ing any convenient known steepness ACE greater than the angle of fric (1[ 183). Note the vert dist A E through which U descends in sliding any dist asAC(AE = ACX sine of A C E) ; also its actual sliding \e\ in ft per sec on reaching C. Calculate the vert dist A D through which it would nave to descend along the plane (from A to B) to acquire that vel if there were no fric. AD == velocity^ in ft p er sec twice the accel g of grav -*)■ Find DE (= A E (E C = A C X cosine ACE A D), and the hor dist E C corresponding to A C i/ac2- A E2;. Then Coeflf of the average fric in sliding from A to C DE ' EC because, if we let A E represent the total sliding force expended (in accelera- tion and in overcoming the fric), then A D represents the portion of A E ex- pended on vel, and D E that expended on fric, and, since C E represents the perpendicular pressure (^ 183), DE EC " friction = coefif. prep pres 186. Or, find sine and tangent of A C E ; and the dist A C ( = time^ in sees X i gf * X sine of A C E) through which U would slide in a given time if there were no fric. Measure the dist A B through which it actually slides in that time ; and find BC = AC — AB. Then coeff of the average } +„„-nn-i^ +o„Amr\/J^ ... ,. ,. - K 4. -D C = ^^^ D C E == tan A O E X -j-?^ fric in sliding from A to B J AC because * g = about 32.2 ft per second per second. 410 FRICTION. (1st) AC:AB:BO :: AS:AD:DS '-''Z iT^irll^Tlroe '' t^« -tual velocity : the frictional retardation sliding force employed the friction, or the sliding : : the total sliding force : in giving the actual ; force required to balance velocity the friction. And, if A E is = the total sliding force, then K C is = the perpendicular pressure, And T) "p = the coefficient of friction = tangent of D E. EC (2nd) Owing to the similarity of the two triangles, A B D and A C E, we have A C : B C : : A E : D E : : — : 5^ : : tangent ACE: tangent D C E. EC EC 187. In 1831 to 1834, Oen'l Artbnr Morin* experimented with pressures not exceeding about 30 lbs per sq in ; and arrived at the following couciusions in regard to sliding fric where the perp pres is considerably less than would be necessary to abrade the surfs appreciably. These were for a long time generally regarded as constituting the tliree fundamental laws of fric. Ist. The ult fric between two bodies is proportional to the total perp force which presses them together; i e, the coeff is independent of t^e perp pres and of its intensity {pres per unit of surf ). Hence 2d. For any given total perp pres, tlie coeff is independent of the area of surf in contact. If upon a hor support we lay a brick, measuring 8X4X2 ins, first upon its long edge (8X2 ins) and then upon its side (8X4 ins), we double the area of contact, while the total pres (the wt of the brick) remains the same, and thus re- duce the pres per sq in by one-half. Consequently (the coeff remaining practically the same) we have only half the fric per sq in. But we have twice as many sq ins of contact, and therefore the same total fric. But if we can increase or diminish the area of contact without affecting the pres per sq in, the total pres will of course vary as the area, and the total fric will vary m the same proportion, for the coeff remains the same. Thus, if we place two similar sheets of paper between the leaves of a book (taking care not lo place both sheets between the same two leaves) and then squeeze the book in a letter- copying press, it will require about twice as much force to pull out both sheets as to pull out only one of them. 3d. Although the coeff of static fric between two bodies is often much greater than their coeff of kinetic fric ; yet the coeff of kinetic fric is inde- pendent of the vel. This applies also (approx) to the/He, and hence to the worifc (in /compounds etc) of overcoming fric through a given dist; for then the work ( == resistce X dist) is independent of the vel. But in a given time, the dist (and consequently the work also) of course varies as the vel. 188. (a) Some kinds of surfaces appear to interlock their projections much more perfectly when at rest relatively to each other, than when in even very slow motion ; and in somie cases the degree of interlocking seems to in- crease with time of contact. Hence there is often a great diff in amount between fric of rest and fric of motion. Thus, Gen'l Morin found that with oak upon oak, fibres of the two pieces at right angles, the resistce to sliding while still at rest, and after being for " some time in contact," was about one eighth greater than when the pieces had a relative vel of from 1 to 5 ft per sec. (b) But experience shows that even very slight jarring suffices to remove this diff; and since all structures, even the heaviest, are subject to occasional jarring, (as a bridge, or a neighboring building, or even a hill, during the passage of a train ; or a large factory by the motion of its machinery ; or in numberless cases, by the action of the wind) it is expedient, in construction, not to rely on fric for stahilUy any further than the coeff for moving fric will justify. When it is to be regarded as a resistce, which we must provide force for overcoming, it should be taken at considerably more than our tabular statement. * See his "Fundamental Ideas of Mechanics", translated hj Jos. Bennett; D. Appl«ton k Ckk, New York. 1860. FRICTION. Table of inoTingr friction, of perfectly smootb, clean, an4 dry, plane surfaces, chiefly from Morin. Materials Experimented -with. Coeffof Fric ; or Propor- tion of Fric to the Pros. Oak ou oak ; all the fibers parallel to the motion " " moving fibres at right angles to the others ; and to the motion. . . ' • " all the fibres at right angles to the motion " " moving fibres on end ; resting fibres parallel to the motion " cast iron, fibres at right angles to motion Elm on oak, fibres all parallel to motion Oak on elm, " " " Elm on oak, moving fibres at right angles to the others, and to motion Ash on oak, fibres all parallel to motion Fironoak, " " " " Beech on oak " *' " " Wrought iron on oak, fibres parallel to motion Wrought iron on elm, '* " " " Wrought iron on cast iron, fibres parallel to motion " "on wrought iron, fibres all parallel to motion Wrought iron (Tn brass W"rought iron on soft limestone, well dressed " " " hard " *' " " " *' *' " '* " wet " " or steel on hard marble, sawed. By the writer about. . *' " •' " " smoothly planed, and rubbed mahogany, fibres par- allel to motion '. " " " " " smoothly planed wh pine Cast iron on oak, fibres parallel to motion " *» " elm, " " " " " " " cast iron << i< li brass Steel on cast iron Steel on steel. By the writer Steel on brass Steel on polished glass. By the writer about. . " quite smooth, but not polished; on perfectly dry planed wh pine, fibres parallel to motion , about. . " quite smooth, but not polished; on perfectly dry planed and smoothed mahogany, fibres parallel to motion about. . Yellow copper on cast iron " " on oak Brass on cast iron " on wrought iron, fibres parallel to motion " on brass " on perfectly dry planed wh pine, fibres parallel to motion about. . " ** '* " " and smoothed mahogany, fibres parallel to mo- tion about. . i*olished marble on polished marble. By the writer average " " on common brick " Common brick on common brick •' Soft limestone well dressed, ou the sam^ Common brick, on well-dressed soft limestone " " " " " hard " Oak across the grain, on soft limestone, well dressed " " " " " hard " " " Hard limestone on hard limestone, both " " " " " soft " " " " Soft " " hard " •' " *' Wood on metal, generally, .2 to .62 mean.. Wood, very smoofA, on the same, generally, .25 to .5 " .. Wood, " " on metal, " .2 to .62 " .. Metal on metal, very smooth, dry " .15 to .22 " .. Masonry and brickwork, dry " ,6 to .7 " .. " " " with wet mortar about.. " " " " slightly damp mortar " .. " on dry clay " .. " " moist" «• .. Marble, sawed ; on the same ; both dry. By the writer •• averaga " .. " " " " " both damp " ..* " " .. " •' on perfectly dry planed wh pine. " ..• " " .. " " on damp planed wh pine " ..* " .. " polished, on perfectly dry planed wh pine •• «' .. White pine, perfectly dry ; planed ; on the same ; all the fibres parallel to motion about.. " " damp, planed ; on the same " .. * But after a few trials the surfaces become so much smoother as to reduce the angles as much a from 2° to 5° ; the sliding blocks weighing about 30 tts w 'h. 412 FRICTION. 189. Recent experiments, with much greater variations of pres and of vel, and with more delicate apparatus for detecting slight changes in the coeff, although giving conflicting results,* show that the three laws in ^ 187 are far from correct for surfs moving at liigli vels, and under great pres; and that they are only approximately correct for ordinary vels and pressures ; for the coeff is found to vary both with the intensity of the pres and with the vel, as also with the temperature.* But in the cases with which the civil engineer has mostly to deal, slight diflfs in the character of the surfs, or even in the dampness of the air, will often cause much greater changes of coeff than tjhose due to any probable changes of pres, vel and temp: so that, within the limits of abrasion, we may generally take Morin's rules as sufficiently cor- rect for such cases. .28 .a7 I 100 90 80 70 60 50 40 30 Telocity of Sliding, in inches per second. Line A shows coeffs at diff vels under a pres of 1.58 l>s per sq in. " B " " " 1.59 " " .< Q « .. <. 1^60 " « « D *' " " 1.61 " " .. E u .« u 417 u u It will be seen that at low vels the coeff decreased when the pres per sq in was almost imperceptibly increased; but this diff disappeared as the vel increased. At vels from 4 to 120 ins per sec, the coeff generally decreased as the vel in- creased ; rapidly at first, but more slowly as the vel became greater. This agrees with other recent expts. But at very low vels (.08 to 5 ins per sec) Prof. Kimball found the coeff (line E) increasing very rapidly with the vel. We have made the scale of coeffs large in order to show their variations, which are so slight that they would otherwise be scarcely perceptible. Less delicate expts would have failed to show them at all. 191. (a) In 1878 Capt. Douglas Chiton and Mr. C^eorse West* tnglioase, Jr., made careful experiments in England to ascertain the effect ol fHction in connection with rail^vay brakes. % The friction and pressure wen 190. Prof. A. S. Kimball, of the Worcester (Mass) Inst of Industrial Science, has made some very delicate experi- A j r - ments upon the fric between surfs of pine wood.f The results are given in Fig 3, merely to show how the coeff varied with vel and pres. Our table gives a coeff of .4 1 ] B / / / / / D for pine on pine. / / / / / / / / / T^. IT is \ /. / y / \ / <> .^ ^ \ ^ 'J> ^ ^ y ^___ 1 ^ ,.^ v^ ^ ^ ^ C- p^ == — =J ■^ ^ » This is not surprising in view of the extent to which the coeff is affected by the nature of the surf. If the shape of the minute projections is such that they fit into each other as perfectly under small pressures as under great ones, and if they are too strong to be broken by the pressures applied^ the coeff, as stated in the 1st law, should be independent of the pres. But if high pres wedges the projections of one body more closely between those of the other, the coeff should increase under such pres. On the other hand, if the higher pres breaks down the projections while the lower ones are unable to do so, the coeff should decrease under the higher pres. The particles thus broken off may either act as a lubricant and thus still further reduce, the fric and its coeff, or (if angular and hard) may increase it. Change of area of contact, under a given total pres, may, by affecting the intensity Of the pres, make changes in the coeff similar to those just mentioned. At high vels the roughnesses have not time to interlock as perfectly as at low vels. Hence we should expect a less coeff at high vels. But high vel generally increases the number of projection!* broken away ; and these may either increase or diminish the coeff, as explained above. High vel often indirectly affects it by increasing the temperature. t Silliman's Journal (American Journal of Science) March 1876 and May 1877. + See Proc, Instn of Mechl Engrs, London, June and Oct 1878 and April 1879; and " Bngineer. lag," London, 18T8 ; vol. 25, pp 432, 469, 490 ; vol. 26, pp 153, 386, 395. FRICTION. 413 antomatically recorded by means of hydraulic gauges. With cast iron brake-blocks and stf el-tired wooden wheels, 43% inches in diameter, they found coeflBcients about as shown in Pig. 4. The points in lines A, B and C show the average brake coeflfs, or coeffs of slid- ing fric between the tread of a rolling wheel and the brake-block. Speed of Car, in miles per hour. 40 35 30 25 20 15 {§.95 "2-20 a |.io. ^.05 X ^ ^:::::zzzt --------- -_ ^ ^£^-«- ^ - ^ . - - ---.--=5=-' : zt .•'--'^ : : : :::'" : - - - ; 3= =■ ^ -lJ- ' " *''' " 5: v^- T - -.--,' „"'.''. "_::: : ,^ x - - - J ___ ..^:__i. +±.g_-_,,,.r— ttj- ^—— |!=:=:::i;;=;==:r?!==:=;=p;;i^^^ !__:5,,.. = =-!:.S!: __^:^ :.05O 55 5U 10 45 40 35 3U 25 20 Speed of Car, in miles per hour. Line A shows brake coeffs obtained immed'y after application of brake " B " " 5 sees " " .i c " " , 15 " *' D shows rail coeffs or coeffs of sliding fric between the tread of a dic- ing or '■'■ skidding''' wheel (held fast by the brake) and the rail. (b) From lines A B and C it appears that the brake coeff obtained at a given length of time after the application of the brake was g:enerally greater at lour than at big^li vels. But where the vel was maintained uniform the brake coeff diminisbed as block &ncl urtaeel remained long^er in contact. Thus, lines A and B show that at 37^ miles per hour the brake coeff was .154 when the brake was first applied (point g)^ but fell to .096 in 5 sees {%). Line A (immed'y after application) shows a higher brake coeff (.132 at/) at 47 J miles than line B (5 sees after application) shows at 37^ miles (.096 at X). The diminution of the rail coeff with length of time of application of brake, was scarcely noticeable. (c) When the brake fric (owing to the reduction of vel and consequent in- crease of coeff) becomes =: the " adhesion " or static fric between the rail and the tire of the rolling wheel, the vel of rotation rapidly falls below that due to the vel of the car ; i e, tbe i¥heel beg-ins to " skid " or slide along the rail: and in from .75 to 3 sees the rotation of the wheel ceases entirely. (d) The rail coeff, line D, is g-enerally much less than the brake coeff, lines A, B and C. The pres on the rail ( = the wt on a wheel) "Was about 5000 ibs per sq in, or greatly in excess of the limit of abrasion. That at the brake was about 200 fcs per sq in. A few expts were made with brake blocks having but \ of the usual area of contact, and therefore 3 times the pres oer sq in under a given total pres. They failed to show conclusively that this caused any marked change in the coeff. (e) The rail coeff, line D, like the brake coeff, increases as the vet diminishes ; slowly at first, but much more rapidly as the speed becomes less; until, at the moment of stopping, it is generally even greater than the brake coeff just before skidding. With steel tires on iron rails at high vels it was somewhat greater than on steel rails, but this diff disappeared as the vel dimin- ished. (f ) liOComotives overcome resistces = from ^ to ^ or more of the wt on all the drivers; % e, they have a coeff of .33 or more, although the experimental coeff for steel on steel in motion at low pres, is only about .15. But the cases are so diff that a similarity in their coeffs could hardly be expected. The great wt, say from 2 to 6 or even 7 tons, on a driver, is concentrated on a surf (where the wheel touches the rail) about 2 ins long X about \ inch wide, or = say 1 sq in. The pres per sq in thus greatly exceeds not only that upon which the tables are based, but also the limit of abrasion. Besides, any point in the tread, during 414 FRICTION. the instant when it is acting as the fulcrum for the steam pres in the cyl, i» stationary upon the rail. Its fric (miscalled " adhesion ") is therefore static. Capt. Galton found that the coeff of *' adhesion " was independent of the vel, and depended only on the character of the surfs in contact. With a four-wheeled car having about 5000 Ihs load on each wheel, it was generally over .20 on dry rails ; in some cases .25 or even higher. On wet or greasy rails, with- out sand, it fell as low as .15 in one case, but averaged about .18. With sand on wet rails it was over .20. Sand applied to dry rails before starting gave .35 and even over .40 at the start, and an average of about .28 during motion ; but sand applied to dry rails while the car was in motion was apt to be blown away by the movement of the car and wheels. (g) Owing to the constancy of the coeflf of " adhesion " under given conditions of tire and rail, the brake fric necessary to *' skid " the wheels in any case was also practically constant for all vels. But at high vels, owing to the lower brake coeff, a higher brake pres was reqd to produce this fixed amount of brake frie The skidding also reqd a longer time than at low speeds. 193. If the pres is sufficient to produce abrasion (indeed, while it is much less) the fric often varies greatly, but no precise law has yet been discov- ered for estimating it. Rennie gives the following table of coefTs of frie of dry surfaces, under pressures g^radually increased up to the limits of abrasion. It will be noticed that in this table the coeff g^enerally increases with the intensity of the pres : Coeffs of friction of dry surfaces, under pressures grrad-^ ually increased up to the limits of abrasion. (By G. Rennie, C E.> Pres. in Lba. Wrought Iron Wrought Iron Steel Brass per on on on on Square Inch, Wrought Iron. Cast Iron. Cast Iron. Cast Iron. .32.5 .140 .174 .166 .157 186 .250 .275 .300 .225 224 .271 .292 .333 .219 336 .312 .333 .347 .215 448 s M5 .354 .208 560 .367 .358 .233 672 .376 .403 .2.33 709 .434 .2.34 784 .232 821 .273 193. (a) Rolling; friction, or that between the circumf of a roll- ing body and the surf upon which it rolls, is somewhat similar to that of a f union rolling upon a rack. In disengaging the interlocking projections, or in ifting the wheel over an obstacle o, Figs 5 and 6, the motive force F, instead of dragging one over the other, as in Fig76, p. 407, acts at the end of a bent lever F E, W Figs 5 and 6, the other end W of which acts in a direction perp to the contact surf; and in practical cases of rolling fric proper the leverage RW of the resisting wt of the wheel and its load is very much less, in proportion to that (FR) of the force F, than in our exaggerated figs. Hence the force F reqd to roll a wheel etc is usually very much less than would be necessary to slide it. (b) There are usually two irays of applying; the force in overcom- ing rolling fric : 1st (Fig 5) at the axis of the rolling body ; as the force of a horse is applied at the axle of a -p, wagon-wheel; or that of a man at the axle of a wheel-barrow : 2d (Fig 6) at the circumf; as when workmen pusli along a heavy timber laid on top of two or more rollers; or as the ends of an iron bridge-truss play backward and forward by contraction and ex- pansion, on top of metallic rollers or balls (p725). In Fig 5 we have, in ad« dition to the rolling fric of the cir- cumf of the wheel on its support, the sliding fric of the axle in its bearing. In Fig 6 we have only rolling fric^ (\ R J ^4 ^v^,^^ m ^ ^ Fig.S Fig. 6 but at both top and bottom of the wheel. (c) When the obstacles o are very small, as in the case of cart-wheels on smooth hard roads, or of car-wheels on iron or steel rails, the leverage FRICTION. 415 (FR) of F becomes, practically, in Fig 5 the radius, and in Fig 6 the diam, of the wheel; while that (RW) of the resistce is very small. Hence, neglecting axle fric in Fig 5, the force F reqd to overcome rolling fric in such cases is directly as the wt W of and on the wheel, and inversely as the diam of the wheel. The few expts that have been made upon the coeffs of rolling fric, apart from axle fric, are too incomplete to serve as a basis for practical rules. (d) The fric (or " acHiesion ") between wheel and rail, which enables a locomotive to move itself and train, or which tends to make a car-wheel revolve notwithstanding the pres of the brake, is a resistce to the sliding of the wheel on the rail ; and is therefore not rolling but sliding fric ; static when the wheels either stand still or roll perfectly on the rails ; and kinetic when they slip or *• skid ". 194, The IViction of liquids moving in contact with solid bodies Is independent of ttie pressure, because the "lifting" of the particles of the fluid over the projections on the surf of the solid body, is aided by the pres of the surrounding particles of the liquid, which tend to occupy the places of those lifted. Hence we have, for liquids, no coeff of fric corresponding with that (= resistce -^ pres) of solids. The resistce is believed to be directly as the area of surf of contact. Recent researches indicate that Resistce = a coeff X area pf surf X ^el» in which both n and the coeff depend upon the vel and upon the character of the surf; and that at low vels n = 1, but that at a certain "critical" vel (which varies with the circumstances) n suddenly becomes = 2, owing to the breaking up of the streatn into marked counter currents or eddies. The resistance of fluid fric arises principally from the counter currents thus set in motion, and which must be brought into compliance with the direction of the force which is urging thp stream forward. 195. Table of coefficients of moving^ friction of smootli plane surfaces, when kept perfectly lubricated. (Morin.) Oak on oak, fibres parallel to motion " " " fibres perpendicular to motion " on elm, fibres parallel to motion " on cast iron, fibres parallel to motion " on wrought iron, " " " Beech on oak, fibres *' " " Elm on oak, " " " '• " on elm, " , '♦ " " •* cast iron, " " " " ■Wrought iron on oak, fibres parallel, greased and wet, .256. " " " " fibres parallel to motion " " on elm, " '* " " " " on cast iron, " ". " " " on wrought iron, " " " *' •' on brass, fibres " " " Cast iron on oak, fibres parallel to motion " •' " " " " " *' greased and wet, .218 " " on elm, " " " " " " on cast iron, with water, .314 " " on brass Copper on oak, fibres parallel to motion Yellow copper on cast iron Brass on cast iron " on wrought iron " on brass Steel on cast iron " on wrought iron " on brass Tanned oxhide on cast iron, greased and very wet, .365 " " on brass " " on oak, with water, .29 Dry Soap. Olive Oil. Tal- low. Lard. .164 .075 .067 .083 .072 .136 .••;; .073 .080 .098 .0.55 .066 .137 .070 .060 .139 .066 .214 .085 .055 .078 .076 .066 .103 .076 .070 .082 .081 .078 .103 .075 .189 .075 .078 .075 .061 .077 .197 .064 .100 .070 .078 .103 .069 .075 .066 .072 .068 .077 .086 .072 .081 .058 .079 .105 .081 .093 .076- .053 .056 .133 .159 .191 .241 .091 .056 The launching friction of the wooden frigate Princeton \ras found by a committee of the Franklin. In8ti^u^B in 1844, to average about .067 or one-fifteenth of the pressure during the first .75 of a second and .022 or one forty-fifth for the next 4 seconds of her motion. The slope of the ways was 1 in 13, or 4 degrees 24 minutes. They were heavily coated with tallow. Pressure on them = 15.84 lbs. per square inch, or 2280 ft)8. per square foot. In the first .75 of a second the vessel slid 2.5 inches; in the next 4 seconds 16 feet 6.5 inches; total for 4.75 seconds 15.7* feet 416 FRICTION. 196. The ftrictlon of lubricated surfaces varies greatly with the character of the surfs and with that of the lubricant and the manner of Ita application. If the lubricant is of poor quality, and scantily and unevenly ap- plied under great pres, it may wear away in places and leave portions of the dry surfs in contact. The conditions then approximate to those of unlubricated surfaces. But if the best lubricants for the purpose are used, and supplied reg- ularly and in proper quantity, so as to keep the surfs always perfectly separated, the case becomes practically one of liquid friction, and the resistce is very small. Between these two extremes there is a wide range of variations (see table, ^ 197 (d)), the coeflF being affected by the smallest change in the condi- tions. Where any degree of accuracy is reqd, we would refer the reader to the experimental results given in Prof. Thurston's very exhaustive work,* devoted exclusively to this intricate subject. 197. (a) Expts by Mr. Arthur M. Wellington upon the fric of lubricated journals t gave a gradual and continuous increase of coeff as the vel of revolution diminished from 18 ft per sec ( = a car speed of 12 miler per hour) to a stop. This increase was very slight at high vels, but much more rapid at low ones ; as in Figs 3 and 4. At vels from 2 to 18 ft per sec the coeff was much less under high pressures than under low ones; but at starting there was little diff in this respect. The coeff increased rapidly as the tempera- ture rose from 100° to 120° and 150° Fahr. (b) Prof. Thurston, also experimenting with lubricated journals, t found that at starting, the coeff increased with increase of pres, as it did also when in motion, if the pres greatly exceeded the max (say 500 to 600 lbs per sq in) allowable in machinery. He also found that at high vels the coeff increased very slowly (instead of continuing to decrease) as the vel increased. (c) Prof. Thurston gives the following approx formulae for journal friction at ordinary temperatures, pressures and speeds, with journal and bearing in good condition and well lubricated: Coelf for starting: = (.015 to .02) X if^pres in Bbs per sq in. Coeff when the shaft , ^, , -^, ^^ l^ vel in ft per min is revolving - (-02 to .03) X -%=z=— ~zzz=zr V^pres in ros per sq in. At pressures of about 200 Rs per sq in : Temperature of minimum ._ ^^ 3. — , . ., r- fric ; in Fahr degs — 15 X l/vel in ft per mm Caution. Tlie leverage, with which journal fric Resists motion, in- creases \¥itli tbe diam of the journal. (d) Tbe following figures, selected from a table of experimental results ^ven by Profo Thurston, merely show the extent to wbich the coen of journal fric is aflfected by pres, vel and temperature; and hence the risk incurred in rigidly applying general rules to such cases. In these expts the character of journal and bearing, the lubricant and its method of application, remained the same throughout. Where these vary, still further, and much greater, variations in the coeff may occur. Steel journal in bronze bearing, lubricated with standard sperm oil. fa U a 130° 90° Speed of revolution ) feet per minute 1 100 feet per minute 1500 ft per mm|l200 ft per min 200 100 I 4 lbs per sq in Coeff .0160 .0056 Coeff .0044 .0031 Coeff .125 Pressures. 200 I 100 I 4 lbs per sq in Coeff .0087 .0040 Coeff .0019 .0019 Coeff .0630 .0630 200 100 lbs per sq in 200 100 lbs per sq in Coeff Coeff .0053 .0037 .0075 .0061 Coeff .0065 .0100 Coeff .0075 .0150 * Friction and Lost Work in Machinery and Mill Work. John Wiley & Sons, New York, t Traua Amer See of Civil Engrs, New York, Dec. 1884. t Journal of the Franklin Institute. Hoi. lil*» FRICTION. 417 (e) Where the force is applied first on one side of the jour- nal and then on the opposite side, as in crank pins, the fric is less than where the resultant pres is always upon one side, as in fly-wheel shafts; because in the former case the oil has time to spread itself alternately upon both sides of the journal. (f ) Friction rollers. If a journal J, in- stead of revolving on ordinary bearings, be sup- ported on friction rollers R, R, the force required to make J revolve will be reduced in nearly the same proportion that tbe diam of the axle o or of the rollers, is less than the diam of the rollers themselves. Mr. Wellington experimented with a patent bearing on this principle, invented by Mr. A. Higley. Diam of rollers RR, 8 ins; of their axles If ins ; of the journal c, 3^ ins. Here, theoretically, . . . ^ diam of axles If ins fric of patent journal = fric of 3^ in journal X ^^^^^1^-^^^^- Yi^s or as 1 to 4.6. Under a load of 279 lbs per sq in, Mr. Wellington found it about as 1 to 4 when starting from rest; and about as 1 to 2 at a car speed of 10 miles per hour. 198. (a) Resistance of railroad rolling stock. This con- sists of rolling fric between the treads of the wheels and the rails (the treads also sometimes slide on the rails, as in going around curves) ; of sliding fric be- tween the journals and their bearings, and between the wheel flanges an<3 the rail heads; of the resistce of the air; and of oscillations and concussions, which consume motive power by their lateral and vert motions, and also increase the wheel and journal fries. Its amount depends greatly upon the condition of the road-bed and rails (as to ballast, alignment, surf, spaces at the joints, dryness etc); upon that of the rolling stock (as to wt carried, kind of springs used, kind and quantity of lubri- cant, condition and dimensions of wheels and axles etc) ; upon grades and curv- ature; upon the direction and force of the wind; and upon many minor con- siderations. Experiments give very conflicting results. (b) During the summer of 1878, Mr. TVelling-ton experimented with loaded and empty box and flat freight cars, passenger and sleeping cars, and at speeds varying from to 35 miles per hour. The cars were started rolling (by grav) down a nearly uniform grade of .7 foot per 100 feet, or 36.5 feet per mile, and 6400 ft long. Their resistces were calculated as in ^ 185. "The rails were of iron, 60 lbs per yd, and the track was well ballasted and in good line and surf, but not strictly first class." The following approx figures are deduced from Mr. Wellington's expts upon cars fitted with ordinary journals:* Car Resistance in pounds per ton (2340 lbs) of iveight of train, on straight and level track in good condition. Empty cars Loaded cars Speed of train in Oscilla- Oscilla- miles per Axle, tion Axle, tion hour tire and flange and con- cuss' n Air Total tire and flange and con- cuss'n Air Total 14 0- 14 18 18 10 6 .6 .4 7 4 .6 .4 5 20 6 2.7 1.3 10 4 2. 1. 7 30 6 5.3 2.7 14 4 1 4.7 2.3 11 (c) With the Higley patent anti-fric roller journal, the resistce to ^/aWin^r was but about 4 lbs per ton. (d) About midway in the track experimented upon, was a curve of 1° de- flection angle (5780 ft rad) 3000 ft long, with its outer rail elevated 3 to 4 ins * Transactioiis> American Society of Civil En^'ineers. Feb 1879. 27 418 FRICTION. above the inner one. The rise of the outer rail was begun on the tangent, about 500 ft before reaching the curve. In the first 500 ft of the curve the resistce was greater than that encountered just before reaching the curve, by from .6 to 2.1 (average 1.1) lbs per ton. In the last 500 ft of the curve this excess had diminished to from .2 to .9 (average .6) lbs per ton. Owing to the continuance of the down grade on the curve, the vel increased as the train traversed the curve ; but it does not clearly appear whether the decrease in curve resistce was due to the increase in vel, or to the fact that the oscillations caused by enteriag the curve gradually ceased as the train went on. (e) Mr. P. H. Dudley, experimenting with his " dynagraph "* ob- tained results from which the following are deduced: Train Resistance in ponnds per ton (2240 lbs) of weig^lit otf train, including^ g^rades. description of train Loaded cars 29 37 25 Empty jWeighttons cars (2240 lbs) Trip Toledo to Cleve- land. 95 miles Cleveland to Erie 95.5 miles Erie to Buffalo. 88 miles Average speed. Miles per hour 20 20 20 Average resist- ance. 8.34 7.67 8.89 ** With the long and heavy trains of the L. S. & M. S. Ry, of 600 to 650 tons, it reqd less fuel with the same engine to run trains at 18 to 20 miles per hour than it did at 10 to 12 miles per hour", owing to the fact that at the higher speeds steam was used expansively to a greater extent, and hence more economically. 199. The work, in ft-lbs, reqd to overcome fric through any dist, is = the fric in lbs X the dist in ft. In order that a body, started slid- ing or rolling freely on a hor plane and then left to itself, may do'this work ; ie, may slide or roll through the given dist, its kinetic energy ( =' its wt in lbs X its vel2 in ft per sec -f- 2gf) must = the fir.st-named prod. Conversely, tlie dist in ft through which such a body will slide or roll on a hor plane, is its kinetic energy in ft-lbs, at start fric in lbs wt of bod y in lbs X Initial vel^ i n ft per sec initial vel ^ in ft per sec *" wt of body in lbs X coeff of fric X 2gf~ ^ coeffoffric X 2 ^ ^ Tbe time reqd, in sees, is == ' dist in ft mean vel, in ft per sec J initial vel in ft per sec Suppose two similar locomotives, A and B, each drawing a train on a level straight track ; A at 10 miles, and B at 20 miles, per hour. The total resistce of each eng and train (which, for convenience, we suppose to be independent of vel) is 1000 lbs. Hence the force, or total steam pres in the two cyls reqd tQ balance the fric and thus maintain the vel, is the same in each eng.' In travel- ing ten miles this force does the same amount of work (1000 lbs X 10 miles = 10000 pound-miles) in each eng, and with the same expenditure of steam in each ; although B must supply steam to its cyls ttvice as fast as A, in order to maintain in them the same pres. Id one hour the force in A does 10000 lb-miles as before, but that in B does (1000 lbs X 20 miles = ) 20000 lb-miles, and with twice A's expenditure of steam. But in fact the resistce of a given train is much greater at higher vels. See table, If 198 (b) And even if we still assumed the resistce to be the same at both vels, B must exert more force than A in order to acquire a vel of 20 miles per hour while A is acquiring 10 miles per hour. * An inst for measuring the strain on the draw-bar of a locomotive, or tbe foroc which the lattetl exerts upon the train, t y= acceleration of gravity = say 32.2 ; 2g= say 64.4. LEVERS. 419 200. Natural Slope. When granular materials, as sand, earth, grain, etc., are deposited loosely, as when they are shoveled from a cutting or dumped from a cart, the angle, formed between a level plane and the sloping surface of the pile of material, is called the natural slope. This angle de- pends upon the friction and adhesion between the separate particles of the material, and often varies, in one and the same material, from time to time, with changes in weather conditions, etc., especially with dampness. Fig:. 85. 201. Any force, p. Fig. 85, acting upon a body, B, will suffice to move the body (see foot-note (*), H 1), provided it exceeds the sum, S, of all resist- ances, including friction between B and the surface upon which B rests, or if it forms, with any other force or forces, P, a resultant, R, greater than S. If, before the application of p, the body is already in uniform motion, P is = S ; and any force, p, however small, will suffice to change the direction of motion. This accounts for the ease with which a revolving shaft may be slid longitudinally in its bearings, and for the fact that a cork may be more easily drawn if we first give it a twisting motion in the neck of the bottle. LEVERS. 202. Classes of tevers. Figs. 86. Levers are classe- according to the relative positions of "power, " * "weight" * and fulcrum iS follows: ^, K )(&) l^ QW Fig:. 86. W (C) Fig. (a), Class 1. Fulcrum R between power w and weight W; " (6), " 2. Weight W between power w and fulcrum R; " (c), " 3. Power W between weight w and fulcrum R. In class 2, the leverage of the power is necessarily greater than that of the weight. In class 3, vice versa. 203. In Fig, 86, taking the moments of the forces about any point at pleasure, as o, we have, for equilibrium : Fig. (a), W . Zw — R ?R + 1^ . Zw = 0; Fig. (6), W.Zw — R^R-w?.^w = 0; Fig.- (c), W .lw — B,ln-}- / / / ' ^^^ ;^ 4//J^//////M ^ — h — > ^ / w / («) (6) Fig-. 95. 218. Work of Overturning. In Figs. 95 (a) and (6), let the shaded portion of each figure be of lead, and the remainder of wood, and let the center of gravity of the entire body, in each case, be at G. Then, since the weight, W, is the same in both cases, as is also its leverage of stability, about o, = y, the moment of stability, = — 6.W, is the same in both cases, as is also the force, P, required to balance that moment when applied at a given elevation, e. As overturning proceeds, the weight, W, remaining un- changed, the leverage and moment of stability, and the overturning moment required, decrease, becoming = when the bodies reach the positions STABILITY. 423 shown by the dotted lines. If the elevation, e, remains constant, the force, P, required for overturning, decreases in the same proportion as the lever- age, etc. 319. But in order that the bodies may be overturned by the force of grav- ity alone, they must be brought into the positions shown by the dotted lines. This requires that the weights of the bodies be lifted through a height = the distance, h, through which their centers of gravity, G, are raised. Hence work of overturning W.A. Since h is greater in Fig. (6), the work of overturning is greater in that case. In civil engineering we are generally concerned with the amount of the force which will begin overturning, rather than with the amount of work required to complete the overthrow. 220. Stability against overturning is of course affected, and may be in- creased, by forces other than the weight of the body itself. Thus, the stability of a bridge pier is ordinarily increased by the weight of the bridge itself if this be brought upon the pier symmetrically. Otherwise the weight of the bridge may either increase or diminish the stability of the pier, accord- ing to circumstances. 221. The coefficient of stability, in any given case, is the ratio of the moment of stability to the overturning moment. Or, Coefficient of stability = moment of stability overturning moment' 222. Let the weight, W, of the stone in Fig. 96 be 10 lbs,, G its center of gravity, and o g = 2 feet. Then the moment of the weight about o, or the moment of stability about o, is 10 X 2 = 20 ft.-lbs,; and, if o n = 5 feet, a 20 force P = — = 4 lbs., will just hold in equilibrium the moment of the 5 weight, so that, except at the corner, o, no pressure will be exerted upon the base o m, although the stone remains in contact with the base. If the force P exceeds 4 lbs., the stone will begin to turn about o. If P is less than 4 lbs., the stone will exert a pressure upon the base o m. Let the stone be supported at o and at m only. The leverage of the sup- porting force R, at m, is = the length o m of the base, = I. Let P = 1 and base o m = 4.5 ft. Then, for equilibrium, W . o g — P. on — R.om = 0; 20 ft.-lbs. — 1 X 5 = 4.5 X R; 20-5 ^ ~ -475" - = 3.33 lbs. In other words, a vertical upward force, R, of 3.33 .. . lbs., at m, will maintain equilibrium. -^^ ^ o/ ff W m. K r — > If Fig. 96. 223. In Fig. 97 (6), let g be the center of gravity of the load W and the W eg table, combined. Fig. 97 (a). Then, upward reaction of 6 = — r-^r^. Those of a and c may be similarly found. ih 424 STATICS. 224. In Fig. 98, let h be the horizontal force exerted at the crown by the left-hand half of the arch, against the half-arch shown, and e its leverage about o. Let W be the weight of the half-arch with its spandrel, acting as a single rigid body, and I its leverage about o. Then, for equilibrium, we have h . e = W . I ; OT h = W .1 Stability on Inclined Planes. 225. Stability on Inclined Planes. Fig. 99. Here, as in If 213, if the re- sultant, R, of the force P and weight W, falls beyond the base, — i. e., if the overturning moment exceeds the moment of stability, — the body will over- turn. If not, it will stand. The force, P, in any given direction, required to prevent overturning, is = the anti-resultant. A, of weight W and reaction II ; and reaction II = anti-resultant of force P and weight W. 226. Neglecting friction, as in Fig. 99 (a), R will be normal to the plane. Taking friction into account. Fig. 99 (b), R may form, with a normal, N, to the plane, an angle, a, not exceeding the angle of friction between the body and the plane. R may be either uphill or downhill from N. 227. In Fig. 100, the body B has less stability against overturning about its toe, a, than has the similar body. A, when the force, n, tends to upset it downhill ; but a greater stability than A against overturning about c under the action of a force tending to upset it uphill. 228. The body C, which would upset if upon a horizontal base, would be stable against overturning if placed upon an inclined plane, as at D. Assum- ing ao = tc,a given upward vertical force would have the same overturning Fig. 100. moment, whether applied at a or at c. But a given horizontal force, applied at any given height, as at g, has a greater leverage, g o, when pushing down- hill than when pushing uphill. In the latter case its leverage is only g t. 229. Structures built upon slopes are liable to slide. This may be ob- viated by cutting the slope into horizontal steps, as at d y. Fig. E; but the vertical faces of such steps break the bond of the masonry ; and, moreover, the joints being more numerous, and the mortar therefore in greater quan- tity, on the deeper side, s d, than cJn the shallower uphill side, e y, the struc- ture is liable to unequal settlement, the downhill side settling most and tend- ing to split away from the uphill portion, as might be the case with a founda- THE CORD. 425 tion firm in some parts and compressible in others. Hence, when circum- stances permit, it is preferable to level off the foundation, as at d v; or, if the structure has to withstand downhillward pressures, v shguld be lower than d, and the courses of masonry laid with a corresponding inclination. THE CORD. 230. The Cord. Figs. 101 (a) and (6) and 102 (a) and (6). In f H 230 to 239 we deal with cords supposed to be perfectly flexible, inextensible, frietionless, weightless and infinitely thin. Figr. 101. 331. Let P be the external force applied to the cord at the knot or pin, o, and let R be the resultant of the stresses, 8\ and s^, or o a and o b, in the two segments, o m and o n, of the cord. Then, for equilibrium, R must be equal to and colinear with P. 833. Knowing the amount of P ( = R), the tensions S\ and 82 may be found by means of If 36; and, vice versa, given si and S2, we may find R ( = P) by 1 35. Or see H 40. 233. If, as in Figs. 101 (a) and (6), the force P be applied to the cord, at o, by means of a fixed knot, incapable of sliding along the cord, so that the seg- ments, o m and o n, of the cord, are of fixed lengths, and the angle, x -\- y, between them, of fixed magnitude, then the force may be applied in any direction, as P or P', passing between the two segments of the cord ; and the components, 8\ and sa, will be equal only when R (P produced) forms equal angles, x and y, with the two segments of the cord. If the direction of the force, as P'', coincides with either segment, as o n, of the cord, that segment transmits the entire force, P'', and the other segment none. Flff. 102. 234. But if, as in Figs. 102 and 103, the force P be applied to the cord by rneans of a frietionless ring, slip-knot, pin or pulley, etc., then, for equilib- rium, the two stresses, s\ and so, must be equal, as must also the two angles, X and y; and, if we suppose the direction of the force P to be changed, as to P', the pin and the cord will readjust themselves, as indicated by the dotted lines in Fig. 103, until the pin finally comes to rest at that point, o', where the angles, x' and y', are equal, and also the stresses, si' and s^'. 426 STATICS. ^U a'\ .+-.--' ^-rK"^ Fig. 103. 335. Even though the pin or pulley be rigidly fixed to some external ob- ject, as at o. Fig. 104, yet, if there is no friction at its axle, or between it and the cord, the components, Si and so, will still be equal, and their resultant, R, will bisect the angle, x -\- y, between them. In other words, the angles, X and y, will be equal. Fig. 104. Fig. 105. ^ 336. When the pin is movable. Figs. 102 and 103, to find the position, o. Fig. 105, which it will assume. From the end, n, of one of the segments, o n, of the cord, draw n v parallel to P. From the end, m, of the other segment, with radius = m o -j- o n, = length of cord, describe an arc, cutting nvind. Bisect ndine. Draw e o normal to n v, intersecting mdino. Then o is the required point. 337. Whether o be a fixed knot or a movable pin or pulley, it is always in the circumference of an ellipse whose foci are at the ends, m and n, of the cord. Fig. 106 338. From the foregoing it follows that, if o, Fig. 106, be a fixed knot, and if the other pins or pulleys, etc., are frictionless, the stress a o, or 8\, will be transmitted uniformly throughout the left segment of the cord, from o to its end at m; and h o, or sa, throughout the right segment, from o to n. FUNICULAR MACHINE. 427 239. Caution. Note that, in Fig, 107 (6), the stresses in all the cords are twice as great as the stresses in the corresponding cords in Fig. 107 (a), although each Fig. shows a load = 4 suspended from the pulley. Thus, if the weight be that of a man, hanging by the rope, and if the rope, in Fig. (a), be just sufficiently strong to hold, it will break if he gives one end of the rope to another man to hold, or makes it fast, as in Fig. (6). v////////////////^^^^^^ 2 4 (a) Fig. 107 (b) The Funicular Machine. 240. When the angles, x and y, Figs. 101, etc., are very great, a very small force, P, will balance a very great stress, siors^, in the cord. When x = y = 90°, we have cos x = cos y = 0, and si = s^ = infinity, however small P may be. If a line, m n, joining the ends of the cord, is horizontal or inclined, the weight of the cord itself acts as a force P. Hence "There is no force, however great, can stretch a cord, however fine, into a horizontal line that shall be absolutely straight." 241. The funicular machine takes advantage of the fact that, when the total angle, x -j- y, between the two segments of the cord, approaches 180°, a small force, P, may balance great stresses, si and 82. Thus, in Fig. 108, let W represent a heavy boat (seen in plan) which is to be hauled ashore. One end of a rope being made fast to the bow of the boat, the rope is passed around one smooth post, n, to another, m, around which it is given one or more whole turns ; and a man stands at the end, e, to take in the slack ; while others, taking hold of the rope between m and n, pull it, in the direction of P, into a position m o n. If the two angles, x and y, are equal, the component in the segment o n exceeds P, so long as the angle x exceeds 60°, and a pull, equal to this component (except in so far as it is reduced by the rigidity of the rope and by its friction against the post n), is exerted upon the boat at W, drawing it a short distance up the beach. The rope is then straightened again, from m to n, by taking in the slack at e, and the operation is repeated as often as may be necessary. yp Fig. 108. The Toggle Joint. 242. The toggle joint. Fig. 109, is simply an inversion of the funicular machine with a fixed knot, the force P and the components, S\ and So, being pushes or compressions, instead of pulls or tensions. The joint being unable to move along the arms, the force P may be applied in any direction at pleas- ure, but it is usually exerted in a direction forming approximately equal angles with the two arms. 428 STATICS. The Pulley. 343. Figs. 110 show the relations of stresses and weights in several ar- rangements of fixed and movable pulleys. Thus, in (a), 1 lb. balances 1 lb., in (6) 2 lbs., in (c) and in {d) 4 lbs. In each case, if the bodies or weights be set in motion, their velocities are inversely as their masses. See ^ 206. Fisr. 110. 244. The simple pulley. Fig. 110 (a), is used simply for convenience of changing direction of stress, for the forces at the two ends of the cord are equal; but in the compound pulley, Figs. 110 (6), (c), (d), a small force (the "power"), moving rapidly, at one part of the rope, balances a greater force (the "weight"), moving slowly, at another part. Hence, the compound pulley is used for the purpose of overcoming great resistances slowly, by means of small forces, moving rapidly. 245. To set such a system in motion * (i. e., to raise the "weight") re- quires that the equilibrium be disturbed by making the "power" exceed the stress in the cord due to the "weight." But the motion, once generated, will continue indefinitely if the "power" is made sufficiently greater than th« "weight" to balance the resistances of friction, etc. The Loaded Cord or Chain. 246. In Figs. Ill the principle of the cord polygon, f ^ 86, etc., is applied to the case of a flexible cord or chain, sustaining four loads, pi . . . p^, at fixed points, and exerting a horizontal t puU, H, at its lower end, and an inclined pull, R, at its upper end. The loads, pi . . . p^, are represented by the vertical line, 0-4, Fig. Ill (a) ; the horizontal pull, H, by 0-c; the amount and direction of the inclined pull, R, at the upper end of the cord, by 4-c, and the tensions in the segments, 1-2, 2-3 and 3-4, by the rays, 1-c, 2-c and 3-c, respectively. 247. The horizontal tension, H ( = the horizontal component of the ten- sion in each segment), is uniform throughout the cord; but the vertical component of the tension in any segment is equal to the sum of the loads between that segment and the pulley, m. Thus, the vertical component * See foot-note (*). H 1. t See foot-note (*), If 249. LOADED CORD. 429 (0-2, Fig. a) of the tension, c-2, in segment 2-3, is = pi + P2 ; that in seg- ment 3-4 is 0-3 = Pi -t- P2 + P3. etc. 248. If all the loads (including W) be increased in the same proportion, as indicated by the dotted lines in Fig. Ill (a), or diminished in the same pro- portion, the new triangles, c' 4' 0, etc., Fig. (a), will be similar to the old, and the profile of the cord. Fig. (6), will remain unchanged, although the stresses in its segments will of course be increased or diminished in the same proportion. 349. In Fig. Ill we make the weight, W, which is necessarily equal to the horizontal* pull, H (see The Cord, 1[*[f 230, etc.), equal also to the Fiff. 112. sum of the loads, pi . . . p4. When this is the case, the cord segment, a-4, next to the support, a, and the corresponding line, c-4. Fig. (o), will be in- clined 45° to the vertical. 250. But if, while the loads, pi . . . P4, remain unchanged, we raise the pulley m, so as to keep H horizontal,* we shall obtain a flatter curve, as in Fig. 112; and, for equilibrium, H (= W) must be made greater than the sum of pi . . . P4. On the other hand, if we place the pulley, m, lower than in Fig. Ill (still keeping H horizontal), we obtain a deeper curve, as in Fig. 113; and H ( = W) must be made less than the sum of pi . . . P4. * In Figs. Ill, 112 and 113 we suppose the weight, W, and the position of the pulley, m, to be so adjusted, relatively to the support, a, that the pull, H, shall remain horizontal. 430 STATICS. ABCHES, DAMS, ETC. THRUST AND RESISTANCE LINES. The Arch. In ^1^ 251 to 257 are given the elements of the commonly accepted theory of the arch. For practical considerations, see Kl 258 to 266, and Stone Bridges. 251. If Figs. Ill, 112 and 113 be inverted, the cord segments will repre- sent struts, sustaining compression, as do the stones of a masonry arch, Fig. 114. Thrust Line. Resistance Line. 353. In the case of an arch. Fig. 114, assuming * that the horizontal thrust H, at the crown, m, and the reaction, R, of the skewback, a, act at the center (or at some other definite point) of crown and of skewback, respectively, their amounts, and the direction of the reaction, R, may be found by means of the Force Triangle, t 51, or by Moments, 1[ 224. (See If 257.) We then suppose the half -arch and its spandrel to be divided, by vertical planes, * Fig. Fi^. 114. 114 (6), into a number of segments, as shown; and, finding the weight and the center of gravity of each such segment (see 1f^ 257 and 266), we treat these segments as we treated the loads, pi . . . p4, of Figs. Ill to 113, lay- ing them off from to 6, Fig. 114 (a), and laying off 0-c horizontal and == H. The rays, c-1, c-2, etc., then give, theoretically,* the directions and amounts of the pressures exerted by the segments, 1, 2, etc., respectively. The broken line, m d e f . . . .a. Fig. 114 (6), thus formed, is called the thrust line, or line of resultants. It corresponds with the cord polygons of Figs. Ill (6), etc.* 353. The resistance line is a broken line joining the points where the several resultants, forming the thrust line, cut the respective joints between the arch stones. 354. When the planes, by which the arch is supposed to be divided into segments, are vertical,* as in Fig. 114 (6), and, indeed, in most actual arches, the thrust and resistance lines. Fig. 115, practically coincide; but if these planes are far from vertical, as in Fig. 115, the two lines separate, the resistance line being always the outer one. Thus, in Fig. 115 (where the thrust line is shown solid, and the resistance line dotted), noticing where resultant a cuts joint A, where resultant h cuts joint B, etc., it will be seen that the two lines practically coincide as far as to joint C, where they begin to diverge. * See Practical Considerations, ^^ 258, etc. ARCHES. 431 255* In ^ 252 we assumed that the arch and its spandrel are divided into vertical segments, incapable (except in the arch ring) of exerting other than vertical pressures. The theoretical resistance line, thus obtained, may, especially in deep arches, pass from the thickness of the arch ring in places ; so that, if no other forces were acting, the arch would open at such places ; on the intrados when the resistance line cuts the extrados, and vice versa; Fig:. 115. but such opening is usu^tlly prevented by other forces, such as the horizontal or inclined pressures of the spandrels. The actual resistance line is thus confined within the thickness of the arch ring. In general, the actual resist- ance line. Fig. 116, approaches the extrados at the crown, and the intrados at the haunches, so that the arch tends to sink at the crown (opening there on the intrados), and to rise at the haunches (opening there on the extrados), as shown. Fig. 116. 256. In order to avoid any tendency of the joints to open at either side, the arch should be so designed that the actual resistance line shall every- where be within the middle third (see tH 145, etc.) of the depth of the arch ring. 257. In general, the design of an arch is reached by a series of approxima- tions. Thus, a form of arch and spandrel must be assumed in advance, in order to find their common center of gravity for the purpose of determining the horizontal thrust, H, and the skewback reaction, R, as in ^ 252; and, if it is afterward found necessary to modify the form first assumed, in order to satisfy the requirements of ^ 256, or for other reasons, we may have to re- compute H and R, again modifying the design, and so on. 432 STATICS. Practical Considerations. 358. While the theoretical thrust and resistance lines, based upon the foregoing assumptions, are easily found, much uncertainty exists as to the positions of the actual thrust and resistance lines in a masonry arch. 359. In the first place, we do not know through what points in the crown and skewback, respectively, the resultants, H and R, pass. 260. Again, we have assumed that the loads on the arch, like those on the cord. Figs. Ill to 113, are incapable of acting otherwise than vertically; whereas the spandrel walls and filling, which form a large portion of the load on a masonry arch, may offer resistances acting in other directions. If the loading were a liquid, like water, its pressures upon the arch ring would be radial, like those of the particles of steam, in a boiler, upon the boiler tubes ; and this condition is probably more or less closely approximated in the case of a loading of clean dry sand; and, less closely, in the case of earth filling. Hence, although the determination of the theoretical thrust and resistance lines in an arch is facilitated by the assumption that the arch is correctly represented by the inverted cord, the distinction between the two cases must be borne in mind when drawing practial conclusions from the lines so found. 361. Thus, in many cases, the theoretical thrust and resistance lines cut the intrados or the extrados in places, thus passing entirely out of the arch ring; so that this would inevitably fall (see ^ 255), were it not for horizontal or inclined resistances exerted by the upper parts of the abutments through the spandrel walls and filling. 363. Hence, in order to determine the actual resistance line, we should not only have to know through what points, in crown and in skewback respectively, the resultants, H and R, pass, but we should also have to ascer- tain and take into account the possible horizontal and inclined resistances of the spandrel walls and filling. But, as this is ordinarily impracticable, we content ourselves either with determining the theoretical thrust and resist- ance lines, as directed above, and then estimating, as well as may be, the resistances of the spandrels, or with reasoning by analogy from the behavior of actual structures. See Stone Bridges. 363. If the inverted cord correctly represented the actual thrust line in a masonry arch, the arch stones, in elliptic or in tieep segmental arches, twould have to be made inordinately deep, in order that the resistance line should nowhere leave the middle third of their depth (see ft 145, etc.); and it might therefore appear rational to make the profile of the arch corre- spond approximately with the thrust line, which usually approaches a para- bola. But, owing to the spandrel resistances, the actual thrust line, even in semicircular arches, probably seldom greatly oversteps the middle third. 364. With a wall or a deep continuous filling, over an arch, if the arch were to settle, or were to be removed, the wall and the filling above it would form an arch, as indicated by the broken lines in Fig. 117; and only that portion Fig. 117. below this arch would fall out. Hence, only this portion can properly be regarded as pressing upon the arch. 365. Neglecting the strength of the mortar, the inclination of each joint between two arch stones must of course be such that the angle, between the thrust, at any joint, and a normal to that joint, shall be less than the angle of friction. See i^ 183, 184. DAMS. 433 266. It is often the case that the spandrels or the spandrel filling are of less specific gravity than the arch ring. In such cases, in order to facilitate the finding of the lines of gravity of the segments, we may, before dividing the half -arch and its spandrels into vertical segments (^ 252), consider the lighter structure of the spandrels as being reduced to an equivalent depth of material having equal specific gravity with the arch. The areas of the several segments, as seen in profile, and as thus reduced, may then be taken as representing their weights. Thus, in Fig. 118, where t 1 1 represents the Fig. 118. top of the spandrels, the curved line e e e represents the top of a filling of equal weight per foot run with the spandrels, but of equal specific gravity with the arch ring. When, as in Fig. 119, the spandrels consist of a series of transverse arches, we may assume that the main arch carries a series of loads concentrated at the piers of these transverse arches. Fig. 119. The Masonry Dam. 267. A dam must be secure against sliding, on its base or on any plane , within the body of the dam, against overturning, and against crushing of the material at any point and consequent opening of a seam at either face of the dam. 268. The dam will be secure against sliding if the resultant of all the pres- sures, upon any surface, forms, with a normal to that surface, an angle less than the angle of friction of the surface. See ![•[[ 183, etc. In practice, the base of the dam is let wpU down into the rock foundation, as indicated in Fig. 122 (a), and continuity of joints is avoided by making all the stones break joints. The angle of friction thus becomes, in effect, 90°, and sliding cannot occur without shearing the stones themselves. 269. If the material is sufficiently strong to resist crushing, under the maximum unit stresses brought upon it, and if the resultant of all the forces acting upon any section falls within the body of the dam, the dam will be secure against overturning. But see H 270. 270. For a given total pressure upon any section, the maximum unit pressure in the section would be least when the resultant cut the middle point of the section: See Center of Pressure, HH 133, etc. It is generally impracticable to secure this ; but the dam must be so designed that, under the maximum unit pressure, the given material shall not be taxed beyond its safe crushing strength. If this is done, and if, under all conditions, the center of pressure is kept within the middle third (see T[ 150) of each hori- zontal section throughout the dam, there will be no tendency to open on either face of the dam. 28 434 STATICS. 271. Let Fig. 120 represent a stone block, resting upon a solid foundation and intended to sustain the pressure, p, of quiet water on one side. Through the center of gravity, g, of the block, draw o' N vertically, to represent the weight, W, of the block. Then the point, 8, where o' N meets the foundation, is the center of pressure for the block alone, i. e., when the water is removed. 372. Let h be the depth of water back of the block, and let the block be one foot in length, measured normally to the paper. Then the amount, in pounds, of the water pressure, against the vertical back, a 6, is p=62.5 AX>2^ and its center of pressure is at a depth, d ^ % A, below the water surface. 273. Combining p with W (^ 35) we obtain R as their resultant, and r as the center of pressure upon the foundation when the block is sustaining the water pressure. 274. Let Fig. 121 represent several such blocks superposed. Let Qi = cen of grav, pi = cen of water pres, for block 1 ; 02 = " " " P2 = " " " " " blocks 1 and 2 combined; 03 = " " " Pa = " " " " " *- 1. 2, and 3 combined; etc. Then, finding ri and si, r2 and s^, rs and S3, etc., for joints 1-2, 2-3, 3-4, etc., as before, we have the points rj, ro, rs, etc., in the resistance line for full dam, and the points si, 82, S3, etc., in the resistance line for empty dam. Fig. 121. 275o In Fig. 122 (a) the curves, u u and d d, indicate the up-stream and down-stream limits, respectively, of the middle third of the plane separating each two blocks, assuming the profile with vertical back ; and the points ri. . . . rj and aj s-j are points in the corresponding resistance lines for full dam and for empty dam respectively. 276. While theory would require the cross-section of the dam to terminate in a sharp angle at the top, it is, of course, always made heavier in practice, as indicated in Fig. 122 (a). 277. For joint 2-3 we may suppose that block 2 had at first been designed rectangular, as shown by dotted line c a (Fig. 122 (c), showing blocks 1 and 2 enlarged) ; but this makes the center of pressure, r\ for the full dam, fall beyond the middle third of the narrow base, a b. We therefore try the trape- zoidal shape cib, with its wider base, i b, and. find that, with this, the center of pressure, ro, although further down-stream than before, falls within the middle third of said wider base. The remainder of the profile is determined by similar trials. DAMS. 435 278. Graphic Method. Suppose the cross-section to be divided, by horizontal sections, into numerous blocks, 1, 2, 3. 4, etc., of a depth approxi- mately = the top width of the dam. In Fig. 122 (6), draw 0-1, 1-2, 2-3, 3-4, etc., vertically, to represent, by scale, the weights of the several blocks, respeatively, and 0-1', l'-2', 2'-3', 3'-4', etc., horizontally, to represent the water pressures against said blocks respectively; and draw I'-l, 2'-2, 3'-3, 4'-4, etc., representing, in amount and in direction, the total inclined pressures upon joints 1-2, 2-3, 3-4, etc., and upon the base, respectively. Thus, 2'-2 represents the resultant of the water pressure (see Hydrostatics) upon blocks 1 and 2, and the combined weight of those blocks. From the points, oi, 02, etc.. Fig. 122 (c), where these two forces meet in each case, draw oi ri, 02 Ti, etc., parallel respectively to I'-l, 2''-2, etc.. Fig. 122 (6), to the corresponding joint. We thus obtain points, ri . . . Tq, in the resistance line for the case where the dam is filled to the assumed depth. The foregoing refers to the diagram for a dam already completed or de- signed. In designing de novo, we of course begin at the top, and lay off the lines 0-1, 0-1' and I'-l in Fig. (6) for the first block; then lines 1-2, l'-2' and 2'-2, for the second block, and so on; making necessary changes, as in ^ 277. 01' 2' 3' 4' («) (C) Ol 1 c 1 1 1 1*1 / * >2 / / h 1 c i ih *r' 6 \ 279. In order that the resistance line, n . . . ro, for the full dam, may be brought well within the middle third, it may sometimes be necessary to adopt a somewhat unwieldy cross-section; but, in view of the imminent danger involved in the smallest opening on the up-stream side of the dam, (see H 281), it is well here to err on the safe side. 280. As this process is carried further down, the angle formed between the down-stream face and the vertical becomes considerable ; and the middle third, in each of the lower joints, is thus brought further down-stream. The centers of pressure, s\ . . . se, for empty dam, may then fall beyond the middle third, on the up-stream side, as indicated at joints 5-6 and 6-7. To obviate this, the up-stream face is sometimes given a curved profile, as at mn. 436 STATICS. Practical Considerations. 281. The assumption of ideal conditions is particularly danger- ous in the case of masonry dams. Thus, any compression of the material at the down-stream face may open seams on the up-stream face; and water, entering these seams, will exert a wedge-like action, shifting the resistance line further down-stream, thus still further increasing the tendency to crush- ing on the down-stream face and to opening on the up-stream face. Again, if any relatively smooth joints have been left, the water, thus penetrating into or under the dam, increases the tendency to slide, not only by diminish- ing the effective weight of the upper portions, but also by acting as a lubri- cant upon the seam where it penetrates. It has been suggested that failures of dams may have been occasioned, in part at least, by vacuum, formed in front of the down-stream face, by the action of the sheet of water falling in front of that face. 283. Theoretically, the deflections of arches, dams and other structures composed of blocks, may be found by means of the formulas in ^^ 162-167 of Trusses; but, owing to uncertainty as to the values of the moduli of elasticity, E, of building stones and of mortar, and to the relative inaccu- racy of finish in masonry work, the formulas are of but little practical value in such cases. THE SCREW. 283. The screw is a spiral inclined plane. The force (or "power") de- scribes a spiral, at the end of a lever arm, while the resistance (or "weight") moves along the axis of the screw. During the time in which the force makes one revolution, the resistance traverses the "pitch," or distance be- tween the centers of two adjacent threads. 284. Hence, if P = power, w = weight, d == pitch, I = lever arm, v = rectilinear velocity of weight, and V = linear (circular) velocity of power, ive have, theoretically :* ^ = Y. = l,""! P V d ' * Neglecting friction, which, however, very greatly modifies the result. EQUILIBRIUM OF BEAMS. 437 FORCES ACTING UPON BEAMS AND TBUSSES. Conditions of Equilibrium. 285. In beams and trusses, for equilibrium, it is necessary and sufficient that the resisting forces, exerted by the material of the structure, and the moments of those forces, shall balance the external or destructive forces and their moments. We here discuss chiefly the destructive forces. For the resisting forces, see Stresses, under Trusses, and Beams or Transverse Strength, under Strength of Materials. 286. The destructive forces are (1) the loads upon the structure, includ- ing its own weight, "live" or moving loads, wind, etc., and (2) the reactions of the supports. We shall here discuss the action of vertical loads only, in- cluding (a) the dead load, or the weight of the structure itself, together with the roadway, etc., and (b) the live, moving or extraneous load of vehicles, trains, persons, etc. The action of horizontal loads (wind, centrifugal force, etc.) is governed by similar laws, and is discussed under Stresses, in Trusses. 287. Let Fig. 123 (a) represent a cantilever, resting upon a support, b, and bearing a load, W, at its outer end, a. The cantilever is prevented from turning about b, by the tension, T, of a horizontal chain, and by the compres- sion, C, in a horizontal strut.* Neglecting the weight of 'the cantilever -J >}ooooop Fl^. 123. itself, the cantilever is acted upon by four external forces, forming two couples; one couple consisting of two vertical forces — viz., the load, W, and the reaction, R', of the support ; the other couple consisting of two horizontal forces — viz., the tension, T, near the top, and the compression, C, near the bottom. Were it not for the reaction, R', of the support, b, the load, W, would pull the cantilever downward, as indicated in Fig. 123 (6) . 288. In Fig. 123 (a) we have: Algebraic sum of vertical forces = R' — W = ; " " horizontal " = T — C = 0; " *t u moments, about any point, as o, W.w R'.r + T.^ + C.c 0. * In Figs. 123 to 127, inclusive, and Figs. 132 and 133, showing cantilevers, beams and part beams, acted upon by loads, by reactions, by pulls of chains and by pushes of struts, the arrows denote forces acting upon the cantilever or beam, or upon its segments, and not forces acting upon the load, the sup- ports, or the connecting chains or struts. Thus, the tension in a chain tends to draw together the two bodies which it connects. Hence, in these cases, the corresponding arrows point toward each other. On the other hand, the com- pression in a strut tends to separate the two bodies between which it acts. Hence its two arrows point away from each other. 438 STATICS. 389. If, as in Fig. 124, the horizontal forces are exerted at the end farthest from the support, and at the same distance apart as before, their amounts and senses must remain respectively the same as before ; but we now have compression, C, at the top, and tension, T, below. Or, if Fig, 123 be in- verted, R' acts as the load, and W as the upward reaction ; and we have, as in Fig. 124, compression, C, at top, and tension, T, below. Thus, Fig. 124 is practically Fig. 123 inverted. 390. The condition described in ^ 289, Fig. 124, represents also the condi- tion in each segment. A, B, of a beam. Figs. 125 (a) and (6) or Figs. 126 (a) and (6), supported at both ends and bearing a concentrated load, W + W, Fig. 125, or W + w, Fig. 126. Figr. 125. 391. Suppose the beam. Fig. 125 (a) or Fig. 126 (a), to be divided into two cantilevers, or part beams, as in Fig. 125 (b) or Fig. 126 (b) ; each part sustaining, at its end, a part of the original load. (See 1 292.) The stresses in the strut and chain. Figs. (6), take the place of stresses in the ma- terial (situated in the dotted line) of the truss or beam, Figs. (a). In a truss, these forces are exerted by the chords : in a beam, by the particles or fibers throughout the section. jjr CJ^, Fi^. 126. 393. If, as inFig. 125 (a), the load is at the center of the span, the spans, X and y, of the cantilevers. Fig. 125 (6), are equal, as are also the loads, W = W, carried by them. But if, as in Fig. 126 (a), the load, W + w, on the beam, is not at the center of the span, the partial loads, W and w, sup- posed to be supported at the ends of the* two cantilevers, or part beams, re- spectively, Fig. 126 (6), are unequal, and inversely proportional to their leverages about their respective supports. Hence, the moments of the two opposite couples are equal. The reaction of each support is equal to the weight carried by the cantilever resting upon it. END REACTIONS. 439 End Reactions. 393. In a cantilever, Fig. 127, there is but one vertical support ; the reac- tion, R', of that support, is = the sum of all the loads, including the weight of the cantilever itself; and the reaction due to each partial load is = suck partial load. Thus, if B = weight of cantilever, R' = W + m; + B. Q^a Fig. 127. Tig. 128. W 294. The reaction, R', must not be confounded with other vertical forces. Thus, a cantilever is often supported as in Fig. 128 (a). The couple, com- posed of two horizontal forces, T and C, Fig. 127, is then replaced by a couple composed of two vertical forces, V and V, Fig, 128 (b) ; one of which, V, co- incides with the reaction, R'. Here, R' + V', acting upward, is the anti- resultant of W, w, B and V, acting downward. 295. In a beam, Fig. 129, the sum of the two end reactions is = the sum of all the loads, including the weight of the beam itself. 296. The reaction, R, of the left support, a, Fig. 129, due to the load, W, alone, is R =^W . ^ (see ^ 17), and the reaction, R', of the right support, 6, is = W — R = W . -y . If the load is central, T^~i^~2' ^^^ W R = R' = |. i m Figr. 129. Tig. 130. 297. Graphically, Fig. 156, suppose a concentrated load, W (not shown), to be placed on the beam at any point, as c. Draw a' a" anid b' b", vertical and each = W. Join a" b'-, also join a' b", and draw g h vertically through c'. Then the ordinate, c' g, to the upper line, a" b', and the ordinate, c' h, to the lower line, a' b'\ give the left and the right end reactions, R and R', respectively. 298. Where, as in Fig. 130, there are two or more loads (in which the weight of the beam may or may not be included), the reactions due to each load may be separately obtained, the sums of these reactions giving the total reactions ; or, the common center of gravity, G, of all the loads may first be found (see ^i[ 125, etc.), and then the reactions found as for a single load, W, Fig. 129; the combined weight of the loads, whose center of gravity is at G, being supposed concentrated there. 440 STATICS. 299. In a beam, Fig. 131, under a load, W, uniformly distributed over any part of the span, let G be the center of gravity of the load, and let x and y be the segments of the span, Z, to the left and right of G respectively. Then, neglecting the weight of the beam, R = W y ; and R' = W — R = W — . W J 1 Fig. 131. 300. If the load is uniformly distributed over the entire span, its center of gravity is at the center of the span, and we have : - » f - -I and R - R'= -^ . Moments and Shears. 301. In order to determine what internal stressed are required, at any point in the span, to maintain equilibrium, we may suppose the cantilever or beam to be cut in two by a section, c c. Fig. 132 or Fig. 133, at such point, and inquire what forces must be applied, in the section, in order to main- tain equilibrium and hold in position the two segments, E and F, into which the section, c c, divides the span, Fig. 132, or that part of the span between the load and a support, Fig. 133. The forces, so ascertained, are evidently equivalent to those actually exerted, for the same purpose, by the material of the beam itself. 302. In Figs. 132 and 133, moments of loads and of reactions, or extern nal or bending moments, are indicated by arrows below the cantilever and beam respectively ; while the resisting moments of the internal forces are indicated by arrows within the body of the cantilever or beam respec- tively. 303. In the cantilever. Fig. 132, the load, w, = 4 lbs., distant 6 ft. from the section, e c, produces there a left-hand or negative moment of6w~ 6 X 4 == 24 ft. -lbs. Hence, for equilibrium, the horizontal strut and chain, at c c, must exert a right-hand or positive resisting moment of 24 ft.-tt)s. ; and, being 2 ft. apart, they must exert a tension, T, and compression, 24 C, of — = 12 lbs. each. At the support, moment of load = 9w = 9X4i'= 36 ft.-lbs.; and T' - C = ^® = 18 lbs. 304. But, considering only the forces thus far discussed, we should find the right segment, F, acted upon, at c c, by a left-hand couple, = rf X T == rfXC = 2X12<= 24 ft.-lbs.; and, at the support, by a right-hand couple, = rf X T' = d X C = 2 X 18 = 36 ft.-lbs. In other words, there would be an unbalanced excess of right-hand moment, =36 — 24 = 3 R' = 3 X 4 = 12 ft.-lbs., acting upon F. F also receives, at the support, the upward reaction, R', = 4 lbs., of that support. Similarly, the couple, rf X T = d X C, at c c, exerts, upon the left segment, E, an apparently unbalanced right-hand moment of2X 12 = 6iy = 6X4==24 ft.-lbs., and E receives, from the load, w, a downward pull = 4 lbs. 305. For equiliorium, therefore, the vertical chain at c c must exert a tension = S = w?=R' = 4 lbs., pulling F downward, and E upward. The downward tension, — S, acting on F at c c, forms, with the reaction, R', of the support, a left-hand couple = 3R' = 3X4=12 ft.-lbs., balancing the excess of right-hand moment acting upon F; while the upward tension, 4- S, acting on E at c c, forms, with the weight, w, a left-hand couple, = 6v> = 6 X 4 = 24 ft.-lbs., balancing the excess of right-hand mom. acting on E. MOMENTS AND SHEARS. 441 306. Similarly, if we suppose the cantilever cut through by a section at any other point, we shall find that a vertical force, = 8 = w == R\ acting upward upon the left segment and downward upon the right segment, is required in order to maintain equilibrium and to transmit the load, w^ to the support, so that the two segments may act unitedly as a single cantilever. This force, S, is called a shear. See m 325, etc. Without it, section E would fall, as in Fig. 123 (6). 307. In the beam, Fig. 133. the total load is 16 lbs. ; and, its distances, 3 ft. and 9 ft., from the left and from the right support respectively, being as 1 to 3, the end reactions (^H 293, etc.) are as 3 to 1 ; or R = 16 X i = 12 lbs.; R' = 16 X i = 4 lbs. We therefore regard the beam as be- ing cut by a section at the load (as well as at c c), and the total Joad of 16 lbs. as divided into two portions; one, W = R = 12 lbs., attached to the end segment, M; and the other, w = R' = 4: lbs., supported by the mid- dle segment, E. Here, as in Fig. 132, segments E and F together form a cantilever, 9 ft. long, loaded with a weight, w, of 4 lbs., at its end; but, Fl§r. 132. Wig. 133. in Fig. 133, the horizontal resisting forces, T' and C, by which the entire cantilever (E + F) is upheld, are exerted, not at the support, as in Fig. 132, but at the end farthest from the support. 308. , We have, therefore, in Fig. 133 : — at c c. Bending moment, positive, Acting onE = 2T' — 6w; = 2X 18 — 6X4 = 36 — 24 =12 ft.-lbs. Acting on F = 3R' = 3X4=12 ft.-lbs. Resisting moment, negative, =2T = 2C = 2X6 = 12 ft.-lbs. Hence, T = C = 6 lbs.; and shear, S = to = R' = 4 lbs. 309. In Fig. 132 or in Fig. 133, considering the segment extendiilg from the load to either support (in Fig. 132 there is but one such segment), it will be seen that, at the free end of any such segment, the horizontal stresses are zero, and that they increase uniformly to a maximum at the other end of the segment. Thus, in Fig. 132, they increase uniformly from 0, at the loaded or free end, to 18 lbs., at the support; while, in Fig. 133, they increase uni- formly from 0, at each support, or free end, to 18 lbs., at the load. 442 STATICS. Moments in Cantilevers. 310. In a cantilever, Fig. 134, each load exerts, about any point between itself and the support, a moment = its weight X the horizontal distance of its center of gravity from such point ; and the total moment, at any point, is the sum of the moments of the several loads about that point. Thus, neglecting the weight of the cantilever itself, we have : aliout b, moment = A . x -\- B . y; " c, " = A . m + B . n; *' d, " = A . z; " A, or any point beyond A, moment = 0. rig. 134. Figr. 135. 311. In a cantilever. Fig. 135, the maximum leverage of any load, W, is evidently its distance, I, from the support, b. Hence, the maximum bend- ing moment of any load upon a cantilever is at the support, and is = W.l. From this maximum, the moment diminishes uniformly to zero, at the load. See H 309. 312. Draw 6' m, Fig. 135 (6), to represent the maximum moment by scale, and join m W. Then, for any point, c, moment = ordinate at c' to line m W. 313. In a cantilever. Fig. 136 (a), with two or more concentrated loads, W and w, let 6' w. Fig. 136 (6), = moment of W at the support; 6' m', Fig. 136 (6), = moment of w, at the support. Then, for both loads, W and w, neglecting the weight of the beam, at d, moment = moment of W alone, = ordinate at d'; at c, moment = sum of moments of W and w, = sum of two ordinates, c' n and c' n', at c'. Fig. 136. Fig. 137. 314. In a cantilever, Fig. 137 (a), under a load, W, uniformly distributed ©ver a length, Z, beginning at the supoort, b. the maximum moment, at the support, b, is = W . "2 . MOMENTS. 443 In Fig. 137 (6), make 6' m = said max moment, and draw a semi-parabola m k', with apex at k'. Then, at any other section, c, the moment is repre- sented by the ordinate, c', of said parabola, and is = to . — , where w = the weight of that portion of W beyond c, and x = the length of that portion. At k, or at any point beyond k, moment = 0. 315. In Fig, 138, neglecting the weight of the cantilever itself, let W repre- sent the weight of the whole load, and w, that of the shaded portion, concen- trated at their respective centers of gravity, G and g. Then, about 6, moment = W . x; " c, " = W , y; " d, " = w . v; " k, or any point beyond k, moment ■■ W 0. n Fig. 138. Moments in Beams. 316. In a beam. Fig. 129, the upward reaction of each abutment exerts; about any point, a moment = reaction X distance of support from such point ; but any load, between such point and the support, exerts a contrary moment = load X distance of load from such point. Thus, about c, moment = R' . e = R (Z — z) — W (y — z). At each support, the moment is 0. 317. In a beam. Fig. 129, carrying a single concentrated load, W, the moment, R'.^;, at any point, c, is = Wz = l^-j . s = R (Z — z) — W (y — z) I ' - z) — W (2/ — 2) . At a point, as c, not under the load, the moment, Wz, is evidently less than the moment, R'.j/, about the point, o, under the load. In other words, the maximum moment is at the point, o, under the load. = w.f a- IF *» ? r K — a> — >}< y > \ 1 Fig. 129 (repeated). Fig. 139. 318. From the point, o', Fig. 139 (6), corresponding to the point, o. Fig. (a), where the load is applied, erect an ordinate, o' m, equal by scale to the (maximum) moment, = R' . 2/ = R . a;, at that point. Join a'm 6'. Then the ordinate to a'm, or to m b\ at any point, c', d\ e', etc., represents by scale the moment at the corresponding point, c, d, e, etc., in the span. 44 STATICS. 319. When the load, W, is at the center of the span; I, Fig. 140, each end W action is == — . Hence, the moment, s', at any point, s, distant y from a ipport, as b, is . W moment — -x- .y. At the center of the span (i. e., at the point under the central load, W) e have: maximum moment, M, = — . -?r = — i — . 2 2 4 Fig:. 139 (repeated). Tig, 140. In order that the maximum moment (at o, Fig. 139) due to an eccen- ic load, W, may be equal to the maximum moment (at center of span, I) le to a given center load, C, we must have wf.s, = c or W = C = c(iy xy d, e. Fig. 141. m', m", repre- Make the long 330. When there are two or more concentrated loads, 3at each load as in Fig. 139, making each short ordinate, w, nt the maximum moment of its single load, c, d or e, alone. „ dinates, M, M' and M'' = the sums of the separate moments, as measured c\ at d', and at e', respectively. Then the ordinate to a' M M' M" h', at ly point, represents the total moment at that point, due to the several ads combined. Fig. 141. Fig. 142. 331. In a beam. Fig. 142, under a uniform load, W, covering the span, Z» le maximum moment is at the center of the span, and is moment = ^■^~ ^-^ W J^ 22 ' 24 W J^ 24 W.Z 8 • MOMENTS. 445 Make o'M, by scale, = the maximum moment; and draw the parabola, a' M b\ with vertex at M. Then the moment, at any section, as s, is repre- sented by the corresponding ordinate, s\ to that parabola. Let w and v — the portions of W to the left and right of s, respec- tively. Then, moment a,ts=--y* = -^x* = ~ moment due to whole load, W, concentrated at s.* At either support, moment = 0. In Fig. 131, at a point, c, under the center ot gravity of a load, W, uni- formly distributed over a portion, s, of the span, neglecting the weight of the beam, ♦ . 13 W s ^ W.s „, W.s moment = R.x — o" ' T "" ^'^ — s, ^ ^ ^ — . 323. Let W = the total load, whether concentrated or uniform, and let I = the span. Then the maximum moment, M, is as given below: = WZ; 2 • WZ . 4 • WJ. 8 • W_Z 8 ' W Z 12 • 323. In the inclined beam. Fig. 143, the inclined distances may be used, instead of the horizontal distances, in finding the reactions. Thus, Cantilever. Load, W, at end. M at support M " *' uniform. " " " M Supported beam.f - •* at center. " at center M .. " •• uniform. " " " M Fixed beam. J •• " at center. " " " or support M ♦• •• " uniform. '* " support M reaction R' w.f = w.-. But, in finding moments of vertical forces, we must of course use the horizontal, not the inclined, distances. Thus, at c, moment R'c; not R'c'. Fig. 143. y W * Moment ats = R' y — v -^ = -^ y _, X W w W — w = Rx-w- = -a:--^x: 2~ "" ^ With W concentrated at «, moment at s W -J- y = wy ■■ W t Beam supported at each end, but not fixed. I Beam fixed at each end. 446 STATICS. 324. In curved beams, the same principles apply as in straight beams. Thus, Fig, 144, at s, moment = W . I. Again, in Fig. 145 (a), reaction R' = — y- , and at s, moment = R\y. Or, as in Fig. 145 (6), from o, where the load is applied, draw o a and o b, to the two supports respectively, and, by means of the force parallelogram, find the components, p and q, of W. Then, at a, moment = n . n. ^ Cb) Tig. 145. Shear. 325. In the beam, a b. Fig. 146 (a), consider the segments, a c and c b, to the left and to the right respectively of the plane n n. Besides the horizontal forces acting across the plane n n, we have seen (T[ 305) that we require also, for equilibrium, a vertical force, = the left end reaction. R, acting down- ward upon the left segment, a c, and forming a couple with R; and, at the same time, acting upward on the right segment, c 6, being = the load, W, minus the right end reaction, R'. This force is called the shear, S, in the section n n. It may be regarded as the transmission of the vertical forces from loads to supports or vice versa. 326. The two segments, a c and c b, thus tend to slide vertically past each other, the right segment, c b, tending downward, owing to the preponder- ance of the load, W, over the right end reaction, R'; and this tendency is resisted by the shear, S, which is = the left end reaction, R. The same ten- dency exists uniformly between W and a, and is resisted throughout by a shear = S = R. 327. Between the load, W, and the right support, 6, also, a uniform shear exists; but here the shear, S', is = the right end reaction, R', = R — W; and, whereas the shear, S, to the left of the load was right-handed or clockwise (the portion to the right of any section, n n, receiving the downward force), and is called positive, or +, the shear on the right of the load is Ze/^-handed or counterclockwise (the portion to the left of any section receiving the down" ward force), and is called negative, or — . 328. The shears, S and S', to the left and to the right of the load, W, are Vepresented by the diagrams in Fig. 146 (6) ; that, S, on the left of the load being drawn above the zero line, a' b\ to indicate a positive shear, and vice versa. 329. Comparing Figs. 146, 147 and 148, notice that, between the left sup- port, a, and the load, W, Fig. 146, we have positive shears, S = 90, Fig. 146, and 8 = 15, Fig. 147; so that, in Fig. 148, where both loads, W and w, are placed upon the same beam, we have, between a and W, a total positive shear of S + s = 90 4- 15 = 105. Between the right support, b, and the load, ly. Fig. 147, we have negative shears, S' = — 30, Fig. 146, and s' = — 45, Fig. 147 ; so that, in Fig-. 148, between b and w, we have a total negative shear = S' + s' = — 30 — 45 = — 75. But, between the points of appli- cation of W and of w, we have S' = — 30, Fig. 146, and s = -f 15, Fig. 147; leaving, between W and w, Fig. 148. s + S' = 15 — 30 = — 15. If the total right end reaction, R' -f r', exceeds w, as we here suppose, the shear, at any point between the two loads, W and w, Fig. 148, is negative, as indi- cated ; and vice versa. SHEAR. 447 330. In any section, the shear is = the reaction at either end, minus any loads between that end and the given section. 331. If, as in Fig. 149, the right end reaction, R' + r', is = the load, w, then the left end reaction, R + r, is = the load, W ; and there is no shear at any point between the two loads. In other words, if the beam be cut by a section at any point between W and w, horizontal forces alone will pre- serve equilibrium, no vertical forces being required, since the two segments have no tendency to slide vertically past each other. 332. A similar condition exists in any section where the sign of the shear changes from -f to — or vice versa. Thus, if the beam be cut by a section immediately under W, Fig. 146 or 148, or under w. Fig. 147, horizontal forces equivalent to the fiber stresses in the beam, will suffice to preserve equilib- rium, without a vertical force, or shear ; there being no tendency of the two segments to slide past each other. Also, when, as in Fig. 149, under W and under w, the shear changes, in amount, from any value, on one side of a section, to 0, on the other side, the shear in the section itself is = 0. 120 L«^ ^ fy,^^9&^ 90 m M-301 Cb) \w60 w 6| '■mr^is r-4sW 15 1 8=- -45 Fig. 146. Fig. 147. 12o( 60 ( 1 W 5= 90 8+S'= "15 -4S S^ -30 Fif?. 148. Figr. 149. 333. But in the section under w, Fig. 148, where the shear changes in amount, although not changing sign from + to — or vice versa, there is a shear = the less of the two shears on the opposite sides of the section, for this is the amount of the shear transferred through the section, or is the tendency of either segment to slide past the other. 334. With any number of loads, if that portion of the total load to the left of any section be called X, and that portion to the right of the same sec- tion be called Y, it will be found that the shear in the section is equal to the difference between that part of X which goes to the right support, h, and that part of Y which goes to the left support, a. 335. With a load, W, Fig. 150 (a), uniformly distributed over the entire W span, the maximum shear, = R = R' = -g, is at each support, a and 6. The minimum shear, = 0, is at the center, c, of the span, which is also the point of maximum bending moment, see ^ 321 and Fig. 142. At any point. 448 STATICS. d, the shear is given by the corresponding ordinate, d', Fig. 150 (6). See Relation between Moment and Shear, til 359, etc. 336. With a load, W, uniformly distributed over any part, y, of the span, Fig. 151 (a), find the end reactions, R and R', as in ^ 299. Then between a and d, shear = S = R ; " eand h, " = S' = R'; at c, " = 0. J R R' d c = y . ^-j^j\ z = y--x = ce = y.. W • W Fig. 150. Figr. 151. < U V i9f{|i||i||jj| I« I I c '"""'' '""'^|W ■""%n , — 1 — 1 1 s\ It \at "\ ^!^. i^) r V ^ Fig. 152. Fig. 153. 837. When the loaded portion, y, of the span, begins at one of the sup- ports, b, Fig. 152 (a), then since R = W-^ = ^'oJ' ^® ^^^^ W^ = d c ■■ y . ry_ R _ ^^ _ y_^ y^ ' • W ^ W ^2Z 2Z* 338, When a concentrated load, W, Fig. 153, is added to a load uniformly distributed over the entire span, or over a part of it, each load produces the same shears as if it alone were upon the span. Those due to W are repre- sented in Fig. 153 (6), while those due to the uniform load are represented in Fig. 153 (c). The resultant shear, due to both loads combined, is repre- sented in Fig. 153 (d). Note that, between v and r, the addition of W, with its positive shear, reduces the negative shear due to the uniform load, and that, between r and z, the addition of W reverses the negative shear; also that it shifts the zero point from z to r. For Continuous Beams, see Beams, under Strength of Materials. INFLUENCE DIAGRAMS. 449 Influence Diagrams. 339. The end reactions, due to a given load, and consequently the moments, shears and stresses, produced, at any given point in a span, by such load, vary as the position of the load, relatively to the supports, is changed. A diagram. Figs. 154 (6), 155 (6), 156 (6), showing the changes thus produced at any given point, is called an influence diagram, or influ- ence line.* Influence Diagram for Moments. 340. Thus, in Fig. 154 (6), a'Mb' is the moment influence diagram for the point, c, under a single concentrated load, W.f Fig. 154. 341. In Fig. 154, let I be the span, x the variable distance of the load, W,t from the right support, h, and y the constant distance, a c, of a given point, c, from the left support, a. Then, for any position of W, the left end reac- tion, R, is = W . y ; and the moment of that reaction about c, = R . j/, = W . y . 2/. The right end reaction is R' = W — j — , and its moment, about c, is R' a — y) = W^-^ (l — y). So long as W is between b and c, the moment at c is = R . j/ = W. y- . y. 342. Since W, y and I are constant, the moment, at c, while W is between b and c, is proportional to the variable distance, x, of the load from b. It therefore increases uniformly, from 0, when W is at b, to its maximum value, M, when W is at c. See ^ 317. Hence, if the ordinate, c' M, be made equal, by scale, to the maximum moment, M, then the moment, at c, for any position, d, e or /, of W, between c and 6, is given by the corresponding ordi- nate, d', e' or /', to the line b' M. Similarly, the moments, at c, for any positions of W between c and a, are given by the ordinates to the line a' M. 343. For the moment, at c, for any number of loads, in any positions, find the moment, at c, for each load separately, as above, and take their sum. 344. It is customary to construct the moment influence diagram for the moments of a load, W, = unity (1 ton, 1 pound, 1 thousand kilograms, etc.). Each ordinate must then be multiplied by its corresponding load, measured in the corresponding unit, in order to obtain the required moment. 345. When W is at the point, c, we have x = I — y. Hence, ordinate, c' M, = maximum moment, = W . — j— . y; or, if W = 1, c' M = — ^ . y. The area of the diagram, a'Mb', is ■■ c'M 1 ^ziy 2 • I -.2/ I (I — y) y * See "Calculation of the Stresses in Bridges for Actual Concentrated Loads," by Prof. Geo. F. Swain, "Trans. Am. Soc. C. E.," vol. xvii, July, 1887. t Inasmuch as the load, in this discussion, occupies different positions at different times, it is not shown in Fig. 154. 29 450 STATICS. 346. If a load, — 1, be distributed over a length, = 1, at c. Fig. 155 (a), the resulting moment, at c, may be represented by the area of the rectangle standing on c^ Fig. (6), the height of said rectangle being the ordinate, c' M, and its length = 1. Similarly, the moment, at c, due to a uniformly dis- tributed load, e /, of 1 per unit length. Fig. (a), may be represented by the sum of the areas of the rectangles between e' and f\ Fig. (b) ; and, if we sup- pose the load, e /, Fig. (a), of 1 per unit length, to be divided into a very large number of very narrow vertical strips, the resulting moment, at c, may be taken as represented by the area of the shaded trapezoid over e' /', Fig. 155 (6) . The moment, at c, due to a load of p (lbs., tons, etc.) per unit length, and occupying the same length, e/, is = p X area of trapezoid over e' /', Fig. 155 (6). 347. Hence, the maximum moment, at c, due to a uniform load of p (lbs., tons, etc.) per unit of length, occurs when that load covers the entire span. This maximum moment is = p X area a'M6', = p jl — y) y See H 345. f b' Fig. 155. Fii:. 156. Influence Diagram for Shear. 348. Under a single concentrated load, the shear, at any point between the load and either support, is '= the reaction of that support. See m 326 and 327. 349. In the shear influence diagram. Fig. 156, as in the moment influence diagram. Fig. 154, let I be the span ; x the variable distance of the load, W, from the right support, 6, and y the constant distance, a c, of a given point, c, from the left support, a. Then, for any position of W, the left end reac- tion, R, or the shear, S, at any point between the load and the left support, is = W . -7- ; and the right reaction, R', or the shear, S', at any point between l — x I the load and the right support, is = W . I 350. The influence line for shear, like that for moments, 1[ 344, is usually constructed for a load = unity, so that S = R = -y; and S' = R' = I X — -J — . Each ordinate of the shear diagram must then be multiplied by W, in order to obtain the required shear, 351. Since W ( = 1) and I are constant, R and S vary directly (and R' and S' inversely) with x. Thus, when W ( = 1) is at fc, we have x = ; S = R = 0, and S' = R' = W = 1. When W ( = 1) is at a, we have x = Z; S = R = 1, and S' = R' = 0. Draw a' a" and h' h'\ each = W ( = 1), and join a" b' and a' 6". Then, with W at c, the (positive) shear, S, at each point, as /, between c and a, is given by the ordinate, c' g, to the line a'' 6' ; while the ordinate, c' h, to the line, a' b", gives the (negative) shear at each point, as c, between c and 6. 353. Similarly, with W at e, the ordinate, e' t, gives the (positive) shear at each point, as c, between e and a; while e^ p gives the (negative) shear at each point between e and b. INFLUENCE DIAGRAMS. 451 353. It will be noticed that, as the load, W, passes from one side to the other of any point, as c, the shear at that point is reversed, tl^p total change in shear being = hc'-\-c'g = hg = the load, W. 354. With a load, W ( = 1), at h, the shear at c is = 0. See H 351. As the load advances from h toward a, the positive shear, S = R = -y-i ** ^' ^^* creases in proportion to the ordinates to the line 6' g, becoming = c^ g — l — y I when W is just to the right of c. With W just to the left of c, we have, eam,inlbs.j X in inches . . , p) lbs. per sq. inch 40 v deflection, ^ moment of inertia '^ in inches ^ *'• BS _ (WJlMJ^* ... (8) 48 Al If the beam is rectangular ^ this becomes Modulna of ( ^^^ 4- H weight of clear \ ^ cube of span elasticity, in =- Un lbs. -^ span of beam, in IhaJ ^ in inches . . . (i| lbs per square inch a y deflection, ^ breadth, ^ cube of depth * '^ in inches ^ inches ^ in inches Corresponding formulae for modulus of elasticity in beams otherwise sup- ported and loaded, may be readily deduced from those for deflection (f ) If equal additions of stress could produce equal additional stretches in a body to an indefinite extent, both within and beyond the elastic limit, then a stress equal to the Modulus of Elasticity would double the lengWi of a bar when applied to it in tension, or would shorten it to zero when applied in cow»pr«s8to». In other words, if equation (5), total stretch. = original length ^ stress per square inch in inches -^ modulus of elasticHy ' held good beyond the elastic limit, as it does (approximately) within that Umlt| and if we could make the stress per square inch equal to the rsodulas of elasticity, we should have total stretch = original length. 458 STRENGTH OF MATERIALS. For example, a one-inch square bar of wrought iron will, within the limit ol elasticity, stretch or shorten, on an average, about t'^^jtu of its length undei each additional load of 2240 lbs. If it could continue to stretch or shorten indefinitely at this rate, it is evident that 12000 times 2240 lbs., or 26 880 000 Ibs^ (which is about the average modulus of elasticity for such bars) could either stretch the bar to double its length or reduce it to zero. If equal additional stresses applied to a bar could indefinitely produce stretches, each bearing a constant proportion to the increased length of the bar^ if in tension; or to the diminished length, if in compression; then the same load which would double the length of the bar if applied in tension, would reduce it to half its length, if applied in compression. is) ^6 gi^6 below a table of average Moduli of Blastlclty, in round numbers, for a few materials ; remarking, by way of caution, that, even in the case of ductile materials, the stretches produced by stresses within the elastic limit are so small, and (owing to difierences in the character of the material) so irregular, that a satisfactory average can be arrived at only by comparing many experiments ; while, in the case of materials, such as stone, brick, etc., where almost no perceptible stretch takes place before rupture, it is scarcely worth while to give any values as representing the actual moduli. Thus, eighteen experiments upon a single brand of neat cement for the St. Louis bridge, indi* cated a Modulus varying from 800 000 to 6 930 000 (!) pounds per square inch in tension, and from 500 000 to 1 600 000 in compression. (h.) Owing to the fact that the stretches within the elastic limit are seldom, if ever, exactly proportional to the stresses, but only approximately so, the modulus of elasticity, as found by experiment for a given material, will generally vary somewhat with the stress at which the stretch is taken. Art. 4: (a) The stress beyond which the stretches in any body increase per* ccptibly faster than the stresses^ is called the limit of elasticity of that body. Owing to the irregularity in the behavior of different specimens of the same material, and to the extreme smallness of the distortions caused in most materials by moderate loads, and because we often cannot decide just when the stretch begins to increase faster than the load, the elastic limit is seldom, if fever, determinable with exactness and certainty.* But by means of a large number of experiments upon a given material we may obtain useful average or minimum values for it, and should in all cases of practice keep the stresses well within such values ; since, if the elastic limit be exceeded (through mis- calculation, or through subsequent increase in the stress or decrease in the Strength of the material) the structure rapidly fails. The table, below, gives approximate average elastic limits for a few materials. (1») Brittle materials, such as stones, cements, bricks, etc., can scarcely be said to have an elastic limit ; or, if they have, it is almost impossible to determins it ; since rupture, in such bodies, takes place before any stretch can be sati» factorily measured. Thus, in the 18 specimens of one brand of cement^ referred to in Art. Sg above, the experiments indicated an elastic limit varyinij between 16 and 104 (!) pounds per square inch in tension, and from 424 to 15(« in comprtssion, (c) Experiments show that a small permanent **set'* (stretch) probably takes place in all cases of stress even under very moderate loads ; but ordinarilt it first becomes noticeable at about the time when the elastic limit is exceeded, Many writers define the elastic limit as that stress at which the first marked permanent set appears. (d) The elastic ratio of a material is the quotient, elastic limit j| ultimate strength 1$ usually expressed as a decimal fraction. ♦ The U. S. Board appointed to test Iron, Steel, Ac., found a variation of nearly 4000 lbs. per square inch in the elastic limit of bars of one make of rolled iroi^ prepared with great care and having very uniform tensile strength ; and, in another very carefully made iron, a difference of over 30 per cent, between twe bars of the same sise. Report, 1881, Vol. 1, p. 31. STRENGTH OF MATERIALS. (e) Table of Moduli of Klastlclty and of Blastlc Limits for dlffereitt materials. The values here given are approximate averages compiled from many sources. Authorities differ considerably in their data on this subject. See Art. 3 (g) and (h), and Art. 4 (a) and (6). Modulus or Coeff Elasticity. Stretch or Compression in a length of 10 ft, .under a load of Approx elas 1000 fts per 1 ton per limit. sq in. sq in. Ins. Ins. B>s per sqin. .075 .168 4500 .092 .207 4000 .086 .192 5000 .013 .029 6000 .009 .019 16000 .120 .269 4600 .007 .015 6300 .007 .015 10000 .120 .269 2000 .015 .034 8200 .010 .022 4600 to to to .005 .012 8000 .007 .015 6250 .006 .015 20000 to to to .003 .007 40000 .004 .009 30000 .005 .010 27000 .008 .018 13000 .109 .244 2300 .167 1100 .120 1100 .086 .192 2700 .120 .269 to to .060 .134 .179 .080 3S00 .075 .168 S300 .008 .018 3700 .075 .168 S300 .004 .009 34000 to to to .003 .006 44000 .003 .007 3900O .120 .269 4000 .060 .134 5000 .026 1500 Ash Beech Birch Brass, cast *' wire Chestnut Copper, cast " wire Elm Olass Iron, cast '* " average " wrought, in either t bars, sheets or plates < " " " average •' wire, hard " wire ropes Larch Lead, sheet *• wire Mahogany Oak ' ' average Pine, white or yellow , Slate Spruoe Steel bars " " average , Sycamore Teak , Tin, cast •• R)s per sq in. 1 600 000 1 300 000 1 400 000 9 200 000 14 200 000 1 000 000 18 000 000 18 000 000 1 000 000 8 000 000 12 000 000 to 23 000 000 17 500 000 18 000 000 to 40 000 000 29 000 000 26 000 000 15 000 000 1 100 000 720 000 1 000 000 1 400 000 1 000 000 to 2 000 000 1 500 000 1 600 000 14 500 000 1 600 000 29 000 000 to 42 000 000 35 500 000 1 000 000 2 000 000 4 600 000 (f) Yield point. In testing specimens of iron and steel, it is commonly found that, after passing the elastic limit, see (Art 4 a), a point is reached where the ratio between the stretch and the stress suddenly begins to increase more rapidly. This is commonly called the yield point. See Fig. D. 460 STRENGTH OF MATERIALS. Art. 5 (») Resilience is a name given to the work (as in inch-pounds) which must be done in order to produce a certain stretch in a given body. This work is equal to resilience _ said stretch y mean stress in pounds employed in producing in inch-pounds in inches ^ the stretch. The total resilience is the work done in causing rapture. The elastic resilience (frequently called, simply, the resilience) is that done in causing the greatest stretch possible within the elastic limit, (b) Suddenly applied loads. Place a weight of 4 lbs. in a spring balance, but let it be upheld by a string fastened to a firm support in such a way that the scale of the balance shall show only 1 lb. By now cutting thia string with a pair of scissors, we suddenly apply 4 — 1=3 lbs. ; and the weight will descend rapidly, until, for an instant, the scale shows about 1 + twice 3 = 7 lbs. In other words, the load of 3 lbs. applied suddenly (but without jar or shock) has produced nearly twice the stretch that it could produce if addsd grain by grain, as in the shape of sand. For, when the load is first applied, the inherent forces, as noticed in Art. 2 (a), are insufl&cient to counteract its stress. Hence the load begins to stretch the spring. The work thus done is equal to the product, suddenly applied weight of 3 lbs. X the stretch of the spring ; and it has been expended (except a small portion required to counteract friction) in bringing the resisting forces into action, thus storing in the spring potential energy nearly sufficient to do the same work ; i. e., to lift the weight (3 lbs.) to the point (1 lb. on the scale) from which it started. But a portion of this energy has to work against friction and the resistance of the air. Therefore the weight does not rise quite to its original height. The shortening of the spring nearly to its original length has now reduced its inherent forces almost to zero ; and the weight again falls, but not so far aa before. It thus vibrates through a less and less distance each time, and finally comes to rest at a point (4 lbs. on the scale) midway between its highest and lowest positions (1 lb. and 7 lbs.) Thus, within the limit of elasticity, a load applied suddenly (though without shock) produces temporarily a •tretcli nearly equal to t\>rlce tlutt wliicl& It could produce It applied gradually ; i. c, twice that which it can maintain after it comes to rest. Renkark. If the load is added in small instalments, each applied suddenly^ then each instalment produces a small temporary stretch and afterward main- tains a stretch half as great. Thus, under the last small instalment, the bar Stretches temporarily to a length greater than that which the total load can maintain, by an amount equal to half the small temporary stretch produced by the sudden application of the last small instalment. (c) The Modulus of E21astlc Resilience (often called, simply, th# Modulus of Resilience) of a material, is the work done upon one cubic inch of it by a gradually applied load equal to the elastic limit. Or, Modulus stretch in inches mean stress ivf TwtriioTino "= P*^ ^^<^^ ^f l^i^gif^ X in lbs. per square inch oi resuience ^^ ^^^ elastic limit causing that stretch If^ as is usually done, we assume this mean stress to be >^ the elastic limilk then, by formula (6) Modulus elastic limit ^ ly rf„«*,v KmM Of resiUence - ^.odulus of elasticity ^ ^ ^^^^'^ "^ ^^ ly square of elastic limit * modulus of elasticity The elastic resilience of any piece is then Eesilience = modulus of resilience X volume of piece in cubic incheisr STRENGTH OF MATERIALS. 461 The modulus of resilience of a material is a measure of its capacity for resist- ing shocks or blows. Elastic Ratio. The elastic ratio of a material is the ratio between its elastic limit (Art. 4 a, and its ultimate strength (Art. 2 c. Thus, if the ultimate tensile strength of a steel bar be 70,000 pounds per square inch, and its elastic limit in tension 39,900 pounds per square inch, its elastic ratio is 39,900 ^ ^^ ■70:000 = ^•^^•. - Inasmuch as it is now generally conceded that the permissible working load of a material should be determined by its elastic limit rather than by its ulti- mate strength, it follows that, other things being equal, a high elastic ratio is in general a desirable qualification ; but, on the other hand, it is possible, by modifying the process of manufacture, to obtain material of high elastic ratio, but deficient in " body " or in resilience — i. e., in capacity to resist the etfect of blows or shocks, or of sudden application or fluctuation of stress. In the manufacture of steel it is found that the elastic ratio is increased by increasing the reduction of area in hammering or rolling, and that the rate of increase of elastic ratio with reduction of area increases rapidly as the reduc- tion becomes very great. This is indicated by the following experiments by Kirkaldy on steel plates :* Plates 1 inch thick, mean elastic ratio 0.53 ' "' " " " 0.53 " 0.54 " 0.61 ^ * Annual Report of the Secretary of the Navy, Washington, 1885, Vol. I. p. 499; and Merchant Shipping Experiments on Steel, Parliamentary Papar, d 2897, London, 1881. 462 STRENGTH OF MATERIALS. Art. 6. A section may be weakened by increasing its \¥idtli. On pp 400, etc., we cousidered the case where the width o^ ...he base is fixed and where the point of application of the resultant of the forces acting upon it is shifted to different positions along the base. We will now notice the case where the resultant is applied at a constant distance from one end of the base, but where the base is of varying width, so that this constant distance may be equal to, or greater or less than, the half width of the base. Let Fig. E represent a side view of a bar of uniform thickness = 1,* but (as Scale of Unit Pressures m i g a C e 1.75 1.50 \ \ F te-i p. \ y \ "■ — 1.00 0.75 0.50 V] / "*>-^. ^1 ^i-e !«tti- 'Sftl -lov ?s ige_ K ■V^ / / \ \ ^ .^ / / \ ^ _^ lent ^un. t pr ;g8U •es \^ ^x^ .^^ Ujdt prcj T»PP ere Ige- O.ftO 1 i »sur< .sat 0.25 0.50 0.7$ 1.00 1.25 1.50 2.00 2.50 S.OO i.00 Widths (ab=-l) shown) of varying width, and subjected to pressures, the resultant P of which is = 1,* and passes through the center of that section ab whose width is 1.* f * We adopt the value 1 for the pressure P, the width a b, and the thickness merely in order to facilitate the explanation. It is not essential to the applica- tion of the principle. t We here suppose ourselves dealing with a perfectly rigid and homogeneous material. In practice, these values would be more or less modified by yielding of the particles under stress, by unevennesses in the surfaces of the supposed cross sections, etc. Nevertheless, the general principles here laid down hold good. STRENGTH OF MATERIALS. 463 The pressure per unit of area of cross section (or "unit stress") in the section j> p a 6 is then ~rTT^ = --= P= 1*, and may be assumed to be uniformly dis- tributed over it. But at other sections of the bar the resultant is nearer to one edge than to the other, and the unit stress can then no longer be assumed to be uniformly dis- tributed over the cross section, but, as explained on pp. 400 to 403 is a'max- imum at the edge nearest to the resultant, and diminishes gradually and uni- formly to a minimum at the farther edge. This is indicated by the shaded" triangles, etc., in Fig. E and' by the curves in Fig. F, which show, for the several sections in Fig. E, the mean unit stress* and the unit stresses at the upper and lower edges respectivelv, calculated by the rules on pp. 231 a and 231 b. These stresses are also given in the following table: P IJiiit {Stresses in Fig. E ; the unit stress -- in section a b being taken as l.f ab ° Stress per unit of area of cross section. Width Mean. At lower edge m e. At upper edge nf. 4.00 0.25 0.8125 — 0.3125 t ef 3.00 Vb 1.00 -Vz 2.50 0.40 1.12 -0^32 2.00 0.50 1.25 — 0.25 cd 1.50 % 1^ 1.25 0.80 1.28 0.32 ab 1.00 1.00 1.00 1.00 gh 0.75 1^ 2% ikt 0.50 2.00 - n 8.00 mn^ 0.25 4.00 — 32 J 40.00 It is important to notice that for a given force P, and for widths less than 3 a 6, the strongest section of this bar is not the widest one, but that (a 6) at which the resultant P passes through the center of the section. In other words, a bar may be weakened by additions to its cross section if those additions are such as to cause the resultant of the pressures to pass else- where than through the center of any cross section. This fact is entirely inde- pendent of the weight of the added portion. Among the sections wider than ab, the weakest is that {cd) whose width is = 1.5 a 6. At that section the lower edge we has its maximum unit stress ( = 1/^ X — 7 ) while at d in the upper edge there is no pressure. Beyond c d the upper edge w/is in tensionX and the unit pressure along me decreases, be- p coming again = —r at ef, where the width e/ is = 3 a 6, and decreasing still further with further increase in width. * In the case discussed on pp. 400 to 403, the mean pressure un= — , uv remained constant so long as the entire surface uv was called into play. Here, on the contrary, the area of the section varies. Hence the mean unit pressure varies also, and inversely as the area. t See foot-note *, p. 462. X In the present discussion, as well as in that on pp. 400 to 403, we have assumed cases of coru,pression for illustration, but the principle involved applies equally to cases where the force applied is tennile. In such cases, however, the terms "pressure " and " tension " are of course reversed. ^ The unit stresses at the edges in section i k are too great to be shown con- veniently in either figure ; while those in section m n (as the table shows) far p exceed the limits of the figures. The pressure at k would be — = oc (infinity) were it not for the tensions in the lower part of the section. 464 STRENGTH OF MATERIALS. When the width becomes less than that at ab, as at g h, etc., the upper edge o the bar comes nearer to the resultant than the lower edge, and hence receive the maximum pressure. When the width = %ab, as at g h, the distance of the resultant from th upper edge is V^ the width of the section. The pressure at the lower edge i PI P then = ; the mean pressure in a A is — - X 7rir~ — 1/^ X ~t> ^^^ ^^^ pressure a ah v.io ao p the upper edge is twice the mean pressure in g A, or 2% X ~^. When the width becomes less than % a &, as at i A; and m n, the pressure at th lower edge me becomes negative or tensile.* Thus, when, as at iA;, the widt; is — }/^ab, and the resultant passes through the upper edge, the unit pressur P P at that edge is = S x , , while the lower edge sustains a tension of 4 X — ; anc ab ab as the section is further reduced, these stresses are still further and very rapidl increased. The condition of those sections (such as m/i) where the line of the resultan passes outside the section, is similar to that of the section mn of a bent hoo' sustaining a load, as in Fig, G. Figr. o. Messrs. William Sellers & Co., of Philadelphia, had occasion to test a numbc of cast-iron beams, each having a large circular opening, as in the annexe figure. These beams broke, not at the smallest section directly under the cente of the opening, but a little to one side, where the section was deeper, as indicate in Fig. H . STRENGTH OF MATERIALS. 465 Fatig^ue of Materials. In the following articles on Strength of Mate- rials, the ultimate or breaking load is that which will, during its first application, rupture the given piece within a short time. But Wohler's and Spangenberg's experiments show that a piece may be ruptured by repeated appliea- tions of a load much less than this ; and that tbe oftener the load is applied the less it needs to be in order to produce rupture. Thus, wrought iron which re- quired a tension of 53000 lbs per sq inch to break it in 800 applications, broke with 35000 lbs per sq inch applied about 10 million times ; the stress, after each application, returning to zero in both cases. The di/f between the maximum and minimum tension in a piece subjected to tension only, or between the max and min compression in a piece subjected to eomp only; or the sum of the max tension and max comp in a piece subjected alternately to tension and comp ; is called the ran^e of stress in the piece. Stresses alternating between and any point within the elastic limit may be repeated many million times without producing rupture.* For a given number of applications, the load required for rupture is least when the range of stress is greatest. If the stress is alternately comp and tension, rupture takes place more readily than if it is always comp or always tension. That is, it takes place with a less range of stress applied a given number of times, or with a less number of applications of a given range of stress. For a given range of stress and given number of applications, the most unfavorable condition is where the tension and comp are equal. The above facts are now generally taken into consideration in designing members of important structures subject to moving loads. For instance, Mr. Jos. M. Wilson, C. E., Mem. Inst. C. E. (London Eng.), Mem. Am. Soc. C. E., uses the following formulae for determining the "permissible stress" in iron bridges, in lbs per sq inch ; in order to provide the proper area of cross section for each member. , For pieces subject to one hind of stress only (all comp or all tension) , /, mm stress in the piece\ k = wf ( IH 1 ■ ^. . — I V max stress m tbe piece/ piece/ For a piece subject alternately to comp and tension, find the max comp and th« max tension in the piece. Call the lesser oi these two maxima "max lesser", and the other or greater one, " max greater ". Then , [ ^ max lesser \ a =» uf 1 1 I \ 2 max greater/ For a piece whose max comp and max tension are equal, this becomes »-«t(.-4)-|- The above a is the permissible tensile stress in lbs per sq inch on any mem- ber; but the permissible compressive stress is found by " Gordon's formula" for pillars, p 495, using a (found as above) as the numerator, instead of/. For a in the divisor or denominator of Gordon's formula (which must not be confounded with the a of the foregoing formulae) Mr. Wilson uses for wrought iron : when both ends are fixed 36000 when one end is fixed and one hinged 24000 when both ends are hinged 18000 Experiments show that materials may fail under a long: continued stress of much less intensity than that produced by the ult or bkg load. * This does not always hold in cases where the elastic limit has been artificially raised by jM-ocees of manufacture, etc. Oft-repeated alternations between tension and compres- sion below such a limit reduce it to the natural one. A slight flaw may cause rupture under comparatively few applications of a range of stress but little greater, or even less, than the elastic Hmit. Rest between stresses increases the resisting power of a piece. In many cases, stresses a little beyond the elastic limit, even if oft- repeated, raise that limit and the strength, but render the piece brittle and thus more liable to rupture from shocks ; and a little further increase of stress rapidly lessens, or may entirely destroy, the elasticity. A tensile stress above the elastic limit greatly lowers, or may even destroy, the compressive elasticity, and vice versa. If a tensile stress, by stretching a piece, reduces its resisting area, it may thus reduce its total strength, even though the strength per $q in has increased. Mr. B. Baker finds that hard steel fatigues much faster under re- peated loads than soft steel or iron. t n = 6500 lbs per sq inch for rolled iron in compression = 7Q00 lbs " " " tension (plates or shapes). «7500B>9 '* for double rolled iron in tension (Hnks or rods). 30 466 STRENGTH OF MATERIALS. TRANSVERSE STRENGTH. 1. In Statics, ^^ 285, etc., we discuss the action of external or destnn tive forces upon cantilevers, beams and trusses. We here discuss the reai tion of the internal or resisting forces (stresses) in solid cantilevers ar beams, in order to determine their loads. See also Trusses. 2» Unless otherwise stated or apparent, we assume that the stresses in a parts of the cantilever or beam are within the elastic limit. Conditions of Equilibrium. 3. For equilibrium, the internal forces, and their moments, must balani the external forces and their moments. In other words, if the cantilever < beam be supposed cut by a section at any point, we must have (1) 2 vertical forces = (2) 2 horizontal forces = (3) 2 moments — Or: (1) Algebraic sum of the internal vertical stresses == algebraic sum of tl: external vertical forces on either side of the section ; (2) Sum of horizontal tensile stresses = sum of horizontal compressv stresses; and (3) Algebraic sum of the moments of the internal stresses =» algebra sum of moments of external forces on either side of the section. 4. Cantilevers and beams of uniform cross-section have usually a supe abundance of strength against shearing, and fail (if at all) near tl middle, where the bending moment is greatest. Hence the discussion ^ their resistance turns principally upon equilibrium of moments. For the resistance to vertical shear, see Statics, ^% 325, etc., and p. 499. See ah Horizontal Shear, ^ If 51 to 53, below. 5. For equilibrium, therefore, the resisting moment, R ( = the sum of tl resisting moments, r, of all the particles in any cross-section of the cant lever or beam, Fig. 1 or 2), must be equal to the bending moment, M, or algi braic sum of the moments of all the external forces on either side of ti section. Reactions of Fibers. 6. In a truss or framed beam (see Trusses) the resistance of each of i1 two chords is regarded as acting in a line passing through the centers of gra^ ity of the cross-sections of the chord; but, in a solid cantilever. Fig. 1, c beam. Fig. 2, the total resisting moment is the sum of the separate resistin moments of the several fibers throughout the cross-section. Neutral Surface. Neutral Axis. 7. When a cantilever (or beam) bends, the fibers in the upper (or lowei part of each cross-section are extended, while those in the lower (or uppei TRANSVERSE STRENGTH. 467 part are compressed (see Figs. 1 and 2) ; the extension and compression being greatest at the top and bottom of the section, and thence decreasing uniformly inward toward a surface, n n. Figs, (a), near the center of the cross-section. In this surface, which is called the neutral surface, the fibers are neither extended nor compressed. The line, o o. Figs. (6), formed by the intersection of the neutral surface with any cross-section of the canti- lever or beam, is called the neutral axis of that section. 8. In order that the algebraic sum of all the horizontal stresses in the cross-section may be zero, as required for equilibrium, the neutral axis must pass through the center of gravity of the section. Hence, the neutral sur- face passes through the centers of gravity of all the cross-sections. 9. The neutral axis may be found by balancing the section (cut out of cardboard) over a knife-edge. Or see Center of Gravity, under Statics, |If ^ 125, etc. Every section has an indefinite number of neutral axes, all passing through its center of gravity in as many different directions. The axis re- quired, in any given case, is that one which is normal to the plane of the bending moment under consideration. ^ In the following discussion, we assume that the neutral axis of the sec- tion is normal to the line of action (usually vertical) of the load", as it gen- erally is. For other cases, as, for instance, the case of roof purlins, see *' The Determination of Unit Stresses in the General Case of Flexure,*' by Prof. L. J. Johnson, Boston Soc. of Civil Engineers, in Jour. Ass'n of Eng'ng Socs., vol. xxviii. No. 5, May, 1902. Resisting Moment. Unit Stress. 10. It is assumed that the extension or compression of each fiber, and therefore the resisting force actually exerted by it, is proportional to its vertical distance, t, above or below the neutral axis. Fig. 3. In Fig. 3. let T = IM the distance from the neutral axis, o o, to the fiber farthest from that axis, either above or below the axis; the unit stress in said farthest fiber; the distance from the neutral axis to any given fiber; the unit stress in said given fiber; the area of said given fiber; the total stress in said given fiber; the resisting moment of said given fiber about the neutral axis ; bending moment at the cross-section under consideration; R = the resisting moment of the entire cross-section ; = 2 r = the sum of the resisting moments of all the fibers ; I = the moment of inertia of the cross-section. See *|^ 14, etc.; = 2 f2 a = the sum, for all the fibers, of t- a; I R X = the section modulus, = 7= = -. See t^ 25, etc. Then the unit stress, in any given fiber, is = « = S ;p I its total stress; t ^ F, is = a s = S a 7=; and its resisting moment, r, is == F < == S a 7^. Henoe^ the resisting moment, R, of the entire section, is 2^0 = -J.I. 468 STRENGTH OF MATERIALS. Hence, also. ^ MT MT SI Since, Q rp ^ « " = --, it follows that ^ =^ Y* ^ and R = M= |.I = y.L The resisting moment, R, is = S X, and the moment of inertia, I, =» TX. When beams are tested to destruction, the value attained by S is called the Modulus of Rupture. 11. It will be noticed that the strengths of similar beams of any shape, and those of rectangular beams, whether similar or not, are directly propor- tional to the product, width X square of depth. See % 63. 12. When the stress, S, upon the extreme fibers, is = the elastic limit of the material, failure is imminent. The permissible unit stress is usually taken as not more than half the elastic limit, and the safe load is that under which S does not exceed the permissible unit stress. 13. The same quantity of material that composes a solid beam, Fig. 2, would present greater resistance to bending or breaking if it were cut in two lengthwise along the neutral surface, n n, and converted into top and bot- tom chords of a truss; because, first, the leverage with which the resistance acts is thus greatly increased ; and, second, the depths of the chords are so small, compared with their distances from the neutral axis, that their fibers may be assumed to act unitedly and equally. Hence, practically, all the fibers in the upper chord must be crushed, or all those in the lower pulled apart, at the same instant, before the truss can give way; whereas, in the solid beam, the extreme upper or lower fibers yield first ; then those next to them, and so on, one after the other. Moment of Inertia. 14. Unlike the moment of a force, which is the product of a force and a distance, the moment of inertia, being the sum of the products of areas of fibers by the squares of their distances from the neutral axis, is a purely geometrical quantity. Thus, the moment of inertia of a given section de- pends solely upon the dimensions and shape of that section, and is inde- pendent of the material and the span of the beam and of the manner in which it is supported or loaded. Unit of Moment of Inertia. The moment of inertia of a figure being the product of an area by the square of a distance, its unit is the fourth power of a unit of length. Thus, in a rectangle 3 ins. wide and 4 ins. deep, I == -j-— = — ~ — ■ = -—^ = 16 biquadratic inches = 16 inch*. 1 X 6 3 In a rectangle 1 inch wide and 6 ins. deep, I = — ^ — = 18 inch*. 15. Comparing similar sections of any shape, their moments of inertia (like their strengths, see 1[ 11) are proportional to the product, breadth X cube of depth. 16. The following illustrated table, pp. 469-471, gives, for several figures of frequent occurrence, (1) I = the moment of inertia = ^ t- a; (2) T = the distance from the neutral axis to the farthest fiber; (3) X == the section modulus = -_, = — =- == "5^ = a"J (4) A = the area of the cross-section. 17. In sections where the distance from the neutral axis to the lower- most fiber, and the corresponding section modulus, differ from those (T Jmd X) pertaining to the uppermost fiber, those corresponding to the owermost fiber are distinguished as T' and X' respectively. 18. In each figure the neutral axis is indicated by a horizontal line «t)ssing the section. MOMENTS OF INERTIA. 469 Moments of Inertia^ etc. I Moment of inertia. Distance frora neutral axis to farthest fiber T Section, modulus. ■Til BD3 12 6 ^~ ^ 1 B (D3_d3) B (D^-d^) U—n-J -11 12 B3 12 B ^B« 12 = 0.118 B ^ U i^^ B4_6* 12 B^-&^ GB B^b^ 12 BPS 36 V2 3 12 B B^-b-^ BD- 21 470 STRENGTH OF MATERIALS. eT* c 1 g" Q «L "11 II li t=l ^ t= ^ "^ § "^ eo 1 Q eo «• g- Q S « « CO t: 00 *~ k "Ml J 1 5 o s; M « 3 t5 II « '^ f ® a s t 1.^^ « P3 2 g« «|o« o|« § 1 i i "ill « 1 1 i S Erf H fil« o « e ■« "« 5 «« M< ■* 00 « Q 1:^ o § 1 i 1 O o b II 1; t il II ;« ^[^ -.n 1» 1 w— S^-> ^ Ij fi i' r *N >, 1 1 ^! s 1 \*-^-^ h-hJ !*5.^h>l k^H-*!*^* 00 c Id s ; -1 MOMENTS OF INERTIA. 471 O o II 5 ^ II o "^ P '53 '^ S O 0) ftl" k-h-W h-^^— >i RS3 I »]0 B$l ^^ l^ts-» 472 STRENGTH OF MATERIALS. 19. The moment of inertia of any figure, about its neutral axis, is the sum of the moments of inertia of its several parts, about that same axis. 30. Let I = the moment of inertia of the entire figure about its neutral axis, o o; i = the moment of inertia of any part, about the neutral axis, o o, of the entire figure; m == the moment of inertia of that part, about its own neutral axis; 'a = the area of that part ; t = the distance of its center of gravity from the neutral axis, o o, of the entire figure. Then 1 = 2 i; and i = m +at^, 21. Thus, in Fig. 4, ti = ^ + B D h^t and I = 1*1 + ^. 12 *-&-^ ^ ".rz.vi*?.. [< -B ^ IC Tig. 4. Figr. 5. 22. Hence, in any hollow section, as in the hollow rectangle, Fig. 5, let I' =■ the moment of inertia of the whole figure (including both the shaded and the unshaded rectangles), i = that of the missing or unshaded rectangle, and I = that of the shaded portion; all referred to the neutral axis, o o, of the shaded portion. Then 1 = 1' — i. Figr. 23. In the case of an irregular section, as Fig. 6, let the section be divided into numerous strips, parallel to the neutral axis and narrow enough to be considered as rectangular; and proceed as in m[ 19 to 21. TRANSVERSE STRENGTH. 473 34. The narrower the strips are taken; the less m becoiies. If the strips be taken so narrow (relatively to the depth of the section) that m may be neglected, then I = 2 i'-^ a, as in H 10. The strips need not be of uniform width. The Section Modulus. S 35« Definition. If the resisting moment, R = -^ . 2 <2 ^^ be divided by the unit stress, S, in the extreme fibers, the quotient, X = -^ = „ = 7p, is called the Section Modulus. This, like the moment of inertia, ^^ 14, etc., is a purely geometrical quantitj> depending solely upon the dimen- sions and shape of the section, and beLig independent of the material, of the span, and of the manner of loading. 36. Having the section modulus, X, we have orjy tc multiply it by the unit stress, S, in the extreme fibers, in order ta obtain the resisting moment, R; or R =- S X. 37. Multiplying the section modulus, X, by the distance,^ T, from the neutral axis to the farthest fibers, we obtain the moment of inertia, I; or, I =TX. 38» The section modulus is usually given in tables of rolled beams, chan- nels and shapes. See tables of Carnegie Beams, etc. Lioi.ding. Strength. 39. The following illustrated tablt gives the maximum moment, M, corresponding to a given load, W, and th" load, W,* corresponding to a given imit stress, S, for different conditions of support and of loading. In this table, M — maximum bending moment ; W = the total extraneous load * on the beam, whether concentrated' at one point (as shown) or uniformly distributed over the span ; I =a the span; S = the unit stress, in the fibers farthest from the neutral axis, due to the extraneous load, W; * T =« the distance from the neutral axis to the farthest fibers; I = the moment of inertia. In rectangular beams, h = breadth; d = depth; I == moment of inertia =» -y^" » "" " S6d2- Of the two diagrams under each loading, the first represents the mo- ments, and the second the shears, in the several parts ^f the span. 30. If S = the permissible unit fiber stress, then, in the foregoing formulas, W = the permissible extraneous load.* 31. It will be noticed that the strengths of similar beams are proportional to their values of — y- ; i. e., the strengths of beams of similar cross-sections are directly proportional to their breadths and to the squares of their depths, and inversely proportional to their spans. * The beam is here supposed to be without weight. See ^f 42, etc. 474 STRENGTH OF MATERIALS. For symbols, see opposite page. Maximum bendlnff moment. General [u rectangular beams 1 T/ '■'/// ^^^-^^iSiill 1 iiiiiiiiiiijiiiiininiiin 1 At Support. I r i At jj.._WZ Support. 2 W=2-y- ^ ~ 4 b_df bd^ A M-^ _2S . Ad^ TF^ "^^^i^ At Center. I !5a_ ^^ iiiiiiiiiiiiiiiiii i At Center and at Support. ly—s 2S bd^ 3 * Ti 4S bd 3 { 1 *^-«iiiinu[llUI« Support. 12 bd' 11 bd' ' I TRANSVERSE STRENGTH. 475 Symbols in Table Opposite. S =» unit stress, due to W, in the extreme fibers; T = distance from the neutral axis to the extreme fibers; I = moment of inertia; M = maximum bending moment; W == load; . I = span. In rectangular beams, b = breadth, „ _ _W_L . ' W ~n^^ d - depth, ^ S6d2' w-7ib^. Beam 1 Inch Square, 1 Foot Span. 32. In a beam, 1 inch square and 1 foot (12 inches) span, supported at both ends, we have, for the extraneous center load : * W = S.-3^ ^— = S.^3^ = -; and S = 18 W. 33. For any other rectangular beam, let L = the span in feet. Then the extraneous center load,* W, required to produce the same unit stress, S, in the extreme fibers, is 34. Thus, for yellow pine, let S — the permissible unit stress « ~ the 3240 elastic limit == — ^ = 1620 lbs. per sq. in. Then, for a beam 1 inch square, 1 foot span, supported at both ends and loaded at center, the permissible load,* W, is ^, nS S 1620 -_^ ^ ==-12"r8'=-T8 =^^^^" and, for a joist, 3 X 12 ins., 20 ft. span; the permissible extraneous ♦ center load is W = W'^ = 90 X ^ ^Q ^^ « 19441bs.; and the permissible extraneous uniform load is = 2 W = 2 X 1944 = 3888 lbs. 35. If the load, W. the span, L, and the coefficient, W, are given, we W L have h (P ^ -^n-- Thus, in the case of the yellow pine joist, mentioned W in ^ 34, of 20 ft span, with a uniform extraneous * load, where 2 W = W T 1 Q44 y 20 2 X 1944 = 3888 lbs.,- we have h d^ = -^ - s,?^-^ = 432. w yo 36. Then, if either 6 or d is given, the other is easily found. If not, assign, to either of them, an arbitrary value. Thus, if 6 = 6, we have

      < '^^*^^ ^°ff ^° ?>^- . . Breadth in inches X Square of depth m inches The constant, for -vrooden beams, may be had, near enough for common practice, by taking one third of the breaking constants in the tabU, page 476. Said constant, thus calculated, is the elastic limit of a beam of the given shape and material, 1 inch broad, 1 inch deep, and of 1 foot span, supported at both ends and loaded at the center. To obtain from it the elastic limit of any other beam of the same design J and the same material, similarly supported and loaded, but of other dimensions, Clastic ^ constant X ^^^'^^^ ^^ inches X square of depth in inches ; **"*** span in feet Multiply the If the beam is result by supported at both ends and loaded at center, 1 « it II « ti a uniformly, 2 fixed t " " " " " at center, 2 " " " " " " uniformly, 3 " ** one end " ** at other end, JiT " *' " ** " " uniformly, 3^ ♦ Of course, in practice, it is frequently difficult to ascertain with precision, when, or under what load, the deflections actually do begin to increase more rapidly than the successive loads. For although hy theory the deflections are practically equal for equal loads, until the elastic limit is reached, yet in fact they are subject to more or less irregularity; for no material composing a Ijeam is perfectly uniform throughout in texture and strength. Hence, instead of regular increase of deflec- tion, we shall have an alternation of larger and smaller ones. Therefore, Home judg- ment is required to determine the final point; in doing which, it is better, in case otf doubt, to lean to the side of safety. It is assumed always that the load is not subject to jars or vibrations. These would increase the deflections. t A beam is said to be "fixed " at either end when the tangent to the longitudinal axis of the deflected beam at that end remains always horizontal. X The shapes of the two beams need not be similar. For instance, the constant deduced from experiments upon any rectangular beam is applicable to any other rectangular beam, whether square or oblong. DEFLECTIONS. 483 The Elastic Curve. 59. When a cantilever. Fig. 1, 9r a beam, Fig any manner, bends, under the action of any load forms a curve, such that, at any section, E_I ^ I^ M M k' 2, supported or fixed in the neutral surface, n n. where R M I the radius of curvature, at the section ; the bending moment, at the section; the moment of inertia of the section : q the elasticity coefficient of the material, = -77 ; any unit stress within the elastic limit ; the unit "stretch" (elongation or compression) produced by S in the given material. (a) (b) Fig. 1 (repeated). Fig. 3 (repeated). The Deflection Coefiicient. 60. Definition. The deflection coefficient, for any given material, is the deflection, in inches, of a beam, of that material, 1 inch square and of 1 foot span, supported at each end, and carrying, at its center, an extraneous load of 1 lb. — f w', where w' = weight of clear span of beam alone, in lbs. 61. Let y = the deflection coefficient for any given material. Then, in any rectangular beam, of the same material, with center load or uniform load, let b = the breadth, in inches; d = the depth, in inches; L = the span, in feet; w = the weight, in lbs., of the clear span of the beam itself; W = center load + i w; = f (uniform load + w). Then, in the given beam, W L3 Deflection = Y = y - ' Breadth* = 6 = W. Load W Depth * ds.Y* 3/wT. '^l'bY 63. The deflection coefficient, y, for any given material, is obtained by experiment, thus: At the center of any rectangular beam, of the given mate- rial, placed horizontally upon two supports, at any convenient and known distance apart, place any load that is within the elastic limit, and measure the resulting deflection, Y, Let W = the extraneous center load + ^ w, where w = the weight, in lbs., of the clear span of the beam itself. Then the deflection coefficient is 2/ = Y . ;j • W . L3' where b and d = the breadth and depth, in inches, and L feet, of the experimental beam. ■ the span, in * In calculating the breadth or the depth, if it is necessary to provide for the weight of the beam itself, we first let W = the extraneous load only, and then proceed by successive approximations, as in ^1i[ 45 and 46, remem- bering, however, that in the case of deflections, 5-8 of the weight of each additional section is to be taken as equivalent center load, and not 1-2 as in the case of strengths. 484 STRENGTH OF MATERIALS. 63. The ratio between any two homologous lines, in any two similar figures, is constant. Hence, in determining or using coefficients, whether for strength or for deflection, by comparing beams of similar sections but of different sizes, we may use any two homologous lines in place of the two breadths, or in place of the two depths, or the same line may be taken in place of both breadth and depth. Thus, in Figs. 11, (a) Fig. 11. B _D _R _ T ~ d~ T ~ ^' Hence, B D2 _ 384 _ _ _ _3 _ R R2 _ 1000 _ ^ _ «,, Td^ -1^ ~^~ ^ - Tt^-'^ - -125' - 8 - 2*. Also, B3 D 1728 _ ,. _ .4 _ R3 R _ 10,000 _ i^ _ 04 64. Deflection coefficients, being the deflections, y, in inches, at the centers, of beams 1 inch square and of 1 foot span, supported at each end and loaded at center with extraneous loads of 1 lb. — 5-8 the weight of the clear span of the beam itself. Average Cast iron, 0.000018 to 0.000036 0.000027 Rolled bar iron, 0.000012 to 0.000024 0.000018 Rolled tool steel, 0.000010 to 0.000020 0.000015 White oak, well seasoned 0.00023 Best Southern pitch pine, well seasoned, \ q oo027 White ash, well seasoned, J Hickory, well seasoned, 0.00016 White pine, wpII seasoned, > Ordinary yellow pine, well seasoned. Spruce, well seasoned, I ^ 000^2 Good, straight-grained hemlock, well sea- j ^.yi\}\3- II n a n g O « 5. f I CQ 4^ S 5 e la! . 1 1 ! .1 o S p g 'S i .• i I i ^ ^ I 1 1 i| =•§ ! iS -^ -a ^H -5 O 9 i «' w *See foot-note (t), p. 493. STRENGTH OF PLATES. 493 100. Rectangular plate, uniformly loaded. Let w = the UDiform load per unit of surface; other letters as in ^ 99. Then, according to Grashof,* 1 L2Z2 W, For a square plate, h = I. Hence, 101. Value of C. For uni- For cen- If the plate is form load. tral load. merely supported along its four edges, C = 1.125 C = 2.00 firmly secured along its four edges, C = 0.75 C = 1.75 103. Circular plate, uniformly loaded. Let w = the load per unit of surface ; S = the unit fiber stress in the material ; E = its elasticity coeflBcient = — r- — - — — -: ; unit stretch r = the radius of the unsupported portion of the plate; t = the thickness of the plate ; Y = the deflection at the center. Then, according to F. Reuleaux.t if the plate is merely supported, If the plate is firmly secured, V — ^ ^ '^ ' S* 6E<3* i^/<\2 J2 w ^ wr* 3 , For strengths of cylinders, pipes, etc., see Hydrostatics, Art. 17. TRANSVERSE AND LONGITUDINAL STRESS COMBINED. 103. Although the combination of longitudinal and transverse stress in the same piece is objectionable, it is often unavoidable. Thus, in a timber roof, the rafters generally act both as columns and as beams. In such cases, the total unit stress, S, in the extreme fibers, is the sura of the uniform stress, Sc, due- to direct compression or tension, and the extreme fiber stress, Sb, due to bending moments only, under the action of the transverse and longitudinal loads combined. Or S = Sc + Sb- Let Mb = the bending moment due to the transverse load; Mc = the bending moment due to longitudinal load, P; and M = the total resultant bending moment, = Mb — Mc when the longitudinal load is tensile; = Mb + Mc when the longitudinal load is compressive. But Mc = P d, where P = the longitudinal load, and d = its leverage, = the deflection of the beam, due to all causes ; and (see ^ 57) d = ^— rr.- J ill L C where I = span, Sb = unit stress in extreme fibers, due to bending; E = modulus of elasticity; T = distance from neutral axis to extreme fibers, and c = a coefficient, whose values, for different cases, are given in ^57. Hence, Mc = P J^^ ; and resultant moment M - Mb + P ^-^ . The h, L c — hi L c resisting moment, R (see ^ 10), is = Sb ^; and, for equilibrium, R = M. I Z^ S Hence, Sb. ws = Mb + P --, ^^ ; whence we derive, for the extreme fiber 1 — ■ hi i. c stress, Sb. due to bending only, under the action of the transverse and longi- tudinal loads combined, * "Theorie der Elasticitat und Festigkeit," Berlin, 1878. t "Der Konstrukteur." Braunschweig, 1889. "The Constructor," trans- lated by H. H. Suplee, Philadelphia, 1893. 494 STRENGTH OF MATERIALS. S _ ^ I where the longi- I Sb = ^ | where the longi- p p >■ tudinal stress is I P Z^ f tudinal stress is I + -pT" I tensile I I — .= — i compressive Besides this we have the unit stress, Sc, due directly to the longitudinal p load, P, and = -r, where A is the area of cross-section of the beam. Hence, A for the total unit stress, S, in the extreme fibers, we have P MbT S = Sc + Sb = -r + . M T When the deflection, d, is negligible, M = ; Sb = — y- , as in 1 10 ; and P M T ^ = A + ~r- It is often assumed that the resultant unit stress, S, in the extreme fibers, is equal to the sum of the longitudinal and transverse unit stresses, and the piece is then so designed that the resultant unit stress, so obtained, shall not exceed the permissible unit stresg. STRENGTH OF PILLARS. 495 STRENOTH OF PII^IiARIS. The foregoing remarks on crushing or compressive strength refer to that of pieces so short as to be incapable of yielding except by crushing proper. Pieces longer in proportion to their diameter of cross section are liable to yield by bending sideways. They are called pillars or columns. The law governing the strength of pillars is but imperfectly understood; and the best formulae are rendered only approximate by slight unavoidable and un- suspected defects in the material, straightness and setting of the column. A very slight obliquity between the axis of a pillar and the line of pressure may reduce the strength as much as 50 per cent; and differences of 10 per cent or more in the bkg load may occur between two pillars which to all appearances are precisely similar and tested under the same conditions. Hence a liberal factor of safety should be employed in using any formulae or tables for pillars. In our following remarks on this subject, the pillars are supposed to sustain a constant load; and the ultimate or breaking load referred to is that one which would, during its first application, cripple or rupture the pillar in a short time. But struts in bridges etc often have to endure stresses which vary greatly in amount from time to time. Their ultimate load is then less. Long pillars with rounded ends, as in Fig 1, have less strength than those with flat ends, whether free or firmly fixed. In steel bridges and roofs, the ends of the struts are frequently sustained by means of pins or bolts passing through (across) them, at either one or both ends. These we will call liing-ed Figr. 1. ends. Pillars so fixed are about intermediate in strength between ' those with flat and those with round ends. There is much uncer- tainty about this and all such matters. The strength of a given ■n-mm hinged-end pillar is increased to an important extent by increas- ^ '^^ ing the diameter of the pin. W/%^ The formula in most general use for the strength of pillars, is that attributed to Prof. I^ewis Oordon of Glasgow, and called by his name. With the use of the proper coeflScients for the given case, it gives results agreeing approximately with averages obtained in practice with pillars of such lengths (say from 10 to 40 diams) as are commonly used. It is as follows Breaking load in fts per sqinch „ / of area of cross section of pillar /2 in which r^a f is a coefficient depending upon the nature of the material and (to some extent) upon the shape of cross section of the pillar. It is often taken, approximately enough, as being the ult crushing strength of short blocks of the given material. For good American wrought iron, such as is used for pillars, 40000 is generally used ; for cast iron 80000. Mr. Cleeman* found for mild steel (.15 per cent carbon) 52000 ; and for hard steel (.36 per cent carbon) 83000 lbs. Mr, C. Shaler Smith gives 5000 for Pine. a, for wrought iron, is usually taken as follows: a = when both ends of the pillar are flat or fixed 36000 to 40000 when both ends of the pillar are hinged 18000 to 20000 when one end is flat or fixed, and the other hinged... 24000 to 30000 For cast iron about one eighth of these figures is generally used ; and for pine about one twelfth. 1 is the length of the pillar. If the pillar has, between its ends, supports which prevent it from yielding side-ways, the length is to be measured between such supports. r is the least radius of gyration of the cross section of the pillar. I and / must be in the same unit ; as both in feet, or both in inches. * Proceedings Engineers' Club of Phila, Nov 1884. 496 STRENGTH OF PILLARS. Radius of g^yration. Suppose a body free to revolve around an axis which passes through it in any direction ; or to oscillate like a pendulum hung from a point of suspension. Then suppose in either case, a certain given amount of force to be applied to the body, at a certain given dist from the axis, or from the point of sus- pension, so as to impart to the body an angular vel ; or in other words, to cause it to describe a number of degrees per sec. Now, there will be a certain point in the body, such that if the entire wt of the body were there concentrated, then the same force as before, applied at the same dist from the axis, or from the point of suspen- sion as before, would impart to the body the same angular motion as before. This point is the center of gyration ; and its dist from the axis, or from the point of sus- pension, is the Radius of gyration, of the body. In the case of areas, as of cross* sections of pillars or beams, the surface is supposed to revolve about an imaginan^ axis ; and, unless otherwise stated, this axis is the neutral axis of t^e area, whick passes through its center of gravity. Then Radius of g'y ration = l/Mon^ent of inertia -7- Area Square of radius of g'y ration = Moment of inertia t- Area In a circle, the radius of gyration remains the same, no matter in what direc- tion the neutral axis may be drawn. In other figures its length is diflerent for the different neutral axes about which the figure may be supposed to be capable of revolving. Thus, in the I beam, page 893, the radius of gyration about the neutral axis X Y is much greater than that about the longer neutral axis A B. In rules for pillars the least radius of gyration must be used. The following formulae enable us to find the least radius of gyration, and the square of the least radius of gyration, for such shapes as are commonly used for pillars. Shape of cross section of pillar Solid square liCast radius Square of M^v o.^»a«iAn least radius of gyration of gyration side2 Hollow square of uniform thickness Solid rectangle side 1/12* least side 1/~I2~* 12 D2 + < least side2 r — A- l^2^22z^2^T Hollow rectangle of uniform thickness \ 12 (C A — c a) 12 (C A — e a) v; C8A — c3a C3A — c3a Solid circle diameter diameterj^ 16 J.-D-H Hollow circle of uniform thickness ~l6~ D2 + (j2 16 * >/l2 = about 3.4641. STRENGTH OF PILLARS. 497 The followiiig^ are oikly approximate : Shape of cross section I^ea«t radius , Square of of pillar of gyration least radius of gyration -F-^- Phoenix column. Carnegie Z-bar column. I beam. D X 0.3636 D2 X 0.1322 B X 0.590 B2 X 0.348 F 4.58 F2 21 Channel. F 3.54 F2 12;5 Deck beam. F 6 F2 36.5 Angle, with equal legs F 5 F2 25 ^. Angle, -with unequal legs Fj F2/2 2.6 (F+/) 13 (F2 + /2) f h^id 32 T, with F = / Cross, with F — / F 4.74 F2 22.5 498 STRENGTH OF IRON PILLARS. The young* engineer must bear in mind that the breakg and the safe loads per sq inch, of pillars of any given material, are not constant quantities ; but diminish as the piece becomes longer in proportion to its diam. If a very long piece or pillar be so braced at intervals as to prevent its bending at those points, then its length becomes virtually diminished, and its strength increased. Thus, if a pillar 100 ft long be sufficiently braced at intervals of 20 ft, then the load sustaine*' may be that due to a pillar only 20 ft long. Therefore, very long pillars used ^.c. bridge piers, &c, are thus braced; as are also long horizontal or inclined pieces, exposed to compression in the form of upper chords of bridges ; or as struts of any kind in bridges, roofs, or other structures. Mistakes are sometimes made by assuming, say 5 or 6 tons per sq inch, as the safe compressing load for cast iron ; 4 tons for wrought; 1000 pounds for timber; without any regard to the length of the piece. But although the final crushing loads, as given in tables of strengths of materials, are usually those for pieces not more than about 2 diams high, they will not be much less for pieces not exceeding 4 or 5 diams. Cautions. Remember a heavily loaded cast-iron pillar is easily broken by a side blow. Cast-iron ones are subject to hidden voids. All are subject to jars and vibrations from moving loads. It very rarely happens that the pres is equally dis- tributed over the whole area of the pillar; or that the top and bottom ends have per- fect bearing at every part, as they had in the experimental pillars.f Cast pillars are seldom perfectly straight, and hence are weakened. Hollow pillars intended to bear beavy loads should not be cast with such mouldings as a a ; or with very projecting bases or caps such as g. Fig 19. It is plain that these are weak, and would break off under a much less load than would injure the shaft of the piliar. When such projecting ornaments are required, they should be cast separately, and be at- tached to a prolongation of the shaft, as cd, by iron pins or rivets. Ordinarily, it is better to adopt a more simple style of base and cap, which, as at 6, can be cast in one piece with the pillar, without weakening it. Fi^. 19. Fig. 20. When a flat-ended pillar, Fig. 20, is so irregularly fixed, that the pressure upon it passes along its diagonal a a, it loses much of its strength. Hence the necessity for equalizing, as far as possible, the pressure over every part of the top and bottom of the pillar ; a point very diflacult to secure in practice. t In important cases both ends should be planed perfectly true. SHEARING AND TORSION. 499 ISHEARINO STRENGTH. Shearing or detrnsion occurs when a body is acted upon by two opposite forces in parallel and closely adjacent planes, tending to slide some of the particles over the others. In Fig 1, the twa forces are (1) the downward pressure of the weight, W, and (2) the upward reaction of the support, A. In sing'le shear. Fig 1, the shearing area, a, = the section ^r^. In double sbear. Fig 1, a = gg^oo = 1y^gg. In Fig 3, a = 6 X cross section of piece. In Fig 4 (single shear), a = section c c. In punching rivet holes, a =i circumference of hole X thickness of plate. In any case, if S = the ultimate unit shearing stress, Shearing strength = S a. Fig. 1 Fig. % Fig. 4 Ultimate miit shearing stress, S, in ibs per sq inch. The following figures indicate the range of values of S in metals and in timber. Metals. Wrought iron, 35,000 to 55,000; cast iron, 20,000 to. 30,000; steel, 45,000 to 75,000 ; copper, 33,000. „r-.u .r. • t^. . * With the grain. Fig 4. Across the grain. " Spruce 250 to 500 3,250 White pine " 2,500 Timber : ) Hemlock ' * * * From our experiments : '^ Yellow pine 4,300 to 5,600 Oak 400 to 700 White oak 4,400 TORSIONAIi JSTRENOTH. Torsion occurs when a body is acted upon by two couples or moments of contrary sense and in different planes. Thus, torsion takes place in a brake axle when we try to turn it while its lower end is held fast by the brake chain ; and in shafting, when it transmits the motive power of an engine to tools. Sup- pose such a body to be divided, by cross sections, into layers. Then each layer tends to shear across from those next to it. Hence, in order to maintain equi- librium, each two adjacent layers must exert, in the cross section between them, an internal resisting moment equal to one of the two external and contrary torsional moments. Resisting inomeiit in a circular cross section of a cylindrical shaft. Let P = the torsional force of one of the two external moments in pounds ; I = its leverage, = its distance from the axis of the shaft in inches ; M = P Z = external or torsional moment in inch-pounds; T = distance from axis to farthest fibers, = radius of shaft in inches,* D = diameter of shaft, = 2 T in inches; S = unit shearing stress in farthest fibers in pounds per sq inch ; t = distance from axis to any given fiber in inches ; s = unit stress in said given fiber in pounds per square inch ; a = area of said given fiber in square inches ; F = total stress in said fiber in pounds ; r = resisting moment of said fiber about the axis in inch-pounds ; R = internal resisting moment of the entire cross section =^ 2 r = sum of resisting moments of all the fibers in inch-pounds; Ip= polar moment of inertia* of the cross section = moment of inertia of cross section about the axis of the shaft = 2 ^2 a^ ^ the sum, for all the fibers, of f^ a in inches. *Tn any figure, the polar moment of inertia, Ip, is = the sum of the greatest and the least moments of inertia of the same figure, about two axes lying in the figure and intersecting in its center. In a solid circle, each of these is a moment oif inertia about a diameter, and is = tt T* -i- 4. Hence, in such a circle, I = TT T4 ^ 2. 500 STRENGTH OF MATERIALS. Then the unit stress, in any given fiber, is 5 = S / -i- T ; its total stress, F, is = as = Sat-7-T; and its resisting moment,- r, is == F t ^= S a t^ -r- T. Fof equilibrium, the internal resisting moment, R, of the entire section, must be =s the external torsional moment, M. Hence, for the Internal Keisisting' moment, R, we have : R = M == 2 r =- 2 S a r^ -- T = (S -- T) ^ t^ a = {S -r- T) Ip. Hence, also, S = M T ^ Ip ; M = S Ip -^ T ; and P = M ^ Z = S Ip -4- (T Z). In a solid circle, Ip = tt T* -r- 2. Hence, S = 2MT-=-(7rT4)=i 2 M -V- (tt T3) ; M = S TT T3 -- 2 ; R = S tt T3 -^ (2 /) ; and Diameter, D, --xV^^'=V'-^ = x.,.Vf. For approximate ultimate values of S, for torsion, use the values for shearing, p 499, with safety factors from 5 to 10. Horse power of shafting-. In one revolution, the force, P Bbs, de- scribing a circle with radius = I ins, does a work = 2 tt Z P inch-ibs, and, in n revolutions, work = 2 ir I F n inch- lbs. If n be the number of revolutions per mimcle, the horse power is : H = 27rZPri-=-(12x 33,000) == 2 tt M w -f- (12 X 33,000) ; or, since F I = M = R == S Ip -i- T, we have : II == S TT 71 Ip -^ (12 X 16,500 T) ; and S = 12 X 16,500 T II -4- (ttw Ip). In a solid cylindrical shaft, Ip = tt T^ ^ 2. Hence, H = S TT n TT T4 -4- (12 X 33,000 T) = S n'^n T3 ^ (12 X 33,000) ; S = 12 X 33,000 H -- (7r2 71 T3) = 12 X 3,843 H ^ (w T3) ; n = 321,000 H -r- (D3 S) : and l>lame*«r, D = 2 T = 2 x - !'1^X. (^ nN. This will be seen by referring to the pressures figured on the left side >X,,^;^ ^^^ ^^ ^^S ^' where, as stated in Art 2, the surf of plank 1, exposed to \S"ri^V "■ >. the pres on the left side, is 20 sq ft ; that of planks 1 and 2, 40 sq ft; ^ that of planks 1, 2, and 3, 60 sq ft, &c. All these surfs commence at the level of the water; and all of them being vert, are of course at the same inclination with the water surf; but their depths are re- spectively 1, 2, and 3 ft. The pres against the surf of 1, is 625 fts; that against the surf of 1, 2, is 6254-1875 = 2500; and that against the surf of 1, 2, 3, is 625 + 1875 + 3120=5625. But 2500 is/oMrtimes 625 ; and 5625 is nine times 625. And the pres against the entire surf 8 5, (which is 5 times as deep as plank 1,) is 25 times as great as that against plank 1 ; or 625 X 25 = 15625 fts = the sum of all the pressures marked on the left side of Pig 5. This follows, from the Rule in Art 1 ; for twice the area of surf, mult by twice the vert depth of the cen of grav below the surf, must give 4 times the pres ; three times the area, by three times the depth, must give 9 times the pres, &c. It follows, also, that at any particular point, or against any given area placed at various depths, the pres will increase simply as the vert depth : thus, if there be three areas, each one sq ft, placed in the same positions, but with their centers of grav respectively 8, 16, and 24 ft below the surf, the pres against them will be respectively as 8, 16, and 24; or as 1, 2, and 3. Art. 4. The pressure of quiet w^ater, in any one g-iven di- rection, against any given plane surface, whether vertical, horizontal, or inclined, is equal to the weight of a prismatic column of water, the area of whose section, parallel to its base, is equal to the area of the projection of the given surface taken at right angles to the given direction, and whose height is equal to the vertical depth of the center of gravity of the given surface below the upper surface of the water. Hence the Rule. To find the pres in lbs, mult together the area in sq ft of the projection taken at right angles to the given direction ; the vert depth in ft of the cen of grav of the pressed surf below the upper surf of the water ; and the constant 62.5 fts wt of a cub ft of water. Ex. Let m, c s n. Fig 7, be an inclined surf, sustaining the pres of water which is level with its top m c. Then the total pres against m c s n, and at right angles to it, as found by the rule in Art 1, is an illustration of the pres- ent rule ; because the projection of mcsn, taken at right angles to the give* direction, or parallel to m c » n, is in fact mcsn itself, or equal to it. Henee the rule in Art 1 is merely a simple modification of the present one, appli- cable to the case of total pres against any surf. But if it be reqd to find only the vert or downward pres against mcsn, in pounds, mult together the area of the hor projection aocm in sq ft; the vert depth in ft of the cen of grav of m es n below* the surf; and 62.5. Or if only the Jior pres against mcsn be sought, mult together the area of the vert projection aosn; the vert depth of the cen of grav of to c « n; and 62.5. In Pig 8 also, the total pres against efghis found by rule in Art 1 ; while the hor and vert pressures against it are found as in Pig 7, by using the projec- tions e/ki, and ki g h. In Pig 7 the vert pres is downward ; while in Pig 8 it is upward ; but this circumstance in no respect affects the rule. Rkm. 1. At any given dppth, the pres, perp to anv given surf, is the same In all directions; but Pigs 7 and 8 show that the total pres oblique to a given nni. Q~ surf will be less than the perp one at the same depth ; because an oblique pro- j:1C1 O jection of a surf must be less than the surf itself, which last is the projectioo D when the pres is perp to it. Thus, in a reservoir, the total pres perp to a sloping side, asmns c. Pig 7, is greater than either the vert or the hor pres upon it. HYDROSTATICS. 505 E^9 a f ^^ \o \ > N^ \ Jl m. ^\"i k V \^ EolO ; -with the same depth ; and 62.5 Again, let Fig 9 represent a conical vessel full of ivater; Ma base 6 c, 2 ft diam ; its vert height a n, 3 ft ; then the circumf of the base will be 8.2832 ft; the area of the base 3.1416 sq ft; the length of its slant side a b or a c, 3.16 ft; the area of its curved slanting sides will be : — rr 9.93 sq ft; and the Tert depth of the cen of grav of the slanting sides will be at two-thirds of the vert height a n from the apex a, or 2 ft. Here, to find the total pres against the base, we have by rule in Art I, 3.1416 X 3 X 62.5 = 589.05 lbs. For the total pres against the slant sides, by the same rule, 9.93 X 2 X 62.5 = 1241.25 fi)s. For the vert pres upward against the entire area of the slant sides, we have given the area of the base (which is here the hor projection of the slant sides) r: 3.1416; and the vert depth of the cen of grav of the slant sides, 2 ft. Therefore, 3.1416 X 2 X 62.5 — 392.7 tt)s, the upward vert pres. Finally, for the hor pres in any given direction against the slant sides of one half of the cone, we have the vert projection of that half, represented by the triangle a b c, with its base 2 ft, and its perp height 3 ft ; and consequently, with an arta of 3 sq'ft. The depth of its cen of grav is 2 ft ; therefore, 3 X 2 X 62.5 = 375 »)s, the reqd hor pre«.* In Fig 10, which represents a vessel full of water, the total pres against the semi-cylindrical surf avemdk, and perp to it, must be also hor, because the surf is vert; but inasmuch as the surf is curved, this total pres, as found by rule in Art 1, acts against it in many di- rections, which might be represented by an infinite number of radii drawn from o as a center. But let it be reqd to find the hor pres in B)8, in one direction only, say parallel to o e, or perp to a d; which ■would be the force tending to tear the curved surf away from the flat sides a b nv, and d c t k, hj producing fractures along the lines a v and d k; or which would tend to burst a pipe or other cylinder. In this case, mult together the area of the vert projection ad kv in sq ft J the depth of the cen of grav of the curved surf in ft ; (which, in the semi-cylinder would be half of e m, or of o i ;) and 62.5. Since the resulting pres is resisted equally by the strength of the vessel along the two lines a v and d k, it is plain that each single thickness along those lines need only be sufficient to resist safely one half of it ; and so in the case of pipes, or other cylinders, such as'hooped cisterns or tanks. See Art 17. Should the pres against only one half of the curved surf, &s edmk be sought, and in a direction parallel to o d, tending to produce frac- tures along the lines e m, and d k, then use the vert projection oen ' as before. It follows, that if the face of a metallic piston be made oonoave or convex, no more pres will be reqd to force the piston through any dist, than if it were flat ; for the pres against the face of the piston, in the direction in which it moves, must be measured by the area of a projection of that face, taken at right angles to said direction ; and the area of said projection will be the same in all three cases. Rem. 2. If a bridg^e pier, or otlier construction. Fig 10 3^, be founded on sand or g^ravel, or on any kind of foundation through which water may find its way underneath, even in a very thin sheet, then the upward pres of the water will take effect upon the pier ; and will tend to lift it, with a force equal to the wt of the water displaced by the pier; (Arts 18, 19. In other words, the effective wt of the submerged portion of the pier, will be reduced 623.^ lbs per cub ft; or nearly the half of the ordinary wt of masonry. But if the foundation be on rock, covered with a layer of cement to prevent the infiltration of water beneath the masonry, no such effect will be produced ; but on the contrarv, the vert pres downward, afforded by the bat- tering sides of the pier, and bv its offsets, will tend to hold it down, and thus increase its stability ; which, in quiet water, will then actually be greater than on land. Art. 5. To divide a rectangular surf, ivlietlier vert an ab c d, or inclined as mnop, Fig 11, whose top ab or mti is level with the surf of the water, by a hor line x 2, such that the total pres against the part above .said hor line, shall equal that against the part be- low it. Rule. Mult one half cf the length of b c, or mp, as the case may be, by the constant number 1.4142; the prod will be b 2, or mx. Ex. Let & c = 12 ft. Then 6 X 1.4142 = 8.4852 ft ; or 6 2. Let mp = 16 ft. Then 8 X 1.4142 = 11.3136 ft, or m x. Bkm. The line x 2, thus found, must not be confounded with the cen of pres, which is entirely diff. See Art 8. Art 6. In a rectangular surf, whether vert as abed, or in- clined n» mnop, Fig 11, whose top a b or mn coincides with the surf of the water, to find any number of points, as 1,2, «tc, through which if hor lines, as 1 «, 2a;, Ac, be drawn, they will divide the given surf into smaller rectangles, all of which shall sustain equal pressures. EuLK. First fix on the number of small rectangles reqd. Then for point 1 from the top. mult l^e number 1 , by this number of rectangle-s. Take the sq rt of the prod. Mult this sq rt by the entire length Firfiol * In a sphere filled with a fluid the total inside pres = 3 times wt of fluid. 506 HYDROSTATICS. h COT mp, aa the case may be. Div the prod by the uumber of rectangles. The quot will be the dist 6 1. orn 1, as the case may be. For the dist b 2, or n 2, proceed in precisely tiie same wayj only instead of the number 1, use the number 2 to be mult by the number of rectangles: and so use successively the numbers 3, 4, 5, &c, if it be reqd to find that number of points. Ex. Let 6 c = 10 ft ; and let it be reqd to find 2 points, 1 and 2, for dividing the rectangular surf abed into 3 rectangular parts, which shall sustain equal pressures. Here we have for point 1, 1X3 = 3. Thesqrt of 3 = 1.732. And 1.732 X 10 (or 6 c) = 17.32. And _-H^'L__rr5.773ft=:JL. For point 2, we have 3 rectangles 24.49 2X3 = 6. Thesqrt of 6=2.449. And 2.449 X 10 (or 6 c) = 24.49. And — = 8.163 ft = 6 2. 3 rectangles And so for any number of points. Rem. 1. This rule will be found useful in spacing- tbe cross- bars of loek-gates; tbe boops around cylin^, this pressure p is of course less than when it is battered; and is also horizoyital; and it tends to over- throw the wall, by making it revolve around its outer toe, or edge t. The center of pressure is at c; c s being 3^ the vertical depth on; in other words, the entire pressure of the water, so far as regards overthrowing the wall as one mass, may be considered as concentrated at the point c; where it acts with an overthrowing leverage 1 1. The pressure in lbs, multiplied by this leverage in feet, gives the moment in foot-ft)s of the overturning force. The wall, on the other hand, resists in a vertical direction g a, with a moment equal to its weight (supposed to be concentrated at its center of gravity g), multiplied by the horizontal distance a i, which constitutes the leverage of the weight with respect to the point ^ as a fulcrum. If the moment of the water is greater than that of the wall, the latter will be overthrown ; but if less, it will stand. In Fig 21 the overturning moment of the water is equal to its calculated pres- sure p X its leverage 1 1 ; while the moment of stability of the wall is equal to its weight X its leverage a t. By aid of a drawing to a scale, we may on this principle ascertain whether any proposed wall will stand. For we have only to calculate the pressure jo, then apply it at c, and at right angles to the back ; pro- . long it to Z ; measure 1 1 by the same scale. Then calculate the weight of wall ; find its center of gravity g ; draw g a vertical, and measure the leverage a t. We then have the data for calculating the two moments. If the water, instead of being quiet, is liable to waves, the wall should be made thicker. HYDROSTATICS. 509 Fig. 24. Art. 11. To find tbe tbickness at base of a wall required to b< safe against, overturning under tiie pres of quiet water level with its top, and pressing against its entire vert back. Caution. See Art. 13. (1st) Vertical wall. Fig 22. Thickness in feet ' Height ' in feel ^ V Factor of safety * 3 X sp grav of wall Height the proper decimal ' in feet X in following table (2d) Rigrbt angeled triangular wall. Fig 23. Thickness at base = in feet Height f Mn feet X -^ Factor of safety * 2 X sp grav of wall thickness, m o, of vertical wall X 1.225. Height the proper decimal in feet X in following table Notwitlistanding their greater thickness at base, such triangular walls con- tain, as seen by the fig, not much more than half the quantity of masonry reqd for vert ones of equal stability. This is owing to the fact that their cent of grav is thrown farther back; thus increasing the leverage by which the wtof the wall resists overthrow. (3d) Wall witb vertical back and sloping: face. Fig 24. / (Ht2, ft X factor of safety *) + (batter A n^, ft X sp grav of wall) Thickness at base in feet 3 X specific gravity of wall = Height in feet X th« proper decimal in the following table. Fig. 22. Sp. Gr. Lbs per Cub Ft. Resist = 1.5 pres. Resist = 2 prei. Resist = 3 pres. Dressed Granite. . . Dressed Sandstone Mortar Rubble Brickwork Fig. 23. Dressed Granite... Dressed Sandstone Mortar Rubbl<3 Brickwork . .. '2.5 2.2 2. 1.8 2.5 2.2 156 137 125 112 156 137 125 112 .447 .477 .500 .527 .548 .584 .613 .646 .51« .550 .578 .609 .633 .675 .707 .746 .633 .674 .707 .746 .775 .82tt .866 .913 Resist = 1.5 pres. Resist = 2 pres. Fig. 24. Batter 1 in. to a foot. Batter 2 ins. to a foot. Batter 4 ins. to a foot. Batter 6 ins. to a foot. Batter 1 in. to a foot. Batter 2 ins. to a foot. Batter 4 ins. to a foot. "7551" .583 .609 .640 Batt«r 6 ins. t* a foot. Dressed Granite... Dressed Sandstone Mortar Rubble Brickwork 2.5 2.2 2. 1.8 156 137 125 112 .449 .480 .502 .530 .458 .488 .510 .539 .487 .515 .536 .562 .532 .558 .578 .602 .519 .552 .571 .610 .526 .560 .586 .618 .593 .622 .646 .674 • Factor of safety : . Req uired moment of stability of wall overturning moment of water 510 HYDROSTATICS. Art. 12. Table showing Iiow tlie stability of a wall sustain- ing- water is affected by a cbange in tbe form of tbe w^all ; the quantity of masonry remaining the same. Rem, When the base of a tri- angular wall, of sp grav 2, is less than ^ the ht, the stability is greatest when the water presses the vert side ; but if the base exceeds ^ the ht, the stability Is greatest with the water on the battered side. Caution. See Art. 13. All these ivalls contain precisely the same quantity of masonry. The masonry is supposed to be mortar rubble, weighing 125 lbs per cubic foot; or twice as much as water; or about the same as ordinary rough mortar rubble. If the sp gr of the masonry is actually greater or less than this, the safety also will be greater or less, in precisely the same proportion. Vertical wall Face vertical : back batters one-tenth height " " " " one-fifth " " " " *• one-fourth " " " " " one-third " " " " " four-tenths " " " " " one-half " Back vertical ; face batters one- tenth height " " " *' one-fifth " " " " " one-fourth " " " " " one- third " " " " " four-tenths " " " " " one-half " Back and face, each batter one-tenth height " " " one-fifth " " •• " " oue-fourth *' '* •' " " one-third " " " " " four- tenths " Base in r^eXt°5 parts of height. wall. .5 1.5 .55 1.8 .6 2.2 .625 2.6 .667 3.5 .7 4.9 .75 14.0 .55 1.8 .6 2.1 .625 2.2 .667 2.4 .7 2.6 .75 2.9 .6 2.2 .7 3.4 .75 4.6 .833 9.0 .9 36.0 rbx Art. 13. Liiability of wall or foundation to crusli under unequal distribution of pressure. Arts 11 and 12 apply only to the stability of a rigid wall resting upon a rt^'/d base, and therefore incapable of failure except by overturning as a ^vhole. They show that the stability is greatest when the water presses against the sloping side. But in practice the point where the resultant of all the pressures on the base of the wall cuts the base, must not be so near to either toe as to endanger a crushing of wall or of foundation. This consideration often makes it best to let the water press against the vert back, notwithstanding the consequent loss in stability. Art. 14. Fig. 25 shows, to scale, a dam wall at Poona, India, designed' by Mr. Fife, C. E., of England. It is of mortar rubble, of 150 lbs per cub ft. Its total, vert height is 100 ft; thickness w v at base, 60 ft 9 ins ; at top, rx, 13 ft 9 ins. The front ru slopes 42 ft in 100 ft; and the back xv, 5 ft in 100 ft. Its foundation is 7 feet deep; but we here assume that the water presses against its ew/ire back xv. Through tl)e cen of grav G draw G* vert. From c, where the direction of the pres P of the water strikes G*, lay oflT en by scale = 139.6 tons (of 2240 lbs) water pres against 1 ft in length of xv; and c/ = 249.4 tons wt of 1 ft length of wall. Complete the parallelogram cnmt of forces. ^^ Its diag c m represents the resultant of all the pressures ^ upon the base uv,and cuts the base at a, 20 ft back from the toe u. Doing the same with the 151.4 tons pres p against ru, we get the resultant oy, which is greater than cm, and cuts the base (at i) only 12.7 ft from the toe V, or 7.3 ft less than a is from u. Hence, when the water presses against xv the wall is less liable to fracture or crushing, and the earth foun- dation w V is more evenly loaded, and hence less liable to yield unequally so as to cause cracks in the wall. On this account xv is made the back of the wall, although the moment of stability of the wall is then only 2.2 (calling the overturning moment of the water 1), while if the water pressed against rw it would be 3, or 36 per cent greater. HYDROSTATICS. 511 Art. 15. The points a and i, Fig 25, are called centeri^ of pressure upon the base, or centers of resistance of the base. If similar points, as d and z, be found in the same way for other lines, as f h, by treating a part (a8 rxhf) of the wall as if it were an entire wall; a slightly curved line joining these points is called the line of pressure. Thus, &a is the line of pres- sure when the water presses against xv. Each point, as d, in 6a, shows where any joint, as, fh, drawn through that point, is cut by the resultant of all the forces acting upon said joint, bi is the line of pres when the water presses against ru. These lines do not show the direction of the resultants. Thus, at a, the latter is em, not ha. The angle between the direction of the resultant and a line at right angles to the bed or joint, must be less than the angle of friction of the materials forming the joint. If from the end m or y of the resultant of the pressures upon any joint, we draw m2 or yZ hor, then c2ot ol (as the case may be) measures the entire vert pres on that joint; and m2 ovyl measures the hor pres against the back of the wall, which tends to cause sliding at the same joint. If the direction of the re- sultant comes within the limit stated in the preceding paragraph, m 2 or yl will be less than the frictional resistance to sliding, which last is = c2 (or o/) X the coeflf of friction for the surfaces forming the joint. Hence sliding cannot take place. Sliding never occurs in the masonnj of walls of ordinary forms.^ Good mortar, well set aids to prevent sliding, but it is better not to rely upon it. But entir ' walls have sliddeu on slippery foundations. Art. 16. In California is this dam of a mining reservoir, built of rough stone without mortar, founded on rock. Height, 70 feet; base, 50; top, 6; water-slope, 30 feet; outer-slope, 14. To prevent leaking the water-slope is only covered with 3-inch plank bolted horizon- tally to 12 by 12 inch strings, built into the stone-work. All laid with some care by hand, except a core of about one-fifth of the mass, wiiich was roughlyHhrown in. Cost about S3 per cubic yard. It has been in use since 1860. Rem. If a dam is compactly baclied with earth at its natural slope, and in sufficient quantity to prevent the water from reaching the dam, the pressure against the dam will not be increased. Art. 17. To find the thickness of a cylinder to resist safely the pressure of water, steam, &c, against its interior. If riveted, see next page. liVhere the thicliness is less than one-thirtieth of the radius, as it is in most cases, the usual formula Thickness pressure ^^ ,. ^ (1> "'inches - safe strength Xradins* Is employed. It regards the material as being subjected only to a direct tensile strain, which is sufficiently correct in such thin shells. For somewhat g-reater pressures and thicknesses. Professor F. Keuleaux (Der Konstrukteur, p 52) gives Thickness _ pressure (^ , pressure \ .. ^ (2) in inches - ^e strength I "^ 2 X safe strength/ ^ ^ '^* For very great pressures and thicknesses, as in hydraulic presses, cannons, Ac, Professor Reuleaux (Konstrukteur, p 53) gives Lamp's formula : Thickness (^) in inches • (V^ X radius.* I safe strength -}- pressure safe strength — pressure The three formulae give results as follows, pressures and strengths in lbs per square inch : -) Diameter. Radius. Pressure. Safe tensile strength. Thickness, inches. Formula (1). Formula (2). Formula (3). 20 inches. 10 inches. 50 500 5000 10000 .05 .50 5.00 .050125 .5125 6.25 .05 .513 7.33 The thicknesses given by the formulas appropriate to the several pressures are Erinted in heavy type. It will be seen that in these cases the results differ ut slightly, except for very great pressures. * In all three formulae take the radius in inches, and the pressure and strength in pounds per square inch. 512 HYDROSTATICS. Rem. 2. Want of uniformity in tbe coolings of thick castings make!> them proportionally weaker than thin ones, so that in order to reduce thickness in important cases we should use only best iron remeited 3 or 4 times, by which means an ult cohesion of about 30000 lbs per sq inch may be secured. But even with this precaution no rule will apply safely in practice to cast cylinders whose thickness exceeds either about 8 to 10 ins, or the inner rad however small. Under a pres of 8000 lbs per sq inch, water will ooze tbrong^li cast iron 8 or 10 inS thick ; and under but 250 lbs per sq inch, through .5 inch. Table of tliiekn esses of sing-le-riveted wrought iron pipes, tanks, standpipes, &c, by the above rule, to bear with a safety of 6 a quiet pressure of 1000 ft head of water, or 434 lbs ye- sq inch ; the ult coh of fair quality plate iron being taken at 48000 fiis per sq inch, or at 8000 lbs for a safety of 6 ; which is farther reduced to 8000 X .56 = 4480 fts, to allow for weakening by rivet holes; for sing-le-riveted cyls have but about ,56 of the strength of the solid sheet; and double- riveted ones about .7. With the above pres and other data, the rule here leads to thickness = .1016 X inner rad in ins. Dl. Ths. Di. Ths. Di. Ths. Di. Ths. Di. Ths. Di. Di. Ths. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. ;Ft. Ins. .5 .025 5 .254 16 .813 30 1.52 60 3.05 120 10 • 6.09 1.0 .051 6 .305 18 .914 33 168 66 3.35 132 11 6.70 1.5 .076 8 .406 20 1.016 36 1.83 72 3.66 144 12 7.31 '2.0 .102 10 .508 22 1.117 42 2.13 84 4.27 192 16 9.75 3.0 .152 12 .609 24 1.219 48 2.44 96 4.88 240 20 12.19 4.0 .203 14 .711 27 L371 54 2.74 108 5.49 288 24 14.63 For a less head or pressure, or for any safety less than 6, it is safe axA near enough in practice, to reduce the thickness of wrought iron cyls in the same proportion as s^d head, pres, or safety is less than the tabular one. Double-riveted cylinders, Fairbairn says, are about 1.25 times as strong as single-riveted. Hence they may be one-fifth part thinner. Liap-welded ones are nearly 1.8 times as strong as single-riveted; and hence may be only .56 as thick. Many continuous miles of double- riveted pipes in California have been in use for years with safetys of but 2 to 2.6. In one case the head is 1720 ft, with a pres of 746 fts per sq inch ; diam 11.5 ins ; thickness, .34 inch. Cast iron city water pipes must be thicker than required by formula (1), in order to endure rough handling and the effects of "water-ram" ^due to sudden stoppage of flow, see second Rem, p513), and to provide against irregularity of casting and the air bubbles or voids to which all castings are more or less liable. In the following table the ultimate tensile strength of cast iron is taken at 18,000 Bbs per square inch. Column A gives thicknesses by Mr. J. T. Fanning's formula (Hydraulic Engineering, p 454). -100)X bore, ins / b ore, ins X .4 X ultimate tensile strength \ 100 / These correspond with average practice. The addition of 100 ft)s to the pres is made in order to allow for water-ram. Column B gives thicknesses by formula (1), taking coefficient of safety = 8 (thus making safe tensile strain = 2250 ft)s per square inch) angl adding three-tenths of an inch to each thickness given by the formula: Thickness ) _^ (pres, R)s per sq in - in inches J ™ ^~- ~ Head in feet 50 100 200 300 500 1000 Pressure, 21.7 43 4 86.8 130 217 434 K)s per sq in Bore, ins. Thickness of pipe, in Indies. A B A B A B A B A B A B 2 .36 .31 .37 .32 .38 .34 .39 .36 .42 .40 .48 .51 3 .37 .31 .38 .33 .40 .35 .42 .40 .45 .45 .54 .60 4 .39 .32 .40 .34 .42 .38 .45 .42 .50 .50 .61 .72 6 .41 .33 .43 .36 .47 .42 .50 .48 .57 .60 .75 .94 8 .45 .34 .47 .38 .52 .47 .57 .55 .66 .70 .90 1.15 10 .47 .35 .50 .40 .56 .50 .62 .60 .74 .81 1.04 1.35 12 ,49 .36 .53 .42 .60 .54 .67 .66 .82 .91 1.18 1.57 16 .55 .38 .60 .46 .70 .62 .79 .77 .98 1.10 1.46 2.00 18 .57 .39 .63 .48 .74 .65 .85 .84 1.06 1.21 1.60 2.20 20 .61 .40 .67 .50 .79 .68 .91 .90 1.15 1.31 1.75 2.50 24 .66 .42 .73 .53 .87 .77 1.02 1.01 1.30 1.51 2.03 2.84 30 .74 .82 .45 .47 .83 .93 .59 .65 1.01 .89 1.15 1.01 1.19 1.19 1,55 1.82 2.12 2.46 2.88 3.47 36 1.36 1.37 1.80 4.11 48 .98 .53 1.13 .77 1.42 t.24 1.70 1.73 2.28 2.73 3.73 5.38 HYDROSTATICS. 613 Table of thickness of lead pipe to bear internal pressures with a safety of 6; taking the ultimate cohesion of lead at 1400 Sts per sq inch. Rem. Although these thicknesses are safe againstquiet pressures,they might not resist shocks caused by too sudden closing of stop-cocks against running water. Heads in Feet. Heads in Feet. a 100 200 300 1 400 1 500 1- a a 100 1 200 1 300 1 400 500 Pres in lbs per sq inch. Pres in lbs per sq inch. 1 43.4 1 86.8 130 174 1 217 43.4 1 86.8 1 130 174 1 217 Thickness in Inches. Thickness in Inches. ^ .026 .055 .089 .128 .171 1 .102 .221 .357 .511 .682 % .038 .083 .134 .192 .256 1J< .127 .276 .447 .639 .853 ^ .051 .111 .179 .256 .341 1^ .153 .332 .536 .767 1.02 H .064 .138 .223 .320 .427 m .178 .387 .626 .895 1.20 ' H .076 .166 .268 .383 .512 2 .204 .442 .714 1.02 1.36 K .089 .193 1 .313 .447 .597 Rem. The valves of water-pipes must be closed slowly, and the necessity for this precaution increases with their diams. Otherwise the sud- den arresting of the momentum of the running water will create a great pressure against the pipes in all directions, and throughout their entire length behind the gate, even if it be many miles ; thus endangering their bursting at any point. Hence stop-gates are shut by screws. Fig. Art. 18. Buoyancy. When a body is placed in a liquid, whether it float or sink, it evidently displaces a bulk of the liquid equal to the bulk of the im- mersed portion of the body ; and the body, in both cases and at any depth, and in any position whatever, is buoyed up by the liquid with a force equal to the weight of the liquid so displaced. Thus, if we immerse entirely in water a piece of cork, c, c, Fig 26, or any body of less specific gravity than water, the cork will, by its weight, or force of gravity, tend to descend still deeper; but the upward buoyant force of the water, being greater than the downward force of gravity of the cork, will compel the latter to rise. In this case, the cork receives a total downward pressure equal to the weight of the vertical column of water above it, shown by the vertical lines in vessel 1 ; and a total upward pressure equal to the weight of the column shown in vessel 2. The difference between these two columns is evidently (from the figs) equal to the bulk of the cork itself; therefore the difference between their weights or pressures (or, in other words, the buoyancy of the water) is equal to the weight or pressure of the water which would have occupied the place of the cork; or, in other words, of the water which is displaced by the cork. This difference, or buoyancy, will plainly be very nearly the same at any depth whatever of entire immersion. It increases slightly with the depth, owing to increase in the density of the water; but, on the other hand it is diminished by compression of the cork. Now the cork, if left to itself, will continue to rise until a portion of it reaches above the surface, as in vessel 3. The downward pressing column then ceases to exist ; and the cork is pressed downward only by its own weight. But, as it now remains stationary, the upward pressure of the water must be equal to the weight of the cork. But the upward pressure of the water arises only from the sliaded column shown in vessel 3; and this column is (as in the case of total immersion) equal to the bulk of water displaced. Therefore, in all cases, the buoyancy is eqtial to the weight of water displaced ; and when the body floats on the'surface, the buoyancy, or the weight of water displaced, is also equal to the weight of the body itself. If the body be of a substance heavier tlian ivater, its weight is greater than the buoyancy of the displaced water, and the body therefore sinks, with a force equal to the difference between the two. Thus, a cubic foot of cast iron weighs 450 lbs., and a cubic foot of water 62.5 lbs., so that the iron sinks with a force of 450 — 62.5 = 387.5 lbs. The same principle applies to other fluids. Thus, light bodies, such as Bmoke, a balloon, etc., in air, all tend, like a cork in water, to fall ; but the air, being heavier, crowds them out of the lower positions which they tend to assume, and pushes them upward. Althougli a pound of lead and a pound of feathers, weighed in the air, balance each other, yet in a vacuum the feathers will outweigh the lead, by as much as the bulk of the air displaced by them outweighs that displaced by the iron. 514 BUOYANCY, FLOTATION, METACENTER, ETC. The downwd forc6 of grar may be regarded as concentrated at the cen of grav G of a floating body. The upwd pres, or buoyancy,! of the water may similarly be regarded as acting at the cen of gr W of the displaced water.* W is also called the center of pressure, or of buoyancy , of the water ; and a vert line drawn through it is called the axis, or vertical, of buoyancy, or of flo* tation. Ordinarily,! W shifts its position with every change in that of the body. Thus in L it is at the cen of gr of the rectangle 0066; and in N at that of the tii» angle a a v. When a floating t, body, L, P or R, is at rest, and undisturbed by any third force, as F, it is said to be in e^iuilibrium, and G and W are then ^ in the same vert line 1 1 Figs L and R, or e eFigP; which line is called the axis, or vertical, of equilibrium.^ When a third force, as F, causes the axis of equilib to lean, as in Figs N, and S, then if a vert line be drawn upwd from the cen W of buoy, the point M where said line cuts said axis, is called the metacenter of the body. | G and W are then no longer in the same vert line; J and the two opp and vert forces, grav and buoy, act- ing upon those points respectively, form a "couple" and, when the third force F is removed, they no longer hold the body in equilib, but cause it to rotate. If (as in Figs and S) the positions of G and W are then such that the metacenter M is above the cen of gr G, this rotation will tend to restore the body to its former position, and the body is said to have been (before the application of the third force F) in stable equilibrium.^ But if (as in N) M is below G the direc- tion of rotation is such as to upset the body, by causing it to depart further from its former position, and the body is said to have been in unstable equilibrium.:]: The tendency ormomentin ft-lbs of a floating body either to upset or to right itself, is, _ the wt of the body (or the equal ^ the hor dist between "W M and G H, upwd pres of the water) in lbs ^ Figs N, O and S, in ft. The third force P may of course be so great as to overpower the tendency of the body to right it» self. Thus, a ship may upset in a hurricane, although judiciously loaded and ballasted for ordlaarj Arinda. A hor section ef a body at water-line is called its plane of flotation. * Tbe body is in fact acted upon by otber forces, such as the hor pressures of the water against its immersed portions ; but as all of these in any one given direction are balanced by equal ones in the opposite direction, they have no effect upon the forces G and W. It is also acted upon by the air, which presses it downwards with a force of 14.75 lbs per sq inch ; but this is balanced by an equal pres of the surrounding air upon the surface of the water. t This buoyancy is made up of the parallel upward pressures of the innumerable vert filaments of the displaced water as shown by Fig 26, and the axis of flotation is their resultant, as in the case of paraUel forces* } The shape of a body (as that of a sphere or cylinder U) may be such that the position of Its cen of buoy W, relatively to that of its cen of gr G, is not changed by the rotation of the body about a given axis (as any axis of the sphere or the longitudinal axis of the cyl), but remains constantly in the same vert line with G, so that the body, in rotating, remains in equilib.* Such a body is said to be in indifferent equilibrium about said axis. But if a cyl U be made to rotate tibout its transverse axis xx, It plainly comes under the remarks on Figs R and S, and may (before rotating) be in either stable or unstable equilib about that axis according to the way in which Its wt is distributed. II This metacenter shifts its position on the line 1 1 according to the inclination of the latter. g Uneven loading*, instead of a third force, may cause a vessel at rest to lean as at P ; and yet the vessel so leaning may be in equilib ; for its axis e e of equilib may be vert, although not coinciding with the axis of symmetry of the vessel, as it does at ttiuL,. ^ In floating bodies, this may sometimes (as in Figs R and S) be the case even when the cen ^ huoy W (not the metacenter) is below the cen of gr G ; because, when the body is forced to lean, w moves to another point In it, and this point maj be such aa to bring H above O. W is always below G in bodies of uniform density, floating at rest, if any part of the iaody is above water. When such bodies are entirely submerged, W and G coincide. HYDROSTATICS. 515 Art. 19. A body ligrtiter than water, if placed at tlie bottom of a vessel coiitaining' virater, w^ill not rise unless the w^ater can g-et under it, to buoy it, or press it upward, as the air presses a balloon of Sinoke upward. Thus, if one side of a block of light wood, perfectly flat and smooth, be placed upon the similarly flat and smooth bottom of • vessel, and held there until the vessel is filled with water, the downward pres wiU keep it in its place, until water insinuates itself beneath through the pores of tha wood. But if the wood be smoothly varnished, to exclude water" from its pores, it will remain at the bottom. On the other hand, a piece of metal may be pre« vented from sinking: in water, by subjecting it to a suffi< cient upward pres only, while the downward pres is excluded. Thus, if the bottonv of an open glass tube, t, Fig 27, and a plate of iron m, be made smooth enough to b« water-tight when placed as in the fig; and if in this position they be placed in « vessel of water to a depth greater than about 8 times the thickness of the iron, the upward pres of the water will hold the iron in its place, and prevent its sinking; because it is pressed upward by a column of water heavier than both the column of air, and its own Fin 27 weight, which press it downward. Pid 28 On this principle iron ships float. Rem. 1. A retaining- wall, as in Fig' 28, founded on piles, may be strong enough to re- sist the pres of the earth e behind it, in case water does not find its way underneath ; and yet may be overthrown if it does ; or even if the earth « s around the heads of the piles becomes sata* . rated with water so as to form a fluid mud. In either case, tha upward pres of the water against the bottom of the wall will Tir» tually reduce the wt of all such parts as are below the water surf, to the extent of 6'2}4 H>s per cub ft ; or nearly one-half of the or- dinary wt of rubble masonry in mortar. Rem. 2o Although the piles under a wall, as in Fig 28, may be ^::j abundantly sufficient to sustain the wt of the wall ; and the wall equally strong in itself to resist the pres of the backing e; yet if the soil 8 8 around the piles be soft, both they and the wall may be pushed outward, and the latter overthrown by the pres of the backing e. From this cause the wing-walls of bridges, when built on piles in very soft soil, are frequently bulged outward and disfigured. In such cases, the piling, and the wooden platform on top of it, should extend over the whole space between the walls; or else some other remedy be applied. Art. 20, Draugrh t of vessels. Since a floating body displaces a wt of liquid equal to the wt of the body, we may determine the wt of a vessel and its cargo, by ascertaining how many cub ft of water they displace. The cub ft, mult by 62%, will give the reqd wt in lbs. Suppose, for instance, a flat-boat, with vert sides, 60 ft long, 15 ft wide, and drawing unloaded 6 ins, or .5 of a ft. In this case it displaces 60 X 15 X .5 = 450 cub ft of water ; which weighs 450 X 623^ = 28125 fts ; which consequently is the wt of the boat also. If the cargo then be put in, and found to sink the boat 2 ft rruyre, we have for the wt of water displaced by the cargo alone, 60 X 15 X 2 X 62J^ == 112500 B)s ; which is also the wt of the cargo. So also, knowing beforehand the wt of the boat and •argo, and the dimensions of the boat, we can find what the draught will be. Thus, if the wt as befora 140625 be 140625 fts. and the boat 60 X 15, we have 60 X 15 X 62J^ = 56250; and := 2.5 ft the required 56250 draught. In vessels of more complex shapes, as In ordinary sailing vessels, the caloalation of the amount of displacement becomes more tedious ; but the principle remains the same. 516 HYDRAULICS. HTDKAULIOS. Flow of Water tbrodgb Pipes. Much of the theory of hydraulics is still matter of dispute. This, and the unavoidable imperfections of actual work, render it advisable to use liberal safety factors iu applying hydraulic formulas. Even new pipes are liable to tuberculations, which materially diminish the flow, and these sometimes greatly increase uuder the action of chemicals in the water. Air in the pipes also diminishes the flow. The term HEAD or TOTAIi HEAB of water, as applied to the fiowage of water through canals, pipes, or openings in reservoirs, &c, means the vert dist it; or p o, Fig 1, from the level surf, mi, of the water in the reservoir, or source of supply, to the center (or more properly t« thecen of grav) o, of the orifice (whether the end of a pipe, r o,to, v o,zo, lo; ov any other kind of opening) through which the disch takes place freely, into the air; or the vert dist a u, or fg, from the same surf, m i, to the level surf, g u, of the water in the lower reservoir ; when the disch takes place under water. Thus, in the case of disch into the air, the vert dist i v or po, is the total head for either of the pipes ro, t o, v o, z a, or I o; and ik is the head for the orifice, k, in the side of the reservoir. And for disch under water, au, or fg, is the head for either the pipe j, or the opening n; without any regard whatever to their depths below the surf of the lower water ; which, according t« the older authorities, do not at all affect their disch. A portion of a pipe may have a head greater than the total head of the entire pipe. Thai th« point 6 in the pipe lo, has a head 61; while the eutire pipe has only the he&d p o. Both in theory and in praotice it is immaterial as reg-ards the vet, and tSae quantity ol" water disehar$sed. whetlier the pipe is inclined downward, as ro. Fig 1; or hor, as t^o; or in- clined upward, as ^o; provided the total head po, and also the leni^th of the pipe, remain nnchang-ed. If one pipe is longer than another, its sides will evidently present more friction against the water, and thus diminish the •vel and the quantity of disch. The' inclined pipes, r o, lo, being of course a little longer than the hor one vo, will therefore each disch a trifle less water; but if the hor one were extended slightly fceyond o, so as to give it the same length as the others, then each of the three would disch the same quantity iu the same time. Art. 1 a. nivisions of the Total Head. In any pipe, as so.r o, t o, V o, z o,or I o. Fig 1, the total head has three distinct duties to perform; 1st, to overcome the resistance to entry at s, r, t. v, z, or I; '2d, to overcome the resistances within the pipe; and, 3d, to ^ive to the water, entering the pipe, the uniform velocity with which it actually flows. For convenience, we regard the total head as divided into three portions, corresponding to these duties; namely, 1st, the entry head; 2d, the resistance, or friction, head; and, 3d. the velocity bead. Art. 1 6. The velocity head is the height through which a body must fall, in vacuo, to acquire the vel with which the water actually flows into the pipe. It is th«refore =: -^, in which v is the vel in ft per sec ; and g is the acceleration of gravity, or 32.2 Art. 1 c. Experiment shows that, with the usual sharp-edged entry, the en- try head is, near enough for practice, = half the vel head. If the entry is shaped Ake Fig 7, scarcely any entry head will be required. But, in pipes longer than about 1000 diameters, the entry head bears so slight a proportion to the total head, that this advantage is of but little importance. It becomes more apparent in shorter pipes. Art. I d'. In Fig 1 we will assume that for any of the pipes, is representa the sum of the vel and entry beads. Then the remainder s v. or to o, of the total head, is the JTriction head; or the head which is just suflBcient to balance the friction and other resistances within the pipe ; and, since the entry head balances the resistance at the entrance to the pipe, the velocity head has only to give velocity to the water iu the vessel, causing it to ent«r HYDRAULICS. 517 the pipe as rapidly as it flows through it, and thus Iceeping the pipe supplied. If, by shortening the pipe, or by smoothing its inner surf, we diminish the total friction, then a less friction head will be required ; but the vel will, at the same time, be increased, and this will require a greater vd head, and entry head, so that the three together make up the total head, as before. Since the friction is equal to the force or head reqd to overcome it, it also is represented by wo. Art. 1 e» The friction head maj' as in v o, « o, and Z o, Fig 1, be all above the entrance to the pipe, and therefore outside of the pipe ; or, as in a pipe laid from « to o, it may be all helore the entrance, and ivithin the pipe; or, as in ro and to, it may be partly above, and partly below, the entrance; and therefore partly within, and partly without, the pipe. The vel and disch, after the pipe is filled, are not affected by this difference in position of the entry end ; but the pressures m the pipe, and the vels while the water is filling an empty pipe, are affected by it, as explained in Arts 1 2 and 1 o. Art. 1/*. But it is necessary that the entry end of the pipe should be placed so far helO'W the surf m i, that thero shall be left, above the cen of grav of the entry end, at least a head, i s, sufficient to perform the duties of the entry and vel heads. If the entry end of any of the pipes be raised above 6, a portion of the vel head will be m the pipe. In other words, the head in the pipe will be more than sufficient to overcome the resistances in the pipe ; and the surplus will act as vel head, and will give greater vel to the water . r > Let a = the area of cross section, and F=? the velocity, of the stream issuing through the short pipe beyond F. Fis called the velocity of efflux. Let Ai, Ao, etc., be the different areas of cross section of b F, and let Vi, v^, etc., be the velocities at those cross sections respectively. Then Q = a V = AiVj^ = Ai Vq, etc.: or F = — , v\ = ~, vo = ■-^, etc". In other words, the veloci- ^ *" ' a' Ai A^ ties are inversely as the areas of cross section. Also, a = %, Ai = — , Ao = -, etc. The losses of pressure, due to the velocities, respectively, are di _vr .d2- t'22 2g' ^ 2g' etc.; as represented by the ordinates between the line o o', of static pressure, and the diagram, ol23456i^, of actual pressures. The difference, due to velocity, between the pres heads at any two points, as c\ and c>i^ where the velocities are 2^ • static head in Vi and vo respectively, is j92 — i'l = "l — ^2 — o ^ ' The remaining pressure head, pi, jso* etc., at any point, is = reservoir — velocity head at the point,"= H — d\, H — d^, etc. The loss of pressure head, at F^ is (6 i^) = j^a = IT — d^; and the pressure drops to zero ; i.e., to the atmospheric pressure. Art. 1 a. Open piezometers. If the lower ends of vertical or inclined tubes, open at both ends, be inserted into a pipe, b F, Fig. 1 J5 i>, as at Cj, C2, etc., the water surface, in these tubes, will stand at heights, 7? i, joo, etc., corre- sponding to the pressure heads at the points where the tubes are inserted. Such tubes are called open piezometers. In order that the water level may be observed, they are of glass, at least in those portions where that level is likely to be found. An obstruction, in the pipe, between Co and F, would raise the level in a piezometer at Cg ; while an obstruction between b and c^ would lower it. * In Art. 1 m-r, for simplicity, we neglect all resistances, including those due to the abrupt enlargements and contractions of the pipe. HYDRAULICS. 519 Fig.l F Art. 1 f . If we imagine any pipe, full of water, to be supplied with a nurobM of piezometers, then a line, joining the tops of the cohimns of water in the several piezometers, is called the bydraulic grade line. Art. 1 «*. In a straight tube of uniform diara tliroughout, as ro, v o, or I o, Fig 1 runuing full and discharging freely into the air, the hyd grade line is a straight line drawn from its disch end o to a point s immediately over the entry end of the pipe, and at a depth below the surf equal to the sum of the vel and entry heads. If the orifice at o be contracted, the hyd grade line must be drawn from « to some point, as e, immediately over o, and depending, for its height, upon the amount of contraction at o. But in this case the point s will also be higher than before, because the vel in the pipe is reduced by the contraction ; and the sum i s of the vel and entry head* will be less. If the disch at o is nnder ivater, the effect upon the position of the grade line will be the same as that of a con- traction of the orifice at o. The point e will be on the surf of the lower water, and immediately over o. Art. 1 V. If the pipe, of uniform diam, (whether discharging freely or through a con- tracted opening at o, whether into the air or under water), is bent or cnrved, the hyd grade line will still be straight, provided the resistances are equal in each equal division of the hor length of the pipe, as in Fig 1 E, where equal divisions vw,to X, &c, of the total length, correspond with equal divisions v a, a b, &c, of the hor length. But in Fig 1 F, the hyd graOe line will take the shape 8 ao. For if, in accordance with Art. 1 G, we divide s o into two equal parts, s m, m o, correspond- ing with the two equal parts vr,ro, of the length of the pipe, we obtain m c = a e for the head consumed in the resistances In v r, leaving only r a for the pres head at r. Art Xw, In a very large vessel, the total head upon any point at the level of the entrance I to a pipe I o o' Fig 1 G, is represented by i /, as already ex- plained but of this total head a portion, as i s, is required to act as velocity head and entry head for the entrance at I, leaving only s I as the pres- sure head upon a point in the pipe, immediately to the right of I. Thus while the pressure, in pounds per square inch, in the vessel at I, is p = ilX 0.434 . that in the pipe at I is p = slX 0.434. But now a portion, as sv, of si, is expended in Zo in balancing or "overcoming" the resistances throughout that portion of the pipe ; and, in doing this work, it gradually diminishes from sv (at /) to nothing (at o) as indicated by the dotted line se. Thus, at the point 6, a portion = bc has already been expended in overcoming the resistances in the pipe between I and 6, leaving c 6 as the pressure head at 6, of which c in must still be expended against resist- ances in the wide pipe between 6 and o, leaving in& = vl = eoas the pressure head for a point just to the left of the contraction at o. The pressure in to is thus gradually diminished from si (at I) to eo = vl (at o). Now a portion es' of eo is required to act as ve- locity and entry head for the entrance o to the narrower portion o o' of the pipe ; because we need at o not only an additional entry head to overcome the resist- ance due to the square shoulder formed by the contraction, but also an addi- tional velocity head to give the increase of velocity which must take place as tlie water passes from the wide pipe lo to the narrower one oo'; for, so long as a pipe runs ftcll and the discharge remains constant, the velocity in each part of the pipe must be inversely as the area of cross section of that part ; because in each second the same quantity of water passes each point; and this constant quantity is = area X velocity. Hence, as the area diminishes, the velocity increases. There remains, therefore, s' o as the pressure head upon a point in the narrow part just to the right of o; and this m turn gradually diminishes to nothing at 520 HYDRAULICS. the end o' of the pipe, as indicated by the dotted line s o\ being all expended in overcoming the resistances in o o'. We thus have, for the hydraulic gradient in Fig 1 G, the broken line ises' o'. When tlie pressure is thus diminished by overcoming resistances, or by ac- celerating velocity, the diminution is called loso is measured vertically from the surface mi in the reser- voir to the center of gravity of the outlet o, as in Fig 1 ; the hydraulic gradient (with the restriction named in Art 1 v) is, as before, a straight line r *ro drawn from the foot * JE JiSM J?- — — jP of the combined entry and '■ velocity heads to the end o; and the velocity and discharge are the same as they would be if all parts of the pipe were brought below sro. But see cau- tions 1 and 2, below. The pressure at any point, g, n or y, is then given by a vertical line, gv, nr or yv, drawn from the point in question to sro\ but for points, as n, situated above sro, this pressure is negative or inuxird ; while at points where sro and the pipe are at the same level, as at /and e, there is neither pressure nor vacuum. Caution 1. But if the water be admitted to the empty pipe at a, while the end is open, the pipe will not form a true syphon. The part agn will then run full, and will have 5 en as its hydraulic gradient; but upon reaching, at w, a portion no of the pipe with a much steeper grade, the water will run off, in no, with a velocity greater than that with which it arrives from a n. Hence the stream in no will have a less area of cross section than in an, and therefore can- not fill no, but will run off in it as in an open gutter. Caution 2. The tendency to vacuum at points above sro causes an accu- mulation, at n, of particles of air that have been carried into the syphon by the water or have found their way in through imperfect joints, etc' ; and these bring about a condition approaching that described in Caution 1; for their expansive force, by reducing the negative pressure or vacuum n r at n, diminishes the total head // r of the part agn, while, by practically reducing the cross-sec- tion of the syphon at n, they require that a portion of the remaining head he used at n, as entry head to overcome the resistance caused by the contraction, and as velocity head to give the increase of velocity needed for passing the nar- rowed section at n. Now since the friction head required for the part agn re- mains about the same, the velocity head in the reservoir is considerably dimin- ished, and the water arrives at n too slowly to keep n o filled. The accumulation of air at n thus retards the flow and disturbs the distribution of the pressures, so that these are no longer correctly indicated by vertical lines drawn to sro. At Blue Ridge Tunnel, Virginia, Col. C. Crozet constructed a drainage syphon 1792 ft long of cast iron faucet pipes 3 ins bore, 9 ft long. Its summit was 9 ft above the surface of the water to be drained ; and its discharge end was 20 ft below said surface, thus giving it a head of 20 ft. At the summit 570 ft from the inlet, was an ordinary cast iron air-vessel with a chamber 3 ft high and 15 ins inner diam. In the stem connecting it with the syphon was a cut-ofT stop- cock ; and at its top was an opening 6 ins diam, closed by an air tight screw lid. At each end of the syphon was a stopcock. To start the flow these end cocks are closed, and the entire syphon and air-vessel are filled with water through the opening at top of air-vessel. This opening is then closed airtight, and the two end cocks afterwards opened ; the cut-off cock remaining open. The flow then begins, and theoretically it should continue without diminution, except so far as the head diminishes by the lowering of the surface level of the pond. But in practice with very long syphons this is not the case, for air begins at once to disengage itself froni the water, and to travel up the syphon to the summit, where it enters the air-vessel, and rising to the top of the chamber gradually drives out the water. If this is allowed to continue the air would first fill the en- tire chamber, and then the summit of the syphon itself, where it would act as a wad completely stopping the flow. The -water-level in the ai r chamber can be detected by the sound made by tapping against the outside with a hammer. 522 HYDRAULICS. To prevent this stoppag>e, the cut-off at the foot of the chamber is closed before the water is all driven out; and the lid on top being removed the chamber is refilled with water, the lid replaced, and the cut-oflf again opened. The flow in the meantime continues uninterjupted, but still gradually diminish- ing notwithstanding the refilling of the chamber; and after a number of refill- ings it will cease altogether, and the whole operation must then be repeated by filling the whole syphon and air chamber with water as at the start. At Col. Crozet's syphon at first owing to the porosity of the joint-caulking, which was nothing but oakum and pitch, air entered the pipes so rapidly as to drive all the water from the chamber and thus require it to be refilled every 5 or 10 minutes; but still in two hours the syphon would run dry. The joints were then thoroughly recaulked with lead, and protected by a covering of white and red lead made into a putty with Japan varnish and boiled linseed oil. But even then the chamber had to be refilled with water about every two hours ; and after six hours the syphon ran dry, and the whole had to be refilled. In this way it continued to work. Care in making the joints air-tight, and an outside and inside coating of the pipes and air-vessel with coal pitch varnish are important precautions. Art. 2. Approximate formulse for the velocity of water in straight, smooth, cylindrical iron pipes, as ro, v o, lo, Fig. 1. Having the total headp o, and the length and diameter of the pipe. Approx mean vel in ft per sec 1 coefficient I ^ = m X a/; J as below >' diam in ft X total head in ft total length in ft + 54 diams in ft Table of coefficients " m ". Diam of pipe, m l>iam of pipe, m feet inches feet inches 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 12.0 23 30 34 37 39 42 44 46 47 48 1.5 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 10.0 18 24 30 36 42 48 60 72 84 120 53 57 60 62 64 66 68 70 72 77 For heads not less than 4 feet per mile, this formula gives results practically corresponding with those by Kutter's formula (p. 523) with coefficient n of roughness = 0.012. But slight differences, as to roughness, etc., may cause con- siderable variations of velocity, especially in small pipes ; for, in such pipes, a given roughness of surface bears a greater proportion to the whole area of surface than in a pipe of large diameter. Extreme accuracy is not to be expected in such matters. As in a river the velocity half way across it, and at the surface, is usually greater than at the bottom and sides, so in a pipe the velocity is greater at the center of its cross section than at its circumf. The niean velocity referred to in our rules is an assumed uniform one which would give the same discharge that the actual ununiform one does. Hence Discharge _^ Mean velocity w in cub ft per sec '°" in ft per sec "^ Area of cross section of pipe in sq ft. 1 cubic foot = 7.48052 U. S. gallons 1 U. S. gallon = .13368 cubic foot = 231 cubic inches. » For intermediate diametera, eto, take intermediate coeffioieat« from the table by simple prv portion. HYDRAULICS. 523 In the case of long pipes with low heads, the sum of the velocity and entry heads is frequently so small that it may be neglected. Where this is the case, or where their amount can be approximately ascertained, Kut- ter's formula, although designed for open channels, may be used. This formula is the joint production of two eminenl Swiss engineers, E. Ganguillet and W. R. Kutter, but for convenience it is usually called by the name of the latter.* It is, properly speaking, a formula for finding the coefficient c in the well known formula, mean velocity = c "j/mean radius X slope ..V diameter X slope According to Kutter, For Sng^lisb measnre. For metric measure. C = slope n slope n 77n .00281 \ ~ ' 7~~~m\b5 \ l/meau rad in feet x^mean rad in metres See also tables of c, pp 566 etc. Tlie mean radius is the quotient, in feet or in metres, obtained by divid- ing the area of wet cross section, in square feet or in square metres, by the wet perimeter (see below) in feet or in metres. In pipes running full, or exactly half full, and in semicircular open channels running full, it is equal to one-fourth of the inner diameter. The wet perimeter is the sum, ab co Figs 28, 29, 30, of the lengths, ^b,bc,co, in feet or in metres, found by measuring (at right angles to the length !>f the channel) such parts of its sides and bottom as are in contact, with the >ater. In pipes running full, it ia of course equal to the inner circumference. The slone is ^ JHcHonheBdwoF^ length of pipe measured in a straight line from end to end. = sine of angle wso, Fig 1. In open channels, this becomes , __ fall of water surface in any portion of the length of the channel "" length of that portion = fall of water surface per unit of length of channel = sine of the angle formed between the sloping surface and the horizon. The number indicating the slope in any given case is plainly the same for English, metric and all other measures. *' n " is a " coefficient of roughness " of wet perimeter, and of course depends chiefly upon the character of the inner surface of the pipe. For iron pipes in good order and from 1 inch to 4 feet diameter, n may be taken at from .010 to .012; the lower figures being used where the pipe is in exceptionally good condition. If the diameter, or the mean radius, is in feet, metres etc, the velocity will be in feet, metres etc, per second. . * See " Flow of Water," translated from Ganguillet and Kutter, by Rudolph Hering and John C. Trautwine, Jr., New York, John Wiley & Sons, 1889. $4.00. 524 HYDRAULICS. The diameter or the slope, required for a given velocity, may be found by trial as follows: assume a diameter, or a slope, as the case may be ; take the corresponding c from tables, pp 566, etc. Then say Approx l>iain required ^ mean ^ 4 ^ / velocity \ for the given vel radius ^ \ cT/sl6pe) Approx Slope required _ / velocity y^ / velocity \2 for the given vel Vci/raean radius/ Ul/r^m5i/* With the approximate diameter (or slope) and c, thus obtained, say v' = cv'raean radius X slope. If •y' is near enough to the given velocity, the assumed diameter (or slope) is the proper one. If not, try again, assuming a grecUer di&meter or slope than before if v' is less than the required velocity, and vice versa. Curves and bends do not greatly affect the discharge, so long as the total heads, and total actual lengths of the pipes remain the same ; provided the tops of all the curves be kept below the hydraulic grade line ; and provision be made for the escape of air accumulating at the tops of the curves. Relation between area, velocity, and discharge. Let g ^ rate of discharge (as in cubic feet per second), V = mean velocity (as in feet per second), a = area of cross section (as in square feet). Then : q = av; v = - ; a = -. a V Relation of discharge to diameter *_and slope. If we assume velocity = c ^/mean radius X slope, or v = c }/ r s (page 523); and if the pipe be of circular cross section, we have, for the rate, Q, of discharge through a pipe of diameter, d, and area. A, of cross section, running full: — Q = Av= — ,c-s^ = g ; or : Q is proportional to the ^/g power (square root of fifth power) of the diam- eter, and to the ^ power (square root) of the slope. For tables of fifth powers, and of square roots of fifth powers, see pp 67-69. Effect of resistances. The pressure head of running water, upon any point in a pipe between the orifice and the reservoir, is : r +v,o Vir.o/1 t^^ head consumed 1 (the total ) ^ PtnthP *^^ in overcoming re- = -l head on V minus \ ^pI Vt + entry + sistances in the pipe V I that point) Ju V' • * liead between the reservoir l^ 81.79925 50 851.1889 89 2696.907 ^% 4.474062 16 87.16174 51 885.5769 90 2757.852 ^% 4.787938 16^" 92.69447 52 920.6459 91 2819.478 3% 5.112453 17 98.39744 53 95(5.3958 92 2881.785 4 5.447609 17K 104.27064 54 992.8267 93 2944.773 ^K 6.149840 18 110.31408 55 1029.9386 94 3008.442 4i| 6.894630 18M 116.52776 56 1067.7314 95 3072.792 7.681980 19 122.91168 57 1106.2051 96 3137.823 5 8.511889 193^ 129.46583 58 1145.3598 97 3203.535 5^ 9.384358 20 136.19022 59 1185.1954 98 3269.927 ^y% 10.299386 21 150.14972 60 1225.7120 99 3337.001 m 11.256973 22 164.79017 61 1266.9096 100 3404.756 Tlie 'we^glat of water in a given length (as one foot) of any pipe or other circular cylinder is in proportion to the square of the bore or inner diameter. Hence the weight of water in 1 foot length of any cylinder of other diameter than those in the table can be found by multiplying that for a 1 inch pipe, 0.340475558, by the square of the inner diameter of the given cylinder in Inches. Thus, for a cylinder 120 inches diameter : diameters = 120^ = 14400, and weight of water in 1 foot depth = 0.340475558 X 14400 = 4902.848 lbs. Or, weight for 120 ins. diam. = 100 X weight for 12 ins. diam. = 100 X 49.02848 = 4902.848 lbs. Similarly, {^^) 2 _ _4^9_ _ 0.191406, and 0.340475558 X 0.191406 = 0.065169 lb. = weight in 1 foot of ^"g- inch pipe. Here, also, -f-^ = half of | ; hence, weight for 3?^ inch = one-fourth of weight for f inch = one-fourth of 0.260677 = 0.065169. ll'eight of one square inch of water 1 foot hig^h, at 62.425 fts. per cubic foot = 62.425 -=- 144 = 0.433507 K). ♦Actual. See nominal and actual diameters, foot note, p 526. 526 HYDRAULICS. TABIiE 3. Areas and Contents of Pipes ; and square roots of ]>ianis. (Original.) Correct. * Area in * Area in * Area in Diam. Diam. in Feet. sq ft, also cub ft, Sq. rt. of Diam. Diam. sq ft, also cub ft. Sq. rt. of Diam. Diam. sq ft, also cub ft, Sq. rt. of ll?8. in 1 foot length of Diam. in Ft. in Ins. in Feet. in 1 foot length of Diam. in Ft. in Ins. in Feet. in 1 foot length of Diam. in Ft. Pipe. Pipe. Pipe. M .0208 .0003 .145 4. .3333 .0873 .579 15. 1.250 1.227 1.118 5-16 .0260 .0005 .161 H .3438 .0928 .588 H 1.271 1.268 1.127 Va .0313 .0008 .177 k- .3542 .0985 .596 ^ 1.292 1.310 1.136 7-16 .0365 .0010 .191 .3646 .1040 .604 H 1.313 1.353 1.146 H .0417 .0014 .204 }4 .3750 .1104 .612 16. 1.333 1.396 1.155 9-16 .0469 .0017 .217 % .3854 .1167 .621 K 1.354 1.440 1.163 % .0521 .0021 .228 % .3958 .1231 .629 34 1.375 1.485 1.172 11-16 .0573 .0026 .2;)9 % .4063 .1296 .637 H 1.396 1.530 1.181 % .0625 .0031 .250 5. .4167 .1363 .645 17. 1.417 1.576 1.190 13-16 .0677 .0036 .260 M, .4271 .1433 .653 1/ 1.437 1.623 1.199 % .0729 .0042 .270 yi .4375 .1503 .660 34 1.458 1.670 1.207 15-16 .0781 .0048 .280 % .4479 .1576 .669 ^ 1.479 1.718 1.216 1. .0833 .0055 .289 34 .4583 .1650 .677 18. 1.5 1.767 1.224 1-16 .0885 .0062 .297 % .4688 .1725 .685 H 1.542 1.867 1.241 % .0938 .0069 .305 H .4792 .1803 .693 19. 1.583 1.969 1.258 3-16 .0990 .0077 .314 % .4896 .1878 .700 ^ 1.625 2.074 1.274 M .1042 .0085 .322 6. .5 .1964 .707 20. 1.667 2.182 1.291 5-16 .1094 .0094 .330 H .5208 .2131 .722 }i 1.708 2.292 1.307 % .1146 .0103 .338 ^ .5417 .2304 .736 21. 1.750 2.405 1.323 7-16 .1198 .0113 .346 H . .5625 .2485 .750 34 1.791 2.521 1.339 14 .1250 .0123 .354 1. .5833 .2673 .764 22. 1.833 2.640 1.354 9-16 .1302 .0133 .361 Va, .6042 .2867 .777 34 1.875 2.761 1.369 % .1354 .0144 .368 34 .6250 .3068 .791 23 1.917 2.885 1.384 11-16 .1406 .0155 .375 % .6458 .3276 .803 ^ 1.958 3.012 1.399 % .1458 .0167 .382 8. .6667 .3491 .817 24. 2.000 3.142 1.414 13-16 .1510 .0179 .389 M .6875 .3712 .829 25. 2.083 3.409 1.443 K .1563 .0192 .395 >i .7083 .3941 .841 26. 2.166 3.687 1.472 15-16 .1615 .0205 .402 M .7292 .4176 .854 27. 2.250 3.976 1.500 2. .1667 .0218 .408 9. .75 .4418 .866 28. .2 333 4.276 1.528 1-16 .1719 .0232 .414 34 .7708 .4667 .879 29. 2 416 4.587 1.555 H .1771 .0246 .420 3^ .7917 .4922 .890 .30. 2.500 4.909 1.581 3-16 .1823 .0260 .427 K .8125 .5185 .902 31. 2.584 5.241 1.607 M .1875 .0276 .433 10. .8333 .5454 .913 32. 2.666 5.585 1.633 5-16 .1927 .0291 .440 Va, .8542 .5730 .924 33. 2.750 5.940 1.658 % .1979 .0308 .445 .8750 .6013 .935 34. 2.834 6.305 1.683 7-16 .2031 .0324 .451 H .8958 .6303 .946 35. 2.916 6.681 1.708 H .2083 .0341 .457 11. .9167 .6600 .957 36. 3.000 7.069 1.732 9-16 .2135 .0358 .462 H .9375 .6903 .968 38. 3.166 7.876 1.779 % .2188 .0375 .467 34 .9583 .7213 .979 40. 3.333 8.727 1.825 11-16 .2240 .0394 .473 H .9792 .7530 .990 42. 3.500 9.621 1.871 H .2292 .0412 .478 12. 1. .7854 1.000 44, 3.666 10.56 1.914 13-16 .2344 .0432 .484 H 1.021 .8184 1.010 48. 4.000 12.57 2.000 % .2396 .0451 .489 H 1.042 .8522 1.020 54. 4.500 15.90 2.121 15-16 .2448 .0471 .495 H 1.063 .8866 1.031 60. 5.000 19.63 2.236 3. .2500 .0491 .500 13. 1.083 .9218 1.041 66. 5.500 23.76 2.345 H .2604 .0532 .510 H 1.104 .9576 1.051 72. 6.000 28.27 2.449 .2708 .0576 . .520 H 1.125 .9940 1.060 78. 6.500 33.18 2.550 % .2813 .0621 .530 H 1.146 1.031 1.070 84. 7.000 38.48 2.646 ^ .2917 .0668 .540 14. 1.167 1.069 1.080 90. 7.500 44.18 2.739 % .3021 .0716 .550 H 1.187 1.108 1.090 96. 8.000 50.27 2.828 H .3125 .0767 .560 H 1.208 1.147 1.099 % .3229 .0819 .570 H 1.229 1.18f7 1.110 * Caution. In the tables on pages 525 and 526, the diameters or bores are the actual ones, as measured in inches. Wrought-iron steam, gas, and water pipes are commonly designated by fictitious or " nominal" diameters, which are mere arbitrary names for the pipes. In the smaller sizes especially, the use of these nominal diameters tends to mislead. Thus, the pipe whose " womma^ " inner diameter is one-eighth inch, has an actual inner diameter of full quarter inch. HYDRAULICS. 527 Art. 3. To findwtlie total bead required for a g^iven velocity, ot fiveii discliarg-e, through a straight, smooth, cylindrical iroa pipe of nown diam and length. If the discharge is given, first find ' mean velocity discharge in cubic feet p er second in feet per second "" area of cross section of pipe^iu square feet Then Vdiam X head _ mean velocity in feet per second length + 54 diams ~ the proper divisor as foUowa diam of pipe in ft .05 .10 .50 1 1.5 2 3 4 divisor 40 43 46 48 51 64 58 61 (for intermediate diams, take intermediate divisors by guess.) From table Art 2, take the coefficient m corresponding to this value of i diam X head ^ ^^^ ^^ ^^^ ^.^^^ ^.^^^ ^^^^^ length + 54 diams' Total _ Igfeet^'perler X (^"°g^^ ^° ^^ + ^^ ^'^"^^ ^° ^^ in^ftft" m^Xdiaminfeet To find the Friction bead. Weisbach^s formula. .01716 \ I^ength Vel^in (01716 \ i^eDgin vei-m .0144 + Iw in feet ^ ft per sec ^velinft l-^ Diam 64.4 per sec I ,„ foot Friction head _ in feet " per sec f in feet For tbe total bead, we have only to add together, the friction bead so found, the velocity bead, taken from the next table, or from Table 10, opposite the given velocity, and tbe entry bead (= say half the velocity head). The sum of the velocity head and entry head rarely amounts to a foot. TABliE 4 Of tbe vel, and discbarge of water through straight, smooth, cylindrical cast-iron pipes; with tbe friction bead required for each 100 feet in length; and also tbe velocity bead. Calculated by means of Weisbach's formula, by James Thompson, A M ; and George Fuller, C E, Belfast, Ireland. Tbe vel bead remains tbe same for any length of pipe ; being dependent only on tbe velocity of th« water in tbe pipe. Tbe entry bead is equal to about half the vel bead. 628 HYDRAULICS. TABIiE 4 Vel. head in Feet. Diauk in Inches Yel. in Feet 3 3H 4 4H 5 per Sec. Prhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Prhead Ft per 100 ft. Cub ft per Min Prhead Ft per 100 ft. Cub ft per Min 2.0 .062 .659 5.89 .565 8.02 .494 10.4 .439 13.2 .395 16.3 2.2 .075 .780 6.48 .669 8.82 .585. 11.5 .520 14.6 .468 18.0 2.4 .090 .911 7.07 .781 9.62 .683 12.5 .607 15.9 .547 19.6 2.6 .105 1.05 7.65 .901 10.4 .788 13.6 .701 17.2 .631 21.3 2.8 .122 1.20 8.24 1.03 11.2 .900 14.6 .800 18.5 .720 22.9 3.0 .140 1.35 8.83 1.16 12.0 1.02 15.7 .905 19.8 .815 24.5 3.2 .160 1.52 9.42 1.31 12.8 1.14 16.7 1.02 21.2 .915 26.2 3.4 .180 1.70 10.0 1.46 13.6 1.27 17.8 1.13 22.5 1.02 27.8 3.6 .202 1.89 10.6 1.62 14.4 1.41 18.8 1.26 23.8 1.13 29.4 3.8 .225 2.08 11.2 1.78 15.2 1.56 19.9 1.39 25.2 1.25 31.0 4.0 .250 2.28 11.8 1.96 16.0 1.71 20.9 1.52 26.5 1.37 32.7 4.2 .275 2.49 12.3 2.14 16.8 1.87 22.0 1.66 27.8 1.50 34.3 4.4 .302 2.71 12.9 2.33 17.6 2.03 23.0 1.81 ,29.1 1.63 36.0 4.6 .330 2.94 13.5 2.52 18.4 2.21 24.0 1.96 304 1.76 37.6 4.8 .360 3.18 14.1 2.72 19.2 2.38 25.1 2.12 31.8 1.91 39.2 5.0 .390 3.43 14.7 2.94 20.0 ?.57 26.2 2.28 33.1 2.05 40.9 5.2 .422 3.68 15.3 3.15 20.8 2.76 27.2 2.45 34.4 2.21 42.5 5.4 .455 3.94 15.9 3.38 21.6 2.96 28.2 2.63 35.8 2.37 44.2 5.6 .490 4.22 16.5 3.61 22.4 3.16 29.3 2.81 37.1 2.53 45.8 5.8 .525 4.50 17.1 3.85 23.2 3.37 30.3 3.00 38.4 2.70 47.4 6.0 .562 4.78 17.7 4.10 24.0 3.59 31.4 3.19 39.7 2.87 49.1 6.2 .600 6.08 18.2 4.36 24.8 3.81 32.4 3.39 41.0 3.05 50.7 6.4 .640 5.39 18.8 4.62 25.6 4.04 33.5 3.59 42.4 3.23 52.3 6.6 .680 5.70 19.4 4.89 26.4 4.28 34.5 3.80 43.7 3.42 54.0 6.8 .722 6.02 20.0 5.16 27.3 4.52 35.6 4.01 45.0 3.61 55.6 7.0 .765 6.35 20.6 5.45 28.0 4.77 36.6 4.24 46.4 3.81 57.2 Vel- head in Feet. Diam. in Inches. Vel. in Feet 5 7 8 9 10 per Sec. Prhead Ft per 100 ft. Cub ft per Min Prhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft pwMin 2.0 .062 .329 23.5 .282 32.0 .247 41.9 .220 53.0 .198 65.4 2.2 .075 .390 25.9 .334 35.3 .293 46.1 .260 58.3 .234 72.0 2.4 .090 .456 28.2 .390 38.5 .342 50.2 .304 63.6 .273 78.5 2.6 .105 .526 30.6 .450 41.7 .394 54.4 .350 68.9 .315 85.1 2.8 .122 .600 32.9 .514 44.9 .450 58.6 .400 74.2 .360 91.6 3.0 .140 .679 35.3 .582 48.1 .509 62.8 .453 79.5 .407 98.2 3.2 .160 .763 37.7 .654 51.3 .572 67.0 .508 84.8 .458 105 3.4 .180 .851 40 .729 54.5 .638 71.2 .567 90.1 .510 111 3.6 .202 .943 42.4 .808 57.7 .707 75.4 .629 95.4 .566 118 3.8 .225 1.04 44J .892 60.9 .780 79.6 .693 101 .624 124 4.0 .250 1.14 47.1 .979 64.1 .856 83.7 .761 106 .685 131 4.2 .275 1.25 49.5 1.07 67.3 .935 87.9 .832 111 .748 137 4.4 .302 1.35 51.8 1.16 70.5 1.02 92.1 .905 116 .814 144 4.6 .330 1.47 54.1 1.26 73.7 1.10 96.3 .981 122 .883 150 4.8 .360 1.59 56.5 1.36 76.9 1.19 100 1.06 127 .954 157 5.0 .390 1.71 58.9 1.47 80.2 1.28 105 1.14 132 1.03 168 5.2 .422 1.84 61.2 1.58 83.3 1.38 109 1.23 138 1.10 170 5.4 .455 1.97 63.6 1.69 86.6 1.48 113 1.31 143 1.18 177 5.6 .490 2.11 65.9 1.81 89.8 1.58 117 1.40 148 1.26 183 5.8 .525 2.25 68.3 1.93 93.0 1.68 121 1.50 154 1.35 190 6.0 .562 2.39 70.7 2.05 96.2 1.79 125 1.59 159 1.43 196 6.2 .600 2.54 73.0 2.18 99.4 1.90 130 1.69 164 1.52 203 6.4 .640 2.69 75.4 2.31 102 2.02 134 1.79 169 1.61 209 6.6 .680 2.85 77.7 2.44 106 2.14 138 1.90 175 1.71 216 6.8 .722 3.01 80.1 2.58 109 2.26 142 2.01 180 1.81 222 7.0 .765 3.18 82.4 2.72 112 2.38 146 2.12 185 1.90 229 HYDRAULICS. 529 TABI.E 4 - - (Continued., Vel- head in Feet. Diam. in Inches. Vel. in Feet 11 12 1 13 14 15 per Sec. Frhead Ft per 100 ft. Cub ft per iMin Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min 2.0 .062 .180 79.2 .165 94.2 .152 110 .141 128 .132 147 2.2 .075 .213 87.1 .195 1(13 .180 121 .167 141 .156 162 2.4 .090 .248 95.0 .228 113 .210 133 .195 154 .182 176 2.6 .105 .287 103 .263 122 .242 144 .225 167 .210 191 2.8 .122 .327 111 .300 132 277 156 .257 179 .240 206 3.0 .140 .370 119 .339 141 .313 166 .291 192 .271 221 3.2 .160 .416 127 .381 151 .352 177 .327 205 .305 235 3.4 .180 .464 134 .425 160 .393 188 .365 218 .340 250 3.6 .202 .514 142 .472 169 .435 199 .404 231 .377 265 3.8 .225 .567 150 .520 179 .480 210 .446 243 .416 280 4.0 .250 .623 158 .571 188 .527 221 .489 256 .457 294 4.2 .275 .680 166 .624 198 .576 232 .534 269 .499 309 4.4 .302 .740 174 .679 207 .626 243 .582 282 .543 324 4.6 .330 .803 182 .736 217 .679 254 .631 295 .589 339 4.8 .300 .867 190 .795 226 .734 265 .682 308 .636 353 5.0 .390 .935 198 .857 235 .791 276 .734 321 .685 368 5.2 .422 1.00 206 .920 245 .850 287 .789 333 .736 383 6.4 .455 1.07 214 .986 254 .910 298 .845 346 .789 397 5.6 .490 1.15 222 1.05 264 .973 309 .903 359 .843 412 6.8 .525 1.22 229 1.12 273 1.04 321 .964 372 .899 427 6.0 .562 1.30 237 1.19 283 1.10 332 1.02 385 .967 442 6.2 .600 1.38 245 1.27 292 1.17 343 1.09 397 1.01 456 6.4 .640 1.47 253 1.35 301 1.24 354 1.15 410 1.08 471 6.6 .680 1.55 261 1.42 311 1.31 365 1.22 423 1.14 486 6.8 .722 1.64 269 1.50 320 1.39 376 1.29 436 1.20 500 7.0 .765 1.73 277 1.59 330 1.46 387 1.36 449 1.27 615 Vel- head in Feet. Diam. in Inches , Vel. in Feet 16 17 18 19 20 perSec. Frhead Ft per 100 ft. Cub ft per Min foo^ft'iperMin Fr head Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min 2.0 .062 .123 167- .116 189 .110 212 .104 236 .099 262 2.2 .075 .146 184 .138 208 .130 233 .123 260 .117 288 2.4 .090 .171 201 .161 227 .152 254 .144 283 .137 314 2.6 .105 .197 218 .185 246 .175 275 .166 307 .158 340 2.8 .122 .225 234 .212 265 .200 297 .189 331 .180 366 3.0 .140 .255 251 .240 2S4 .226 318 .214 354 .204 393 3.2 .160 .286 268 .269 302 .254 339 .241 378 .229 419 3.4 .180 .319 284 .300 321 • .283 360 .269 401 .255 445 3.6 .202 .354 301 .333 340 .314 382 .298 425 .283 471 3.8 .225 .390 318 .367 .359 .347 403 .328 449 .312 497 4.0 .250 .428 335 .403 378 .380 424 .360 472 .342 623 4.2 .275 .468 352 .440 397 .416 445 .394 496 .374 660 4.4 .302 .509 368 .479 416 .452 466 .429 619 .407 676 4.6 .330 .552 385 .519 435 .490 48S .465 643 .441 602 4.8 .360 .596 402 .561 454 .^0 509 .502 667 .477 628 5.0 .390 .642 419 .605 473 .571 530 .541 690 .514 664 5.2 .422 .690 435 .650 •492 .614 551 .581 614 .562 680 5.4 .455 .740 452 .696 511 .657 672 .623 638 .592 707 6.6 .490 .791 469 .744 529 .703 594 .666 661 .632 733 6.8 .525 .843 486 .793 548 .749 615 .710 685 .674 769 6.0 .562 .897 502 .844 567 .798 636 .755 709 .718 786 6.2 .600 .953 519 .897 586 .847 657 .802 732 .762 811 6.4 .640 1.01 636 .951 605 .898 678 .851 756 .808 838 6.6 .680 1.07 653 1.01 624 .950 700 .900 780 .855 864 6.8 .722 1.13 569 1.06 643 1.00 721 .951 803 .904 896 7.0 .765 1.19 586 1.12 662 1.06 74.'' 1.00 827 .953 1 Plfi 34 530 HYDRAULICS. TABLE 4 — (Continued.) Vel- head in Feet. Diam^iu laches. Vel. in Feet 22 24 26 28 30 perSec. Prhead Ft per 100 ft. Cub ft per Min Prhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Min Frhead Ft per 100 ft. Cub ft per Mia 2.0 .062 .090 316 .082 377 .076 442 .070 513 .066 589 2.2 .075 .106 348 .097 414 .090 486 .083 564 .078 648 2.4 .090 .124 380 .114 452 .105 531 .097 616 .091 707 2.6 .105 .143 412 .131 490 .121 575 .112 667 .105 766 2.8 .122 .164 443 .150 528 .138 619 .128 718 .120 824 3.0 .140 .185 475 .170 565 .157 663 .145 770 .136 883 3.2 .160 .208 507 .191 603 .176 708 .163 821 .152 942 3.4 .180 .232 538 .213 641 .196 752 .182 872 .170 1001 3.6 .202 .257 570 .236 678 .218 796 .202 923 .189 1060 3.8 .225 .284 601 .260 716 .240 840 .223 974 .208 1119 4.0 .250 .311 633 .285 754 .263 885 .244 1026 .228 1178 4.2 .275 .340 665 .312 791 .288 929 .267 1077 .249 1237 4.4 .302 .370 697 .339 829 .313 973 .290 1129 .271 1296 4.6 .330 .401 728 .368 867 .339 1017 .315 1180 .294 1355 4.8 .360 .434 760 .397 905 .367 1062 .341 1231 .318 1414 6.0 .390 .467 792 .428 942 .395 1106 .367 1283 .343 1472 5.2 .422 .502 823 .460 980 .425 1150 .394 1334 .368 1531 5.4 .455 .538 855 .493 1018 .455 1194 .423 1385 .394 1590 5.6 .490 .575 887 .527 1055 .486 1239 .452 1437 .422 1649 5.8 .525 .613 918 .562 1093 .519 1283 .482 1488 .450 1708 6.0 .562 .652 950 .598 1131 .552 1327 .513 1539 .478 1767 6.2 .600 .693 982 .635 1168 .586 1371 .544 1590 .508 1826 6.4 .640 .735 1013 .673 1206 ;622 1416 .577 164-2 .539 1885 6.6 .680 .778 1045 .713 1241 .658 1460 .611 1693 .570 1943 6.8 722 .821 1077 .753 1282 .695 1504 .645 1744 .602 2003 7.0 '.765 .867 1109 .794 1319 .733 1548 .681 1796 .635 2061 HYDRAULICS. 531 Art* 4 To find tbe discbarg^e throiig'h a compound pipe r o, Fig 1 H, composed of any number of pipes, a, 6, c, z, of different diaiu* eterS) which decrease from the reservoir toward the outflow o. ' ^ '^ u ^^ 1 Fio.iH First find what part of the total head H is employed in forcing the water through the last pipe z alone^ thus : Let La, lib, L c, and Lz, be the lengths in ft of the pipes a, 6, c, and z, respectively; D a, D 6, D c, and D z their diameters in ft ; and A a, A 6, A c, and A z the areas of their cross sections in sq ft. Then The head in in forSng^he ( _ The total head H in feet waterthrough f - r A z^ X ^ Z ^ /La-h(54XDa) , L 6-f (54XD fc) . L o-f (54XD c)\ H thejast pipe ^ 1+ ll ^^^r,,^j, ^^ X (, a a3 X D a + ^aTT^TdT + Ac^XdTJA However many divisions the pipe may have, proceed in tlie same way as above, using Area* X Diam - ^, . ^ i. j. • • j Length -f 54 Diams for the last or narrowest division; and Length + 64 Diams * Area^ x Diam for each of the others. Then, by the formulae. Art. 2, find the velocity in ft per second, and discharge in cub ft per second, of the last pipe z, using its actual diameter, length, and croee sectional area, and the head just found. Said discharge is evidently the discharge for the compound pipe. For the velocity in any portion^ as 6, say area of cross section , area of cross , , velocity , velocity in the of the given portion • section of a • • iu z • given portion. For the above rule and formula, we are indebted to Mr, Howard Murphy, C B, of Phila; and for the opportunity of testing it experimentally, to Messrs Morris, Tasker & Co, Limited, Pascal Iron Works, Philadelphia. 532 WATER-PIPES. Art. 4 a.* The Ventari Meter is designed for the measurement of the flow of liquids in pipes of large dimensions, running full. Tlie meter proper, patented by Clemens Herschel, consists essentially of a mere constriction in the area of cross-section of the pipe, with openings in the pipe opposite its normal and its constricted diameters, for measuring, by piezometers or pressure-gauges, the pressures at those points; while the register, patented by Messrs. Frederick N. Connet and Walter W. Jackson, is an elaborate mechanism, provided with clock-work and dials. Theory .t Let Figs. 1 to 3 represent a Venturi meter tube, Fig. 1. with three piezometers in place, viz.: No. 1, over the tube up-stream from the constriction ; No. 2, over the constriction itself; and No. 3, over the tube down-stream from the constriction. Let the unshaded area W in Figs. 1 to 3, represent the depths at which the water stands above any assumed hori- zontal datum plane 0-0; and let the shaded area A represent the uniform pressure of the atmos- ^ " phere, which, for convenience, we may suppose to be converted into some liquid of the specific gravity of water, but distinguishable, by its appearance, from the water. The vertical distance, between the upper boundary of this latter area and any given point in the tube, represents the combined pressure of air and water at such point. The velocities in the meter tube, at any instant, are of necessity inversely proportional to the areas of cross section; and, as the heads corresponding to the several velocities are proportional to the squares of those velocities, the remaining or pressure heads must vary also, the smallest or lowest pressure head standing over the throat, where the velocity is greatest. The increase of velocity, ac- FiG. 2. quired by the fluid in passing from section 1 to section 2, is again given up in passing from section 2 to section 3; and, in the case of a perfect fluid, the pressure lost between sections 1 and 2 would be perfectly restored in passing from section 2 to section 3. In practice, a small total loss occurs. This loss is greater with high than with low velocities. For a given head in piezometer No. 1 and given diameter of pipe at section 1, the expenditure of head in velocity between sections 1 and 2 increases as the area of the throat is diminished and as the throat Telocity is thereby increased.^ In Fig. 2 is shown the case where all of the water head above the top of the throat is required to maintain the velocity through the throat. In Figs, 1 and 2 the head, If, expended in the increase of velocity between sections 1 and 2 is represented by the difference in level between the tops of the two water columns 1 and 2, or between the tops, of the two corresponding air columns. In Fig. 2 this difference is equal to the io^a^ vertical height of the water column at section 1 above the top of the throat at section 2. * Abridged from a description prepared by the writer as Chairman of a Com- mittee of the Franklin Institute. Journal of the Franklin Institute, February, 1899. t The Venturi meter, apart from its merits as a measuring device, embodies important hydraulic principles. Hence its theory is here stated niore fully than would otherwise be necessary. X In a given Venturi tube the pressure and velocity at the throat may be varied also by modifying those at sectons 1 and 3, as by regulating the openings of the valves of influx to and of efflux from the meter tube, by changing the total head on the system, etc. WATER-PIPES. 533 Fig. 3. If now (Fig. 3) the throat sec- tion be still further reduced (the other conditions remain- ing as before), the throat veloc- ity will thereby be still further increased ; for the total pressure available for increase of veloc- ity between sections 1 and 2 con- sists not merely in the^depth of water above the tube, but also in the atmospheric pressure, rep- resented by the shaded area A above the water W. In Fig. 3 all the water has dis- appeared from piezometer 2, and q even a portion of the liquid representing the air has also disappeared, leaving only a portion of the latter to represent Such pressure as now remains in the throat. In other words, the pressure within the throat is now less than the atmospheric pressure. In Fig. 3, the loss of head, due to increase of velocity between sections 1 and 2, is H= hw ■\-ha = the entire available head of water, hw, plus a portion, ha, of the atmospheric pressure. The latter portion, ha, is frequently called "the vacuum." The top of the water column having now disappeared below the top of the throat, it is no longer feasible to ascertain the loss of head by taking the differ- ence between the levels of the water surfaces in piezometers 1 and 2. The degree of " vacuum " may be Fig. 4. found, as shown in Fig. 4, by using, in place of the piezometers, a glass tube bent over and led downward into an open vessel con- taining water or mercury. The height to which the water (or the mercury, converted into feet of water) rises in this tube, shows the extent of the vacuum, or the portion, A«, of the air pressure which has been called into service in producing the high velocity through the throat. By adding this to hw, we obtain, as above, the total loss of head H between sections 1 and 2. When the reduction of area at the throat has proceeded so far that the entire available pressure of water and air at section 1 is required, in order to maiu- FiG. 5. tain tlie corresponding velocity through the throat (t. e., when the line repre- senting the upper surface of the air falls to the level of the top of the throat), 534 WATER-PIPES. no further increase of throat velocity can be secured (with a given total head over section 1) by still further narrowing the throat. If the throat is further narrowed, the velocity through it will remain the same ; and, the rate of dis- charge being thus diminished, the velocity through section 1 will be neces- sarily reduced. In other words, throttling begins. Let vi be the velocity in section 1, above the throat, and V2 the "throat veloc- ity," or velocity in the throat or section 2. Referring to Fig. 5, the velocity head at section 1, measured from an assumed datum represented by the upper horizontal lines, is and that at section 2 is '^-t' 2g Neglecting resistances to flow, the loss of head, between sections 1 and 2, or " the head on the Venturi," is equal to the increase in the velocity head, ot to the loss in pressure, between a^, and 02, or Hence, h.^ = '^ = B. + h-^ and tbroat velocity = vg ^2^(H + AJ = ^j2g{H+^)^ In other words, the velocity at the throat is that corresponding to the " head H on the Venturi," plus the head corresponding to the velocity of approach Vi in section 1. But, since the velocities are inversely as the areas of cross-section a^ and a^, ■ ~ '■" ""^ "^ ~ ai2 ^- - - "- - 2, and throat velocity = vg = . ^ -V2g H, The ratio — ^» between the area a^ of cross-section at the throat, and that, Oj, at the upper end of the up-stream cone, is called the tbroat ratio. For a ratio of 1 : 9 we have • «! _ 9 _ _9- _ /si or V2 = 1.0062)^2^7 i/. The Tenturi tube, for pipes not over 60 inches in diameter, is formed of 8«^veral short sections of cast iron pipe, having the required taper, and fur- * By Bernouilli's theorem, j?i + ^1 = Pa + ^2- WATER-PIPES. 5e35 nished with flanges, by means of which the sections are bolted together to form the two truncated cones required. In the smaller sizes, the shorter cone is generally in one section and the longer cone in two or more sections. The throat section is generally made in a separate piece, and is either made of bronze or lined with that metal. The ends of the Venturi tube are furnished with either bell, spigot, or flanged ends, according to the character of the pipe in which the tube is to be used. For still iarjser streams, such as those in masonry conduits or riveted flumes, the Venturi tube may be made of wooden staves,' sheet steel, cement concrete, brick or other suitable material, metal being used for the throat piece and where required by the pressure. The tliroat piece is surrounded by an annular chamber called the press- ure ctiamber. which communicates with the interior of the throat by means of several holes drilled radially through the walls of the latter at equal or nearly equal distances around the circumference. A similar pressure chamber is provided at the larger end of the short cone for observing the pressure in the normal section up-stream from the throat ; and, if it is desired to ascertain the final loss of head due to the passage of the water through the Venturi, a similar chamber must be provided at the larger end of the longer or down-stream cone. In designating- tlie size of the meter, the diameter of the pipe of which it forms a part is used, and not the throat diameter. Thus, a meter for use in a 6-inch pipe is called a 6-inch meter. The reg'ister gives periodic registrations, usually every ten minutes, in which the head ff = h2— hi, existing &t the instant of registry, is recorded in terms of the total discharge in cubic feet since the last registry and as an in- crease in the total number of cubic feet registered. In other words, the registry involves the assumption that the average velocity, during the period between two registrations, is equal to the velocity at the instant of the following registration. The register may be placed at a considerable distance (not exceeding, say, 500 feet) from the Venturi tube. It must be placed at such a depth below the hydraulic grade line that the pressures existing in the Venturi tube shall at all times be transmitted to the register. The pipe lines, connecting the Venturi with the register, must be covered, and a shelter from weather and frost must be provided for the register. The size and cost of the register are independent of the size of the Venturi. Behavior. From experiments by Mr. Herschel,* f ^ by the Bureau of Water, Philadelphia,f and by others,! it appears that the Venturi meter may ordi- narily be depended upon to give results within 3 per cent, of the true discharge. With a 48 inch Venturi, Mr. Herschel ^ found a total loss of head, due to the passage of the water through the Venturi tube, of about 10.6 per cent, of the head H on the Venturi. With two 54 inch Venturis, Professors Marx, Wing, and Hoskinsgf found a loss of 14.9 per cent., part of which, no doubt, was due to the presence of a 42 inch gate valve in the down-stream cone. This last result would add about 1.12 feet to the head required in pumping 20,000,000 gallon^ daily through a 48 inch main and a Venturi having a throat ratio of 1 : 9. The Venturi meter has been found to give perfectly satisfactory results in measuring the flow of brine and very hot water. Venturi tubes are made with throat ratios ranging from 1:4^ (or 2: 9) to 1: 16. The former are adapted to high, and the latter to low velocities; for, where the velocity in the pipe is low, it is necessary to accelerate it greatly in the throat in order to obtain sutficient loss of pressure to secure reliable' in- dications in the register. These cannot be obtained where the throat velocity is less than about 3 feet per second. With a throat ratio of 1 : 16, this would give a pipe velocity of {^e ^^^^ P^i* second. On the other hand, a meter with a high throat ratio, adapted to low velocities, would, with high velocities, exceed the upper limit of the register. Owing to its unobstructed channel, free from moving parts, the Venturi meter is far less liable to clogging than the forms of meter in common use. The prices of the principal sizes of the Venturi meter are as follows: — on board cars at Providence, R. 1. 6 inch ^600.00 24 inch ^1,130.00 48 inch $3,060.00 12 inch 770.00 36 inch 1,680.00 60 inch 4,890.00 These prices include the register, which, in the smaller sizes, constitutes the principal item of cost. Discount, 1901, 10 per cent. * Trans. Am. Soc. Civil Engrs., Nov., 1887, Vol. XVII., page 228. fJournal of the Franklin Institute. Feb., 1899. 1[ Journal New England Waterworks Assn., Vol. VIII., No. 1, Sep., 1893. 2 Trans. Am. Soc. Civil Engrs,, Vol. XL., Dec, 1898, pp. 471, etc. 536 WATER-PIPES. Fig. 6. 7i = ftt>» Art. 4 b. Tlie Ferris-Pitot meter, invented and patented by Mr. Walter Ferris, of Philadelphia, in designed to measure the flow of liquids in pipes running full. It consists of a device for the registration of the results obtained by the Pitot tube, described on pages 561 and 562, and of special devices to prevent the clogging of the tubes and to permit their examination while in use. In Fig. 6 let P represent the level at which the water stands in the straight Pitot tube, 5. Then h^ki^, or the difference in level between the columns in the two tubes, is the head (theoretically =— ) due to the velocity of the water in the pipe as it impinges against the open up-stream end of the bent tube, c. For a given velocity, v, this ditference, //, is constant, and is Independent of the pressure represented by P. The Ferris register, like that of the Venturi meter, records the velocity (existing ai the instant of registra- tion) in terms of the total discharge since the last regis- try and as an increase in the total number of cubic feet registered. The registry thus involves the assumption that the average velocity, during the period between registrations, is equal to the velocity at the end of that period. In the Ferris meter the registration is made every two minutes. Evidently the instrument measures the velocity at only one point in the cross-section of the pipe, arid it may thus be used to de- termine successively the velocities at any number of such points, but the ve- locity at such a point may or may not be equal to the mean velocity in the entire cross-section. The instrument is therefore usually calibrated by reference to some accepted standard, and the coeflicient or coefficients thus obtained are used in subsequent observations. The recording mechanism is operated by a small hydraulic motor, driven by means of the flow of the water in the pipe itself. For this purpose a second pair of Pitot tubes, is inserted into the pipe; and the current, flowing through these tubes, drives the motor without loss of water, the water used for power being returned to the pipe. If the velocity in the pipe is less than 3 feet per second it must be increased by means of a " reducer." Experiments made by Mr. Ferris and by the Bureau of Water, Philadelphia, indicate that the Ferris-Pitot meter will ordinarily register within 3 per cent, of the true discharge. In general, the size and cost of the registering apparatus are independent of the size of the pipe. HYDRAULICS. 537 Art. 5. Resistance of curves and bends in water pipes. Much uncertainty exists respecting these matters. Weisbach's form- ula,* for the resistance due to a circular curve, Figs. 2 and 3, is A = C A 180 il = [0.131+ 1.8.7 (^) J] A 180 '^9 •where h = head in feet required to overcome resistance due to curve or bend^ C = experimental coeflQicient, A = angle of deflection, in degrees, V = mean velocity of flow in pipe, in feet per second, g = acceleration of gravity = 32.2 ft per sec per sec, ^9 = head theoretically due to velocity v, D = inside diameter of pipe, in feet, r = inside radius of pipe, in feet, R = radius of axis of curve, in feet. IfrH-R= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0:9 1.0 then C = 0.131 0.138 0.158 0.206 0.294 0.440 0.661 0.977 1.408 1.978 Fig. a. Fig. 2. Figr. 4. (See next page.) According to this formula, the resistance due to curvature decreases rapidly as R increases from J^ D to 2 D ; and but little further decrease occurs beyond R = 5 D ; but, from very careful and elaborate experiments on city water mains, from 12 to 30 ins diameter, in Detroit, Mich.,f the investigators conclude that a line of pipe with a curve of short radius R (down to a limit of R = 2>^ D) causes less resistance than does a line of equal length and equal total angle A, with a curve of longer radius R. Their results were approximately as follows, where H = resistance due to a section of 80 diameters in length, with a curve of A = 90° at mid-length, h = resistance in a tangent of length = 80 diameters. If R -f- D = 1 2 2.5 3 4 5 10 15 20 25 then B.-irh = 1.35 1.14 1.13 1.14 1.18 1.24 1.50 1.66 1.80 1.95 They found also that the loss of bead, due to a curve, occurs not only in the curve itself, but that head continues to be lost in the following tangent, for some distance down stream from the curve. Their experiments led to the inference that even very sliffbt deflections, A, in the line, cause material losses of bead, and that»care in securing a straight alignment is therefore highly advisable. For bends, see next page. *Der Ingenieur, pp. 444, 445. t Paper by Gardner S. Williams, Clarence W. Hubbell, and George H. Fenkell, Transactions, American Society of Civil Engineers, 1901. 538 HYDRAULICS. For abrupt ang'les. Fig. 4, Weisbach gives : Resistance, in feet of head =— h = c^ == (0.95 sin2 J^ A +,2.05 sin* y^ A) ^ \iy^K== 10° then c = 0.03 20° 0.14 30° 0.36 40° 0.74 45° 0.98 50° 1.26 55° 1.56 60° 1.86 65° 2.16 2.43 Figr. 4. In addition to the resistance offered to flow, curves and bends in- volve additional labor and expense in manufacture and in laying; and vertical bends and curves lead to the formation of pockets of sediment at the feet of slopes, and of air cushions at their summits. a h Art. 6. Although, in Fig. 5, the static pressures upon the equal bases, a b and a' &', of the two pipes are equal (see Hydrostatics, Art. 1) ; yet, in order to pump water through either pipe, at a given velocity, an additional force is required, in order to overcome resistances to flow ; and these resistances and the additional force required in order to overcome them, will be greater in the longer than in the shorter pipe. HYDRAULICS. 539 Art. 7. Flow throngli orifices. Theoretically the velocity, v, of a fluid flowiug through a small orifice iu the side or bottom of a very large vessel, is equal to that acquired by a body falling freely in vacuo through a height equaUto the head, h, or depth, measured vertically from the level surface of the fluid iu the vessel, to the center of gravity of the orifice ; or, ^l/2gh = |/ 64.4 A = 8.03 ^Z h\ and h = 2^' 64.4 = 0.0155 v\ This law applies equally to all flnids. Thus, theoretically, mer- cury, water, air, etc., all flow with equal velocities from a given orifice under a given head. For deviations in practice from this theoretical law, see Art. 9, etc. Table 10. Velocities theoretically due to given heads. Head Vel. Head Vel. Head^Vel. Head Vel. Head Vel. Head Vel Head Vel. Feet. Ft per Feet. Ft per sec. Feet. Ft per sec. Feet. Ft per sec. Feet. Ft per sec. Feet. Ft per sec. Feet. Ft per sec. .005 .57 .29 4.32 .77 7.04 1.50 9.83 7. 21.2 28 42.5 76 69.9 .010 .80 .30 4.39 .78 7.09 1.52 9.90 .2 21.5 29 43.2 77 70.4 .015 .98 .31 4.47 .79 7.13 1.54 9.96 .4 21.8 30 43.9 78 70.9 .020 1.13 .32 4.54 .80 7.18 1.56 lO.O .6 22.1 31 44.7 79 71.3 .025 1.27 .33 4.61 .81 7.22 1.58 10.1 .8 22.4 32 45.4 80 71.8 .030 1.39 .34 4.68 .82 7 26 1.60 10.2 8. 22.7 S3 4(5.1 81 72.2 .035 1.50 ..35 4.75 .83 7.31 1.65 10.3 .2 23.0 34 46.7 82 72.« .040 1.60 .36 4.81 .84 7.35 1.70 10.5 .4 23.3 35 47.4 83 73.1 .045 lw70 .37 4.87 .85 7.40 1.75 io:6 .6 23.5 36 48.1 84 73.5 .050 1.79 .38 4.94 .86 7.44 1.80 10.8 .8 23.8 37 48.8 85 74.0 .055 1.88 .39 5.01 .87 7.48 1.85 10.9 9. 24.1 38 49.5 86 74.4 .060 1.97 .40 5.07 .88 7.53 1.90 11.1 .2 24.3 39 50.1 87 74.8 . .065 2.04 .41 5.14 .89 7.57 1.95 11.2 .4 24.6 40 50.7 88 75.3 .070 2.12 .42 5.20 .90 7.61 2. 11.4 .6 24 8 41 51.3 89 75.7 .075 2.20 .43 5.26 .91 7.65 2.1 :i.7 .8 25.1 42 52.0 90 76.1 .080 2.27 .44 5.32 .92 7.70 2.2 11.9 10. 25.4 43 52.6 91 76.5 .085 2.34 .45 5 38 .93 7.74 2.3 12.2 .5 26.0 44 53.2 92 76.9 .090 2.41 .46 5.44 .94 7.78 2.4 12.4 11. 26.6 45 53.8 93 77.4 .095 2.47 .47 5.50 .95 7.82 2.5 12.6 .5 27.2 46 54.4 94 77.8 .100 2.54 .48 5.56 .96 7.86 2.6 12.9 12. 27.8 47 55.0 95 78.2 .105 2.60 .49 5.62 .97 7.90 2.7 13.2 .5 28.4 48 55.6 96 78.6 .110 2.66 .50 5.67 .98 7.94 2.8 13.4 13. 28.9 49 56.2 97 79.0 .115 2.72 .51 5.73 .99 7.98 2.9 13.7 .5 29.5 50 56.7 98 79.4 .120 2.78 .52 5.79 IFt. 8.03 3. 13.9 14. 30.0 51 57.3 99 79.8 .125 2.84 .53 5.85 1.02 8.10 3.1 14.1 .5 30.5 52 57.8 100 80.3 .130 2.89 .54 5.90 1.04 8.18 3.2 14.3 15. 31.1 53 58.4 125 89.7 .135 2.95 .55 5.95 106 8.26 .3.3 14.5 .5 31.6 54 59.0 150 98.8 .140 3.00 .56 6.00 1.08 8.34 3.4 14.8 16. 32.1 55 59.5 175 106 .145 3.05 .57 6.06 1.10 8.41 .3.5 15. .5 32.6 56 60.0 200 114 .150 3.11 .58 6.11 1.12 8.49 3.6 15.2 17. 33.1 57 60.6 225 120 .155 3.16 .59 6.17 1 14 8.57 3.7 15.4 .5 33.6 58 61.1 250 126 .160 3.21 .60 6.22 1.16 8.64 3.8 15.6 18. 34.0 59 61.6 275 133 .165 3.26 .61 6.28 1.18 8.72 3.9 15.8 .5 34.5 60 62.1 300 139 .170 3.31 .62 6.32 1 20 8.79 4. 16.0 19. 35.0 61 62.7 350 150 .175 3.36 .63 6.37 1.22 8.87 .2 16.4 .5 35.4. 62 63.2 400 160 .180 3.40 .64 6.42 1.24 8.94 .4 16.8 20. 35.9 63 63.7 4.50 170 .185 3.45 .65 6.47 1.26 9.01 .6 17.2 .5 36.3 64 64.2 500 179 .190 3.50 .66 6.52 1.28 9.08 .8 17.6 21. 36.8 65 64.7 550 188 .195 3.55 .67 6.57 1.30 9.15 5. 17.9 .5 37.2 66 65.2 600 197 .200 3.59 .68 6.61 1.32 9.21 .2 18.3 22. 37.6 67 65.7 700 212 .21 3.68 .69 6.66 1.34 9.29 .4 18.7 .5 38.1 68 66.2 800 227 .22 3.76 .70 6.71 1.36 9.36 .6 19. 23. 38.5 69 66.7 900 241 .23 3.85 .71 6.76 1.38 9.43 .8 19.3 .5 ,38.9 70 67.1 1000 254 .24 3.93 .72 6.81 1.40 9.49 6. 19.7 24. 39.3 71 67.6 .25 4.01 .73 6.86 1.42 9.57 .2 20.0 .5 39.7 72 68.1 .26 4.09 .74 6.91 1.44 9.63 .4 20.3 25 40.1 73 68.5 .27 4.17 .75 6.95 1.46 9.70 .6 20.6 26 40.9 74 69.0 .28 4.25 .76 6.99 1.48 9.77 .8 20.9 27 41.7 75 69.5 540 HYDRAULICS. Fig. 6. Art. 8. On the flow of water tbroug-ta vertical opeiiiii^STS fur- nisbed with short tubes. When water flows from a reservoir, Fig 6, through a vert partition mm a a, the thickness a m of which is about m or 3 times the least transverse dimension of the opening, (whether that dimension be its breadth, or its height;) or when, if the partition be very thin, as n n, the water flows through a tube, as at t, the length of which is about 2 or 3 times its least transverse dimension, then the eflBuent stream will entirely flll the opening, or the tube, as shown in Fig 6 ; or, in technical language, will run with a full flow; or a full bore ; and will disch more water in a given time, than if the tube were either materially longer or shorter. For if longer than 3 times the least transverse dimension, the flow will be impeded by the increased friction against the sides of the tube ; and if shorter than about twice the least transverse dimension, the water will not flow in a full stream, but in a contracted one, as shown by Fig 11. This will be the case whether the tube be circular, or rectilinear, in its cross-section. To find approximately the actual vel. and disch into the air, throug-h a tube, or opening-, either circular or recti- linear in its outline, or cross-section ; and whose leng-th c i, or c e, in the direction of the flow, is about 2}4 or :i times its least transverse dimension ; when the surface-level, s, Fig 6, remains constantly at the same heig-ht; and which height must not be below the upper edge of the tube, or opening. RoTLE 1. Take out the theoretical vel from Table 10, corresponding to the head measured vert from the center (or more properly, the cen of grav) c, of the opening, to the level water surf*. Mult it by the coeff' of disch .81. The prod will be the reqd vel, in ft per sec. Mult this actual vel by the transverse area of the opening, in sq ft. If circular, knowing its diam, this area will be found in Table 3. The prod will be the quantity of water dischd, in cub ft per sec; within, probably, S or 4 per cent. RoxE 2. Find the sq rt of the head in ft. Mult this sq rt by 6.5. The prod will be the actual vel in ft per sec. Ex. An opening c • ; or box-shaped tube c (, Fig 6, is 3 feet vride, by .25 of a ft high ; and its length in the direction ci or c e in which the water flows is about .62 of a ft, or about 2^^ times its least transverse dimension, or its height. The head from the cen of grav c, of the opening, to the constant surf-level s, is 4 feet. What will be the vel of the water ; and how much will be dischd per sec ? By Rule 1. The theoretical vel (Table 10, ) corresponding to a head of 4 ft is 16 ft per sec. And 16 X .81 = 12.96 ft per sec, the actual vel reqd. Again, the transverse area of the opening, or of the tube, is 3 ft X .25 ft = .75 sq ft. And .75 X 12.96 = 9.72 cub ft; the quantity dischd per sec. By Rule 2. The sq rt of 4 is 2. And 2 X 6.5 = 13 ft per sec. the reqd vel, as before ; the very slight diff' being owing to the omission of small decimals in the coeffs. Rem. 1. If the short tube * projects partly inside of the vert partition ti n, the disch will be diminished about }/g part. In that case, use .71 or .7 instead of the .81 of Rule 1 ; or 5.7 instead of the 6.5 of Rule 2. Rem. 2. When the thickness a m of the vert partition m m a a; or the length c e of the tube t, Fig 6, is increased to about 4 times the least transverse dimension of the opening ; or of the diam, when circular; then the additional friction against its sides begins appreciably to lessen the vel and disch. In that case, or for still greater lengths, up to 100 diams, they may be found approximately, by nsing instead of the coefif of disch .81 iu Rule 1, the following coeffs, by which to mult the theoretical vels »f Table 10. TABL.E 11. Length of Length of Pipe Coeff. Pipe Coeff. in Diams. in Diams. 4 .80 40 .62 6... ... .76 50... ... .60 10 .74 60 .57 15... ... .71 70... ... .55 20 .69 80 .52 25... ... .67 90... ... .50 30 .65 100 .48 Rem. 3. When the length of the opening or tube, in the direction in which the water flows, becomes ?.es« than about twice its least transverse dimension, the disch is diminished ; so that for lengths from 1^ times, down to openings in a very thin plate, we may use .61, instead of the .81 of Rule 1. For such openings, however, see Arts 9 and 10. Rem. 4. But on the other hand, the disch through such short openings and tubes as are shown in Fig 6, may be increased to nearly the theoretical ones of Table 10, by merely rounding off neatly the edges of the entrance end or mouth, as in Fig 7 ; which is the shape, and half actual size of one with ■which Weisbach obtained .975 of the theoretical vel and discharge, when the head was 10 ft; and .958 HYDRAULICS. 541 ■with a head of one foot ; so that in similar cases, .975, and .958 may be used instead of the coeflf .8\ in Rule 1. Figr.a As much as .92 to .94 may be obtained by widening the opening, m n, toward its outer mouth, o«. Fig, 8, making the divergence, or angle o, about 5° ; or by widening it toward its inner mouth, as at t c, Fig 9 ; but increasing the angle of divergence, at h, to from 11° to 16°. In all cases, we consider the small end as being the opening whose area must be multiplied by the vel to get the discharge. In some experiments made with larg^e pyramidal wooden trotig'hs 9.5 ft long, with an inner mouth of 3.'2 X 2.4 ft, and a discharging one of .62 X -4^ ft; and under a head of 93^ feet, the discharge was .98 of the theoretical one, due to the smaller end. Therefore, .98 may be used in such cases, instead of the .81 of Rule 1. Rem. 5. By using an adjutage shaped as in Fig 10, the discharge may be increased to several times that due to the head above the center of gravity s of the orifice mn: because in such cases, as explained in Art 1 w, the true head at s, or the head causing the rapid flow through the nar- rowest portion mn, may be much ^ greater than the head above e. See the Venturi Meter, Art 4 a. Fig. 10. Art. 9. On the disch of water throag:h opening^s in thin vert partitions, with plane or flat faces, ee, or n n, Fig 11.* If the face e e, or n n, instead of being plane, and vert, should be curved, or inclining in diflF dir«ctions toward the opening, then the disch will be altered. When water flows from a reservoir. Fig 11, through a vert plane plate or partition nn, which is not thicker than about the least transversedimension of the opening, whether thatdimension be its breadth, or its height o o ; t or when, i' the partition e e itself is much thicker, we give the opening the shape shown at b, (which evidently amounts to the same thing,) then the eSiuent stream will not pass out with a, full flow, as in Fig 6, but will assume the shape shown in Fig 11 ; forming, just outside of the opening, what is called the vena contracta, or contracted vein. In order that this contraction may take place to its fullest extent, or become complete, the inner sharp edges of the opening must not approach either the surf of the water, or the bottom or sides of the. reservoir, nearer than about l}4 times the least transverse dimension of the opening. The contracted vein occu/s at a dist of about half the smallest di- mension of the orifice, from the orifice itself. In a circular orifice, at about half the diam dist; and ordinarily its area is about .62. or nearly % that of the orifice itself^ At this point the actual mean vel of the stream is very nearly (about .97) the theoretical vel given by Table 10, and hence the actual discha are but .62, or nearly % of the theoretical ones. Case 1. To find the actual disch into air.J thron^h either a circular or rectilinear g opening; in a thin vert plane parti- Fig. 13. ♦We believe that these rules for thin plate are also sufiiciently approximate for most practical purposes, if the opening be in the bottom of the reservoir; or in an inclined, instead of a vert side. t When the side of a reservoir, or the edge of a plank, &c, over which water flows, has no greater thickness than this, the water is said to flow through, or over, thih plate, or thin partition. t Should the disch take place under water, as in Fig 12. hoth surf-levels re- maining constant, then the head to be used is the vert diff ao.of the two levels. After making the calculation with this head, we should, according to "Weisbach, deduct the y^ part; inasmuch as he states that the disch is that much less when under water, than when it takes place freely into the air. Other experimenters, however, assert that it is precisely the same in both cases. § If the shape of the opening is oval, triangular, or irregular, the head must be measured vert from its cen of gray. 542 HYDRAULICS. ^ion, when tlie contraction is complete ; an€l when the snrf- level, .v, remains constantly at the same height; water being supplied to the reservoir as last »»s it runs out at the open- ing.* Rule 1. When the head, measured vert from the center (or rather from the cen of grav) c, of the opening, to the surf-level s of the reservoir, is not less than 1 ft. nor more than 10 ft ; and when the least transverse dimension of the opening is not less than an inch, mult the theoretical vel in ft per sec due to the head. (Table 10, ) by the coefficient of disch Q-z. The prod will be the acMial mean vel of the water throuch the opening. Mult this vel by the area of the opening in sq ft; tbe prod will b6 the disch in cub ft per sec, approximately. "When the head is greater than 10 ft, use .6, instead of .62. Rule 2. Find the sq rt of the head in ft. Mult this sq rt by 5 j the prod will be the vel in ft pei sec ; which mult by the area as before for the disch. Ex. What will be the disch through an opening in complete contraction, whose dimensions are ft ins, or .5 ft vert ; and 4 ft hor ; the vert head above the cen of grav of the opening being constantly «feet? . By Rule 1. The theoretical vel (Table 10, ) corresponding to 6 ft head, is 19.7 ft per sec. And 19.7 X .62 = 12.214 ft, the reqd vel. Again, the area of the opening = .5 X 4 = 2 sq ft ; and 12.214 X 2 r; 24.428 cub ft per sec ; the disch. By Rule 2. The sq rt of 6 = 2.45 : and 2.45 X 5 = 12-.25 ft per sec, the reqd vel ; and 12.25 X 2 ::^ 24.5 cub ft per sec, the disch. Both Tery approx even if the orifice reaches to the surface of the issuing water. Rem. 1. The coef .62 Is a mean of results of many old experimenters In 1874 Genl. T. G. Ellis of Massachusetts conducted an elaborate series (Trans Am Soc C E, Feb 1876) on a large scale, the general results of which, within less than 1 per ct, are given in the follow, ing table. See also Rem 3. The sharp edged oriflces were in iron plates .25 to .5 inch thick. Orifice. Head above Center. Coef. 2 ft sq. 2. to .H.5 ft. .60 to .61 2 " long, 1 ft high 1.8 to 11.3 " .60 to .61 2 " long. .5 high 1.4 to 17.0 " .61 to .60 2 " diam. . l.S to 9.6 " .59 to .61 Rem. 2. Extreme care is reqd to obtain correct results; but for manj purposes of the engineer an error of 5 to 10 per ct Is unimportant. It will rarely happen that greater accuracy is required than njay be obtained by the foregoini rules; but when such does occur, aid may be derived from the following table dedUCCu from the experiments of Lrcsbros and Poncelet. on openings 8 ins wide, of diff heights, and with difif heads. Use that coeff in the table which applies to the case, in- stead of the .62 of Rule 1. In some of the cases in this table, the upper edge of the opening is nearer the surf-level of the reservoir than l}4 times its least transverse dimension. TABIiE 12. Coefficients for rectang^ular opening-s in thin vertical partitions in full contraction.* Head Head The breadth in all the openings rr 8 inches. above cen. of grav. of above cen. of grav. of HEIGHT OF OPENING. opening opening Ins. Ins. Ins. Ins. Ins. Ins. Ins. in Feet. in Inches. 8 6 4 3 2 1 .4 .033 .4 .8 1 13^ .70 .69 .68 .68 .0666 .65 .64 .64 .0833 .125 .61 ,1666 2 .60 .62 .64 .68 .2083 ^H .59 .61 .62 .64 .67 .250 • 3 .60 .61 .62 .64 .67 .2917 3>^ .57 .60 .61 .62 .64 .66 .3333 4 .58 .60 .61 .63 .64 .66 .3750 4)^ .56 .59 .60 .61 .63 .64 .66 .4167 5 .57 .59 .61 .62 .63 .64 .66 .6666 8 .59 .60 .61 .62 .63 .64 .65 1 12 ,60 .60 .61 .62 .63 .63 .64 3 36 .60 .60 .61 .62 .62 .63 .63 5 60 .60 .60 .61 .61 .62 .62 .62 10 120 .60 .60 .60 .60 .60 .61 .61 Rem. 3. Careful experiments on openingrs 4^>^ ft wide, and 1^ ins hi$::h. under heads of from 6 to 15 ft, show that the coeff .62 will give results correct within -Jjr- part, for openings of that size also, under large heads; although the thickness oa ibe partition varied on its diff sides, from 12 to 20 ins. It must be recollected, however, that nothinf, more than close approximations are to be attained in such matters. Rem. 4. It has been asserted by some writers, that when tw^o or more cowtiguons opening's are discharging; at the same time from the same reser- voir, they disch less in proportion than when only one of them is open. Other experiments, how* ever, seem to show that this is not the case ; it is therefore probable, at least, that the difif, if any, is but trifling. * See first footnote on preceding page. HYDRAULICS. 543 Ca&e 2. Tlie cliscliarg-e throngrh tbiii vert partitions in coin« plete contraction, when tlie surface-level, niy Figr 13, descends as the w^ater lloivs out into the air. In this case, if -the reservoir is prismatic, that is, if its iior sections are everywhere equal ; audit' uo water is flowing into the reser- voir, to supply the place of that which flows out, then, to find the time reqd to disch the reservoir. Bulb. Inasmuch as the time in which such a reservoir entirely discharges itself, is twice that iu wkich the same quantity would flow out under a constant head, as in Case 1, therefore, cal- culate the disch iu cub ft per sec by Rule 1, Art 9 ; div the number of cub ft con- tained in the reservoir, above the level g of the bottom of the opening, Fig 13, by this disch ; the quot will be the number of sec in which a volume equal to that in the reservoir, to the depth g, wouW run out iu Case 1, of a constant head. And twice this number will be the seconds reqd to empty the reservoir iu Case 2, of a varying head. Rem. If it should be reqd to find the time in which such a prismatic reservoir would partly empty itself, as, for instance, from m to n, Fig 13, first calculate, by the above rule, the sees necessary to empty it if it had only been filled to n ; and afterward calculate as if it had been filled to m. The difif between the two times will evidently be the time reqd to empty It from m to n. If the opening is not iu complete contraction, see Arts 11, &c. If the disch is into a lower reservoir, w^hose surf-level remains constant, proceed in tlie same manner; only use the diCf of level of the two surfs as the head, and afterward (according to Weisbach) increase the time -J-^ part. Art. 10. Disch from a reservoir R, Fig" 14, the surf-level, «, of which remains constantly at the same height; through an opening, o. in thin vert partition; and in complete con- traction; but entirely under water; and into a prismatic reservoir, m. Fifif. 13. R e Fig. 14. Seconds req Uired / height a c ^ hor area of to discharge a quantity = ^ jn ft '^ m in sq ft cda, the level c remaining = constant. ,^/height ac ^ hor area of ^ „ Seconds required __ in ft ^ m i n sq ft ^ * to raise level in wi from c to a ' a of opening , o in sq ft Seconds required to raise level in m from c to = any other level, d. V both in ft j "^ ^° «q f^ ^' Area of opening ,^ ko v s •« o in sq ft ^ -"^ ^ "•*' Rem. 1. If it should be reqd to find the time of filling m, from its bottom c, up to d, we may do so very approximately by calculating by the first rule in Art 9, the time reqd from e to the center of the opening o, as if all that portion of the disch took place into air; and afterward, from the center of the opening to d, by the rule just given. This case is similar to that of filling a lock from the canal reach above, in which the surf- level may be considered constant. Rem. 2. If the bottom of the opening o, should coincide with the bottom of the reservoir, then the coeff will become greater than .62. See Art 11, for obtaining coeffs for imperfect contraction. Rem. 3. If the opening, instead of being in complete con- traction, is of any of the shapes Figs 6 to 9, then a reference to Art 8 will show what coefiF must be substituted for .62. Case, 3. l>isch from one prismatic reservoir. Fig 15, 1¥, into another, X, of any comparative sizes whatever, through an opening o, in a plane thin vert partition, and in complete contraction; when the water rises in X, while it falls in W. To find the time in which the water, flowing from W into X, through 0, will fall through the dist as, so as to stand at the same level s c, in both reservoirs. In this case, the water reqd to fill X from c to d, (d being the bottom of the opening o,) flows out into the air ; and the time necessary for it to do so, must be calculated separately from that reqd above d, which flows into water. Rule. First from e to d. Find the hor area of each reservoir, in sq ft. Mult the hor area of X, by the vert depth de in ft, for the cub ft contained in that portion. Div these cub ft by the hor area of W. The quot will be the dist am, in feet, through which the water in W must descend, in order to fill X to d. FifiT. 15. Seconds r«- quired to low- er from a to m, and raise from eio d. Twice the / /yT. hor area of X I "V W iu aq ft ^ z ^/head mn\ ~^ in ft J Area of opening ^ «., ^ c o iu sq ft A .oz X t 544 HYDRAULICS. Seeoiids required to lower from m to «, and raise from d to c. (Very approx) Hor area of ^ twice the hor area y. ,^/head m n Y in en ff '^ nf W in an ff. ^ ^^ «n ff of W in sq ft Area or ,/iaoT area nor area\ opening X ( of W + of X ) X .62 X 8.03 oin sq ft \in sq ft in sq ft/ Fig. 16. Ex. Let the hor area of "W be 100 sq ft : and that of X, 60 sq ft. Let an be 20 ft ; and mn 16 ft ; and the area of the opening o, 3 sq ft. In what time will the water descend from a to s, and rise from e to c ? Inasmuch as the method of finding the time for filling from e tor^ M ^ 3 » /^^^^<-^:;^^====^=f^ i j cu. ft. < r M^S^^^i^ 0.029 O.OSBJ^ ^ (^persec. \^^^ '^-^^^i^-^::=^^=^ J 0.627 > ^ ] .8^ j cu. ft. [per mln r^==^ ^^:rr:^::^^^;:::;--^^ o. m^ ^ ^ CC^-^ t^^;;:;^^ ao25^ ^^ \a^ ^ XL -'^^ 0.024 J ^ i.( ^ j cu. ft. 0.023^ y^ 2,. m ^ (per day 0.^ o.cm^ M. ^ 2,2«0 2,80< 2,400^ 0.21^ D.22 - \ gallOQ9 I per sec. 0.( 19-^^'^ 1. i^^ '^ i l,100>"^^ 0.2( ^ j gallous ^per mla ^ -^ 2,000^ 1 3^ ^ 0.018^ 1.2 1,800^ ,900^ 0. J c 18^ ;.iy^ ^ ^ 1^ ,00( ^ j galloas /per day \.y^ 1,700^ 0.11 ^ n ^ 18,000^ "^ 0.16 17, m^ 1,600 ^ 0.15^ 15^ ^ 16,000^^ 0.14^ 0-j ^\h 000^ ^ wm^ Discharge 1 1 , ^^y^ 13,000 _u lucu 2 V , ft. per see. 2 g xj hea d ib ft. 12,000^ 144 3 4 5 6 7 8 $> ^ Head. In laches i Diagram of discliarg'e tliroug*!! an orifice 1 inch square. The thickness of plank, in which the orifice is cut, varies from 1 to 3 ins, and the lower edge of the orifice is often chamfered, flaring outwardly. The bottom of the orifice is sometimes flush with the bottom of the box, but is usually raised 2 or 3 ins above it, as in our Fig. * Condensed from circular of Pelton Water Wheel Co. HYDRAULICS. 547 Art. 14 (a). On the discharge of water over weirs or Over^^ falls. The weir affords a very convenient means for gauging the flow of small streams, for measuring the quantity of water supplied to water-wheels, etc. [b) A measuring weir is always arranged with its baclc, or up-stream side, a &, Fig. 20, vertical, aud as nearly as may be at right angles to the direction of flow of the stream. The ends, a A, a h. Figs. 21 and 22, are vertical, and the crest a a is horizontal. (c) £ii€l eon tractions. When the weir a a extends entirely across the channel of approach, as in Fig. 21, so that its ends a h'a h coincide with, or form portions of, the sides 65 of the channel, contraction (Art. 9, p. 541) takes place only on the top and bottom of the sheet of water passing over the weir, as at m c and at a, Fig. 20, and is entirely " suppressed " at the ends, so that the water flows out as shown in Fig. 21 a. Such a weir is called a suppressed weir, or a weir without end contraction. But when, as in Figs. 22 and 22 a, the ends ah, ah are at a distance from the sides 5 5 of the channel or reservoir, contraction takes place at the ends- of the weir, as shown at a and a, as well as over the crest. Such contraction diminishes the discharge. A weir of this kind is called a weir with end contractions. Other things being equal, the extent of the contraction, and its effect upon the discharge, increase with the head H. When the length a a or L of the weir ex- ceeds about 10 times the head H, the effect of the end contractions upon the dis- charge is nearly imperceptible ; but as the length diminishes in proportion to the head, the effect of the contraction increases rapidly. Mr. P'rancis (Art. 14 m) found that when L = only 4 X H, the discharge was reduced 6 per cent, by com- plete end contractions. In view of the uncertainty as to the effect of end con- tractions, it is better to avoid them and to use weirs, like Fig. 21, where the con- traction is suppressed ; but if end contraction is permitted at all, it must be made complete;^ for the coefficients given do not apply to cases of incomplete contraction, i.e., with contraction only par^/y suppressed. {d) In a w^eir w^ithont end contraction, care must be taken that the air has free access to the space {w, Fig. 20, or 22 h) behind the falling sheet of water. Otherwise a partial vacuum forms there, the sheet is drawn inward toward the weir, and the discharge is greatly modified. At the same time, the sheet should be pre- vented from expanding laterally as it leaves the crest. Both of these objects may be attained by prolonging the upper portion only of both sides of the channel a little way down-stream beyond the crest and the upper part of the falling sheet, as in Fig. 22 h. Mr. Francis found that such projections, by confining the sheet laterally, diminished the discharge about 0.4 per cent. (e) Ordinarily the crest is *' in thin plate" or "in thin partition" (see foot- note t, p. 541), so that the sheet passing over the weir touches it only at the very corner, a. Fig. 20, A rounded corner increases the discharge, as does the round- ing of the edges of an orifice (Art. 8, p. 541), and a crest sufficiently wide to de- flect the falling sheet diminishes the discharge (see coeflBcients for this case in Table 15, p. 554), but both forms introduce much uncertainty, and should there- fore be avoided. * The contraction is said to be "complete" when it is practically as great as it could be made by any further increase of the distance a s, Figs. 22 and 22 a ; and thlg is believed to be attained when a s is made equal to the head H. Fig.22b 548 HYDRAULICS. (/) The lengtli Ia of the crest. Figs. 21 to 22 a, should be at least three times thr! head H, in order to reduce the effect of friction of the sides ss and that of end contractions where such exist. The heig'ht jj, i^'ig. 20, of the vertical back a b in contact with the water should be not less than twice the head H; for, in order to reduce the velocity of approach (see Art. 14 u), the cross-section of the channel leading to the weir should be large in propor- tion to that of the stream ac. The cross-section of the channel of approach should be as ri^gaiar as possible. ((/) The weir should be stoutly built, as vibrations of the structure may seriously modify the discharge. (Ji) Theoretically, the head is the vertical distance H', Fig. 24, from the crest a to a point o' where the water is perfectty still, and the surface therefore hori.zoritaL But in fact the head is usually measured from the crest a to a point o a few feet back from the weir, where the water is only comparatively still, the velocity of approach being perceptible. (See Art. 14, u.) The difference between the head H actually measured and the head H' to still water is usually very slight. It is greatly exaggerated in the figure. The correct measurement of the head is a delicate matter, the dis- charge being increased or diminished about l^per cent, by 1 per cent, of in- crease or diminution of the head. Waves or ripples and other disturbances of the surface, and capillary attraction, are the chief sources of error. (/) To avoid the latter difficulty, the hook-g-au^e is used for measuring the height of the water surface in important cases. This consists of a long grad- uated rod, provided at its foot with an upturned hook or point, and sliding vertically (by means of a screw motion) in a fixed support, to which is attached a vernier indicating on the scale the height of the point. The sliding rod is first run down until the point is well below the surface, and then gradually raised by means of the screw until the point just reaches the surface, which is indicated by the first appearance of a "pimple" in the water surface imme- diately over the hook. Under favorable circumstances a good hook-gauge may be read within from .0002 to .0005 foot. {j) To avoid inaccuracies due to the distnrbance of the surface by the current, by wind, etc., the level is sometimes taken (with the hook-gauge or otherwise) in a side chamber which communicates with the main channel of approach. The surface in the chamber maintains the same level as that in the channel itself, but is comparatively free from disturbance. Or a bucket com- municating with the channel by means of a pipe, can be made to serve in the same way. Either may of course be sheltered from the wind. Caution. Messrs. Fteley and Stearns found that when the bucket or chamber communicated with the water n,mr the bottom and close behivd the weir,\he head thus obtained was generally somewhat greater than that found by measurement near the surface and (i feet back from the weir. But Mr. Francis found the difference scarcely perceptible. (k) Great care is necessary in adjusting* the hook-gauge for the height of the crest; for any error in this affects all the subsequent experi- ments. The hook is usually adjusted to the height of the surface when the latter just reaches the level of the crest ; but this method is rendered inaccurate by capillary attraction at the crest. A more accurate method is to have, in addition to the hook-gauge, a stout jized hook, pointing upward, the level of which, rela- tively to that of the crest, may be ascertained by means of an engineer's level, holding the rod on the crest and also on the point of the fixed hook. The water surface is then allowed to fall slowly until a " pimple " just appears over the fixed hook. It is then kept at that level and the hook-gauge adjusted accordingly. Or if the gauge-hook is a stout one, the levelling rod may be set at once upon its point without having recourse to a fixed hook. It is better to adjust the hook- gauge so as to read zero for the crest level, which is thus made the datum ; for the reading of the hook-gauge for the water surface then gives the head H at once, and without subtracting the height of the crest. HYDRAULICS. 549 (1) Formalse for weir cliscbargre. Let Q, ^ the actual discharge over the weir, in cubic feet per second ; * Q,' -= the theoretical discharge over the weir, in cubic feet per second', H =H' t = the vertical distance or head a vi, Fig. 24, p. 556, in feet,* measured from the crest a to the korizonlai surface o' of still water up-stream from the weir ; li = the length a a of the weir, in feet,* Figs. 21 to 22 a ; ^ g = the acceleration of gravity = say 82.2 feet* per second per second*, ^ , . „ ,. , actual discharge Q c = coefhcient of discharge = -r r^ — \ — t-- — ^ = ^, J ^ theoretical discharge Q' 2 ', =--o}/2g=my2g= say 5.36 c = say 8.025 m. Then, tor the tlieoretical dischargee, we have 2 ^ „ ,^_,-, a'= 3 LHv'25'H;^ and for the actual '2TH7 area am 3 ^ ^ Z T^ig.23 or two thirds of the theoretical hori= ' zontal velocity a a' of the particles passing immediately over the weir. As in the case of orifices (Art. 9, p. 541), the actual vel. at the %mallek section of the sheet after passing the weir (corresponding to the "vena contracta ") is probably very nearly equal to this theoretical velocity. * "Lowell Hydraulic Experi- ments," Van Nostrand, New York, 1883. t In Messrs. Fteley and- Stearns' experiments this figure was not constant at 0.10, but varied between 0.061 and 0.124, generally increasing as the head decreased. % We here suppose the head to be measured to the surface of still water, so that H and H' (see Art. 14 h, p. 548) are the same. See Velocity of Approach, Art. 14 (m). ^ Since 1 meter = 3.2808 ft., the value of a: for metric measure corresponding to Mr. Francis' 3.33, is = 3.33 -r- v/3.2808 = 3.33 s- 1.8113 = 1.83& ^2a-B:—^, HYDRAULICS. 551 Table 13.* I>iscliarg-e in cubic feet per second for eacli foot in leng-th of weir in thin plate and without end contraction, by the Francis formula: Disehar;^e, Q -- 3.33 L fit = 3.33 L H yK. Very approximate also when there is end con'iraction, provided that L is at least = 10 H ; and but about 6 per cent* in excess of the truth if L = 4 H. Mr. Francis limits the formula to heads H from 0.5 foot to 2.0 feet, but no serious error will result from using the table for any of the heads given. For weirs of other leng-ths than 1 foot, multiply the tabular discharge by the actual length in feet. .01 foot = 0.12 inch = scant % inch. Head, H, Cub. ft. Head, H, Cub ft. Head, H, Cub. ft. Head, H, Cub. ft. Head, H. Cub. ft. in ft. per sec. 0.003 in ft. .51 per sec. 1.213 in ft. 1.01 per sec. in ft. per sec. in ft. per sec. .01 3.380 1.51 6.179 2.01 9.489 .02 0.009 .52 1.249 1.02 3.430 1.52 6.240 2.02 9.560 .03 0.017 .53 1.285 1.03 3.481 1.53 6.302 2.03 9.631 .04 0.027 .54 1.321 1.04 3.532 1.54 6.364 2.04 9.703 .05 0.037 .55 1.358 1.05 3.583 1.55 6.426 2.05 9.774 .06 0.049 .56 1.395 1.06 3.634 1.56 6.488 2.06 9.846 .07 0.062 .57 1.433 1.07 3.686 1.57 6.551 2.07 9.917 .08 0.075 .58 1.471 1.08 3.737 1.58 6.613 2.08 9.989 .09 0.090 .59 1.509 1.09 3.790 1.59 6.676 2.09 10.062 .10 0.105 .60 1.548 1.10 3.842 1.60 6.739 2.10 10.134 .11 0.121 .61 1.586 1.11 3.894 1.61 6.803 2.11 10.206 .12. 0.138 .62 1.626 1.12 3.947 1.62 6.866 2.12 10.279 .13 0.156 .63 1.665 1.13 4.000 1.63 6.930 2.13 10.352 .14 0.174 .64 1.705 1.14 4.053 1.64 6.994 2.14 10.425 .15 0.193 .65 1.745 1.15 4.107 1.65 7.058 2.15 10.49S .16 0.213 .66 1.786 1.16 4.160 1.66 7.122 2.16 10.571 .17 0.233 .67 1.826 1.17 4.214 1.67 7.187 2.17 10.645 .18 0.254 .68 1.867 1.18 4.268 1.68 7.251 2.18 10.718 .19 0.276 .69 1.909 1.19 4..323 1.69 7.316 2.19 10.792 .20 0.298 .70 1.950 1.20 4.377 1.70 7.381 2.20 10.866 .21 0.320 .71 1.992 1.21 4.432 1.71 7.446 2.21 10.940 .22 0.344 .72 2.034 1.22 4.487 1.72 7.512 2.22 11.015 .23 0.367 .73 2.077 1.23 4.543 1.73 7.577 2.23 11.089 .24 0.392 .74 2.120 1.24 4.598 1.74 7.643 2.24 11.164 .25 0.416 .75 2.163 1.25 4.654 1.75 7.709 2.25 11.239 .26 0.441 .76 2.206 1.26 4.710 1.76 7.775 2.26 11.314 .27 0.467 .77 2.250 1.27 4.766 1.77 7 842 2.27 11.389 .28 0.493 .78 2.294 1.28 4.822 1.78 7.908 2.28 11.464 .29 0.520 .79 2.338 1.29 4.879 1.79 7.975 2.29 11.540 .30 0.547 .80 2.383 1.30 4.936 1.80 8.042 2.30 11.615 .31 0.575 .81 2.428 1.31 4.993 1.81 8.109 2.31 11.691 .32 0.603 .82 2.473 1.32 5.050 1.82 8.176 ■ 2.32 11.767 .33 0.631 .83 2.518 1.33 5.108 1.83 8.244 2.33 11.843 .34 0.660 .84 2.564 1.34 5.165 1.84 8.311 2.34 11.920 .35 0.690 .85 2.610 1.35 5.223 1.85 8.379 2.35 11.996 .36 0.719 .86 2.656 1.36 6.281 1.86 8.447 2.36 12.073 .37 0.749 .87 2.702 1.37 5.340 1.87 8.515 2.37 12.150 ,38 0.780 .88 2.749 1.38 5.398 1.88 8.584 2.38 12.227 .39 0.811 .89 2.796 1.39 5.457 1.89 8.652 2.39 12.304 .40 0.842 .90 2.843 . 1.40 5.516 1.90 8.721 2.40 12.381 .41 0.874 .91 2.891 1.41 5.575 1.91 8.790 2.41 12.459 .42 0.906 .92 2.939 1.42 5.635 1.92 8.859 2.42 12.536 .43 0.939 .93 2.987 1.43 5.694 1.93 8.929 2.43 12.614 .44 0.972 .94 3.035 1.44 5.754 1.94 8.998 2.44 12.692 .45 1.005 .95 3.083 1.45 5.814 1.95 9.068 2.45 12.770 .46 1.089 .96 3.132 1.46 5.875 1.96 9.138 2.46 12.848 .47 1.073 .97 3.181 1.47 5.935 1.97 9.208 2.47 12.927 .48 1.107 .98 3.231 1.48 5.996 1.98 9.278 2.48 13.005 .49 1.142 .99 3.280 1.49 6.057 1.99 9.348 2.49 13.084 .50 1.177 1.00 3.330 1.50 . 6.118 2.00 9.419 2.50 13.163 * Table 13 is an extension of the " original " table published in our first edition, 1872. Most of the values now given are taken, by permission, from a table published by Messrs, A. W. Huuking and Frank S Hart, of Lowell, Mass., in May, 1884. 552 HYDRAULICS. (n) Messrs. A. Fteley and F. P. ^Stearns * experimented at Boston, Mass., in 1877-79, upon weirs 5 feet and 19 feet long, 3 feet 2 inches and 6 feet 63^ inches high, and under heads from U.8 inch to 19 inches. For weirs in thin par- tition and without end contraction, with a rectangular and uniform channel of approach and under heads greater than 0.07 foot or 0.84 inch (other conditions as specified in (6) and (tZ;), their formula is: ^t (6). Bischarge, Q = 3.31 L H2 + 0.007 L = 0.4125 L H 1/27H + 0.007 L In their experiments, the heads were measured six feet back from the weir. The total variation in the values of the coeflBcients obtained was about 2^^ per cent. Compare foot-note § below. (o) M. Baziut- experimented at Dijon, France, in 1886-88, with weirs from t^ about 13^ to 6]4 feet long, from about 9 inches to 3 feet 9 inches § high, and under heads from 2}4 to 21 inches. The top of the weir is shown in Fig, 23 a. The weirs were placed at diflferent points in a rectangular and regular canal TOO feet long, smoothly lined with cement. Compare foot-note ^ below. While Mr. Francis and Messrs. Fteley and Stearns provide for the effect of velocity of approach (see Art. 14 10 and v} hj modifying the measured head H, M. Bazin includes it in th^ cojfficient m in the formula Q = m L H ]/2 g H. After compar- ing his experiments with those of Messrs. Fteley and Stearns, ^(Art. 14 n and v), he gives, for m, in cases where there is no 'velocity of approach, the values M in the last column of Table 14, or, very approximately, o /^'o velocity of approach, \ _, ^ -iac , ^-^^^ ( H and H' identical ) M = 0.405 + ^jp. When velocity of approach is to be taken into account: Fig. 23 a. Measurements in meters. :M [— (wf^n- (7); where H is the head actually measured to running watei^ and p is the height a & of the weir, Fig. 20. H and p must of course both be measured in the same unit, as both in meters, or both in feet, etc. M. Bazin believes that except in the case of very low weirs (which should be avoided) the values of m given by formula (7) and in Table 14 calculated from it, will be found within 1 per cent, of the truth for weirs in thin partition ancl without end contraction, if the conditions of his experiments are exactly repro- duced, and provided especially that the sheet of water is not allowed to expand laterally after passing the crest (Art. 14 (d)) and that the air has free access to the space lo, Fig. 20, behind the falling sheet of water. For heads between 4 inches and 1 foot, M. Bazin gives, as sufficiently ap- proximate, when there is no velocity of approach, M = 0.425,g and, to allow for velocity of approach, m = 0.425 -f 0.21 ( ^t-^ — )• \±1 -\-p/ * Transactions, American Society of Civil Engineers, Jan., Feb. and March, 1883. + See correction for Telocity of Approach, Art. 14- (ii). X Experiences nouvelles sur I'ecoulement en deversoir. Extrait des Anuales des Fonts et Chaussees, Oct., 1888. Paris, Vve Ch. Dunod, 1888. Translation by A. Marichal and John C. Trautwine, Jr., presented to Engineers' Club of Philadelphia, in 1889, for publication in its Proceedings. g This would make x = 3.41 (since x = m y2'g = 8.025 m) ; whereas Mr. Francis gives X = 3.33, which agrees very well with Messrs. Fteley and Stearns, within the limits of Art. 14 (m). Yet M. Bazin measured the head 16 feet back from the weir, while the other experimenters measured it only 6 feet back, and the slight increase of head thus obtained by M. Bazin would of itself have made his coefficient loiver than theirs. Its excess may be largel}' due to the character of the channel of approach, which in his case was from 50 to 700 feet long, rectangular and regular in cross-section, and smoothly coated with cement. In the other experiments it wa» much less regular. HYDRAULICS. 553 Table 14. Tallies of m, in the formula Q = m L H }/2 glL ... or in^!ib!^ftTe?1ec. = '^ X ^^"S^^' ^" ^*- X ^^^^'^ .^^^^ ^^ ^t* X i/2yH7Tt; by M. Bazin's formula (7) : m = Mf 1 + 0.55 ( =j r . For M, see last column of table. Or, very approximately, M = 0\405 + ~.^ . Note. The coefficient to, for any given case, remains the same for English, metric or other measure ; provided the head, the length and g are all measured in the same unit, and the discharge in the cube of that unit ; for m is simply 2 2 actual discharge 3 "~ 3 theoretical discharge * It will be noticed that below the heavy lines the head H is greater than »^ height j3, and thus exceeds the limit laid down in (f ) and (m). Head H, Fig. 24, p. 556. approximate feet. inches. .05 .06 .07 .08 .09 .164 .197 .230 .262 .295 1.97 2,36 ?.76 3.15 3.54 .10 .328 3.94 .12 .14 .394 .459 4.72 5.51 .16 .18 .525 .591 6.30 7.09 .20 .656 7.87 .22 .722 8.66 .26 .28 .853 .919 10.24 11.02 .30 .984 11.81 .32 .34 .36 .38 1.050 1.116 1.181 1.247 12.60 13.39 14.17 14.96 .40 1.312 15.75 .42 44 .46 .48 1.378 1.444 1.509 1.575 16.54 17.32 18.21 18.90 .50 1.640 19.69 .52 .54 .56 .58 .60 1.706 1.772 1.837 1.903 1.969 20.47 21.26 22.05 22.83 23.62 Height, p, Pig. 20, of crest of \reir above bed of up-stream channel. meters 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.50 2.00 " o ^feet 0.656 0,984 1,312 1.640 l.%9 2.624 3.280 4.920 6.560 .i n. finches 7,87 11.81 15.75 19.69 23.62 31.50 39,38 59,07 78,76 in .458 .456 .455 .458 .457 m .453 .450 .448 .447 .447 ni .451 ,447 .445 .443 .442 m .450 .445 .443 .441 .440 ni .449 ,445 .442 .440 .438 m .449 .444 .441 .438 .436 m .449 .443 .440 .438 .436 m .448 .443 .440 .437 .435 in .448 .443 .439 ,437 .434 .4481 .4427 .4391 .4363 .4340 .459 .447 .442 .439 .437 .435 .434 .433 .433 .4322 '.462 1 .4661 .448 .450 .442 .443 .438 .438 .436 .435 .4.S3 .432 .432 .430 .430 .428 .430 .428 .4291 .4267 .471* .475 .453 .456 .444 .445 .438 .439 .435 .435 .431 .431 .429 .428 .427 .426 .426 .425 .4246 .4229 .480 .459 .447 .440 .436 .431 .428 .425 .423 .4215 .484 .488 .462 .465 HI .442 .444 .437 .438 .431 .432 .428 .428 .424 .424 .423 .422 .4203 .4194 .492 .496 .468 .472 .455' .457 ,446 ,448 .440 .441 .432 .433 .429 .429 .424 .424 .422 .422 .4187 .4181 .500 .475 .478 .481 .4^3 .486 .460 .462 .464 .467 .499 .450 .452 .454 .466 .458 .443 .434 ,436 .437 .438 .439 .430 .430 .431 .432 .432 .424 .424 .424 .424 .424 ,421' .421 .421 .421 .421 .4174 ^ .444 .446 ,448 .449 .4168 .4162 .4156 .4150 zz .489 .491 .494 ,496 .472 .474 .476 .478 .480 .482 ,483 .485 .487 ,489 .490 .459 .461 .463 .465 .467 .468 .470 .472 .473 ,475 .476 .451 .452 ,454 .456 .457 .459 .460 .461 .463 .464 .466 .440 .433 ,434 .435 .435 .436 ,437 .424 .425 .425 .425 425 .426 .426 .426 .427 .427 .427 .421 ,421 .421 .421 ,421 .421 .421 .421 .421 ,421 .421 .4144 ^ .441 ,442 .443 .444 .445 .446 ,447 .448 .449 .451 .4139 .4134 .4128 .4122 .4118 ,438 .439 ,440 .441 ,4112 4107 4101 ::::: .4096 .4092 Owing to the wide range of the head H and of the height p in these experi- ments, we find in them a urider diverg-ence in the values of the coefficient than resulted from the earlier investigations. Thus, the smallest value of to above the heavy lines is 0.4092, or about one nineteenth less than the mean, 0.4325; and the greatest is 0.459, or about one sixteenth more than 0.4325. * In these experiments, the head H was measured at a point 5 meters (16,4 ft ) back from the weir. The correction for velocity of approach is contained in the coefficient m. t M is the value of m when there is no velocity of approach ; i. e., where the cross- •lectlon of the channel of approach is indefinitely great compared with that of the stream of water passing over the weir. 654 HYDRAULICS. (2>) From a comparison of a number of experimental data, the author deduced the following Table 15. Approximate values of the coefficient m in the formula : ^ = mLHi/2yH, for weirs of several different shapes and thicknesses. (Original.) 3 ft. thick ; Head, H. Sharp Edge.* 2 Inches thick. smooth ; slop- ing outward and downward, 3 ft. thick ; smooth ; an4 level. Feet. Inches. from 1 in 12 to 1 in 18. in m m m .0833 1 .41 .37 .32 .27 .1666 2 .40 .38 .34 .30 .25 3 .40 .39 .34 .31 .8333 4 .40 .41 .35 .31 .4166 5 .40 .41 .35 .32 .5 6 .39 .41 .35 .33 .5833 7 .39 .41 .35 .32 .6666 8 .39 .41 .34 .31 .8333 10 .38 .40 .34 .31 1. 12 .38 .40 .33 .31 2. 24 .37 .39 .32 .30 3. 36 .37 .39 .32 .30 (q) To find the head H, approximately ; having the discharge Q. According to formulae (3) and (4), Art. 14 (/). Hence Q = m LH V/27H = a; LH /H = m L v^|/H3, =a;LyH3, ^ =Vm«L«2^=" ^|i^^^ ..... (8) Head, H, approximately _^ /square of discharge of stream, in cub, ft. per sec. ^ \ m2 X length^ X 64.4 |sq. of discharge^ *^ 'V ^2~xlength2"' The coefficient m, or x itself varies somewhat with the head ; but the formula may be usefully employed as an approximation by taking, for sharp-crested weirs, m = 0.415 (m^ = 0.172) or a; = 3.33 (x^ = n). For other shapes, see Table 15, above. {r) Submerg-ed -weirs. Fig. 23 6, are those in which the surface of the dotrn-stream water at h, after the construction of the weir, is higher than the crest a. In a weir discharging freely into the air, as in Fig. 20, Mr. Francis found that with a head of 1 foot the discharge was diminished only about one thousandth part by placing a solid horizontal floor about 6 inches below and in front of the crest of the weir for the water to fall upon. Also, when the head was 10 inches, and the water fell freely through the air into water of considerable depth (as in Fig. 20), the quantity discharged was the same whether the surface of the down-stream water was about 3 inches or about 13 inches below the crest a. In experiments by Mr. Francis and by Messrs. Fteley and Stearns, with air freely admitted underneath the falling sheet of water just below the crest a, th« discharge was not appreciablv affected by a submergence of h = from 0.017 H to 0.023 H. When air was only partially admitted, the discharge was affected {in- creased) by less than one per cent, while h remained less than 0.15 H. Fij§:.23b * These values are lower than those g.iven in Art. 14 (w) and (h), and much lowei than those in (o). HYDRAULICS. 555 Bnbnat's formula for submergred weirs. Let Bl and /* = the heads measured vertically from the crest a of the weir to the surface of still water * up-stream and down-stream from the weir, respectively. ci = H— A = their difference = the difference iri level between the up-stream and down-stream surfaces of still-water ;* : coefficient of discharge = ; actual discharge Then theoretical discharge* Q = cl(a-|-| d) l/273';t (9)or^ Actual discliarse _ ^ x ^^?g^^ ?f X 8.025 v/rfmlt.X f A in ft. -f | (f in ft.) . in cub. ft. per sec. weir in ft. ^ \ 3 / («) MessrSi Fteley and iStearns % experimented at Boston in 1877 with submerg;ed iveirs under up-stream heads H from about 4 to 10 inches ; and Mr. Francis I at Lowell in 1883 under heads from about 1 foot to 2 feet 4 inches. From these experiments we deduce the following Table 16, of approximate values of the coefficient c in the formula for discharge over submerg^ed weirs. Deduced from experiments by Fteley and Stearns and by J. B. Francis. In Mr. Francis' experiments, the value of c for a given value of A -j- H generally increased as H increased. Fteley and Stearns. J. B. Francis. (H = 0.325 to 0.815 feet.) (H = 1 to 2.32 feet.) ^-*-H c c .05 .623 to .632 .10 .625 to .635 .620 to .630 .20 .618 to .628 .610 to .625 .30 .600 to .610 .598 to .615 .40 .590 to -600 .586 to .610 .50 .585 to .595 .5^5 to .607 .60 .583 to .593 .585 to .607 .70 • .580 to .590 .585 to .607 .80 .681 to .591 .685 to .607 .90 .590 to .600 .95 .610 to .615 * For velocity of approach, see Art. 14 (m) etc. t In deducing this formula, the water that passes over the weir between c and 6 is assumed to flow as over a weir with its crest at b, and with free discharge into the air, as over the crest a in Fig. 20 ; and for this portion, by formula (2) in Art. 14 (i), the discharge would be : Qj = cL|d]/27d; while the water that passes through the lower portion between h and a is regarded a^i flowing through a submerged vertical orijice whose height is 6 a = ^, under a hsed = d. For this lower portion, therefore, the discharge would be : Q^ = c L 7i yWgH. It is assumed that the coefficient of discharge c is the same for the upper section c 6 as for the lower one a b. Hence,'adding these two discharges together, we obtain, for the entire disharge : 1 Transactions, American Society of Civil Engineers, March, 1883, p. 101, etc. g Transactions, American Society of Civil Engineers, Sept., 1884, p. 295, etc. 556 HYDRAULICS. (t) Mr. Clemens Herscliel,* comparing these experiments with some earlier ones by Mr. Francis, gives the following : Having ascertained the depths H and h of the crest below the still-water levels up-stream and down-stream respectively, divide h by H. Find the quotient, as nearly as may be, in the column headed A -4- H in Table 17. Take out the cor- responding coeflScient a, and multiply it by the up-stream head H.f The product a H is the head which would cause the given weir to discharge the same quantity freely into the air, as in Fig. 20. Find the discharge into air over the given weir with the -bead a H ; and this discharge will be approximately the same as that of the actual submerged weir under the up-stream head H and against the down-stream head h ; or (H being the actual up-stream head on the submerged weir) the discharge is Q = m L a H \/TgaR = x L a H ^aBT. (10). TABI.I: 17. A-H a ^-H a .72 .74 .76 .78 .80 a .10 .20 .25 .80 .35 .40 1.000 to 1.010 0.975 to 0.995 0.960 to 0.984 0.945 to 0.973 0.928 to 0.960 0.912 to 0.946 .45 .50 .55 .60 .65 .70 0.894 to 0.930 0.874 to 0.910 0.853 to 0.889 0.829 to 0.863 0.803 to 0.833 0.775 to 0.799 0.762 to 0.784 0.747 to 0.769 0.732 to 0.752 0.713 to 0.733 0.693 to 0.713 (u) Telocity of approach. si "^ See Fig. 24. It is generally impracticable to measure the head H' to perfectly still water o' up-stream. The head is usually measured at a point o, from 2 or 3 to 6 or 8 feet or more up-stream from the weir, according to the size of the latter. At f^X^^'P'- ^' '^~>-.:, ' v [ such points the velocity is generally ap- — ^^Jj- C J'^v.^ ■ Sv^. \ preciable, and the surface therefore a -'js. i v '^^ \ little lower than at o'. Hence a formula I . ^s^^ _> ^^.^u^ ^ ^ using the smaller head H so measured, '' insteadofH', and coefficients based upon H', will give too small a discharge. Mr. Francis found that a current of 1 foot per second, or nearly 0.7 mile per hour, at the point o to which the head was measured, increased the discharge but about 2 per cent, when the head was 13 inches; and a current of 6 inches per second increased the discharge about 1 per cent, when the head was 8 inches. If, however, the velocity of approach is such as to require consideration, pro- ceed as follows: For the approximate mean velocity of approach, we have: Fig.24: a' approxi mate discharge 3.33 L H^ ' area of entire cross section of stream at o and, for the head due to this velocity, ^ = s- Then, for all practical purposes, we may say : H' = H -f- A ; or Q = m L (H + A) ]/2 ^(H + h) = x L (H -f- A)2 (11) although, strictly speaking, the difference of level between o' and o is really (as shown in Fig. 24) somewhat greater than h^ or than v^ -ir2g, because some head is lost in friction between o' and o. * Transactions, American Society of Civil Engineers, May, 1885, pp. 189, etc. t Mr. Herschers table, from which ours is condensed, gives a for every 0.01 foot of A -^ H ; but the values of a intermediate of those we have selected may be taken from our table almost exactlv by simple proportion. The range in the coefficient a in the table for each value ck h ^Yi is that indicated by the experiments, which varied similarly. We are not instructed how to select between these extremes; but in most cases their mean value is probably nearest right. HYDRAULICS. 557 (v) Messrs. Fteley and Stearns make H' = H + 1.5 // for suppressed weirs, and H' = H -f- 2.05 h for weirs with complete end contractions, as averages ; and ^Ir. Hamilton Smith, Jr.,* after comparing their experiments with others by Lesbros, Castel and Mr. Francis, gives H.' = K ^ll/^h, and H'= H + ].4/i,for the two cases respectively. i'w) On the other haud, Mr. Francis' formula, as modified for velocity of approach, Q = x L t (YlETW— yW) = m L t ]/2^~(v/(H + h)^ — y/W) t . . . (12), makes the effect of H' less than that of H + A. (x) Messrs. A. 1*\ Hunking- and Frank S. Hart, Civil and Hy- draulic Engineers, have substituted for the expression (]/(H -f h)^ — yii^) in formula (12), the equivalent one K v H^, in which K is a coefficient deduced from the former expression, and therefore depending upon the relation between H and h, or, ultimately, upon that between the cross-section a s Fig. 24 at the weir and the entire cross-section of the stream at o. Having found the area of cross-section at o, divide it by ( L — n ^ ] , which is the length of the weir corrected for contraction. See Art. 14 (m). Call the quotient D.§ Divide the measured head H by D. Find this last quotient in the column ^_ of the table. Multiply the approximate discharge, Q = 3.33 /l — 71 ^ I H2,by the corresponding coefficient K; or Actual Discharge Q = 3.H3 K (h — n~) hI. . . Table 18. Coefficient K in formula (13), (13) H H H H H K K K K K D D D D D .01 1.0000 .09 1.0020 .17 1.0072 .24 1.0143 .31 1.0239 .02 1.0001 .10 1.0025 .18 1.0081 .25 1.0155 .32 1.0254 .03 1.0002 .11 1.0030 .19 1.0090 .26 1.0168 .33 1.0271 .04 1.0004 .12 1.0036 .20 1.0100 .27 1.0181 .34 1.0287 .05 1.0006 .13 1.0042 .21 1.0110 .28 1.0195 .35 1.0305 .06 1.0009 .14 1.0049 .22 1.0121 .29 1.0209 .36 1.0322 .07 1.0012 .15 1.0056 .23 1.0132 .30 1.0i£4 .37 1.0»41 .08 1.0016 .16 1.0064 * *' Hydraulics," John Wiley & Sons, New York, 1886. f If there are end contractions, L here becomes f L — n r^ I • See Art. 14 (m). X This formula is deduced as follows : Let the area of the parabolic segment a s a'. Fig. 24, represent the theoretical discharge over a weir one foot long (as explained in foot-note ^ p. 549) under the measured head H = as, as though there were no current at o. Let ms = h = v^ -i-2 g. The theoretical velocities of the particles passing the oblique plane o a under their actual heads, will now be represented by horizontal lines 8s", a a", etc., etc., drawn from every point in s a to the outer curve .s" a" ; the line s s" representing V = velocity of approach = \/2 gh, and a a" representing y/2 g (H~+~fe). Then, area s s" a" a 2 2 = area ma" a — area m s" s = - - area of rectangle an — — - area of rectangle s k = -|-(H-f ?i) y2g (ii + /i)_-|^ y2Th, = -|y2g (l/(H + ;.)3 -Yh- and the actual discharge is Q = c X length of weir X area s s" a" a = c L — y2j ( V(H -f h)^ — }/P j = m L Y2J (yiW+liY^ — fhA = a: L (y'iRT'hJ'' — j/p)- § In a weir without end contraction, D = H 4- 2>. )' 558 HYDRAULICS. K is very approximately = 1 -f — ( -— j . Hence = [3.33 + 0.83(«y](L-nf)Ht. . (14) See Journal of the Franklin Institute, Philadelphia, August, 1884, from which we condense the above table. (y) M. Bazin, see Art. 14 (o), provides for the velocity of approach by modi- (H \* — J ; while by Messrs. Hunking and Hart's method (based upon Mr. Francis' experi- / H \ 2 ments) m becomes ^^^^ 0.415 + 0.10 I y 1 • Art. 15. Inclined -weirs. If the up-stream face of the weir, instead of being vertical, as in Fig. 25, is inclined up-stream, as in Fig. 25 a, or down- stream, as in Fig. 25 &, the character and amount of the discharge are modified. With an up-stream inclination (Fig. 25a) the lower side of the sheet of water passing over the weir leaps higher, and tends more and more up-stream as the Fig. 25 a. Inclined np'Stream Fig. 25 6. Inclined down-stream. Inclination is increased. With a down-stream inclination (Fig. 25 b), on the con- trary, as the inclination increases the upward leap of the sheet decreases, its profile becomes more and more flattened, and the curve of the upper surface, due to^the fall, extends farther up-stream from the crest of the weir. An up-stream inclination (Fig. 25 a) decreases, and a down-stream one (Fig. 25 b) increases, the discharge, as is indicated by the following coefficients ob- tained by M. H. Bazin :* For the y means of its velocity. Select I a.place where the cross-section remains, for "•-^^^^ jl. U I ^ J^ . ^ a short distance, tolerably uniform, and free from counter-currents, eddies, still water, or other irregularities. Prepare a careful cross-section, as Fig. 27. By means of poles, or buoys, n, n, divide the stream into sections, a, 6, c, i&c. Plant two range- poles, R, R, at the upper end, and two others at the lower end, of the distance through which the floats are to pass ; for observing by a seconds watch, or a pendu- lum, the time which they occupy in the passage. Then measure the mean velocity of each section a. h, c, &c., separately, and directly, by means of long floats, as F L, reaching to near the bottom : and projecting a little above the surface. The floats may be tin tubes, or wooden rods; weighted in either case, at the lower end, until they will float nearly vertical. They must be of different lengths, to suit the depths of the different sections. For this purpose the float may be made in pieces, with screw- joints. The area of each separate ^ection of the stream in square feet, being multi- plied by the observed mean velocity of its water in feet per second, will give the discharge of that section in cubic feet per second. And the discharges of all the separate sections thus obtained, when added together, will give the total discharge of the stream. And this total didcharge, divided by the entire area of cross-section of the stream in square feet, gives the mean velocity of all the water of the stream, in feet per second. Rem. If the eliaiiiiel is in common earth, especially if sandy, the loss bysoakage into the soil, and by evaporation, will frequently abstract so much water that the disch will gradually beLorue less and less, the farther down stream it is measured. Long canal feeders thus generally deliver into the canal but a small proportion of the water that enters their upper ends. The double float is used for ascertaining vels at diff" depths. It consists of a float resting upon the surface of the water, and of a heavier body, or " lower float", which is suspended from the upper float by means of a cord. The depth )f the lower float of course depends upon the length of the suspending cord ^which may be increased or diminished at pleasure until the lower float is be- lieved to be at that depth for which the vel is wanted), and upon its straight- ness, which is more or less afiected by the current. Owing to this latter circum- stance, it is difficult to know whether the lower float is really at the proper depth. Moreover it is uncertain to what extent the two floats and the string interfere with one another's motions. In deep water the string may oppose a greater area to the current than the lower float itself does. It thus becomes oubtful to what extent the vel of the upper float can be relied upon as indicat- ing that of the water at the depth of the lower one. Art. 19. Castelli's quadrant, or hydrometric pendulum, consisted of a metallic ball suspended by a thread from the center of a graduated are. The instrument was placed in the current, with the arc parallel to the direction of flow ; and the vel was then calculated from the angle formed be- tween the thread and a vert line. Oauthey's pressure plate was a sheet of metal suspended by one of its ands, about which it was left free to swing. The plate was immersed in the stream, with its face at right angles to the current. The vel was estimated by means of the weight required to make the plate hang vert in opposition to the force of the current. Pitot's tube was originally a simple glass tube. Fig. 27 A, open at both ends and bent in the shape of the letter L. One leg of the L was held horizontal under water, with its open end facing the current ; and the velocity v at the point o where it was placed was measured by the vertical height h i theoretically = — j to which the water rose in the other leg above the surface of the stream. 3e 562 HYDRAULICS. As developed by M. l>arcy and by Prof. S. "W. Robinson,* and rudely indicated in Fig. 27 B, Pilot's tube consists essen- tialh^ of two horizontal glass or metal tubes a and b, of very small bore, placed side by side in the current and pointed up-stream. Tube a receives the current in its open up-stream end, while b is closed at its up-stream end, and has small lateral openings only. The other end of each tube communicates, by means of small metal or rubber piping, with one leg of an inverted U-shaped glass gauge fixed in a boat or on shore. For convenience, the iwo flexible pipes may be joined together into one double pipe. -By sucking through a stop-cock T at the top, water is drawn up to any convenient height in the two legs of the gauge. When there is no current, the two columns of course stand at the same height ; but in a current, the dif- ference h in their heights is such that v == \"1 g A, no cor- rective coefficient being required. The instrument is re- markably simple and accurate, and can be used in very narrow and shallow streams of water or of gas. It meas- ures velocities as low as 4 inches per second. In practice, a and b are fixed together in one piece, and placed, when in use, In a metal frame which slides vertically, either upon a wire passing through it and provided with a plummet which rests upon the bottom and keeps the wire stretched, or (in streams shallower than about 20 feet) upon a vertical wooden rod, the lower end of which holds in the bed of the stream. In the former case, the frame is provided with a long vane for keeping the instrument h«aded up- stream. In either case, means are provided for showing the depth to which the instrument is submerged. By making the gauge scale adjustable vertically, and placing it (at each change of depth of instrument) with its zero opposite the top of the lower column, we obviate the necessity of observing the height of both columns at each reading ; for the reading of the upper column alone then gives the head h at once. Art. 20. The wheel meter consists of a wheel which is turned by the current, and which communicates its motion, by means of its axle and gearing, to indices which record the number of revolutions. The instrument may be clamped to any part of a long pole reaching to the bottom of a stream, and thus maj be used at any depth. The observer, by means of a wire, rod or string reaching down to the instrument, throws the registering apparatus first into, and then out of, gear with the wheel (applying a brake to the former at the instant it is thrown out of gear), and carefully noting the times when he does so. The instrument is then raised, the number of revolutions in the measured time is read ofl^ from the indices, and from it the velocity is calculated. But the meter is often made self-reg'istering' ; the wheel, at each revolution, auto- matically breaking and re-establishing a galvanic current generated by a bat- tery. The wire carrying this current is thus made to operate Morse telegraphic registering apparatus placed in a boat or on shore. A number of meters, so arranged, can be attached at different points on the same pole at the same time, and thus simultaneous observations of velocities at different depths may be made and registered. Meters are usually so arranged as to swing freely about the long vertical pole to which they are clamped, and are provided each with a vane or tail similar to that of a windmill, for keeping the wheel in the proper position as regards the current. The wheels are generally made like those of a windmill; i. «., with blades set at such an angle as to present a sloping surface to the current ; and -with the axis of the wheel parallel to the direction of flow. The axis runs in agate bearings. When desired, the rim of the wheel is furnished with an air- chamber, which just counterbalances the weight of the wheel, and thus removes journal friction due to it. Meters provided with electrical registering apparatus sometimes have the gearing and indices, etc., enclosed in a glass case, to prevent them from becoming clogged by weeds, sediment, etc. A wheel meter is rated by moving it at a known velocity through still water, and noting the eflfect produced. In this way a coeflftcient is obtained for each meter, which, when multiplied by the number of revolutions recorded in any given case, gives the velocity for that case. *See Van Nostrand's Magazine. March, 1878, and August, 1886. HYDRAULICS. 663 Art. 21. Kutter's formula for the mean vel of water flowing in open channels of uniform cross section and slope throughout. Cautioo. The use of all such formulae is liable to error arising from the difl&culty of ascertaining the exact condition of the stream as regards roughness of bed, surface slope,* etc. Bem. 1. Care must be taken tbat the bottoni Tel is not so g^reat as to ^vear anvay the soil. If there is any such danger artificial means must be applied to protect the channel-way; or it may be advisable to reduce the rate of fall, and increase the cross section of the channel ; so as to secure the same disch, but with less vel. A liberal increase sliould also be madf in the dimensions of such channels, to compensate for obstructions to the flow, arisiug from the growth of aquatic plants, or deposits of mud from rain- washes, etc ; or even from very strong winds blowing against the current. Rem. 2. l¥ater running in a channel with a horizontal bed, or bottoni, cannot have a uniform vel, or depth, throug-h- out its course; because the action of gravity due to the inclined plane of » sloping bottom, is wanting in this case; and the water can flow only bv forming its surface into an inclined plane; which evidently involves a diminution of depth at every successive dist from the reservoir. Theory of floiv. It is generally held that the resistances to the flow ot water in a pipe or channel are directly proportional to the area of the bed sur. face with which the water comes in contact (i e, to the product of the " wetted perimeter" as a &co Figs 28, 29, 30 mult by the length of the channel, or of the portion of it under consideration) ; and to the square of the vel of the flowing water; and, inasmuch as the resistance at any given point in the cross section appears to be inversely as the dist of that point from the bottom or sides, we conclude that the total resistances are inversely as the area of the cross section ; because the greater that area, the greater would be the mean dist of all the par. tioles from the bottom and sides. The resistance is independent of the pre^mre. In short, the resistances are assumed to be in proportion to vei2 X wet perimeter X length v^pl area of cross section a and the head h'' in feet or in metres etc, required to overcome those resist, ances, is resistance a coefacient, vel2 X wet perimeter X length ,,. ^v'''pt "*''*" C area of wet cross section a from which we have '' = ^cpl' ^' ^^i-V^^'\ area of wet resistce - ah" , ll a A" , / 1 vv'\/ cross section head «.= _. and v = ^--- or vel -^-^ X \ ^^^ perimeter >ii .15 122 105 93 83 75 62 52 42 31 24 20 17 .15 S- .2 130 114 100 90 81 67 57 46 34 27 22 19 .2 r2^ - .3 143 125 111 100 90 76 64 52 39 31 25 22 .3 "o 0^ .4 151 133 119 107 98 82 70 57 44 35 29 24 .4 ^ & .6 162 143 129 116 106 90 77 64 49 39 33 28 .6 'S'§ .8 170 151 135 123 112 95 82 68 53 43 35 31 .8 p^ 1 175 156 141 i 128 117 99 87 72 56 45 88 33 1 sS 1.5 185 165 149 136 125 107 94 79 62 51 43 37 1.5' ^s ■ 2 191 171 155 142 130 112 99 83 66 55 46 40 2 © 11 3 199 179 162 149 138 119 105 89 71 59 51 45 3 3.28 201 181 164 151 139 121 106 91 72 60 52 46 3.28 II o 4 204 184 167 154 142 123 1 109 93 76 63 55 48 4 6 210 190 173 160 148 129 115 99 81 68 59 52 6 10 217 196 180 166 154 136 121 105 86 74 65 58 10 ft^ 20 225 204 187 173 161 143 128 112 93 80 71 64 20 2" 50 231 210 194 181 168 150 135 119 100 87 78 71 50 S 100 235 214 197 1 184 1 172 153 139 122 1 104 1 91 82 1 75 100 For slopes steeper than .01 per unit of length, = 1 in 100 = 52.8 feet per mile, c remains practically the same as at that slope. But the velocitp (being = c X l/mean radius X slope) of course continues to increase as thfl 8k)pe becomes steeper. HYDRAULICS. 569 Table of coefficient c, for mean radii in metres. Mean Coefficients n of roughness. Mean 5 radR radR s metres .009 c .010 C .011 .012 013 015 017 .020 .025 .030 .035 .040 metres «^. c c c c c c c c c c o .025 34 29 25 22 20 17 14 11 9 7 6 5 .025 *s -.05 44 38 33 30 27 22 19 16 12 9 8 7 .05 ^o .1 58 50 44 40 36 30 26 21 16 13 11 9 .1 »-o .2 72 63 56 51 46 39 34 28 21 18 15 13 .2 ^i ^ 82 72 64 58 53 45 39 33 25 21 17 15 .3 tta ■ .4 89 79 71 64 59 60 44 37 29 23 20 17 .4 .6 .99 88 80 72 67 57 50 42 33 28 23 20 .6 1. 111 100 90 83 77 67 59 50 40 33 28 25 1. 1.50 121 109 100 92 85 74 66 57 46 38 33 29 1.50 e 2 127 115 106 98 91 80 71 61 50 42 37 32 2 1 3 136 124 114 106 99 87 78 68 56 48 42 37 3 4 142 130 120 111 104 93 83 73 61 52 46 41 4 6 149 137 127 119 111 100 90 80 67 58 51 46 6 10 158 145 135 127 120 108 98 88 75 66 69 53 10 i 15 164 151 141 133 126 114 104 94 81 72 64 59 15 20 167 155 145 137 130 118 108 98 85 75 68 62 20 . 30 172 160 1 150 142 135 123 113 103 90 81 74 68 30 r .025 40 35 30 26 24 20 17 13 10 8 7 6 .025 O .05 52 44 39 34 31 26 22 18 13 11 9 7 .05 .1 65 57 50 44 40 34 29 24 18 14 12 10 .1 *S . .2 79 69 62 55 51 43 37 30 23 19 16 13 .2 .3 87 77 69 62 67 48 42 35 27 22 18 16 .3 .4 93 83 74 67 62 53 46 38 30 25 21 18 .4 p,c^ .6 102 90 82 74 69 69 52 43 34 28 24 21 .6 i- 1. 111 100 90 83 77 67 59 50 40 33 28 25 1. 1.5 118 107 97 90 83 73 65 55 45 38 33 28 1.5 O it 2 123 111 102 94 87 77 68 59 48 41 35 31 2 •^ 3 129 117 108 100 93 83 74 64 53 45 40 35 3 yu 4 133 121 112 104 97 86 77 68 56 49 43 38 4 «s 6 138 126 117 t 109 102 91 82 72 61 53 47 42 6 t^ 10 143 1 131 122 i 114 107 96 87 78 66 58 52 47 10 15 147 1 135 126 118 111 100 91 82 70 62 56 51 15 20 150 137 128 120 113 103 94 84 72 64 58 53 20 30 152 140 131 123 116 105 97 87 76 68 62 57 30 o r .025 47 40 f 35 31 28 22 19 15 11 9 7 6 .025 .05 59 50 44 40 35 29 25 20 15 12 10 8 .05 '= o .1 72 62 55 50 45 37 32 26 19 16 13 11 .1 3 o .2 84 74 66 60 54 46 39 82 25 20 17 14 .2 «s .3 91 81 73 66 60 51 44 37 28 23 19 17 .3 *a .4 97 86 77 70 64 55 48 40 61 25 21 18 .4 ©^ .6 104 92 83 76 70 60 53 45 35 29 25 21 .6 1. 1.5 111 117 100 105 90 96 83 88 77 82 67 72 59 64 50 40 33 37 28 32 25 28 1. 54 44 1.5 «l 2 120 109 100 92 85 75 67 57 47 40 34 30 2 4 128 116 107 99 92 82 73 64 53 46 40 36 4 6 131 119 110 102 96 85 77 67 56 49 43 39 6 10 135 123 114 106 100 89 81 71 60 53 47 43 10 15 137 126 116 109 102 92 83 74 63 55 50 46 15 s 30 141 129 120 112 106 95 87 78 67 69 64 60 30 =«§ f .025 52 45 40 35 31 25 21 17 12 9 8 6 .025 as g.2 .050 63 55 48 43 39 32 27 21 16 12 10 8 .050 .1 75 66 69 53 48 40 34 27 21 16 13 11 .1 .2 87 77 69 62 57 48 41 34 26 21 17 15 .2 2^ .4 99 88 80 72 66 57 49 41 32 26 22 19 .4 •I .6 104 93 84 77 71 61 53 45 36 29 25 22 .6 II .fl" 1 111 100 90 83 77 67 59 60 40 33 28 25 1 4^^ 2 118 107 98 90 84 74 65 56 46 39 34 30 2 ft^ 4 124 113 104 97 90 79 71 62 51 44 39 35 4 ©i^ 10 130 119 110 102 96 85 77 67 57 60 45 40 10 S'o 30 135 124 114 107 100 90 82 73 62 55 50 46 30 670 HYDRAULICS. Table of coefficient c, for mean radii in wip^re*.— Continued. 5 Mean Coefacients n of roughness. Mean feo radR radB .2 meters .009 .010 .011 .012 .013 .015 .017 .020 .025 .030 .035 .040 metres © c C c c c c c c c c c c {3 .025 55 47 41 37 33 27 22 17 13 10 8 7 .025 .050 66 68 61 45 40 33 28 23 17 13 11 9 .050 .1 78 68 61 55 50 42 35 28 21 17 14 12 .1 fi- .2 90 80 70 64 69 49 42 35 27 22 18 15 .2 .3 95 85 76 70 63 64 47 39 30 24 21 17 .3 ©TH .4 99 89 80 73 67 67 60 42 32 27 22 20 .4 .6 105 94 85 78 72 62 54 45 36 30 25 22 .6 1 111 100 90 83 77 67 69 60 40 33 28 25 1 II 2 117 106 97 89 83 73 65 56 46 38 34 30 2 i 4 123 111 102 95 88 78 70 61 50 43 38 34 4 1 6 125 114 105 97 91 81 72 63 63 46 40 36 6 10 128 117 108 100 93 83 75 66 65 48 43 39 10 s 30 132 121 112 104 98 87 79 70 60 52 48 43 I 30 ^ .025 67 60 43 38 34 28 23 18 13 11 9 7 .025 9d . .050 69 59 52 47 42 34 29 23 17 13 11 9 .060 ^§ .1 80 70 63 66 60 42 36 30 22 17 14 12 .1 o,^ .2 90 80 72 65 60 50 43 35 27 22 18 16 .2 .3 96 86 77 70 64 54 47 39 30 25 21 18 .3 .4 100 89 81 74 67 58 60 42 33 27 23 19 .4 .6 104 94 85 78 72 62 64 46 36 30 25 22 .6 •pSf 1 111 100 90 83 77 67 59 50 40 33 28 25 1 n i* 2 116 106 97 90 83 72 64 55 45 38 33 29 2 *g 4 121 111 102 94 87 77 69 60 50 42 37 33 4 ft^ 6 124 113 104 97 90 80 71 62 52 45 40 36 6 5o 10 127 115 106 99 92 82 73 64 54 47 42 38 10 3 30 130 119 110 102 96 86 77 68 58 51 46 42 30 .025 59 50 44 39 35 28 24 19 14 10 9 7 .025^ ^ . .05 69 60 53 48 43 35 29 24 18 14 11 9 .05 S| .1 81 71 63 57 51 43 36 30 22 18 15 12 .1 fc. ^ .2 91 81 72 65 60 50 44 36 27 22 18 16 .2 a-= .3 97 86 77 71 65 55 48 40 31 25 21 18 .3 .4 101 90 81 74 68 58 50 42 33 27 23 20 .4 o II .6 106 95 86 78 72 62 54 46 36 30 25 22 .6 11 a 1. 111 100 90 83 77 67 59 50 40 33 28 25 1. 1.5 115 104 94 87 80 70 62 53 43 36 31 27 1.5 2 117 105 96 89 83 72 64 55 45 38 33 29 2 4 121 110 101 93 87 76 68 59 49 42 37 33 4 S^ 10 126 114 105 98 91 81 73 64 53 46 41 37 10 30 129 118 108 101 95 84 77 67 57 50 45 41 30 For slopes steeper tlian .01 per unit of length, = 1 in 100, the eo' efficient c remains practically the same as at that slope. The velocity, however, being = cX l/mean radius X slope, continues to increase as the slope becomes steeper. To construct a diagram, fig 30 A, from which the values g^iven by Kutter's formula may be taken by inspection. Draw xz hor, and say from 2"to 4 ft long; and oy vert at any point o within say the middle third o\ xz. On oy lay off, as shown on the left, the values of c for which the diagram will probably be used. If a scale of .05 inch, or .002 metre, per unit of c be used, and be made to include c = 250 for English meas- ure, or 150 for metric measure, oy will be about 1 ft long. For the sake of clearness we show only the larger divisions in this and in what follows. On oz lay off, as shown on its upper side, the square roots of all the values of the mean rad R for which the diagram is to be used. One inch per ft, or .06 metre per metre, of sq rt, is a convenient scale. Ma7'k the dividing points with the respective values of the mean radii themselves. Having decided upon the flattest slope to be embraced in the diagram, say w = 41.6 + .0028 flattest slope per unit of length for English measure. HYDRAULICS. 571 iFig.SOA. 50 13 w — 23 + - for metric meaaure. flattest slope per unit of length For each value of n to be embraced in ttie diagram, say y — «,-= -J — for English measure ; or p — w = for metric measure. ' n n To each value of y — w, add w, thus obtaining values of y. We take .000026 per unit of leugth as the flattest slope,* and .01, .02, .03 and .04 for n.f Henc9 .0028 (using English measure) w = 41.6 + .qqqqo^ 1.811 1.811 1.811 1.811 y — tc — - .01 * .02 .04 =- 41.6 + 112 = 153.6. ; = 181.1, 90.5, 60.4, 45.8 respectively, * This is about as flat as is likely to occur in practice. t In most cases many intermediate ones would be used. 572 HYDRAULICS. and y — 181.1 + 153.6, 90.5 + 153.6, 60.4 + 153.6 and 45.3 -f 153.6; or 334.7, 244.1, 214.0 and 198.9 respectively. Lay off these values of y on oy in pencil, as at y, y', y", and y'", using the scale already laid otf for c on oy. From each point, y^y' etc, draw a hor pencil line yt. y' t' etc, and mark on it, in pencil, the value of n used in determining its height oy etc. Next say x = w? X greatest value of n. Make o a; = x by the scale of sq rts of R on z. In our case o i = 153.6 X .04 = 6.144 by the scale of sq rts of R, or = 6.144* «= 37.75 by the scale of R. Divide ox into as many equal spaces (4 in our case) as .01 is contained in greatest n. Mark the dividing points with the values of w, as in our Fig. From each dividing mark on ox erect a perpendicular, {xt'" etc) in pencil, to cut that hor line {y'" f' etc) which corresponds to the same value ol n. The interseciions are points in a hyperbola. Join them by straight lines t'" i'\ t" t\ t' t etc. From r iu oz (corresponding to a mean rad of 3.28 ft, or 1 metre) draw radial lines, rt, rt\ rt" etc. Mark them "w = .01 ", "71 = .02" etc, the same as their corresponding lines yt, y' t' etc. For each slope (S) to be used in the diagram (except the flattest, for which this has already been done) say (0028 \ 41.6 + -j — ) X greatestn, for English measure. z\ x" etc == ( 23 4- '—, — '— 1 X greatest n, for metric measure. ' \ slope / Thus, our slopes are = .000025, .00005, .0001 and .01 per unit of length. Hence, ;^"_(41.6+^)X. 04 =1.675. Lay off each value of x', x" etc from oy on a separate hor pencil line 0' x' etc, using the scale of sq rts of R as on oz. Mark each line 0' x' etc in pencil with the slope used in fixing its length. Divide each disto'x'etc into the same number of equal parts as ox. From the dividing points (which, like thoseof ox, represent the values of w) erect perps to cut the radial lines rt'", rt" etc, each perp cutting that radial line which cor- responds to the value of n represented by the point at the foot of the perp. The intersections corresponding to each line 0' x' etc form a hyperbolic curve. Mark each curve with the slope of its corresponding line, ox, 0' x' etc. The drawing is now in the shape proposed by Mess Ganguillet and Kutter, and is ready for use in finding either c, », R or S when the other three are given. Thus : Ist. Having R, S and n, to find c. For example let R = 20 ft, S = .00005, n = .03. From the intersection d of slope curve .00005 and radial line n = .03, draw* (i-20 to the point (20) in oz corresponding to the given R. At e, where d-20 cuts oy, is the reqd c, = 96 in this case. 2d. Having R, S and c, to find n. For example let R = 20 ft, S = .00005, c = 96. Through the points R = 20 in oz, and c = 96 in oy, draw* d-20 to cut curve .00005. n (= .03) is found by m«ans of the radial lines nearest to the in- tersection, d. 3d. Having S, w and c, to find R. For example let S = .00005, n = .03, c = 96. Find curve .00005 and radial line n = .03. From tlieir intersection d draw d-'ik through the point eshowingc = 96. Its intersection with oz shows the reqd R, 20 in this case. * Instead of drawing these lines, we may use a fine black thread with a loop at one end. Drive a needle either into one of the points R or into one of the intersections, d etc. Slip the loop over the needle. The other end of the thread is held between the fingers, and the thread is made to cut the other points as reqd. The diagram should lieperfectly flat, and the string be drawn tight at each ob- serration, in order that friction between string and paper may not prevent the string from forming* ■traight line. Or the free end of the string may rest on a pamphlet or other object about }^ inch thick, to keep the string clear of the diagram. Special care must then be taken to have the eye perp over the point observed. HYDRAULICS. 673 4tli. Having R, c and n, to find S. For example let R = 20 ft, c = 96, n '-= .03. Through R = 20 and c = 96 draw d-20. S (.00005) is found by means of the curves nearest to the point d of intersection of d-20 with radial line n = .03. The following addition to Kutter's diagram, proposed by Mr Rudolph Hering, Civil and Sanitary Engineer, Philadelphia,* enables us to read tlie veloc- ity from the diag-ram. Fi nd the sq rt of the recipr ocal of each slope to be embraced in the diagram = -x -. .^ - , 77 . Lay off these sq rts on the right of oy, using \ slope per unit of length "^ ^ & ^. » the scale of c already laid off on its left. In our fig we have so proportioned the two scales that — - == -^ — . Mark the dividing points with the slopei l/recip of S ^ per unit of length. On oz lay off the vels to be embraced in the diagram, using the scale of sq rts of R already laid off on oz, and making — ^ = — ^ 1/R . l/recip of S 1st. Having R, S andw; to find v. For example let R = 20 ft, S -= .00005, % = .03. From R = 20 draw d-20 to the intersection d of curve .00005 with radial line w == .03. d-20 cuts oy at e, where c = 96. With a parallel ruler join R — = 20 with S = .00005 on o y. Draw a parallel line through c = 96. It cuts o 2 at m, giving the reqd vel, 3.03 ft per sec. 2d. Having R, S and v; to find n. For example let R = 20 ft, S = .00005, t; — = 3.03 ft per sec. With a parallel ruler join R = 20 and slope .00005 on oy. Draw a parallel line through v = 3.03. It cuts oy at e, where c = 96. Through R = 20 and c = 96, draw d-20 to cut curve .00005. The point dof intersection, being on radial line n = .03, shows .03 to be the proper value of n. Any line drawn to the curves from R = 3.28 ft or 1 metre, is one of the radial lines used in making the diagram. It therefore necessarily cuts all the slope surves at points showing the same value of n. 3d. Having S, n and v; to find R. For example, let S = .00005, n = .03, V = 3.03 ft per sec. Assume a value of R, say 10 ft. Find curve .00005 and radial line n = .03. Join their intersection d with R = 10 ft. The connecting line cuts oy at c = 82. With a parallel ruler join c == 82 with v = 3.03. Draw a parallel line through slope = .00005 on oy. It cuts o z at R = 27.3, showing that a new trial is necessary, and with an assumed R greater than 10 ft. If R thus found is the same as the assumed one, the latter is correct. If they are nearly equal, their mean may be taken. 4tli. Having R, n and v ; to find S. For example, let R = 20 ft, w = .08, V = 3.03 ft per sec. Assume a slope (say .0001). Find its curve, and radial line n = .03. Join their intersection with R = 20, and note the value (89) of c where the connecting line cwts oy. With a parallel ruler join c = 89 with v = 3.03. Draw a parallel line through R = 20. It cuts oy zi slope .000058, showing that a new trial is necessary, and with an assumed S flatter than .0001. If R is 3.28 ft, or 1 metre, the diagram gives the corrects at the first trial, no matter what S was assumed at starting. With any other R, if the diagram gives the Kime S as that assumed, the latter is correct. If the two differ but slightly, we may take their mean. • Trantftctiona of the Auerican Society of Civil Engineers, January ll7t. 574 VELOCITIES IN SEWERS. Table of vels in Circular Brick Sewers when running full, by Kutter's formula, but taking n at .015 instead of his .013, in consideration of the rough character of sewer brickwork generally. When riinniii$ir only balf full the vel will be the same as when full, but this is not the case at any other depth whether greater or less. At greater ones it increases until the depth equals very nearly .9 of the diam, when it U about 10 per cent greater than when either full or half full. From depth of .9 of the diam the vel decreases whether the depth becomes greater or less. At depth of .25 diam the vel is about .78 of that when full ; and then diminishes muoll more rapidly for less depths. All this applies also to pipes. The vel for any fall or diam intermediate of those in the table can be found by simple proportion. Original. FaU in ft Diameters in feet. m ft per mile. 2 3 4 6 8 12 16 20 per 100 ft Yelocitiea in feet per second. .1 .19 .27 .35 .50 .64 .89 1.10 1.34 .0018 .2 .30 .42 .53 .74 .93 1.26 1.56 1.84 ,0031 .4 .46 .65 .80 1.08 1.39 1.81 2.20 2.60 .007« .6 .59 .81 1.00 1.35 1.70 2.22 2.70 3.18 .0114 .8 .69 .95 1.17 1.57 1.94 2.56 3.08 3.60 .0161 1.0 .79 1.07 1.32 1.77 2.16 2.84 3.43 3.96 .0189 1.25 .89 1.21 1.49 1.98 2.42 3.17 3.8 4.5 .0287 1.50 .98 1.33 1.64 2.18 2.64 3.5 4.2 4.9 .0284 1.75 1.06 1.44 1.78 2.34 2.85 3.8 4.5 5.3 .0881 2.0 1.15 1.55 1.91 2.53 3.1 4.0 4.8 5.6 .0379 2.5 1.32 1.78 2.18 2.85 3.5 4.5 5.4 6.3 .0473 3.0 1.44 1.94 2.38 3.2 3.8 5.0 6.0 6.9 .0568 3.5 1.58 2.10 2.58 3.4 4.1 5.3 6.5 7.4 .0662 4. 1.68 2.2 2.7 3.6 4.4 ■ 5.7 6.9 7.9 .0768 5. 1.90 2.5 3.1 4.1 4.9 6.3 7.6 8.7 .0947 6. 2.06 2.7 3.3 4.4 5.4 6.9 8.3 9.6 .1136 7. 2.2 3.0 3.6 4.8 5.8 7.5 9.0 10.4 .1326 8. 2.4 3.2 3.8 5.1 6.2 8.0 9.7 11.1 .1514 9. 2.5 3.4 4.1 5.4 6.6 8.5 10.3 11.8 .1703 10. 2.7 3.5 4.3 5.7 6.9 9.0 10.8 12.5 .1894 12. 2.9 3.9 4.8 6.3 7.6 9.9 11.9 13.6 .2273 15. 3.3 4.4 5.4 7.1 8.5 11.0 13.3 15.3 .2841 18. 3.6 4.8 5.9 7.7 9.3 12.1 14.5 16.7 .3409 21. 3.9 5.1 6.3 8.4 10.0. 13.0 15.7 17.9 .3975 24. 4.2 5.5 6.8 8.9 10.8 13.9 16.8 19.2 .4546 27. 4.5 5.9 7.2 9.5 11.4 14.8 17.9 20.4 .5109 30. 4.7 6.2 7.5 9.9 12.0 15.6 18.8 21.5 .5682 35. 5.0 6.7 8.2 10.8 13.0 16.8 20.4 23.2 .6629 40. 5.4 7.1 8.7 11.5 13.9 18.0 21.7 24.8 .7576 45. 5.6 7.5 9.2 12.2 14.8 19.1 23.0 26.3 .8523 50. 5.9 8.0 9.7 12.8 15.5 20.1 24.2 27.7 .9470 60. 6.5 8.7 10.7 14.1 17.0 22.1 26.5 30.3 1.136 70. 7.0 9.4 11.5 15.2 18.4 23.9 28.5 32.8 1.326 80. 7.4 10.1 12.3 16.2 19.7 25.5 31.0 35.0 1.515 90. 7.9 10.7 13.1 17.2 20.9 27.0 32.3 37.1 1.705 100. 8.4 11.3 13.8 18.2 22.0 28.5 34.1 39.1 1.894 A vel of 10 ft per sec = 600 ft per minute = 36000 ft, or 6.818 miles per hour. About 5 ft per sec is as great as can be adopted in practice to prevent the lower parts of the sewers from wearing away too rapidly by the debris carried along by the water. HYDRAULICSc 575 Art. 23. The rate at ivliich rain ^i^ater reaches a sewer or culvert, etc. Burkli-Zieg-ler Formula. See "European Sewerage Systems," by Rudolph Hering, C. E., iu Trans. Am. Soc. C. E., Nov. 1881. ^l^^"/h'?/ -'^ coef Av. cub. ft. of rainfajl ac?e"i each- = . according X per second per acre. to judgment during heaviest fall. 1 4 /Av. si \ No.of slope of ground feet per 1000 ft. lug sewer ""•' — ^^ — *= » xNo.of acres drained Coefficient, for paved streets, 0.75 ; for ordinary cases, 0.625 ; for suburbs with gardens, lawns, and macadamized streets, 0.31. Note that 1 iuch of rainfall per hour may be taken as eqiiivalent to 1 cubic foot per second per acre. See Conversion Tables, pp. 235, etc. Example. If an area of 3100 acres (nearly 5 square miles), with an average slope of 5 feet per 1000 feet, receives a maximum rainfall of 3 inches per hour, then, assuming a coefficient of 0.5, the rate at which the water would reach the mouth of a sewer at the lower end of the 3100 acres would be 0.5 X 3 X ^-^yq-^ = 0.5 X 3 X 0.203 = 0.305 cubic feet per second per acre; or 0.305 X 3100 = 945.5 cubic feet per second, total. Let the grade of the intended sewer be say 4 feet per mile; and, to avoid excessive wear of its brickwork by debris swept along by the water, let its velocity be limited to 6.3 feet per second, which may be permitted on occasions as rare as rains of 3 inches per hour, althougli, for tolerably constant flow, where liable to debris, it should not exceed about 5 feet per second. Find, in table opposite, the diameter, 14 ft., corresponding, as nearly as may be, to a velocity of 6.3, and to a grade of 4 feet per mile. The area is 154 square feet. Hence, 154 X 6.3 = 970 cubic feet per second == capacity of sewer. To allow for deposits in the sewer, make the diameter say 14.5 or 15 feet. Table of least Telocities and g-rades for drain-pipes and sevi^ers in cities, in order that they may under ordinary circumstances keep themselves clean, or free from deposits. (Wicksteed.) Grade. Grade. Diam. Vel. in ft. Grade, Feet per Diam. Vel. in ft. Grade, Feet per In Inches. per Min. lin Mile. in Inches. per Min. lin Mile. 4 240 36 146.7 18 180 294 18.0 6 220 65 81.2 21 180 3J3 15.4 7 220 76 69.5 24 180 392 13.5 8 220 . 87 60.7 30 180 490 10.8 9 220 98 53.9 36 180 588 9.0 10 210 119 44.4 42 180 686 7.7 11 200 145 36.7 48 180 784 6.8 12 190 175 30.2 54 180 882 6.0 15 180 244 21.6 60 ISO 980 5.4 Weight i)er foot run of g-1aze .0507 .0621 .0819 .135 -2079 .4788 .9621 0728 .0902 .1104 .14.56 .240 .3696 .8412 1.710 1137 .1410 .1725 .2275 .375 .5775 1.320 2.672 1638 .2030 .2484 .3276 .540 .8316 1.915 3,848 4550 .5640 .6901 .9100 1.50 2.310 5.280 10.69 2.326 4.144 6 475 9.304 25.9 TABTiE Increased velocities produced at and by rounded, or pointed obstructions. If square, these vels must, accord- ing to Nicholson, be increased }/^ part. Original Vel. of Stream.* Proportion of Area of Water-way, occnpied by the Obstrnctions. tV 1 tV 1 i 1 4 1 i 1 i 1 4 1 f 1 f Per Per Sec. Hour. Ins, Pt. Miles. Velocity prodnced at the Obstrnction in Feet per Second. 3 1,^ .170 .28 ,29 .30 .32 .35 .394 .52 .7 1.05 6 1^ .341 .56 .58 ,60 .64 .70 .788 1.05 1.4 2,1 12 1 .681 1.13 1.16 1,20 1.26 1.40 1.58 2.1 2.8 4.2 24 2 l..'i6 2.27 2.33 2,40 2.52 2.80 3.16 4.2 5.6 8.4 36 8 2.04 3.39 3.48 3,60 3.78 4.20 4.74 6.3 8.4 12.6 48 4 2.72 4.64 4.66 4,80 5.04 5.60 6.32 8.4 11.2 16.8 60 5 3.41 5.60 5.80 6,00 6.40 7.00 7.88 10.5 14.0 21.0 72 6 4.09 6.78 6.96 7.20 7.56 8.40 9.48. 12.6 16.8 25.2 120 10 6.81 11,3 11.6 12.0 12.6 14.0 15.8 21.0 28.0 42.0 * A very vague expression. Does it refer to the greatest surface vel at mid-ohannel ; er to the mean vel of the entire cross- section? 37 578 HYDRAULICS. Art. 26. The resistance of water a$;aiiist a flat surface moT- ing' throns'h it at risj;-lit aiig'les, is nearly as the squares of the Tel; and, according to Hutton, its amount in fibs per sq ft approx = Square of vel in ft per sec. Or like tlie pres of a riinning- stream against a perp fixed flat Barfaee, it is = wt of a col of water whose base =: pressed surf, and whose ht=:head due to the vel The resist of a sphere ia to that of its great circle about as 1 to 2.9. When the moTing surf. Instead of being at right angles to the direction in which it moves, forms another angle with it, the resistance becomes less in about the following proportions. Therefore, Ivhen the surf ia inclined, first calculate the resistance as if at right angles : and then mult bj the following decimals ©pposite the angle of inclination : 90°.. ..1.00 . 60°... . .88 *0».. . .58 20°.. . .16 80 .. .. .98 55 .. . .83 35 .. . .46 15 .. . .10 70 .. .. .95 50 ... . .76 30 .. . .34 10 .. . .06 65 .. .. .92 45 ... . .68 25 .. . .24 5 .. . .02 Ttie scour, or abrading* power of mioving: water is considered U be as the square of its vel. Art. 27. To calculate the horse-power of falling* w^ater, on the ordinary assumption that a horse-power is equal to 33000 lbs lifted 1 foot vert per min. That of average horses is really but about % as much, or 22000 S)s, 1 foot high per min. Mult together th« number of cub ft of water which fall per min ; the vert height or head in feet, through which it falls; and the number 62.3, (the wt of a cub ft of water in lbs :) and dir the prod by SSOOOT Or, by formula, cub ft ^ vert ^ lbs The number of ^ per min ^ height in ft ^ 62.3 horse.powers ^^ Ex. Over a fall 16 ft in vert height, 800 cub ft of water are dischd per min. How many horse powers does the fall afford ? cub ft ft a* „ 800 X 16 X 62.3 797440 Here. = •= 24.17 h-now. ' 33000 33000 *^ TFater-WheelB do not realize all the power inherent in the water, as found by out rule. Thus, underjinots realize but from >i to J^ ; breast-wheels, }4 ; overshots, from % to Ji ; tur- bines, Ji to .85 of 14; ; according to the skill of design, and the perfection of workmanship. Even when the wheel revolves in a close-fitting casing, or breast, elbow buckets give considerably more power than plain radial or center-buckets. Of the power actually received by a wheel, part is expended in friction, &c; while the remainder does the useful or paying network of raising water, grinding grain, sawing. &c. Observations by Oenl Haupt, in 1866, gave the following results for a small hydraulic ram. Head of water to ram = 8.812 ft ; diam of drive-pipe = l^ ins; length 15 ft. Diam of delivery-pipe — % inch; length 200 ft. Vert height to which the water was raised by the delivery-pipe, 63.4 feet. Strokes of ram per min, 170. Quantity of water which worked the ram =: 768 cub ins, =: 3.31 galls, = 27.73 fi)8 per min. Quantity raised 63.4 ft high tba water ft ft-Qw per min, = 48 cub ins, = 1.736 fts. Hence the power expended per min, waa 27.73 X 8.812 = 244.35. lbs water ft ft-tti8 And the useful effect, was 1.736 X 63.4 = 110.06. Hence the ratio which the useful effect bears to the power in this instance, is '- — , or .45. The actual power of the ram is, however, greater than this, inasmuch as it has to overcome the friction of the water along the delivery-pipe.* To find the horse-power of a running stream. Water-wheels •with simple float- boards, t instead of buckets, are sometimes driven by the mere force of the ordinary natural current of a stream, without any appreciable fall like that in the foregoing case. In such eases, we must substitute the virtual or theoretic head ; which is that which would impart to it the game vel which it actually has. This virtual head may be taken at ouce from i aide p. 539. Thus, a stream has a vel of 2.386 miles per hour; or 210 ft per min ; or 3}^ ft per sec ; and in the column of heads in Table 10, opposite to 3.5 vel per sec, we find the reqd head .190 of a ft. Having thus found the head, we must now find the quantity of water which passes any given area of the stream in a min. Thus, suppose that the tmmerscci part of a float when vert is 5 ft long, and 1 ft wide or deep; then the area of this part which receives the force of the current, is 5 X 1 = 5 square feet. Hence, area vel 5 sq ft X 210 = 1050 cub ft per min. Having now the cub ft per min, and the vert height or head, the number of horse-powers of the stream of the given area, is found by the foregoing rule, or formula. * A committee of the Franklin Institute, in 1850. gave .71 as the coeffloient for a ram at the Girard College, in which the diam of drive-pipe was 2^ ins ; its length, 160 ft; fall, 14 ft. Delivery-pipe, 1 inch diam ; 2260 ft long : vert rise, or height to which the water was raised. 93 ft. No details of the experiment are given. Some large rams in France give a useful effect of from .6 to .65 of the whole power expended. It is an excellent machine for many purposes; and is sometimes used for filling railway tanks at water stations. t Such wheels, for floating- mills, in £urope, rarely exceed 15 ft fLiam. Whatever the diam, they may have about 18 to 20 floats. The floats are from 8 to 16 ft long ; and about 4 to -r^ as deep as the diam of the wheel. They should not dip their entire depth into the water, but nearly so. They should not be in the same straight line with the radii ; but should incline from them 30° up stream, to produce their full effect. All these remarks apply to wheels moving freely in a wide or indefinite channel ; as in the case of a floating mill, built on a scow, and anchored out in a stream : but not to wheels for which the water is dammed up, and acts with a prac- tical fall. No great exactness is to be expected in rules on this subject. The best vel for the wheel is abe«t .4 that of the stream. HYDRAULICS. 579 cub ft per min ^ vert ht in/t ^ lbs •^__ Xo of « 1050 ^ .190 ^ 6-2.3 12429 **'"• H. Pow. 33000 ■ = 33000 = -^'^ ^f ^ ^- ^^• But in practice the wheels actually realize but about 0.4 of this power of the stream. Therefore, the actual power of our wheel will be but .377 X .4 ^ 0.1508 of a horse power; or 33000 X .1508 = 4976 ft-ft)s per minute. Making a rough allowance for the friction of the machine at its journals, &c, we should have about 4400 ft-ft)s of iiseful power per minute ; that is, the wheel would actually raise about 440 fts 10 ft high ; or 44 lbs 100 ft high, &c, pei min. The vel of the stream must not be measured at the surface : but at about }4 of the depth to which the floats are to dip, or be immersed. This, however, is necessary chiefly in shallow streams, in which the depth of the float bears a considerable ratio to that of the water. This power of a running- stream, (for any given area of transverse seetionO increases as tSie enbes of the vels: for, as we have seen, the power in ft-ft>s per min is found by mult together the weight of water which passes through the section in a min, and the virtual head in ft ; and since tnis weight increases as tlie vel, and this head as the square of the vel, the prod of the two (or the power) must be as the cube of the vel. Therefore, if the vel in the foregoing case had been 10.5 ft per sec, or 8 times 3.5 ft, the power of the wheel would have been 27 times as great, or .1508 X 27 = 4.07 horse powers. 580 DREDGING. DEEDGING. I>REDQiNG is generally done by skilled contractors, who own the requisite machinee, ■C0W3 or lighters, &c ; and who make it a specialty. It is necessary to specify whether the dredged material is to be measured in place before it is loosened: or after being deposited in the scow: because it occupies more bulk after being dredged. It was found, in the extensive dredgings for deepening the River St Lawrence through the Lake of St Peter, that on an average a cub yd of tolerably stiff mud in place, makes 1.4 yds in the scow ; or 1 in the scow, makes .715 in place. Also stipulate whether the removal of bowlders, sunken trees, &c, is to constitute an extra. These often require sawing and blasting under water. The cost per cub yd for dredging varies much With the depth of water : the quantity and character of the material : the dist to which it has to be removed; whether it can be at once discharged from the machine by means of projecting side-shoots or slides : or must be discharged into scows, to be re- moved to a short dist by poling, or to a greater dist by steam tugs ; whether it can be dropped or dumped into deep water by means of flap or trap doors in the bottom of the hoppers of the scows ; or must be shoveMed from the scows ivto xhalloto water, (at say 4 to 8 cts per yd ;) cr upon land, (at say from 6 to 10 or 20 cts for the shovelling ftlone, or shovelling and wheeling, as the case may be;) whether much time must be consumed in moving the machine forward frequently, as when the excavation is narrew, and of but little depth; as in deepening a canal, Ac ; whether many bowl- ders and sunken trees are to be lifted ; whether interruptions may occur from waves in storms; whether fuel can be readily obtained, &c, &c. These considerations may make the cost per cub yd in one case from 2 to 4 times as great as in another. The actual cost of deepening a ship-channel through Lake St Peter, to 18 ft, from its orig- inal depth of 11 ft, for several miles through moderately stiff mud, was 14 cts per cub yd in place, or 10 cts in the scows ; including removing the material by steam tugs to a dist of about i^ a mile, and dropping it into deep water. 'Ibis includes re- pairs of plant of all kinds, but no profit. It was a favorable case. When the buckets work in deep water they do not become so well filled as when the water is shallower, because they have a more vertical movement, and, therefore, do not scrape along as great a distance of the bottom. Hence one reason why deep dredging costs more per yard; in addition to having to be lifted through a greater height. Perhaps the following table is tolerably approximate for large works in ordinary mud, sand, or gravel ; assuming the plant to hare been paid for by the company ; and that common labor costs $1 per day. Table of actual cost of dredging- on a large scale; inclnd- iiig- dropping- the material into scows, along-side; or into side-shoots, on board. €onanion labor S^l per day. Repairs of plant are inclwded ; but no protit to contractor. (Uriginal.J Depth in Ft. Cts per Yard, in place. Cts per Yard, in .scow. Depth in Ft. Cts per Yard, in place. Cts per Yapd, in scow. L»ss than 10 10 to 15 15 to 20 20 to 25 8.4 9.8 11.2 U.O 6 7 8 10 25 to 30 30 to 35 35 to 40 18.2 25.2 35.0 13 18 25 For towing of the SCOWS by steam tugs to a distance of ^ mile, and dropping the mud into deep water, add 4 cts. per yard in the scow ; for ^ mile, 6 cts. ; for % mile, 8 cts. ; for 1 mile, 10 cts. Add profit to contractor. On a small scale work is done to a less advantage ; and a corresponding increase must be made in these prices. Also, if the contractor himself furnishes the dredgers and plant, a still further addition must be made. It is evident that the subject admits of no great precision. Small Jobs, even in favora- ble material, but in inconvenient positions, may readily cost two or three times as much per yard as the above : and in very hard material, as in cemented gravel and clay, four or five times as much for the dredging. The cost of towing, however, will remain as before, if wages are the same. The cost of dredgers, tugs, &c., will vary of course with their capabilities, strength of construction, style of finish, whether having accommodations for the men to live on board or not, &c. When for use in salt water, the bottoms of both dredgers and scows should be coppered, to protect them from , sea-worms ; and if occasionallv exposed to high waves, both should be J extra strong. The most powerful machines on the St. Lawrence cost about i DREDGING. 581 $45,000 each ; and removed in 10 working hours on an average about 1800 cubic yards in place, or 2520 in the scows. Good machines, capable, under similar circumstances, of doing as much, may, however, be built for about ^25,000 to $30,000. To remove this quantity to a distance of 3^ to 1 mile, would require two steam tugs, costing about ^000 to $10,000 each ; and 4 to 6 scows (some to be loading while others are away), holding from 30 to 60 cubic yards each, and costing frf)m $800 to $1500 eacli at the shop. Scows with two hoppers are best. Such a dredger would require at least 8 or 10 men, including cap- tain, engineer, fireman, and cook ; each tug 4 or 5 men ; and each scow 2 men. The engineer should be a blacksmith ; or a blacksmith should be added. In certain cases a physician, clerk, assistant engineer, &c., may be needed. Dredgers are often built on the principle of the Yankee Excavator, with but a single bucket or dipper, of from 1 to 2 cubic yards capacity. Hull about 25 by 60 feet. Draft 3 feet. Cylinder about 7 or 8 inches diameter ; 15- to 18-incli stroke ; ordinary working' pressure 50 to 80 lbs. per square inch, according to hardness of material. Cost $8000 to $12,000. Will raise as an average day's work (10 hours) from 200 to TjOO yards in place, or 280 to 700 in the scow, according to the deptb, nature of the material, &c. Require 5 or 7 men in all aboard, including cook. Burn 3^^ to 1 ton of coal daily. Tolerably large bowlders and sunken logs can be raised by the dipper. =!= When the material is hard and compacted, the buckets of dredgers should be armed with strong steel teeth projecting from their cutting edges. On arriving at such material every alternate bucket is sometimes unshipped. By arranging the buckets so as to dredge a few feet in advance of the hull, low tongues of dry land may be cut away : the machine thus digging its own channel. The daily work in such cases w ill not average half as much as in wet soil. On small operations, clred^er» worked by two or more horses, instead of by steam, will answer very well in soft material ; or even in moderately hard, by reducing the size and number of the buckets. A two- horse machine will raise from 50 to 100 yards of ordinary mud in place, or 70 to 140 in the scow, per day, at from 12 to 15 feet depth. Soft material in small quantity, and at moderate depth, may be removed by the slow and expensive mode of the toag:-scoop, or ba^-spoon. This is simply a bag B, made of canvas or leather, and having its mouth surrounded by an oval iron ring, the lower part of which is sharpened to form a cutting edge. It has a fixed handle h, and a swivel handle i. One man pushes the bag down into the mud by A, while another pulls it along by the. rope g ; and when filled, another raises it by the rope c, and empties it. If the bag is large, a wind- lass may be used for raising it. The men may work from a scow or raft properly anchored. Or a long- handled metal spoon, shaped like a deeply-dished hoe, may be used by only one man ; or a larger spoon may be guided by -a man, and dragged forward and backward by a horse walking in a circle on the scow, &c., (fee. The w^eig-ht of a cubic yard of wet dredged mud, pure sand, or gravel, averages about 1^^ tons; say 111 !bs. per cubic foot ; muddy gravel, full l]^ tons ; sav 125 ibs. per cubic foot. Pure sand or gravel dredges easily ; also beds of shells. Wet dredged clay will slide down a shoot inclined at from 5 to 1, to 3 to 1, according to its freedom from sand, &c. ; but wet sand or gravel will not slide down even 3 to 1, without a free flow of water to aid it ; otherwise it requires much pushing. * The writer has seen cases in which a circnlar saw for logs in deep water would have been a very useful addition to a dredger. It should be worked by steam ; and be adjustable to different depths. It would cost but about $500. 582 FOUNDATIONS. FOUNDATIONS. A VOLUME might be occupied by this important subject alone. We have space for ©lily a few general hints ; leaving it to the student to determine liow far they may be applicable in any given case. In ordinary cases, as in culverts, retaining walls, ice, if excavations, or wells, &c, in the vicinity, have not already proved that the soil is reliable to a considerable depth, it will usually be a suflBcient precaution, after having dug and levelled off the foundation pita or trenches to a depth of 3 to 5 ft, to test it by an iron rod, or a pump-auger ; or to sink holes, in a few spots, to the depth of 4 to 8 ft farther ; (depending upon the weight of the intended structure ;) to ascer- tain if the soil continues firm to that distance. If it does, there will rarely be any risk in proceeding at once with the masonry; because a stratum of firm soil, from 4 to 8 ft thick, will be safe for almost any ordinary structure ; even though it should be underlaid by a much softer stratum. If, however, the firm upper stratum is ex- posed to running water, as in the case of a bridge-pier in a river, care must be taken to preserve it from gradually washing away; or from becoming loosened and broken up by violent freshets; especially if they bring down heavy masses of ice, trees, and other floating matter. These are sometimes arrested by piers, and accumulate so as to form dams extending to the bottom of the stream ; thus creating an increase of velocity, and of scouring action, that is very dangerous to the stability both of the bottom and of the structure. When the testing has to be made to- a considerable depth, it may be necessary to drive down a tube of either wrought or cast iron, to prevent the soil from falling into the unfinished hole. If necessary, this tube may be in short lengths, connected by screw joints, for convenience of driving ; and the •arth inside of it may be removed by a small scoop with a long handle.* Boring's in common soils or clay may be made 100 feet deep in a day or two by a common wood auger 1}4 inches diameter, turned by Iavo to four men with 3 feet levers. This will bring up samples. in starting the masonry, the largest stones should of course be placed at the bot- tom of the pit, so as to equalize the pressure as much as possibly; and care should be taken to bed them solidly in the soil, so as to have no rocking tendency. The next few courses at least should be of large stones, so laid as to break joint thoroughly with those below. The trenches should be refilled with earth as soon as the masonry will permit; so as to exclude rain, which would injui-e the mortar, and soften the foundation. It is well to ram or tread the earth to some extent as it is being deposited. If the tests show that the soil (not exposed to running water) is too soft to support the masonry, then the pits should be made considerablj' wider and deeper ; and after- ward be filled to their entire width, and to a depth of from 3 to 6 or more ft, (de- pending on the weight to be sustained,) with rammed or rolled layers of sand, gravel, or stone broken to turnpike size ; or with concrete in which there is a good propor- tion of cement. On this deposit the masonry may be started. The common practice in such cases, of laying planks or wooden platforms in the foundations, for building upon, is a very bad one. For if the planks are not constantly kept thoroughly wet^ they will decay in a few years ; causing cracks and settlements in the masonry. Some portions of the brick aqueduct f for supplying Boston with water gave a great deal of trouble where its trenches passed through running quicksands and other treacherous soils. Concrete was tried, but the wet quicksand mixed itself with it, and killed it. Wooden cradles, &c, also failed ; and the diflficulty was finally overcome by simply depositing in the trenches about two feet in depth of strong gravel.^ Sand or gravel, when prevented from spreading sideways, forms one of the best of foundations. To prevent this spreading, the area to be built on may be sur- rounded by a wall ; or by squared piles driven so close as to touch each other ; or in less important cases, by short sheet piles only. But generally it is suflBcient simply * Subterranean caverns in limestone regions are a frequent source of trouble, against which it is difficult to adopt precautions. f The Cochituate aqueduct, built 1846-48 ; egg-shape, 6 feet 4 inches X 5 feet, with semicircular invert. X Smeaton mentions a stone bridge built upon a natural bed of gravel only about two feet thick, overlying deep mud so soft that an iron bar 40 feet long sank to the head by its own weight. C)ne of the piers, however, sank while the arches were being turned, and was restored by Smeaton. Although a wretched precedent for bridge-building, this example illustrates the bearing power of a thick layer of well-compacted gravel. FOUNDATIONS. 583 to give the trenches a good width ; and to ram the sand or gravel (which are all the better if wet) in layers; taking care to compact it well against the sides of the trench also. Under heavy loads, some settlenaent will, of course, take place, as is the case in all foundations except rock. If very heavy, adopt piling, &c. See Grillage. Wtien an unreliable soil overlies a firm one, but at such a depth that the excavation of the trenches (Which then must evidently be made wider, as well as deeper) becomes too troublesome and expensive; especially when (as generally happens in that case) water percolates rapidly into the trenches from the adjacent strata, we may resort to piles. When making deep foundation-pits in damp clay, we must remember that this material, being soft, has, to a certain degree, a tendencv to press in every direc- tion, like .water. This causes it to bulge inward at the sides, and upward at the bottom. The excavations for tunnels,' or for vertical shafts, often close in all around, and become much contracted thereby be fore thev can be lined ; there- fore they should be dug larger than would otherwise be nec.essary. Tlie bottoms of canal and railroad excavations in moist clay are frequently pressed upward by the weight o( the sides. I>ry clay rapidly absorbs moisture from the air, and swells, producing effects similar to the foregoing. Its expansion is attended by great pressure ; so that retaining-walls backed with dry rammed clay will be in ^danger of bulging if the clay should become" wet. It is a treacherous material to work in. For concrete foundations, see pp. 946, &c. As to the g-rentest load that may safely be trusted on an earth founda- tion, Kankine advises not to exceed 1 to 1.5 tons per square foot. But experi- ence proves that on good compact gravel, sand, or loam, at a depth beyond atmospheric influences, 2 to 3 tons are safe, or even 4 to n tons if a few inches of settlement may be allowed, as is often the case in isolated structures without tremors. Years may elapse before this settlement ceases entirely. Pure clay, especially if damp, is more compressible, and should not be trusted with more than 1 to 2.5 tons, according to the case. All earth foundations must yield some- what. Equality of pressure is a main point to aim at. Tremor in- creases settlements, and causes them to continue for a longer period, especially in weak soils. Great care must be taken not to overload in such cases, even if piled. Foundations in silty soils will probably settle, in years, at the rate of from 3 to 12 inches per ton (up to 2 tons) per square foot of quiet load, if not on piles. Figure 2 shows an easy mode of obtaining a foundation in certain cases. It is the *'pierre perdue" (lost stone) of the French; in English, "ran- dom stone," or rip-rap. It is merely a deposit of rough angular quarry stone thrown into the water ; the largest ones being at the outside, to resist disturbance from freshets, ice, floating trees, &c. A part of the interior may be of small quarry chips, with some gravel, sand, clay, &c. When the bottom is irregular rock, this process saves the expense of'levelling it off to receive the masonry. For 2 or 3 feet below the surface of the water, the stones may generally be disposed by hand, so as to lie close and firmly. Small spawls packed between the larger ones will make the work smoother, and less liable to be displaced by violence. Cramps or chains may at times be useful for connecting several of the large stones together for greater stability. Rip-rap, however, is apt to settle. If ttie bottom is so yielding as to be liable to wash away in freshets, it may, in addition, be protected, as in Fig 2, by a covering of the same kind of stones, as at c : extend- ing all around the struc- ture. Or the main pile of stones may be extend- ed as per dotted line at d; BO that if the bottom should wash away, as pef dotted line at o, the stones d will fall int« the cavity, and thus pre- vent further damage. Sheet-piles, s s, may be jlriven as an additional precaution. For greater security, the bed of the river may be dredged or scooped under the entire space to be covered by the main deposit as oer dotted lines in Fig 3, to as great a depth as any scouring would be apt to reach ; 584 FOUNDATIONS. this excavation also to be filled with stone. Such foundations are evidently bttl adapted to quiet water. The masonry should rest on a strong platform. Large depoBltn of stone, as In these two figs, greatly increase the velocity, and the scoarlng actJou of the stream around them, especially In freshets ; un- less the bottom on each side from the de- t>OBlt be dredged oat t«> such an extent that the original area of water shall not be reduced. If the bottom isi treacherous, ttls should be done before depositing the isoveriog stones c, Fig 2. Judgment and Experience are necessary In such matters, lis in all others connected with engineer- ing. Mere study will not guard against •Oustant failures. Theory and practice aunt guide each other. Fig .3 is another simple method; and Vfaen It does not create too great an ob- struction to the navigation of the stream, or to the escape of its waters in time of high flresbeta, Is a very effective one. Here lh« piles are first driven into the river bottom, for the support of the pier; then the deposit of stone is thrown in, for the support and protection of the piles; preventing them from bending under their loads; and shielding them from blows from floating bodies. The tops of the Diles being cut off to a level, a strong platform of timber Is laid on top of them, as a base for the mas'onrv. The top of the platform should not be less than about 12 or 18 ins below ordinary low water, to prevent decay. Mitchell's Iron screw pile ; or hollow piles of cast iron, may be used Instead of wooden ones. Figs 4 represent a convenient method of establishing a foundation in water, by means of a timber crib, A A« without a bottom. It should be built of squared timbers, notch- ed together at their crossings, as shown at Fig 5 ; each notch being ^ of the depth of the Btick. By this means each timber is support- ed throughout its entire length by the one below it; and resists pulling in both directions. Bolts also are driven at the intersections; at least in the sides of the crib, to prevent one portion trom being floated off from the other. The crib is thus divided into square or rectangular cells, ft-om 2 to 4 or 6 ffe on a side, according to the requirements of the case. The partitions between the cells are put together in the eame manner as ihose at the sides of the cribs; and consequently, like the latter, form solH wooden wall* The orlb may be ft-araed afloat, at any conveoteut spot; and when finished, may be to«ed to tfm final place, where it is carefully moored In position, and then sunk by throwing stone into a few tells provided with platforms, as at c c. for that purpose. These platforms should be placed a little above the lower edge of the cells, so as not to prevent the crib from settling slightly Into the soil, and thus coming to a full bearing upon the bottom. After it has been sunk, all the cells are filled with rough stone. A stout top platform may be added or not, as the case may be ; also, a protection, 1 1, of random stone, to prevent undermining by the current. If the sides are exposed to abrasion from Ice, Ac, they may be covered in whole or in part with plank, or plate iron ; and the angles strength- ened by iron straps, &c. In deep water, a foundation may be made partly of random stone, as ia Figs 2 and 3 ; and on top of this may be sunk a crib, with Its top about 2 It under low water, as a bas« for the masonry. This Is much safer than random stone alone. On uneven rock bottom it may be necessary to scribe the bottom of the crib to fit the rock; or the crib may first be sunk by means of a loaded platform on its top, or by filling some of its cells, until its lowest timbers are within a short distance above the bottom. Being there kept in a horlrontal position, small stonee may be thrown luto the oells, and allowed to find their way under the timbers of the crib, thus forming a level support for It. The cells may then be ttled; and rip-rap deposited ootside around tbecrlb to prevent the small stones from being diapiaoed. FOUNDATIONS. 685 A crib with only an ontside row of cells for sinking it may fes built i and the ioterior chamber may be filled with concrete under water. The masonry may then rest on the concrete alone. If the crib rests upon a foundation of broken stone, the upper interstices of this stone should first be levelled off by small stone or coarse gravel to receive the concrete of the inner chamber. Or a crib like Fig*. 4 maj' be sunk, and piles be driven in the cells, which may afterward be filled with broken stone or concrete. The masonry may then rest on the piles only, which in turn will be defended by the crib. If the bottoni' is liable to scour, place sheet-piles or rip-rap around the base of the crib. By all means avoid a crib like e, Fig* 5>^, much higher at one part than at another, if the superstructure s is to rest on the timher of the crib instead of ou piles, or on concrete independent of the timber ; for the high part of the crib will compress more under its load than the low part, anfi win thus cause the superstructure to lean or to crack. A crib either straight sided or circular, with only an outer row of cells for pnd- dling; may be used as a cofferdam (see cofferdams, p. 586). The joints between the outer timbers should be well caulked ; and care be taken, by means of outside pile-planks, gravel, &c, to prevent water from entering beueath it. The cast-iron Bridg-e across the Schnylkill at Chestnut St, Phila, Mr. Strickland Kneass, Engineer, aftbrds a striking example of crib foundation. The center pier stands on a crib, an oblong octagon in plan ; 31 by 87 feet at base ; 24 by 80 ft at top ; and (with its platform) 29 ft high. Its timbers are of yellow pine, hewn 12 ins square : and framed as at Fig 5. The lower timbers were carefully cut or scribed to conform to the irregularities of the tolerably level rock upon which it rests. These were ascertained ("after the 8 ft depth of gravel had been dredi?ed off) in the usual manner of mooring above the site a large floating wooden platform, composed of timbers corresponding in position with all those of the lower course of the intended crib, both longitudinal and transverse. Soundings were then taken close together along all these lines of timber. Most of the celH are about 3 by 4 ft on a side, in the clear. A few of them had platforms at the level of the second course from the bottom, for receiving stone for sink, ing the crib ; the others are open to the bottom. The crib was built in the water ; and was kept floating, during its construction, with its unfinished top continually just above water, by gradually loading it with more stone as new timbers were adied. The stone required for this purpose alone was 300 tons. When the crib was towed into position, and moored, 150 tons more were added for sinking it. All the cells were afterward filled with rongh dry stone, and coarse gravel screenings ; making a total of 1666 tons. A platform of 12 by 12 iuch squared timber covered the whole ; its top being 2^ ft below low water. The pier alone, which stands on this crib, weighs 3255 tons; and during its construction it compressed the crib 6^ ins. The weight of superstructure resting on the pier, may be roughly taken at 1000 tons more. An ordinary caisson is merely a strong" scoiv, or a box with- out a lid; and with sides which may at pleasure be readily detached from its bottom. It is built on land, and then launched. The masonry may first be built in it, either in whole or in part, while afloat; and the whole being then towed" into" place, and moored, may be sunk to the bottom of tl^e river, to rest upon a foundation previously prepared for it, either by piling, if necessary; or by merely levelliag off the natural surface, &c. The bottom of the caisson constitutes a strong limber platform, upon which the miisonry rests; and is so arranged, that after it is sunk, the sides may be detached from it, and removed to be rebbttomed for use at another pier, if needed. This detaching may be effected b_v some such contrivance as that shown in Fig 6, where P P m; is the bottom of the caisson , to which are firmly attached at intervals strong iron eyes t; which are taken hold of by hooks d, a* the lower end of long bolts E »i,* reaching to the top timbers S of the crib, where they are confined by screw nuts n. By loosening the nuts n, the hooks d can be detached from the eyes t; and the sides can then be removed from the bottom; there being no other connec- tion between the two. These hooks and eyes are usually placed outside of the caisson: the screw nuts n being sustained by the projecting ends of cross pieces, as tt, Fig 9. The im- proper position given them in our Fig was merely for convenience of illustrating the prin- ciple. It will sometimes be necessary to have one side detachable from the others, in order to float the caisson away clear from the finished pier; unless it be floated away before the masonry has been built so high as to render the precaution use- less. Fig 6 shows one of many ways of constructing a caisson ; with sides consisting of upright corner-posts, I; cap pieces S, on top ; and sills g at bottom, resting on the bottom platform P P w ; intermediate uprights T, framed into the caps and sills; the whole being covered outside by one or two thicknesses of planking B, which, as well as the platform, should be well calked, to prevent leaking. Tarpaulin also may be nailed outside to assist in this. The greatest trouble from leaking is where the sides join the platform. On top of the platform is firmly spiked a timber o o, extending all around it just inside of the inner lower edge of the sides of the caisson. Its use is to prevent the sides from being forced inward by the pressure of the water outside. The details of construction will of course vary with the requirements of the case. In deep caissons, inside cross-braces or struts from side to side, 'as at c c. Fig 7, will be required to prevent the sides from being forced inward by the pressure of the water, as the vessel gradually sinks while the masonry is being built within it. As the masonry is carried up. the struts are removed; and short ones, extending from the sides of the caisson to the masonry, are inserted in their place. When the caisson is shallow, only the upper course of braces will be required, they also support a platform for the workmen and their materials. In deep caissons, in order not to be in the way of the masons, the outer planking of the sides may, in part, be gradually built up as the masonry progresses Ft may sometimes be expedient to build the masonry hollow at first, «"'tb ♦bin trp«sv«rse wails inside to stiffen it if necessary ; and to com- 686 FOUNDATIONS. plete the interior after sinking the caisson. Indeed, masonry or brickwork, in cement, may thus uw built hollow at first, resting on the platform ; the masonry itself forming the sides of the caisson. Or the sides may consist of a water-tight casing of iron, or wood, of the shape of the intended pier, &c. This casing being confined to the platform, becomes, in fact, a mould, in which the pier may be formed, and sunk at the same time by filling it with hydraulic concrete. For concrete foundations, see pp 946 &c. On rock bottom the under timbers of the platform may be cut to suit the irregularities as already itated under " Cribs." Or the bottom may be levelled up by first depositing large stones around the area upon which the caisson is to rest ; and then filling between these with smaller stones andgravai; testing the depth by sounding. Or a level bed of cement concrete may, with care, be deposited in the water. If there are deep narrow crevices in the rock, through which the concrete may escape, they may be first covered with tarpaulin. Diving bells may often^be used to advantage, in all such operations. But in the case of very irregular rock, it will often be better to resort to cof- fer-dams. Valves for the admission of water for sinking the caisson are usually introduced. If, after sinking, it should be necessary to again raise the whole, it is only Accessary to close the valves, and pump out the water. Guide piles may be driven and braced along- side of the caisson, to insure its sinking vertically, and at the proper spot. Or it may be lowered by screws supported by strong temporary framework. Assuming the uprights I, T, &c. Fig 6, to be sufficiently braced, as at cc. Fig 7, the following table will show the thickness of planking necessary for difiFerent distances apart of the uprights, (in the dear,) to insure a safety of six against the pressure of the water at different depths; and at the ■ame time not to bend inward under said pressure, more than xttt P^^^ o^ the distance to which they stretch from upright to upright ; or at the rate of }^ inch in 10 ft stretch ; J^ inch in 5 ft, 4a. Such a table may be of use in other matters. Table of tbickness of white pine plank required not to bend more than -^^-^ part of its clear horizontal stretch, under Clilferent heads of water. (Original.) HEADS IK PEET. Stretch in Ft. 40 30 1 «0 1 10 1 ' Thickness in Inches. 3 SH 3 2H 2H m 4 W2 4 3^ 2% 2M 6 6% 6 5^ 4M 3Ji 8 9 8 7 5>^ ^H 10 ii}4 10 8H 7 634 12 1334 I2}i 10% 8>^ 6% 15 163^ 15 13 10>^ 8H 20 22i4 20 iiH 14 11 Coffer-dams are enclosures from which the water may be pumped out, so aa to allow the work to be done in the open air. Their construction of course varies greatly. In still shallow water, a mere well-built bank of clay and gravel ; or of bags partly filled with those materials when there is "much current, will answer ©very purpose ; or (depending on thfi depth) a single or double row of sheet-piles ; or of squared piles of larger dimensions, driven touching each other; their lower ends a few feet in the soil ; a,nd their upper ones a little above high water, and protected outside by heaps of gravelly soil or puddle, (as at P in Fig 7,) to prevent leaking. The sheet-piles may be of wood; or of cast iron, of a strong form. The sufficiency of a mere bank of irell-packed earth in still ^ater, is shown by the embankments or levees, thrown up in all countries, to pre- vent rivers from overflowing adjacent low lands. The general average of the levees along 700 miles of the Mississippi, is about 6 ft high; only 3 ft wide on top; side- •lopes V^ to 1. In floods the river rises to within a foot or less of their tops ; and frequently bursts through them, doing immense damage. They are entirely too slight. The method of a single row of 12 by 12 inch squared piles, driven in contact with each other, (close pilcs^ a,nd. simply backed by an outer deposit of impervious soil, is very effective; and with the addition of interior cross-braces or struts, like cc, Fig 7, to prevent crushing inward by the outside pressure of the water and puddle when pumped out, has been successfully employed in from 20 to 25 ft depth of water, in which there was not sufficient current to wash away the puddle. The cross-braces are inserted successively, as the water is being pumped out ; beginning, of course, with the upper ones. The ends of these braces may abut on longitudinal timbers, bolted to the piles for the purpose. Another method is a strong: crib, com- posed of uprights framed into caps and sills ; and covered outside with squared timbers or plank, laid touching each other, and well calked ; as in the caisson, Fig 6 ; but without a bottom. Between the opposite pairs of uprights are strong i nterior atruts, as c c, Fig 7, reaching from side to side, to prevent crushing inward. Th« FOUNDATIONS. 687 np'per series of these usually supports a platform for the workmen, windlasses, fte. The crib iiaving been built on land, is launched, taken to its final place, and sunk bj piling stones on a temporary platform resting on the cross-struts ; the bottom of the stream having been previously levelled off, if necessary, for its reception. To prevent leaking under the bottom of the crib, sheet-piles may be driven around it, their heads extending a few feet above its bottom ; or a small deposit of outside puddle may be placed around it, as shown at the stone deposits 1 1, Fig 4. Or a broad flap of tarpaulin may be closely nailed around and a little above the lower edge of the crib ; so arranged that it may be spread out loosely on the river bottom, to a width of a few feet all around the outside of the crib ; and the puddle may be placed upon it. Such a tarpaulin is also very useful in case the river bottom is somewhat irregular, and cannot be levelled off without too great expense ; in which case the crib cannot come to a full bearing apon it; and consequently the water would leak or flow beneath freely. It is especially adapted to uneven rock ; where sheet-piles cannot be driven. An artificial stratum of impervious soil may, how- ever, be deposited on bare roclc ; in which case the sinking of the crib, and the subsequent operations •»ill be the same as on a natural stratum. These expedients are evidently more or less applicable ia other oases, where, to avoid repetition, they are not specially mentioned. Plan a ( one end. Flgr 7 is another crib coflTer-dam : in which the sides, instead of being planked longitudinally, as in the last instance, are sheathed with vortical sheet-piles Sy driven after the crib is sunk. It is much inferior to the last, owing to its greater liability to leak. In one of this description. Fig 7, successfully used in 16 ft water, the dimensions of the crib were 34 ft by 80 ft. Along each long side were 7 uprights t, t, 19 ft long, 12 ins square, 12^ ft apart. Into each opposite pair of these were notched, and held by dog-irons, 6 cross-braces c c, of 12 ins square. The distance between the two upper ones was 3 ft in the clear ; gradually diminishing to 18 ins between the two lower ones, on account of the increased pressure of the water in descending. On the outside of the uprights, and opposite the ends of the braces, were bolted longi* SECTION ludinal timbers to support the out«ide pressure against the 3-inch sheet-piling s». Other longitudinal pieces o o, confine the heads of the sheet-piles to the top of the crib after they are driven. The feet of the sheet-piles were cut to an angle, as at m ; to make them draw close to each other at bottom in driving. The sheet-piles will drive in a far more regular and satisfactory manner, with the arrangement shown in Figs 8. Hereooaretheuprigbts; ccarepairsof longitudinal 588 FOUNDATIONS. pieces, notched and bolted to the uprights, near both their tops and their feet ; and at as many intermediate points as may be desired. The sheet-piles I, are inserted between these ; and of course are guided during their descent much more perfectly than in Pig 7. When the current is too strong to permit the use of outside puddle, P, Fig 7, th« principle of coffer-dam shown in Fig 9, is generally used ; in which both sides of the puddle are protected from washing away. The space to be enclosed by the dam is sur- rounded by two rows of firmly-driven main piles p p, on which the strength chiefly depends. They may be round. In deciding upon their number, it must be remem- bered that they may have to resist floating ice, or accidental blows from vessels, &c. With reference to this, extra /ewder-piles may be driven. A little below the tops of the main piles are bolted two outside longitudinal pieces w w, called wales ; and oppo- site to them two inner ones, as in the fig. The outer ones serve to support cross- timbers 1 1, which unite each pair of opposite piles, and steady them ; and prevent their spreading apart by the pressure of the puddle P. The inner ones act as guides for the sheet-piles s s, while being driven ; after which the heads of the sheet-piles are spiked to them. In deep water these sheet-piles must be very stout, say 12 ins square ; to resist the pressure of the compacted puddle. A g'ang'way m, is often laid on top of the cross-pieces 1 1, for the use of the workmen in wheeling materials, &c. The puddle P is deposited in the water in the space, or boxing, between the sheet-piles. It shonld be put in in layers, and com- pacted as well as can be done without causing the sheet-piles to bulge, and thus open their joints. The bottom of the puddle-ditch should be deepened, as in the fig, in case it consists, as it often does, of loose porous material which would allow water to leak in beneath it and the sheet-piles. This leaking under the dam is frequently a source of much trouble and expense. Water will find its way readily through almost any depth and distance of clean coarse gravelly and pebbly bottom, unmixed with earth. Sand is also troublesome ; and if a stratum of either should present itself ex- tending to a great depth, it will generally be exped'cat to resort to either simple cribs, Fig 4; or to caissons; with or without piles in either case, according to cir- cumstances. But if such open gravel, or any other permeable or shifting material, as soft mud, quicksand, &c, is present in a stratum but a few feet in thickness, and underlaid by stiff clay, or other safe material, leaking* may be prevented, or at least much reduced, by driving the sheeting-piles 2 or 3 ft into this last ; and by deepening the puddle-trench to the same extent. It may sometimes be better, and more con- venient, to dredge away the bad material entirely from all the space to be enclosed by the dam, and for a short distance beyond, before commencing the construction of the latter. If the dam. Fig 9, is (as it should be) well provided with cross-braces. like c c. Fig 7, extending across the enclosed area, the thickness or width o o of the puddle, need not be more than 4 or 5 feet for shallow depths ; or than 5 to 1 ft for great ones ; because its use is then merely to prevent leaking. But if there are no braces, it must be made wider, so as to resist upsetting bodily; and then, with good puddle, o o may, as a rule of thumb, be ^ of the vertical depth o I below high water; except when this gives less than 4 ft ; in which case make it 4 ft ; unless more should be required for the use of the workmen, for depositing materials, kc. Or if the excavation for the masonry is sunk deeper than the puddle, the dam must be wider; elae it may be upset into the excavated pit. PLAN The oxcavated soil may be raised in buckets by windlasses, or by hand, in Bucoessive stages. The pumps may oe worked by hand, or by steam, as the case may require; as also the windlasses generally needed for lowering mortar, stone, &c. More or less leak- ing may always be anticipated, notwithstanding every precaution. Where a coffer-dam is exposed to a violent current, and great danger from ice, &c, the ex- pensive mode shown in Figs 10 may become necessary. The two black rectangles c c, repre- sent two" lines of rough cribs filled with stone, and sunk in position ; one row being enclosed by the other ; with a space several feet wide be- tween them. Sheet-piles p p are then driven around the opposite faces of the two rows of cribs ; and the puddle is deposited within the boxing thus provided for it, as shown in the fig. Where the current is not strong enough to wash away gravel backing, we may. on rock especially, enclose the space to be built on, by a single quadrangle of cribs sunk by stone ; and after adopting precautions to prevent the gravel from being pressed in beneath the cribs, apply the backing.* Figs IQi^ show the plan, outside view, and transverse section, to a scale of 20 ft to an inch, of a coffer-dam on rock, in 8 to 9 ft water, used successfully on the Schuylkill Navigation. » A pure clean coarse gravel is entirely, unfit for such purpose*. A considerable proportion •! MMrth Is essential for preventing leaks. FOUNDATIONS. 689 CoflfeF-dnni on I*OCl<« Uprights b, about 1 ft square, and 10 ft apart from center to centw along the sides of the dam ; and 10 ft in the clear, transversely of the dam, support two lines of hori- zontal stringers, i i ; inside of which are the two lines of sheeting-piles, s s, enclosing between them a width of 7 ft of gravel puddle. Two flat iron bars (t t, of the transverse section) tie together eack pair of uprights b b. These bars are }4 inch thick, by '2}4 insdeep, and 9 ft long. Their hooked endi fit iuto eye-bolts c, which pass through the uprights b ; outside of which they are fastened by keys, *, (■ee detail sketch.) Between the keys and b, were washers. At the corners of the dam (see plan) were additional tie-bars, as shown. A small band of straw, as seen at y, wrapped around the tie- bars just inside of the sheet-piles ; and kept in place by the puddle ; effectually prevented the leaking which generally proves so troublesome in such cases. The stout oblique braces, o o, were merely- spiked to the outside faces of the uprights 6. They are not shown inthe transverse section. This dam was built on shore; in sections 30 to 40 ft long. These were floated into place, and weighted down, •heet-piled, and puddled with gravel. The dam had sluices by which water was admitted when necessary for preventing the outside head from exceeding 9 ft. The lengths of the uprights 6 6 were irat found by careful soundings. b Id OUTSIDE, PLAN The mooring;- of lar^e caissons or cribs, preparatory to sinking them, is sometimes troublesome, especially in strong currents. It may be necea- sary to drive clumps of piles; or to temporarily sink rough cribs filled with stone, ^o which to attach the long guide-ropes by which the manoeuvring into position, &c, js done. Frequently dams are left standing after the work is done ; if not in the way of navigation, or otherwise objectionable ; inasmuch as the materials are rarely worth the expense of removal. But if removed, the piles should not be drawn out of the ground ; but be cut off close to river bottom ; for if drawn, the water entering their holes may soften the soil under the masonry. It is often expedient to drive two rows of piles from the dam to the shore, for supporting a gangway for the workmen; or even for horses and carts ; or for a railway for the easy delivery of large stones, &c. Coffer-dams may be sunk tbrou^h a soft to a firm soil, in shape of a box of cribwork, either rectangular or circular, and without a bottom. This being strongly put together, and provided with proper temporary internal bracing, (to be gradually removed as the masonry is built up,) is floated into place; and after being loaded so as to rest on the soft bottom, is sunk by dredging out the soft material from ins.ide. Additional loading will sometimes be required for over- coming the friction of the soil against the outside ; or it may even become necessary to dredge away some of the outer material also. On rock it may at times be expedient to drill holes in deep water, for receiving the ends of piles, or of iron rods, Ac. This may be done by means of long drill-rods, working in an iron tube or pip© sunk as a guide to the rod; with its lower end over the spot to be bored. Or a diving- bell may be used. Or a cylin^ to 4 ft apart each way, from center to center, depending on the char- acter of the soil, and the weight to be sustained. A treaci-wlieel is more economical than the winch for raising the hammer, when this is done by men. Morin found that the work performed by men working 8 hours per day, was 3900 foot-pounds per man, per minute by the tread-wheel; and only 2600 by a winch. After piles have been driven, and their heads carefully sawed off to a level, if not under water, the spaces between them are in important cases filled up level with their tops with well rammed gravel, stone spawls, or concrete, in order to impart some sustaining power to the soil between the piles. Two courses of stout timbers (from 8 to 12 ins square, according to the weight to be carried) are then bolted or treen ailed to the tops of the piles and to each other, as shown in the Fig, forming what is called a grillage. On top of these is bolted a floor or platform of thick plank for the support of the masonry ; or the timbers of the upper course of the grillage may be laid close together to form the floor. The space below the floor should also, in important cases, be well packed with gravel, spawls, or concrete. If under water, the piles are sawed oflf by a diver, or by a circular saw driven by the engine of the pile-driver, and the grillage is omitted. Instead of it the masonry or concrete may be built in the open air in a caisson, which gradually sinks as it becomes filled; or on a strong platform which is lowered upon the piles by screws as the work progresses. Or a strong caisson may first be sunk entirely under water, and then be filled with concrete, up to near low water; the caisson being allowed to remain. Or the cnisson may form a cofferdam, to be first sunk, and then pumped out. If the ground is liable FOUNDATIONS. 591 to wash away from around the piles, as in the case of bridge piers, &c, defend it by sheet-piles, or rip- rap, or both. The cost of a floating- steam pile driver, scow 24 ft by 50 ft, draft 18 ins, with one engine for driving, and one (to ssve time) for getting another pile ready ; with one ton hammer, is about ^6000 ; and $^500 more will add a cir- cular saw, &c, fur sawing off piles at any reqd depth. Requires engineman, cook, and 4 or 5 others. Will burn about half a ton of coal per day. Driving 20 feet into gravel, and sawing off, will average from 15 to 20 piles per day of 10 hours. In mud about twice as many. On land about half as many as in water. In the g-unpowder pile driver invented by the late Mr. Thomas Shaw, of Philadelphia, the hammer is worked by small cartridges of powdei', placed one by one in a receptacle on top of the pile ; and exploded by the hammer itself. It can readily make 30 to 40 blows of 5 to 10 ft per minute; and, since the hammer does not come into actual contact with the piles, it does not injure their heads at all; thus dispensing with iron hoops, &c, for preserving them. When only a slight blow is required, a smaller cartridge is used. To drive a pile 20 ft into mud averages about one-third of a pound of powder ; into gravel, 4 times as much. This machine does not assist in raising the pile, and placing it in position, as is done by ordinary steam pile drivers ; the latter, however, average but from 6 to 14 blows per minute. Piles have been driven by exploding small charges of dynamite laid upon their heads, which are protected by iron plates. Steam-hammer pile drivers, operating on the principle of that devised by Nasmyth about 1850, are economical in driving to great depths in difficult- soils where there are say 200 or more piles in clusters or rows, so that the machine can readily be moved from pile to pile. The steam cylinder is upright, and is confined between the upper ends of two vertical and parallel I or channel beams about 6 to 12 ft long and 18 ins apart, the lower ends of which confine between tbem a hollow conical *^ bonnet cast* ing"," which fits over the head of the pile. This casting is open at top, and through it the hammer, which is fastened to the foot of the piston-rod, strikes the head of the pile. Each of the vertical beams encloses one of the two upright guide-timbers, or "leaders," of the pile driver, between which the driving apparatus, above do. scribed, is free to slide up or down as a whole. When a pile has been placed in position, ready for driving, the bonnet casting is placed upon its head, thus bringing the weight of the beams, cylinder, hammer, and casting upon the pile. This weight rests upon the pile throughout the driving, the apparatus sliding down between the leaders as the pile descends. The steam is convened from the boiler to the cyl by a flexible pipe. When it ia admitted to the cyl, the hammer is lifted about 30 or 40 ins, and upon its escape the hanamer falls, striking the head of the pile. About 60 blows are delivered per min- ute. The hammer is provided with a trip-piece which automatically admits steam to the cylinder after each blow, and opens a valve for its escape at the end of the up-stroke. By altering the adjustment of this trip-piece, the length of stroke (and thus the force of the blows) can be increased or diminished. The admission and escape of steam, to and from the cyl, can also be controlled directly by the attendant. The number of blows per minute is increased or diminished by regulating the sup' ply of steam. In making the up-stroke, the steam, pressing against the lower cyl liead, of course presses downward on the pile and aids its descent. The chief advantage of these machines lies in the great rapidity with which the blows follow one another, allowing no time for the disturbed earth, sand, &c, to recompact itself around the sides, and under the foot, of the pile. This enables the machines to do work which cannot be done with ordinary pile drivers. They have driven Norway pine piles 42 ft into sand. They are less liable than others to split and broom the pile, so that these may i>e of softer and cheaper wood. The bonnet casting keeps the head of the pile constantly in place, so that the piles do not " dodge " or get out of line. Their heads have, in some cases, been set on fire by the rapidly succeeding blows. These machines consume from 1 to 2 tons of coal in 10 hours, and require a crew of 5 men. They work with a boiler pressure of from 50 to 75 lbs per sq inch. 592 FOUNDATIONS. Rules for the Susiaintng- Power of Piles. ' They diSe: very much. No rule can apply correctly to all conditions. The ground itself bctweei the piles, in most cases, supports a part of the load ; although the whole of it is usually assigned to the piles. Again, in very clayey soils, there is greater liability to sink somewhat with the lapse of time, in consequence of the admission of water between the pile and the clay ; thus diminishing tb» friction between them. The less firm the soil, the more will the piles be affocted by tremors ; whie^ also tend in time to cause siuking. In some cases this sinking will not be that of the piles settling deeper into the earth around them ; but that of the entire compacted mass of piles and earth int» which they were driven, settling down into the less dense mass below them. Piles are sometimes blamed for settlements which are really due to the crushing (flatways) of the timbers which rest immediately upon their heads. lu the fine LiOiiflon bridg'e across the Thames, each pile under some of the piers sustains the very heavy load of 80 tons. They are driven but 20 feet into the stiff, blue London clay ; and are placed nearly 4 ft apart from center to center ; which is too much for such piers and arches. At 3 ft apart scant, they would have had but 45 tons to sustain. They are 1 ft in diam at the middle of their length. Ugly set- tlements, some of them to the extent of about a ft, have occurred under these piers. Blaekfriars bridgpe, in the same vicinity, exhibits the same defect. By some this is ascribed in both cases to the gradual admission of vrater between the clay and the piles, perhaps by capillary action of the piles themselves ; or perhaps by direct leaking. It may, however, be owing in part to the crushing of the platforms on top of the piles ; or to a bodily settlement of the entire mass of piled clay, into the unpiled clay beneath, under the immense load that rests upon it. This here amounts to 5}^ tons per sq foot of area covered by a pier ; and is probably too much to trust upon damp clay, when even the slightest sinking is prejudicial. maj J. Sanders, U. S. Engs, experimented largely at Fort Delaware in river mud ; and gave the following in the Jour. Franklin Inst, Nov 1851. For the safe load for a common wooden pile, driven until it sinks through only small and nearly equal distances, under successive blows, divide the height of the fall in ins, by the small sinking at each blow in ins. Mult the quot by the weight of the hammer, ram, or monkey, in tons or pounds, as the case may be. Divide the prod by 8. He does not state any specific coefficient of safety. Example. At the CiieBtnnt St Bridce« Philada, the greatest weight on any pile is 18 tons. Mr Kneass had the piles driven until they sank ^, or .73 of an inch under each blow from a 1200 ft hammer, falling 20 ft. Was he safe in doing so 7 Here we have the fall in ins = 20 X 12 = 240. And 240 384000 -rg- = 320 ; and 320 X 1200 = 384000 B)s ; and — g = 48000 fts, = 21.4 tons safe load by Maj San- ders' rule. The soil was river mud. Our own rule is as follows. Mult together the cube rt of the fall in ft ; the wt of hammer in lbs ; and the decimal .023. Divide the prod by the last sinking in ins. -\- 1. The quotient will be the extreme load that will be just at the point of causing more sinking. For the safe load take from ene twelfth to one half of this, according to circumstances. Or, as a formula, Cube rt of y Wt of hammer ^ q.,. Extreme load _ fall in feet ^ in pounds ^ ' ' in tons "" Last sinking in inches -f- 1 Example. The same as the foregoing at Chestnut St Bridge. Here the cube rt of 20 ft fan is S.714 ft. Hence we have Extreme load = 2.714 X laoo X .023 ^ U^^ ^^^ ^^^^ in tons .75 + 1 1.75 Or say half of this, or 21.4 tons, the load for a safety of 2. Major Sanders* rule makes the safe load 21.4. The actual one is 18 tons. A safety of 2 is not enough for river mud. But although Major Sanders' rule and our own agree very well in this instance if a safety of 2 fee taken for each, they differ widely in some others. Thus at Neullly Qrldse* France, Perronet's heaviest hammer weighed 2000 fts, fall 5 ft, sinkage .25 of an inch in the last 16 blows ; or say .016 Inch per blow. The piles sustain 47 tons each. Our rule gives 38.8 tons for a safety of 2 ; while San- ders' rule gives 515 tons safe load ! If, as we think probable, there was no actual sinking at the last blow, then our rule gives 39.3 tons for a safety of 2 ; while Sanders' gives infinity. At the Hull Docks, England, piles 10 ins square, driven 16 ft into alluvial mud. by a 1500 E> ham- mer, falling '24 ft, sank 2 ins per blow at the end of the driving. They sustain at least 20 tfons each, or according to some statements 25 tons. Oar rule gives 33.2 tons for the extreme load ; or 16.6 for f safety of only !. Sanders gives for safety 12.06 tons. As before remarked, 2 is not safety enough loi,' jn.ud. In mud, it is not primarily the piles, but the piled goU that settles, bodily, for years. At the Boyal Border Bridge, Ei>gland, piles were very firmly driven from 30 to 40 ft in s&n^ and gravel, in some ca.ses wet. Pine was first tried, but it split and broomed so badly under the hard driving, that American elm was substituted, with success. They were driven until they sank but .05 inch per blow, under a 1700 ft monkey, falling 16 ft. They support 70 tons each. Our rule gives 47 tons for a safety of 2 ; while Sanders gives 364 tons safe load t Itisthe writer's opinion, however, that the piles did not aciually sink, as was (and always is, in such cases) taken for granted by the observers ; but that they were merely compressed or partially crushed by overdriving. Most of the piles were driven until they sank (?) only an inch under 150 blows ; but we doubt whether they were any safer, or farther in the ground, than when they had re- ceived only one of them ; and consider such extreme precaution wor.se than useless. In gome experiments (1873) at Philada, a trial pile was driven 15 ft into soft river mud, by a 1600 ft hammer ; its last sinking being 18 ins under a fall of 36 ft. Only 5 hours after it was driven It was loaded with 6.5 tons ; which caused a sinking of but a very small fraction of an inch. 0"»- rule FOUNDATIONS. 593 ?ive,'? 6.4 tons as the extreme load. Uuder 9 tons it sank .75 of an inch ; ajd under 15 tons, 5 ft. By Maj Sauders' rule its safe load would be 2.14 tons. A U. S. CJovt trial pile, about 12 ins sq, driven 29 ft through layers of silt, sand, and clay, ham- mer 910 lbs, fall 5 ft, last sinking .375 of an inch, bore 26.6 tons ; but sank slowly under 27.9 tons. Our rule gives 26 tons extreme load. Freneh eiie:lueerg consider a pile safe for a load of 25 tons, when it is driven to the refusal of 1344 as, falling 4 ft ; our rule gives 24.2 tons for safety 2. They estimate the refusal by its not sink- ing more than .4 of an inch under 30 blows. In many important bridges &c they drive untU there is ID "sinking under an 800 tt> hammer, falling 5 ft. Our rule here gives 31.5 tons extreme load ; or 15.7 'or safety 2. As to the proper load for safety, we think that not more than one-half the extreme load given ay our rule should be taken for piles thoroughly driven in^rm soils; nor more than one-sixth when in river mud or marsh ; assuming, as we have hitherto done, that their feet do not rest upon rock. If liable to tremors, take only half these loads. Piles may be made of any required size as regards either length or crosa section, by bolt- ing and fishing together sidewise and lengthwise, a number of squared timbers. Piles with blunt ends. At South Sfeet Bridge, Phila, 1200 stout piles of Nova Scotia spruce with blunt ends were driven 15 to 35 ft, partly in strong eravel, by a common steam pile driver, at a total cost (piles and driving) of S7 to $8 each. At Wilmington Harbor, Cal, Mr. C. B. Sears, . (J. S. Army. (Jour. Am. Soc. C. E., Dec 1876) found that in firm compact wet sand, after the first few blows the piles would not penetrate more than .5 to 1.5 ins at a blow, no matter how far the 2400 ft hammer fell. The unpointed ones of which there were many thousands, drove quite as readilv to aver- age depths of 15 ft in this sand as the pointed ones, and with much less tendency to cant, 'as a high Fall had no farther effect than to batter the heads he reduced it to 10 ft. which drove an average of about .72 inch to a blow. To insure straight driving, the ends must be at right angles to the length. Instead of driving piles to moderate depths it mav at times be better to merel} plant them butt lown in holes bored by an auger like Pierce's Well Borer. The ultimate friction of piles even with the bark on, and driven about 3 ft apart fiom cen to cen probably never much exceeds about 1 ton per sq ft even when well driven into dense moist sand or loamy gravel ; nor more than .5 to .75 of a ton in common soils and clays; or than .1 to .2 af a ton in silt or wet river mud dependine on the depth and density. The frletion of cast Iron cylinders seems to be about .3 that of piles. There is a great difference in the penetrability of different lands. Thus, in the Lary bridge, no special difficulty was found in driving piles 35 tt into deep wet land ; while, in other wet localities, piles of very tough wood, well shod with iron. CHnuot be driven 5 ft into sand, without being battered to pieces. The same difference has been found in the case of icrew-piles. At the Brandywine light-house these could not be forced more than 10 ft into the clean wet sand. Stiff wet clay (and clean gravels) also differ very much in this respect. Generally they ire penetrable to any required depth with comparative ease ; but we have seen stout hemlock piles Mattered to pieces in driving 6 ft through wet gravel ; and Mr. Rendel found that at Plymouth he 'could not by any force drive screw-piles more than about 5 ft into the clay, which is not as stiff as the London clay," on which the forementioned new London and Blackfriars "bridges were founded; uad into which even ordinary wooden piles were driven 20 ft without special difficulty. A mixture of mud with the sand or gravel facilitates driving very much ; but before beginning an ixtensive system of piling, a few experimental ones should be driven, to remove doubt as to the rouble and expense that may be anticipated. Mere boring will often be but a poor substitute for this. Afl a general rule, a heavy hammer with a low fall, drives more pleasantly than a light one with a ligh fall. Where a hammer of % ton (1500 fts) falling 25 ft, in a very strong ground, shattered the )iles; one of 2 tons, (4500 fts,) with 7 ft fall, drove them satisfactorily. More blows can be made in he same time with a low fall ; and this gives less time for the soil to compact itself around the piles )etween the blows. At times a pile -may resist the hammer after sinking some distance j but start igain after a short rest; or it may refuse a heavy hammer, and start under a lighter one. It may Irive slowly at first, and more rapidly afterward, from causes that may be difficult to discover. The Iriving of one sometimes causes adjacent ones previously driven, to spring upward several feet. A aile is in the most favorable position when its foot rests upon rock, after its entire length ha.s been Iriven through a firm soil, which affords perfect protection against its bending like an overloaded K»lumn; and at the same time creares great friction against its sides; thus assisting much in sus- laining the load, and thereby relieving the pressure upon the foot. A pile may rest upon rock, and fet be very weak ; for if driven through very soft soil, all the pressure is borne by the sharp point; md the pile becomes merely a column in a worse condition than a pillar with one rounded end. Se« L tg 1. Strength of Iron Pillars. In such soils the piles need very little sharpening; indeed, had better be driven without any ; or even butt end down. The driving of a pile in soft ground or mud will generally cause an adjacent one previously driven, bo lean outwards unless means be taken to prevent it. In piling an area of firm soil it is best to begin at its center and work outwards ; otherwise the soil may become so consolidated that the central ones can scarcely be driven at all. £la*^tic reaction of the soil has been known to cause entire piled areaa to rise, together with the piles, before they were built upon. In very firm soil, especially if stony; or even in soft soil, if the piles are pointed, and are to be driven to rock; their feet should be protected by shoes of either wrought iron, as at a, .e scabbled grranite rubble, such as is generally used ai backing for the foregoing ashlar ; stones averaging about 3^ cub yd each : Cost per liabor at $1 per day* cubjdof masonry. Getting out the stone from the quarry by blasting, allowing J^ for waste in scabbling; li cub yds at $3.00 $3.43 Hauling 1 mile, loading and unloading , 1.20 Mortar ; (2 cub ft, or 1.6 struck bushels quicklime, either in lump or ground ; and 10 cub ft, or 8 struck bushels of sand, or gravel ; and mixing) 1.50 Scabbiing; laying, including scaffold, hoisting machinery, &c 2.50 Neat cost 8.63 Profit to contractor, say 15 per ct 1.30 Total cost .T, 9.93 Common rubble of small stones, the average size being such as two men can handle, costs, to get it out of the quarry, about 80 cts per yard of pile ; or to allow for waste, say $1.00. Hauling 1 mile, $1.00. It can be roughly scabbled, and laid, for $1.20 more ; mortar as foregoing, $1.50. Total neat cost, ^.70 ; or, with 15 per ct profit, $5.40, at ike above wages for labor, W^i th smaller stones, such as one man can handle, we may say, stone 70 cts; hauling $1 i jaying and scaflfold, tools &c, $1; mortar $1.50. Making the neat cost $4.20; or with 15 per ot profit, $4.83. Neat scabbled irregular range- work costs from $2 to $3 more per yd than rubble; accoTding to the charac- ter of the stone &c. The laying of thin walls costs more than that of thick ones, such as abutments &c.'>- The eo«t of plain 8 inch thick ashlar facing^s for dwellings &c in Philada, in 1888, ia about as follows per square foot showing, put up, including everything. Sand- ■tone, $1.50 10 $2.25. Pennsylvania marble, $2.50. New England marble, $2.75 to $3.25. Granite. $2.25 to $2.75. If 6 ins thick, deduct one-eighth part. First ClaSS artificial StOUC could be made and put up at one-third the price. K'orth River blue Stoue flags, 3 ins thick, for footwalks, p«t down, including gravel &c, 70 cts per sq foot. Bclg'ian Street pavement, with gravel, complete, ^3.50 per sq yard in Eastern cities. When dressed ashlar facing is backed by rubble, the expense per cub yard of the entire mass will of course vary according to the proportions of the two. Thus, if ashlar at $12 per yd, is backed by an equal thickness of rubble at $5, the mean cost will be ($12 -f $5) ^ 2 = $8.50 ; or if the rubble is twice as thick as the ashlar then ($12 -f $5 + $5) -> 3 = $7.33, &c. Such compound walls are weak and apt to separate in time, as also walls of cut stone backed by concrete, or by brick ; from unequal settlement of the two parts. At times the contractor must be allowed extra in opening new quarries; in forminc short roads to his work ; in digging foundations ; or for pumping or otherwis* draining them, whe» springs are unexpectedly met with ; for the centers for arches, &c; unless these items are expressly included in the contract per cub yd. RETAINING-WALLS. 603 EETAINING-WALLS. Art. 1. A retaining'-wall is one for sustaining the pressure of earth, sand, or other filling or hacking, deposited behind it after it is built ; in distinction to a face-ivall, which is a similar structure for preventing the fall of earth which is in its undisturbed natural position, but in which a vert or inclined face has been excavated. The earth is then in so consolidated a condition as to exert little or no lateral pres, and therefore the wall may generally be thinner than a retaining one. This, however, will depend upon the nature and position of the strata in which the face is cut. If the strata are of rock, with interposed beds of clay, earth, or sand ; and if they dip or incline toward the ■wall, it may require to be of far greater thickness than any ordinary retaining- wall ; because when the thin seams of earth become softened by infiltrating rain, they act as lubrics, like soap, or tallow, to fa- cilitate the sliding of the rock strata ; and thus bring an enormous pres against the wall. Or the rock may be set in motion by the action of frost upon the clay seams ; or, as sometimes occurs, by the tremor pro- duced by passing trains. Even if "there be no rock, still if the strata of soil dip toward the wall, there will always be danger of a similar result; and addi- tional precautions must be adopted, especially when the strata reach to a much greater height than the wall. A vertical urall has both c o and d a vert. Experience, rather than theo- ry, must be our guide in the building of both kinds of wall. We recommend that the hor thickness a b, Fig 1, at the base of a vert or nearly vert retaining-wall c d b a, which sustains a backing of either sand, gravel, or earth, level with its top c d, as in the fig, should not be less than the following, in railroad practice, when the foundations are not more than about three feet deep. When the backing^ is deposited loosely, as usual, as when, dumped from carts y cars, me of the most prominent writers who give practical rules on this subject. Thus Poncelet, who certainly is at their head, states that his tables, for practical use, give thicknesses of base for sus- taining 1 ^^ times the theoretical pres ; and this he considers amply safe. Yet, for a vert wall of cut granite, his base for sustaining dry sand level with the top, as in Fig 1, is .35 of the vert height; and for brick, .45. But the writer fouud that when not subject to tremor, a wooden model of a vert wall, weighing but 28 lbs per cub ft, aud with a base of .35 of its height, balanced perfectly dry sand ■loping at 114 to 1, and weighing 89 fi»s per cub ft. Now, THK BESISTANCK OF SIMILAR WALLS, OP THE SAME DIMKN8IOWB, ▼AKiES A8 THEIR SPECIFIC GRAVITIES ; and, since granite weighs about 165 lbs per cub foot, or 6 times as much as our model, it follows, we conceive, that a wall of that material, with a base of .35 of its height, must have a resistance of 6 times any true theoretical pres, instead of only 1.8 . times ; and that his brick wall must have about b times the mere bal- ancing resistance. Our experiments were made in an upper room of a •trongly built dwelling ; and we found that the tremor produced by pass- ing vehicles in the street, by the shutting of doors, and walking about the room, sufficed to gradually produce leaning in walls of considerably more than twice the mere balancing stability while quiet; and it appears te us that the injurious effects of a heavy train would be comparatively quite as great upon an actual retaining-wall, supporting so incohesive a material as dry sand. Since, therefore, Poncelet's wall is in this Instance sufficiently stable for practice, it seems to us that his theory, which neglects the effect of tremors, Ac, must be defective. He also gives -i of the height as a suf- ficiently safe thickness for a vert granite wall supporting stiff earth; but ■we suspect that very few engineers would be willing to trust to that pro- portion, when, as usual, the earth is dumped in from carts, or cars ; espe- cially during a rainy period. If deposited, and consolidated in layers, theory could scarcely assign any thickness for the wall ; for the backing thus becomes, as it were, a mass of unburnt brick, exerting no hor thrust; and requiring nothing but protection from atmospberie influence, to insure its stability without any retaining-wa.\l. It is with great diffidence, and distrust in our opinions, that we venture to express doubts respecting the assumptions of so profound an in- vestigator and writer as Poncelet; and we do so only with the hope that the views of more compe- tent persons than ourselves, may be thereby elicited. Our own have no better foundation than ex- periments with wooden and brick models, by ourselves ; combined with observation of actual walls. Art. 3. After a wall ab c o, Fig 3, with a vert back, has been proportioned by our rule in Art 1, it may be converted into one with an offsetted back, as a t n o. This will present greater resistance to overturning; and yet con- tain no more material. Thus, through the center t of the back, draw any line i n; from n draw n s, vert ; divide i .^ 6.90 36 16.2^ 55 37.8 88 96.8 5 .313 % 2.63 24 7.20 37 17.1 56 39.2 90 101.3 ^ .378 15 2.81 }4 7.50 38 18.1 57 40.6 92 105.8 6 .450 ^ 3.00 25 7.81 39 190 58 42.1 94 110.5 J^ .528 16 3.20 J4 8.13 40 20.0 59 43.5 96 115.2 7 .613 3.40 26 8.45 41 21.0 60 45.0 98 120.1 3^ .703 17 3.61 ^A 8.78 42 22.1 62 48.1 100 125.0 8 .800 34 3.83 27 9.12 43 23.1 64 51.2 102 130.1 J^ .903 18 4.05 H 9 45 44 24.2 66 54.5 104 135.2 9 1.01 H 4.28 28 9.80 45 25.3 68 57.8 106 140.5 M 1.13 19 4.51 >^ 102 46 26.5 70 61.3 10 1.25 H 4.75 29 10.5 47 27.6 72 64.8 STONE BRIDGES. Art. 1. In an arch sts, Fig 1, the dist eo is called its span; ia its rise; t its crown ; its lower boundary line, eao, its soffit, or intrados ; the upper one, Vtr, its back, or extrados. The terms soflBt and back are also applied to the entire lower and upper curved surfaces of the whole arch. The ends of an arch, or the showing areas comprised between its intrados and extrados, are its faces ; thus the area sis a is a face. The inclined surfaces or joints, re, ro, upon which the/ee< of the arch rest, or from which the arch springs, are the skei¥backs. Lines level with e and o, at right angles to the faces of the arch, and forming the lower edges of its feet, (see nn. Fig 2^,) are the springing- lines, or spring's. The blocks of which the arch itself is composed, are the areb-stones, or voussoirs. The center one, ta, is the keystone; and the lowest ones, ss, the spring'ers. The term archblock might be substituted for voussoir, and like it would apply to brick or other material, "as well as to stone. The parts tr, tr, are the haunches ; and the spaces tr I, trb, above these, are the spandrels. The material deposited in these spaces is the spandrel filling^ ; it is sometimes earth, sometimes ma- sonry ; or partly of each, as in Fig 1. In large arches, it often consists of several parallel spandrel-walls, tl. Fig 23^, running lengthwise of the roadway, or astraddle of the arch. They are covered at top either by small arches from wall to ■wall, or by flat stones, for supporting the material of the roadway. They are also at times connected together by vert cross-walls at intervals, for steadying them laterally, as at tt, Fig 23^. The parts gpen, gpon, Fig 1, are the abutments of the arch; en, on, the faces; gp, gp, the backs; and pn, pn, the bases of the abuts. The bases are usually widened by feet, steps, or offsets, d d, for dis- tributing the wt of the bridge over a greater area of foundation ; thus diminishing the danger of set- tlement. The distance t a in any arch-stone, is called its depth. The only arches in common use for bridges, are the circular, (often called segmental); and the elliptic. Art. 2. To find the depth of keystone for first-class cut -stone arches, whether cir- cular or elliptic."^ Find the rad c o. Fig 1, which will touch the arch at o, a, and c. Add together this rad, and half the span o t. Take the sq rt of the sum. Div this sq rt by 4. To the quot add 5^^ of a ft. Or by formula, Fitfl * Inasmuch as the rules which we give for arches and abuts are entirely original and novel, it may not be amiss to state that they are not altogether empirical ; but are based upon accurate drawinji 614 STONE BRIDGES. Depth of key _ V Rad -\- half span i q 9 fg^i in feet ~ 4 -r -y For second-class work, this depth may be increased about 3>^th part; or for brick or fair rubble, about J^rd. See table of Keystones. In large arches it is advisable to increase the depth of the archstones toward the springs ; but when the span is as small as about 60 to 80 or 100 feet, this is not at all necessary if the stone is good ; although the arch will be stronger if it is done. Iq practice this increase, even in the largest spans, does not exceed from }^ to 3^ the depth of the key ; although theory would require much more in arches of great rise. Rem. To find tlie rad c o, whether the arch be circular or elliptic. Square half the span e 0. Square the whole rise i a. Add these squares together; div the sum by twice the rise i a. Or it may be found near enough for this purpose by the dividers, from a small arch drawn to a scale. Amount of pressure sustained by arclistones. In bridges of the same width of roadway ; if all the other parts bore to each other the same propor- tion as the spans, the total pres would increase as the squares of the spans, while the' pressure per square foot would increase as the spans. But in practice the depth of the archstones increases much less rapidly than the span ; while the thickness of the roadway material, and the extraneous load per sq ft, remain the same for all spans. Hence the total pressures, at key and at spring, increase less rapidly than the squares of the spans ; but more rapidly than the simple spavs ; as do also the pressures per square foot. Thus in two bridges of the same width, but with spans of 100 and 200 ft, with depths of archstones taken from our table and uniform from key to spring; supposed to be filled up solid with masonry of 160 lbs per cub ft, to a level of about 15 Inches above the crown, (including the stone paving of the roadway); with au extraneous load of 100 lbs per sq ft; the pressures will be approximately as fol- lows: Span 100 ft. 1 Span 200 ft. AT KEY. AT SPRING. II AT KEY. 1 AT SPRING. For 1 ft in width of its entire depth. Per sq ft. For 1 ft in width of its entire depth. Per sq ft. For 1 ft in width of its entire depth. Per sq ft. For 1 ft in width of its entire depth. Per sq ft. Rise. Tons. Tons, Tons. 58 Tons. 18?i Tons. 126 Tons, 29M Tons. 179 Tons. 42 ^ 86% 12^ 57 19 112 '-^7^ 181 44 i 18 11 9 57« 67>^ 20 22>i 25 97 80M 57^ 24}^ 21 15}^ ]88 207 230 47^ 54>^ 61}^ It will be seen that with the same span, the pres at the key becomes less, while that at the spring becomes greater, as the rise increases. Also that when the archstones are of uniform deptii, the pres at either spring of a semicircular arch is about 4 times as great as at the key ; whereas when the rise is but one-sixth of the span, the pres ac spring averages but about one-third greater than at the key. These proportions vary feomewhat in different spans. The greater pres per sq ft at the springs may be reduced by increasing the depth of the archstones towards the springs. This however is not necessary in moderate spans, inasmuch as good stone will be safe even under this greater pres. By usini^ parallel spandrel ivalls, see Fig 2^, or by partly fill- ing with earth instead of masonry, the pres on the archstones may be diminished, say, as a rough average, about \ part. and calculations made by the writer, of lines of pres, &c, of arches from 1 to 300 ft span, and of every rise, from a semicircle to j^^ of the span. Prom these drawings he endeavored to find proportions which, although they might not endure the test of strict criticism, would 3till apply to all the cases •ritli an accuracy sufiBcient for ordinary practical purposes. STONE BRIDGES. 615 Table 1. Of some existing' arches, -with both their actual and their calculated depths (by our rule) of keystone. Where two depths are given in the column of keys the smallest is for first class cut-stone, and the largest for good rubble, or brick. Those also which are not specified are of first-class cut-stone. C stands for circular, E for elliptic. For 2d class work, add about J^th part; and for brick, or fair rubble, about 34tb. ^ ^ i^ K! &ii rrtrri *^ ^ X«O0C«C fc ■* Tjt ■* -* ■* ,-1 "^ M -W! eo •^ c4 eo CO i4 -^ ^. "5 t'*? oq>ONoo a Si Si rv, -*' O C^' X O t-^ £J S ^' '*' © M «' X 'r. ***vft-*N05pH CO « COCO — < rHCOCOi-l rH r- I>« i-l rH « M rt r- I e^oioio^ S ■* e^ ^4 ^ rH »3j^S * t-t-t- t- S « § * tOta lO in ■* -^ eo n OOKHO O K CKO O OOO O OOO B K O O O HO W U O OOK II *^0 so-g a « ill c : 1 : 1 ! c il : p: : [c H c E 'c 1 c t C J! K £f"^ «. .•2 « So o^ s 3 .2 -S. ' . S: p3 ;« CB o> li fl C3 - ti e c3 J? .2 a c « a • fc S £ &^ - S, -S Eh ^ < . S 5 . -S ??S •S .>; K P3 OS M U Sj ^ i § «! Hj Eh * See Experimental Arch at Souppes, France. t Suines bridge. Some authorities give 2 ft 4 ins as the depth at k 616 STONE BRIDGES. Experimental Arcb at Souppes, France. See Table, Span = about 18 X rise. Span. Eise. Radius of intrados. Depth of arch-stones at spring. at key Width. on faces. betw faces. Meters 37.886 124.30 2.125 85 5 1.10 3.61 1.10 80 3 5 Feet 6.97 280.52 3.61 2.624 11.5 Arch of granite. The centers rested (for four months) on sand in 16 cylinders, 1 ft diameter, 1 ft high, of g'^-inch sheet iron. The unloaded arch settled 15 millimeters (0.59 inch) on striking the centers. The additional settlements under extraneous loads were as follows: Extraneous load. Increase of settlement. Kilograms. Pounds. Millimeters. Inches. Distributed 3670fX) 4975 132600 809000 11000 292000 21 0.3 1.2 0.8 Center 0.012 Distributed 0.047 With the distributed load of 367000 kilog, a load of 4975 kilog, falling 0.3 m (11.8 ins) on key, caused vibrations of 2.8 mm (OJl inch). Annates des Fonts et Chaussees, 1866 Part 2, 1868 Part 2. ^ The ureh on the BotrBBOnvAis Bailwat, is probably the boldest;* and ths GABnt Jomr asch, by Capt, now Gen'l M. O. Meigs, U S Army, the grandest stone one in existence. Pomt-t-Pktdd, in Wales, is a common road bridge, of very rude construction ; with a dangerously steep roadway. It was built entirely of rubble, in mortar, by a common country mason, in 1750 ; and is still in perfect condilion. Only the outer, or showing arch-stones, are 2.5 ft deep ; and that depth is made up of two stones. The inner arch-stones are but 1.5 ft deep ; and but from 6 to 9 inches thick. The stone quar« ried with tolerably fair natural beds; and received little or no dressing in addition. The bridge is a flne example of that ignorance which often passes for boldness. Pont Napoleon carries a railroad across the Seine at Paris. The arches are of the uniform depth of 4 ft, from crown to spring. They are composed chiefly of smaU rough quarry chips, or spawls ; well washed, to free them from dir» and dust; and then thoroughly bedded in good cement; and grouted with the same. It is \n fact an arch of cement-concrete. The Pont de Alma, near it, and built in the same way, has elliptic arches of from 126 to 141 ft span ; with rises of i- the span. Key 4.9 ft. These two bridges, considering th« want of precedent in this kind of construction, on so large a scale, must be regarded as very bold; and as reflecting the highest credit for practical science, upon their engineers, Darcel aud Couche. Some trouble arose from the unequal contraction of the different thicknesses of cement. They shoitf what may be readily accomplished in arches of moderate spans, by means of small stone, and good hydraulic cement when large stone fit for arches is not procurable. In Pont Napoleon the depth of arch is less than our rule gives for second class cut-stone. Art. 3. Tbe keystones for larg^e elliptic arches by the best en- gineers, are generally made about 3^3 part deeper than our rule requires ; or than is considered necessary for circular ones of the same span and rise ; in order to keep the line of pres well within the joints ; although the elliptic arch,with its spandrel filling, has slightly less wt ; and that wt ha» a trifle less leverage than in a circular one ; and consequently it exerts less pres both at the key, and at the skew- back. See London, Gloucester, and Waterloo bridges, in the preceding table. Ebm. Young engineers are apt to affect shallow arch-stones ; but it would be far better to adopt the opposite course ; for not only do deep ones make a more stable structure, but a thin arch is as unsightly an object as too slender a column. Accord- ing to our own taste, arch-stones fully 3/3 deeper than our rule gives for first-class cut stone, are greatly to be preferred when appearance is consulted. Especially when an arch is of rough rubble, which costs about the same whether it is bui/t up as arch, or as spandrel filling, it is mere folly to make the arches shallow. Stability and durability should be the objects aimed at; and when they can be attained even to excess, without increased cost, it is best to do so. * Built like that at Souppes. STONE BRIDGES. 617 Table 2. Depths of keystones for arches of first-class cut stone, by Art 2. For second class add full one-eighth part ; and for superior brick one* fourth to one-third part, if the span exceeds about 15 or 20 ft. Original. Hise, in parts of the span. Span. Feet. i i i 4 i ■ * i A Key. Ft. * Key. Ft. Key. Ft. Key. Ft. Key. Ft. Key. Ft. Key. Ft, 2 .55 .56 .58 .60 .61 .64 .68 4 -.70 .72 .74 .76 .79 .83 .88 6 .81 .83 .86 .89 .92 .97 1.03 8 .91 .93 .96 1.00 1.03 1.09 1.16 10 .99 l.Ol 1.04 1.07 1.11 1.18 1.26 15 1.17 1.19 1.22 1.26 1.30 1.40 1.50 20 1.32 1..35 1.33 1.43 1.48 1.59 1.70 25 1.45 1.48 1.53 1.58 1.64 1.76 1.88 30 1.57 1.60 1.65 1.71 1.78 1.91 2.04 35 1.68 1.70 1.76 1.83 1.90 2.04 2.19 40 1.78 1.81 1.88 1.95 2.03 2.18 2.33 50 1.97 200 2.08 2.16 2.25 2.41 2.58 60 2.14 2.18 2.26 2.35 2.44 2.62 2.80 80 2.44 2.49 2.58 2.68 2.78 2.98 3.18 100 2.70 2.75 2.86 2.97 3.09 3.32 3.55 120 2.94 2.99 3.10 3.22 3.35 3.61 3.88 140 3.16 3.21 3.33 3.46 3.60 3.87 4.15 160 3.36 3.44 3.58 3.72 3.87 4.17 180 3.56 3.63 3.75 3.90 4.06 4.S8 200 3.74 3.81 3.95 4.12 4.29 220 3.91 4.00 4.13 4.30 4.48 240 4.07 4.15 4.30 4.48 260 4.23 4.31 4.47 4.66 280 4.38 4.46 4.63 300 4.53 4.62 4.80 Art. 4. To proportion the abuts for an arch of stone or brick, whether circular or elliptic. (Original.) The writer ventures to oflFer the following rule, in the belief that it will be found to combine the requirements of theory with those of economy and ease of applica- tion, to perhaps as great an extent as is attainable in an endeavor to reduce so com- plicated a subject, to a simple and reliable working rule for prac- tical bridg'e-builflers. This is all that he claims for it. Notwithstanding its sirflplicity. it is the result of much labor on his part. It applies equally to the smallest culvert, and to the largest bridge ; whatever may be the proportions of span and rise ; and to any height of ahut whatever. It applies also to all the usual methods of filling above the arch ; whether with solid masonry to the level vf, Fig 2, of the top of the arch; or entirely with earth ; or partly with each, as represented in the fig; or with parallel spandrel-walls extending to the back of the abut, as in Fig 2}^. Altiiough the stability of an abut cannot remain precisely the same under all these conditions, yet the diff of thickness which would follow from a strict investigation of each par- ticular case, is not suflBcient to warrant us in embarrassing a rule intended for popu- lar use, by a multitude of exceptions and modifications which would defeat the very object for which it was designed. We shall not touch upon the theory of arches, except in the way of incidental allusion to it. Theories for arches, and their abuts, omit all consideration of passing loads ; and consequently are entirely inapplicable in practice when, as is frequently the case, (especially in railroad bridges of moderate spans,) the load bears a large ratio to the wt of the arch itself Hence the theoretical line of thrust bas no place in such cases. Our rule is intended for common practice : and we conceive that no error of practical importance will attend its application to any case whatever ; whether the arch be circular or elliptic. It gives a thickness of abut, which, without any backing of earth behind it. is safe in itself, and in all cases, against the pres, w^hen the bridge is unloaded. Moreover, in very large arches, in which the greatest load likely to come upon them in practice is small in comparison with the wt of the arch itself, and tlie filling above it, our abuts would also be safe from the loaded bridge, without any dependence upon the earth behind them ; but as the arches become less, and consequently the wt of the load becomes greater in proportion to that of the arch, and of the filling above it, we must depend more and more upon the resistance of the earth behind the abuts, in order to avoid the neces- sity of giving the latter an extravagant thickness. It will therefore be understood thrnughnut that, excpt luhen parallel spandrel walls are used, oxir rides svppose that after the bridge is fudshed^ earth will he dfposited behind the abuts,