Mechanics' Pocket Memoranda. 7. ,C)t NfCAL SUPPLY COMPANY DRAWING MATERIALS SCRANTON, i'H'.'i - U’CVA AND NEWYORK / Mechanics’ ) Pocket Memoranda A CONVENIENT ENT 3 ' ALL INTE TBOOK FOITULL jpfyqps INTERESTED IN Mechanical EnAperin^f Steam Engineering, Electrical jyg, Railroad Engineering, Hydraulic - ;ineering, Bridge Engineering, Etc. Eng BY INTERNATIONAL CORRESPONDENCE SCHOOLS SCRANTON, PA. 7 th Edition , 277th Thousand , 18 th Impression Scranton, Pa. INTERNATIONAL TEXTBOOK COMPANY Copyright , 1893 , 1594, 1897 , 1898 r 1899 , 1900, The Colliery Engineer Company Copyright , 1994, Oy International Textbook Company N . Entered at Stationers' Hall , London All rights reserved PRINTED BY International Textbook Co. Scranton, Pa. 3256 PREFACE. £ a o The first edition (2,000 copies) of the pocket- book of which this is the outcome was issued in October, 1893, in the form of a notebook contain- ing 74 printed pages, with about the same number ^of blank pages for memoranda, whence the title Mechanics’ Pocket Memoranda. The little book proved so popular that a new edition ( 10,000 copies ) enlarged to 110 pages was issued 8 months latei*. In June, 1897, the blank pages were discarded, the work was entirely recast and enlarged to 318 pages, and the edition (third) consisted of 25,000 copies. Before printing the fifth edition (March, 1898), a large amount of matter relating especially to Plumbing, Heating, and Ventilation and the Build- ing Trades was taken out, replaced by tables of logarithms, trigonometric functions, etc., together with directions for using them, and other new matter, the result being to confine the work more particularly to the different branches of engineer- ing and mechanics. It has been the aim of the publishers, from the first, to present to the public a handbook of a size convenient to carry in the coat or hip pocket — a pocketbook in reality— which w r ould contain rules, formulas, tables, etc. in most common use by iii IV PREFACE. engineers, together with explanations concerning them and practical examples illustrating their use. We have not endeavored to produce a condensed cyclopedia of engineering or of any branch of it, but we have striven to anticipate the daily wants of the user and to give him the information sought in the manner best suited to his needs. Our aim has been to meet the necessities not only of the engineer but of all in any manner interested in engineering, and in accomplishing this we have selected that rule, formula, or process which was, in our opinion, best adapted to the circumstances of the case, describing it fully, giving full direc- tions how and when to use it, and not mention- ing other methods (when such were available); in other words, we have made the selection instead of leaving the choice to the judgment of the user, which is frequently at fault. The exceedingly large sale proves that the idea was popular and has vindicated our judgment. We hope that succeed- ing editions will meet and merit the same approval that has been accorded those preceding. The present (seventh) edition contains the most convenient table of powers, roots, and reciprocals of numbers yet printed. This table was arranged and computed by us and will be of great use to all having occasion to use it. International Correspondence Schools, December 1 , 1903 . INDEX A. Page Absolute pressures 26 Alloys 19 Alternating system, Size of wires for 239 Aluminum and copper, Properties of 250 Ampere 230 Angles or arcs, Measures of 2 Annunciator system 248 Anode of an electric battery 263 Arc lamps, Connections for 253 Arcs or angles, Measures of 2 Areas and circumferences of circles 82-90 Irregular 119 of circles, Table of 82 Avoirdupois weight 3 B. Batteries, Storage 267 storage, Regulation of 267 Various chemical 263 -266 Beams, Bending moments in - 152 Cantilever 152 Deflection of 152 fixed at both ends 152 Simple. 152 Bearing of a line 276 of a line, Deduced 283 Bearings for line shafting, Distance apart of 193 Bell wiring 241 Belting 140 Rope 209 Belt pulleys • 204 Bending moments in beams 152 Birmingham wire gauge 249 Blow, Force of a 140 v VI INDEX. Page Blueprint paper, To make 175 Blueprints 175 Boilers 158 (steam) 158 (steam) Foaming and priming of 168 (steam) Horsepower of 168 (steam) Inspection and care of 162 (steam) Prevention of scale in 164 (steam) To develop dome of 158 (steam) To develop slope sheet of 159, 160 Bolts for cylinder heads 217 for steam chests 217 Standard proportions of 22 Booster 267 Brake, Prony 260 Bridge, Wheatstone 270 Briggs, or common, logarithms 32 c. Cables, Carrying capacity of Chain Testing of (electrical) Calendar, Perpetual Candlepower Capacity, Measures of of cables Cathode of an electric battery Center of gravity Centrifugal force Chain cables, Wrought-iron Chains and ropes Change gears Characteristic of a logarithm *. . . Chemical treatment of feedwater Chimneys Formulas for Table of sizes of Chord of circle Circle, Area of Chord of Circumference of Segment of Circles, Tables of circumferences and areas of Circuits, Derived, or shunt Motor Size of wire for arc-light Circular pitch, Formula for pitch, Table of rings, Area of rings, Volume of . . . Circumferences and areas of circles 239 14 271 327 236 6 239 263 121 121 14 33 165 170 171 172 116 113 116 113 115 82-90 ' 234 253 253 228 230 116 117 82-90 INDEX. vii Page Clearance, Piston 216 Coefficient of elasticity 152 Coefficients of expansion 19 Columns, Formulas for strength of 156 Commutator, Sparking at 259 Compass surveying 276-279 Compound-geared lathes, Screw cutting with 182 pulley, Formula for 138 Compression, Table of ultimate strength for 151 Conductivity, Electrical 232 Conductor, Direction of motion of 233 Size of 235-240' Cone, Formulas for 117 Conical frustum, Formulas for 117 Connecting-rods 224 Connections for dynamo-electric machines 252 Copper and aluminum, Properties of 250 Corliss engine crank-shaft 223 engine cylinder 221 Corrosion of boilers 163 Cotters for connecting-rods 226 Couplings, Flange 194 Flexible . : 195 Proportions of 195 Shaft . 194 Course of a line in surveying. 276 Crank-shafts for Corliss engines 223 -shafts for high-speed engines .' . . 223 Cross-over tracks 325 Cube root 105 Cubes and squares 106 Cubic expansion, Coefficient of 19 measure 2 Current, Rules for direction of electrical 232 Strength of 231 Curves, Deflection angles of 286 Degree of 286 Elevation of railroad 311 of saturation * 256 Tangent distance of 287 To lay out with transit 288 To lay out without transit 290 Curving of rails 309 Cylin der hea ds 217 heads, Bolts for 217 Cylinders for Corliss engines 221 Formulas for strength of 157 Proportions of 216-219 Stuffingbox for 228 Surface of 116 Volume of 116 INDEX. viii D. Page Decimals of a foot, Equivalent in inches of 91 Declination of needle 277 Deflected line ' 281 Deflection of beams 152 Deflections, Tangent and chord 297 tangent and chord, Formulas for 297 Density - 25 Derived circuits 234 Designs of machine details 192 Development of boiler dome 158 of boiler slope sheet 159 Diagram, Slide-valve 188 Diametral pitch, Formula for 229 pitch, Table of 230 Differential pulley 138 Division by logarithms 42 Dome of boiler, To develop 158 Double movable pulley 137 Draft of chimneys, Formulas for 171 Drills, Speed of twist 177 Dry measure 3 Duty of pumps 144 Dynamo design 254 -electric machines, Connections for 252 machines 246 wiring, Underwriters’ rules for 245 Dynamos and motors 253 Faults of 258 E. Earthwork, Calculation of 306 Eccentric 227 Efficiency, Lamp 240 Motor 253-260 Elastic limit, Table of 152 Electric gas lighting 269 motors, Application of 261 Electricity. 230-275 Electrodeposition — . ' 269 Electrolyte of an electric battery 263 Electromagnet, Polarity of 233 Electromotive force 230 force, Formula for 254 Elements, Table of chemical 16 Elevation of railroad curves 311 Ellipse, Formulas for 115 Emery wheels, Speed of 176 Engine horsepower, Formula for 185 English and metric measures, Conversion tables of 7 INDEX. ix Page Equivalent decimal parts of one foot 91 decimal parts of one inch 91 Evolution by logarithms 46 Table method of 103 Exhaust heating 173-174 ports, Dimensions of 217 Expansion, Coefficients of 19 Exponents 32 External inspection of boilers 164 F. Factors of safety, Table ot 151 Prime 80 Failure of dynamos 258 Falling bodies 120 Feedwater heaters 166 Methods of purifying 165 Testing of 164 Field magnet 255 magnet, Reversal of 258 Filtration of feedwater 165 Flange coupling 194 Flanges, Pipe 215 Flexible coupling 195 Flexure, Ultimate strength of 151 Flow of water in pipes 147 Fluxes for soldering and welding 24 Foaming of boilers 168 Foot, Decimals of a 91 Force, Formula for electromotive 254 Magnetizing 255 of a blow 140 Forces, Resultant of 137 Formulas 93-302 How to use 93 Frog (railroad work) 312 Angle 313 Crotch or middle 324 distance 314 Frustum of cone, Formulas for 117 of pyramid , Formulas for 118 Fusion, Latent heat of 18 Temperature of 18 G. G 2 , Values of 153 Galvanometer 271 Gases, Weights of 14 Gas lighting, Electric 269 Gauge, Birmingham wire 249 B.&S. wire 248 INDEX.' Page Gauge, sizes of wire, with equivalent sectional areas 248 Gearing, Formulas for 228 Gears, Change, for screw cutting 178 To calculate speed of 142 Gibs for connecting-rods 224 Gland 228 Grade lines 296 Rate of 296 Gravity, To find center of < 121 Grindstone, Speed of 176 Gyration, Square of least radius of 153 To find radius of , 131 To find radius of, experimentally 133 H. Hangers, Shaft 202 Heat 19 Latent, of fusion. 18 of liquid 25 Heating by exhaust steam 173 of dynamos 259 surface, Square feet of, per horsepower 168, 169 surface, Ratio of, to grate area 168 Helix, Formula for 116 To construct a 116 High-speed engines. Crank-shaft for 223 Horsepower of belts 140 of boilers 168 of electrical currents 232 of engines 185 of pumps . ._ 184 of rope belting 210 Theoretical 184 Hydrokinetics 145 Hydromechanics 144 Hydrostatics 144 Hyperbolic logarithms 32 I. I, Values of ' 153 Incandescent lamp data 240 wires.. Underwriters’ regulations for 245 Inch, Equivalent decimal parts of 91 Inches and parts thereof in decimals of one foot. ... 91, 92 Inclined planes, Formula for 138 Incrustation in boilers 164 Indicated horsepower of engines, Formula for 185 Inertia, To find moment of 125 To find moment of, experimentally.. 133 To find moment of, for various sections 153 Inspection of boilers 162 INDEX. xi Page Insulation, Test of 272 Interior wiring 235 Involution by logarithms 44 Iron bars, Weight of round and square 21 Irregular areas 119 J. Joint coupling, Universal 195 Journal box, Design of 195 K. Kerosene in boilers 167 Keys for shafting, Proportions of 194 Kilowatt 232 L. Lamps, Efficiency of 240 Incandescent, data 240 in series (electric light) • 253 Lap, Inside and outside 187 Latent heat of fusion 18 heat of vaporization 18 Lathe, Change gears of 178 Compound-geared 182 Simple-geared 178 Law, Ohm’s . 231 Lead of valve 187 Lead, Weight of sheet 21 Leakage, Magnetic 258 Leclanche cell 244,269 Legal ohm 230 Length, Measures of 5 Leveling, Direct 292 Grade lines in 296 notes, How to check and keep 294 Profiles in 296 Levers 136 Linear expansion, Coefficient of 18 measure 1 Line shafting > y . . . . 193 Lines of force, Leakage of 257 of force, Number of 254 Lining for seats 216 Liquid, Heat of 25 measures. 4 Liquids, Weights of 13 Locknuts 192 Logarithmic table 50-67 table, Use of 34 Logarithms 32 Long-ton table 3 INDEX. xii M • Page Machine design 175 tools, Cutting speeds for 176 tools, Motors for 263 Magnetic meridian 277 permeability 257 Manila rope belting • 209 rope belting, Weight of '. . . . 209 Mantissa of a logarithm. 33 Materials, Strength of - 150 Mean effective pressure 185 Measure, Cubic 2 Dry 3 Linear 1 Liquid 4 Surveyor’s 1 Surveyor’s square 2 Measures and weights 1-4 and weights, Metric 5 of angles or arcs. 2 of capacity 6 of length 5 *of surface (not land) 5 of volume 5 of weight 6 Mechanical powers 136 Mechanics 120, 149 Mensuration 113 Meridian, Magnetic. 277 True 277 Metals, Weights of 10 Metric and English measures, Conversion table for. . 7 system 5 Mil .'. 2.35 Mil, Circular 235 Miscellaneous table 4 Moment of inertia defined 125 of inertia of various sections 153 of inertia, To find, experimentally 133 of resistance defined 134 of resistance of various sections 133 Moments, Bending 152 Mo to r ci rcuits 253 efficiency, Approximate. . .. 253 Motors, Application of electric 261 for machine tools 263 Output and efficiency of 260 Polarity of 234 Underwriters’ rules for 247 Multiple arc, Lamps in -. -. . . . 253 Multiplication by logarithms 41 INDEX. xiii N. Page Needle, Declination of 277 Neutral axis 135 Numbers, Prime 79 Nuts, Proportions of 22 o. Oblique fixed pulley. Formula for. . Ohm, Legal Ohm’s law Oil cup Oscillation, To find center of To find radius of, experimentally. Output of motors 138 230 231 198 127 133 260 P. Packing rings 224 Paper, To make blueprint. 175 Parallel, Lamps in 253 Parallelogram 114 of forces, Explanation of 137 Passage, Steam 217 Pedestals, Design and proportions of 195-202 Percussion, To find center of 130 Permeability, Magnetic 257 Perpetual calendar 327 Pipe flanges 215 Weight of cast-iron. . .• 23 Pipes and cylinders, Strength of 157 Flow of water in 147 Sizes of wrought-iron 24 Piston clearance 216 Pistons 223 Pitch, Formula for circular 228 Table of circular 230 Formula for diametral 228 Table of diametral 230 of bolts in cylinder heads 217 of bolts in steam-chest covers 217 Polarity of a dynamo, To determine 233 of an electromagnet, To determine 233 Polishing wheels, Speed of 176 Polygon of forces, Explanation of 137 Polygons, Regular 119 Port, Exhaust 217 Steam 217 Power transmitted by leather belting 141 transmitted by rope belting 209 Powers, Mechanical 136 roots, and reciprocals, Table of 110 Pressure, Mean effective 185 XIV INDEX. Page Pressures, Absolute 26 Prime factors, Table of 80 numbers 79 Priming of boilers 168 Prismoidal formula 306 Profiles in leveling 296 Projectiles 120 Prony brake 260 Properties of aluminum and copper 250 of saturated steam 29 Proportions of belt pulleys 204-209 of flange couplings 194 of journal boxes 195-202 of keys 193 of rope-pulley rims 211 of shaft hangers 202 Pulleys, Belt 204 Differential 138 Double movable 137 Formula for compound 138 Proportions of 204-209 Quadruple movable 138 Rope 211 Single fixed 137 Single movable 137 Speed of 142 Pumps, Discharge of 143 Duty of 144 Horsepower of » 143 Pyramid Formulas for ' 118 Formulas for frustum of 118 Q. Quadruple movable pulley, Formula for 138 R. R, Values oi 153 Radiating surface in exhaust-steam heating 174 Radii and deflections. Table of 298-300 Radius of gyration, To find. 131 of gyration, To find, experimentally 133 of gyration, To find, for various sections 153 of oscillation, To find, experimentally 133 Rate of transmission of electricity 231 Reciprocal of a number 108 Rectangle, Formula for 114 Regular polygons. 119 Resistance, Electrical 231 Moment of 134 Moment of, for various sections 153 of copper wire 251-253 INDEX. xv Page Resistance of derived circuit 234 Resultant of forces 137 Retaining walls 300 walls, Resistance of, to overturning 303 Reversal of field 258 Ribs for piston 224 for steam-chest cover 220 Ring, Formula for circular 117 Formula for 116 Root, Cube 105 Square 103 Roots, Method of extracting 103 Table of 110 Rope belting 209 belting, Pulleys for 211 Weight of manila 209 Ropes and chains, Strength of 157 Wire 212 wire, Pulleys for 213 s. Safety, Table of factors of valves Saturated steam, Properties of 'Saturation curves (electrical) Scale in boilers, Prevention of Screw cutting, Change gears for Formulas for threads, Proportion of Seats, Lining for Sector, Formula for Segment, Formula for Series. Lamps in Shaft couplings hangers Shafting, Formulas for Line Shafts, Crank Shearing strengths, Table of Sheaves for rope gearing Sheet lead, Weight of Shunt circuit Simple-geared lathe, Screw cutting with Single fixed pulley, Formula for movable pulley, Formula for Size of copper wire for circuits Slide valve valve diagram Slope sheet of boiler, To develop Soldering, Fluxes for Solders 151 173 29 256 164 178, 139' 22 216 115 115 253 194 202 157 193 223 151 213 21 234 178 137 137 235-252 187 188 160 24 20 XVI INDEX. Sparking at commutator Specific gravity, Table of heat, Table of volumes Sp>eed, Cutting ' ... . of emery wheels of gears, To calculate of grindstones of polishing wheels of pulleys, To calculate of twist drills Sphere, Formula for Spiral, Length of Square measure root Squares and cubes Standard pipe flanges Steam chest chest bolts chest covers Heating by exhaust port area Properties of saturated - tables Velocity of, through ports Steel, Tempering of Stone, Weight of Storage batteries Strands in wire rope Strap, Eccentric and end of connecting-rod Strength of materials Stroke of engine Stuffingbox Surcharged walls, Pressure on. . . . Surface expansion, Coefficients of Measures of (not land) Surveying with compass with transit Surveyor’s measure. square measure Switch Point or split Stub stub, To lay out Systems, Annunciator Page 259 10 18 25 176 176 142 176 176 142 177 117 118 2 103 106 215 219 220 221 173, 174 219 29 25 219 6 10 263 212 226 224 150 216 228 305 19 5 276-326 276 280 1 315 315 315 320 243 T. Table, Long- ton 3 Miscellaneous 4 INDEX. xvii Page Table of chemical elements 16 of powers, roots, and reciprocals 110 Tables, Steam 25 Wire 247-251 Teeth of wheels 228 Temperature of fusion 18 of vaporization 18 Tempering steel 6 Tensile strength of materials 151 Tension of rope belting, Formula for 210 Testing of cables (electrical) 271 Threads, Cutting screw 178 Proportions of screw 22 Three-wire system, Edison 253 Ton, Long 3 Tools, Cutting speeds for machine 176 cutting, Motors for 263 Torque 261 Tracks, Cross-over 325 Trackwork 309 Transit notes, How to keep 291 surveying 279 Trapezium and trapezoid, Formula for 115 • Triangles, Formulas for 114 Triangulation 283 Trigonometric functions, Directions for use of table of 68 functions, Table of 74-78 Troy weight 3 Tunnel sections 306 Turnouts 312 Type metals 15 U. Ultimate strength of materials 151 Underwriters’ line wire. . . . 247 rules for incandescent wire 245 Units, Electrical 230 Universal joint coupling '. 195 Useful tables 1-92 V. Valve diagram 188 Valves, Safety 173 Slide 187 Vaporization, Latent heat of 18, 25 Temperature of . . . .* 18 Vapors, Weights of 14 Velocity of steam through ports 219 Vernier 279 Volt 230 Volume, Measures of 5 Volumes, Specific 25 INDEX. xviii w. Page Water, Testing of feed Flow of, in pipes Watt, The unit .' Wedge, Formula for Weight, Avoirdupois Measures of of bar iron, round and square of copper wire of manila rope of sheet lead of various substances Troy Weights and measures and measures, Metric system of Welding fluxes . Wheatstone bridge Wheel and axle Wheels, Speed of emery Speed of polishing Wheel work, Formulas for Width of belts. Formulas for Wire, copper, Sizes for circuit copper, Weight of gauges, Sizes of B. & S. and Birmingham rope, Steel. rope, Strands in tables Underwriters’ line Wires, Equivalent areas of, B. & S. gauge. . Wiring, Bell.- Interior Work, Definition of Wristpin brasses Wrought-iron pipe, Sizes of 165 147 231 117 3 6 21 238 209 21 10 3 1-4 5-6 24 270 136 176 176 136 140 . 235-240 238 249 212 212 . 247-251 247 248 241 235 139 224 24 8 88*. 8 8 Mechanics’ Pocket Memoranda USEFUL TABLES. WEIGHTS AND MEASURES. 12 inches (in.) . LINEAR MEASURE. — 1 foot 3 feet = 1 yard 5.5 yards = 1 rod 40 rods = 1 furlong 8 furlongs = 1 mile ... in. ft. yd. rd. fi 36 = 3 = 1 198 = 16.5 - 5.5 = 1 7,920 II 05 o 11 to 8 II o II 63,360 = 5,280 = 1,760 = 320 = ...ft. .yd. ..rd. .fur. .mi. SURVEYOR’S MEASURE. inches = 1 link links = lrod rods ) links > = 1 chain feet j chains = 1 mile. 1 mi. = 80 ch. = 320 rd. = 8,000 li. = 63,360 in. ...li. .rd. .ch, .mi. 1 2 USEFUL TABLES. SQUARE MEASURE. 144 square inches (sq. in.). 9 square feet 30£ square yards 160 square rods 640 acres = 1 square foot sq. ft. = 1 square yard sq. yd. = 1 square rod sq. rd. = 1 acre A. = 1 square mile sq. mi. sq. mi. A. sq. rd. sq. yd. sq. ft. sq. in. 1 = m = 102,400 = 3,097,600 = 27,878,400 = 4,014,489,600 SURVEYOR'S SQUARE MEASURE. 625 square links (sq. li.) = 1 square rod sq. rd. 16 square rods = 1 square chain sq. ch. 10 square chains = 1 acre A. 640 acres = 1 square mile sq. mi. 36 square miles (6 mi. square) = 1 township Tp. 1 sq. mi. = 640 A. = 6,400 sq. ch. = 102,400 sq. rd. = 64,000,000 sq. li. The acre contains 4,840 sq. yd., or 43,560 sq. ft., and is equal to the area of a square measuring 208.71 ft. on a side. CUBIC MEASURE. 1,728 cubic inches (cu. in.) = 1 cubic foot .. 27 cubic feet = 1. cubic yard... 128 cubic feet = 1 cord 24£ cubic feet = 1 perch 1 cu. yd. = 27 cu. ft. = 46,656 cu. in MEASURE OF ANGLES OR ARCS. 60 seconds (") * = 1 minute ' 60 minutes — 1 degree ° 90 degrees = 1 rt. angle or quadrant □ 360 degrees .. = 1 circle cir. 1 cir. = 360° = 21,600' = 1,296,000" ..cu. ft. .cu. yd. cd. P. WEIGHTS AND MEASURES. 3 AVOIRDUPOIS WEIGHT. 437.5 grains (gr.) = 1 ounce oz. 16 ounces = 1 pound lb. 100 pounds = 1 hundredweight cwt. 20 cwt., or 2,0001b = 1 ton T. IT. = 20 cwt. = 2,0001b. = 32,000 oz. = 14,000,000 gr. The avoirdupois pound contains 7,000 grains. 16 ounces LONG TON TABLE. .... — 1 pound lb. 112 pounds — 1 hundredweight .. ....cwt. 20 cwt., or 2,240 lb. — 1 ton T. 24 grains (gr.) TROY WEIGHT. ....pwt. 20 pennyweights s=3 1 ounce oz* 12 ounces — 1 pound lb* lib. = 12 oz. = 240 pwt. = 5,760 gr. 2pints(pt.) DRY MEASURE. qt. 8 quarts ..: = 1 peck pk. 4 pecks = 1 bushel bu. lbu. = 4pk. = 32 qt.= 64 pt. The U. S. struck bushel contains 2,150.42 cu. in. = 1.2444 cu. ft. • By law, its dimensions are those of a cylinder 18£ in. in diameter and 8 in. deep. The heaped bushel is equal to 1£ struck bushels, the cone being 6 in. high. The dry gallon contains 268.8 cu. in., being £ of a struck bushel. For approximations, the bushel may be taken at 1£ cu. ft.; or a cubic foot may be considered £ of a bushel. The British bushel contains 2,218.19 cu, in. = 1.2837 cu. ft. = 1.032 U. S. bushels. 4 USEFUL TABLES. LIQUID MEASURE. 4 gills (gi.) =*1 pint Pt. 2 pints = 1 quart -v qt. 4 quarts = 1 gallon gal. 31£ gallons = 1 barrel bbl. 2 barrels, or 63 gallons .... s= 1 hogshead hhd. 1 hhd. = 2 bbl. = 63 gal. = 252 qt. = 504 pt. = 2,016 gi. The U. S. gallon contains 231 cu. in. = .134 cu. ft., nearly; or 1 cu. ft. contains 7.481 gal. The following cylinders con- tain the given measures very closely: Diam. Height. Diam. Height. Gill If in. 3 in. Gallon . 7 in. 6 in. Pint .... 3£in. 3 in. 8 gallons Min. 12 in. Quart ... .... 3* in. 6 in. 10 gallons 14 in. 15 in. When water is at its maximum density, 1 cu. ft. weighs 62.425 lb. and 1 gallon weighs 8.345 lb. For approximations, 1 cu. ft. of water is considered equal to 7i gal., and 1 gal. as weighing 8| lb. The British imperial gallon, both liquid and dry, con- tains 277.274 cu. in. = .16046 cu. ft., and is equivalent to the volume of 10 lb. of pure water at 62° F. To reduce British to U. S. liquid gallons, multiply by 1.2. Conversely, to convert U. S. into British liquid gallons, divide by 1.2; or, increase the number of gallons MISCELLANEOUS TABLE. 12 articles = 1 dozen. 20 quires = 1 ream. 12 dozen = 1 gross. 1 league = 3 miles. 12 gross = 1 great gross. 1 fathom = 6 feet. 2 articles = 1 pair. 1 hand = 4 inches. 20 articles = 1 score. 1 palm = 3 inches. 24 sheets = 1 quire. 1 span = 9 inches. 1 sea mile (U. S.) = 6,080 ft. = 1£ statute miles (roughly). 1 meter = 3 feet 3| inches (nearly). THE METRIC SYSTEM. 5 THE METRIC SYSTEM. The metric system is based on the meter, which, according to the U. S. Coast and Geodetic Survey Report of 1884, is equal to 39.370432 inches. The value commonly used is 39.37 inches, and is authorized by the U. S. government. The meter is defined as one ten-millionth the distance from the pole to the equator, measured on a meridian passing near Paris. There are three principal units— the meter, the liter (pro- nounced lee-ter), and the gram, the units of length, capacity, and weight, respectively. Multiples of these units are obtained by prefixing to the names of the principal units the Greek words deca (10), hecto (100), and kilo (1,000); the submulti- ples, or divisions, are obtained by prefixing the Latin words deci ( T V), centres), and milli (nrW)- These prefixes form the key to the entire system. In the following tables, the abbreviations of the principal units of these submultiples begin with a small letter, while those of the multiples begin with a capital letter; they should always be written as here printed. MEASURES OF LENGTH. 10 millimeters (mm.) — 1 centimeter cm. 10 centimeters = 1 decimeter dm. 10 decimeters = 1 meter m. 10 meters — 1 decameter Dm. 10 decameters = 1 hectometer Hm. 10 hectometers = 1 kilometer Km. MEASURES OF SURFACE (NOT LAND). 100 square millimeters (mm 2 .) = 1 square centimeter cm 2 . 100 square centimeters = 1 square decimeter dm 2 . 100 square decimeters = 1 square meter m 2 . MEASURES OF VOLUME. 1,000 cubic millimeters (mm 3 .) = 1 cubic centimeter cm 3 . 1,000 cubic centimeters = 1 cubic decimeter dm 3 . 1,000 cubic decimeters »= 1 cubic meter m 3 . 6 USEFUL TAELES. M EASURES OF CAPACITY. 10 milliliters (ml.) = 1 centiliter. cl. 10 centiliters = 1 deciliter dl 10 deciliters = 1 liter 1. 10 liters — 1 decaliter Dl. 10 decaliters . — 1 hectoliters HI. 10 hectoliters = 1 kiloliters Kl. Note. — T he liter is equal lo the volume that is occupied by 1 cubic decimeter. MEASURES OF WEIGHT. 10 milligrams (mg.) = 1 centigram eg. 10 centigrams . = 1 decigram dg. 10 decigrams = 1 gram g. 10 grams ,= 1 decagram Dg. 10 decagrams = 1 hectogram Hg. 10 hectograms = 1 kilogram Kg. 1,000 kilograms = 1 ton T. Note.— The gram is the weight of 1 cubic centimeter of pure distilled water at a temperature of 39.2° F. ; the kilogram is the weight of 1 liter of water; the ton is the weight of 1 cubic meter of water. TEMPERING OF STEEL. The following colors may be made use of in tempering steel-cutting tools: Corresponding Temperature F. Lancets Razors All kinds of wood-cutting tools Screw taps Chipping chisels, hatchets, and saws All kinds of percussive tools y Springs j Pale yellow 430° Straw yellow 450° Darker straw yellow 470° Yellow 490° Brown yellow 500° Brown (slightly tinged purple) 520° Light purple 530° Clear black 570° Dark blue 600° THE METRIC SYSTEM. 7 1,828.8 121.92 21.336 .3048 .2438 1,972.6046 CONVERSION TABLES. By means of the tables on pages 8 and 9, metric measures can be converted into English, and vice versa , by simple addi- tion. All the figures of the values given are not required, four or five digits being all that are commonly used; it is only in very exact calculations that all the digits are necessary. Using table, proceed as follows: Change 6,471.8 feet into meters. Any number, as 6,471.8, may be regarded as 6,000 + 400 + 70 + 1 + .8; also, 6,000 = 1,000 X 6; 400 = 100 X 4, etc. Hence, looking in the left-hand column of the upper table, page 8, for figure 6 (the first figure of the given number), we find opposite it in the third column, which is headed “Feet to Meters,” the number 1.8287838. Now, using but five digits and increasing the fifth digit by 1 (since the next is greater than 5) , we get 1.8288. In other words, 6 feet = 1.8288 meters; hence, 6,000 feet = 1,000 X 1.8288 = 1,828.8, simply moving the decimal point three places to the right. Likewise, 400 feet = 121.92 meters; 70 feet = 21.336 meters; 1 foot = .3048 meter, and .8 foot = .24C" meter. Adding as shown above, we get 1,972.6046 meters. Again, convert 19.635 kilos into pounds. The work should be perfectly clear from the explana- tion given above. The result is 43.2875 pounds. The only difficulty m applying these tables lies in locating the decimal point; it may always be found thus: If the figure considered lies to the left of the decimal point, count each figure in order, beginning with units (but calling unit’s place zero), until the desired figure is reached, then move the decimal point to the right as many places as the figure being considered is to the left of the unit figure. Thus, in the 'first case above, 6 lies three places to the left of 1, which is in unit’s place; hence, the decimal point is moved three places to the right. By exchanging the words “right ” and “ left,” the statement will also apply to decimals. Thus, in the second case above, the 5 lies three places to the right of unit’s place; hence, the decimal point in the number taken from the table is moved three places to the left 22.046 19.8416 1.3228 .0661 .0110 43.2875 8 USEFUL TABLES. Conversion Table— English Measures Into Metric. English. Metric. Metric. Metric. Metric. Inches to Meters. Feet to Meters. Pounds to Kilos. Gallons to Liters. 1 .0253998 .3047973 .4535925 3.7853122 2 .0507996 .6095946 .9071850 7.5706244 3 .0761993 .9143919 1.3607775 11.3559366 4 .1015991 1.2191892 1.8143700 15.1412488 5 .1269989 1.5239865 2.2679625 18.9265610 6 .1523987 1.8287838 2.7215550 22.7118732 7 .1777984 2.1335811 3.1751475 26.4971854 8 .2031982 2.4383784 3.6287400 30.2824976 9 .2285980 2.7431757 4.0823325 34.0678098 10 .2539978 3.0479730 4.5359250 37.8531220 [ Conversion Table— English Measures Into Metric. English. Metric. Metric. Metric. Metric. Square Inches to Square Meters. Square Feet to Square Meters. Cubic Feet to Cubic Meters. Pounds per Square Inch to Kilo per Square Metgr. 1 .000645150 .092901394 .028316094 703.08241 2 .001290300 .185802788 .056632188 1,406.16482 3 .001935450 .278704182 .084948282 2,109.24723 4 .002580600 .371605576 .113264376 2,812.32964 5 .003225750 .464506970 .141580470 3,515.41205 6 .003870900 .557408364 .169896564 4,218.49446 7 .004516050 .650309758 .198212658 4,921.57687 8 .005161200 .743211152 .226528752 5,624.65928 9 .005406350 .836112546 .254844846 6,327.74169 10 .006451500 .929013940 .283160940 7,030.82410 THE METRIC SYSTEM. 9 Conversion Table— Metric Measures Into English. Metric. English. English. English. English. Meters to Inches. Meters to Feet. Kilos to Pounds. Liters to Gallons. 1 39.370432 3.2808693 2.2046223 .2641790 . 2 78.740864 6.5617386 4.4092447 .5283580 3 118.111296 9.8426079 6.6138670 .7925371 4 157.481728 13.1234772 8.8184894 1.0567161 5 196.852160 16.4043465 11.0231117 1.3208951 6 236.222592 19.6852158 13.2277340 1.5850741 7 275.593024 22.9660851 15.4323564 1.8492531 8 314.963456 26.2469544 17.6369787 2.1134322 9 354.333888 29.5278237 19.8416011 2.3776112 10 393.704320 32.8086930 22.0462234 2.6417902 Conversion Table— Metric Measures Into English. Metric. English. English. English. English. Square Meters to Square Inches. Square Meters to Square - Feet. Cubic Meters to Cubic Feet. Kilos per Square Meter to Pounds per Square Inch. 1 1,550.03092 10.7641034 35.3156163 .001422310 2 3,100.06184 21.5282068 70.6312326 .002844620 3 4,650.09276 32.2923102 105.9468489 .004266930 4 6,200.12368 43.0564136 141.2624652 .005689240 5 7,750.15460 53.8205170 176.5780815 .007111550 6 9,300.18552 64.5846204 211.8936978 .008533860 7 10,850.21644 75.3487238 247.2093141 .009956170 8 12,400.24736 86.1128272 282.5249304 .011378480 9 13,950.27828 96.8769306 317.8405467 .012800790 10 15,500.30920 107.6410340 353.1561630 .014223100 i 10 USEFUL TABLES. SPECIFIC GRAVITY. The specific gravity of a body is the ratio between its weight and the weight of a like volume of distilled water at a temperature of 39.2° F. For gases, air is taken as the unit. One cubic foot of water at 39.2° F. weighs 62.425 pounds. Name of Substance. Specific Gravity. Weight per Cu. In. Pounds. Metals. Platinum, rolled 22.009 .819 Platinum, wire 21.042 .760 Platinum, hammered 20.337 .735 Gold, hammered 19.361 .699 Gold, pure cast 19.258 .696 Gold, 22 carats fine 17.486 .632 Mercury, solid at — 40° F 15.632 .565 Mercury, at + 32° F. 13.619 .492 Mercury, at 60° F. 13.580 .491 Mercury, at 212° F 13.375 .483 Lead, pure 11.330 .409 Lead, hammered 11.388 .411 Silver, hammered 10.511 .380 Silver, pure 10.474 .378 Bismuth 9.746 .352 Copper, wire and rolled 8.878 .321 Copper, pure 8.788 .317 Bronze, gun metal 8.500 .307 Brass, common 8.500 .307 Steel, cast steel 7.919 .286 Steel, common soft 7.833 .283 Steel, hardened and tempered 7.818 .282 Iron, pure 7.768 .281 Iron, wrought and rolled 7.780 .281 Iron, hammered 7.789 .281 Iron, cast 7.207 .260 Tin, from Bohmen 7.312 .264 Tin, English 7.201 .263 Zinc, rolled 7.101 .260 Antimony 6.712 .242 Aluminum 2.660 .096 Stones and Earths. Emery 4.000 .145 Limestone 2.700 .098 Asbestos, starry 3.073 .111 SPECIFIC GRAVITY. 11 Table— ( Continued) * Name of Substance. Specific Gravity. Weight per Cu. In. Pounds. Glass, flint Glass, white Glass, bottle Glass, green Marble, Parian ... Marble, African... Marble, Egyptian Mica 3.500 2.900 2.732 2.642 2.838 .1260 .1050 .0987 .0954 .1025 2.708 2.668 2.800 .0978 .0964 .1012 Chalk 2.784 .1006 Coral, red Granite, Susquehanna. Granite, Quincy Granite, Patapsco Granite, Scotch I Marble, white Italian Marble, common | Talc, black Quartz Slate Pearl, oriental Shale Flint, white Flint, black Stone, common Stone, Bristol Stone, mill Stone, paving Gypsum, opaque Grindstone Salt, common Saltpeter Sulphur, native Common soil I Rotten stone Clay Brick Niter Plaster Paris 1 Ivor^ J Sand Phosphorus | Borax | Coal, anthracite 2.700 2.704 2.652 2.640 2.625 2.708 2.686 2.900 2.660 2.800 2.650 2.600 2.594 2.582 2.520 2.510 2.484 2.416 2.168 2.143 2.130 2.090 2.033 1.984 1.981 1.900 2.000 1.900 1.872 2.473 1.822 2.650 1.770 1.714 1.640 1.436 .0975 .0977 .0958 .0954 .0948 .0978 .0970 .0105 .0961 .1012 .0957 .0939 .0937 .0933 .0910 .0907 .0897 .0873 .0783 .0774 .0769 .0755 .0734 .0717 .0716 .0686 .0723 .0686 .0676 .0893 .0659 .0957 .0639 .0619 .0592 .0519 12 USEFUL TABLES. Table— ( Continued). Name of Substance. Specific Gravity. Weight per Cu. In. Pounds. Coal, Maryland 1.355 .0490 Coal, Scotch 1.300 .0470 Coal, Newcastle 1.270 .0459 Coal, bituminous 1.350 .0488 Earth, loose 1.360 .0491 Lime, quick 1.500 .0542 Charcoal .* .441 .0159 Woods (Dry). Alder .800 .0289 Apple tree .793 .0287 Ash, the trunk .845 .0305 Bay tree .822 .0297 Beech .852 .0308 Box, French .960 .0347 Box, Dutch 1.328 .0480 Box, Brazilian red 1.031 .0372 Cedar, wild .596 .0215 Cedar, Palestine .613 .0221 Cedar, American .561 .0203 Cherry tree .672 .0243 Cork .250 .0090 Ebony, American 1.220 .0441 Elder tree *. .695 .0251 Elm .560 .0202 Filbert tree .600 .0217 Fir, male .550 .0199 Fir, female .498 .0180 Hazel .600 .0217 Lemon tree .703 .0254 Lignum-vitse 1.330 .0481 Linden tree .604 .0218 Logwood Mahogany, Honduras .913 .0330 .560 .0202 Maple .790 .0285 Mulberry .897 .0324 Oak .950 .0343 Orange tree .705 .0255 . Pear tree .661 .0239 Poplar .383 .0138 Poplar, white Spanish .529 .0191 Sassafras .482 .0174 Spruce .500 .0181 Spruce, old .460 .0166 SPECIFIC GRAVITY. 13 Table— ( Continued). Name of Substance. Specific Gravity. Pine, southern Pine, white .... Walnut .720 .400 .610 Liquids. Acid, acetic ....... Acid, nitric .7 Acid, sulphuric Acid, muriatic - : Acid, phosphoric Alcohol, commercial Alcohol, pure ..... Beer, lager Champagne Cider , Ether, sulphuric Egg Honey Human blood - Milk Oil, linseed Oil, olive Oil, turpentine Oil, whale Proof spirit Vinegar Water, distilled (62.425 lb. per cu. ft.) Water, sea Wine 1.062 1.217 1.841 1.200 1.558 .S33 .792 1.034 .997 1.018 .739 1.090 1.450 1.054 1.032 .915 .870 .932 .925 1.080 1.000 1.030 .992 Miscellaneous. Beeswax Butter India rubber Fat Gunpowder, loose Gunpowder, shaken Gum arabic Lard Spermaceti Sugar Tallow, sheep Tallow, calf Tallow, ox Atmospheric air .965 .942 .933 .923 .900 1.000 1.452 .947 .943 1.605 .924 .934 .923 .0012 Weight per Cu. In. i Pounds. .0260 .0144 .0220 .0384 .0440 .0665 .0434 .0563 .0301 .0286 .0374 .0360 .0368 .0267 .0394 .0524 .0381 .0373 .0340 .0331 .0314 .0337 .0334 .0390 .0361 .0372 .0358 .0349 .0340 .0337 .0333 .0325 .0361 .0525 .0342 .0341 .0580 .0334 .0337 .0333 14 USEFUL TABLES. Table — ( Continued ) . Name of Substance. Specific Gravity. Weight per Cu. Ft. Grains. Gases and Vapors. At 32° and a tension of 1 atmosphere. Atmospheric air 1.0000 565.11 Ammonia gas .5894 333.1 ' Carbonic acid 1.5201 859.0 Carbonic oxide .9673 546.6 Light carbureted hydrogen .5527 312.3 Chlorine 2.4502 1,384.6 Olefiant gas .9672 546.6 Hydrogen .0692 39.1 Oxygen 1.1056 624.8 Sulphureted hydrogen 1.1747 663.8 Nitrogen .9713 548.9 Vapor of alcohol 1.5890 898.0 Vapor of turpentine spirits 4.6978 2,654.8 Vapor of water .6219 351.4 Smoke of bituminous coal .1020 57.6 Smoke of wood .9000 508.6 ' Steam at 212° F .4880 275.8 The weight of a cubic foot of any solid or liquid is found by multiplying its specific gravity by 62.425 lb. avoirdupois. The weight of a cubic foot of any gas at atmospheric pres- sure and at 32° F. is found by multiplying its specific gravity by .08073 lb. avoirdupois. WROUGHT-IRON CHAIN CABLES. The strength of a chain link is less than twice that of a straight bar of a sectional area equal to that of one side of the link. A weld exists at one end and a bend at the other, each requiring at least one heat, which produces a decrease in the strength. The report of the committee of the U. S. Testing Board, on tests of wrought-iron and chain cables, contains the following conclusions: “That beyond doubt, when made of American bar iron, with cast-iron studs, the studded link is inferior in strength to the unstudded one. CHAIN CABLES. 15 “That, when proper care is exercised in the selection of material, a variation of 50 to 170 of the strongest may be expected in the resistance of cables. Without this care the variation may rise to 250. “ That with proper material and construction the ultimate resistance of the chain may be expected;to vary from 1550 to 1700 of that of the bar used in making the links, and show an average of about 1630. “That the proof test of a chain cable should be about 500 of the ultimate resistance of the weakest link.” From a great number of tests of bars and unfinished cables, the committee considered that the average ultimate resistance and proof tests of chain cables made of the bars, whose diameters are given, should be such as are shown in the accompanying table. Ultimate Resistance and Proof Tests of Chain Cables. Diam. of Bar. Inches. Average Resist. = 1630 of Bar. Pounds. Proof Test. Pounds. Diam. of Bar. Inches. Average Resist. = 1630 of Bar. Pounds. Proof Test. Pounds. 1 71,172 33,840 1* i k ih 162,283 77,159 \p. 79,544 37,820 174,475 82,956 88,445 42,053 187,075 88,947 IP 97,731 46,468 200,074 95,128 107,440 51,084 ! lit 213,475 101,499 V* 117,577 55,903 227,271 108,058 1% 128,129 60,920 241,463 114,806 IP 139,103 150,485 66,138 71,550 2 256,040 121,737 TYPE METALS. Name. Proportions. Smallest type 3 X, 1 A Small type 4 1 A Medium type 5 X, 1 A Large type 6 X, 1 A Largest type 7 X, 1 A In the above table, X represents the lead, and A the anti- mony in the alloy. 16 USEFUL TABLES. TABLE OF ELEMENTS. Aluminum Antimony (stibium) Arsenic Barium Beryllium Bismuth Boron Bromine Cadmium Caesium Calcium Carbon Cerium Chlorine Chromium Cobalt Columbium Copper (cuprum) Didymium Erbium Fluorine Gallium Germanium Gold (aurum) Hydrogen Indium Iodine Iridium Iron (ferrum) Lanthanum Lead (plumbum) Lithium Magnesium Mercury (hydrargyrum) Manganese....'. Molybdenum Nickel Niobium Nitrogen Osmium Oxygen rmbol. Atomic Weight/ Al 27.04 Sb 119.96 As 74.9 Ba 136.9 Be 9.08 Bi 207.5 B 10.9 Br 79.76 Cd 111.7 Cs 133.0 Ca 39.91 C 11.97 Ce 141.2 Cl 35.37 Cr 52.45 Co 58.6 Cb 93.7 Cu 63.18 D 147.0 E 169.0 F 19.06 G 69.8 Ge 72.32 Au 196.2 H 1.0 In 113.4 I 126.54 Ir 196.7 Fe 55.88 Lqi 139.0 Pb 206.39 Li 7.01 Mg 23.94 Mg 199.8 Mn 54.8 Mo 95.6 Ni 58.6 94.0 14.01 Os 198.6 0 15.96 * Principally from the 16th edition Des Ingenieurs Taschen- buch. The names of the non-metals are printed in heavy type. TABLE OF SPECIFIC HEATS. 17 Table — ( Continued). Symbol. Atomic Weight. Palladium Phosphorus Platinum Potassium (kalium) Rhodium Rubidium Ruthenium Scandium Selenium Silicon Silver (argentum) ... Sodium (natrium) ... Strontium Sulphur Tantalum Tellurium Thallium Thorium Tin (stannum) Titanium Tungsten (wolfram) Uranium Vanadium Ytterbium Yttrium Zinc Zirconium Pd 106.2 P 30.96 Pt 194.43 K 39.04 Rh 104.1 Rb 85.2 Ru 103.5 Sc 44.04 Se 78.00 Si 28.00 Ag 107.66 Na 23.0 Sr 87.3 S 31.98 Ta 182.0 Te 128.0 Tl 203.6 Th 231.5 Sn 117.35 Ti 48.0 W 183.6 U 240.0 V 51.2 Yb 93.0 Y 172.6 Zn 64.88 Zr 90.0 TABLE OF SPECIFIC HEATS. Solids. Copper 0951 Cast iron 1298 Gold 0324 Wrought ir,on 1138 Lead 0314 Platinum 0324 Steel (soft) 1165 Steel (hard) 1175 Zinc 0956 Silver 0570 Tin 0562 Ice 5040 Brass .0939 Glass 1937 Sulphur 2026 Charcoal 2410 18 USEFUL TABLES. Liquids. Water 1.0000 Alcohol 7000 Mercury 0333 Benzine 4500 Glycerine .5550 Lead (melted) 0402 Sulphur (melted) 2340 Tin (melted) 0637 Sulphuric acid 3350 Oil of turpentine 4260 Gases. Air 23751 Superheated steam .... ... .4805 Oxygen 21751 Carbonic oxide (CO).... .. .2479 Nitrogen 24380 Carbonic acid ( C0 2 ) ■■■■ ... .2170 Hydrogen 3.40900 Olefiant gas ... .4040 TEMPERATURES AND LATENT HEATS OF FUSION AND OF VAPORIZATION. Substance. Temperature of Fusion. Temperature of Vaporization. Latent Heat of Fusion. Latent Heat of Vaporization. Water 32° 212° 142.65 966.6 Mercury -37.8° 662° 5.09 157 Sulphur. 228.3° 824° 13.26 Tin 446° 25.65 Lead 626° 9.67 Zinc 680° 1,900° 50.63 493 Alcohol Unknown 173° 372 Oil of turpentine ... 14° 313° 124 Linseed oil 600° Aluminum 1,400° Copper 2,100° Cast iron 2,192° 3,300° Wrought iron 2,912° 5,000° Steel 2,520° Platinum 3,632° Iridium 4,892° Example.— How many units of heat are required to melt 10 lb. of zinc from a temperature of 60° F.? COEFFICIENTS OF EXPANSION. 19 Solution.— The specific heat of zinc is found from the table to be .0956. Hence, the number of heat units necessary to raise it to the melting point is 10 X (680 — 60) X .0956 = 592.72. Latent heat of fusion = 50.63 heat units. Hence, the total number of heat units required is 592.72 + 10 X 50.63 = 1,099.02. HEAT. Coefficient of Expansion for a Number of Substances. Name of Substance. Linear Expansion. Surface Expansion. Cubic Expansion. Cast iron .00000617 .00001234 .00001850 Copper .00000955 .00001910 .00002864 Brass .00001037 .00002074 .00003112 Silver .00000690 .00001390 .00002070 Bar iron .00000686 .00001372 .00002058 Steel (untempered) .00000599 .00001198 .00001798 Steel (tempered) .00000702 .00001404 .00002106 Zinc .00001634 .00003268 .00004903 Tin .00001410 .00002820 .00003229 Mercury .00003334 .00006668 .00010010 Alcohol .00019259 .00038518 .00057778 Gases .00203252 Example.— A wrought-iron bar 22 ft. long is heated from 70° to 300°. How much will it lengthen? Solution— 22 X (300 - 70) X .00000686 = .0347116 ft. = .41654 in. ALLOYS. Note. — A = Antimony, B = Bismuth, C = Copper, G = Gold, I = Iron, L = Lead, N = Nickel, S = Silver, T = Tin, Z = Zinc. Name. Proportions. Brass, common yellow 2 C, 1 Z Brass, to be rolled 32 C, 10 Z, 1.5 T Brass castings, common ... 20 C, 1.25 Z, 2.5 T Brass castings, hard 25 C, 2 Z, 4.5 T Brass propellers 8 C, .5 Z, 1 T Gun metal 8 C, 1 T 20 USEFUL TABLES. Alloys — ( Continued). Name. Proportions. Copper flanges 9 C, 1 Z, .26 T Muntz’s metal 6 C, 4Z Statuary 91.4 C, 5.53 Z, 1.7 T, 1.37 L German silver 2 C , 7.9 N, 6.3 Z, 6.5 1 Britannia metal 50 A, 25 T, 25 B Chinese silver 65.1 C, 19.3 Z, 13 N, 2.58 S , 12 1 Chinese white copper 20.2 C, 12.7 Z, 1.3 T, 15.82V Medals 100 C, 8 Z Pinchbeck . 5 C, 1 Z Babbitt’s metal 25 T, 2 A, .5 C Bell metal, large 3 C, 1 T Bell metal, small 4 C f 1 T Chinese gongs 40.5 C, 9.2 T Telescope mirrors 33.3 C , 16.7 T White metal, ordinary 3.7 C, 3.7 Z, 14.2 T, 28.4 A White metal, hard 35 C, 13 Z, 2.2 T Sheeting metal 56 C, 45 Z, 12 arsenic Metal, expands in cooling 75 L, 16.7 A, 8.3 B ALLOYS FOR SOLDERS. Name. Proportions. Melting Point. Newton’s fusible 8 B, 5 L, 3 T, 212° Rose’s fusible 2 B, 1 L, 1 T, 201° A more fusible 5 B, 3 L, 2 T, 199° Still more fusible 12 T, 25 L, 50 B, 13 cadmium, 155° For tin solder, coarse, 1 T, 3 X, 500° For tin solder, ordinary 2 T, 1 L, 360° For brass, soft spelter 1 C, 1 Z, 550° B ard, for iron 2 C, 1 Z, 700° 1 or steel 19 S, 3 C, 1 Z For fine brasswork 1 S, 8 C, 8Z Pewterer’s soft solder 2 B, 4X, 3 T Pewterer’s soft solder 1 B, 1 L, 2 T Gold solder 24 G, 2S,1C Silver solder, hard 4 S, 1 C Silver solder, soft 2 S, 1 brass wire For lead 16 T, 33 X ROLLED IRON. 21 WEIGHT OF ROUND AND SQUARE ROLLED IRON. From ^ in. to 9 % in- in Diameter, and 1 ft. in Length. Side or Diam. Inches. Weight. Lb. per ft. Side or Diam. Inches. Weight. Lb. per ft. Round. Square. Round. Square. i h .010 .013 3% 39.864 50.756 % .041 .053 4 42.464 54.084 .093 .118 4% 45.174 57.517 k .165 .211 4 y 47.952 61.055 s| .373 .475 4% 50.815 64.700 .663 .845 4*4 53.760 68.448 % 1.043 1.320 4^1 56.788 72.305 7a 1.493 1.901 59.900 76.264 /8 2.032 2.588 m 63.094 80.333 1 2.654 3.380 5 66.350 84.480 V/s 3.359 4.278 5 % 69.731 88.784 V4 4.147 5.280 534 73.172 93.168 1% 5.019 6.390 5% 76.700 97.657 5.972 7.604 534 80.304 102.240 i% 7.010 8.926 5% 84.001 106.953 ■ M 8.128 10.352 534 87.776 111.756 9.333 11.883 91.634 116.671 2 10.616 13.520 6 95.552 121.664 2% 11.988 15.263 103.704 132.040 2/i 13.440 17.112 112.160 142.816 2^1 14.975 19.066 6?| 120.960 154.012 16.588 21.120 7 130.048 165.632 2tI 18.293 23.292 7^ 139.544 177.672 2M 20.076 25.560 7>| 149.328 190.136 2% 21.944 27.939 7^ 159.456 203.024 3 23.888 30.416 8 169.856 216.336 3% 25.926 33.010 834 180.696 230.068 3K 28.040 35.704 8/1 191.808 244.220 n 30.240 38.503 3/4 203.260 258.800 32.512 41.408 9 215.0-iO 273.792 3 P 34.886 44.418 227.152 289.220 3/4 37.332 47.534 9>1 239.600 305.056 WEIGHT OF SHEET LEAD. Thickness. Inches. W£ht. Thickness. Inches. W’ght. Lb. Thickness. Inches. W’ght. .017 1 .085 5 .152 9 .034 2 .101 6 .169 10 .051 3 .118 7 .186 11 .068 4 .135 8 .203 • 12 PROPORTIONS OF THE UNITED STATES STANDARD SCREW THREADS, NUTS, AND BOLT HEADS. Notation of letters. All dimensions in inches. D — outside diameter of screw;. d = diameter of root of thread, or of hole in the nut; p = pitch of screw; t = number of threads per inch; / = flat top and bottom; © = outside diameter of hexagon nut or bolt head; < = inside [diameter of hexagon, or side of square nut or bolt head; s = diagonal of square nut or bolt head; h = height of rough or unfinished bolt head. The height of finished nut or bolt head is made equal to the diameter D of the screw. V = vu d = D- D + 10 — 2.909 1 16.64 ' ~ p 1.299 . 3 Z> 1 “• l = ^r + 8- 0= t = 1.414 i. 1.155 i. /=|. CAST-IRON PIPE. 23 WEIGHT OF CAST-IRON PIPE PER FOOT IN POUNDS. These weights are for plain pipe. For hautboy pipe add 8 in. in length for each joint. For copper add for lead, g ; for welded iron, add T V, or multiply by 1.0667. Thickness of Pipe in Inches. liore. Inches. 34 % 'X % % Vs 1 V4 1% 1 3.07 5.07 7.38 1M 3.69 6.00 8.61 4.30 6.92 9.84 1% 4.92 7.84 11.10 2 5.53 8.76 12.30 16.2 2 M 6.15 9.69 13.50 17.7 2 0 6.76 10.60 14.80 19.2 24.0 2/4 7.37 11.50 16.00 20.8 25.9 3 7.98 12.50 17.20 22.3 27.7 33.4 3 % 9.21 14.30 19.70 25.4 31.4 37.7 4 10.30 16.10 22.20 28.5 35.1 42.0 4 % 11.70 18.00 24.60 31.5 38.8 46.3 5 12.90 19.80 27.10 34.6 42.5 50.6 5% 14.20 21.70 29.50 37.7 46.1 54.9 6 15.40 23.50 32.00 40.8 49.8 59.2 68.9 6% 16.60 25.40 34.50 43.8 53.5 63.5 73.8 84.4 7 17.80 27.20 36.90 46.9 57.2 67.8 78.7 89.4 7^ 19.10 29.10 39.40 50.0 60.9 72.1 83.7 95.5 108 8 20.30 30.90 41.80 53.1 64.6 76.4 88.6 101.0 114 127 8% 21.50 32.80 44.30 56.1 68.3 80.7 93.5 107.0 120 134 9 22.80 34.60 46.80 59.2 72.0 85.1 98.4 112.0 126 140 9 >4 24.00 36.40 49.20 62.3 75.7 89.3 103.0 118.0 132 147 10 25.10 38.30 51.70 65.3 79.4 93.6 108.0 123.0 138 164 11 27.60 42.00 56.60 71.5 86.7 102.0 118.0 134.0 151 168 12 30.00 45.70 61.50 77.7 94.1 111.0 128.0 145.0 163 181 13 32.50 49.40 66.40 83.8 102.0 120.0 138.0 156.0 175 195 14 35.00 53.10 71.40 89.4 109.0 128.0 148.0 168.0 188 208 15 37.40 56.70 76.30 96.1 116.0 137.0 158.0 179.0 200 222 16 39.10 60.40 81.20 102.0 124.0 145.0 167.0 190.0 212 235 17 42.30 64.10 86.10 108.0 131.0 154.0 177.0 201.0 225 249 18 44.80 67.80 91.00 115.0 139.0 163.0 187.0 212.0 237 262 19 47.30 71.50 96.00 121.0 146.0 171.0 197.0 223.0 249 276 20 49.70 75.20 101.00 127.0 153.0 180.0 207.0 234.0 261 289 22 54.60 82.60 111.00 139.0 168.0 196.0 227.0 256.0 286 316 24 59.60 89.90 121.00 152.0 183.0 214.0 246.0 278.0 311 343 26 64.50 97.30 131.00 164.0 198.0 231.0 266.0 300.0 335 370 28 69.40 105.00 140.00 176.0 212.0 249.0 286.0 323.0 360 397 30 74.20 112.00 150.00 188.0 227.0 266.0 305.0 345.0 384 424 24 USEFUL TABLES. TABLE OF STANDARD DIMENSIONS OF WROUGHT- IRON WELDED PIPES. Nominal Diameter. External Diameter. Thickness. Internal Diameter. I Internal Circum- ference. External Circum- ference. Length of Pipe per Sq. Ft. of Inter- nal Surface. Length of Pipe per Sq. Ft. of Exter- nal Surface. Internal Area. Weight per Foot. No. of Threads per Inch of Screw. In. In. In. In. In. In. Ft. Ft. In. Lb. Vs .40 .068 .27 .85 1.27 14.15 9.440 .057 .24 27 17 .54 .088 .36 1.14 1.70 10.50 7.075. .104 .42 18 78 .67 .091 .49 1.55 2.12 7.67 5.657 .192 .56 18 .84 .109 .62 1.96 2.65 6.13 4.502 .305 .84 14 /A 1.05 .113 .82 2.59 3.30 4.64 3.637 .533 1.13 14 1 1.31 .134 1.05 3.29 4.13 3.66 2.903 .863 1.67 IK 1.66 .140 1.38 4.33 5.21 2.77 2.301 1.496 2.26 11 % i % 1.90 .145 1.61 5.06 5.97 2.37 2.010 2.038 2.69 11 % 2 2.37 .154 2.07 6.49 7.46 1.85 1.611 3.355 3.67 11 % 2% 2.87 .204 2.47 7.75 9.03 1.55 1.328 4.783 5.77 8 3 3.50 .217 3.07 9.64 11.00 1.24 1.091 7.388 7.55 8 4.00 .226 3.55 11.15 12.57 1.08 0.955 9.887 9.05 8 4 4.50 .237 4.03 12.65 14.14 .95 0.849 12.730 10.73 8 4 ^ 5.00 .247 4.51 14.15 15.71 .85 0.765 15.939 12.49 8 5 5.56 .259 5.04 15.85 17.47 .78 0.629 19.990 14.56 8 6 6.62 .280 6.06 19.05 20.81 .63 0.577 28.889 18.77 8 7 7.62 .301 7.02 22.06 23.95 .54 0.505 38.737 23.41 8 8 8.62 .322 7.98 25.08 27.10 .48 0.444 50.039 28.35 8 9 9.69 .344 9.00 28.28 30.43 .42 0.394 63.633 34.08 8 10 10.75 .366 10.02 i 31.47 33.77 .38 0.355 78.838 40.64 8 FLUXES FOR SOLDERING OR WELDING. Iron Borax Tinned iron Resin Copper and brass Sal ammoniac Zinc Chloride of zinc Lead Tallow or resin Lead and tin pipes Resin and sweet oil Steel.— Pulverize together 1 part of sal ammoniac and 10 parts of borax and fuse until clear. When solidified, pul- verize to powder. STEAM TABLES. 25 STEAM TABLES. Whenever the pressure of saturated steam is changed, there are other properties that change with it. These prop- erties are the following: 1. The temperature of the steam, or, what is the same thing, the boiling point. 2. The number of B. T. U. required to raise a pound of water from 32° (freezing) to the boiling point corresponding to the given pressure. This is called the heat of the liquid. 3. The number of B. T. U. required to change the water at the boiling temperature into steam at the same tempera- ture. This is called the latent heat of vaporization , or, simply, the latent heat. 4. The number of heat units required to change a pound of water at 32° to steam of the required temperature and pressure. This is called the total heat of vaporization , or, simply, the total heat. It is plain that the total heat is the sum of the heat of the liquid and the latent heat. That is, total heat = heat of liquid + latent heat. 5. The specific volume of the steam at the given pressure; that is, the number of cubic feet occupied by a pound of steam of the given pressure. 6. The density of the steam; that is, the weight of 1 cubic foot of the steam at the given pressure. All the above properties are different for different pres- sures. For example, if steam boils under atmospheric pres- sure, the temperature is 212°; the heat of the liquid is 180.531 B. T. U.; the latent heat, 966.069 B. T. U.; the total heat, 1,146.6 B. T. U. A pound of steam at this pressure occupies 26.37 cu. ft., and a cubic foot of the steam weighs about .037928 lb. When the pressure is 70 lb. per sq. in. above' vacuum, the temperature is 302.774°; the heat of the liquid is 272.657 B. T.-U.; the latent heat is 901.629 B. T. U.; the total heat is 1,174.286 B. T. U. A pound of the steam occupies 6.076 cu. ft., and a cubic foot of the steam weighs .164584 lb. These properties have been determined by direct experi- ment for all ordinary steam pressures. They are given in the table of the properties of saturated steam, pages 29-31. 26 USEFUL TABLES. Explanation of the Table. Column 1 gives the pressures from 1 to 300 lb. These pres- sures are above vacuum. The steam gauges fitted on steam boilers register the pressure above the atmosphere. That is, if the steam is at atmospheric pressure, 14.7 lb. per sq. in., the gauge registers 0. Consequently, the atmospheric pressure must be added to the reading of the gauge to obtain the pres- sure above vacuum. In using the table, care must be taken not to use the gauge pressures without first adding 14.7 lb. per sq. in. Pressures registered above vacuum are called absolute pressures. The pressures given in column 1 are absolute. Absolute pressure per square inch = gauge pressure per square inch -f 14.7. Column 2 gives the temperature of the steam when at the pressure shown in column 1. Column 3 gives the heat of the liquid. It will be noticed that the values in column 3 may be obtained approximately by subtracting 32° from the temperature in column 2. If the specific heat of water were exactly 1.00, it would, of course, take exactly 212 — 32 = 180 B. T. U. to raise a pound of water from 32° to 212°. But experiment shows that the specific heat of water is slightly greater than 1.00 when the temper- ature of the water is above 62°, and it therefore takes 180.531 B. T. U. to raise a pound of water from 32° to 212°. Column 4 gives the latent heat of vaporization , which is seen to decrease slightly as the pressure increases. Column 5 gives the total heat of vaporization. The values in column 5 may be obtained by adding together the corre- sponding values in columns 3 and 4. Column 6 gives the weight of a cubic foot of steam in pounds. As would be expected, the steam becomes denser as the pressure rises, and weighs more per cubic foot. Column 7 gives the number of cubic feet occupied by 1 pound of steam at the given pressure. It will be noticed that the corresponding values of columns 6 and 7 multiplied together always produce 1. Thus, for 31.3 pounds pressure, gauge, .11088 X 9.018 = 1.000, nearly. Column 8 gives the ratio of the volume of a pound of STEAM TABLES. 27 steam at the given pressure, and the volume of a pound of water at 39.2°. The values in column 8 may be obtained by dividing 62.425, the weight of a cubic foot of water at 39.2°, by the numbers in column 6. Examples on the Use of the Steam Table. Example 1.— Calculate the heat required to change 5 lb. of water at 32° into steam at 92 lb. pressure above vacuum. Solution.— From column 5, the total heat of 1 lb. at 92 lb. pressure is 1,180.045 B. T. U. 1,180.045 X 5 = 5,900.225 B. T. U. Example 2.— How many heat units are required to raise lb. of water from 32° to 250° F.? Solution.— Looking in column 3, the heat of the liquid of 1 lb. at 250.293° is 219.261 B. T. U. 219.261 — .293 = 218.968 B. T. U. = heat of liquid for 250°. Then, for 8£ lb. it is 218.968 X 84- = 1,861.228 B. T. U. Example 3.— How many foot-pounds of work will it require to change 60 lb. of boiling water at 80 lb. pressure, absolute, into steam of the same pressure? Solution.— Looking under column 4, the latent heat of vaporization is 895.108; that is, it takes 895.108 B. T. U. to change 1 lb. of water at 80 lb. pressure into steam of the same pressure. Therefore, it takes 895.108 X 60 = 53,706.48 B. T. U. to perform the same operation on 60 lb. of water. 53,706.48 X 778 = 41,783,641.44 ft.-lb. Example 4.— Find the volume occupied by 14 lb. of steam at 30 lb., gauge pressure. Solution. — 30 lb., gauge pressure = 30 + 14.7 = 44.7, abso- lute pressure. The nearest pressure in the table is 44 lb., and the volume of a pound of steam at that pressure is 9.403 cu. ft. The volume of a pound at 46 lb. pressure is 9.018 cu. ft. 9.403 — 9.018 = .385 cu. ft., the difference in volume for a 385 difference in pressure of 2 lb. = .1925 cu. ft., the differ- ence in volume for a difference in pressure of 1 lb. .1925 X .7 = .135 cu. ft., the difference in volume for a difference in pressure of .7 lb. Therefore, 9.403 — .135 = 9.268 cu. ft. is the volume of 1 lb. of steam at 44.7 lb. pressure. The .135 cu. ft. 28 USEFUL TABLES. is subtracted from 9.403 cu. ft., since the volume is less for a pressure of 44.7 lb. than for a pressure of 44 lb. 9.268X14 = 129.752 cu. ft. Example 5.— Find the weight of 40 cu. ft. of steam at a temperature of 254° F. Solution.— The weight of 1 cu. ft. of steam at 254.002°, from the table, is .078839 lb. Neglecting the .002°, the weight of 40 cu. ft. is, therefore, .078839 X 40 = 3.15356 lb. Example 6.— How many pounds of steam at 64 lb. pressure, absolute, are required to raise the temperature of 300 lb. of water from 40° to 130° F., the water and steam being mixed ? Solution.— The number of heat units required to raise 1 lb. from 40° to 130° is 130 — 40 = 90 B. T. U. (Actually a little more than 90 would be required, but the above is near enough for all practical purposes.) Then, to raise 300 lb. from 40° to 130° requires 90 X 300 = 27,000 B. T. U. This quantity of heat must necessarily come from the steam. Now, 1 lb. of steam at 64 lb. pressure gives up, in condensing, its latent heat of vaporization, or 905.9 B. T. U. But, in addi- tion to its latent heat, each pound of steam on condensing must give up an additional amount of heat in falling to 130°. Since the original temperature of the steam was 296.805° F. (see table), each pound gives up by its fall of temperature 296.805 — 130 = 166.805 B. T. U. Therefore, each pound of the steam gives up a total of 905.9 + 166.805 = 1,072.705 B. T. U. It will, therefore, take = 25.17 lb. of steam to 1,072.705 accomplish the desired result. With the steam tables a reliable thermometer may be used for ascertaining the pressure of saturated steam or for testing the accuracy of a steam gauge. The temperature of the steam being measured by the thermometer, the corresponding abso- lute pressure is found from the steam tables; the gauge pres- sure is then found by subtracting 14.7 from the absolute pressure. Thus, the temperature of the steam in a condenser being 142°, we find from the steam tables that the correspond- ing absolute pressure is 3 lb. per sq. in., nearly. STEAM TABLES. 29 The Properties of Saturated Steam. Pressure Above Vacuum in Pounds per Square Inch. Temperature, Fahrenheit Degrees. Quantity of Heat in British Thermal Units. Weight of a Cubic Foot of Steam in Pounds. Volume of a Pound of Steam in Cubic Feet. Ratio of Vol. of Steam to Vol. of Equal Weight of Dist. Water at Temp, of Maximum Density. Required to Raise Tem- perature of the Water From 32° to t°. Total Latent Heat at Pressure p. Total Heat Above 32°. 1 2 3 4 5 6 7 8 P t Q L H W V R 1 102.018 70.040 1,043.015 1,113.055 .003027 330.4 20,623 2 ! 126.302 94.368 1,026.094 1,120.462 .005818 171.9 10,730 3 : 141.654 109.764 1,015.380 1,125.144 .008522 117.3 7,325 4 153.122 121.271 1,007.370 1,128.641 .011172 89.51 5,588 5 162.370 130.563 1,000.899 1,131.462 .013781 72.56 4,530 6 170.173 138.401 995.441 1,133.842 .016357 61.14 3,816 7 176.945 145.213 990.695 1,135.908 .018908 52.89 3,302 8 182.952 151.255 986.485 1,137.740 .021436 46.65 2,912 9 188.357 156.699 982.690 1,139.389 .023944 41.77 2,607 10 193.284 161.660 979.232 1,140.892 .026437 37.83 2,361 11 197.814 166.225 976.050 1,142.275 .028911 34.59 2,159 12 1 202.012 170.457 973.098 1,143.555 .031376 31.87 1,990 13 I 205.929 174.402 970.346 1,144.748 .033828 29.56 1,845 14 209.604 178.112 967.757 1,145.869 .036265 27.58 1,721 14.69 : 212.000 180.531 966.069 1,146.600 .037928 26.37 1,646 15 213.067 181.608 965.318 1,146.926 .038688 25.85 1,614 16 j 216.347 184.919 963.007 1.147.926 .041109 24.33 1,519 17 ! 219.452 188.056 960.818 1,148.874 .043519 22.98 1,434 18 222.424 191.058 958.721 1,149.779 .045920 21.78 1,359 19 j 225.255 193.918 956.725 1,150.613 .048312 20.70 1,292 30 USEFUL TABLES. Table— ( Continued). 1 2 3 4 5 6 7 8 V t Q L H W V It 20 227.964 196.655 954.814 1,151.469 .050696 19.730 1,231.0 22 233.069 201.817 951.209 1,153.026 .055446 18.040 1,126.0 24 237.803 206.610 947.861 1,154.471 .060171 16.620 1,038.0 26 242.225 211.089 944.730 1,155.819 .064870 15.420 962.3 28 246.376 215.293 941.791 1,157.084 .069545 14.380 897.6 30 250.293 219.261 939.019 1,158.280 .074201 13.480 841.3 32 254.002 223.021 936.389 1,159.410 .078839 12.680 791.8 34 257.523 226.594 933.891 1,160.485 .083461 11.980 948.0 36 260.883 230.001 931.508 1,161.509 .088067 11.360 708.8 38 264.093 233.261 929.227 1,162.488 .092657 10.790 673.7 40 267.168 236.386 927.040 1,163.426 .097231 10.280 642.0 42 270.122 239.389 924.940 1,164.329 .101794 9.826 613.3 44 272.965 242.275 922.919 1,165.194 .106345 9.403 587.0 46 275.704 245.061 920.968 1,166.029 .110884 9.018 563.0 48 278.348 247.752 919.084 1,166.836 .115411 8.665 540.9 50 280.904 250.355 917.260 1,167.615 .119927 8.338 520.5 52 283.381 252.875 915.494 1,168.369 .124433 8.037 501.7 54 285.781 255.321 913.781 1,169.102 .128928 7.756 484.2 56 288.111 257.695 912.118 1,169.813 .133414 7.496 467.9 58 290.374 260.002 910.501 1,170.503 .137892 7.252 452.7 60 292.575 262.248 908.928 1,171.176 .142362 7.024 438.5 62 294.717 264.433 907.396 1,171.829 .146824 6.811 425.2 64 296.805 266.566 905.900 1,172.466 .151277 6.610 412.6 66 298.842 268.644 904.443 1,173.087 .155721 6.422 400.8 68 300.831 270.674 903.020 1,173.694 .160157 6.244 389.8 70 302.774 272.657 901.629 1,174.286 .164584 6.076 379.3 72 304.669 274.597 900.269 1,174.866 .169003 5.917 369.4 74 306.526 276.493 898.938 1,175.431 .173417 5.767 360.0 76 308.344 278.350 897.635 1,175.985 .177825 5.624 351.1 78 310.123 280.170 896.359 1,176.529 .182229 5.488 342.6 80 311.866 281.952 895.108 1,177.060 .186627 5.358 334.5 82 313.576 283.701 893.879 1,177.580 .191017 5.235 326.8 84 315.250 285.414 892.677 1,178.091 .195401 5.118 319.5 86 316.893 287.096 891.496 1,178.592 .199781 5.006 312.5 88 318.510 288.750 890.335 1,179.085 .204155 4.898 305.8 STEAM TABLES. 31 Table— ( Continued ) . 1 2 3 4 5 6 7 8 p t Q L H W V R 90 320.094 290.373 889.196 1,179.569 .208525 4.796 299.4 92 321.653 291.970 888.075 1,180.045 .212892 4.697 293.2 94 323.183 293.539 886.972 1,180.511 .217253 4.603 287.3 96 324.688 295.083 885.887 1,180.970 .221604 4.513 281.7 98 326.169 296.601 884.821 1,181.422 .225950 4.426 276.3 100 327.625 298.093 883.773 1,181.866 .230293 4.342 271.1 105 331.169 301.731 881.214 1,182.945 .241139 4.147 258.9 110 334.582 305.242 878.744 1,183.986 .251947 3.969 247.8 115 337.874 308.621 876.371 1,184.992 .262732 3.806 237:6 120 341.058 311.885 874.076 1,185.961 .273500 3.656 228.3 125 344.136 315.051 871.848 1,186.899 .284243 3.518 219.6 130 347.121 318.121 869.688 1,187.809 .294961 3.390 211.6 135 350.015 321.105 867.590 1,188.695 .305659 3.272 204.2 140 352.827 324.003 865.552 1,189.555 .316338 3.161 197.3 145 355.562 326.823 863.567 1,190.390 .326998 3.058 190.9 150 358.223 329.566 861.634 1,191.200 .337643 2.962 184.9 160 363.346 334.850 857.912 1,192.762 .358886 2.786 173.9 170 368.226 339.892 854.359 1,194.251 .380071 2.631 164.3 180 372.886 344.708 850.963 1,195.671 .401201 2.493 155.6 190 377.352 349.329 847.703 1,197.032 .422280 2.368 147.8 200 381.636 353.766 844.573 1,198.339 .443310 2.256 140.8 210 385. / 59 358.041 841.556 1,199.597 .464295 2.154 134.5 220 389.736 362.168 838.642 1,200.810 .485237 2.061 128.7 230 393.575 366.152 835.828 1,201.980 .506139 1.976 123.3 240 397.285 370.008 833.103 1,203.111 .527003 1.898 118.5 250 400.883 373.750 830.459 1,204.209 .547831 1.825 114.0' 260 404.370 377.377 827.896 1,205.273 .568626 1.759 109.8 270 407.755 380.905 825.401 1,206.306 .589390 1.697 105.9 280 411.048 384.337 822.973 1,207.310 .610124 1.639 102.3 290 414.250 387.677 820.609 1,208.286 .630829 1.585 99.0’ 300 417.371 390.933 818.305 1,209.238 .651506 1.535 95.8 32 USEFUL TABLES. LOGARITHMS. EXPONENTS. By the use of logarithms, the processes of multiplication, division, involution, and evolution are greatly shortened, and some operations may be performed that would be impossible without them. Ordinary logarithms cannot be applied to addition and subtraction. The logarithm of a number is that exponent by which some fixed number, called the base, must be affected in order to equal the number. Any number may be taken as the base. Suppose we choose 4. Then the logarithm of 16 is 2, because 2 is the exponent by which 4 (the base) must be affected in order to equal 16, since 4 2 = 16. In this case, instead of reading 4 2 as 4 square, read it 4 exponent 2. With the same base, the logarithms of 64 and 8 would be 3 and 1.5, respect- ively, since 4 3 = 64, and 4 1 - 5 = 4 5 = 8. In these cases, as in the preceding, read 4 3 and 4 1 - 6 as 4 exponent 3, and 4 expo- nent 1.5, respectively. Although any positive number except 1 can be used as a base and a table of logarithms calculated, but two numbers have ever been employed. For all arithmetical operations (except addition and subtraction) the logarithms used are called the Briggs, or common, logarithms, and the base used is 10. In abstract mathematical analysis, the logarithms used are variously called hyperbolic, Napierian , or natural loga- rithms, and the base is 2.718281828+. The common logarithm of any number may be converted into a Napierian logarithm by multiplying the common logarithm by 2.30258509+, which is usually expressed as 2.3026, and sometimes as 2.3. Only the common system of logarithms will be considered here. Since in the common system the base is 10, it follows that, since 10 1 = 10, 10 2 = 100, 10 3 = 1,000, etc., the logarithm (ex- ponent) of 10 is 1, of 100 is 2, of 1,000 is 3, etc. For the sake of brevity in writing, the words “ logarithm of” are abbreviated to “log.” Thus, instead of writing logarithm of 100 = 2, write log 100 = 2. When speaking, however, the words for which “ log” stands should always be pronounced in full. LOGARITHMS. 33 From the above it will be seen that, when the base is 10, since 10° = 1, the exponent 0 = log 1; since 10 1 = 10,. the exponent 1 = log 10; since 10 2 = 100, the exponent 2 = log 100; since 10 3 = 1,000, the exponent 3 = log 1,000; etc. Also, since 10- 1 = = .1, the exponent — 1 = log .1; since 10~ 2 = = .01, .the exponent — 2 = log .01; since 10~ 3 = = *001, the exponent — 3 = log .001; etc. From this it will be seen that the logarithms of exact powers of 10 and of decimals like .1, .01, and .001 are the whole numbers 1, 2, 3, etc. and —1, —2, —3, etc., respectively. Only numbers consisting of 1 and one or more ciphers have whole numbers for logarithms. Now, it is evident that, to produce a number between 1 and 10, the exponent of 10 must be a fraction; to produce a number between 10 and 100, it must be 1 plus a fraction; to produce a number between 100 and 1,000, it must be 2 plus a fraction; etc. Hence, the logarithm of any number between 1 and 10 is a fraction; of any number between 10 and 100, 1 plus a fraction; of any number between 100 and 1,000, 2 plus a fraction, etc. A logarithm, therefore, usually con- sists of two parts: a whole number, called the characteristic , and a fraction, called the mantissa. The mantissa is always expressed as a decimal. For example, to produce 20, 10 must have an exponent of approximately 1.30103, or 10 1 . 30103 = 20, very nearly, the degree of exactness depending on the num- ber of decimal places used. Hence, log 20 = 1.30103, 1 being the characteristic, and .30103, the mantissa. Referring to the second part of the preceding table, it is clear that the logarithms of all numbers less than 1 are nega- tive, the logarithms of those between 1 and .1 being —1 plus a fraction. For, since log .1 = —1, the logarithms of .2, .3, etc. (which are all greater than .1, but less than 1) must be greater than —1; i. e., they must equal —1 plus a fraction. For the same reason, to produce a number between .1 and .01, the logarithm (exponent of 10) would be equal to —2 plus a fraction, and for a number between .01 and .001, it would be equal to —3 plus a fraction. Hence, the logarithm 34 USEFUL TABLES. of any number between 1 and .01 has a negative character- istic of 1 and a positive mantissa; of a number between .1 and .01, a negative characteristic of 2 and a positive mantissa; of a number between .01 and .001, a negative characteristic of 3 and a positive mantissa; of a number between .001 and .0001, a negative characteristic of 4 and a positive mantissa, etc. The negative characteristics are dis- tinguished from the positive by the — sign written over the char- acteristic. Thus, 3 indicates that 3 is negative. It must be remembered that in all cases the mantissa is posi- tive. Thus^the logarithm 1.30103 means +1 + .30103, and the logarithm 1.30103 means —1 + .30103. Were the minus sign written in front of the characteristic, it would indicate that the entire logarithm was negative. Thus, —1.30103 = —1 -.30103. Rule for Characteristic.— Startingfrom the unit figure, count the number of places to the first (left-hand) digit of the given number, calling unit’s place zero; the number of places thus counted will be the required characteristic. If the first digit lies to the left of the unit figure, the characteristic is positive; if to the right, negative. If the first digit of the number is the unit figure, the characteristic is 0. Thus, the charactertisic of the logarithm of 4,826 is 3, since the first digit, 4, lies in the 3d place to the left of the unit figure, 6. The characteristic of the logarithm of 0.0000072 is —6 or 6 , since the first digit, 7, lies in the 6th place to the right of the unit figure. The char- acteristic of the logarithm of 4.391 is 0, since 4 is both the first digit of the number and also the unit figure. TO FIND THE LOGARITHM OF A NUMBER. To aid in obtaining the mantissas of logarithms, tables of logarithms have been calculated, some of which are very elaborate and convenient. In the Table of Logarithms, the mantissas of the logarithms of numbers from 1 to 9,999 are given to five places of decimals. The mantissas of logarithms of larger numbers can be found by interpolation. The table contains the mantissas only; the characteristics may be easily found by the preceding rule. LOGARITHMS. 35 The table depends on the principle, which will be explained later, that all numbers having the same figures in the same order have the same mantissa, without regard to the position of the decimal point, which affects the charac- teristic only. To illustrate, if log 206 = 2.31387, then, log 20.6 = 1.31387; log .206 = 1.31387; log 2.06 = .31387; log .0206 = 2.31387; etc. To find the logarithm of a number not* having more than four figures: Rule . — Find the first three significant figures of the number whose logarithm is desired , in the left-hand column; find the fourth figure in the column at the top (or bottom) of the page; and in the column under (or above) this figure, and opposite the first three figures previously found, will be the mantissa or decimal pari of the logarithm. The characteristic being found as pre- viously described, write it at the left of the mantissa, and the resulting expression will be the logarithm of the required number. Example.— F ind from the table the logarithm (a) of 476; (b) of 25.47; (c) of 1.073; (d) of .06313. Solution.— (a) In order to economize spaoe and make the labor of finding the logarithms easier, the first two figures of the mantissa are given only in the column headed 0. The last three figures of the mantissa, opposite 476 in the column headed N (N stands for number), are 761, found in the column headed 0; glancing upwards, we find the first two figures of the mantissa, viz., 67. The characteristic is 2; hence, log 476 = 2.67761. Note.— Since all numbers in the table are decimal frac- tions, the decimal point is omitted throughout; this is cus- tomary in all tables of logarithms. (b) To find the logarithm of 25.47, we find the first three figures, 254, in the column headed N, and on the same hori- zontal line, under the column headed 7 (the fourth figure of the given number), will be found the last three figures of the mantissa, viz., 603. The first two figures are evidently 40, and the characteristic is 1; hence, log 25.47 = 1.40603. (c) For 1.073; in the column headed 3, opposite 107 in the column headed N, the last three figures of the mantissa are found, in the usual manner, to be 060. It will be noticed 36 USEFUL TABLES. that these figures are printed *060, the star meaning that instead of glancing upwards in the column headed 0, and taking 02 for the first two figures, we must glance downwards and take the two figures opposite the number 108, in the left-hand column, i. e., 03. The characteristic being 0, log 1.073 = 0.03060$ or, more simply, .03060. (d) For .06313; the last three figures of the mantissa are found opposite 631, in column headed 3, to be 024. In this case, the first two figures .occur in the same row, and are 80. Since the characteristic is 2, log .06313 = 2.80024. If the original number contains but one digit (a cipher is not a digit), annex mentally two ciphers to the right of the digit; if the number contains but two digits (with no ciphers between, as in 4,008), annex mentally one cipher on the right before seeking the mantissa. Thus, if the logarithm ot 7 is wanted, seek the mantissa for 700, which is .84510; or, if the logarithm of 48 is wanted, seek the mantissa for 480, which is .68124. Or, find the mantissas of logarithms of num- bers between 0 and 100, on the first page of the tables. The process of finding the logarithm of a number from the table is technically called taking out the logarithm. To take out the logarithm of a number consisting of more than four figures, it is inexpedient to use more than five figures of the number when using five-place logarithms (the logarithms given in the accompanying table are five-place). Hence, if the number consists of more than five figures and the sixth figure is less than 5, replace all figures after the fifth with ciphers; if the sixth figure is 5 or greater, increase the fifth figure by 1 and replace the remaining figures with ciphers. Thus, if the number is 31,415,926, find the logarithm of 31,416,000; if 31,415,426, find the logarithm of 31,415,000. Example.— Find log 31,416. Solution.— Find the mantissa of the logarithm of the first four figures, as explained above. This is, in the present case, .49707. Now, subtract the number in the column headed 1, opposite 314 (the first three figures of the given number), from the next greater consecutive number, in this case 721, in the column headed 2. 721 — 707 = 14; this number is called the difference. At the extreme right of the page will be found a LOGARITHMS. 37 secondary table headed P. P., and at the top of one of these columns, in this table, in bold-face type, will be found the difference. It will be noticed that each column is divided into two parts by a vertical line, and that the figures on the left of this line run in sequence from 1 to 9. Considering the difference column headed 14, we see opposite the number 6 (6 is the last or fifth figure of the number whose logarithm we are taking out) the number 8.4, and we add this number to the mantissa found above, disregarding the decimal point in the mantissa, obtaining 49,707 + 8.4 = 49,715.4. Now, since 4 is less than 5, w r e reject it, and obtain for our complete mantissa .49715. Since the characteristic of the logarithm of 31,416 is 4, log 31,416 = 4.49715. Example.— Find log 380.93. Solution.— Proceeding in exactly the same manner as above, the mantissa for 3,809 is 58,081 (the star directs us to take 58 instead of 57 for the first two figures); the next greater mantissa is 58,092, found in the column headed 0, opposite 381 in column headed N. The difference is 092 — 081 = 11. Look- ing in the section headed P. P. for column headed 11, we find opposite 3, 3.3; neglecting the .3, since it is less than 5, 3 is the amount to be added to the mantissa of the logarithm of 3,809 to form the logarithm of 38,093. Hence, 58,081 + 3 = 58,084, and since the characteristic is 2, log 380.93 = 2.58084. Example— Find log 1,296,728. Solution.— Since this number consists of more than five figures and the sixth figure is less than 5, we find the loga- rithm of 1,296,700 and call it the logarithm of 1,296,728. The mantissa of log 1,296 is found to be 11,261. The difference is 294 — 261 = 33. Looking in the P. P. section for column headed 33, we find opposite 7, on the extreme left, 23.1; neg- lecting the .1, the amount to be added to the above mantissa is 23. Hence, the mantissa of log 1,296,728 = 11,261 + 23 = 11,284; since the characteristic is 6, log 1,296,728 = 6.11284. Example.— Find log 89.126. Solution.— Log 89.12 = 1.94998. Difference between this and log 80.13 = 1.95002 — 1.94998 = 4. The P. P. (propor- tional part) for the fifth figure of the number 6 is 2.4, or 2. Hence, log 89.126 = 1.94998 + .00002 = 1.95000. 38 USEFUL TABLES. Example.— Find log .096725. Solution.— Log .09672 = 2.98552. Difference = 4. P. P. for 5 = 2 Hence, log .096725 = 2.98554. To find the logarithm of a number consisting of five or more figures: Rule. — I. If the number consists of more than five figures and the sixth figure is 5 or greater , increase the fifth figure by 1 and write ciphers in place of the sixth and remaining figures. II. Find the mantissa corresponding to the logarithm of the first four figures , and substract this mantissa from the next greater mantissa in the table; the remainder is the difference. III. Find in the secondary table headed P. P. a column headed by the same number as that just found for the difference , and in this column , opposite the number corresponding to the fifth figure {or fifth figure increased by 1) of the given number {this figure is always situated at the left of the dividing line of the column ), will be found the P. P. {proportional part) for that number. The P. P. thus found is to be added to the mantissa found in II, as in the preceding examples , and the result is the mantissa of the logarithm of the given number , as nearly as may be found with five-place tables. TO FIND A NUMBER WHOSE LOGARITHM IS GIVEN. Rule. —I. Consider the mantissa first. Glance along the differ- ent columns of the table which are headed 0, until the first two figures of the mantissa are found. Then, glance down the same column until the third figure is found {or 1 less than the third figure) . Having found the first three figures, glance to the right along the row in which they are situated until the last three figures of the mantissa are found. Then, the number that heads the column in which the last three figures of the mantissa are found is the fourth figure of the required number, and the first three figures lie in the column headed N, and in the same row in which lie the last three figures of the mantissa. II. If the mantissa cannot be found in the table, find the mantissa that is nearest to, but less than, the given mantissa, and which call the next less mantissa. Subtract the next less mantissa LOGARITHMS. 39 from the next greater mantissa in the table to obtain the difference. Also, subtract the next less mantissa from the mantissa of the given logarithm, and call the remainder the P. P. Looking in the secondary table headed P. P. for the column headed by the difference just found, find the number opposite the P. P. just found (or the P. P. corresponding most nearly to that just found); this number is the fifth figure of the required number ; the fourth figure will be found at the top of the column containing the next less mantissa, and the first three figures in the column headed N and in the same row that contains the next less mantissa. III. Having found the figures of the number as above directed, locate the decimal point by the rules for the character- istic, annexing ciphers to bring the number up to the required number of figures if the characteristic is greater than 4. Example.— Find the number whose logarithm is 3.56867. Solution.— The first two figures of the mantissa are 56; glancing down the column, we find the third figure, 8 (in con- nection with 820), opposite 370 in the N column. Glancing to the right along the row containing 820, the last three figures of the mantissa, 867, are found in the column headed 4; hence, the fourth figure of the required number is 4, and the first three figures are 370, making the figures of the required number 3,704. Since the characteristic is 3, there are three figures to the left of the unit figure, and the number whose logarithm is 3.56867 is 3,704. Example.— Find the number whose logarithm is 3.56871. Solution.— The mantissa is not found in the table. The next less mantissa is 56,867; the difference between this and the next greater mantissa is 879 — 867 = 12, and the P. P. is 56,871 — 56,867 = 4. Looking in the P. P. section for the column headed 12* we do not find 4, but we do find 3.6 and 4.8. Since 3.6 is nearer 4 than 4.8, we take the number opposite 3.6 for the fifth figure of the required number; this is 3. Hence, the fourth figure is 4; the first three figures 370, and the figures of the number are 37,043. The charac- teristic being 3, the number is 3,704.3. Example.— Find the number whose logarithm is 5.95424. Solution. — The mantissa is found in the column headed 0, opposite 900 in the column headed N. Hence, the fourth 40 USEFUL TABLES. figure is 0, and the number is 900,000, the characteristic being 5. Had the logarithm been 5.95424, the number would have been .00009. Example.— Find the number whose logarithm is .93036. Solution.— The first three figures of the mantissa, 930, are found in the 0 column, opposite 852 in the N column; but since the last two figures of all the mantissas in this row are greater than 36, we must seek the next less mantissa in the preceding row. We find it to be 93,034 (the star directing us to use 93 instead of 92 for the first two figures), in the column headed 8. The difference for this case is 039 — 034 = 5, and the P. P. is 036 — 034 = 2. Looking in the P. P. section for the column headed 5, we find the P. P., 2, opposite 4. Hence, the fifth figure is 4; the fourth figure is 8; the first three figures 851, and the number is 8.5184, the characteristic being 0. Example.— Find the number whose logarithm is 2.05753. Solution.— The next less mantissa is found in column headed 1, opposite 114 in the N column; hence, the first four figures are 1,141. The difference for this case is 767 — 729 = 38, and the P. P. is 753 — 729 = 24. Looking in the P. P. section for the column headed 38, we find that 24 falls between 22.8 and 26.6. The difference between 24 and 22.8 is 1.2, and between 24 and 26.6 is 2.6; hence, 24 is nearer 22.8 than it is to 26.6, and 6, opposite 22.8, is the fifth figure of the number. Hence, the number whose logarithm is 2.05753 is .011416. In order to calculate by means of logarithms, a table is absolutely necessary. Hence, for this reason, we do not explain the method of calculating a logarithm. The work involved in calculating even a single logarithm is very great, and no method has yet been demonstrated, of which we are aware, by which the logarithm of a number like 121 can be calculated directly. Moreover, even if the logarithm could be readily obtained, it would be useless without a complete table, such as that which is here given, for the reason that «,fter having used it, say to extract a root, the number corresponding to the logarithm of the result could not be found. LOGARITHMS. 41 MULTIPLICATION BY LOGARITHMS. The principle upon which the process is based may be illustrated as follows: Let X and Y represent two numbers whose logarithms are x and y. To find the logarithm of their product, we have, from the definition of a logarithm, 10* = X, (1) and 10 y = Y. (2) Since both members of (1) may be multiplied by the same quantity without destroying the equality, they evidently may be multiplied by equal quantities like 10 y and Y. Hence, multiplying (1) by (2), member by member, 10*X10 V = 10 v + y = XY, or, by the definition of a logarithm, x + y = log X Y. But X Y is the product of X and Y, and x + y is the sum of their logarithms; from which it follows that the sum of the loga- rithms of two numbers is equal to the logarithm of their product. Hence, To multiply two or more numbers by using logarithms: Rule . — Add the logarithms of the several numbers , and the sum will be the logarithm of the product. Find the number corre- sponding to this logarithm, and the result will be the number sought. Example.— M ultiply 4.38, 5.217, and 83 together. Solution.— Log 4.38 = .64147 Log 5.217 = .71742 Log 83 = 1.91908 Adding, 3.27797 = log (4.38 X 5.217 X 83). Number corresponding to 3.27797 = 1,896.6. Hence, 4.38 X 5.217 X 83 = 1,896.6, nearly. By actual multiplication, the productls 1,896.5818, showing that the result obtained by using logarithms was correct to five figures. When adding logarithms, their algebraic sum is always to be found. Hence, if some of their numbers multiplied together are wholly decimal, the algebraic sum of the char- acteristics will be the characteristic of the product. It must be remembered that the mantissas are always positive. Example.— M ultiply 49.82, .00243, 17, and .97 together. 42 USEFUL TABLES. Solution — Log 49.82 = 1.69740 Log .00243 = 3.38561 Log 17 = 1.23045 Log .97 = 1.98677 Adding, 0.30023 = log (49.82 X .00243 X 17 X .97) . Number corresponding to 0.30023 = 1.9963. Hence, 49.82 X .00243 X 17 X .97 = 1.9963. In this case the sum of the mantissas was 2.30023. The integral 2 added to the positive characteristics makes their sum = 2 + 1 + 1 = 4; sum of negative characteristics = 3 + 1 = 4, whence 4 + (— 4) = 0. If, instead of 17, the number had been .17 in the above example, the logarithm of .17 would have been 1.23045, and the sum of the logarithms would have been 2.30023; the product would then have been .019963. It can now be shown why all numbers with figures in the same order have the same mantissa, without regard to the decimal point. Thus, suppose it were known that log 2.06 = .31387. Then, log 20.6 = log (2.06 X 10) = log 2.06 + log 10 = .31387 + 1 = 1.31387. And so it might be proved with the decimal point in any other position. DIVISION BY LOGARITHMS. As before, let X and Y represent two numbers whose loga- rithms are x and y. To find the logarithm of their quotient, we have, from the definition of a logarithm, 10* = X, (1) and 10 y = Y. (2) Dividing (1) by (2), ltf~ y =» or, by the definition of a X X logarithm, x — y = log — . But -y is the quotient of X-r- Y, and x — y is the difference of their logarithms, from which it follows that the difference between the logarithms of two numbers is equal to the logarithm of their quotient. Hence, to divide one number by another by means of logarithms : Rule . — Subtract the logarithm of the divisor from the logarithm of the dividend , and the result will be the logarithm of the quotient. LOGARITHMS. 43 Example.— Divide 6,784.2 by 27.42. Solution.— Log 6,784.2 = 3.83150 Log 27.42 = 1.43807 difference = 2.39343 = log (6,784.2-=- 27.42). Number corresponding to 2.39343 = 247.42. Hence, 6,784.2 -4- 27.42 = 247.42. When subtracting logarithms, their algebraic difference is to be found. The operation may sometimes be confusing, because the mantissa is always positive, and the character- istic may be either positive or negative. When the logarithm to be subtracted is greater than the logarithm from which it is to be taken, or when negative characteristics appear, subtract the mantissa first , and then the characteristic, by changing its sign and adding. Example.— Divide 274.2 by 6,784.2. Solution.— Log 274.2 = 2.43807 Log 6,784.2 = 3.83150 2.60657 First subtracting the mantissa .83150 gives .60657 for the mantissa of the quotient. In subtracting, 1 had to be taken from the characteristic of the minuend, leaving a charac- teristic of 1. Subtract the characteristic 3 from this, by changing its sign and adding 1 — 3 = 2, the characteristic of the quotient. Number corresponding to 2.60657 = .040418. Hence, 274.2 -4- 6,784.2 = .040418. Example.— Divide .067842 by .002742. Solution.— Log .067842 = ^2.83150 Log .002742 = 3.43807 difference = 1.39343 Since .83150 — .43807 = .39343 and — 2 + 3 = 1, number cor- responding to 1.39343 = 24.742. Hence, .067842 -4- .002742 = 24.742. The only case that is likely to cause trouble in subtract- ing is that in which the logarithm of the minuend has a nega- tive characteristic, or none at all, and a mantissa less than the mantissa of the subtrahend. For example, let it be re- quired to subtract the logarithm 3.74036 from the logarithm 44 USEFUL TABLES. 3.55145. The logarithm 3.55145 is equivalent to— 3 + .55145. Now, if we add both +1 and —1 to this logarithm, it will not change its value. Hejice, 3.55145 = —3 — 1 + 1 + .55145 = 4 + 1.55145. Therefore, 3.55145 — 3.74036 = 4 + 1.55145 3 + .74036 difference — 7 + .81109 = 7.81109. Had the characteristic of the above logarithm been 0 instead of 3, the process would have been exactly the same. Thus, .55145 = 1 +• 1.55145^ hence, 1 + 1.55145 3+ .74036 difference = 4 + .81109 = 4.81109. Example.— Divide .02742 by_67.842. Solution.— Log .02742 = 2.43807 = 3 + 1.43807 Log 67.842 = 1.83150 = 1 + .83150 difference = 4 + .60657 = 4.60657. Number corresponding to 4.60657 = .00040417. Hence, .02742 -f- 67.842 = .00040417. Example. — What is the reciprocal of 3.1416? Solution.— Reciprocal of 3.1416 = 0 * and log — ^ o.141d 0.1410 = log 1 - log 3.1416 = 0 — _.49715. Since 0 = -1 + 1, 1 + 1.00000 .49715 difference = 1 + ^50285 = 1.50285. Number w T hose logarithm is 1.50285 = .31831. INVOLUTION BY LOGARITHMS. If X represents a number whose logarithm is x , we have, from the definition of a logarithm, 10* = X. Raising both numbers to some power, as the nth, the equation becomes l 0 *n _ x n But X n is the required power of X, and xn is its logarithm, from which it follows that the logarithm of a number LOGARITHMS. 45 multiplied by the exponent of the power to which it is raised is equal to the logarithm of the power. Hence, to raise a number to any power by the use of logarithms: Rule.— Multiply the logarithm of the number by the exponent that denotes the power to which the number is to be raised , and the result will be the logarithm of the required power. Example.— What is (a) the square of 7.92? (6) the cube of 94.7? (c) the 1.6 power of 512, that is, the value of 512 1 * 6 ? Solution.— ( a) Log 7.92 = .89873; exponent of power = 2. Hence, .89873 X 2 = 1.79746 = log 7.92 2 . Number correspond- ing to 1.79746 = 62.727. Hence, 7.92 2 = 62.727, nearly. (6) Log 94.7 = 1.97635; 1.97635 X 3 = 5.92905 = log 94.73. Number corresponding to 5.92905 = 849,280, nearly. Hence, $4.7* = 849,280, nearly. (c) Log 512!-6 = 1.6 X log 512 = 1.6 X 2.70927 = 4.334832, or 4.33483 (when using five-place logarithms) = log 21,619. Hence, 512 1 * 6 = 21,619 nearly. If the number is wholly decimal, so that the characteristic is negative, multiply the two parts of the logarithm separately by the exponent of the number. If , after multiplying the mantissa , the product has a characteristic , add it , algebraically , to the neg- ative characteristic multiplied by the exponent , and the result will be the negative characteristic of the required power. Example. — Raise .0751 to the fourth power. Solution.— Log .0751 4 = 4 X log .0751 = 4 X 2.87564. Mul- tiplying the parts separately^ 4X2 = 8 and 4 X .87564 = 3.50256. Adding the 3 and 8, 3 + (— 8) = — 5; therefore, log .0751 4 = 5.50256. Number corresponding to this = .00003181. Hence, .0751* = .00003181. A decimal may be raised to a power whose exponent con- tains a decimal as follows: Example.— Raise .8 to the 1.21 power. Solution.— Log .8 1 - 21 = 1.21 X 1.90309. There are several ways of performing the multiplication. First Method.— Adding the characteristic and mantissa algebraically, the result is —.09691. Multiplying this by 1.21 gives —.1172611, or —.11726, when using five-place logarithms. To obtain a positive mantissa, add +1 and —1; whence, log . 8 b 2 i = — ! + 1 — .11726 = 1.88274. 46 USEFUL TABLES. Second Method .— Multiplying the characteristic and man- tissa separately gives —1.21 + 1.09274. Adding characteristic and mantissa algebraically, gives —.11726; then, adding +1 and -1, log .8 1 - 21 = 1.88274. Third Method— Multiplying the characteristic and man- tissa separately gives —1.21 + 1.09274. Adding the decimal part of the characteristic to the mantissa gives —1 + (—.21 -i- 1.09274) = 1.88274 = log .8 1 -* 1 . The number corresponding to the logarithm 1.88274 = .76338. Any one of the above three methods may be used, but we recommend the first or the third. The third is the most elegant and saves figures, but requires the exercise of more caution than the first method does. Below will be found the entire work of multiplication for both .8 1 - 21 and .8- 21 . 1.90309 1.21 1.90309 .21 90309 180618 90309 1.0927389 — 1.21 90309 180618 -F 1.1896489 —1 — .21 1.9796489, or 1.97965. 1.8827389, or 1.88274. In the second case, the negative decimal obtained by multiplying —1 and .21 was greater than the positive decimal obtained by multiplying .90309 and .21; hence, +1 and — 1 were added, as shown. EVOLUTION BY LOGARITHMS. If X represents a number whose logarithm is x, we have, from the definition of a logarithm, 10* = X. Extracting some root of both members, as the nth, the equation becomes 10” = i/x But \/ AXs the required root of X, and ^ is its logarithm, from which it follows that the logarithm of a number divided LOGARITHMS. 47 by the index of the root to be extracted is equal to the logarithm of the root. Hence, to extract any root of a number by means of logarithms: Rule. — Divide the logarithm of the number by the index of the root; the result will be the logarithm of the root. Example.— Extract (a) the square root of 77,851; ( b ) the cube root of 698,970; (c) the 2.4 root of 8,964,300. Solution— (a) Log 77,851 = 4.89127; the index of the root is 2; hence, log \/ 77,851 = 4.89127 -r- 2 = 2.44564; number corresponding to this = 279.02. Hence, i/ 77,851 = 279.02, nearly. ( b ) Log #" 698,970 = 5.84446^- 3 = 1.94815 = log 88.746; or, # / 698,970 = 88.747, nearly. (c) Log 8,964,300 = 6.95251 -- 2.4 = 2.89688 = log 788.64; or, 2 y / 8,964,300 = 788.64, nearly. If it is required to extract a root of a number wholly deci- mal, and the negative characteristic will not exactly contain the index of the root, without a remainder, proceed as follows: Separate the two parts of the logarithm; add as many units ( or parts of a unit) to the negative characteristic as will make it exactly contain the index of the root. Add the same number to the mantissa, and divide both parts by the index. The result will be the characteristic and mantissa of the root. Example.— Extract the cube root of .0003181. Solution.— Log # / .0003181 _ log .0003181 _ 4.50256 3 3 * (4 + 2 = 6)_+ (2 + .50256 = 2.50256). (6-4-3 = 2) + (2.50256 -- 3 = .83419); or, log #^ .0003181 = 2.83419 = log .068263. Hence, #".0003181 = .068263. Example.— Find the value of 1 ‘v / . 0003181. 0 T 1.41/—-- - log .0003181 4.50256 Solution.— Log y .0003181 = — = — -. If —.23 be added to the characteristic, it will contain 1.41 exactly 3 times. Hence, [-4 + (— .23) = —4.23] + (.23 + .50256 = .73256). ( — 4.23 -f- 1.41 = 3) + (.73256 -f- 1.41 = .51955); or, log x ' v / . 0003181 - 3.51955 = log .0033079. Hence, .0003181 = .0033079. 48 USEFUL TABLES. Example. — Solve this expression by logarithms: 497 X .0181 X 762 3, 300 X. 6517 ~~ ‘ Solution.— Log 497 = 2.69636 Log .0181 = 2.25768 Log 762 = 2.88195 Log product = 3.83599 Log 3,300 = 3.51851 Log .6517 = 1.81405 Log product == 3.33256 Hence, 3.83599 - 3.33256 = .50343 = log 3.1874. 497 X. 0181X762 3,300 X .6517 „ „ , 3 / 504,203X507 , . Example.— S olve 7 b y logarithms. Solution.— Log 504,203 = 5.70260 Log 507 = 2.70501 Log product = 8.40761 Log 1.75 = .24304 Log 71.4 = 1.85370 Log 87 = 1.93952 Log product = 4.03626 8.40761 - 4.03626 = 1.45712 = log 28.65. Hence, % 504,203 X 507 = 28.65. 1.75 X 71.4 X 87 Logarithms can often be applied to the solution of equations. Example.— Solve the equation 2.43s 5 = v / .0648. Solution.— 2.43s 5 = v / .0648. ■^^648 Dividing by 2.43, s 5 = • 2.43 Taking the logarithm of both numbers, 5 X log s = lQg ^ Q6 f 8 — log 2.43; LOGARITHMS. 49 5 log x = 2.81158 6 .38561 = Jl. 80193 - .38561 = 1.41632. Dividing by 5, log x = 1.88326; whence, x = .7643. Example.— Solve the equation 4.5* = 8. Solution.— Taking the logarithms of both numbers, x log 4.5 = log 8, log 8 _ .90309 .65321* whence, ~~ log 4.5 Taking logarithms again, log x = log .90309 - log .65321 = 1.95573 — 1.81505 = .14068; whence, x = 1.3825. Remark. — Logarithms are particularly useful in those cases when the unknown quantity is an exponent, as in the last example, or when the exponent contains a decimal, as in several instances in the examples given on pages 45-49. Such examples can be solved without the use of logarithms, but the process is very long and somewhat involved, and the arithmetical work required is enormous. To solve the exam- ple last given without using the logarithmic table and obtain the value of x correct to five figures would require, perhaps, 100 times as many figures as were used in the solution given, and the resulting liability to error would be correspondingly increased; indeed, to confine the work to this number of figures would also require a good knowledge of short-cut methods in multiplication and division, and judgment and skill on the part of the calculator that can only be acquired by practice and experience. Formulas containing quantities affected with decimal exponents are generally of an empirical nature; that is, the constants or exponents or both are given such values as will make the results obtained by the formulas agree with those obtained by experiment. Such formulas occur frequently in works treating on thermodynamics, strength of materials, machine design, etc. 50 USEFUL TABLES. COMMON LOGARITHMS. N. L. 0 1 2 3 4 5 6 7 8 9 P. P. 400 00 000 043 087 130 173 217 260 303 346 “389 101 432 475 518 561 604 647 “689 “732 ~ m > “817 44 43 42 102 860 903 945 988 *030 *072 *115 *157 *199 *242 1 4.4 4.3 4.2 103 01 284 326 368 410 452 494 536 578 620 662 2 8.8 8.6 8.4 104 703 745 787 828 870 912 953 995 *036 *078 3 13.2 12.9 12.6 105 02 119 160 202 243 284 325 366 407 4 19 490 4 17.6 17.2 16.8 106 531 572 612 653 694 735 776 816 857 898 5 22.0 21.5 21.0 107 938 979 *019 *060 *100 *141 *181 *222 *262 *302 6 26.4 25.8 25.2 108 03 342 383 423 463 503 543 583 623 663 703 7 30.8 30.1 29.4 109 743 782 822 862 902 941 981 *021 *060 *100 81 35.2 34.4 33.6 4io; 04 139 179 218 258 297 336 376 415 454 493 9 39.6 38.7 37.8 111 532 571 610 650 689 727 766 805 844 883 41 40 39 112 922 961 999 *038 *077 *115 *154 *192 *231 *269 1 4.1 4.0 3.9 113 05 308 346 385 423 461 500 538 576 614 652 2 8.2 8.0 7.8 114 690 729 767 805 843 881 918 956 994 *032 3 12.3 12.0 11.7 115 06 070 108 145 183 221 258 296 333 371 408 4 16.4 16.0 15.6 116 446 483 521 558 595 633 670 707 744 781 5 20.5 20.0 19.5 117 819 856 893 930 967 *004 •nil *078 *115 *151 6 24.6 24.0 23.4 118 07 188 225 262 298 335 372 408 445 482 518 7 28.7 28.0 27.3 119 555 591 628 664 700 737 773 809 846 882 8 32.8 32.0 31.2 420 918 954 990 *027 *063 *099 *135 *m *207 *243 9 36.9 36.0 35.1 121 08 279 314 350 386 422 458 493 529 565 600 38 37 36 122 636 672 707 743 778 814 849 884 920 955 1 3.8 3.7 3.6 123 991 *026 *061 *096 *132 *167 *202 *237 *272 *307 2 7.6 7.4 7.2 124 09 342 377 412 447 482 517 552 587 621 65(3 3 11.4 11.1 10.8 125 691 726 760 795 830 864 899 934 968 *003 4 15.2 14.8 14.4 126 10 037 072 106 140 175 209 243 278 312 346 5 19.0 18.5 18.0 127 380 415 449 483 517 551 585 619 653 687 6 22.8 22.2 21.6 128 721 755 789 823 857 890 924 958 992 *025 7 26.6 25.9 25.2 129 11 059 093 126 160 l:»:; 227 261 294 327 361 8 9 30.4 34.2 29.6 33.3 28.8 32.4 430 394 428 461 . 494 528 561 594 628 661 694 131 727 760 793 826 860 893 926 959 992 *024 35 34 33 132 12 057 090 123 156 189 222 254 287 320 352 1 3.5 3.4 3.3 133 385 418 450 483 516 548 581 613 646 67 S 2 7.0 6.8 6.6 134 710 743 775 808 840 872 905 937 969 *001 3 10.5 10.2 9.9 135 13 033 066 098 130 162 194 226 258 290 322 4 14.0 13.6 13.2 136 354 386 418 450 481 513 545 577 (309 640 5 17.5 17.0 16.5 137 672 704 735 767 799 830 862 893 925 956 6 21.0 20.4 19.8 138 988 *019 *051 *082 *114 *145 *176 *208 *239 *270 7 24.5 23.8 23.1 139 14 301 333 364 395 426 457 489 520 551 582 8 28.0 31.5 27.2 30.6 26.4 29.7 440 613 644 675 706 737 768 799 829 860 891 9 141 922 953 983 *014 *045 *076 *106 *137 *168 *198 32 31 30 142 15 229 259 200 320 351 381 412 442 473 503 1 3.2 3.1 3.0 143 534 564 594 625 655 6s 5 715 746 776 806 2 6.4 6.2 6.0 144 836 866 897 927 957 987 *017 *047 *077 *107 3 9.6 9.3 9.0 145 16 137 167 197 227 256 286 316 346 376 406 4 12.8 12.4 12.0 146 435 465 495 524 554 584 613 643 673 702 5 16.0 15.5 15.0 147 732 761 791 820 850 879 909 938 967 997 6 19.2 18.6 18.0 148 17 026 056 085 114 143 173 202 231 260 289 7 22.4 21.7 21.0 149 319 348 377 406 435 464 493 522 551 580 8 25.6 24.8 24.0 28.8 27.9 27.0 450 609 638 667 696 725 754 782 811 840 ”869 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. LOGARITHMS. 51 Table — ( Continued ). N. L. 0 1 2 3 4 5 6 7 8 9 P . P. 150 17 609 638 667 696 725 754 “782 811 840 869 151 898 '926 955 “984 #013 *041 *070 *099 *127 *156 29 28 152 18 184 213 241 270 298 327 355 384 112 441 l 2.9 1 2.8 153 469 498 526 554 583 611 639 667 696 724 2 5.8 5.6 154 752 780 808 837 865 893 921 949 977 *005 3 8.7 8.4 155 19 033 061 089 117 145 173 201 229 257 285 4 11.6 1 11.2 156 312 340 368 396 424 45 L 479 507 535 562 5 14.5 ; 14.0 157 590 618 645 673 700 728 756 783 811 838 6 17 '.4 16.8 158 866 893 921 948 976 *003 *030 *058 *085 *112 7 20.3 19.6 159 20 140 167 194 222 249 276 303 330 358 385 8 23.2 22.4 160 412 439 466 493 520 548 “575 “602 629 “656 9 26.1 25.2 161 683 710 737 763 790 817 844 871 898 925 27 26 162 952 978 *005 *032 *059 *085 *112 *139 *165 *192 1 2.7 1 1 2.6 163 21 219 245 272 299 325 352 378 405 431 458 2 5.4 1 5.2 164 484 511 537 564 590 617 643 669 696 722 3 8.1 | 7.8 165 748 775 801 827 854 880 906 932 958 985 4 10.8 | 10.4 166 22 Oil 037 063 089 115 141 167 194 220 246 5 13.5 i 13.0 167 272 298 324 350 376 401 427 453 479 505 6 16.2 i 15.6 168 531 557 583 608 634 660 686 712 737 763 7 18.9 1 18.2 169 789 814 840 866 891 917 943 968 994 *019 8 21.6 1 20.8 i 23.4 170 23 045 070 096 121 147 172 198 223 249 274 171 300 325 350 376 401 426 452 477 502 528 25 172 553 578 603 629 654 679 704 729 754 779 1 2.5 173 805 830 855 880 905 930 955 980 ■ini:, *030 2 5.0 174 24 055 080 105 130 155 180 204 229 254 279 4 7.5 175 304 329 353 378 403 428 452 477 502 527 4 10.0 176 551 576 601 625 ()50 674 699 724 748 773 5 12.5 177 797 822 846 871 895 920 944 969 993 *018 6 15.0 178 25 042 066 091 115 139 164 188 212 237 261 7 17.5 179 285 310 334 358 382 406 431 455 479 503 8 20.0 180 527 551 575 600 624 648 672 696 720 744 £4 *9 181 768 792 816 840 864 888 912 935 959 983 24 23 182 26 007 031 055 079 102 126 150 174 198 221 1 2.4 1 2.3 183 245 269 293 316 340 364 387 411 435 458 2 4.8 4.6 184 482 505 529 553 576 600 623 647 670 694 3 7.2 6.9 185 717 741 764 788 811 834 858 881 905 928 4 c >.6 9.2 186 951 975 998 *021 *045 *068 *091 *114 *138 *161 5 12.0 | 11.5 187 27 184 207 231 254 277 300 323 346 370 393 6 14.4 ! 13.8 188 416 439 462 485 508 531 554 577 600 623 7 16.8 ! 16.1 189 646 669 692 715 738 761 784 807 830 852 8 19.2 18.4 190 875 ”898 "921 “944 “967 989 *012 *035 *058 *081 9 21.6 [ 20.7 191 28 103 ~126 ~149 “m 194 “217 “240 “262 285 “307 22 21 192 330 353 375 398 421 443 466 488 511 533 1 2.2 2.1 193 556 578 601 623 646 668 691 713 735 758 2 4.4 4.2 194 780 803 825 847 870 892 914 937 959 981 3 < 5.6 6.3 195 29 003 026 048 070 092 115 137 159 181 203 4 8.8 8.4 196 226 248 270 292 31 1 336 358 380 403 425 5 11.0 10.5 197 447 469 491 513 535 557 579 601 623 645 6 13.2 12.6 198 667 68> 710 732 754 776 798 820 842 863 7 15.4 14.7 199 885 907 929 951 973 994 *016 *038 *060 *081 8 r i .6 16.8 200 30 103 “ l 25 “ l 46 “ l 68 “ l 90 “2 U “233 “255 “276 “298 9 13.8 18.9 N. L.O 1 2 3 4 5 6 7 8 9 P. P. 52 USEFUL TABLES. Table — ( Continued). N. L.O 1 2 3 4 5 6 7 8 9 r, . r 200 30 103 125 146 168 190 211 233 255 276 298 201 320 341 363 384 406 428 449 471 492 "514 22 21 202 535 557 578 600 621 643 664 685 707 728 1 2.2 2ul 203 750 771 792 814 835 856 878 899 920 942 2 4.4 4.2 204 963 984 *006 *027 *048 *069 *091 *112 *133 *154 3 1 5.6 6.3 205 31 175 197 218 239 260 281 302 323 345 366 4 8.8 8.4 206 387 408 429 450 471 492 513 534 555 576 5 11.0 10.5 207 597 618 639 660 681 702 723 744 765 785 6 13.2 12.6 208 806 827 848 869 890 911 931 952 973 994 7 15.4 14:7 209 32 015 035 056 077 098 118 139 160 181 201 8 17.6 16.8 210 222 243 263 284 305 ~325 "346 ~366 "387 "408 9 19.8 18.9 211 428 449 469 490 510 531 552 572 593 613 20 212 634 654 675 695 715 736 756 777 797 818 1 2.0 213 838 858 879 899 9l9 940 960 980 *001 *021 2 4.0 214 33 041 062 082 102 122 143 163 183 203 224 3 6.0 215 244 264 284 304 325 345 365 385 405 425 4 8.0 216 445 465 486 506 526 546 566 586 606 626 5 10.0 217 646 666 686 706 726 746 766 786 806 826 6 12.0 218 846 866 885 905 925 945 965 985 *005 *025 7 14.0 219 34 044 064 084 104 124 143 163 183 203 223 8 16.0 220 242 262 282 301 321 341 361 380 400 420 18.0 221 439 459 479 498 518 537 557 577 596 616 19 222 635 655 674 694 713 733 753 772 792 811 1 1.9 223 830 850 869 889 908 928 947 967 986 *005 2 3.8 224 35 025 044 064 083 102 122 141 160 180 199 3 5.7 225 218 238 257 276 295 315 334 353 372 392 4 7.6 226 411 430 449 468 488 507 526 545 564 583 5 9.5 227 603 622 641 660“ 679 698 717 736 755 774 6 11.4 228 793 813 832 851 870 889 908 927 946 965 7 13.3 229 984 *003 *021 *040 *059 *078 *097 *116 *135 *154 8 15.2 230 36 173 192 211 229 248 267 286 305 324 342 9 17.1 231 361 380 399 418 436 455 474 493 511 530 18 232 549 568 586 605 624 642 661 680 698 717 1 1.8 233 736 754 773 791 810 829 847 806 884 903 2 3.6 234 922 940 959 977 996 *014 *033 •051 *070 *088 3 5.4 235 37 107 125 144 162 181 199 218 236 254 273 4 7.2 236 291 310 328 346 365 383 401 420 43s 457 5 9.0 237 475 493 511 530 548 566 585 603 621 639 6 10.8 238 658 676 694 712 731 749 767 785 803 822 7 12.6 239 840 858 876 894 912 931 949 967 985 *003 8 14.4 240 38 021 039 057 075 093 112 130 148 166 "184 9 16.2 241 202 220 238 256 274 292 310 328 346 364 17 242 382 399 417 435 453 471 489 507 525 543 1 1.7 243 561 578 596 614 632 650 668 686 703 721 2 3.4 244 739 757 775 792 810 828 846 863 881 899 3 5.1 245 917 934 952 970 987 *005 *023 '041 *058 *076 4 6.8 246 39 094 111 129 146 164 182 199 217 235 252 5 8.5 247 270 287 305 322 340 358 375 393 410 428 6 10.2 248 445 463 480 498 515 533 550 568 585 602 7 11.9 249 620 637 655 672 690 707 724 742 759 777 8 9 13.6 250 794 811 829 846 863 881 898 915 933 950 N. L.O 1 2 3 4 5 6 7 8 9 P. P. LOGARITHMS. 53 Table— ( Continued). N. L. 0 1 2 3 4 5 6 7 8 9 P . P. 250 39 794 811 829 846 863 881 898 915 933 950 251 967 985 *002 *019 *037 *054 *071 *088 *106 *123 18 252 40 140 157 175 192 209 226 243 261 27 * 295 1 1.8 253 312 329 346 364 381 398 415 432 449 466 2 3.6 254 483 500 518 535 552 569 586 603 620 637 3 5.4 255 654 671 688 705 722 739 756 773 790 807 4 7.2 256 824 841 858 875 892 909 926 943 960 976 i 5 9.0 257 993 *010 *027 *044 *061 *078 *095 *111 * 12 S *145 6 10.8 258 41 16 ^ 179 196 212 229 246 263 280 296 313 7 12.6 259 33 (f 347 363 380 397 414 430 447 464 481 8 9 14.4 260 497 514 531 547 564 581 597 614 631 647 16.2 261 664 681 697 714 731 747 764 780 797 814 17 262 830 847 863 880 896 913 929 946 963 979 1 1.7 263 996 *012 *029 *045 *062 *078 *095 *111 *127 •144 2 3.4 264 42 160 177 193 210 226 243 259 275 292 308 1 3 5.1 265 325 341 357 374 39 C 406 423 439 455 472 4 6.8 266 488 504 521 537 553 570 586 602 619 635 5 8.5 267 651 667 684 700 716 732 749 765 781 797 6 10.2 268 813 830 846 862 878 894 911 927 943 959 7 11.9 269 975 991 *008 *024 *040 *056 *072 *088 *104 *120 8 13.6 270 43 136 152 169 185 201 217 233 249 265 281 9 15.3 271 297 313 329 345 361 377 393 409 425 441 16 272 457 473 489 505 521 537 553 569 584 600 1 1.6 273 616 632 648 664 680 696 712 727 743 759 2 3.2 274 775 791 807 823 838 854 870 886 902 917 3 4.8 275 933 949 965 9 M *012 *028 *044 *059 *075 4 6.4 276 44 091 107 122 138 154 170 185 201 217 232 5 8.0 277 248 264 279 295 311 326 342 358 373 389 6 9.6 278 404 420 436 451 467 483 498 514 529 545 7 11.2 279 560 576 592 607 623 638 654 669 685 700 8 12.8 280 716 731 747 762 778 793 809 824 840 855 9 14.4 281 871 886 902 917 932 948 963 979 994 *010 15 282 45 025 040 056 071 086 102 117 133 148 163 1 1.5 283 179 194 209 225 240 255 271 286 301 317 2 3.0 284 332 347 362 378 393 408 423 439 454 469 3 4.5 285 484 500 515 530 545 561 576 591 606 621 4 6.0 286 637 652 667 0,-2 697 712 728 743 758 773 5 7.5 287 788 803 818 834 849 864 879 894 909 924 6 9.0 288 939 954 969 984 *000 *015 *030 *045 *060 1 *075 7 10.5 289 46 090 105 120 135 15 U 165 180 195 210 225 8 9 12.0 13.5 290 240 255 270 285 300 315 330 345 359 374 291 389 404 419 434 449 464 479 494 509 523 14 292 538 553 568 583 598 613 627 642 657 672 1 1.4 293 687 702 716 731 746 761 776 790 805 820 2 2.8 294 835 850 864 879 894 909 923 938 953 967 3 4.2 295 982 997 *012 *026 *041 *056 *070 *085 *100 *114 4 5.6 296 47 129 144 159 173 188 202 217 232 246 261 5 7.0 297 276 290 305 319 334 349 363 378 392 407 6 8.4 298 422 436 451 465 480 494 509 5241 538 553 7 9.8 299 567 582 596 611 625 640 654 669 683 698 8 11.2 300 712 727 741 756 770 784 799 813 828 842 9 12.6 N. L. 0 1* 2 3 4 5 1 6 7 8 9 P. P. 54 USEFUL TABLES. Table — ( Continued). LOGARITHMS. 55 Table— ( Continued). N. L. 0 1 2 3 4 5 6 7 8 9 P. P. 350 851 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 N. 54 407 419 “543 667 790 913 035 157 279 400 522 432 “555 679 802 925 047 169 291 413 534 444 "568 691 814 937 060 182 303 42a 546 456 5si ) 704 827 949 072 194 315 437 558 469 “593 716 839 962 (tsi 2H6 328 449 570 481 “605 728 851 974 096 218 340 461 582 494 “617 741 864 986 108 230 352 473 594 “506 “630 753 876 998 121 242 364 485 606 518 “642 765 888 *011 133 255 376 497 618 13 1 1.3 2 2.6 3 3.9 4 5.2 5 6.5 6 7.8 7 9.1 8 10.4 9 11.7 12 1 1.2 2 2.4 3 3.6 4 4.8 5 6.0 6 7.2 7 8.4 8 9.6 9 10.8 II 1 1.1 2 2.2 3 3.3 4 4.4 5 5.5 6 6.6 7 7.7 8 8.8 9 9.9 10 1 1.0 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8 0 9 9.0 531 654 777 900 55 023 145 267 388 509 630 642 654 666 678 691 703 715 727 739 “859 979 *098 217 336 455 573 691 808 751 871 991 56 110 229 348 467 585 703 763 883 *003 122 241 360 478 597 714 775 895 *015 134 253 372 490 608 726 787 907 *027 146 265 384 502 620 738 799 919 *038 158 277 396 514 632 750 811 931 *050 170 289 407 526 644 761 823 943 *062 182 301 419 538 656 773 835 955 *074 194 312 431 549 667 785 847 967 *086 205 324 443 561 679 797 820 832 844 855 867 879 891 902 914 926 937 57 054 171 287 403 519 634 749 864 949 066 183 299 415 530 646 761 875 961 078 194 310 426 542 657 772 887 972 089 206 322 438 553 669 784 898 984 101 217 334 449 565 680 795 910 996 113 229 345 461 576 692 807 921 *008 124 241 357 473 588 703 818 933 *019 136 252 368 484 600 715 830 944 *031 148 264 3S0 496 611 726 841 955 *043 159 276 392 507 623 738 852 967 978 58 092 206 320 433 546 659 771 883 995 990 104 218 331 444 557 670 782 894 *006 *001 115 229 343 456 569 681 794 906 *017 *013 127 240 354 467 580 692 805 917 *028 *024 138 252 365 478 591 704 816 928 *040 *035 149 263 377 490 602 715 827 939 *051 *047 161 274 388 501 614 726 838 950 *062 *058 172 286 399 512 625 737 850 961 *073 *070 184 297 410 524 636 749 861 973 *084 *081 195 309 422 535 647 760 872 984 *095 59 106 118 129 140 151 162 173 184 195 207 218 329 439 550 660 770 879 988 60 097 229 340 450 561 671 780 890 999 108 240 351 461 572 682 791 901 *010 119 251 362 472 583 693 802 912 *021 130 262 373 483 594 704 813 923 *032 141 273 384 494 605 715 824 934 *043 152 284 395 506 616 726 835 945 *054 163 295 406 517 627 737 846 956 *065 173 306 417 528 638 748 857 966 *076 184 318 428 539 649 759 868 977 *086 195 206 L. 0 217 228 239 249 260 271 282 293 304 1 2 3 4 5 6 7 8 9 P. P. 56 USEFUL TABLES. Table — ( Continued). N. L.O 1 2 3 4 *5 6 7 8 9 P. P. 400 60 206 217 228 239 249 260 271 282 293 304 401 314 325 336 347 369 “379 "390 ^401 “412 402 423 433 444 455 466 477 487 498 509 520 403 531 541 552 563 574 584 595 606 617 627 404 638 649 660 670 681 692 703 713 724 735 405 746 756 767 778 788 799 810 821 831 842 406 853 863 874 885 895 906 917 927 938 949 II 407 959 970 981 991 *002 *013 *023 *034 *045 *055 1 1.1 408 61 066 077 087 098 109 119 130 140 151 162 2 2.2 409 172 183 194 204 215 225 236 247 257 3 3.3 410 278 289 300 310 321 331 342 352 363 374 4 5 4.4 5.5 411 384 395 405 416 426 437 448 458 469 479 6 6.6 412 490 500 511 521 532 542 553 563 574 584 J7 7.7 413 595 606 616 627 637 648 658 669 679 690 8 8.8 414 700 711 721 731 742 752 763 773 784 794 9 9.9 415 805 815 826 836 847 857 868 878 888 899 416 909 920 930 941 951 962 972 982 993 *003 417 62 014 024 034 045 055 066 076 086 097 107 418 118 128 138 149 159 170 180 190 201 211 419 221 232 242 252 263 273 284 294 304 315 420 325 335 346 356 366 377 "387 “397 ~408 ~418 10 421 428 439 449 459 469 480 490 500 511 521 422 531 542 552 562 572 583 503 603 613 624 1 1.0 423 634 644 655 665 675 685 696 706 716 726 2 2.0 424 737 747 757 767 778 788 798 808 818 829 3 3.0 425 839 849 859 870 880 890 900 910 921 931 4 4.0 426 941 951 961 972 982 902 *002 *012 *022 *033 5 5.0 427 63 043 053 063 073 083 094 104 114 124 134 6 6.0 428 144 155 165 175 185 195 205 215 225 236 7 7.0 429 246 256 266 276 286 296 306 317 327 337 8 g 8.0 9.0 j 430 347 357 367 377 387 397 407 417 428 ~438 431 448 458 468 478 488 498 508 518 528 538 432 548 558 568 579 589 599 609 619 629 639 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 435 849 859 869 879 889 899 909 919 929 939 436 949 959 969 979 988 998 *008 *018 *028 *038 9 437 64 048 058 068 078 088 098 108 118 128 137 1 0.9 438 147 157 167 177 187 197 207 217 227 237 2 1.8 439 246 256 266 276 286 296 306 316 326 335 3 2.7 440 345 "355 “365 ~375 385 "395 ~404 414 424 ~434 4 5 3.6 441 444 454 464 473 483 493 503 513 523 532 6 5.4 442 542 552 562 572 582 591 601 611 621 631 7 6.3 443 640 650 660 670 680 689 699 709 719 729 g 7.2 444 738 748 758 768 777 787 797 807 816 826 g 8.1 445 836 846 856 865 875 885 895 904 914 924 446 933 943 953 963 972 982 992 *002 *011 *021 447 65 031 040 050 060 070 079 089 099 108 118 448 128 137 147 157 167 176 186 196 205 215 449 225 234 244 254 263 273 283 292 302 312 450 321 331 341 350 360 369 379 389 398 408 N. L.O 1 2 3 4 5 6 7 8 9 P. P. LOGARITHMS. 67 Table— ( Continued). N. L. 0 1 2 3 4 5 6 7 8 9 450 65 321 331 341 350 360 ”369 ”379 ”389 ”398 408 451 418 427 437 447 456 "466 175 ”485 "495 ”504 452 514 523 533 543 552 562 571 581 591 600 453 610 619 629 639 648 658 667 677 686 696 454 706 715 725 734 744 753 763 772 782 792 455 801 811 820 830 839 849 858 868 877 887 456 896 906 916 925 935 944 954 963 973 982 457 992 *001 *011 *020 *030 *039 *049 *058 *068 *077 458 66 087 096 106 115 124 134 143 153 162 172 459 181 191 200 210 219 229 238 247 257 266 460 276 285 295 304 314 323 332 342 351 ”361 461 370 380 389 398 408 417 427 436 445 455 462 464 474 483 492 502 511 521 530 539 549 463 558 567 577 586 596 605 614 624 633 642 464 652 661 671 680 689 699 708 717 727 736 465 745 755 764 773 783 792 801 811 820 829 466 839 848 857 867 876 885 894 904 913 922 467 932 941 950 960 969 978 987 997 *006 *015 468 67 025 034 043 052 062 071 080 089 099 108 469 117 127 136 145 154 164 173 182 191 201 470 210 219 228 237 247 256 265 274 284 293 471 302 ~ 3U 321 330 339 “348 ~357 ”367 "376 ”385 472 394 403 413 422 431 440 449 459 468 477 473 486 495 504 514 523 532 541 550 560 569 474 578 587 596 605 614 624 633 642 651 660 475 669 679 688 697 706 7 ] 5 724 733 742 752 476 761 770 779 788 797 800 815 825 834 843 477 852 861 870 879 888 897 906 916 925 934 478 943 952 961 970 979 988 997 *006 *015 *024 479 68 034 043 052 061 070 079 088 097 106 115 480 124 133 142 151 160 169 178 187 196 205 481 215 224 233 242 251 260 269 278 287 296 482 305 314 323 332 341 350 359 368 377 386 483 395 404 413 422 431 440 449 458 467 476 484 485 494 502 511 520 529 538 547 556 565 485 574 583 592 601 610 619 628 637 646 655 486 664 673 681 690 699 708 717 726 735 744 487 753 762 771 780 789 797 806 815 824 833 488 842 851 860 869 878 886 895 904 913 922 489 931 940 949 958 966 975 984 993 *002 *011 490 69 020 028 037 046 055 064 073 082 090 099 491 108 117 126 135 144 152 161 170 179 188 492 197 205 214 223 232 241 249 258 267 276 493 285 294 302 311 320 329 338 346 355 364 494 373 381 390 399 408 417 425 434 443 452 495 461 469 478 487 496 504 513 522 531 539 496 548 557 566 574 583 592 601 609 618 627 497 636 644 653 662 671 679 '688 697 705 714 498 723 732 740 749 758 767 775 784 793 801 499 810 819 827 836 845 854 862 871 880 888 500 897 ”906 914 ”923 ”932 ”940 ”949 ”958 ”966 “975 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. 10 1 1.0 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8.0 9 9.0 9 1 0.9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 8 1 0.8 2 1.6 3 2.4 4 3.2 5 4.0 6 4.8 7 5.6 8 6.4 9 7.2 P. P. 58 USEFUL TABLES. Table— ( Continued ). N. L. 0 1 2 - 3 4 5 1 6 1 7 8 1 9 P. P. 500 69 897 906 914 923 932 940 949 958 966 975 501 984 992 *001 ,*010 *018 *027 *036 *044 *053 *062 502 70 070 079 088 096 105 114 122 131 140 148 503 157 165 174 183 191 200 209 217 226 234 504 243 252 260 269 278 286 295 303 312 321 505 329 338 346 355 364 372 381 389 398 406 506 415 424 432 441 449 458 467 475 484 492 9 507 501 509 518 526 535 544 552 561 569 578 1 0.9 508 586 595 603 612 621 629 638 646 655 663 2 1.8 509 672 680 689 697 706 714 723 731 740 749 3 2.7 510 757 766 '774 783 791 800 “808 ”817 “825 ”834 4 5 3.6 4.5 511 842 851 859 868 876 885 893 902 910 919 g 5.4 512 927 935 944 952 961 969 978 986 995 *003 6.3 513 71 012 020 029 037 046 054 063 071 079 088 g 7.2 514 096 105 113 122 130 139 147 155 164 172 9 g|l 515 181 189 198 206 214 223 231 240 248 257 516 265 273 282 290 299 307 315 324 332 341 517 349 357 366 374 383 391 399 408 416 425 518 433 441 450 458 466 475 483 492 500 508 519 517 525 533 542 550 559 567 575 584 592 520 600 609 617 625 634 642 650 ”659 667 675 521 684 692 700 709 717 725 734 742 7 50 759 8 522 767 775 784 792 800 809 817 825 834 842 1 0.8 523 850 858 867 875 883 892 900 908 917 925 2 1.6 524 933 941 950 958 966 975 983 991 999 *008 3 2.4 525 72 016 024 032 041 049 057 066 074 082 090 4 3.2 526 099 107 115 123 132 140 148 156 165 173 5 4.0 527 181 189 198 206 214 222 230 239 247 255 6 4.8 528 263 272 280 288 296 304 313 321 329 337 7 5.6 529 346 354 362 370 378 387 395 403 411 419 8 6.4 530 428 ~436 "444 “452 “460 “469 “477 485 “493 ”501 9 7.2 531 509 "518 “526 ~534 "542 “550 “558 “567 “575 583 532 591 599 607 616 624 632 640 648 656 665 533 673 681 689 697 705 713 722 730 738 746 534 754 762 770 779 787 795 803 ! 811 819: 827 535 835 843 852 860 868 876 884 892 900 908 536 916 925 933 941 949 957 965 ! 973 981 989 7 • 537 997 *006 *014 *022 *030 *038 *046 *054 *062 *070 1 0.7 538 73 078 086 *094 102 111 119 127 135 143 151 2 1.4 539 159 167 175 183 191 199 207 215 223 231 3 2.1 540 239 247 255 263 272 280 288 [”296 “304 ”312 4 2.8 541 320 ”328 336 ”344 “352 “360 “368 “376 ”384 ”392 g 4.2 542 400 408 416 424 432 440 448 456 464 472 4.9 543 480 488 496 | 504 512 520 528 536 544 552 8 5.6 544 560 568 576 ! 584 592 600 608 616 624 632 9 6.3 545 640 648 656 ! 664 672 679 687 695 703 711 546 719 727 735 1 743 751 759 767 775 783 791 547 799 807 815 823 830 838 846 8o4 862 870 548 878 886 894 ! 902 910 918 926 933 941 949 549 957 965 973 981 989 997 *005, *013 *020 *028 550 ?4 036 t 044 “052 060 068 “076 | 084 | 092 099 107 N. L.O 1 2 i 3 i 4 5 1 6 7 8 9 l P. P. N. 50 551 552 553 554 555 556 557 558 559 60 561 562 563 564 565 566 567 568 569 70 571 572 573 574 575 576 577 578 579 80 581 582 583 584 585 586 587 588 589 90 591 592 593 594 595 596 597 598 599 00 N. LOGARITHMS. 59 Table— ( Continued). 2 3 4 5 6 7 8 9 P. P. 052 060 068 076 084 092 099 107 131 139 147 155 162 170 178 186 210 218 225 233 241 249 257 265 288 296 304 312 320 327 335 343 367 374 382 390 398 406 414 421 445 453 461 468 476 484 492 500 523 531 539 547 554 562 570 578 601 609 617 624 632 640 648 656 679 687 695 702 710 718 726 733 757 764 772 780 788 796 803 811 834 842 850 858 865 873 881 889 912 920 927 935 943 950 958 966 8 989 997 *005 *012 *020 *028 *035 *043 1 0.3 066 074 082 089 097 105 113 120 2 1.6 143 151 159 166 174 182 189 197 3 2.4 220 228 236 243 251 259 266 274 4 3.2 297 305 312 320 32* 335 343 351 5 4.0 374 381 289 397 404 412 420 427 6 4.8 450 458 465 473 481 488 496 504 7 5.6 526 534 542 549 557 565 572 580 8 9 6.4 7.2 603 610 618 626 633 641 648 656 679 686 694 702 709 717 724 732 755 762 770 778 785 793 800 808 831 838 846 853 861 868 876 884 906 914 921 929 937 944 952 959 982 989 997 *005 *012 *020 "627 *035 057 065 072 080 087 095 103 110 133 140 148 155 163 170 178 185 208 215 223 230 238 245 253 260 283 290 298 305 313 320 328 335 358 365 373 380 388 395 403 410 433 440 448 455 462 470 477 485 7 507 515 522 530 537 545 552 559 1 0.7 582 589 597 604 612 619 626 634 2 1.4 656 664 671 678 686 693 701 708 3 2.1 730 738 745 753 760 768 775 782 4 2.8 805 812 819 827 834 842 849 856 5 3.5 879 886 893 901 908 916 923 930 6 4.2 953 960 967 975 982 989 997 *004 7 4.9 026 034 041 048 056 063 070 078 8 5.6 100 107 115 122 129 137 144 151 9 6.3 173 181 188 195 203 210 217 225 247 254 262 269 276 283 291 298 320 327 335 342 349 357 364 371 393 401 408 415 422 430 437 444 466 474 481 488 495 503 510 517 539 546 554 561 568 576 5831 590 612 619 627 634 641 648 656 | 663 685 692 699 706 714 721 728 735 757 764 772 779 786 793 801 808 830 837 844 851 Q 59 866 873 880 2 3 4 5 6 7 8 9 P. P. 60 N. 60CT 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 N. USEFUL TABLES. Table— ( Continued). 2 3 4 5 6 7 8 9 P . P. 830 837 844 851 859 866 873 880 902 909 916 924 931 938 "945 952 974 981 988 996 #003 #010 #017 *025 046 053 061 068 075 082 089 097 118 125 132 140 117 154 161 168 190 197 204 211 219 226 233 240 262 269 276 283 290 297 305 312 8 333 340 347 355 362 369 376 383 1 0.8 405 412 419 426 433 440 447 455 2 1.6 476 483 490 497 504 512 519 526 3 2.4 547 554 561 569 576 583 590 “597 4 3.2 618 625 633 640 647 654 661 668 5 g 4.0 4 8 689 696 704 711 718 725 732 739 7 5.6 760 767 774 781 789 796 803 810 g 831 838 845 852 859 866 873 880 902 909 916 923 930 937 944 951 7.2 972 979 986 993 #000 #007 *014 *021 043 050 057 064 071 078 085 092 113 120 127 134 141 148 155 162 183 190 197 204 211 218 225 232 253 260 267 274 281 288 295 302 323 330 337 344 "351 “358 "365 “372 7 393 400 407 414 421 428 435 442 1 0.7 463 470 477 484 491 498 505 511 2 1.4 532 539 546 553 560 567 574 581 3 2.1 602 609 616 623 630 637 644 650 4 2.8 671 678 685 692 699 706 713 720 5 3.5 741 748 754 761 768 775 782 789 6 4.2 810 817 824 831 837 844 851 858 7 4.9 879 886 893 900 906 913 920 927 8 5.6 948 955 962 969 975 982 989 996 9 6.8 017 024 030 037 044 -051 058 065 085 092 099 106 113 120 127 134 154 161 168 175 182 188 195 202 223 229 236 243 250 257 264 271 291 298 305 312 318 325 332 339 359 366 373 380 387 393 400 407 6 428 434 441 448 455 462 468 475 1 0.6 496 502 509 516 523 530 536 513 2 1.2 564 570 577 584 591 598 604 611 3 1.8 632 638 645 652 659 665 672 679 4 2.4 699 706 713 720 726 733 740 747 5 g 3.0 3.6 767 774 781 787 794 801 808 814 7 4.2 835 841 848 855 862 868 875 882 g 4.8 902 909 916 922 929 936 943 949 9 5*4 969 976 983 990 996 #003 *010 *017 037 043 050 057 064 070 077 084 104 111 117 124 131 137 144 151 171 178 184 191 198 204 211 218 238 245 251 258 265 271 278 285 305 311 318 325 331 338 345 351 2 3 4 5 6 7 8 9 P. P. N. 550 651 652 653 654 655 656 657 658 659 >60 661 662 663 664 665 666 667 668 669 170 671 672 673 674 675 676 677 678 679 80 681 682 683 684 685 686 687 688 689 90 691 692 693 694 695 696 697 698 699 00 N. LOGARITHMS. 61 Table— ( Continued). L. 0 1 2 3 4 5 6 7 8 ! 9 F *. P. 81 291 298 305 311 318 325 331 ”338 345 351 358 365 371 378 385 391 398 405 ”4U “418 425 431 438 445 451 458 465 471 478 485 491 498 505 511 518 525 531 538 544 551 •558 564 571 578 584 591 598 604 611 617 624 631 637 644 651 657 664 671 677 684 690 697 704 710 717 723 730 737 743 750 757 763 770 776 783 790 796 803 809 816 823 829 836 842 849 856 862 869 875 882 889 895 902 908 915 921 928 935 941 948 954 961 968 974 981 987 994 *000 *007 *014 32 020 027 033 040 046 053 060 066 073 i 079 7 086 092 099 105 112 119 125 132 138 ! 145 l 0.7 151 158 164 171 178 184 191 197 1 204 1 210 2 1 .4 217 223 230 236 243 249 256 ' 263' 269 | 276 3 2.X 282 289 295 302 308 315 321 j 328' 334 1 341 4 2.8 347 354 360 367 373 580 387 393! 400 406 5 3.5 413 419 426 432 439 445 452 ; 458 465 471 6 4.2 478 484 491 497 504 510 517 523! 530 536 7 4.9 543 549 556 562 569 575 582 : 588 j 595 601 8 9 5.6 6.3 607 614 620 627 633 640 646 653 j 659 666 672 679 685 692 698 705 711 ; 718 i ”m ^30 737| 743 750 756 763 769 ! 776 ' 782 : 789 795 802, 808 814 821 827 834 840 ! ! 847: 853 860 866 872 ■ 879! 885 898 905 1 1 911 918 924 930 837; 943 950 956 963 969! 975' 982 988 995 1*001 *008 *014 *020 *027 *033 *040' *046 *052 33 0591 065 072 078 085 091 097, 104! 110 117 123 129 136 142 149 155! 161: 168 174 181 187: 193 200 o 213 219, 225 1 232 238 245 251 1 257, 264 i 270. 276 283 ( 289 296 302 308 315 321 327 334 340 347 353 1 359 366 372 6 378 385 i 391 398 404 410 417J 423 429, 436 1 0.6 . 442 448 455 461 1 467 474 480' 487 493, 499 2 1.2 506: 512 518 525! 531 537 544, 5501 556 563 3 1.8 569| 575; 582 588 594 601 607 1 613; 620 626 4 2.4 632i 639 i 645 651 i 658 664 670 677! 683 689 5 3.0 696 702 708 715 721 727 734! 740 ! 746 753 6 3.6 759 765 771] 778, 784 790 797 1 803 ; 809 816 7 4.2 822 828 835 841 j 847 853 860 j 866 872, 879 8 4.8 885 ”891 “897 "904j 910 916 ”923 ”929 ; ”935! 942 9 5.4 948 "954 ”960 “967! ”973 ”979 “985 “992 ”998, *004 34 Oil 017 023 029! 036 042 048 055 061 067 073 080 086 092 098 105 111 117 123 130 136 142 148 155 161 167 173 180 186 192 198 205 211 217 223 230 236 242 248 255 261 267 273 280 286 292 298 305 311 317 323 330 336 342 348 354 361 367 373 379 386 392 398 404 410 417 423 429 435 442 448 454 460 466 473 479 485 491 497 504 510 516 522 528 535 541 547 553 559 566 L. 0 1 2 3 4 5 6 7 8 9 P . P. 62 N. 70CT 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 N. USEFUL TABLES. Table — ( Continued). 2 3 4 5 6 7 8 9 P >. P. 522 528 535 541 547 553 559 566 584 590 597 603 609 615 “621 "628 646 652 658 665 671 677 683 689 708 714 720 726 733 739 745 751 770 776 782 788 794 800 HOT 813 831 837 844 850 856 862 868 874 893 899 905 911 917 924 930 936 7 954 960 967 973 979 985 991 997 l 0.7 016 022 028 034 040 046 052 058 2 1.4 077 083 089 095 101 107 114 120 3 2.1 138 144 150 156 163 169 175 181 4 2.8 199 205 211 217 224 230 236 242 5 g 3.5 4.2 260 266 272 278 285 291 297 303 7 4.9 321 327 333 339 345 352 358 364 g 5.6 382 388 394 400 406 412 418 425 9 443 449 455 461 467 473 479 485 503 509 516 522 528 534 540 546 564 570 576 582 r.s< 594 600 606 625 631 637 643 649 655 661 667 685 691 697 703 709 715 721 727 745 751 757 763 769 775 781 788 806 812 818 824 830 836 842 848 6 866 872 878 884 890 896 902 908 1 0.6 926 932 938 944 950 956 962 968 2 1.2 986 992 998 *004 *010 *016 *022 *028 3 1.8 046 052 058 064 070 076 082 088 4 2.4 106 112 118 124 130 136 141 147 5 3.0 165 171 177 183 189 195 201 207 6 3.6 225 231 237 243 249 255 261 267 7 4.2 285 291 297 303 308 314 320 326 8 4.8 e a 344 350 356 362 368 374 380 386 9 0.4 404 410 415 421 427 433 ~439 “445 463 469 475 481 487 493 499 504 522 528 534 540 546 552 558 564 581 587 593 599 605 611 617 623 641 646 652 658 664 670 676 682 700 705 711 717 723 729 735 741 5 759 764 770 776 782 788 794 800 1 0.5 817 823 829 835 841 847 853 859 2 1.0 876 882 888 894 900 906 911 917 3 1.5 “935 “941 "947 “953 “958 964 “970 “976 4 2.0 O R ~994 999 *005 *011 *017 *023 *029 *035 5 g 2.5 3.0 052 058 064 070 075 081 087 093 7 3 . 5 ' 111 116 122 128 134 140 146 151 g 4.0 169 175 181 186 192 198 204 210 9 4.5 227 233 239 245 251 256 262 268 286 291 297 303 309 315 320 326 344 349 355 361 367 373 379 384 402 408 413 419 425 431 437 442 460 466 371 477 483 489 495 500 518 “523 529 535 “541 “5 + 7 “552 "558 2 3 4 5 6 7 8 9 P . P. . 50 751 752 753 754 755 756 757 758 759 60 761 762 763 764 765 766 767 768 769 70 771 772 773 774 775 776 777 778 779 80 781 782 783 784 785 786 787 788 789 90 791 792 793 794 795 796 797 798 799 too N. LOGARITHMS. 63 Table— ( Continued). 2 3 4 5 6 7 8 9 P. P. 518 523 529 535 541 547 552 558 576 581 587 593 599 604 610 • ill, 633 639 645 651 656 662 668 674 691 697 703 708 714 .720 726 731 749 754 760 766 772 777 783 789 806 812 818 823 829 835 841 846 864 869 875 881 887 892 898 904 921 927 933 938 944 950 955 961 978 984 990 996 *001 *007 *013 *018 036 Oil 047 053 058 064 070 076 093 098 104 110 116 ~121 127 133 150 156 161 167 173 178 184 190 6 207 213 218 224 230 235 241 247 l 0.6 264 270 275 281 287 292 298 304 2 1.2 321 326 332 338 343 349 355 360 3 1.8 377 383 389 395 400 406 412 417 4 2.4 434 440 446 451 457 463 468 474 5 3.0 491 497 502 508 513 519 525 530 6 3.6 547 553 559 564 570 576 5*1 587 7 4.2 604 610 615 621 627 632 638 643 8 4.8 9 5.4 660 666 672 677 683 689 694 700 717 722 728 734 739 "745 "750 "756 773 779 784 790 795 801 807 812 829 835 840 846 852 857 863 885 891 897 902 908 913 919 925 941 947 953 958 964 969 975 981 997 *003 *009 *014 *020 *025 *031 *037 053 059 064 070 076 081 087 092 109 115 120 126 131 137 143 148 165 170 176 182 187 193 198 204 "m 226 232 "237 243 "248 "254 "260 "276 282 "287 "293 ~298 “304 "310 315 5 332 337 343 348 354 360 365 371 1 0.5 387 393 39* 404 409 415 421 426 2 1.0 443 448 454 459 465 470 476 481 3 1.5 498 504 509 515 520 526 531 537 4 2.0 553 559 564 570 575 581 586 592 5 2.5 609 614 620 625 631 636 642 647 6 3.0 664 669 675 680 686 691 697 702 7 3.5 719 724 730 735 741 746 752 757 8 4.0 774 779 785 790 796 801 807 812 9 4.5 "829 "834 840 "845 851 "856 “862 867 883 889 894 900 905 911 916 922 938 944 949 955 960 966 971 977 993 998 *004 *009 *015 *020 *026 *031 048 053 059 064 069 075 080 086 102 108 113 119 124 129 135 140 157 162 168 173 179 184 189 195 211 217 222 227 233 238 244 249 266 271 276 282 287 293 298 304 320 "325 331 "336 "342 “347 "352 "358 2 3 4 5 6 7 8 9 P. P. 64 USEFUL TABLES. Table— ( Continued). N. L. 0 1 2 3 4 5 6 7 8 9 I \ P. 800 90 309 314 320 325 331 336 “342 "347 352 358 801 363 369 274 380 385 390 396 401 407 412 802 417 423 428 434 439 445 450 455 461 466 803 472 477 482 488 493 499 504 509 515 520 804 526 531 536 542 547 553 558 563 569 574 805 580 585 590 596 601 607 612 617 623 628 806 634 639 644 650 655 660 666 671 677 682 807 687 693 698 703 709 714 720 725 730 736 808 741 747 752 757 763 768 773 779 784 789 809 795 800 806 811 816 822 827 832 838 843 810 849 854 859 865 870 875 "881 “886 891 "897 811 902 907 913 918 924 929 “934 940 “945 950 1 6 812 956 961 966 972 977 982 988 993 998 *004 0.6 813 91 009 014 020 025 030 036 041 046 052 057 2 1.2 814 062 068 073 078 084 089 094 100 105 110 3 1.8 815 116 121 126 132 137 142 148 153 158 164 4 2.4 816 169 174 180 185 190 196 201 200 212 217 5 3.0 817 222 228 233 238 243 249 254 259 265 270 6 3.6 818 275 281 286 291 297 302 307 312 318 323 7 4.2 819 328 334 339 344 350 355 360 365 371 376 8 9 4.8 5.4 820 381 387 392 397 403 408 "413 “418 424 429 821 434 440 445 450 '455 461 *466 *471 “477 “482 822 487 492 498 503 508 514 519 524 529 535 823 540 545 551 556 561 566 572 577 582 587 824 593 598 603 609 614 619 624 630 635 640 ' 825 645 651 656 661 666 672 677 682 687 693 826 698 703 709 714 719 724 730 735 740 745 827 751 756 761 766 772 777 782 787 793 798 828 803 808 814 819 824 829 834 840 845 850 829 855 861 866 871 876 882 887 892 897 903 830 908 “913 "918 924 “929 ”934 “939 "944 “950 "955 831 960 "965 “971 ~976 “981 "986 "991 "997 *002 •HI) 7 5 832 92 012 018 023 028 033 038 044 049 054 059 1 0.5 833 065 070 075 080 085 091 096 101 106 111 2 1.0 834 117 122 127 132 137 143 148 153 158 163 3 1.5 835 169 174 179 184 189 195 200 205 210 215 4 2.0 836 221 226 231 236 241 247 252 257 262 267 5 2.5 837 273 278 283 288 293 298 304 309 314 319 6 3.0 838 324 330 335 340 345 350 355 361 366 371 7 3.5 839 376 381 387 392 397 402 407 412 418 423 8 4.0 840 428 "433 ~443 "449 “454 “459 “464 469 “474 9 4.5 841 480 ”485 "490 ”495 “500 “505 "5ll “516 “521 “526 842 531 536 542 547 552 557 562 567 572 578 843 583 588 593 59s 603 609 614 619 624 629 844 634 639 645 650 655 660 665 670 675 681 845 686 691 696 701 706 711 716 722 727 732 846 737 742 747 752 758 763 768 773 77s 783 847 788 793 799 804 809 814 819 824 829 334 848 840 845 850 855 860 865 870 875 881 886 849 891 996 901 906 911 916 921 927 932 937 850 942 “947 “952 957 “962 “967 “973 978 “983 “988 N. L.O 1 2 3 4 5 6 7 8 9 P . P. . i50 851 852 853 854 855 856 857 858 859 60 861 862 863 864 865 866 867 868 869 70 871 872 f«73 874 875 876 877 878 879 80 881 882 883 884 885 886 887 888 889 90 891 892 893 894 895 896 897 898 899 00 N. LOGARITHMS. 65 Table— ( Continued). 2 3 4 5 6 7 8 9 P. P. 952 957 962 967 973 978 983 988 *003 *008 *013 *018 *024 *029 *034 *039 054 059 064 069 075 080 085 090 105 110 115 120 125 131 136 141 156 161 166 171 176 181 186 192 207 212 217 222 227 232 237 242 258 263 268 273 278 283 288 293 6 308 313 318 323 328 33 1 339 344 1 0.6 359 364 369 374 379 384 389 394 2 1.2 409 4-4 420 425 430 435 440 445 3 1.8 460 465 470 4 75 480 485 490 495 4 2.4 510 515 520 526 ”53l "536 "541 546 5 6 3.0 3.6 561 566 571 576 581 586 591 596 7 4.2 611 616 621 626 631 636 641 646 8 4.8 661 666 671 676 682 687 892 897 9 5.4 712 717 722 727 732 737 742 747 762 767 772 777 782 787 792 797 812 817 822 827 832 837 842 847 862 867 872 877 882 887 892 897 912 917 922 927 932 937 942 947 962 967 972 977 982 987 992 ”997 012 017 022 027 ~032 "037 "042 "047 5 062 067 072 077 082 086 091 096 1 0.5 111 116 121 126 131 136 141 146 2 1.0 161 166 171 176 181 186 191 196 3 1.5 211 216 221 226 231 236 240 245 4 2.0 260 265 270 275 280 285 290 295 5 2.5 310 315 320 325 330 335 340 345 6 3.0 359 364 369 374 379 384 389 394 7 3.5 409 414 419 424 429 433 438 443 8 4.0 "458 "463 "468 "473 "478 ~483 ”488 ^93 9 4.5 507 "512 "517 “522 "527 ~532 *537 542 557 562 567 571 576 581 586 591 606 611 616 621 626 630 635 640 655 660 665 670 675 680 685 689 704 709 714 719 724 729 734 738 753 758 763 768 773 778 783 787 4 802 807 812 817 822 827 832 836 1 0.4 851 856 861 866 871 876 880 885 2 0.8 900 905 910 915 919 924 929 934 3 1.2 "949 "954 "959 "963 "968 "973 "978 ”983 4 1.6 998 *002 *007 *012 *0l7 *022 *027 ■■■032 5 g 2.0 2.4 D46 051 056 061 066 071 075 080 7 2.8 095 100 105 109 114 119 124 129 8 3.2 143 148 153 158 163 168 173 177 9 3.6 192 197 202 207 211 216 221 226 240 245 250 255 260 265 270 274 289 294 299 303 308 313 318 323 337 342 347 352 357 361 366 371 386 390 395 -400 405 410 415 419 434 439 444 448 453 1 458 463 468 2 3 4 5 6 1 7 8 9 P . P. 66 USEFUL TABLES. Table— ( Continued). N. L. 0 1 2 3 4 5 6 7 8 9 P. P. 900 95 424 429 434 439 444 448 453 458 463 468 901 472 477 482 487 492 497 501 506 511 516 902 521 525 530 535 540 545 550 554 559 564 903 569 574 578 583 588 593 598 602 607 612 904 617 622 626 631 636 641 646 650 655 660 905 665 670 674 679 684 6.-9 694 698 703 708 906 713 718 722 727 732 737 742 746 751 756 907 761 766 770 775 780 785 789 794 799 804 908 809 813 818 823 828 832 837 842 847 852 909 856 861 866 871 875 880 885 890 895 899 910 904 909 914 918 923 928 933 938 942 ”947 911 952 957 961 966 971 976 “980 “985 ”990 995 5 912 999 *004 *009 *014 *019 *023 *028 *033 *038 *042 1 0.5 913 96 047 052 057 061 066 071 076 080 085 090 2 1.0 914 095 099 104 109 114 118 123 128 133 137 3 1.5 915 142 147 152 156 161 166 171 175 180 185 4 2.0 916 190 194 199 204 209 213 218 223 227 232 5 2.5 917 237 242 246 251 256 261 265 270 275 280 6 3.0 918 284 289 294 298 303 308 313 317 322 327 7 3.5 919 332 336 341 346 350 355 360 365 369 374 8 9 4.0 920 379 384 388 393 398 402 407 ”412 "417 ”421 4.5 921 426 431 435 440 445 '450 ”454 ”459 464 “468 922 473 478 483 487 492 497 501 506 511 515 923 520 525 530 534 539 544 548 553 558 562 924 567 572 577 581 586 591 595 600 605 609 925 614 619 624 628 633 638 642 647 652 656 926 661 666 670 675 680 685 689 694 699 703 927 708 713 717 722 727 731 736 741 745 750 928 755 759 764 769 774 778 783 788 792 797 929 802 806 811 816 820 825 830 834 839 844 930 848 "853 858 “862 "867 ”872 ”876 ”881 ”886 ”890 931 895 "900 "904 “909 914 ”918 “923 ”928 “932 ”937 4 - 932 942 946 951 956 960 965 970 974 979 984 1 0.4 933 988 993 997 *002 *007 *011 *016 *021 *025 ’030 2 0.8 934 97 035 039 044 049 053 058 063 067 072 077 3 1.2 935 081 086 090 095 100 104 109 114 118 123 4 1.6 936 128 132 137 142 146 151 155 160 165 169 5 2.0 937 174 179 183 188 192 197 202 206 211 216 6 2.4 938 220 225 230 234 239 243 248 253 257 262 7 2.8 939 267 271 276 280 285 290 294 299 304 308 8 3.2 940 313 ~317 322 “327 "331 ”336 “340 ”345 ”350 ”354 9 3.6 941 359 “364 ”368 373 “377 “382 “387 391 ”396 ”400 942 405 410 414 419 424 428 433 437 442 447 943 451 456 460 465 470 474 479 483 488 493 944 497 502 506 511 516 520 525 529 534 539 945 543 548 552 557 562 566 571 575 580 585 946 589 594 598 603 607 612 617 621 626 630 947 635 640 644 649 653 658 663 667 672 676 948 681 685 690 695 699 704 708 713 717 722 949 727 731 736 740 745 749 754 759 763 768 950 772 777 782 786 791 795 800 804 809 813 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. LOGARITHMS. 67 T able— ( Continued ) . N. L. 0 1 2 3 4 5 6 7 I 8 9 P . P. 950 97 772 ~777 782 “786 "791 705 800 “804 N 111 “813 951 818 823 827 832 836 841 *845 “850 ”855 859 952 864 868 873 877 882 886 891 896! 900 905 953 909 914 918 923 928 932 937 941 946 950 954 955 959 964 868 973 978 982 987 991 996 955 98 000 005 009 014 019 023 028 032 037 041 956 046 050 055 059 064 068 073 078 082 087 957 091 096 100 105 109 114 118 123 127 132 958 137 141 146 150 155 159 164 168 173 177 959 182 186 191 195 200 204 209 214 218 223 960 227 232 236 241 245 250 “254 “259 "263 "268 961 272 277 281 286 290 295 "299 “304 "308 “313 5 962 318 322 327 331 336 340 345 349 1 354 358 1 0.5 963 363 367 372 376 381 385 390 1 394; 399 403 2 1.0 964 408 412 417 421 426 430 435 j 439! 444 448 3 1.5 965 453 457 462 466 471 475 4801 484 j 489 493 4 2.0 966 498 502 507 511 516 520 525 529i 534 538 5 2 .5 967 543 547 552 556 561 565 570 574| 579 583 3.0 968 588 592 597 601 605 610 614 l 619! 623 628 7 3.5 969 632 637 641 646 650 655 659 i 664; 668 673 8 9 4.0 4.5 970 677 682 686 691 695 700, "704 1 709 1 Tl3 “717 971 722 726 731 735 740 744 “749 !“753l "758 “762 972 7671 771 776 780 784 789 793 i 798 802 807 973 811! 816 820 825 829 834 838 ! 843! 847 851 974 8561 860 865 869 874 878 883 1 887; 892 896 975 900 905 i 909 914 918 923 927 932 i 936 941 976 945 i 949 954 958 963 967 972 1 976 1 981 985 977 9891 j 994 998 *003 *007 *012 *016 *021 *025 *029 978 99 034 038 043 1 047 j 052 056 1 061 ! 065 069 074 979 078 1 083 087 092 ! 096 100 105 j 109 I 114 118 980 123 hm Hi 136 "lio 145 "149 154 “l58 ~162 981 167 171 176 180 185 189 193 i 198 1 202 207 4 982 211 216 220 224 229 233 238 '242 247 251 1 0.4 983 255 260 264 269 273 277 282 ! 286 291 295 2 0.8 984 300 304 308 313 317 322 326 1 330 335 339 3 1.2 985 344 348 352 357 361 366 370 374 379 383 4 1.6 986 388 392 396 401 405 410 414 419 423 427 5 2.0 987 432 436 441 445 449 454 458 463 467 471 6 2.4 988 476 480 484 489 493 498 502 506 511 515 7 2.8 989 520 524 528 533 537 542 546 550 555 559 8 3.2 990 564 “568 “572 “577 “581 "585 "590 “594 "599 ”603 9 3.6 991 607 “612 “616 "621 "625 “629 “634 “638 “642 “647 992 651 656 660 664 669 673 677 682 i 686 691 993 695 699 704 708 712 717 721 726 730 734 994 739 743 747 752 756 760 765 769 ! 774 778 995 782 787 791 795 800 804 808 813 i 817 822 996 826 830 835 839 843 848 852 856 861 865 997 870 874 878 .883 887 891 896 900 904 909 998 913 917 922 926 930 935 939 944 948 952 999 957 961 965 970 974 978 983 987 991 996 1000 00 000 004 “009 1)13 "017 “022 “026 “030 “035 ”039 N. L. 0 1 2 3 4 5 6 7 8 9 P . P. 68 USEFUL TABLES. TRIGONOMETRIC FUNCTIONS.. DIRECTIONS FOR USING THE TABLE. The table given on pages 74-78 contains the natural sines, cosines, tangents, and cotangents of angles from 0° to 90°. Angles less than 45° are given in the first column at the left- hand side of the page, and the names of the functions are given at the top of the page; angles greater than 45° appear at the right-hand side of the page, and the names of the func- tions are given at the bottom. Thus, the second column con- tains the sines of angles less than 45° and the cosines of angles greater than 45°; the sixth column contains the cotangents of angles less than 45° and the tangents of angles greater than 45°. To find the function of an angle less than 45°, look in the column of angles at the left of the page for the angle, and at the top of the page for the name of the function; to find a function of an angle greater than 45°, look in the column at the right of the page for the angle and at the bottom of the page for the name of the function. The successive angles differ by an interval of 10'; they increase downwards in the left-hand column and upwards in the right-hand column. Thus, for angles less than 45° read down from top of page, and for angles greater than 45° read up from bottom of page. The third, fifth, seventh, and ninth columns, headed d , contain the differences between the successive functions; for example, in the second column we find that the sine of 32° 10' is .5324 and that the sine of 32° 20' is .5348; the difference is .5348 —.5324 = .0024, and the 24 is written in the third column, just opposite the space between .5324 and .5348. In like man- ner the differences between the successive tabular values of the tangents are given in the fifth column, those between the cotangents in the seventh column, and those for the cosines in the ninth column. These differences in the functions cor- respond to a difference of 10' in the angle; thus, when the angle 32° 10' is increased by 10', that is, to 32° 20', the increase of the sine is .0024, or, as given in the table, 24. It will be observed that in the tabular difference no attention is paid to the decimal point, it being understood that the difference is TRIGONOMETRIC FUNCTIONS. G9 merely the number obtained by subtracting the last two or three figures of the smaller function from those of the larger. These differences are used to obtain the sines, cosines, etc. of angles not given in the table; the method employed may be illustrated by an example. Required, the tangent of 27° 34'. Looking in the table, we see that the tangent of 27° 30' is .5206, and (in column 5) the difference for 10' is 37. Differ- ence for 1' is 37 -f- 10 = 3.7, and difference for 4' is 3.7 X 4 = 14.8. Adding this difference to the value of the tan 27° 30', we have tan 27° 30' = .5206 difference for 4' = 14.8 tan 27° 34' = .5220.8 or .5221, to four places. Since only four decimal places are retained, the 8 in the fifth place is dropped and the figure in the fourth place is increased by 1, because 8 is greater than 5. To avoid multiplication, the column of proportional parts, headed P. P., at the extreme right of the page, is used. At the head of each table in this column is the difference for 10', and below are the differences for any intermediate number of minutes from 1' to 9'. In the above example, the differ- ence for 10' was 37; looking in the table with 37 at the head, the difference opposite 4 is 14.8; that opposite 7 is 25.9; and so on. For want of space, the differences for the cotangents for angles less than 45° (or the tangents of angles greater than 45°) have been omitted from the tables of proportional parts. The use of these functions should be avoided, if possible, since the differences change very rapidly, and the computa- tion is therefore likely to be inexact. The method to be employed when dealing with these functions may be shown by an example: Required, the tangent of 76° 34'. Since this angle is greater than 45°, w T e look for it in the column at the right, and read up; opposite the 76° 30', we find, in sixth col- umn, the number 4.1653, and corresponding to it in seventh column is the difference 540. Since 540 is the difference for 10', the difference for 4' is 540 X ^ = 216. Adding this difference: tan 76° 30' = 4.1653 difference for 4' = 216 tan 76° 34' = 4.1869 70 USEFUL TABLES. When the angle contains a certain number of seconds, divide the number by 6, and take the whole number nearest to the quotient; look out this number in the table of propor- tional parts (under the proper difference ), and take out the number that is opposite to it. Shift the decimal point one place to the left, and then add it to the partial function already found. Find the sine of 34° 26' 44". sine 34° 2 O' = .5640 difference for 6' = 14.4 difference for 44" = 1.7 Difference for 10' = 24. sine 34° 26' 44" = .5656 -b* = 7}. Look out in the P. P. table the number under 24 and opposite 7. It is 16.8. Shifting the decimal point one place to the left, we get 1.68, or, say, 1.7. The tangent is found in the same way as the sine. To find the cosine of an angle: As the angle increases, the value of the cosine decreases, so that, instead of adding the values corresponding to 6' and 44" to the function already found, we subtract them from it. Thus, find cos 34° 26' 44". cos 34° 20' = .8258 Difference for 10' = 17. difference for 6' = 10.2 difference for 44" = 1.2 total difference = 11.4 .8247 The number under the 17 and opposite the 7, in the P. P. table, is 11.9. Therefore, take 1.19, or, say, 1.2. Therefore, cos 34° 26' 44" = .8258 - .0011 = .824°. Only four decimal places are kept; therefore, the figure of the difference following the decimal point is dropped before subtracting. The cotangent is found in the same manner. We will now consider angles greater than 45°. Find the sine of 68° 47' 22". In obtaining the difference , it must be remembered to choose the one between the sine of 68 u 40' and the next angle above it, namely, 68° 50'. TRIGONOMETRIC FUNCTIONS. 71 sine 68° 40' = .9315 difference for 7' = 7 difference for 19" = .4 Difference for 10' = 10. sine 68° 47' 22" = .9322 * = 3f, say 4. Under the 10 and opposite the 4 is the number 4.0; shifting the deci- mal point, we get .4. As usual, only four decimal places are kept. The tangent is found in the same manner. Find cos 68° 47' 22". As before, the cosine decreases as the angle increases; therefore, we subtract the successive sine values correspond- ing to the increments in the angle.. cos 68° 40' = .3638 difference for 7' = 18.9 difference for 22" = 1.1 total difference = Difference for lO' = 27. 20 .3618 Under the 27 and opposite the 4 is the number 10.8; there- fore, take 1.08 in this case, or, say, 1.1. Therefore, cos 68° 47' 22" = .3638 - .002 = .3618. The cotangent is found in the same way. In finding the functions of an angle, the only difficulty likely to be encountered is to determine whether the differ- ence obtained from the table of proportional parts is to be added or subtracted. This can be told in every case by observing whether the function is increasing or decreasing as the angle increases. For example, take the angle 21°; its sine is .3584, and the following sines, reading downwards, are .3611, .3638, etc. It is plain, therefore, that the sine of say 21° 6' is greater than that of 21°, and that the difference for 6' must be added. On the other hand, the cosine of 21° is .9336, and the following cosines, reading downwards, are .9325, .9315, etc.; that is, as the angle grows larger the cosine decreases. The cosine of an angle between 21° and 21° 10', say 21° 6', must therefore lie between .9325 and .9315; that is, it must be smaller than .9325, which shows that in this case the difference for 6' must be subtracted from the cosine of 21°. We will now consider the case in which the function, i. e., the sine, cosine, tangent, or cotangent, is given and the cor- responding angle is to be found. 72 USEFUL TABLES. Find the angle whose sine is .4943. The operation is arranged as follows: .4943 Difference for .4924 = sin 29° 30'. 1st remainder 19 18.2 = difference for 7'. 2d remainder .8 .78 = difference for .3' or 18". .4943 = sin 29° 37' 18". Looking down the second column, we find the sine next smaller than .4943 to be .4924, and the difference for 10' to be 26. The angle corresponding to .4924 is 29° 30'. Sub- tracting the .4924 from .4943, the first remainder is 19; looking in the table of proportional parts, the part next lower than this difference is 18.2, opposite which is 7'. Subtracting this difference from the remainder, we get .8, and, looking in the table, we see that 7.8 with its decimal point moved one place to the left is nearest to the second difference. This is the difference for .3' or 18". Hence, the angle is 29° 30' + 7' ■f 18 = 29° 37' 18". Find the angle whose tangent is .8824. .8824 Difference for 10' = 51. .8796 = tan 41° 20'. 1st remainder 28 25.5 = difference for 5'. 2d remainder 2.5 2.55 = difference for .5' or 30". .8824 = tan 41° 25' 30". In the two examples just given, the minutes and seconds corresponding to the 1st and 2d remainders are added to the angle taken from the table. Thus, in the first example, an inspection of the table shows that the angle increases as the sine increases; hence, the angle whose sine is .4943 must be greater than 29° 30', whose sine is .4924. For this reason the correction must be added to 29° 30'. The same reasoning applies to the second example. TRIGONOMETRIC FUNCTIONS. 73 Find the angle whose cosine is .7742. .7742 Difference for 10' = 18. .7735 = cos 39° 20'. 1st remainder 7 5.4 = difference for 3'. 2d remainder 1.6 1.62 = difference for .9' or 54". 39° 20' — 3' 54" = 39° 16' 6", which is the angle whose cosine is .7742. Looking down the eighth column, headed cos, the next smaller cosine is .7735, to which corresponds the angle 39° 20'. The difference for 10' is 18. Subtracting, the remain- der is 7, and the next lower number in the table of propor- tional parts is 5.4, which is the difference for 3'. Subtracting this from 1st remainder, 2d remainder is 1.6, which is nearest 16.2 of table of proportional parts, if the decimal point of the latter is moved to the left one place. Since 16.2 corresponds to a difference of 9', 1.62 corresponds to a difference of .9% or 54". Hence, the correction for the angle 39° 20' is 3' 54", From the table, it appears that, as the cosine increases, the angle grows smaller; therefore, the angle whose cosine is .7742 must be smaller than the angle whose cosine is .7735, and the correction for the angle must be subtracted. Find the angle whose cotangent is .9847. .9847 Difference for 10' = 57. .9827 = cos 45° 30'. 1st remainder 20 17.1 = difference for 3'. 2d remainder 2.9 2.85 = difference for .5' or 30". 45° 30' — 3' 30" = 45° 26' 30", the angle whose cotangent is .9847. In finding the angle corresponding to a function, as in the above examples, the angles obtained may vary from the true angle by 2 or 3 seconds; in order to obtain the number of seconds accurately, the functions should contain six or seven decimal places. o 0 I 2 3 4 5 6 7 8 9 ).0058 ).0087 ).0116 ).0145 >.0175 >.0204 ).0233 ).0262 ).0291 ).0320 ).0349 >76378 ).0407 ).0437 ).0466 ).0495 ).0524 ).0553 ).0582 >.0612 >.0641 >.0670 >.0699 >.0729 >.0758 >.0787 1.0816 1.0846 >.0875 76904 1.0934 >.0963 1.0992 (.1022 (.1051 U 080 (.1110 .1139 .1169 .1198 0228 0257" 1287 1317 .1346 0376 .1405 .1435 .1465 .1495 .1524 0554 0584 .1257 Cot. infinit . 343.7737 171.8854 114.5887 85.9398 68.7501 57.2900 49.1039 42.9641 38.1885 34.3678 31.2416 28.6363 26.4316 24.5418 22.9038 21.4704 20.2056 19.0811 18.0750 17.1693 16.3499 15.6048 14.9244 14.3007 13.7267 13.1969 12.7062 12.2505 11.8262 11.4301 11.0594 10.7119 10.3854 10.0780 9.7882 9.5144 9.2553 9.0098 8.7769 8.5555 8.3450 ~~ 8.1443 7.9530 7.7704 7.5958 7.4287 7.2687 7.1154 6.9682 6.8269 6.6912 6.5606 6.4348 81861 61398 47756 38207 31262 26053 22047 18898 16380 14334 12648 11245 10061 905’ 8194 7451 6804 6237 5740 5298 4907 4557 4243 3961 3707 3475 3265 3074 2898 2738 2591 2455 2329 2214 2105 2007 1913 1826 1746 1671 1600 1533 1472 1413 1357 1306 1258 1210 Cot. d. Tan. I d. Cos. 1.0000 1.0000 .0000 1.0000 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993 0.9992 0.9990 0.9989 0.9988 0.9986 0.9985 0.9983 0.9981 0.9980 0.9978 09976 679974 0.9971 0.9969 0.9967 0.9964 09962 679959 0.9957 0.9954 0.9951 0.9948 0.9945 09942 0.9939 0.9936 0.9932 0.9929 09925 0.9922 0.9918 0.9914 0.9911 0.9907 0.9903 P . P. <> 90 50 30 40 30 1 3.0 20 2 6.0 10 3 : 9.0 0 89 4' 5 12.0 15.0 50 6 18.0 40 7 21.0 30 8 24.0 20 10 0 88 9 27.0 29 50 •1 ; 2.9 40 2 j 5.8 30 3 | 8 .T 20 4 ! 11.6 10 5 ' 14.5 0 87 6 17.4 50 40 30 7 8 1 20.3 ! 23.2 9 ; 26.1 20 10 0 86 1 28 2.8 50 2 5.6 40 3 8.4 30 4 11.2 20 5 14.0 10 6 16.8 0 85 7 8 19.6 22.4 50 9 25.2 40 30 20 5 10 1 0.5 0 84 2 1.0 1.5 2.0 50 4 40 5 2.5 30 6 3.0 20 7 3.5 10 8 4.0 0 83 9 4.5 50 40 4 30 20 1 2 0.4 0.8 10 3 1.2 o 82 4 1.6 50 5 2.0 40 6 2.4 30 7 2.8 8 3.2 — 9 3.6 t o P . P. 0 10 •20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 ,32 Cot. 6.3138 6.1970 6.0844 5.9758 5.8708 5.7694 5.6713 5 .57 64 5.4845 5.3955 5.3093 5.2257 5.1446 5^0658 4.9894 4.9152 4.8430 4.7729 4/7046 4.6382 4.5736 4.5107 4.4494 4.3897 4.3315 4.2747 4.2193 4.1653 4.1126 4.0611 4.0108 3.9617 3.9136 3.8667 3.8208 3.7760 3.7321 jT .6891 3.6470 3.6059 3.5656 3.5261 3.4874 3^4495 3.4124 3.3759 3.3402 3.3052 3.2709 3.2371 3.2041 3.1716 3.1397 3.1084 3.0777 Cot. d. Tan. d. Sin. d. Cos. 0.9877 0.9872 0.9868 0.9863 0.9858 0.9853 0.9848 0.9793 0.9787 0.9696 0.9689 0.9681 0.9674 0.9667 0.9659 ( L 9652 0.9644 0.9636 0.9628 0.9621 0^9613 0.9605 0.9596 0.9588 0.9580 0.9572 0.9563 0.9555 0.9546 0.9537 0.9528 0.9520 0.9511 0 81 50 0 80 50 10 0 79 50 0 78 10 077 50 40 30 20 10 0 76 50 40 30 20 10 0 75 50 o 74 0 73 o 72 P, . P. 32 31 30 1 3.2 3.1 3.0 2 6.4 6.2 6.0 3 9.6 9.:'. 9.0 4 12 8 12.4 12.0 5 16.0 15.5 15.0 6 19.2 18.6 18.0 7 22.4 21.7 21.0 8 24.8 24.0 9 28.8 27.9 27.0 29 28 27 1 2.9 2.8 2.7 2 5.8 5.6 5.4 3 8.7 8.4 8.1 4 11.6 11.2 10.8 5 14.5 14.0 13.5 6 17.4 16.8 16.2 7 20.3 19.6 18.9 8 23.2 22.4 21.6 9 26.1 25.2 24.3 9 8 0.9 0.8 1.8 1 1.6 2.72.4 3.63.2 4.5 4.0 5.4 4.8 6.3 1 5.6 7.2 6.4 8.1 1 7.2 7 6 1 0.7 0.6 2 1.41.2 3 2.11.8 4 2.8 j 2.4 5 3.5 3.0 6 4.23.6 7 4.9 4.2 8 5.6 4.8 9 6.3 i 5.4 5 I’ 4 1 0.50.4 2 1 . 0 ! 0.8 3 1.51.2 4 2.0 ! 1.6 5 2.5'2.0 6 3.0 2.4 7 3.5 j 2.8 8 4.0 3.2 9 4.5, 3.6 P. P. / ~0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 Tan. 0.5095 Cot. d. Cot. 3.0777 3.0475 3.0178 2.9887 2.9600 2.9319 2.9042 2.8770 2.8502 2.8239 2.7980 2.7725 2.7475 2.7228 2.6985 2.6746 2.6511 2.6279 2.6051 2.5826 2.5605 2.5386 2.5172 2.4960 2.4751 2.4545 2.4342 2.4142 2.3945 2.3750 2.3559 2.3369 2.3183 2.2998 2.2817 2.2637 2.2460 2.2286 2.2113 2.1943 2.1775 2.1609 2.1445 2.1283 2.1123 2.0965 2.0809 2.0655 2.0503 2.0353 2.0204 2.0057 1.9912 1.9768 1.9626 Tan. Cos. 0.9511 0.9502 0.9492 0.9483 0.9474 0.9465 0.9455 0.9*46 0.9436 0.9426 0.9417 0.940' 0.9397 0.9387 0.9377 0.9367 0.9356 0.9346 0.9336 0.9325 0.9315 0.9304 0.9293 0.9283 0.9272 0.9261 0.9250 0.9239 0.9228 0.9216 0.9205 0.9194 0.9182 0.9171 0.9159 0.9147 0.9135 0.9124 0.9112 0.9100 0.9088 0.9075 •0.9063 0.9051 0.9038 0.9026 0.9013 0.9001 0.8988 0.8975 0.8962 0.8949 0.8936 0.8923 0.8910 d. Sin. d. o 72 50 o 71 o 70 50 0 69 0 67 0 66 0 65 50 0 64 50 o 63 I ’. P . 37 36 35 1 3.7 3.6 3.5 2 7.4 7.2 7.0 3 11.1 10.8 10.5 4 14.8 14.4 14.0 5 18.5 18.0 17.5 6 22.2 21.6 21.0 7 25.9 25.2 24.5 8 29.6 28.8 '28.0 9 33.3,32.4 .31.5 34 33 32 1 3.4 3.3 3.2 2 6.8 6.6 6.4 3 10.2 9.9 9.6 4 13.6 13.2 12.8 5 17.0 16.5 16.0 6 20.4 19.8 19.2 7 23.8 23.1 [22.4 8 27.2 26.4 25.6 9 30.6 29.7128.8 28 27 26 1 2.8 2.7 2.6 2 5.6 5.4 5.2 3 8.4 8.1 7.8 4 11.2 10.8 10.4 5 14.0 13.5 13.0 6 16.8 16.2 15.6 7 19.6 18.9 18.2 8 22.4 21.6 20.8 9 25.2, ,24.3,23.4 10.4 9.6 11.7;10.8 II I0| 9 1.0 j 0.9 2.0 1.8 3.02.7 4.0 3.6 5.04.5 6.0 j 5.4 7.0 6.3 8.0 7.2 9.9,9.08.1 P . P . p . p . \ 44 143 CM 1 4.4 4.3 4.2 2 8.8 8.6 8.4 3 13.2 12.9 12.6 4 17.6 17.2 16.8 5 22.0 21.5 21.0 , 6 26.4 25.8 25.2 5 7 30.8|30.1 29.4 8 35.2 34.4 ! 33.6 9 39.6,38.7,37.8 41 40 39 1 4.1 4.0 3.9 2 8.2 8.0 7.8 1 3 12.3 12.0 11.7 4 16.4 16.0 15.6 5 20.5 20.0 19.5 6 24.6 24.0 23.4 7 28.7 28.0 27.3 8 32.8 32.0 31.2 9 36.9 36.0 35.1 38 3.8 7.6 11.4 15.2 19.0 22.8 26.6 30.4 34.2 37 3.7 7.4 11.1 14.8 18.5 22.2 25.9 29.6 33.3 26 25 24 1 2.6 2.5 2.4 2 5.2 5.0 4.8 3 7.8 7.5 7.2 4 10.4 10.0 9.6 4 13.0 12.5 12.0 6 15.6 15.0 14.4 7 18.2 17.5 16.8 8 20.8 20.0 19.2 9 23.4 22.5 21.6 23 17 16 1 2.3 1.7 1.6 2 4.6 3.4 3.2 3 6.9 5.1 4.8 4 9.2 6.8 6.4 5 11..') 8.5 8.0 6 13 .S 10.2 9.6 7 16.1 11.9 11.2 8 18.4 13.6 12.8 9 20.7 15.3 14.4 15 14 13 1 1.5 1.4 1.3 2 3.0 2.8 2.6 3 4.5 4.2 3.9 4 6.0 5.6 5.2 5 7.5 7.0 6.5 6 9.0 8.4 7.8 7 10.5 9.8 9.1 8 12.0 11.2 10.4 9 13.5 12.6 11.7 P . P . Sin. j 0.5878 0.5901 ) 0.5925 ; o ) 0.5948 24 - 23 ) 0.5972 ) 0.5995 Tan. 0.7265 0.7310 0.7355 0.7400 0.7445 0.7490 0.7536 0.7581 0.7627 0.7673 0.7720 0.7766 0.7813 0.7860 0.7907 0.7954 0.8002 o.8o;o 0.8098 0.8146 0.8195 0.8243 0.8292 0.8342 0.8391 0.8441 0.8491 0.8541 0.8591 0.8642 0.8693 0-8744 0.8796 0.8847 0.8899 0.8952 0.9004 Cos. d. 0.9057 0.9110 53 0.9163 54 0.9217 0.9271 54 55 0.9325 0.9380 0.943 j 0.9490 0.9545 0.9601 0.9657 0.9713 0.9770 0.9827 Cot. 1.3764 1.3680 1.3597 1.3514 1.3432 1.3351 1.3270 1.3190 1.3111 1.3032 1.2954 1.2876 1.2799 1.2723 1.2647 1.2572 1 .2497 1.2423 1.2349 1.2276 1.2203 1.2131 1.2059 1.1988 1.1918 1.1847 1.1778 1.1708 1.1640 1.1571 1.1504 1.1436 1 .1369 1.1303 1.1237 1.1171 1.1106 1.1041 1.0977 1.0913 1.0850 1.0786 1.0724 1.0661 1 .0599 1 .0538 1.0477 1.0416 1.0355 1.0295 1.0235 1.0176 1.0117 1.0058 1.0000 , Tan. Cos. i 108090 0.8073 0.8056 0.8039 0.8021 0.8004 0.7986 0.7969 0.7951 0.7934 0.7916 0.7898 0.7880 0.7862 0.7844 0.7826 0.7808 0.7790 0.7771 0.7753 0.7735 0.7716 0.7698 0.7679 0.7660 0.7642 0.7623 0.7604 0.7585 0.7566 0.7547 ! 0.7528 ; 0.7509 ; 0.7490 , 0.7470 0.7451 0.7431 . 0.7412 : 0.7392 i 0.7373 . 0.7353 ’ 0.7333 ; 0.7314 ! 0.7294 ' 0.7274 0.7254 0.7234 0.7214 0.7193 0.717:! , 0.7153 , 0.7133 , 0.7112 0.7092 1 0.7071 d. Sin. d. ' ° 0 54 50 0 53 0 52 50 o 51 0 50 0 49 50 0 48 50 10 0 47 50 0 46 50 0 45 P. P. 58 5.8 11.6 17.4 23.2 29.0 34.8 40.6 46.4 52.2 54 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 50 55 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 51 5.1 10.2 15.3 20.4 10.4 1 15.6 1 20.8 2 26.5 26.0 25.5 31.8:31.2 30.6 37.1 36.4 35.7 42.4! 4 1.6 40.8 47.7,46.8 45.9 49 1 48 4.9 4.8 9.8 l 9.6 14.714.4 19.6 19.2 5.0 24.0 28.8 33.6 38.4 43.2 45 14.1 18.8 23.5 23.0 22.5 28.2 27 .6 j 27 .0 32.9 32.2 31.5 37.6 36.8'36.0 42.3 41.4] 40.5 24 23 22 21 1 2.4 2.3 2.2 2.1 2 4.8 4.6 4.4 4.2 3 7.2 ' 6.9 6.6 6.3 4 9.6 9.2 8.8 8.4 5 12.0 11.5 11.0 10.5 6 14.4 13.8 13.2 12.6 7 16.8 16.1 15.4 14.7 8 19.2 18.4 17.6 16.8 9 21.6 20.7 19.8 18.9 20 19 18 17 1 2.0 1.9 1.8 1.7 2 4.0 3.8 3.6 3.4 3 6.0 5.7 5.4 5.1 4 8.0 7.6 7.2 6.8 5 10.0 9.5 9.0 8.5 6 12.0 11.4 10.8 10.2 7 14.0 13.3 12.6 11.9 8 16.0 15.2 14.4 13.6 9 18.0 17.1 |l6.2 15.3 P. P. PRIME NUMBERS. 79 PRIME NUMBERS. Every prime number is an odd number and has for its unit figure 1, 3, 7, or 9; any odd number that has 5 for its unit fig- ure is divisible by 5, and is not a prime number. The prime factors of any number less than 1,000 may be found from the following table. If the number is odd and does not end with 5, the factors are given directly; thus, the prime factors of 357 are 3, 7, and 17; those of 931 are 7, 7, and 19, the exponent 2 of the 7 indicating that 7 is used twice as a factor. If a number is a prime number, the space beside it is blank; thus, 317 and 859 are prime numbers. To find the prime factors of an odd number that has 5 for the unit figure, divide by 5 until a quotient is obtained which does not have 5 for a unit figure; the factors of this quotient are then found from the table, and with the 5’s already used as divisors constitute the prime factors. For example, to find the prime factors of 5,775 proceed as follows: 5,775-4-5 = 1,155; 1,155-4-5 = 231; from the table, 231 = 3 X 7 X 11; hence, 5,775 = 3 X 5 X 5 X 7 X 11. If the number is even, divide it by 2, the quotient by 2, and so on until an odd quotient is reached; then find the prime factors of the quotient from the table. The process of finding the prime factors of 936 is as follows: 936 -4- 2 = 468; 468 -4- 2 = 234; 234 2 = 117; 117 = 3 2 X 13, from table. Hence, 936 = 2' X 3 2 X 13 = 2 X 2 X 2 X 3 X 3 X 13. FACTORS OF 3.1416. Not Regarding Decimal Point, 3.1416 = 2 X 15708 22 X 1428 68 X 462 3 X 10472 24 X 1309 77 X 408 4 X 7854 28 X 1122 84 X 374 6 X 5236 33 X 952 88 X 357 7 X 4488 34 X 924 102 X 308 8 X 3927 42 X 748 119 X 264 11 X 2856 44 X 714 132 X 238 12 X 2618 51 X 616 136 X 231 14 X 2244 56 X 561 154 X 204 17 X 1848 21 X 1496 66 X 476 168 X 187 BO USEFUL TABLES. PRIME FACTORS. Prime Factors of All Odd Numbers From 1 to 1,000 That Are Not Divisible by 5. 1 101 201 3-67 301 7-43 401 3 103 203 7-29 303 3-101 403 13-31 7 107 207 3-23 307 407 11-37 9 32 109 209 11-19 309 3-103 409 11 111 3-37 211 311 411 3-137 13 113 213 3-71 313 413 7-59 17 117 3 2 T3 217 7-31 317 417 3-139 19 119 7.17 219 3-73 319 11-29 419 21 3-7 121 112 221 13T7 321 3-107 421 23 123 3-41 223 323 17-19 423 32-47 27 3 3 127 227 327 3-109 427 7-61 29 129 3-43 229 329 7-47 429 3T1-13 31 131 231 3-7-11 331 431 33 3T1 133 7-19 233 333 3 2 -37 433 37 137 237 3-79 337 437 19-23 39 3T3 139 239 339 3T13 439 41 141 3-47 241 341 '11-31 441 32.72 43 143 11-13 243 35 343 73 443 47 147 3-72 247 13-19 347 447 3-149 49 72 149 249 3-83 349 449 51 3T7 151 251 351 33-13 451 11-41 53 153 3 2 -17 253 11-23 353 453 3151 57 319 157 257 357 3-7T7 457 59 159 3-53 259 7-37 359 459 3 3 -17 61 161 7-23 261 3 2 -29 361 192 461 63 32.7 163 263 363 3-11 2 463 67 167 267 3-89 367 467 69 3*23 169 132 269 369 32-41 469 7-67 71 171 3 2 '19 271 371 7-53 471 3-157 73 173 273 3-7T3 373 473 11-43 77 7*11 177 3-59 277 377 13-29 477 32-53 79 179 279 32-31 379 479 81 34 181 281 381 3-127 481 13-37 83 183 3-61 283 383 483 3-7-23 87 3-29 187 11-17 287 7-41 387 32-43 487 89 189 3 3 -7 289 172 389 489 3-163 91 7-13 191 291 3-97 391 17-23 491 93 3*31 193 293 393 3-131 493 17-29 97 197 297 3 3 -U 397 497 7-71 99 32-11 199 299 13-23 399 3-7T9 499 PRIME FACTORS. 81 Prime Factors of All Odd Numbers From 1 to 1,000 That Are Not Divisible by 5. ( Continued ). 501 3167 601 701 801 3 2 -89 901 17-53 503 603 3 2 '67 703 19-37 803 11-73 903 3-7-43 507 3-13 2 607 707 7-101 807 3-269 907 509 609 3-7-29 709 809 909 3 2 -101 511 7-73 611 13-47 711 32-79 811 911 513 3 3 T9 613 713 23-31 813 3-271 913 11-83 517 11-47 617 717 3-239 817 19-43 917 7131 519 3-173 619 719 819 3 2 -7-13 919 521 621 3 3 ‘23 721 7-103 821 921 3-307 523 623 7-89 723 3-241 823 923 13-71 527 17-31 627 3T1T9 727 827 927 32-103 529 23 2 629 17-37 729 3 e 829 929 531 3 2 '59 631 731 17-43 831 3-277 931 72-19 533 13-41 633 3-211 733 833 7 2 -17 933 3-311 537 3-179 637 7 2 -13 737 11-67 837 3 3 "31 937 539 7 2 T1 639 3 2 -71 739 839 939 3-313 541 641 741 3-13-19 841 29 2 941 543 3181 643 743 843 3-281 943 23-41 547 647 747 3 2 -83 847 7-11 2 947 549 3 2 *61 649 11-59 749 7-107 849 3-283 949 13-73 551 19-29 651 3-7-31 751 851 23-37 951 3-317 553 7-79 653 753 3-251 853 953 557 657 3 2 -73 757 857 957 3-11-29 559 13-43 659 759 3-11-23 859 959 7137 561 3-11-17 661 761 861 3-7-41 961 312 563 663 3T3-17 763 7*109 863 963 32-107 567 3 4 *7 667 23-29 767 13-59 867 3 -17 2 967 569 669 3-223 769 869 11-79 969 3T7-19 571 671 11-61 771 3-257 871 13-67 971 573 3-191 673 773 873 32-97 973 7-139 577 677 777 3-7-37 877 977 579 3-193 679 7.97 779 19-41 879 3-293 979 11-89 581 7-83 681 3-227 781 11-71 881 981 32-109 583 11-53 683 783 3 3 ‘29 883 983 587 687 3-229 787 887 987 3-7-47 589 19-31 689 13-53 789 3-263 889 7T27 989 23-43 591 3-197 691 791 7T13 891 3 4 "11 991 593 693 32-7-11 793 13*61 893 19-47 993 3-331 597 3-199 697 17-41 797 897 3-13*23 997 599 699 3-233 799 17-47 899 29-31 999 3 3 *37 82 USEFUL TABLES. CIRCUMFERENCES AND AREAS OF CIRCLES FROM 1-64 TO 100. Diam. Circum. Area. Diam. Circum. Area. .0491 .0002 4% 13.7445 15.0330 .0982 .0008 4K 14.1372 15.9043 .1963 .0031 4% 14.5299 16.8002 Vs .3927 .0123 4/4 4J| 14.9226 17.7206 .5890 .0276 15.3153 18.6555 li .7854 .0491 5 15.7080 19.6350 £ .9817 .0767 5% 16.1007 20.6290 % 1.1781 .1104 5M 16.4934 21.6476 £ 1.3744 .1503 5/4 16.8861 22.6907 % 1.5708 .1963 534 17.2788 23.7583 £ 1.7671 .2485 5^4 17.6715 24.8505 1.9635 .3068 5^4 18.0642 25.9673 £ 2.1598 .3712 5% 18.4569 27.1086 2.3562 .4418 6 18.8496 28.2744 2.5525 .5185 6/4 19.2423 29.4648 Pa 2.7489 .6013 6 M 19.6350 30.6797 if 2.9452 .6903 (% 20.0277 31.9191 1 3.1416 .7854 634 20.4204 33.1831 lVs 3.5343 .9940 6% 20.8131 84.4717 3.9270 1.2272 6/4 21.2058 35.7848 1% 4.3197 1.4849 6j| 21.5985 37.1224 4.7124 1.7671 7 21.9912 38.4846 1% 5.1051 2.0739 734 22.3839 39.8713 1/4 1% 5.4978 2.4053 7)4 22.7766 41.2826 5.8905 2.7612 23.1693 42.7184 2 6.2832 3.1416 7 )f 23.5620 44.1787 2% 6.6759 3.5466 7 5 /| 23.9547 45.6636 2 y* 7.0686 3.9761 7 % m 24.3474 47.1731 2ya 7.4613 4.4301 24.7401 48.7071 7.8540 4.9087 8 25.1328 50.2656 2&s 8.2467 5.4119 8)4 25.5255 51.8487 8.6394 5.9396 834 25.9182 53.4563 2,4 9.0321 6.4918 8/4 26.3109 55.0884 3 9.4248 7.0686 8)4 26.7036 56.7451 334 9.8175 7.6699 f/a 27.0963 58.4264 334 10.2102 8.2958 8% 27.4890 60.1322 Wa 10.6029 8.9462 8/4 27.8817 61.8625 33 4 10.9956 9.6211 9 28.2744 63.6174 3 % 11.3883 10.3206 9)4 934 28.6671 65.3968 3/4 11.7810 11.0447 29.0598 67.2008 3% 12.1737 11.7933 9% 29.4525 69.0293 4 12.5664 12.5664 9)4 29.8452 70.8823 434 12.9591 13.3641 9*1 30.2379 72.7599 434 13.3518 14.1863 tfi 30.6306 74.6621 TABLE OF CIRCLES. 83 Table— ( Continued ) . Diam. Circum. Area. Diam. Circum. Area. 9% 31.0233 76.589 15% 49.0875 191.748 10 31.4160 78.540 15% 49.4802 194.828 10% 31.8087 80.516 15% 49.8729 197.933 io k 32.2014 82.516 16 50.2656 201.062 10% 32.5941 84.541 16% 50.6583 204.216 io% 32.9868 86.590 16% 51.0510 207.395 10% 33.3795 88.664 1&\ 51.4437 210.598 10% 33.7722 90.763 16% 51.8364 213.825 10% 34.1649 92.886 16% 52.2291 217.077 11 34.5576 95.033 16% 52.6218 220.354 11 % 34.9503 97.205 16% 53.0145 223.655 11 % 35.3430 99.402 17 53.4072 226.981 ii% 35.7357 101.623 17% 53.7999 230.331 n% 36.1284 103.869 17% 54.1926 233.706 n% 36.5211 106.139 17% 54.5853 237.105 n% 36.9138 108.434 17% 54.9780 240.529 11 % 37.3065 110.754 17% 55.3707 243.977 12 37.6992 113.098 17% 55.7634 247.450 12% 38.0919 115.466 17% 56.1561 250.948 12% 38.4846 117.859 18 56.5488 254.470 12% 38.8773 120.277 18% 56.9415 258.016 12% 39.2700 122.719 18% 57.3342 261.587 12% 39.6627 125.185 18% 57.7269 265.183 12% 40.0554 127.677 18% 58.1196 268.803 12% 40.4481 130.192 18% 58.5123 272.448 13 40.8408 132.733 18% 58.9050 276.117 13% 41.2335 135.297 18% 59.2977 279.811 13% 41.6262 137.887 19 59.6904 283.529 13% 42.0189 140.501 19% 60.0831 287.272 3-3% 42.4116 143.139 19% 60.4758 291.040 13% 42.8043 145.802 19% 60.8685 294.832 13 % 43.1970 148.490 19% 61.2612 298.648 13% 43.5897 151.202 19% 61.6539 302.489 14 43.9824 153.938 19% 62.0466 306.355 14% 44.3751 156.700 19% 62.4393 310.245 14% 44.7678 159.485 20 62.8320 314.160 14% 45.1605 162.296 20% 63.2247 318.099 1434 45.5532 165.130 20% 63.6174 322.063 3-4% 45.9459 167.990 20% 64.0101 326.051 14% 46.3386 170.874 20% 64.4028 330.064 14% 46.7313 173.782 20% 64.7955 334.102 15 47.1240 176.715 20% 65.1882 338.164 15% 47.5167 179.673 20% 65.5809 342.250 15% 47.9094 182.655 21 65.9736 346.361 15% 48.3021 185.661 21% 66.3663 350.497 15% 48.6948 188.692 21% 66.7590 354.657 84 USEFUL TABLES. Table — ( Continued). Diam. Circum. Area. Diam. Circum. Area. 21% 67.1517 358.842 27% 85.2159 577.870 91 1/ 67.5444 363.051 27% 85.6086 583.209 21^| 67.9371 367.285 • 27% 86.0013 588.571 21% 68.3298 371.543 27% 86.3940 593.959 21% 68.7225 375.826 27fl 86.7867 599.371 22 69.1152 380.134 27% 87.1794 604.807 22% 69.5079 384.466 27% 87.5721 610.268 2234 69.9006 388.822 28 87.9648 615.754 70.2933 393.203 28% 88.3575 621.264 2234 70.6860 397.609 28% 88.7502 626.798 2g| 71.0787 402.038 28% 89.1429 632.357 71.4714 406.494 28% 89.5356 637.941 22^ 71.8641 410.973 28% 89.9283 643.549 23 72.2568 415.477 28% 90.3210 649.182 23% 72.6495 420.004 28% 90.7137 654.840 93% 73.0422 424.558 29 91.1064 660.521 23|| 73.4349 429.135 29% 91.4991 666.228 73.8276 433.737 29% 91.8918 671.959 23% 74.2203 438.364 29% 92.2845 677.714 23% 74.6130 443.015 29% 92.6772 683.494 23% 75.0057 447.690 29^| 93.0699 689.299 24 75.3984 452.390 29% 93.4626 695.128 24% 75.7911 457.115 29% 93.8553 700.982 24% 76.1838 461.864 30 94.2480 706.860 24% 76.5765 466.638 30% 94.6407 712.763 24% 76.9692 471.436 30% 95.0334 718.690 24% 77.3619 476.259 30% 95.4261 724.642 24% 77.7546 481.107 30% 95.8188 730.618 24% 78.1473 485.979 30% 96.2115 736.619 25 78.5400 490.875 30% 96.6042 742.645 25% 78.9327 495.796 30% 96.9969 748.695 25% 79.3254 500.742 31 97.3896 754.769 25% 79.7181 505.712 31% 97.7823 760.869 25% 80.1108 510.706 31% 98.1750 766.992 25*| 80.5035 515.726 31% 98.5677 773.140 25% 80.8962 520.769 31 y 98.9604 779.313 25% 81.2889 525.838 31^1 99.3531 785.510 26 81.6816 530.930 31% 99.7458 791.732 26% 82.0743 536.048 31% 100.1385 797.979 26% 82.4670 541.190 32 100.5312 804.250 26% 82.8597 546.356 32% 100.9239 810.545 26% 83.2524 551.547 32% 101.3166 816.865 26% 83.6451 556.763 32% 101.7093 ' 823.210 26% 84.0378 562.003 32% 102.1020 829.579 26% 84.4305 567.267 32% 102.4947 835.972 27 84.8232 572.557 32% 102.8874 842.391 TABLE OF CIRCLES. 85 Table — ( Continued). Diam. Circum. Area. Diam. Circum. Area. 32% 103.280 848.833 38% 121.344 1,171.731 33 103.673 855.301 38% 121.737 1,179.327 33 % 104.065 861.792 38% 122.130 1,186.948 33% 104.458 868.309 39 122.522 1,194.593 33% 104.851 874.850 39% 122.915 1,202.263 33 % 105.244 881.415 39% 123.308 1,209.958 33% 105.636 888.005 39% 123.700 1,217.677 33% 106.029 894.620 39% 124.093 1,225.420 33% 106.422 901.259 39% 124.486 1,233.188 34 106.814 907.922 39% 124.879 1,240.981 34% 107.207 914.611 39% 125.271 1,248.798 34lJ 107.600 921.323 40 125.664 1,256.640 34% 107.992 928.061 40% 126.057 1,264.510 34% 108.385 934.822 40l| 126.449 1,272.400 34% 108.778 941.609 40% 126.842 1,280.310 34% 109.171 948.420 40% 127.235 1,288.250 34% 109.563 955.255 40% 127.627 1,296.220 35 109.956 962.115 40% 128.020 1,304.210 35% 110.349 969.000 40% 128.413 1,312.220 35% 110.741 975.909 41 128.806 1,320.260 35^1 111.134 982.842 41% 129.198 1,328.320 35% 111.527 989.800 41% 129.591 1,336.410 35% 111.919 996.783 41% 129.984 1,344.520 35% 112.312 1,003.790 41% 130.376 1,352.660 112.705 1,010.822 41% 130.769 1,360.820 36 113.098 1,017.878 41% 131.162 1,369.000 36% 113.490 1,024.960 41% 131.554 1,377.210 36% 113.883 1,032.065 42 131.947 1,385.450 36% 114.276 1,039.195 42% 132.340 1,393.700 36% 114.668 1,046.349 42% 132.733 1,401.990 36% 115.061 1,053.528 423% 133.125 1,410.300 36% 115.454 1,060.732 42% 133.518 1,418.630 36% 115.846 1,067.960 42% 133.911 1,426.990 37 116.239 1,075.213 42% 134.303 1,435.370 37% 116.632 1,082.490 42% 134.696 1,443.770 37% 117.025 1,089.792 43 135.089 1,452.200 37% 117.417 1,097.118 43% 135.481 1,460.660 37% 117.810 1,104.469 43% 135.874 1,469.140 37 % 118.203 1,111.844 43% 136.267 1,477.640 37% 118.595 1,119.244 43% 136.660 1,486.170 37% 118.988 1,126.669 43% 137.052 1,494.730 38 119.381 1,134.118 43% 137.445 1,503.300 38% 119.773 1,141.591 43% 137.838 1,511.910 38% 120.166 1,149.089 44 138.230 1,520.530 38% 120.559 1,156.612 44% 138623 1,529.190 38% 120.952 1,164.159 44% 139.016 1,537.860 86 USEFUL TABLES. Table — ( Continued). Diam. Circum. Area. Diam. Circum. Area. 8 139.408 1,546.56 50% 157.473 1,973.33 139.801 1,555.29 50% 157.865 1,983.18 140.194 1,564.04 50% 158.258 1,993.06 44 % 44% 140.587 1,572.81 50% 158.651 2,002.97 140.979 1,581.61 50% 159.043 2,012.89 45 141.372 1,590.43 50% 159.436 2,022.85 45% 141.765 1,599.28 50% 159.829 2,032.82 45% 142.157 1,608.16 51 160.222 2,042.83 45% 142.550 1,617.05 51% 160.614 2,052.85 4^1 45% 142.943 143.335 1,625.97 1,634.92 if 51% 51% 161.007 161.400 2,062.90 2,072.98 143.728 1,643.89 161.792 2,083.08 144.121 1,652.89 162.185 2,093.20 46 144 514 1,661.91 51% 162.578 2,103.35 46% 144.906 1,670.95 51% 162.970 2,113.52 4fil| 145.299 1,680.02 52 163.363 2,123.72 46% 145.692 1,689.11 52% 52% 163.756 2,133.94 46% 146.084 1,698.23 164.149 2,144.19 46% 146.477 1,707.37 52% 164.541 2,154.46 46k 146.870 1,716.54 52% 164.934 2,164.76 46% 147.262 1,725.73 52% 165.327 2,175.08 47 147.655 1,734.95 52% 165.719 2,185.42 47 % 148.048 1,744.19 52% 166.112 2,195.79 SB 148.441 1,753.45 53 166.505 2,206.19 148.833 1,762.74 53% 166.897 2,216.61 47% 149.226 1,772.06 II 167.290 2,227.05 47k 149.619 1,781.40 167.683 2,237.52 47 ^ 150.011 1,790.76 53% 168.076 2,248.01 47% 150.404 1,800.15 53^ 168.468 2,258.53 48 150.797 1,809.56 53^1 53j| 168.861 2,269.07 48% 151.189 1,819.00 169.254 2,279.64 48% 151.582 1,828.46 54 169.646 2,290.23 48^ 151.975 1,837.95 54% 170.039 2,300.84 152.368 1,847.46 54% 170.432 2,311.48 48% 152.760 1,856.99 54% 170.824 2,322.15 48% 153.153 1,866.55 54% 171.217 2,332.83 48% 153.546 1,876.14 54% 171.610 2,343.55 49 153.938 1,885.75 54% 172.003 2,354.29 49% 154.331 1,895.38 54% 172.395 2,365.05 .SB 154.724 1,905.04 55 172.788 2,375.83 155.116 1,914.72 55% 173.181 2,386.65 SB St 155.509 1,924.43 55% 173.573 2,397.48 155.902 1,934.16 55% 173.966 2,408.34 156.295 1,943.91 55% 174.359 2,419.23 156.687 1,953.69 55% 174.751 2,430.14 50 157.080 1,963.50 55% 175.144 2,441.07 TABLE OF CIRCLES. 87 Table— ( Continued). Diam. Circum. Area. Diam. Circum. Area. 55% 175.537 2,452.03 61% 61% 193.601 2,982.67 56 175.930 2,463.01 193.994 2,994.78 176.322 2,474.02 61% 194.386 3,006.92 5634 176.715 2,485.05 62 194.779 3,019.08 56% 177.108 2,496.11 62% 195.172 3,031.26 56% 177.500 2,507.19 62% 195.565 3,043.47 56% 177.893 2,518.30 62% 195.957 3,055.71 sak 178.286 2,529.43 62% 196.350 3,067.97 56% 178.678 2,540.58 62% 196.743 3,080.25 57 179.071 2,551.76 62% 197.135 3,092.56 57% 179.464 2,562.97 62% 197.528 3,104.89 57% 179.857 2,574.20 63 197.921 3,117.25 57$ 180.249 2,585.45 63^ 198.313 3,129.64 57k 180.642 2,596.73 198.706 3,142.04 57% 181.035 2,608.03 63^1 199.099 3,154.47 57% 181 427 2,619.36 63% 199.492 3,166.93 57JI 181.820 2,630.71 63% 199.884 3,179.41 58 182.213 2,642.09 63% 200.277 3,191.91 58% 182.605 2,653.49 63% 200.670 3,204.44 58% 182.998 2,664.91 64 201.062 3,217.00 58|| 183.391 2,676.36 64% 201.455 3,229.58 183.784 2,687.84 64% 201.848 3,242.18 58% 184.176 2,699.33 64% 202.240 3,254.81 58% 184.569 2,710.86 64% 202.633 3,267.46 58% 184.962 2,722.41 64% 203.026 3,280.14 59 185.354 2,733.98 64% 203.419 3,292.84 59% 185.747 2,745.57 64% 203.811 3,305.56 59% 186.140 2,757.20 65 204.204 3,318.31 59% 186.532 2,768.84 65% 204.597 3,331.09 59% 186.925 2,780.51 65% 204.989 3,343.89 59% 187.318 2,792.21 65% 205.382 3,356.71 59% 187.711 2,803.93 65% 205.775 3,369.56 59% 188.103 2,815.67 65% 206.167 3,382.44 60 188.496 2,827.44 65% 206.560 • 3,395.33 60% 188.889 2,839.23 65% 206.953 3,408.26 60% 189.281 2,851.05 66 207.346 3,421.20 60% 189.674 2,862.89 66% 207.738 3,434.17 60% 190.067 2,874.76 66% 208.131 3,447.17 60% 190.459 2,886.65 66% 208.524 3,460.19 60% 190.852 2,898.57 66% 208.916 3,473.24 60% 191.245 2,910.51 66% 209.309 3,486.30 61 191.638 2,922.47 66% 209.702 3,499.40 61% 192.030 2,934.46 66% 210.094 3,512.52 61% 192.423 2,946.48 67 210.487 3,525.66 61% 192.816 2,958.52 67% 210.880 3,538.83 61% 193.208 2,970.58 67% 211.273 3,552.02 88 USEFUL TABLES. Table — ( Continued). Diam. Circum. Area. Diam. Circum. Area. 67% 211.665 3,565.24 73% 229.729 4,199.74 67% 212.058 3,578.48 73% 230.122 4,214.11 67% 212.451 3,591.74 73% 230.515 4,228.51 67% 212.843 3,605.04 73% 230.908 4,242.93 67% 213.236 3,618.35 73% 231.300 4,257.37 68 213.629 3,631.69 73% 231.693 4,271.84 68% 214.021 3,645.05 73% 232.086 4,286.33 68% 214.414 3,658.44 74 232.478 4,300.85 68% 214.807 3,671.86 74% 232.871 4,315.39 68% 215.200 3,685.29 74% 233.264 4,329.96 68|| 215.592 3,698.76 74% 233.656 4,344.55 Ws 215.985 3,712.24 74% 234.049 4,359.17 216.378 3,725.75 74% 234.442 4,373.81 69 216.770 3,739.29 74% 234.835 4,388.47 69Ys 217.163 3,752.85 74% 235.227 4,403.16 69% 217.556 3,766.43 75 235.620 4,417.87 69% 217.948 3,780.04 75% 236.013 4,432.61 69% 218.341 3,793.68 75% 236.405 4,447.38 69% 218.734 3,807.34 75% 236.798 4,462.16 69% 219.127 3,821.02 75% 237.191 4,476.98 69% 219.519 3,834.73 75% 237.583 4,491.81 70 219.912 3,848.46 75% 237.976 4,506.67 70% 220.305 3,862.22 75% 238.369 4,521.56 70^1 220.697 3,876.00 76 238.762 4,536.47 70% 221.090 3,889.80 76% 239.154 4,551.41 70% 221.483 3,903.63 76% 239.547 4,566.36 70% 221.875 3,917.49 76% 239.940 4,581.35 70% 222.268 3,931.37 76% 240.332 4,596.36 70% 222.661 3,945.27 76% 240.725 4,611.39 71 223.054 3,959.20 76% 241.118 4,626.45 71% 223.446 3,973.15 76% 241.510 4,641.53 71% 223.839 3,987.13 77 241.903 4,656.64 71% 224.232 4,001.13 77% 242.296 4,671.77 71% 224.624 4,015.16 77% 242.689 4,686.92 71% 225.017 4,029.21 77% 243.081 4,702.10 71% 225.410 4,043.29 77% 77% 243.474 4,717.31 71% 225.802 4,057.39 243.867 4,732.54 72 226.195 4,071.51 77% 77% 244.259 4,747.79 72% 226.588 4,085.66 244.652 4,763.07 72% 226.981 4,099.84 78 245.045 4,778.37 72% 227.373 4,114.04 78% 245.437 4,793.70 72% 227.766 4,128.26 78% 245.830 4,809.05 72% 228.159 4,142.51 78% 246.223 4,824.43 72% 228.551 4,156.78 78% 246.616 4,839.83 72% 228.944 4,171.08 78% 247.008 4,855.26 73 229.337 4,185.40 78% 247.401 4,870.71 TABLE OF CIRCLES. 89 Table— ( Continued). Diam. Circum. Area. Diam. Circum. Area. 78k 247.794 4,886.18 84V 265.858 5,624.56 79 248.186 4,901.68 84V 266.251 5,641.18 79k 248.579 4,917.21 84k 266.643 5,657.84 79V 248.972 4,932.75 85 267.036 5,674.51 79k 249.364 4,948.33 85k 267.429 5,691.22 79V 249.757 4,963.92 85V 267.821 5,707.94 79% 250.150 4,979.55 85k 268.214 5,724.69 79% 250.543 4,995.19 85 V 268.607 5,741.47 79j| 250.935 5,010.86 85k 268.999 5,758.27 80 251.328 5,026.56 85V 269.392 5,775.10 80k 251.721 5,042.28 85k 269.785 5,791.94 80V 252.113 5,058.03 86 270.178 5,808.82 80k 252.506 5,073.79 86k 270.570 5,825.72 80V 252.899 5,089.59 86V 270.963 ',842.64 80 % 253.291 5,105.41 86k 271.356 5,859.59 lo^l 253.684 5,121.25 86k 271.748 5,876.56 254.077 5,137.12 86k 272.141 5,893.55 81 254.470 5,153.01 86V 272.534 5,910.58 81^ 254.862 5,168.93 86j| 272.926 5,927.62 255.255 5,184.87 87 273.319 5,944.69 81k 255.648 5,200.83 87k 273.712 5,961.79 8 iv 256.040 5,216.82 87k 274.105 5,978.91 81k 256.433 5,232.84 87k 274.497 5.996.05 81V 256.826 5,248.88 87V 274.890 6,013.22 81k 257.218 5,264.94 87^| 275.283 6,030.41 82 257.611 5,281.03 87V 275.675 6,047.63 82k 258.004 5,297.14 87j| 276.068 6,064.87 82V 258.397 5,313.28 88 276.461 6,082.14 82k 82V 258.789 5,329.44 88k 276.853 6,099.43 259.182 5,345.63 88V 277.246 6,116.74 82k 259.575 5,361.84 88k 277.629 6,134.08 82V 259.967 5,378.08 88V 278.032 6,151.45 82% 260.360 5,394.34 88^ 278.424 6,168.84 83 250.753 5,410.62 88V 278.817 6,186.25 83k 261.145 5,426.93 88k 279.210 6,203.69 83V 261.538 5,443.26 89 279.602 6,221.15 83k 261.931 5,459.62 89k 279.995 6,238.64 83V 262.324 5,476.01 89V 280.388 6,256.15 83 f 8 262.716 5,492.41 89k 280.780 6,273.69 263.109 5,508.84 89k 281.173 6,291.25 83k 263.502 5,525.30 89k 281.566 6,308.84 84 263.894 5,541.78 89V 281.959 6,326.45 84V 264.287 5,558.29 89k 282.351 6,344.08 264.680 5,574.82 90 282.744 6,361.74 84/ls 265.072 5,591.37 90k 283.137 6,379.42 84k 265.465 5,607.95 90k 283.529 6,397.13 90 USEFUL TABLES. T a b le — ( Continued ) . Diam. Circum, Area. Diam. Circum. Area. 90% 283.922 6,414.86 95% 299.237 7,125.59 90% 284.315 6,432.62 95% 299.630 7,144.31 90% 284.707 6,450.40 95% 300.023 7,163.04 90 % 285.100 6,468.21 95% 300.415 7,181.81 285.493 6,486.04 95% 300.808 7,200.60 91 285.886 6,503.90 95% 301.201 7,219.41 91% 286.278 ” 6,521.78 96 301.594 7,238.25 91% 286.671 6,539.68 96% 301.986 7,257.11 91ft 287.064 6,557.61 96% 302.379 7,275.99 287.456 6,575.56 96% 302.772 7,294.91 91% 287.849 6,593.54 96% 303.164 7,313.84 91% 288.242 6,611.55 96% 303.557 7,332.80 91% 288.634 6,629.57 96% 303.950 7,351.79 92 289.027 6,647.63 96% 304.342 7,370.79 92% 289.420 6,665.70 97 304.735 7,389.83 92% 289.813 6,683.80 97% 305.128 7,408.89 92% 290.205 6,701.93 97% 305.521 7,427.97 92% 290.598 6,720.08 97% 305.913 7,447.08 92% 290.991 6,738.25 97% 306.306 7,466.21 92% 291.383 6,756.45 97% 306.699 7,485.37 92% 291.776 6,774.68 97% 307.091 7,504.55 93 292.169 6,792.92 97% 307.484 7,523.75 93% 292.562 6,811.20 98 307.877 7,542.98 93% 292.954 6,829.49 98% 308.270 7,562.24 93% 293.347 6,847.82 98% 308.662 7,581.52 293.740 6,866.16 98|| 309.055 7,600.82 93% 294.132 6,884.53 309.448 7,620.15 93% 294.525 6,902.93 98% 309.840 7,639.50 93% 294.918 6,921.35 98% 310.233 7,658.88 94 295.310 6,939.79 98% 310.626 7,678.28 94% 295.703 6,958.26 99 311,018 7,697.71 94% 296.096 6,976.76 99% 311.411 7,717.16 94% 296.488 6,995.28 99% 311.804 7,736.63 94% 296.881 7,013.82 99% 312.196 7,756.13 94% 297.274 7,032.39 99% 312.589 7,775.66 94% 297.667 7,050.98 99% 312.982 7,795.21 94% 298.059 7,069.59 99% 313.375 7,814.78 95 298.452 7,088.24 99% 313.767 7,834.38 95% 298.845 7,106.90 100 314.160 7,854.00 The preceding table may be used to determine the diameter when the circumference or area is known. Thus, the diameter of a circle having an area of 7,200 sq. in. is, approximately, 95$ in. DECIMAL EQUIVALENTS. 91 DECIMAL EQUIVALENTS OF PARTS OF ONE INCH. 1-64 .015625 17-64 .265625 1-32 .031250 9-32 .281250 3-64 .046875 19-64 .296875 1-16 .062500 5-16 .312500 5-04 .078125 21-64 .328125 3-32 .093750 11-32 .343750 7-64 .109375 23-64 .359375 1-8 .125000 3-8 .375000 9-64 .140625 25-64 .390625 5-32 .156250 13-32 .406250 11-64 .171875 27-64 .421875 3-16 .187500 7-16 .437500 13-64 .203125 29-64 .453125 7-32 .218750 15-32 .468750 15-64 .234375 31-64 .484375 1-4 .250000 1-2 .500000 33-64 .515625 49-64 .765625 17-32 .531250 25-32 .781250 35-64 .546875 51-64 .796875 9-16 .562500 13-16 .812500 37-64 .578125 53-64 .828125 19-32 .593750 27-32 .843750 39-64 .609375 55-64 .859375 5-8 .625000 7-8 .875000 41-64 .640625 57-64 .890625 21-32 .656250 29-32 .906250 43-64 .671875 59-64 .921875 11-16 .687500 15-16 .937500 45-64 .703125 61-64 .953125 23-32 .718750 31-32 .968750 47-64 .734375 63-64 .984375 3-4 .750000 1 1 DECIMALS OF A FOOT FOR EACH 1-32 OF AN INCH. Inch. 0" 1" 2" 3" 4" 5" 0 0 .0833 .1667 .2500 .3333 .4167 .0026 .0859 .1693 .2526 .3359 .4193 A .0052 .0885 .1719 .2552 .3385 .4219 V? .0078 .0911 .1745 .2578 .3411 .4245 V & .0104 .0937 .1771 .2604 .3437 .4271 & .0130 .0964 .1797 .2630 .3464 .4297 A .0156 .0990 .1823 .2656 .3490 .4323 & .0182 .1016 .1849 .2682 .3516 .4349 M .0208 .1612 .1875 .2708 .3542 .4375 .0234 .1068 .1901 .2734 .3568 .4401 .0260 .1094 .1927 .2760 .3594 .4427 II .0286 .1120 .1953 .2786 .3620 .4453 h .0312 .1146 .1979 .2812 .3646 .4479 .0339 .1172 .2005 .2839 .3672 .4505 TV .0365 .1198 .2031 .2865 .3698 .4531 d .0391 .1224 .2057 .2891 .3724 .4557 % .0417 .1250 .2083 .2917 .3750 .4583 & .0443 .1276 .2109 .2943 .3776 .4609 TJS .0469 .1302 .2135 .2969 .3802 .4635 If .0495 .1328 .2161 .2995 .3828 .4661 % .0521 .1354 .2188 .3021 .3854 .4688 u .0547 .1380 .2214 .3047 .3880 .4714 H .0573 .1406 .2240 .3073 .3906 .4740 it .0599 .1432 .2266 .3099 .3932 .4766 92 USEFUL TABLES. Table— ( Continued). Inch. 0" 1" 2" 3" 4" 5". % .0625 .1458 .2292 .3125 .3958 .4792 if .0651 .1484 .2318 .3151 .3984 .4818 H .0677 .1510 .2344 .3177 .4010 .4844 27 .0703 .1536 .2370 .3203 .4036 .4870 /I .0729 .1562 .2396 .3229 .4062 .4896 §7 .0755 .1589 .2422 .3255 .4089 .4922 || .0781 .1615 .2448 .3281 .4115 .4948 §7 .0807 .1641 .2474 .3307 .4141 .4974 DECIMALS OF A FOOT FOR EACH 1-32 OF AN INCH, Inch. 6" 7" 8" 9" 10" 11" 0 .5000 .5833 .6667 .7500 .8333 .9167 A .5026 .5859 .6693 .7526 .8359 .9193 .5052 .5885 .6719 .7552 .8385 .9219 ' ft .5078 .5911 .6745 .7578 .8411 .9245 .5104 .5937 .6771 .7604 .8437 .9271 .5130 .5964 .6797 .7630 .8464 .9297 ’re .5156 .5990 .6823 .7656 .8490 .9323 .5182 .6016 .6849 .7682 .8516 .9349 u .5208 .6042 .6875 .7708 .8542 .9375 & .5234 .6068 .6901 .7734 .8568 .9401 V re .5260 .6094 .6927 .7760 .8594 .9427 1 1 32 .5286 .6120 .6953 .7786 .8620 .9453 % .5312 .6146 .6979 .7812 .8646 .9479 M re .5339 .6172 .7005 .7839 .8672 .9505 .5365 .6198 .7031 .7865 .8698 .9531 1 6 32 .5391 .6224 .7057 .7891 .8724 .9557 .5417 .6250 .7083 .7917 .8750 .9583 §7 .5443 .6276 .7109 .7943 .8776 .9609 re .5469 .6302 .7135 .7969 .8802 .9635 .5495 .6328 .7161 .7995 .8828 .9661 .5521 .6354 .7188 .8021 .8854 .9688 14 .5547 .6380 .7214 .8047 .8880 .9714 re .5573 .6406 .7240 .8073 .8906 .9740 ?§ .5599 .6432 .7266 .8099 .8932 .9766 % .5625 .6458 .7292 .8125 .8958 .9792 §1 .5651 .6484 .7318 .8151 .8984 .9818 .5677 .6510 .7344 .8177 .9010 .9844 S .5703 .6536 .7370 .8203 .9036 .9870 .5729 .6562 .7396 .8229 .9062 .9896 .5755 .6589 .7422 .8255 .9089 .9922 .5781 .6615 .7448 .8281 .9115 .9948 31 52 .5807 .6641 .7474 .8307 .9141 .9974 FORMULAS. 93 FORMULAS. = {+[-:(V / X/h-):-]| = The term formula , as used in mathematics and in techni- cal books, may be defined as a rule in which symbols are used instead of words; in fact, a formula may be regarded as a shorthand method of expressing a rule. Most people having no knowledge of algebra regard for- mulas with distrust; they think that a person must be a good algebraic scholar in order to be able to use formulas. This idea, however, is erroneous. As a rule, no knowledge of any branch of mathematics except arithmetic is required to enable one to use a formula. Any formula can be expressed in words, and when so expressed it becomes a rule. Formulas are much more convenient than rules; they show at a glance all the operations that are to be performed; they do not require to be read three or four times, as is the case with most rules, to enable one to understand their meaning; they take up much less space, both in the' printed book and in one’s note book, than rules; in short, whenever a rule can be expressed as a formula, the formula is to be preferred. In the following pages we purpose to show the reader how to use such formulas as he is likely to encounter in “pocket- books,” or other works of like nature. The signs used in formulas are the ordinary signs indica- tive of operations and the signs of aggregation. All these signs are used in arithmetic, but, to refresh the reader’s memory, we will explain their nature and uses before pro- ceeding further. The signs indicative of operations are six in number, viz.: + » > X, -T-, | , ~V • The sign ( + ) indicates addition, and is called plus; when placed between two quantities, it indicates that the two quantities are to be added. Thus, in the expression 25 + 17, the sign ( + ) shows that 17 is to be added to 25. The' sign (— ) indicates subtraction, and is called minus; when placed between two quantities, it indicates that the 94 FORMULAS. quantity on the right is to be subtracted from that on the left. Thus, in the expression 25 — 17, the sign (— ) shows that 17 is to be subtracted from 25. The sign (x) indicates multiplication, and is read times , or multiplied by; when placed between two quantities, it indi- cates that the quantity on the left is to be multiplied by that on the right. Thus, in the expression 25 X 17, the sign (X) shows that 25 is to be multiplied by 17. The sign (-^) indicates division, and is read divided by; when placed between two quantities, it indicates that the quantity on the left is to be divided by that on the right. Thus, in the expression 25 -f- 17, the sign (^) shows that 25 is to be divided by 17. Division is also indicated by placing a straight line between the two quantities. Thus, 25 1 17, 25/17, and ff all indicate that 25 is to be divided by 17. When both quantities are placed on the same horizontal line, the straight line indicates that the quantity on the left is to be divided by that on the right. When one quantity is below the other, the straight line between indicates that the quantity above the line is to be divided by the one below it. The sign (/) indicates that some root of the quantity to the right is to be taken; it is called the radical sign. To indicate what root is to be taken, a small figure, called the index , is placed within the sign, this being always omitted when the square root is to be indicated. Thus, y / 25 indicates that the square root of 25 is to be taken; ^ 25 indicates that the cube root of 25 is to be taken, etc. Note.— As the term “quantity” is a very convenient one to use, we will define it. In mathematics the word quantity is applied to anything that it is desired to subject to the ordinary operations of addition, subtraction, multiplication, etc., when we do not wish to be more specific and state exactly what the thing is. Thus, we can say “two or more numbers,” or “ two or more quantities.” The word quantity is more general in its meaning than the word number. The signs of aggregation are four in number, viz.: , (), [],and | respectively called the vinculum , the paren- thesis, the brackets , and the brace; they are used when it is desired to indicate that all the quantities included by them FORMULAS. 95 are to be subjected to the same operation. Thus, if we desire to indicate that the sum of 5 and 8 is to be multiplied by 7, and we do not wish to actually add 5 and 8 before indicating the multiplication, we may employ any one of the four signs of aggregation as here shown : 5 + 8 X 7, ( 5 + 8 ) X 7, [5 + 8] X 7, \ 5 + 8 1 X 7. The vinculum is placed above the quantities which are to be treated as one quantity and subjected to the same operations. While any one of the four signs may be used as shown above, custom has restricted their use somewhat. The vincu- lum is rarely used except in connection with the radical sign. Thus, instead of writing 1^(5 + 8), [5 + 8], or 1^5 + 8| for the cube root of 5 plus 8, all of wh ich w ould be correct, the vinculum is nearly always used, ^5 + 8. In cases where but one sign of aggregation is needed (except, of course, when a root is to be indicated), the parenthesis is always used. Hence, (5 + 8) X 7 would be the usual way of expressing the product of 5 plus 8 and 7. If two signs of aggregation are needed, the brackets and parenthesis are used, so as to avoid having a parenthesis within a parenthesis, the brackets being placed outside. For example, [(20 — 5) -i- 3] X 9 means that the difference between 20 and 5 is to be divided by 3, and this result multiplied by 9. If three signs of aggregation are required, the brace, brackets, and parenthesis are used, the brace being placed outside, the brackets next, and the parenthesis inside. For example, \ [(20 — 5) -4-3] X9 — 21 £ -4-8 means that the quo- tient obtained by dividing the difference between 20 and 5 by 3 is to be multiplied by 9; and that 21 is to be subtracted from the product thus obtained, and the result divided by 8. Should it be necessary to use all four signs of aggrega- tion, the brace would be put outside, the brackets next, the parenthesis next, and the vinculum inside. For example, J [(20 — 5 3) X 9 — 21] -4- 8 £ X 12. The reason for using the brace in this last instance will be explained, as it is not generally understood. When several quantities are connected by the various signs indicating addition, subtraction, multiplication, and division, the operation indicated by the sign of multiplication 96 FORMULAS. must always be performed first. Thus, 2 + 3X4 equals 14, 3 being multiplied by 4 before adding to 2. Similarly, 10 -f- 2 X 5 equals 1, since 2X5 equals 10, and 10 -i- 10 equals 1. Hence, in the above case, if the brace were omitted, the Tesult would be + whereas, by inserting the brace, the result is 36. Following the sign of multiplication comes the sign of division in its order of importance. For example, 5 — 9 3 equals 2, 9 being divided by 3 before subtracting from 5. The signs of addition and subtraction are of equal value; that is, if several quantities are connected by plus and minus signs, the indicated operations may be performed in the order in which the quantities are placed. There is one other sign used, which is neither a sign of aggregation nor a sign indicative of an operation to be per- formed; it is (=), and is called the sign of equality; it means that all on one side of it is exactly equal to all on the other side. For example, 2 = 2, 5 — 3 = 2, 5 X (14 — 9) =25. Having described the signs used in formulas, the formulas themselves will now be explained. First consider the well- known rule for finding the horsepower of a steam engine, which may be stated as follows : Divide the continued product of the mean effective pressure in pounds per square inch , the length of the stroke in feet , the area of the piston in square inches , and the number of strokes per minute by 83,000 ; the result will be the horsepower. This is a very simple rule, and very little, if anything, will be saved by expressing it as a formula, so far as clearness is concerned. The formula, however, will occupy a great deal less space, as we shall show. An examination of the rule will show that four quantities ' (viz., the mean effective pressure, the length of the stroke, the area of the piston, and the number of strokes) are multi- plied together, and the result is divided by 33,000. Hence, the rule might be expressed as follows : mean effective pressure v „ stroke (in pounds per square inch) (in feet) area of piston number of strokes ^ (in square inches) (per minute) * ’ FORMULAS. 07 This expression could be shortened by representing each quantity by a single letter, thus: representing horsepower by the letter the mean effective pressure in pounds per square inch by “P,” the length of the stroke in feet by “X,” the area of the piston in square inches by “A,” the number of strokes per minute by “iV,” and substituting these letters for the quantities that they represent, the above expression would reduce to PXLXAXN 33,000 a much simpler and shorter expression. This last expression is called a formula. The formula just given shows, as we stated in the begin- ning, that a formula is really a shorthand method of express^ ing a rule. It is customary, however, to omit the sign of multiplication between two or more quantities when they are to be multiplied together, or between a number and a letter representing a quantity, it being always understood that when two letters are adjacent with no sign between them, the quantities represented by these letters are to be multiplied. Bearing this fact in mind, the formula just given can be further simplified to PLAN 11 ~ 33,000 * The sign of multiplication, evidently, cannot be omitted between two or more numbers, as it w r ould then be impossible to distinguish the numbers. A near approach to this, how- ever, may be attained by placing a dot between the numbers that are to be multiplied together, and this is frequently done in works on mathematics whep it is desired to economize space. . In such ca ses it is usual to put the dot higher than the position occupi ^d by the decimal point. Thus, 2-3 means the same as 2 X 3; o42-749-l,006 indicates that the numbers 542, 749, and 1,006 are to be multiplied together. It is also customary to omit the sign of multiplication in expressions similar to the followi ng: a X l /b + c, 3X (b + c), (b + c) X a, etc., writing them a\/ b + c, 3(b + c), (b + c)a, etc. The sign is not omitted when several quantities are included by a vinculum, and it is desired to indicate that the quantities 93 FORMULAS. so included are t o be multi plied b y ano ther quantity. For example, 3XHc, 6 + cX«, 4 /b + c X a, etc., are always written as here printed. Before proceeding further, we will explain one other device that is used by formula makers, and which is apt to puzzle one who encounters it for the first time. It is the use of what mathematicians call primes and subs., and what printers call superior and inferior characters. As a rule, formula makers designate quantities by the initial letters of the names of the quantities. For example, they represent volume by v, pressure by p, height by h, etc. This practice is to be commended, as the letter itself serves in many cases to identify the quantity that it represents. Some authors carry the practice a little further and represent all quantities of the same nature by the same letter throughout the book, always having the same letter represent the same thing. Now, this practice necessitates the use of the primes and subs, above mentioned when two quantities have the same name, but represent different things. Thus, consider the word pressure as applied to steam at different stages between the boiler and the condenser. First, there is absolute pres- sure, which is equal to the gauge pressure in pounds per square inch plus the pressure indicated by the barometer reading (usually assumed in practice to be 14.7 pounds per square inch, when a barometer is not at hand) . If this be represented by p, how shall we represent the gauge pressure? Since the absolute pressure is always greater than the gauge pressure, suppose we decide to represent it by a capital letter, and the gauge pressure by a small (lower-case) letter. Doing so, P represents absolute pressure, and p gauge pres- sure. Further, there is usually a “drop” in pressure between the boiler and the engine, so that the initial pres- sure, or pressure at the beginning of the stroke, is less than the pressure at the boiler. How shall we represent the initial pressure? We may do this in one of three ways, and still retain the letter p or P to represent the word pressure: First, by the use of the prime mark; thus, p ' or P' (read p prime and p major prime) may be considered to represent the initial gauge pressure or the initial absolute pressure. FORMULAS. 99 Second, by the use of sub. figures; thus, p\ or Pi (readp sub. one andp major sub. one). Third, by the use of sub. letters: thus, pi or Pi (read p sub. i and P major sub. i). Likewise, p' 1 (read p second), p 2 , or p r might be used to represent the gauge pressure at release, etc. Sub. letters have the advantage of still further identifying the quantity represented; in many instances, however, it is not convenient to use them, in which case primes and subs, are used instead. The prime notation may be continued as follows: p'",p iy ,p y , etc.; it is inadvisable to use superior figures, for example, p 1 , p 2 , p 3 , p a , etc., as they are liable to be mistaken for exponents. The main thing to be remembered by the reader is that when a formula is given in which the same letters occur several times, all like letters having the same primes or subs, represent the same quantities, while those that differ in any respect represent different quantities. Thus, in the formula t __ W i Si ti + W 2 So h + w 3 s 3 1 3 ~ w x Si + Wo s 2 -} i v 3 s 3 * Wi, w 2 , and w 3 represent the weights of three different bodies; Si,s 2 , and s 3 their specific heats; and ti,i*>, and t 3 their tem- peratures; while t represents the final temperature, after the bodies have been mixed together. It is very easy to apply the above formula when the values of the quantities represented by the different letters are known. All that is required is to substitute the numeri- cal values of the letters, and then perform the indicated operations. Thus, suppose that the values of Si, and t\ are, respectively, 2 pounds, .0951, and 80°; of w 2 , s 2 , and for 7.8 pounds, 1, and 80°, and of w s , s 3 , and t 3 , 3£ pounds, .1138, and 780°; then, the final temperature t is, substituting these values for their respective letters in the formula, 2 X .0951 X 80 + 7.8 X 1 X 80 + 3i X .1138 X 780 2 X .0951 + 7.8 X 1 + 3i X .1138 15.216 + 624 + 288.483 __ 927.699 .1902 + 7.8 + .36985 ~ 8.36005 — 11U ' y? * In substituting the numerical values, the signs of multi- plication are, of course, written in their proper places; all the multiplications are performed before adding, according to the rule previously given. 100 FORMULAS. The reader should now be able to apply any formula involving only algebraic expressions that he may meet with, not requiring the use of logarithms for their solution. We will, however, call his attention to one or two other facts which he may have forgotten. 160 Expressions similar to — sometimes occur, the heavy line "25 indicating that 160 is to be divided by the quotient obtained by dividing 660 by 25. If both lines were light it would be impossible to tell whether 160 was to be divided by or 160 whether — - was to be divided by 25. If this latter result 660 160 were desired, the expression would be written In every case the heavy line indicates that all above it is to be divided by all below it. 160 In an expression like the following, — , the heavy line tn 66U is not necessary, since it is impossible to mistake the opera- 660 tion that is required to be performed. But, since 7 + -^r 175 -f 660 . , ... , 175 + 660 „ „ , 660 , = — ~ — , if we substitute — ■== — for 7 + — , the heavy 25 ZD ZD line becomes necessary in order to make the resulting expres- sion clear. Thus, 160 160 160 660 “ 175 + 660 “ 835' 7 + 25 25 25 Fractional exponents are sometimes used instead of the radical sign. That is, instead of ind' mating the square, cube, fourth root, etc. of some quantity, as 37 by j/ 37, ^ 37, W, etc. these roots are indicated by 37* 37* 37*, etc. Should the numerator of the fractional exponent be some quantity other than 1, this quantity, whatever it may be, indicates that the quantity affected by the exponent is to be raised to the power indicated by the numerator; the denominator is FORMULAS. 101 always the index of the root. Hence^jnstead of expressing the cube root of the square of 37 as $ 37 2 , it may be expressed 37^, the denominator being the index of the root; in other words, ^ S7 2 = 37 Likewise, (1 + a 2 b ) 3 may also be written (1 + a 2 6)*, a much simpler expression. We will now give several examples showing how to apply some of the more difficult formulas that the reader may encounter. The area of any segment of a circle that is less than (or equal to) a semicircle is expressed by the formula. A 7T r 2 E ~360 in which A = area of segment; 7T = 3.1416; r = radius; E = angle obtained by drawing lines from the center to the extremities of arc of segment; c = chord of segment; h — height of segment. Example.— W hat is the area of a segment whose chord is 10 in. long, angle subtended by chord is 83.46°, radius is 7.5 in., and height of segment is 1.91 in. ? Solution.— A pplying the formula just given, A = r r 2 E c I - 2 (r “ 7i) = 3.1416X7.52X83.46 360 2 v 360 = 40.968 — 27.95 = 13.018 sq. in., nearly. j (7.5-1.91) The area of any triangle may be found by means of the following formula, in which A = the area, and a, 5, and c represent the lengths of the sides: A 4 a H 2 ^)’ Example.— W hat is the area of a triangle whose sides are 21 ft., 46 ft., and 50 ft. long ? Solution.— In order to apply the formula, suppose we let a represent the side that is 21 ft. long; b, the side that is 50 ft. long; and c, the side that is 46 ft. long. Then, substituting in the formula, 102 FORMULAS. -i VH^-)' - -IV* ,441- 441 + 2,500- 100 - 2,11 6 ^ 2 = 25 1/441 - 8.252 = 25 l/441 — 68.0625 = 25 V 372.9375 = 25 X 19.312 = 482.8 sq. ft., nearly. The above operations have been extended much further than was necessary; this was done in order to show the reader every step of the process. The Rankine-Gordon formula for determining the least load in pounds that will cause a long column to break is P = S A 1+? Z in which P = load (pressure) in lb.; S = ultimate strength (in lb. per sq. in.) of material composing column; A = area of cross-section of column in sq. in.; q = a factor (multiplier) whose value depends on the shape of the ends of the column and on the material composing the column; l = length of the column in in.; G = least radius of gyration of cross-section of column. The values of S, q, and G 2 are all given in printed tables on pages 151, 153, and 156. Example. — What is the least load that will break a hollow steel column whose outside diameter is 14 in., inside diam- eter 11 in., length 20 ft., and whose ends are flat? Solution.— For steel, S = 150,000, and q = — for flat- 25,000 ended steel columns; A, the area of the cross-section, = .7854(di 2 - d 2 2 ), cli and do being the outside and inside diam- eters, respectively; l = 20 X 12 = 240 in.; and G 2 = - r j~ Substituting +ese values in the formula, SA 150,000 X .7854(14 2 -112) _ P = — - 1 + q G* i + 1 : ^ x 25,000 A 14 2 + ll 2 150.000 X 58.905 1 + .1163 8,835.750 1.1163 16 7,915,211 lb. INVOLUTION AND EVOLUTION. 103 INVOLUTION AND EVOLUTION. By means of the following table the square, cube, square root, cube root, and reciprocal of any number may be obtained correct always to five significant figures, and in the majority of cases correct to six significant figures. In any number, the figures beginning with the first digit * at the left and ending with the last digit at the right, are called the significant figures of the number. Thus, the num- ber 405,800 has the four significant figures 4, 0, 5, 8; and the number .000090067 has the five significant figures 9, 0, 0, 6, and 7. The part of a number consisting of its significant figures is called the significant part of the number. Thus, in the number 28,070, the significant part is 2807; in the number .00812, the significant part is 812; and in the number 170.3, the significant part is 1703. In speaking of the significant figures or of the significant part of a number, the figures are considered, in their proper order, from the first digit at the left to the last digit at the right, but no attention is paid to the position of the decimal point. Hence, all numbers that differ only in the position of the decimal point have the same significant part. For example, .002103, 21.03, 21,030, and 210,300 have the same significant figures 2, 1, 0, and 3, and the same significant part 2103. The integral part of a number is the part to the left of the decimal point. It will be more convenient to explain first how to use the table for finding square and cube rootg. SQUARE ROOT. First point off the given number into periods of two figures each, beginning with the decimal point and proceeding to the left and right. The following numbers are thus pointed off: 12703, 1'27'03; 12.703, 12.70'30; 220000, 22W00; .000442, .00W42. A cipher is not a digit. 104 SQUARE ROOT. Haying pointed off the number, move the decimal point bo that it will fall between the first and second periods of the significant part of the number. In the above numbers, the decimal point will be placed thus: 1.2703, 12.703, 22, 4.42. If the number has but three (or less) significant figures, find the significant part of the number in the column headed n; th e squ are root will be found in the column headed j/n or |/ 10 n, according to whether the part to the left of the decimal point contains one figure or two figures. Thus, j/ 4.42 = 2.1024, and i/ 22 = 4/ 10 X 2.20 = 4.6904. The decimal point is located in all cases by reference to the original number after pointing off into periods. There will be as many figures in the root preceding the decimal point as there are periods preceding the decimal point in the given number; if the number is entirely decimal , the root is entirely decimal , and there will be as many ciphers following the decimal point in the root as there are cipher periods following the decimal point in the given number. Applying this rule, )/ 220000 = 469.04 and ]/. 000442 = .021024. The operation when the given number has more than three significant figures is best explained by an example. Example.— (a) 4/ 3J.416 = ? (6) 1/ 2342.9 = ? Solution.— ( a) Since the first period contains but one figure, there is no need of moving the decimal point. Look in the column headed n 2 and find two consecutive numbers, one a little greater and the other a little less than the given number ; in the present case, 3.1684 = 1.78 2 and 3.1329 = 1.77 2 . The first three figures of the root are therefore 177. Find the difference between the two numbers between which the given number falls, and the difference between the smaller number and the given number ; divide the second difference by the first difference, carrying the quotient to three decimal places and increasing the second figure by 1 if the third is 6 or a greater digit. The two figures of the quotient thus determined will be the fourth and fifth figures of the root. In the present example, dropping decimal points in the remainders, 3.1684 — 3.1329 = 355, the first difference; INVOLUTION AND EVOLUTION. 105 3.1416 — 3 1329 = 87, the second difference; 87 h- 355 = .245+, or .25. Hence, V 3.1416 = 1.7725. (6) i/*2342.9 = ? Pointing off into periods we get 23'42.90; moving the decimal point we get 23.4290; the first three figures of the root are 484; the first difference is 23.5225 — 23.4256 = 969; the second difference is 23.4 290 — 23.4256 = 34; 34 -f- 969 = .035+, or .04. Hence, = 48.404. CUBE ROOT. The cube root of a number is found in the same manner as the square root, except the given number is pointed off into periods of three figures each. The following numbers would be pointed off thus: 3141.6, 3'141.6; 67296428, 67'296'428; 601426.314, 60P426.314; .0000000217, .000'000'021'700. Having pointed off, move the decimal point so that it will fall between the first and second periods of the significant part of the number, as in square root. In the above num- bers the decimal point will be placed thus: 3.1416, 67.296428, 601.426314, and 21.7. If the given number has but three (or less) significant figures, find the significant part of the number in the column headed w; the cube root will be found in the column headed f/ n , i/l0 n, or ]/l00 n, according to whether one, two, or three figures precede the decimal point after it has been moved. Thus, the cube root of 21.7 will be found oppo- site 2.17, in column headed ^ 10 n, while the cube root of 2.17 would be found in the column headed n, and the cube root of 217 in the column headed ^ 100 n, all on the same line. If the given number contains more than three sig- nificant figures, proceed exactly as described for square root except that the column headed n 3 is used. Example.— (a) ^0000062417 = ? (5) 1 / 50932676 = ? Solution.— (a) Pointing off into periods, we get 000'006'241'700; moving the decimal point, we get 6.2417. The number falls between 6.22950 = 1.843 and 6.33163 = 1.85 s ; the first difference = 10213; the second difference is 106 SQUARES AND CUBES. 6.24170 - 6.22950 = 1220; 1220 -4- 10213 = .119+, or .12, the fourth and fifth figures of the root. The decimal point is located by the rule previously given; hence, 1^.0000(162417 = .018412. (6) ^ 50932676 = ? As the number contains more than six significant figures, reduce it to six significant figures by replacing all after the sixth figure with ciphers, increasing the sixth figure by 1 when the seventh is 5 or a greater digit. In other words, the first five figures of ^ 50932700 and of i^ 50932676 are the same. Pointing off into periods, we get 50'932'700; moving the decimal point, we get 50.9327, which falls between 50.6530 = 3.70 3 and 51.0648 = 3.71 3 ; the first difference is 4118; the second difference is 2797; 2797 -4- 4118 = .679+, or .68. The integral part of the root evidently con- tains three figures; hence, ^ 50932676 = 370.68, correct to five figures. SQUARES AND CUBES. If the given number contains but three (or less) signifi- cant figures, the square or cube is found in the column headed w 2 or n 3 , opposite the given number in the column headed n. If the given number contains more than three significant figures, proceed in a manner similar to that described for extracting roots. To square a number, place the decimal point between the first and second significant figures and find in the column headed \/ n or ]/ 10 n two consecu- tive numbers, one of which shall be a little greater and the other a little less than the given number. The remainder of the work is exactly as heretofore described. To locate the decimal point, employ the principle that the square of any number contains either twice as many figures as the num- ber squared or twice as many less one. If the column headed ■j/ 10 n is used, the square will contain twice as many figures, while if the column headed \/ n is used, the square will contain twice as many figures as the number squared, less one. If the number contains an integral part, the principle is applied to the integral part only; if the number is wholly decimal, there' will be twice as many ciphers following the INVOLUTION AND EVOLUTION. 107 decimal in the square or twice as many plus one as in the number squared, depending on whether |/l0n or j/n column is used. For example, 273.4‘2 2 will contain five figures in the integral part; 4516.2 2 will contain eight figures in the integral part, all after the fifth being denoted by ciphers; .0029453 2 will have five ciphers following the decimal point; .052436 2 will have two ciphers following the decimal point. Example.— (a) 273.42 2 = ? (6) .052436 2 = ? Solution.— ( a) Placing the decimal point between the first and second significant figures, the result is 2.7342; this number occurs between 2.73313 = V 7.47 and 2.73496 = j/ 7.48 in the column headed y n. The first difference is 2.73496 — 2.73313 = 183; the second difference is 2.73420 — 2.73313 = 107; and 107 -r- 183 = .584+, or .58. Hence, 273.42 2 = 74,758, correct to five significant figures. (6) Shifting the decimal point to between the first and second significant figures, we get the number 5.2436, which falls between 5.23450 = i/ 27.4 and 5.24404 = i/ 27.5. The first difference is 954; the second difference is 910; 910 -j- 954 = .953-+ or .95. Hence, .052436 2 = .0027495, to five significant figures. A number is cubed in exactly the same manner, using the column headed ^ n, ^ 10 n, or ^ 100 ?i, according to whether the first period of the significant part of the number contains one, two, or three figures, respectively. If the number con- tains an integral part, the number of figures in the integral part of the cube will be three times as many as in the given number if column headed $ 100 n is used; it will be three times as many less 1 if the column headed $ 10 n is used; and it will be three times as many less 2 if the column headed n is used. If the given number is wholly decimal the cube will have either three times, three times plus one, or three times plus two, as many ciphers following the decimal as there are ciphers following the decimal point in the given number. Example.— (a) 129.6843=? (6) .76442 2 = ?. (c)j .032425* = ? Solution.— ( a) Placing the decimal point between the 108 RECIPROCALS. first and second signifi cant figures, the number 1.29684 is found between 1.29664 = fziS and 1.29862 = f '~2A9. The first difference is 198; the second difference is 20; and 20 -s- 198 = .101+, or .10. Hence, the first five significant figures are 21810; the number of figures in the integral part of the cube is 3 X 3 — 2 = 7; and 129.684 3 = 2,181,000, correct to five sig- nificant figures. (ft) 7.64420 occurs between 7.64032 = 1^446 and 7.64603 == ^ 447. The first difference is 571; the second difference is 388; and 388 -5- 571 = .679+, or .68. Hence, the first five signifi- cant figures are 44668; the number of ciphers following the decimal point is 3 X 0 = 0; and .76442 3 = .44668, correct to five significant figures. (c) 3.2425 falls between 3.24278 = ^ 34.1 and 3.23961 = 1^34.0. The first difference is 317; the second difference is 289; 289 -4- 317 = .911+, or .91. Hence, the first five significant figures are 34091; the number of ciphers following the decimal point is 3 X 1 + 1 = 4; and .032425 3 = .000034091, correct to five significant figures. RECIPROCALS. The reciprocal of a number is 1 divided by the number. By using reciprocals, division is changed into multiplication, since a^-ft = ^ = a X The table gives the reciprocals of all numbers expressed with three significant figures to six significant figures. By proceeding in a manner similar to that just described for powers and roots, tne reciprocal of any number correct to five significant ngures may be obtained. The decimal point in the result may be located as follows* If the given number has an integral part, the number of ciphers following the decimal point in the reciprocal will be one less than the number of figures in the integral part of the given number; and if the given number is entirely decimal, the number of figures in the integral part of the reciprocal will be one greater than the number of ciphers following the decimal point in the given number. For example, the recip- rocal of 3370 = .000296736 and of .00348 = 287.356. INVOLUTION AND EVOLUTION. 109 When the number whose reciprocal is desired contains more than three significant figures, express the number to six significant figures (adding ciphers, if necessary, to make six figures) and find between what two numbers in the column headed ^ the significant figures of the given number falls; then proceed exactly as previously described to deter- mine the fourth and fifth figures. Example.— (a) The reciprocal of 379.426 =? ( b ) - nnn ~L 00 - = ? Solution. — (a) .379426 falls between .378788 = — ^ and .380228 = -i- . The first difference is 380228 — 378788 = 1440; Abo the second difference is 380228 — 379426 = 802; 802 1440 — .557, or .56. Hence, the first five significant figures are 26356, and the reciprocal of 379.426 is .0026356, to five sig- nificant figures. ( 6 ) .469200 falls between .469484 = —^3 an( * -467290 = The first difference i& 2194; the second difference is 284; 284 - 7 - 2194 = .129+, or .13. Hence, — " 1 - ■ = 2131.3, correct to .0004692 five significant figures. 110 POWERS, ROOTS, AND RECIPROCALS. yllOn 1 n n n 2 n 3 Vn A/lU n ^100 1.01 1.0201 1 03030 1.00499 3.17805 1.00332 2.16159 4.65701 .990099 1.02 1.0404 1.06121 1.00995 3.19374 1.00662 2.16870 4.67233 .980392 1.03 1.0609 1.09273 1.01489 3.20936 1.00990 2.17577 4.68755 .970874 1.04 1.0816 1.12486 1.01980 3.22490 1.01316 2.18278 4.70267 .961539 1.05 1.1025 1.15763 1.02470 3.24037 1.01640 2.18976 4.71769 .952381 1.06 1.1236 1.19102 1.02956 3.25576 1.01961 2.19669 4.73262 .943396 1.07 1 1449 1.22504 1.03441 3.27109 1.02281 2.20358 4.74746 .934579 1.08 1.1664 1.25971 1.03923 3.28634 1.02599 2.21042 4.76220 .925926 1.09 1.1881 1.29503 1.04403 3.30151 1.02914 2.21722 4.77686 .917431 1.10 1.2100 1.33100 1.04881 3.31662 1.03228 2.22398 4.79142 .909091 1.11 1.2321 1 36763 1.05357 3.33167 1.03540 2.23070 4.80590 .900901 1.12 1.2544 1.40493 1.05830 3.34664 1.03850 2,23738 4.82028 .892857 1.13 1.2769 1.44290 1.06301 3.36155 1.04158 2.24402 4.83459 .88*956 1.14 1.2996 1.48154 1.06771 3.37639 1.04464 2.25062 4.84881 .877193 1.15 1.3225 1.52088 1.07238 3.39116 1.04769 2.25718 4.86294 .869565 1.16 1.3456 1.56090 1.07703 3.40588 1.05072 2.26370 4.87700 .862069 1.17 1.3689 1.60161 1.08167 3.42053 1.05373 2.27019 4.89097 .854701 1.18 1 .3924 1.64303 1.08628 3.43511 1.05672 2.27664 4.90487 .847458 1.19 1.4161 1.68516 1.09087 3.44964 1.05970 2.28305 4.91868 .840336 1.20 1.4400 1.72800 1.09545 3.46410 1.06266 2.28943 4.93242 .833335 1.21 1.4641 1.77156 1.10000 3.47851 1.06560 2.29577 4.94609 .826446 1.22 1.4884 1.81585 1.10454 3.49285 1.06853 2.30208 4.95968 .819672 1.23 1.5129 1.86087 1.10905 3.50714 1.07144 2.30835 4.97319 .813008 1.24 1.5376 1.90662 1.11355 3.52136 1.07434 2.31459 4.98663 .806452 1.25 1.5625 1.95313 1.11803 3.53553 a l . 07722 2.32080 5.00000 .800000 jl .26 1.5876 2.00038 1.12250 3.54965 1.08008 2.32697 5.01330 .793651 1.27 1.6129 2.04838 1.12694 3.56371 1.08293 2.33310 5.02653 .787402 1.28 1.6384 2.09715 1.13137 3.57771 1.08577 2.33921 5.03968 ,.781250 1.29 1.6641 2.14669 1.13578 3.59166 1.08859 2.34529 5.05277 .775194 1.30 1.6900 2.19700 1.14018 3.60555 1.09139 2.35134 5.06580 .769231 1.31 1.7161 2.24809 1.14455 3.61939 1.09418 2.35735 5.07875 .763359 1.32 1.7424 2.29997 1.14891 3.63318 1.09696 2.36333 5.09164 .757576 1.33 1.7689 2.35264 1.15326 3.64692 1.09972 2.36928 5.10447 .751880 1.34 1.7956 2.40610 1.15758 3.66060 1.10247 2.37521 5.11723 .746269 1.35 1.8225 2.46038 1.16190 3.67423 1.10521 2.38110 5.12993 .740741 1.36 1.8496 .2.51546 1.16619 3.68782 1.10793 2.38696 5.14256 .735294 1.37 1.8769 2.57135 1.17047 3.70135 1.11064 2.39280 5.15514 .729927 1.38 1.9044 2.62807 1.17473 3.71484 1.11334 2.39861 5.16765 .724638 1.39 1.9321 2.68562 1.17898 3.72827 1.11602 2.40439 5.18010 .719425 1.40 1.9600 2.74400 1.18322 3.74166 1.11869 2.41014 5.19249 .714286 1.41 1.9881 2.80322 1.18743 3.75500 1.12135 2.41587 5.20483 .709220 1.42 2.0164 2.86329 1.19164 3.76829 1.12399 2.42156 5.21710 .704225 1.43 2.0449 2.92421 1.19583 3.78153 1.12662 2.42724 5.22932 .699301 1.44 2.0736 2.98598 1.20000 3 79473 1.12924 2.43288 5.24148 .694444 1.45 2.1025 3.04863 1.20416 3.80789 1.13185 2.43850 5.25359 .689655 1.46 2.1316 3.11214 1.20830 3.82099 1.13445 2.44409 5.26564 .684932 1.47 2.1609 3.17652 1.21244 3.83406 1.13703 2.44966 5.27763 .680272 1.48 2.1904 3.24179 1.21655 3.84708 1.13960 2.45520 5.28957 .675676 1.49 2.2201 3.30795 1.22066 3.86005 1.14216 2.46072 5.30146 .671141 1.50 2.2500 3.37500 1.22474 3.87298 1.14471 2.46621 5.31329 .666667 POWERS, ROOTS, AND RECIPROCALS. Ill n 7*3 7 l 3 yin VlO 11 fin ^10 11 1 n ^100?* 1.51 2.2801 3.44295 1.22882 3.88587 1.14725 2.47168 5.32507 .662252 1.52 2.3104 3.51181 1.23288 3.89872 1.14978 2.47713 5.33680 .657895 1.53 2.3409 3.58158 1.23693 3.91152 1.15230 2.48255 5.34848 .653595 1.54 2.3716 3. §5226 1.24097 3.92428 1.15480 2.48794 5.36011 .649351 1.55 2.4025 3.72388 1.24499 3.93700 1.15729 2.49332 5.37169 .645161 1.56 2.4336 3.79642 1.24900 3.94968 1.15978 2.49866 5.38321 .641026 1.57 2.4649 3.86989 1.25300 3.96232 1.16225 2.50399 5.39469 .636943 1.58 2.4964 3.94431 1 .25698 3.97492 1.16471 2.50930 5.40612 .632911 1.59 2.5281 4.01968 1.26095 3.98748 1.16717 2.51458 5.41750 .628931 1.60 2.5600 4.09600 1.26491 4.00000 1.16961 2.51984 5.42884 .625000 1.61 2.5921 4.17328 1.26886 4.01248 1.17204 2.52508 5.44012 .621118 1.62 2.6244 4.25153,1 1.27279 4.02492 1.17446 2.53030 5.45136 .617284 1.63 2.6569 4.33075 1.27671 4.03733 1.17687 2.53549 5.46256 .613497 1.64 2.6896 4.41094 1.28062 4.04969 1.17927 2.54067 5.47370 .609756 1.65 2.7225 4.49213 1.28452 4.06202 1.18167 2.54582 5.48481 .606061 1.66 2.7556 4.57430 1.28841 4.07431 1.18405 2.55095 5.49586 .602410 1.67 2.7889 4.65746 1.29228 4.08656 1.18642 2.55607 5.50688 .598802 1.68 2.8224 4.74163 1.29615 4.09878 1.18878 2.56116 5.51785 .595238 1.69 2.8561 4.82681 1.30000 4.11096 1.19114 2.56623 5.52877 .591716 1.70 2.8900 4.91300 1.30384 4.12311 1.19348 2.57128 5.53966 .588235 1.71 2.9241 5.00021 1.30767 4.13521 1.19582 2.57631 5.55050 .584795 1.72 2.9584 5.08845 1.31149 4.14729 1.19815 2.58133 5.56130 .581395 1.73 2.9929 5.17772 1.31529 4.15933 1.20046 2.58632 5.57205 .578035 1.74 3.0276 5.26802 1.31909 4.17133 1.20277 2.59129 5.58277 .574713 1.75 3.0625 5.35938 1.32288 4.18330 1.20507 2.59625 5.59344 .571429 1.76 3.0976 5.45178 1 .32665 4.19524 1.20736 2.60118 5.60408 .568182 1.77 3.1329 5.54523 1.33041 4.20714 1.20964 2.60610 5.61467 .564972 1.78 3.1684’ 5.63975 1.33417 4.21900 1.21192 2.61100 5.62523 .561798 1.79 3.2041" 5.73534 1.33791 4.23084 1.21418 2.61588 5.63574 .558659 1.80 3.2400 5.83200 1.34164 4.24264 1.21644 2.62074 5.64622 .555556 1.81 3.2761 5.92974 1.34536 4.25441 1.21869 2.62558 5.65665 .552486 1.82 3.3124 6.02857 1.34907 4.26615 1.22093 2.63041 5.66705 .549451 1.83 3.3489 6.12849 1.35277 4.27785 1.22316 2.63522 5.67741 .546448 1.84 3.3856 6.22950 1.35647 4.28952 1.22539 2.64001 5.68773 .543478 1.85 3.4225 6.33163 1.36015 4.30116 1.22760 2.64479 5.69802 .540541 1.86 3.4596 6.43486 1.36382 4.31277 1.22981 2.64954 5.70827 .537634 1.87 3.4969 6.53920 1.36748 4.32435 1 23201 2.65428 5.71848 .534759 1.88 3.5344 6.64467 1.37113 4.33590 1.23420 2.65900 5.72865 .531915 1.89 3.5721 6.75127 1.37477 4.34741 1.23639 2.66371 5.73879 .529101 1.90 3.6100 6.85900 1.37840 4.35890 1.23856 2.66840 5.74890 .526316 1.91 3.6481 6.96787 1.38203 4.37035 1.24073 2.67307 5.75897 .523560 1.92 3.6864 7.07789 1.38564 4.38178 1.24289 2.67773 5.76900 .520833 1.93 3.7249 7.18906 1.38924 4.39318 1.24505 2.68237 5.77900 .518135 1.94 3.7636 7.30138 1.39284 4.40454 1.24719 2.68700 5.78896 .515464 1.95 3.8025 7.41488 1.39642 4.41588 1.24933 2.69161 5.79889 .512821 1.96 3.8416 7.52954 1.40000 4.42719 1.25146 2.69620 5.80879 .510204 1.97 3.8809 7.64537 1.40357 4.43847 1.25359 2.70078 5.81865 .507614 1.98 3.9204 7.76239 1.40712 4.44972 1.25571 2.70534 5.82848 .505051 1.99 3.9601 7.88060 1.41067 4.46094 1.25782 2.70989 5.83827 .502513 2.00 4.0000 8.00000 1.41421 4.47214 1.25992 2.71442 5.84804 .500000 112 POWERS, ROOTS, AND RECIPROCALS. n 7l 2 n 3 Vw A/10 ~n 'iln ylio n ^T00n 1 n 2.01 4.0401 8.12060 1.41774 4.48330 1.26202 2.71893 5.85777 .497512 2.02 4.0804 8.24241 1.42127 4.49444 1.26411 ! 2.72343 5.86746 .495050 2.03 4.1209 8.36543 1.42478 4.50555 1.26619 2.72792 5.87713 .492611 2.04 4.1616 8.48966 1.42829 4.51664 1.26827 2.73239 5.88677 .490196 2.05 4.2025 8.61513 1.43178 4.52769 1.27033 2.73685 5.89637 .487805 2.06 4.2436 8.74182 1.43527 4.53872 1.27240 2.74129 5.90594 .485437 2.07 4.2849 8.86974 1.43875 4.54973 1.27445 2.74572 5.91548 .483092 2.08 4.3264 8.99891 1.44222 4.56070 1.27650 2.75014 5.92499 .480769 2.09 4.3681 9.12933 1.44568 4.57165 1.27854 2.75454 5.93447 .478469 2.10 4.4100 9.26100 1.44914 4.58258 1.28058 2.75893 5.94392 .476191 2.11 4.4521 9.39393 1.45258 4.59347 1 .28261 2.76330 5.95334 .473934 2.12 4.4944 9.52813 1.45602 4.60435 1.28463 2.76766 5.96273 .471698 2.13 4.5369 9.66360 1.45945 4.61519 1.28665 2.77200 5.97209 .469484 2.14 4.5796 9.80034 1.46287 4.62601 1.28866 2.77633 5.98142 .467290 2.15 4.6225 9.93838 1.46629 4.63681 1.29066 2.78065 5.99073 .465116 2.16 4.6656 10.0777 1.46969 4.64758 1.29266 2.78495 6.00000 .462963 2.17 4.7089 10.2183 1.47309 4.65833 1.29465 2.78924 6.00925 .460830 2.18 4.7524 10.3602 1.47648 4.66905 1.29664 2.79352 6.01846 .458716 2.19 4.7961 10.5035 1.47986 4.67974 1.29862 2.79779 6.02765 .456621 2.20 4.8400 10.6480 1.48324 4.69042 1.30059 2.80204 6.03681 .454546 2.21 4.8841 10.7939 1.48661 4.70106 1 . 30256 . 2.80628 6.04594 .452489 2.22 4.9284 10.9410 1.48997 4.71169 1.30452 2.81051 6.05505 .450451 2.23 4.9729 11.0896 1.49332 4.72229 1.30648 2.81472 6.06413 .448431 2.24 5.0176 11.2394 1.49666 4.73286 1.30843 2.81892 6.07318 .446429 2.25 5.0625 11.3906 1.50000 4.74342 1.31037 2.82311 6.08220 .444444 2.26 , 5.1076 11.5432 1.50333 4.75395 1.31231 2.82728 6.09120 .442478 2.27 5.1529 11.6971 1.50665 4.76445 1.31424 2.83145 6.10017 .440529 2.28 5.1984 11.8524 1.50997 4.77493 1.31617 2.83560 6.10911 .438597 2.29 5.2441 12.0090 1.51327 4.78539 1.31809 2.83974 6.11803 .436681 2.30 5.2900 12.1670 1.51658 4.79583 1.32001 2.84387 6.12693 .434783 2.31 5.3361 12.3264 1.51987 4.80625 1.32192 2.84798 6.13579 .432900 2.32 5.3824 12.4872 1.52315 4.81664 1.32382 2.85209 6.14463 .431035 2.33 5.4289 12.6493 1.52643 4.82701 1.32572 2.85618 6.15345 .429185 2.34 5.4756 12.8129 1.52971 4.83735 1.32761 2.86026 6.16224 .427350 2.35 5.5225 12.9779 1.53297 4.84768 1.32950 2.86433 6.17101 .425532 2.36 5.5696 13.1443 1.53623 4.85798 1.33139 2.86838 6.17975 .423729 2.37 5.6169 13.3121 1.53948 4.86826 1.33326 2.87243 6.18846 .421941 2.38 5.6644 13.4813 1.54272 4.87852 1.33514 2.87646 6.19715 .420168 2.39 5.7121 13.6519 1.54596 4.88876 1.33700 2.88049 6.20582 .418410 2.40 5.7600 13.8240 1.54919 4.89898 1.33887 2.88450 6.21447 .416667 2.41 5.8081 13.9975 1.55242 4.90918 1.34072 2.88850 6.22308 .414938 2.42 5.8564 14.1725 1.55563 4.91935 1.34257 2.89249 6.23168 .413223 2.43 5.9049 14.3489 1.55885 4.92950 1.34442 2.89647 6.24025 .411523 2.44 5.9536 14.5268 1.56205 4.93964 1.34626 2.90044 6.24880 .409836 2.45 6.0025 14.7061 1.56525 4.94975 1.34810 2.90439 6.25732 .408163 2.46 6.0516 14.8869 1.56844 4.95984 1.34993 2.90834 6.26583 .406504 2.47 6.1009 15.0692 1.57162 4.96991 1.35176 2.91227 6.27431 .404858 2.48 6.1504 15.2530 1.57480 4.97996 1.35358 2.91620 6.28276 .403226 2.49 6.2001 15.4382 1.57797 4.98999 1.35540 2.92011 6.29119 .401606 2.50 6.2500 15.6250 1.58114 5.00000 1.35721 2.92402 6.29961 .400000 POWERS, ROOTS, AND RECIPROCALS. 112a n 7 l 2 n 3 >fn VlO n 3j— \?l <10 n II 1 i n 2.51 6.3001 15.8133 1.58430 5.00999 1 .35902 2.92791 6.30799 .398406 2.52 6.3504 16.0030 1.58745 5.01996 1.36082 1 2.93179 6.31636 .396825 2.53 6.4009 16.1943 1.59060 5.02991 1.36262 ! 2.93567 6.32470 .395257 2 54 6.4516 16.3871 1.59374 5.03984 1.36441 2.93953 6.33303 .393701 2.55 6.5025 16.5814 1 .59687 5.04975 1.36620 2.94338 6.34133 .392157 2.56 6.5536 16.7772 1.60000 5.05964 1.36798 2.94721 6.34960 .390625 2.57 6.6049 16.9746 1.60312 5.06952 1.36976 2.95106 6.35786 .389105 2.58 6.6564 17.1735 1 .60624 5.07937 1.37153 2.95488 6.36610 .387597 2.59 6.7081 17.3740 1.60935 5.08920 1.37330 2.95869 6.37431 .386100 2.60 6.7600 17.5760 1.61245 5.09902 1.37507 2.96250 6.38250 .384615 2.61 6.8121 17.7796 1.61555 5.10882 1.37683 2.96629 6.39068 .383142 2.62 6.8644 17.9847 1.61864 5.11859 1.37859 2.97007 6.39883 .381679 2.63 6.9169 18.1914 1.62173 5.12835 1.38034 2.97385 6.40696 .380228 2.64 6.9696 18.3997 1 .62481 5.13809 1.38208 2.97761 6.41507 .378788 2.65 7.0225 18.6096 1.62788 5.14782 1.38383 2.98137 6.42316 .377359 2.66 7.0756 18.8211 1.63095 5.15752 1.38557 2.98511 6.43123 | .375940 2.67 7.1289 19.0342 1.63401 5.16720 1.38730 2.98885 6.43928 .374532 2.68 7.1824 19.2488 1.63707 5.17687 1.38903 2.99257 6.44731 .373134 2.69 7.2361 19.4651 1.64012 5.18652 1.39076 2.99629 6.45531 .371747 2.70 7.2900 19.6830 1.64317 5.19615 1.39248 3.00000 6.46330 .370370 2.71 7.3441 19.9025 1.64621 5.20577 1.39419 3.00370 6.47127 .369004 2.72 7.3984 20.1236 1.64924 5.21536 1 .39591 3.00739 6.47922 .367647 2.73 7.4529 20.3464 1.65227 5.22494 1.39761 3.01107 6.48715 .366300 2.74 7.5076 20.5708 1.65529 5.23450 1.39932 3.01474 6.49507 .364964 2.75 7.5625 20.7969 1.65831 5.24404 1.40102 3.01841 6.50296 .363636 2.76 7.6176 21.0246 1.66132 5.25357 1.40272 3.02206 6.51083 .362319 2.77 7.6729 21.2539 1.66433 5.26308 1.40441 3.02571 6.51868 .361011 2.78 7.7284 21.4850 1.66733 5.27257 1.40610 3.02934 6.52652 .359712 2.79 7.7841 21.7176 1.67033 5.28205 1.40778 3.03297 6.53434 .358423 2.80 7.8400 21.9520 1.67332 5.29150 1.40946 3.03659 6.54213 .357142 2.81 7.8961 22.1880 1.67631 5.30094 1.41114 3.04020 6.54991 .355872 2.82 7.9524 22.4258 1.67929 5.31037 1.41281 3.04380 6.55767 .354610 2.83 8.0089 22.6652 1.68226 5.31977 1.41448 3.04740 6.56541 .353357 2.84 8.0656 22.9063 1.68523 5.32917 1,41614 3.05098 6.57314 .352113 2.85 8.1225 23.1491 1.68819 5.33854 1.41780 3.05456 6.58084 .350877 2.86 8.1796 23.3937 1.69115 5.34790 1.41946 3.05813 6.58853 .349650 2.87 8.2369 23.6399 1.69411 5.35724 1.42111 3.06169 6.59620 .348432 2.88 8.2944 23.8879 1.69706 5.36656 1.42276 3.06524 6.60385 .347222 2.89 8.3521 24.1376 1.70000 5.37587 1.42440 3.06878 6.61149 .346021 2.90 8.4100 24.3890 1.70294 5.38516 1.42604 3.07232 6.61911 .344828 2.91 8.4681 ’24.6422 1.70587 5.39444 1.42768 3.07585 6.62671 .343643 2.92 8.5264 24.8971 1.70880 5.40370 1.42931 3.07936 6.63429 .342466 2.93 8.5849 25.1538 1.71172 5.41295 1.43094 3.08287 6.64185 .341297 2.94 8.6436 25.4122 1.71464 5.42218 1.43257 3.08638 6.64940 .340136 2.95 8.7025 25.6724 1.71756 5.43139 1.43419 3.08987 6.65693 .338983 ; 2.96 8.7616 25.9343 1.72047 5.44059 1.43581 3.09336 6.66444 .337838 2.97 8.8209 26.1981 1.72337 5.44977 1.43743 3.09684 6.67194 .336700 2.98 8.8804 26.4636; 1.72627 5.45894 1.43904 3.10031 6.67942 .335571 2.99 8.9401 26.7309 i 1.72916 5.46809 1.44065 3.10378 6.68688 .334448 3.00 9.0000 27.0000 1.73205 5.47723 1.44225 3.10723 6.69433 .333333 1126 POWERS, ROOTS, AND RECIPROCALS. n W 2 n 3 ■V5 VlO n I yfton 1 s © © 1 n 3.01 9.0601 27.2709 1.73494 5.48635 1.44385 3.11068 6.70176 .332226 3.02 9.1204 27.5436 1.73781 ' 5.49545 1.44545 3.11412 6.70917 .331126 3.03 9.1809 27.8181 1.74069 5.50454 1.44704 3.11755 6.71657 .330033 3.04 9.2416 28.0945 1.74356 5.51362 1.44863 3.12098 6.72395 .328947 3.05 9.3025 28.3726 1.74642 5.52268 1 45022 3.12440 6.73132 .327869 3.06 9.3636 28.6526 1.74929 5.53173 1.45180 3.12781 6.73866 .326797 3.07 9.4249 28.9344 1.75214 5.54076 1.45338 3.13121 6.74600 .325733 3.08 9.4864 29.2181 1.75499 5.54977 1.45496 3.13461 6.75331 i .324675 3.09 9.5481 29.5036 1.75784 j 5.55878 1.45653 3.13800 6.76061 .323625 3.10 9.6100 29.7910 1.76068 j 5.56776 1.45810 3.14138 6.76790 .322581 3.11 9.6721 30.0802 1.76352 5.57674 1.45967 3.14475 6.77517 .321543 3.12 9.7344 30.3713 1.76635 5.58570 1.46123 3.14812 6.78242 .320513 3.13 9.7969 30.6643 1.76918 5.59464 1.46279 3.15148 6.78966 .319489 3.14 9.8596 30.9591 1.77200 5.60357 1.46434 3.15484 6.79688 .318471 3.15 9.9225 31.2559 1.77482 5.61249 1.46590 3.15818 6.80409 .317460 3.16 9.9856 31.5545 1.77764 5.62139 1.46745 3.16152 6.81128 .316456 3.17 10.0489 31.8550 1.78045 5.63028 1.46899 3.16485 6.81846 .315457 3.18 10.1124 32.1574 1.78326 5.63915 1.47054 3.16817 6.82562 .314465 3.19 10.1761 32.4618 1 .78606 5.64801 1.47208 3.17149 6.83277 .313480 3.20 10.2400 32.7680 1.78885 5.65685 1.47361 3.17480 6.83990 .312500 3.21 10.3041 33.0762 1.79165 5.66569 1.47515 3.17811 6.84702 .311527 3.22 10.3684 33.3862 1.79444 5.67450 1.47668 3.18140 6.85412 .310559 3.23 10.4329 33.6983 1.79722 5.68331 1.47820 3.18469 6.86121 .309598 3.24 10.4976 34.0122 1.80000 5.69210 1.47973 3.18798 6.86829 .308642 3.25 10.5625 34.3281 1.80278 5.70088 1.48125 3.19125 6.87534 .307692 3.26 10.6276 34.6460 1.80555 5.70964 1.48277 3.19452 6.88239 .306749 3.27 10.6929 34.9658 1.80831 5.71839 1.48428 3.19779 6.88942 .305810 3.28 10.7584 35.2876 1.81108 5.72713 1.48579 3.20104 6.89643 .304878 3.29 10.8241 35.6129 1.81384 5.73585 1.48730 3.20429 6.90344 .303951 3.30 10.8900 35.9370 1.81659 5.74456 1.48881 3.20753 6.91042 .303030 3.31 10.9561 36.2647 1.81934 5.75326 1.49031 3.21077 6.91740 .302115 3.32 11.0224 36.5944 1.82209 5.76194 1.49181 3.21400 6.92436 .301205 3.33 11.0889 36.9260 1.82483 5.77062 1.49330 3.21723 6.93130 .300300 3.34 11.1556 37.2597 1.82757 5.77927 1.49480 3.22044 6.93823 .299401 3.35 11.2225 37.5954 1.83030 5.78792 1.49629 3.22365 6.94515 .298508 3.36 11.2896 37.9331 1.83303 5.79655 1.49777 3.22686 6.95205 .297619 3.37 11.3569 38.2728 1.83576 5.80517 1 .49926 3.23005 6.95894 .296736 3.38 11.4244 38.6145 1.83848 5.81378 1.50074 3.23325 6.96582 .295858 3.39 11.4921 38.9582 1.84120 5.82237 1.50222 3.23643 6.97268 .294985 3.40 11.5600 39.3040 1.84391 5.83095 1.50369 3.23961 6.97953 .294118 3.41 11.6281 39.6518 1.84662 5.83952 1.50517 3.24278 6.98637 .293255 3.42 11.6964 40.0017 1.84932 5.84808 1.50664 3.24595 6.99319 .292398 3.43 11.7649 40.3536 1.85203 5.85662 1.50810 3.24911 7.00000 .291545 3.44 11.8336 40.7076 1.85472 5.86515 1.50957 3.25227 7.00680 .290698 3.45 11.9025 41.0636 1.85742 5.87367 1.51103 3.25542 7.01358 .289855 3.46 11.9716 41.4217 1.86011 5.88218 1.51249 3.25856 7.02035 .289017 3.47 12.0409 41.7819 1.86279 5.89067 1.51394 3.26169 7.02711 .288184 3.48 12.1104 42.1442 1.86548 5.89915 1.51540 3.26482 7.03385 .287356 3.49 12.1801 42.5085 1.86815 5.90762 1.51685 3.26795 7.04058 .286533 3.50 12.2500 42.8750 1.87083 5.91608 1.51829 3.27107 7.04730 .285714 POWERS, ROOTS, AND RECIPROCALS. 112c n 71 2 ? l 3 >I7i VlO n «10 n ^100 n 1 n 3.51 12.3201 43.2436 1.87350 5.92453 1.51974 3.27418 7.05400 .284900 3.52 12.3904 43.6142 1.87617 5.93296 1.52118 3.27729 7.06070 .284091 3.53 12.4609 43.9870 1.87883 5.94138 1.52262 3.28039 7.06738 .283286 3.54 12.5316 44.3619 1.88149 5.94979 1.52406 3.28348 7.07404 .282486 3.55 12.6025 44.7389 1.88414 5.95819 1.52549 3.28657 7.08070 .281690 3.56 12.6736 45.1180 1.88680 5.96657 1.52692 3.28965 7.08734 .280899 3.57 12.7449 45.4993 1.88944 5.97495 1.52835 3.29273 7.09397 .280112 3.58 12.8164 45.8827 1.89209 5.98331 1.52978 3.29580 7.10059 .279330 3.59 12.8881 46.2683 1.89473 5.99166 1.53120 3.29887 7.10719 .278552 3.60 12.9600 46.6560 1.89737 6.00000 1.53262 3.30193 7.11379 .277778 3.61 13.0321 47.0459 1.90000 6.00833 1.53404 3.30498 7.12037 .277008 3.62 13.1044 47.4379 1.90263 6.01664 1.53545 3.30803 7.12694 .276243 3.63 13.1769 47.8321 1.90526 6.02495 1.53686 3.31107 7.13349 .275482 3.64 13.2496 48.2285 1.90788 6.03324 1.53827 3.31411 7.14004 .274725 3.65 13.3225 48.6271 1.91050 6.04152 1.53968 3.31714 7.14657 .273973 3.66 13.3956 49.0279 1.91311 6.04979 1.54109 3.32017 7.15309 .273224 3.67 13.4689 49.4309 1.91572 6.05805 1.54249 3.32319 7.15960 .272480 3.68 13.5424 49.8360 1.91833 6.06630 1.54389 3.32621 7.16610 .271739 3.69 13.6161 50.2434 1.92094 6.07454 1.54529 3.32922 7.17258 .271003 3.70 13.6900 50.6530 1.92354 6.08276 1.54668 3.33222 7.17905 .270270 3.71 13.7641 51.0648 1.92614 6.09098 1.54807 3.33522 7.18552 .269542 3.72 13.8384 51.4788 1.92873 6.09918 1.54946 3.33822 7.19197 .268817 3.73 13.9129 51.8951 1.93132 6.10737 1.55085 3.34120 7.19841 .268097 3.74 13.9876 52.3136 1.93391 6.11555 1.55223 3.34419 7.20483 .267380 3.75 14.0625 52.7344 1.93649 6.12372 1.55362 3.34716 7.21125 .266667 3.76 14.1376 53.1574 1.93907 6.13188 1.55500 3.35014 7.21765 .265957 3.77 14.2129 53.5826 1.94165 6.14003 1.55637 3.35310 7 . 22405 ' .265252 3.78 14.2884 54.0102 1.94422 6.14817 1.55775 3.35607 7.23043 .264550 3.79 14.3641 54.4399 1.94679 6.15630 1.55912 3.35902 7.23680 .263852 3.80 14.4400 54.8720 1.94936 6.16441 1.56049 3.36198 7.24316 .263158 3.81 14.5161 55.3063 1.95192 6.17252 1.56186 3.36492 7.24950 .262467 3.82 14.5924 55.7430 1.95448 6.18061 1.56322 3.36786 7.25584 .261780 3.83 14.6689 56.1819 1.95704 6.18870 1.56459 3.37080 7.26217 .261097 3.84 14.7456 56.6231 1.95959 6.19677 1.56595 3.37373 7.26848 .260417 3.85 14.8225 57.0666 1.96214 6.20484 1.56731 3.37666 7.27479 .259740 3.86 14.8996 57.5125 1.96469 6.21289 1.56866 3.37958 7.28108 .259067 3.87 14.9769 57.9606 1.96723 6.22093 1.57001 3.38249 7.28736 .258398 3.88 15.0544 58.4111 1.96977 6.22896 1.57137 3.38540 7.29363 .257732 3.89 15.1321 58.8639 1.97231 6.23699 1.57271 3.38831 7.29989 .257069 3.90 15.2100 59.3190 1.97484 6.24500 1.57406 3.39121 7.30614 .256410 3.91 15.2881 59.7765 1.97737 6.25300 1.57541 3.39411 7.31238 .255755 3.92 15.3664 60.2363 1.97990 6.26099 1.57675 3.39700 7.31861 .255102 3.93 15.4449 60.6985 1.98242 6.26897 1.57809 3.39988 7.32483 .254453 3.94 15.5236 61.1630 1.98494 6.27694 1.57942 3.40277 7.33104 .253807 3.95 15.6025 61.6299 1 .98746 6.28490 1.58076 3.40564 7.33723 .253165 3.96 15.6816 62.0991 1.98997 6.29285 1.58209 3.40851 7.34342 .252525 3.97 15.7609 62.5708 1.99249 6.30079 1.58342 3.41138 7.34960 .251889 3.98 15.8404 63.0448 1 99499 6.30872 1.58475 3.41424 7.35576 .251256 3.99 15.9201 63.5212 1.99750 6.31664 1.58608 3.41710 7.36192 .250627 4.00 16.0000 64.0000 2.00000 6.32456 1.58740 3.41995 7.36806 .250000 112 d POWERS, ROOTS, AND RECIPROCALS. n W2 n 3 A/lO n *!n | 'V'lO n j^lOOn 1 n 4.01 16.0801 64.4812 2.00250 6.33246 1.58872 3.42280 7.37420 .249377 4.02 16.1604 64.9648 2.00499 6.34035 1.59004 3.42564 7.38032 .248756 4.03 16.2409 65.4508 2.00749 6.34823 1.59136 3.42848 7.38644 .248139 4.04 16.3216 65.9393 2.00998 6.35610 1.59267 3.43131 7.39254 .247525 4.05 16.4025 66.4301 2.01246 6.36396 1.59399 3.43414 7.39864 .246914 4.06 16.4836 66.9234 2.01494 6.37181 1.59530 3.43697 7.40472 .246305 4.07 16.5649 67.4191 2.01742 6.37966 1.59661 3.43979 7.41080 .245700 4.08 16.6464 67.9173 2.01990 6.38749 1.59791 3.44260 7.41686 .245098 4.09 16.7281 68.4179 2.02237 6.39531 1.59922 3.44541 7.42291 .244499 4.10 16.8100 68.9210 2.02485 6.40312 1.60052 3.44822 7.42896 .243902 4.11 16.8921 69.4265 2.02731 6.41093 1.60182 3.45102 7.43499 .243309 4.12 16.9744 69.9345 2.02978 6.41872 1.60312 3.45382 7.44102 .242718 4.13 17.0569 70.4450 2.03224 6.42651 1.60441 3.45661 7.44703 .242131 4.14 17.1396 70.9579 2.03470 6.43428 1.60571 3.45939 7.45304 .241546 4.15 17.2225 71.4734 2.03715 6.44205 1.60700 3.46218 7.45904 .240964 4.16 17.3056 71.9913 2.03961 6.44981 1.60829 3.46496 7.46502 .240385 4.17 17.3889 72.5117 2.04206 6.45755 1.60958 3.46773 7.47100 .239808 4.18 17.4724 73.0346 2.04450 6.46529 1.61086 3.47050 7.47697 .239234 4.19 17.5561 73.5601 2.04695 6.47302 1.61215 3.47327 7.48292 .238664 4.20 17.6400 74.0880 2.04939 6.48074 1.61343 3.47603 7.48887 .238095 4.21 17.7241 74.6185 2.05183 6.48845 1.61471 3.47878 7.49481 .237530 4.22 17.8084 75.1514 2.05426 6.49615 1.61599 3.48154 7.50074 .236967 4.23 17.8929 75.6870 2.05670 6.50385 1.61726 3.48428 7.50666 .236407 4.24 17.9776 76.2250 2.05913 6.51153 1.61853 3.48703 7.51257 .235849 4.25 18.0625 76.7656 2.06155 6.51920 1.61981 3.48977 7.51847 .235294 4.26 18.1476 77.3088 2.06398 6.52687 1.62108 3.49250 7.52437 .234742 4.27 18.2329 77.8545 2.06640 6.53452 1.62234 3.49523 7.53025 .234192 4.28 18.3184 78.4028 2.06882 6.54217 1.62361 3.49796 7.53612 .233645 4.29 18.4041 78.9536 2.07123 6.54981 1.62487 3.50068 7.54199 .233100 4.30 18.4900 79.5070 2.07364 6.55744 1.62613 3.50340 7.54784 .232558 4.31 18.5761 80.0630 2.07605 6.56506 1.62739 3.50611 7.55369 .232019 4.32 18.6624 80.6216 2.07846 6.57267 1.62865 3.50882 7.55953 .231482 4.33 18.7489 81.1827 2.08087 6.58027 1.62991 3.51153 7.56535 .230947 4.34 18.8356 81.7465 2.08327 6.58787 1.63116 3.51423 7.57117 .230415 4.35 18.9225 82.3129 2.08567 6.59545 1.63241 3.51692 7.57698 .229885 4.36 19.0096 82.8819 2.08806 6.60303 1.63366 3.51962 7.58279 .229358 4.37 19.0969 83.4535 2.09045 6.61060 1.63491 3.52231 7.58858 .228833 4.38 19.1844 84.0277 2.09284 6.61816 1.63616 3.52499 7.59436 .228311 4.39 19.2721 84.6045 2.09523 6.62571 1.63740 3.52767 7.60014 .227790 4.40 19.3600 85.1840 2.09762 6.63325 1.63864 3.53035 7.60590 .227273 4.41 19.4481 85.7661 2.10000 6.64078 1.63988 3.53302 7.61166 .226757 4.42 19.5364 86.3509 2.10238 6.64831 1.64112 3.53569 7.61741 .226244 4.43 19.6249 86.9383 2.10476 6.65582 1.64236 3.53835 7.62315 .225734 4.44 19.7136 87.5284 2.10713 6.66333 1.64359 3.54101 7.62888 .225225 4.45 19.8025 88.1211 2.10950 6.67083 1.64483 3.54367 7.63461 .224719 4.46 19.8916 88.7165 2.11187 6.67832 1.64606 3.54632 7.64032 .224215 4.47 19.9809 89.3146 2.11424 6.68581 1.64729 3.54897 7.64603 .223714 4.48 20.0704 89.9154 2.11660 6.69328 1.64851 3.55162 7.65172 .223214 4.49 20.1601 90.5188 2.11896 6.70075 1.64974 3.55426 7.65741 .222717 4.50 20.2500 91.1250 2.12132 6.70820 1.65096 3.55689 7.66309 .222222 POWERS, ROOTS, AND RECIPROCALS. U2e 1 n 71 * n 3 $10 n i 'In \To7i $10071 n 4.51 20.3401 91.7339 2.12368 6.71565 1.65219 3.55953 7.66877 .221730 4.52 20.4304 92.3454 2.12603 6.72309 1.65341 3.56215 7.67443 .221239 4.53 20.5209 92.9597 2.12838 6.73053 1.65462 3.56478 7.68009 .220751 4.54 20.6116 93.5767 2.13073 6.73795 1.65584 3.56740 7.68573 .220264 4.55 20.7025 94.1964 2.13307 6.74537 1.65706 3.57002 7.69137 .219780 4.56 20.7936 94.8188 2.13542 6.75278 1.65827 3.57263 7.69700 .219298 4.57 20.8849 95.4440 2.13776 6.76018 1.65948 3.57524 7.70262 .218818 4.58 20.9764 96.0719 2.14009 6.76757 1.66069 3.57785 7.70824 .218341 4.59 21.0681 96.7026 2.14243 6.77495 1.66190 3.58045 7.71384 .217865 4.60 21.1600 87.3360 2.14476 6.78233 1.66310 3.58305 7.71944 .217391 4.61 21.2521 97.9722 2.14709 6.78970 1.66431 3.58564 7.72503 .216920 4.62 21.3444 98.6111 2.14942 6.79706 1.66551 3.58823 7.73061 .216450 4.63 21.4369 99.2528 2.15174 6.80441 1.66671 3.59082 7.73619 .215983 4.64 21.5296 99.8973 2.15407 6.81175 1.66791 3.59340 7.74175 .215517 4.65 21.6225 100.545 2.15639 6.81909 1.66911 3.59598 7.74731 .215054 4.66 21.7156 101.195 2.15870 6.82642 1.67030 3.59856 7.75286 .214592 4.67 21.8089 101.848 2.16102 6.83374 1.67150 3.60113 7.75840 .214133 4.68 21.9024 102.503 2.16333 6.84105 1.67269 3.60370 7.76394 .213675 4.69 21.9961 103.162 2.16564 6.84836 1.67388 3.60626 7.76946 .213220 4.70 22.0900 103.823 2.16795 6.85565 1.67507 3.60883 7.77498 .212766 4.71 22.1841 104.487 2.17025 6.86294 1.67626 3.61138 7.78049 .212314 4.72 22.2784 105.154 2.17256 6.87023 1.67744 3.61394 7.78599 .211864 4.73 22.3729 105.824 2.17486 6.87750 1.67863 3.61649 7.79149 .211417 4.74 22.4676 106.496 2.17715 6.88477 1.67981 3.61904 7.79697 .210971 4.75 22.5625 107.172 2.17945 6.89202 1.68099 3.62158 7.80245 .210526 4.76 22.6576 107.850 2.18174 6.89928 1.68217 3.62412 7.80793 .210084 4.77 22.7529 108.531 2.18403 6.90652 1.68334 3.62665 7.81339 .209644 4.78 22.8484 109.215 2.18632 6.91375 1.68452 3.62919 7.81885 .209205 4.79 22.9441 109.902 2.18861 6.92098 1.68569 3.63171 7.82429 .208768 4.80 23.0400 110.592 2.19089 6.92820 1.68687 3.63424 7.82974 .208333 4.81 23.1361 111.285 2.19317 6.93542 1.68804 3.63676 7.83517 .207900 4.82 23.2324 111.980 2.19545 6.94262 1.68920 3.63928 7.84059 .207469 4.83 23.3289 112.679 2.19773 6.94982 1.69037 3.64180 7.84601 .207039 4.84 23.4256 113.380 2.20000 6.95701 1.69154 3.64431 7.85142 .206612 4.85 23.5225 114.084 2.20227 6.96419 1.69270 3.64682 7.85683 .206186 4.86 23.6196 114.791 2.20454 6.97137 1.69386 3.64932 7.86222 .205761 4.87 23.7169 115.501 2.20681 6.97854 1.69503 3.65182 7.86761 .205339 4.88 23.8144 116.214 2.20907 6.98570 1.69619 3.65432 7.87299 .204918 4.89 23.9121 116.930 2.21133 6.99285 1.69734 3.65682 7.87837 .204499 4.90 24.0100 117.649 2.21359 7.00000 1.69850 3.65931 7.88374 .204082 4.91 24.1081 118.371 2.21585 7.00714 1.69965 3.66179 7.88909 .203666 4.92 24.2064 119.095 2.21811 7.01427 1.70081 3 66428 7.89445 .203252 4.93 24.3049 119.823 2.22036 7.02140 1.70196 3.66676 7.89979 .202840 4.94 24.4036 120.554 2 22261 7.02851 1.70311 3.66924 7.90513 .202429 4.95 24.5025 121.287 2.22486 7.03562 1.70426 3.67171 7.91046 .202020 4.96 24.6016 122.024 2.22711 7.04273 1.70540 3.67418 7.91578 .201613 4.97 24.7009 122.763 2.22935 7.04982 1.70655 3.67665 7.92110 .201207 4.98 24.8004 123.506 2.23159 7.05691 1.70769 3.67911 7.92641 .200803 4.99 24.9001 124.251 2.23383 7.06399 1.70884 3.68157 7.93171 .200401 5.00 25.0000 125.000 2.23607 7.07107 1.70998 3.68403 7.93701 .200000 112/ POWERS, ROOTS, AND RECIPROCALS. 1 n n n 2 n 3 ■Vio n ^10 n ■ v '100 n 5.01 25.1001 125.752 2.23830 7.07814 1.71112 3.68649 7.94229 .199601 5.02 25.2004 126.506 2.24054 7.08520 1.71225 3.68894 7.94757 .199203 5.03 25.3009 127.264 2.24277 7.09225 1.71339 3.69138 7.95285 .198807 5.04 25.4016 128.024 2.24499 7.09930 1.71452 3.69383 7.95811 .198413 5.05 25.5025 128.788 2.24722 7.10634 1.71566 3.69627 7.96337 .198020 5.06 25.6036 129.554 2.24944 7.11337 1.71679 3.69871 7.96863 .197629 5.07 25.7049 130.324 2.25167 7.12039 1.71792 ■ 3.70114 7.97387 .197239 5.08 25.8064 131.097 2.25389 7.12741 1.71905 3.70358 7.97911 .196850 5.09 25.9081 131.872 2.25610 7.13442 1.72017 3.70600 7.98434 .196464 5.10 26.0100 132.651 2.25832 7.14143 1.72130 3.70843 7.98957 .196078 5.11 26.1121 133.433 2.26053 7.14843 1.72242 3.71085 7.99479 .195695 5.12 26.2144 134.218 2.23274 7.15542 1.72355 3.71327 8.00000 .195313 5.13 26.3169 135.006 2.26495 7.16240 1.72467 3.71566 8.00520 .194932 5.14 26.4196 135.797 2.26716 7.16938 1.72579 3.71810 8.01040 .194553 5.15 26.5225 136.591 2.26936 7.17635 1.72691 3.72051 8.01559 .194175 5.16 26.6256 137.388 2.27156 7.18331 1.72802 3.72292 8.02078 .193798 5.17 26.7289 138.188 2.27376 7.19027 1.72914 3.72532 8.02596 .193424 5.18 26.8324 138.992 2.27596 7.19722 1.73025 3.72772 8.03113 .193050 5.19 26.9361 139.798 2.27816 7.20417 1.73137 3.73012 8.03629 .192678 5.20 27.0400 140.608 2.28035 7.21110 1.73248 3.73251 8.04145 .192308 5.21 27.1441 141.421 2.28254 7.21803 1.73359 3.73490 8.04660 .191939 5.22 2 . 7.2484 142.237 | 2.28473 7.22496 1.73470 3.73729 8.05175 .191571 5.23 27.3529 143.056 [ 2.28692 7.23187 1:73580 3.73968 8.05689 .191205 5.24 27.4576 143.878 2.28910 7.23878 1.23691 3.74206 8.06202 .190840 5.25 27.5625 144.703 . j 2.29129 7.24569 1.73801 3.74443 8.06714 .190476 5.26 27.6676 145.532 ! 2.29347 7.25259 1.73912 3.74681 8.07226 ! .190114 5.27 27.7729 146.363 2,29565 7.25948 1.74022 3.74918 8.07737 .189753 5.28 27.8784 147.198 ! 2.29783 7.26636 1.74132 3.75158 8,08248 .189394 5.29 27.9841 148.036 2.30000 7.27324 1.74242 3.75392 8.08758 .189036 5.30 28.0900 148.877 2.30217 7.28011 1.74351 3.75629 8.09267 .188679 5.31 28.1961 149.721 2.30434 7.28697 1.74461 3.75865 8.09776 .188324 5.32 28.3024 150.569 2.30651 7.29383 1.74570 3.76100 8.10284 .187970 5.33 28.4089 151.419 2.30868 7.30068 1.74680 3.76336 8.10791 . 187617 - 5.34 28.5156 152.273 2.31084 7.30753 1.74789 3.76571 8.11298 .187266 5.35 28.6225 153.130 2.31301 7.31437 1.74898 3.76806 8.11804 .186916 5.36 28.7296 153.991 2.31517 7.32120 1.75007 3.77041 8.12310 .186567 5.37 28.8369 154.854 2.31733 7.32803 1.75116 3.77275 8.12814 .186220 5.38 28.9444 155.721 2.31948 7.33485 1.75224 3.77509 8.13319 .185874 5.39 29.0521 156.591 2.32164 7.34166 1.75333 3.77740 8.13822 .185529 5.40 29.1600 157.464 2.32379 7.34847 1.75441 3.77976 8.14325 .185185 5.41 29.2681 158.340 2.32594 7.35527 1.75549 3.78210 8.14828 .184843 5.42 29.3764 159.220 2.32809 7.36206 1.75657 3.78442 8.15329 .184502 5.43 29.4849 160.103 2.33024 7.36885 1.75765 3.78675 8.15831 .184162 5.44 29.5936 160.989 2.33238 7.37564 1.75873 3.78907 8.16331 .183824 5.45 29.7025 161.879 2.33452 7.38241 1.75981 3.79139 8.16831 .183486 5.46 29.8116 162.771 2.33666 7.38918 1 .76088 3.79371 8.17330 .183150 5.47 29.9209 163.667 2.33880 7.39594 1.76196 3.79603 8.17829 .182815 5.48 30.0304 164.567 2.34094 7.40270 1.76303 3.79834 8.18327 .182482 5.49 30.1401 165.469 2.34307 7.40945 1.76410 3.80065 8.18824 .182149 5.50 30.2500 166.375 2.34521 7.41620 1.76517 3.80295 8.19321 .181818 POWERS, ROOTS, AND RECIPROCALS. 112^ n 7*2 71 * >/l0 n ^10 71 1 ! \ylmn 1 ! 71 5.51 30.3601 167.284 2.34734 7.42294 1.76624 1 3.80526 8.19818 .181488 5.52 30.4704 168.197 2.34947 7.42967 1.76731 3.80756 8.20313 .181159 5.53 30.5809 169.112 2.35160 7.43640 1.76838 3.80986 8.20808 .180832 5.54 30.6916 170.031 2.35372 7.44312 1.76944 3.80115 | 8.21303 .180505 5.55 30.8025 170.954 2.35584 7.44983 1.77051 3.81444 8.21797 .180180 5.56 30.9136 171.880 2.35797 7.45654 1.77157 3.81673 8.22290 .179856 5.57 31.0249 172.809 2.36008 7.46324 1.77263 3.81902 8.22783 .179533 5.58 31.1364 173.741 2.36220 7.46994 1.77369 3.82130 8.23275 .179212 5.59 31.2481 174.677 2.36432 7.47663 1.77475 3.82358 | 8.23766 .178891 5.60 31.3600 175.616 2.36643 7.48331 1.77581 3.82586 8.24257 .178571 5.61 31.4721 176.558 2.36854 7.48999 1.77686 3.82814 8.24747 .178253 5.62 31.5844 177.504 2.37065 7.49667 1.77792 3.83041 8.25237 .177936 5.63 31.6969 178.454 2.37276 7.50333 1.77897 3.83268 8.25726 .177620 5.64 31.8096 179.406 2.37487 7.50999 1.78003 3.83495 8.26215 .177305 5.65 31.9225 180.362 2.37697 7.51665 1.78108 3.83721 8.26703 .176991 5.66 32.0356 181.321 2.37908 7.52330 1.78213 3.83948 8.27190 .176678 5.67 32.1489 182.284 2.38118 7.52994 1.78318 3.84174 8.27677 .176367 5.68 32.2624 183.250 2.38328 7.53658 1.78422 3.84400 8.28164 .176056 5.69 32.3761 184.220 2.38537 7.54321 1.78527 3.84625 8.28649 .175747 5.70 32.4900 185.193 2.38747 7.54983 1.78632 3.84850 8.29134 .175439 5.71 32.6041 186.169 2.38956 7.55645 1.78736 3.85075 8.29619 .175131 5.72 32.7184 187.149 2^39165 7.56307 1.78840 3.85300 8.30103 .174825 5.73 32.8329 188.133 2.39374 7.56968 1.78944 3.85524 8.30587 .174520 5.74 32.9476 189.119 2.39583 7.57628 1.79048 3.85748 8.31069 .174216 5.75 33.0625 190.109 2.39792 7.58288 1.79152 3.85972 8.31552 .173913 5.76 33.1776 191.103 2.40000 7.58947 1.79256 3.86196 8.32034 .173611 5.77 33.2929 192.100 2.40208 7.59605 1.79360 3.86419 8.32515 .173310 5.78 33.4084 193.101 2.40416 7.60263 1.79463 3.86642 8.32995 .173010 5.79 33.5241 194.105 2.40624 7.60920 1.79567 3.86865 8.33476 .172712 5.80 33.6400 195.112 2.40832 7.61577 1.79670 3.87088 8.33955 .172414 5.81 33.7561 196.123 2.41039 7.62234 1.79773 3.87310 8.34434 .172117 5.82 33.8724 197.137 2.41247 7.62889 1.79876 3.87532 8.34913 .171821 5.83 33.9889 198.155 2.41454 7.63544 1.79979 3.87754 8.35390 .171527 5.84 34.1056 199.177 2.41661 7.64199 1.80082 3.87975 8.35868 .171233 5.85 34.2225 200.202 2.41868 7.64853 1.80185 3.88197 8.36345 .170940 5.86 34.3396 201.230 2.42074 7.65506 1.80288 3.88418 8.36821 .170649 5.87 34.4569 202.262 2.42281 7.66159 1.80390 3.88639 8.37297 .170358 5.88 34.5744 203.297 2.42487 7.66812 1.80492 3.88859 8.37772 .170068 5.89 34.6921 204.336 2.42693 7.67463 1.80595 3.89082 8.38247 .169779 5.90 34.8100 205.379 2.42899 7.68115 1.80697 3.89300 8.38721 .169492 5.91 34.9281 206.425 2.43105 7.68765 1.80799 3.89520 8.39194 .169205 5.92 35.0464 207.475 2.43311 7.69415 1.80901 3.89739 8.39667 .168919 5.93 35.1649 208.528 2.43516 7.70065 1.81003 3.89958 8.40140 .168634 5.94 35.2836 209.585 2.43721 7.70714 1.81104 3.90177 8.40612 .168350 5.95 35.4025 210.645 2.43926 7.71362 1.81206 3.90396 8.41083 .168067 5.96 35.5216 211.709 2.44131 7.72010 1.81307 3.90615 8.41554 .167785 5.97 35.6409 212.776 2.44336 7.72658 1.81409 3.90833 8.42025 .167504 5.98 35.7604 213.847 2.44540 7.73305 1.81510 3.91051 8.42494 .167224 5.99 35 . 8801 ' 214.922 2.44745 7.73951 1.81611 3.91269 8.42964 .166945 6.00 36.0000 216.000 2.44949 7.74597 1.81712 3.91487 8.43433 .166667 11 2h POWERS, ROOTS, AND RECIPROCALS. n tt 2 tt 3 AS VTOn fllOn 1 £ 8 1 n 6.01 36.1201 217.082 2.45153 7.75242 1.81813 3.91704 8.43901 .166389 6.02 36.2404 218.167 2.45357 7.75887 1.81914 3.91921 8.44369 .166113 6.03 36.3609 219.256 2.45561 7.76531 1.82014 3.92138 8.44836 .165838 6.04 36.4816 220.349 2.45764 7.77174 1.82115 3.92355 8.45303 .165563 6.05 36.6025 221.445 2.45967 7.77817 1.82215 3.92571 8.45769 .165289 6.06 36.7236 222.545 2.46171 7.78460 1.82316 3.92787 8.46235 .165017 6.07 36.8449 223.649 2.46374 7.79102 1.82416 3.93003 8.46700 .164745 6.08 36.9664 224.756 2.46577 7.79744 1.82516 3.93219 8.47165 .164474 6.09 37.0881 225.867 2.46779 7.80385 1.82616 3.93434 8.47629 .164204 6.10 37.2100 226.981 2.46982 7.81025 1.82716 3.93650 8.48093 .163934 6.11 37.3321 228.099 2.47184 7.81665 1.82816 3.93865 8.48556 .163666 6.12 37.4544 229.221 2.47386 7.82304 1.82915 3.94079 8.49018 .163399 6.13 37.5769 230.346 2.47588 7.82943 1.83015 3.94294 8.49481 .163132 6.14 37.6996 231.476 2.47790 7.83582 1.83115 3.94508 8.49942 .162866 6.15 37.8225 232.608 2.47992 7.84219 1.83214 3.94722 8.50404 .162602 6.16 37.9456 233.745 2.48193 7.84857 1.83313 3.94936 8.50864 .162338 6.17 38.0689 234.885 2.48395 7.85493 1.83412 3.95150 8.51324 .162075 6.18 38.1924 236.029 2.48596 7.86130 1.83511 3.95363 8.51784 .161812 6.19 38.3161 237.177 2.48797 7.86766 1.83610 3.95576 8.52243 .161551 6.20 38.4400 238.328 2.48998 7.87401 1.83709 3.95789 8.52702 .161290 6.21 38.5641 239.483 2.49199 7.88036 1.83808 3.96002 8.53160 .161031 6.22 38.6884 240.642 2.49399 7.88670 1.83906 3.96214 8.53618 .160772 6.23 38.8129 241.804 2.49600 7.89303 1.84005 3.96426 8.54075 .160514 6.24 38.9376 242.971 2.49800 7.89937 1.84103 3.96639 8.54532 .160256 6.25 39.0625 244.141 2.50000 7.90569 1.84202 3.96850 8.54988 .160000 6.26 39.1876 245.314 2.50200 7.91202 1.84300 3.97062 8.55444 .159744 6.27 39.3129 246.492 2.50400 7.91833 1.84398 3.97273 8.55899 .159490 6.28 39.4384 247.673 2.50599 7.92465 1.84496 3.97484 8.56354 .159236 6.29 39.5641 248.858 2.50799 7.93095 1.84594 3.97695 8.56808 .158983 6.30 39.6900 250.047 2.50998 7.93725 1.84691 3.97906 8.57262 .158730 6.31 39.8161 251.240 2.51197 7.94355 1.84789 3.98116 8.57715 .158479 6.32 39.9424 252.436 2.51396 7.94984 1.84887 3.98326 8.58168 .158228 6.33 40.0689 253.636 2.51595 7.95613 1.84984 3.98536 8.58620 .157978 6.34 40.1956 254.840 2.51794 7.96241 1.85082 3.98746 8.59072 .157729 6.35 40.3225 256.048 2.51992 7.96869 1.85179 3.98956 8.59524 .157480 6.36 40.4496 257.259 2.52190 7.97496 1.85276 3.99165 8.59975 .157233 6.37 40.5769 258.475 2.52389 7.98123 1.85373 3 99374 8.60425 .156986 6.38 40.7044 259.694 2.52587 7.98749 1.85470 3.99583 8.60875 .156740 6.39 40.8321 260.917 2.52784 7.99375 1.85567 3.99792 8.61325 .156495 6.40 40.9600 262.144 2.52982 8.00000 1.85664 4.00000 8.61774 .156250 6.41 41.0881 263.375 2.53180 8.00625 1.85760 4.00208 8.62222 .156006 6.42 41.2164 264.609 2.53377 8.01249 1.85857 4.00416 8.62671 .155763 6.43 41.3449 265.848 2.53574 8.01873 1.85953 4.00624 8.63118 .155521 6.44 41.4736 267.090 2.53772 8.02496 1.86050 4.00832 8.63566 .155280 6.45 41.6025 268.336 2.53969 8.03119 1.86146 4.01039 8.64012 .155039 6.46 41.7316 269.586 2.54165 8.03741 1.86242 4.01246 8.64459 .154799 6.47 41.8609 270.840 2.54362 8.04363 1.86338 4.01453 8.64904 .154560 6.48 41.9904 272.098 2.54558 8.04984 1.86434 4.01660 8.65350 .154321 6.49 42.1201 273.359 2.54755 8.05605 1.86530 4.01866 8.65795 .154083 6.50 42.2500 274.625 2.54951 8.06226 1.86626 4.02073 8.66239 .153846 POWERS, ROOTS, AND RECIPROCALS. 112l Wn 1 n n 3 V/l VlO n ^100 n n 6.51 42.3801 275.894 2.55147 8.06846 1.86721 4.02279 8.66683 .153610 6.52 42.5104 277.168 2.55343 8.07465 1.86817 4.02485 8.67127 .153374 6.53 42.6409 278.445 2.55539 8.08084 1.86912 4.02690 8.67570 .153139 6.54 42.7716 279.726 2.55734 8.08703 1.87008 4.02896 8.68012 .152905 6.55 42.9025 281.011 2.55930 8.09321 1.87103 4.03101 8.68455 .152672 6.56 43.0336 282.300 2.56125 8.09938 1.87198 4.03306 8.68896 .152439 6.57 43.1649 283.593 2.56320 8.10555 1.87293 4.03511 8.69338 .152207 6.58 43.2964 284.890 2.56515 8.11172 1.87388 4.03715 8.69778 .151976 6.59 43.4281 286.191 2.56710 8.11788 1.87483 4.03920 8.70219 .151745 6.60 43.5600 287.496 2.56905 8.12404 1.87578 4.04124 8.70659 .151515 6.61 43.6921 288.805 2.57099 8.13019 1.87672 4.04328 8.71098 .151286 6.62 43.8244 290.118 2.57294 8.13634 1.87767 4.04532 8.71537 .151057 6.63 43.9569 291.434 2.57488 8.14248 1.87862 4.04735 8.71976 .150830 6.64 44.0 S 96 292.755 2.57682 8.14862 1.87956 4.04939 8.72414 .150602 6.65 44.2225 294.080 2.57876 8.15475 1.88050 4.05142 8.72852 .150376 6.66 44.3556 295.408 2.58070 8.16088 1.88144 4.05345 8.73289 .150150 6.67 44.48 S 9 296.741 2.58263 8.16701 1.88239 4.05548 8.73726 .149925 6.68 44.6224 298.078 2.58457 8.17313 1.88333 4.05750 8.74162 .149701 6.69 44.7561 299.418 2.58650 8.17924 1.88427 4.05953 8.74598 .149477 6.70 44.8900 300.763 2.58844 8.18535 1.88520 4.06155 8.75034 .149254 6.71 45.0241 302.112 2.59037 8.19146 1.88614 4.06357 8.75469 .149031 6.72 45.1584 303.464 2.59230 8.19756 1.88708 4.06558 8.75904 .148810 6.73 45.2929 304.821 2.59422 8.20366 1.88801 4.06760 8.76338 .148588 6.74 45.4276 306.182 2.59615 8.20975 1.88895 4.06961 8.76772 .148368 6.75 45.5625 307.547 2.59808 8.21584 1.88988 4.07163 8.77205 .148148 6.76 45.6976 308.916 2.60000 8.22192 1.89081 4.07364 8.77638 .147929 6.77 45.8329 310.289 2.60192 8.22800 1.89175 4.07564 8.78071 .147711 6.78 45.9684 311.666 2.60384 8.23408 1.89268 4.07765 8.78503 .147493 6.79 46.1041 313.047 2.60576 8.24015 1.89361 4.07965 8.78935 .147275 6.80 46.2400 314.432 2.60768 8.24621 1.89454 4.08166 8.79366 .147059 6.81 46.3761 315.821 2.60960 8.25227 1.89546 4.08365 8.79797 .146843 6.82 46.5124 ! 317.215 2.61151 8.25833 1.89639 4.08565 8.80227 .146628 6.83 46.6489 318.612 2.61343 8.26438 1.89732 4.08765 8.80657 .146413 6.84 46.7856 320.014 2.61534 8.27043 1 .89824 4.08964 8.81087 .146199 6.85 46.9225 321.419 2.61725 8.27647 1.89917 4.09164 8.81516 .145985 6.86 47.0596 322.829 2.61916 8.28251 1.90009 4.09362 8.81945 .145773 6.87 47.1969 324.243 2.62107 8.28855 1.90102 4.09561 8.82373 .145560 6.88 47.3344 325.661 2.62298 8.29458 1.90194 4.09760 8.82801 .145349 6.89 47.4721 327.083 2.62488 8.30060 1.90286 4.09958 8.83229 .145138 6.90 47.6100 328.509 2.62679 8.30662 1.90378 4.10157 8.83656 .144928 6.91 47.7481 329.939 2.62869 8.31264 1.90470 4.10355 8.84082 .144718 6.92 47.8864 331.374 2.63059 8.31865 1.90562 4.10552 8.84509 .144509 6.93 48.0249 332.813 2.63249 8.32466 1.90653 4.10750 8.84.934 .144300 6.94 48.1636 334.255 2.63439 8.33067 1.90745 4.10948 8.85360 .144092 6.95 48.3025 335.702 2.63629 8.33667 1.90837 4.11145 8.85785 .143885 6.96 48.4416 337.154 2.63818 8.34266 1.90928 4.11342 8.86210 .143678 6.97 48.5809 338.609 2.64008 8.34865 1.91019 4.11539 8.86634 .143472 6.98 ; 48.7204 340.068 2.64197 8.35464 1.91111 4.11736 8.87058 .143267 6.99 48.8601 341.532 2.64386 8.36062 1.91202 4.11932 8.87481 .143062 7.00 , 49.0000 343.000 2.64575 8.36660 1.91293 | 4.12129 8.87904 .142857 112 ; POWERS, ROOTS, AND RECIPROCALS. n W2 W 3 yfn *10 n 7.01 49.1401 344.472 2.64764 8.37257 7.02 49.2804 345.948 2.64953 8.37854 7.03 49.4209 347.429 2.65141 8.38451 7.04 49.5616 348.914 2.65330 8.39047 7.05 49.7025 350.403 2.65518 8.39643 7.06 49.8436 351.896 2.65707 8.40238 7.07 49.9849 353.393 2.65895 8.40833 7.08 50.1264 354.895 2.66083 8.41427 7.09 50.2681 356.401 2.66271 8.42021 7.10 50.4100 357.911 2.66458 8.42615 7.11 50.5521 359.425 2.66646 8.43208 7.12 50.6944 360.944 2.66833 8.43801 7.13 50.8369 362.467 2.67021 8.44393 7.14 50.9796 363.994 2.67208 8.44985 7.15 51.1225 365.526 2.67395 8.45577 7.16 51.2656 367.062 2.67582 8.46168 7.17 51.4089 368.602 2.67769 8.46759 7.18 51.5524 370.146 2.67955 8.47349 7.19 51.6961 371.695 2.68142 8.47939 7.20 51.8400 373.248 2.68328 8.48528 7.21 51.9841 374.805 2.68514 8.49117 7.22 52.1284 376.367 2.68701 8.49706 7 23 52.2729 377.933 2.68887 8.50294 7.24 52.4176 379.503 2.69072 8.50882 7.25 52.5625 381-.078 2.69258 8.51469 7.26 52.7076 382.657 2.69444 8.52056 7.27 52.8529 384.241 2.69629 8.52643 7.28 52.9984 385.828 2.69815 8.53229 7.29 53.1441 387.420 2.70000 8.53815 7.30 53.2900 389.017 2.70185 8.54400 7.31 53.4361 390.4>18 2.70370 8.54985 7.32 53.5824 392.223 2.70555 8.55570 7.33 53.7289 393.833 2.70740 8.56154 7.34 53.8756 395.447 2.70924 8.56738 7.35 54.0225 397.065 2.71109 8.57321 7.36 54.1696 398.688 2.71293 8.57904 7.37 54.3169 400.316 2.71477 8.58487 7.38 54.4644 401 .947 2.71662 8.59069 7.39 54.6121 403.583 2.71846 8.59651 7.40 54.7600 405.224 2.72029 8.60233 7.41 54.9081 406.869 2.72213 8.60814 7.42 55.0564 408.518 2.72397 8.61394 7.43 55.2049 410.172 2.72580 8.61974 7.44 55.3536 411.831 2.72764 8.62554 7.45 55.5025 413.494 2.72947 8.63134 7.46 55.6516 415.161 2.73130 8.63713 7.47 55.8009 416.833 2.73313 8.64292 7.48 55.9504 418.509 2.73496 8.64870 7.49 56.1001 420.190 2.73679 8.65448 7.50 1 56.2500 421.875 2.73861 8.66025 1.91384 1.91475 1.91566 1.91657 1.91747 1.91838 1.91929 1.92019 1.92109 1.92200 1.92290 1.92380 1.92470 1 .92560 1.92650 1.92740 1.92829 1.92919 1.93008 1.93098 1.93187 1.93277 1.93366 1.93455 1.93544 1.93633 1.93722 1.93810 1.93899 1.93988 1.94076 1.94165 1.94253 1.94341 1.94430 1.94518 1.94606 1.94694 1.94782 1.94870 1.94957 1.95045 1.95132 1.95220 1.95307 1.95395 1 .95482 1.95569 1.95656 1.95743 *10 n 4.12325 4.12521 4.12716 4.12912 4.13107 4.13303 4.13498 4.13695 4.13887 4.14082 4.14276 4.14470 4.14664 4.14858 4.15051 4.15245 4.15438 4.15631 4.15824 4.16017 4.16209 4.16402 4.16594 4.16786 4.16978 4.17169 4.17361 4.17552 4.17743 4.17934 4.18125 4.18315 4.18506 4.18696 4.18886 4.19076 4.19266 4.19455 4.19644 4.19834 4.20023 4.20212 4.20400 4.20589 4.20777 4.20965 4.21153 4.21341 4.21529 4.21716 *10071 8.88327 8.88749 8.89171 8.89592 8.90013 8.90434 8.90854 8.91274 8.91693 8.92112 8.92531 8.92949 8.93367 8.93784 8.94201 8.94618 8.95034 8.95450 8.95866 8.96281 8.96696 8.97110 8.97524 8.97938 8.98351 8.98764 8.99176 8.99588 9.00000 9.00411 9.00822 9.01233 9.01643 9.02053 9.02462 9.02871 9.03280 9.03689 9.04097 9.04504 9.04911 9.05318 9.05725 9.06131 9.06537 9.06942 9.07347 9.07752 9.08156 9.08560 1 n .142653 .142450 .142248 .142046 .141844 .141643 .141443 .141243 .141044 .140845 .140647 .140449 .140253 .140056 .139860 .139665 .139470 .139276 .139082 .138889 .138696 .138504 .138313 .138122 .137931 .137741 .137552 .137363 .137174 .136986 .136799 .136612 .136426 .136240 .136054 .135870 .135685 .135501 .135318 .135135 .134953 .134771 .134590 .134409 .134228 .134048 .133869 .133690 .133511 .133333 POWERS, ROOTS, AND RECIPROCALS. lV2k n n 2 7l 3 VlOn ftbOn 1 ^10 n n 7.51 56.4001 423.565 2.74044 8.66603 1.95830 4.21904 9.08964 .133156 7.52 56.5504 425.259 2.74226 8.67179 1.95917 4.22091 9.09367 .132979 7.53 56.7009 426.958 2.74408 8.67756 1.96004 4.22278 9.09770 .132802 7.54 56.8516 428.661 2.74591 8.68332 1.96091 4.22465 9.10173 .132626 7.55 57.0025 430.369 2.74773 8.68907 1.96177 4.22651 9.10575 .132450 7.56 57.1536 432.081 2.74955 8.69483 1.96264 4.22838 9.10977 .132275 7.57 57.3049 433.798 2.75136 8.70057 1.96350 4.23024 9.11378 .132100 7.58 57.4564 435.520 2.75318 8.70632 1.96437 4.23210 9.11779 .131926 7.59 57.6081 437.245 2.75500 8.71206 1.96523 4.23396 9.12180 .131752 7.60 57.7600 438.976 2.75681 8.71780 1.96610 4.23582 9.12581 .131579 7.61 57.9121 440.711 2.75862 8.72353 1.96696 4.23768 9.12981 .131406 7.62 58.0644 442.451 2.76043 8.72926 1.96782 4.23954 9.13380 .131234 7.63 58.2169 444.195 2.76225 8.73499 1.96868 4.24139 9.13780 .131062 7.64 58.3696 445.994 2.76405 8.74071 1.96954 4.24324 9.14179 .130890 7.65 58.5225 447.697 2.76586 8.74643 1.97040 4.24509 9.14577 .130719 7.66 58.6756 449.455 2.76767 8.75214 1.97126 4.24694 9.14976 .130548 7.67 58.8289 451.218 2.76948 8.75785 1.97211 4.24879 9.15374 .130378 7.68 58.9824 452.985 2.77128 8.76356 1.97297 4.25063 9.15771 .130208 7.69 59.1361 454.757 2.77308 8.76926 1.97383 4.25248 9.16169 .130039 7.70 59.2900 456.533 2.77489 8.77496 1.97468 4.25432 9.16566 .129870 7.71 59.4441 458.314 2.77669 8.78066 1.97554 4.25616 9.16962 .129702 7.72 59.5984 460.100 2.77849 8.78635 1.97639 4.25800 9.17359 .129534 7.73 59.7529 461.890 2.78029 8.79204 1.97724 4.25984 9.17754 .129366 7.74 59.9076 463.685 2.78209 8.79773 1.97809 4.26168 9.18150 .129199 7.75 60.0625 465.484 2.78388 8.80341 1.97895 4.26351' 9.18545 .129032 7.76 60.2176 467.289 2.78568 8.80909 1.97980 4.26534 9.18940 .128866 7.77 60.3729 469.097 2.78747 8.81476 1.98065 4.26717 9.19335 .128700 7.78 60.5284 470.911 2.78927 8.82043 1.98150 4.26900 9.19729 .128535 7.79 60.6841 472.729 2.79106 8.82610 1.98234 4.27083 9.20123 .128370 7.80 60.8400 474.552 2.79285 8.83176 1.98319 4.27266 9.20516 .128205 7.81 60.9961 476.380 2.79464 8.83742 1.98404 4.27448 9.20910 .128041 7.82 61.1524 478.212 2.79643 8.84308 1.98489 4.27631 9.21303 .127877 7.83 61.3089 480.049 2.79821 8.84873 1.98573 4.27813 9.21695 .127714 7.84 61.4656 481.890 2.80000 8.85438 1.98658 4.27995 9.22087 .127551 7.85 61.6225 483.737 2.80179 8.86002 1.98742 4.28177 9.22479 .127389 7.86 61.7796 485.588 2.80357 8.86566 1.98826 4.28359 9.22871 .127227 7.87 61.9369 487.443 2.80535 8.87130 1.98911 4.28540 9.23262 .127065 7.88 62.0944 489.304 2.80713 8.87694 1.98995 4.28722 9.23653 .126904 7.89 62.2521 491.169 2.80891 8.88257 1 .99079 4.28903 9.24043 .126743 7.90 62.4100 493.039 2.81069 8.88819 1.99163 4.29084 9.24433 .126582 7.91 62.5681 494.914 2.81247 8.89382 1.99247 4.29265 9.24823 .126422 7.92 62.7264 496.793 2.81425 8.89944 1 1.99331 4.29446 9.25213 .126263 7.93 62.8849 498.677 2.81603 8.90505 1 1.99415 4.29627 9.25602 .126103 7.94 63.0436 500.566 2.81780 8.91067 1.99499 4.29807 9.25991 .125945 7.95 63.2025 502.460 2.81957 8.91628 1.99582 4.29987 9.26380 .125786 7.96 63.3616 504.358 2.82135 8.92188 1.99666 4.30168 9.26768 .125628 7.97 63.5209 506.262 2.82312 8.92749 1 .99750 4.30348 9.27156 .125471 7.98 63.6804 508.170 2.82489 8.93308 1.99833 4.30528 9.27544 .125313 7.99 63.8401 510.082 2.82666 8.93868 1.99917 4.30707 9.27931 .125156 8.00 64.0000 512.000 2.82843 8.94427 2.00000 4.30887 9.28318 .125000 112 1 POWERS, ROOTS, AND RECIPROCALS. » ri 2 W 3 Vw VlOn ^IlOn aToo n 1 n 8.01 64.1601 513.922 2.83019 8.94986 2.00083 4.31066 9.28704 .124844 8.02 64.3204 515.850 2.83196 8.95545 2.00167 4.31246 9.29091 .124688 8.03 64.4809 517.782 2.83373 8.96103 2.00250 4.31425 9.29477 .124533 8.04 64.6416 519.718 2.83549 8.96660 2.00333 4.31604 9.29862 .124378 8.05 64.8025 521.660 2.83725 8.97218 2.00416 4.31783 9.30248 .124224 8.06 64.9636 523.607 2.83901 8.97775 2.00499 4.31961 9.30633 .124070 8.07 65.1249 525.558 2.84077 8.98332 2.00582 4.32140 9.31018 .123916 8.08 65.2864 527.514 2.84253 8.98888 2.00664 4.32818 9.31402 .123762 8.09 65.4481 529.475 2.84429 8.99444 2.00747 4.32497 9.31786 .123609 8.10 65.6100 531.441 2.84605 9.00000 2.00830 4.32675 9.32170 .123457 8.11 65.7721 533.412 2.84781 9.00555 2.00912 4.32853 9.32553 .123305 8.12 65.9344 535.387 2.84956 9.01110 2.00995 4.33031 9.32936 .123153 8.13 66.0969 537.368 2.85132 9.01665 2.01078 4.33208 9.33319 .123001 8.14 66.2596 539.353 2.85307 9.02219 2.01160 4.33386 9.33702 .122850 8.15 • 66.4225 541.343 2.85482 9.02774 2.01242 4.33563 9.34084 .122699 8.16 66.5856 543.338 2.85657 9.03327 2.01325 4.33741 9.34466 .122549 8.17 66.7489 545.339 2.85832 9.03881 2.01407 4.33918 9.34847 .122399 8.18 66.9124 547.343 2.86007 9.04434 2.01489 4.34095 9.35229 .122249 8.19 67.0761 549.353 2.86182 9.04986 2.01571 4.34272 9.35610 .122100 8.20 67.2400 551.368 2.86356 9.05539 2.01653 4.34448 9.35990 .121951 8.21 67.4041 553.388 2.86531 9.06091 2.01735 4.34625 9.36370 .121803 8.22 67.5684 555.412 2.86705 9.06642 2.01817 4.34801 9.36751 .121655 8.23 67.7329 557.442 2.86880 9.07193 2.01899 4.34977 9.37130 .121507 8.24 67.8976 559.476 2.87054 9.07744 2.91980 4.35153 9.37510 .121359 8.25 68.0625 561.516 2.87228 9.08295 2.02062 4.35329 9.37889 .121212 8.26 68.2276 563.560 2.87402 9.08845 2.02144 4.35505 9.38268 .121065 8.27 68.3929 565.609 2.87576 9.09395 2.02225 4.35681 9.38646 .120919 8.28 68.5584 567.664 2.87750 9.09945 2.02307 4.35856 9.39024 .120773 8.29 68.7241 569.723 2.87924 9.10494 2.02388 4.36032 9.39402 .120627 8.30 68.8900 571.787 2.88097 9.11043 2.02469 4.36207 9.39780 .120482 8.31 69.0561 573.856 2.88271 9.11592 2.02551 4.36382 9.40157 .120337 8.32 69.2224 575.930 2.88444 9.12140 2.02632 4.36557 9.40534 .120192 8.33 69.3889 578.010 2.88617 9.12688 2.02713 4.36732 9.40911 .120048 8.34 69.5556 580.094 2.88791 9.13236 2.02794 4.36907 9.41287 .119904 8.35 69.7225 582.183 2.88964 9.13783 2.02875 4.37081 9.41663 .119761 8.36 69.8896 584.277 2.89137 9.14330 2.02956 4.37255 9.42039 .119617 8.37 70.0569 586.376 2.89310 9.14877 2.03037 4.37430 9.42414 .119474 8.38 70.2244 588.480 2.89482 9.15423 2.03118 4.37604 9.42789 .119332 8.39 70.3921 590.590 2.89655 9.15969 2.03199 4.37778 9.43164 .119190 8.40 70.5600 592.704 2.89828 9.16515 2.03279 4.37952 9.43539 .119048 8.41 70.7281 594.823 2.90000 9.17061 2.03360 4.38126 9.43913 .118906 8.42 70.8964 596.948 2.90172 9.17606 2.03440 4.38299 9.44287 .118765 8.43 71.0649 599.077 2.90345 9.18150 2.03521 4.38473 9.44661 .118624 8.44 71.2336 601.212 2.90517 9.18695 2.03601 4.38646 9.45034 .118483 8.45 71.4025 603.351 2.90689 9.19239 2.03682 4.38819 9.45407 .118343 8.46 71.5716 605.496 2.90861 9.19783 2.03762 4.38992 9.45780 .118203 8.47 71.7409 607.645 2.91033 9.20326 2.03842 4.39165 9.46152 .118064 8.48 71.9104 609.800 2.91204 9.20869 2.03923 4.39338 9.46525 .117925 8.49 72.0801 611.960 2.91376 9.21412 2.04003 4.39511 9.46897 .117786 8.50 72.2500 614.125 2.91548 9.21954 2.04083 4.39683 9.47268 .117647 POWERS, ROOTS, AND RECIPROCALS. 112m „ n 2 n :i Vi n ^I^On 'yin <10 n 460 .23 X .50 = .115 .115 X 1.502 = .259 .23 X .50 = .115 .115 X 1.002 = .115 .23 X .50 = .115 .115 X 0.502 = .029 .23 X .25 = .058 .058 X 0.125 2 = .001 1.917 11.722 2 2 A = 3.834 I = 23.444 CENTER OF OSCILLATION. 127 If the web of the beam is divided into areas 1 in. in height (instead of £ in.), the value of I obtained will be 23.46 in. If the section is considered to be of the form indi- cated by the dotted lines in Fig. 1, and to have the same area as the original section, then, by the formula for the moment of inertia of an I-beam given in Table V, page 153, the value of r 3.50 X 6 s — 3.27 X 5.25 3 c „ I - - = 23.57. The true value is almost exactly 23.48 in. Any one of these values would be sufficiently correct for most practical pur- poses. If it is desired to find * the moment of inertia * of a body about a given * axis with reference to * the weight of the body, * the process is substan- * tially the same as in the example given for the plane section, ex- OLU 1 1 1 cept that the weight of each small part of the Fig. 2. body is taken instead of the area of each small part of the section. CENTER OF OSCILLATION. The center of oscillation of a pendulum or other body vibrating or rotating about a fixed axis or center is that point at which, if the entire weight of the body were con- centrated, the body would continue to vibrate in the same intervals of time. When a pendulum, or other suspended body, is oscillating backward and forward, it is plain that those particles that are farther from the point of suspension travel through greater distances, and therefore move with greater velocities than those particles that are nearer the point of suspension. 128 MECHANICS. But there is evidently some point on the pendulum that travels through the same distance and has the same velocity as the average distance and average velocity of all the par- ticles. This point is called the center of oscillation; it is not situated at the center of gravity. It always exists in the ball of a revolving governor or other rotating body. The axis or center around which the body rotates (corresponding to the point of suspension in pendulum) is the axis of rotation. The distance from the axis, or center of rotation, to the center of oscillation is sometimes called the true length of the pendulum; it is also called the radius of oscillation; the latter name is preferable. To find the radius of oscillation: Divide the moment of inertia of the body about the given axis of rotation by the product of the total weight of the body , multi- plied by the distance from the given axis to the center of gravity of the body. The centers of oscillation and of rotation (point of suspen- sion) are interchangeable. If the position of a pendulum is reversed, and suspended from its center of oscillation, the pendulum will vibrate in the same intervals of time. Example. — It is desired to find the position of the center of oscillation of a wrought-iron bar 1 in. square and 12 in. long, axis of rotation perpendicular to the bar at one end: Weight of Each Cu. In. Sq.of Dist. .281 X p Cn II 0.070 .281 X 1.5 2 = 0.632 .281 X 2.52 = 1.756 .281 X 3.5 2 = 3.442 .281 X II fh 5.690 .281 X II ?c 10 8.500 .281 X 6.52 = 11.872 .281 X 7.52 = 15.806 .281 X 8.52 = 20.302 .281 X 9.52 = 25.360 .281 x: 10.52 = 30.980 .281 x: Ll .52 = 37.162 -M- i-m-jr- Jjj ira --- L_U_ u? ... |+b 0: — [1 -- j * 7U .JO -II J-S-5- 2 -~ TF 3.372 161.572 = I CENTER OF PERCUSSION. 129 Solution.— For the purposes of the example it will be sufficiently accurate to find the moment of inertia by con- sidering the bar to be divided into 12 equal cubes, each con- taining 1 cu. in. of metal, as indicated in the figure, and the weight of each cube to be concentrated at its center of gravity. The weight of 1 cu. in. of wrought iron is .281 lb., and of a bar 1 in. square and 1 ft. long it is .281 X 12 = 3.372 lb. Hence, I = .281 X .5 2 + .281 X 1.5 2 + etc. = 161.572. (See page 128.) The exact value of I is 161.856; this shows that the approximate method is very close. According to the rule previously given, if the moment of inertia is divided by the product of the weight of the body, by the distance from the axis of rotation to the center of gravity, the quotient will be the radius of oscillation. Therefore, the distance from the exact center of oscillation of a wrought-iron bar, 1 in. square and 12 in. long, to an axis of rotation perpendicular to the end of the bar, is 161.856 . . 3.372 X 6 8m '’ or two-thirds of the length of the bar. The value of I for a bar of any cross-section, provided it is uniform throughout its length, revolving about an axis per- pendicular to it and passing through its end, is WP 3 ’ in which W is the weight of the bar, and l is its length. Hence, I = WV 3.372 X 12 2 = 161.856. 3 3 If the axis passes through the center of gravity of the bar, I WV 12 ’ CENTER OF PERCUSSION. The center of percussion with respect to a given axis of rotation may be defined as the point of application of the resultant of the forces that cause the body to rotate. It is that point at which if a force is applied, the force will have no effect at the axis of rotation. 130 MECHANICS. Strike anything solid, as an anvil, with a stick. If the ' end of the stick hits the anvil, the opposite end will sting your hand and will jerk in the direction in which the blow is struck; if the center of the stick hits the anvil it will again sting your hand, but you will jerk it in a direction opposite to the movement of the blow. But somewhere between the end and the center of the stick will be a point where it may hit the anvil and not sting your hand at all. This point is the center of percussion. Level off the surface of some wet sand and lay a strip of board upon it (say 18 in. long and 3 in. wide). Strike or press the board near the center and the entire length of the board will be imprinted in the sand; but press it near one end and the opposite end will be raised up from the sand and will make no imprint. Between the center and the end of the board is a point that if pressed upon will cause no movement in the opposite end, i. e., the end of the board will neither press into the sand nor be lifted from it, but the imprint in the sand will diminish to zero at the end of the board. The point pressed or struck will be the center of percussion. If the board is of uniform width, the center of percussion will be at one-third of the distance from one end of the board. Similarly in the preceding illustration, if the stick is of uni- form size and weight, and your hand grasps it at one end, the point at which it can strike the anvil without affecting your hand will be at one-third the distance from the opposite end. In all cases the center of percussion is identical with the center of oscillation , and its position is found in the same manner. Example.— It is desired to find the position of the center of oscillation or percussion of two balls fastened upon a rod. The first, weighing 2 lb., is at a distance of 18 in. from the axis of rotation, and the second, weighing 1 lb., is at a distance of 36 in. from the axis. (See figure.) Solution.— For simplicity, the rod will be assumed to have no weight. Consider the weight of each ball to be concentrated at its center of gravity. RADIUS OF GYRATION. 131 The moment of inertia is found as follows. Sq. of Wt. Dist. 2 X 18 2 = 1 X J36 2 = 648 1,296 1,944 = I. The center of gravity of the two balls is found to be at a distance of 6 in. from the larger, or 24 in. from the axis of rota- tion (see page 124), and the combined weight of the two balls is 2 + 1 = 3 lb. Therefore, the center of percussion is found 1 944 to be at a distance of g ^ — = 27 in. from the axis of rotation. But, in an actual case, the rod would have weight, and its moment of inertia must be considered as w'ell as the moment of inertia of the balls. If we assume that the rod is of steel, f in. in diameter and 86 in. long, it will weigh X .7854 X 36 X -283 = 1.125 lb. .283 lb. is the weight of 1 cu. in. of steel. Using the formula given on page 129, I=J Fg = L 1 2 5X3g = 486- Adding this result to the former, 1,944 + 486 = 2,430 = moment of inertia of rods and balls. The center of gravity of the combination is found by the formula (see page 124) p P + w - Substituting, = 1 T V 24 - l T 7 r = 22 x \ in. = distance from end of rod to center of gravity. Applying the rule given for finding the center of oscilla- tion, the distance of the center of percussion from the end of 2 430 thebarls (1 + 2 + 1 ' 125)x - 22A = 26.34 in., very nearly. RADIUS OF GYRATION. The center of gyration is that point in a revolving body at which, if the entire mass of the body were concentrated, the moment of inertia with respect to a given axis would be the same as in the body. An ounce of cork occupies about 94 times as much space as 132 MECHANICS. an ounce of platinum; but the ounce of platinum can have the same moment of inertia as the ounce of cork, if its center of gyration has the same position with respect to the axis of rotation. The center of gyration is not at the center of gravity , nor >at the center of oscillation , but at some point in a straight line between those centers. The radius of gyration is the distance from the axis of rotation to the center of gyration. The square of the radius of gyration is the average of the squares of the distances from the axis of rotation to each ele- mentary particle of the body, or to each elementary area of the section, as the case may be. But the sum of these squares of distances, multiplied by the weight or area of each ele- mentary part, equals the moment of inertia; therefore, the moment of inertia divided by the weight of the body or area of the section equals the square of the radius of gyration; the square root of this quotient is the radius of gyration. But, according to the rule for finding the radius of oscil- lation, the quotient obtained by dividing the moment of inertia by the weight or area equals the product of the dis- tance from the axis of rotation to the center of gravity, mul- tiplied by the radius of oscillation; and, therefore, the radius of gyration is a mean proportional between these distances. If the distance from the axis of rotation to the center of gravity is known, and the radius of oscillation is known, the radius of gyration may be found by multiplying these two known distances together and extracting the square root of the product. In the example of the I-beam, Fig. 2, page 126, the sum of the areas of the half section of the beam is 1.917, and the area of the entire section is 3.834 sq. in. Therefore, the radius of gyration of this beam about an axis through the center of /2344 gravity perpendicular to the web = = 2.47 in. In the example of the iron bar 12 in. long (see figure, page 128), the distance from the axis of rotation to the center of gravity is 6 in., and the radius of oscillation was found to equal 8 in. Therefore, the radius of gyration about an RADIUS OF GYRATION. 133 axis perpendicular to the bar at one end = j/ 6 X 8 = 6.93 in. Or, the moment of inertia of the bar = 161.586, and the weight of the bar = 3.372 lb. Therefore, the radius of gyra- tion ,'161.586 \ 3.372 6.93 in., very nearly. The radius of gyration is used in determining the strength of columns. The axis must be taken in such a direction that the result will be the least radius of gyration of the column; this condition is usually obtained when the axis is perpen- dicular to the least diameter or side of the column. The various relations between these quantities may be concisely expressed by the following formulas, in which A = area of section (or weight of body if the weight is used); g = distance from axis of rotation to center of gravity; G = radius of gyration; r 0 = radius of oscillation; I = moment of inertia. Then, I — A G 2 . I = Agr 0 . % II /-p I I G = \^‘ 9 ~ Ar 0 ' G 2 c "S II G = v gr 0 . 9 = 7- ' 0 r 0 = - — . y g:G = G: r a . To find the radius of oscillation, radius of gyration, and moment of inertia, experimentally. The connecting-rod of an engine is represented in the figure. It is desired to find the moment of inertia of the rod about an axis of rotation through the center of the crosshead pin A. This may l)e accomplished, experimentally, as follow’s: Suspend the rod from the crosshead pin in such a manner 134 MECHANICS. that it will swing freely; cause it to swing, or oscillate, and note the exact time of the vibrations. Remove the crosshead pin and reverse the rod, but, instead of suspending it by the crankpin, suspend it by a movable pin B f that can be clamped at any desired point upon the rod. C is another view of this pin. There will be a point on the rod from which it may be suspended by means of the movable pin, so that it will vibrate in exactly the same intervals of time as when suspended from the crosshead pin. This point is the center of oscillation , for the center of oscillation and the center of rotation are inter- changeable; the point will be found at about one-third the length of the rod from the crankpin. Find this center of oscillation, experimentally, and carefully measure the dis- tance from the center of the movable pin to the center of the crosshead-pin hole. This distance is the radius of oscillation — r 0 . Next remove the movable pin, and find the center of gravity (lengthwise) of the rod by balancing it across a knife edge, and measure the distance from the center of gravity thus found to the center of the crosshead-pin hole; this dis- tance = g. Finally, weigh the rod. The product of the weight (= A), the radius of oscillation (== r 0 ), and the distance from the center of crosshead pin (axis of rotation) to the center of gravity (= g) will be the moment of inertia. For, by the formula, I = A g r 0 . The radius of gyration G may be found by the formula G = + or G = j / gr Ma MOMENT OF RESISTANCE. If the moment of inertia of the cross-section of a beam is divided by the distance from the neutral axis (see definition on next page) to the extreme fiber, i. e., the fiber that is far- thest from the axis, the quotient will be the quantity known as the moment of resistance. It is evident that, if a beam is strained by a vertical load, the greatest stress will be in the extreme upper and lower fibers of the beam. MOMENT OF RESISTANCE. 135 The intensity of the stress that can be borne by the extreme fibers is the limit of the strength of the beam. The upper fibers are compressed and the lower fibers are stretched, but somewhere along or near the center of a vertical section of the beam, the fibers are neither extended nor compressed; the position of these fibers is called the neutral surface , and the line where this neutral surface inter- sects a right section of the beam is the neutral axis of the section. The neutral axis passes through the center of gravity of the section. If the moment of resistance is multiplied by the amount of stress that may be allowed per square inch upon the extreme fiber, the product will represent the efficiency of the beam to resist bending moment. Example.— Referring to the 6" I-beam, Figs. 1 and 2, pages 126 and 127, for which the moment of inertia of the section has been found, it is desired to ascertain the load that a w r rought-iron beam of the same dimensions as Fig. 1 will carry at the center of a span 8 ft. between supports. Solution.— The moment of resistance for the section = 23 48 — = 7.83. In Table II, page 151, the ultimate strength or fiber stress for wrought iron is given as 50,000 lb. per sq. in., and in Table I, page 151, the factor of safety given for wrought iron under a steady stress is 4; therefore, the safe fiber stress for wrought iron = ~ = 50, ^ - Q - = 12,500 lb. per sq. in., and the moment of resistance multiplied by the safe CJ D fiber stress, or — = 7.83 X 12,500 = 97,875 in.-lb. But l = 8 ft., or 96 in.; equating the bending moment for a load at the center of a beam (- Wl \ 4 > with the moment of resist- _ __ SR Wl 96 W .. ance, or putting M = — — = — — ; then — — = 97,875; there- 4 4 4 fore, W = 4,078 lb., the load that can be safely supported at the center of the beam. 136 MECHANICS. MECHANICAL POWERS. F : W = l : L. FL = Wl. F = Wl w = FL Fa F = W — F— W' > ' II II & .F12 = TFr. Wr TFr E ~ ~F~' R * RF RF r r = ^F* F = Wrr ' ~RR' ' IF = .F1212' n = number of revolutions of large gear. n : n' = r ' : 22. v : v f = rr':R 12'. v = velocity of TF; v' = velocity of .F. TFrr'r" FRR'R" ~RR'R' r W ~ rr'r'- * n:n" = r' r" : 1222'. v:v' = r r' r" : 12 12' 12". r, r', r", etc. = radii of the pinions; R, 22', 12", etc. = radii of the wheels. MECHANICAL POWERS. 137 Let db and qb represent the magnitudes and direc- tions of two forces that act to move the body b. By completing the parallel- ogram there will be obtained a diagonal force fb , whose magnitude and direction are equal to the effect produced by d b and q b. fb is called the resultant of d b and q b. If three or more forces act in different directions to move a body b, find the resultant of any two of them, and consider it as a single force. Between this and the next force find a second resultant. Thus, pb, qb, and r b are magnitudes and directions of the forces, pb qb + rb = gb + rb = fb, the magnitude and direc- tion of the three forces, pb, qb, and r b. A SINGLE riXED PULLEY. F = W. V = V f . v = velocity of IE; v' — velocity of F. A SINGLE MOVABLE PULLEY. F : W = 1 : 2, or F = £ W. If the force F be applied at a and act upwards, the result will be the same. v' = 2 v. v = velocity of W ; v' = velocity of F. A DOUBLE MOVABLE PULLEY,, F : W = 1 : 4, or F = £ W. Let u = number of parts of rope, not counting the free end. F = W -T- u. v : v' = 1 : u. v = velocity of W ; v' — velocity of F. 138 MECHANICS. QUADRUPLE MOVABLE PULLEY. F = I W. F : W = 1 : 8. Let u = number of parts of rope, not counting the free end; then, F — W H- u. v : v' = 1 : u. v = velocity of W; v' = velocity of F. COMPOUND PULLEY. u = number of movable pulleys. W F = ~ W = 2 U F. 2 “ v : v f = 1 : 2“. v = velocity of IT; v' = velocity of F. DIFFERENTIAL PULLEY. 2 PE W = E-r’ AN OBLIQUE FIXED PULLEY. F : IF = 1 : 2 cos z. w INCLINED PLANE. MECHANICAL POWERS. 139 SCREW. P = pitch of the screw; r = radius on which the force jPacts. F : W : : P: 2 nr. F = WP 2 n r W = 2nr F Work is the overcoming of resistance through a distance. The unit of work is the foot-pound; that is, it equals 1 pound raised vertically 1 foot. The amount of work done is equal to the resistance in pounds multiplied by the distance in feet through which it is overcome. If a body is lifted, the resist- ance is the weight or the overcoming of the attraction of gravity, the work done being the weight in pounds multiplied by the height of the lift in feet. If a body moves in a hori- zontal direction, the work done is the friction overcome, or the force needed to move a resistant body or combination of bodies, multiplied by the distance moved. In order to com- pare the different amounts of work done by different systems of forces, time is also considered. One horsepower is 550 ft.-lb. of work in 1 second, or 33,000 ft.-lb. in 1 minute, or 1,980,000 ft.-lb. in 1 hour. The work necessary to be done in raising a body weighing W lb. through a height of h ft. equals Wh ft.-lb. The total work that any moving body is capable of doing in being Wv* brought to rest equals its kinetic energy, or — — , when v is 2 g the velocity in feet per second. Thus, the work that a cannon ball weighing 800 lb. and traveling with a velocity of 1,200 ft. per sec. could do, is If stopped in 1 min., the horsepower would be 17,910,447 -s- 33,000 = 542.8, nearly. 140 MECHANICS. FORCE OF A BLOW. In order to determine the force of a blow, the velocity of the object at the instant of striking must be known, and also the time required to bring the body to rest. It is a very difficult matter to determine the exact time, but a close approximation to the striking force may be obtained by dividing the kinetic energy of the body at the instant of stri- king by the average amount of penetration or compression produced by the striking body. Let F — striking force in pounds; W = weight of striking body in pounds; v = velocity of striking body in feet per second; R — distance penetrated or amount of compres- sion = the distance through which the resist- ance acts, in feet; t — time required to bring the body to rest; h = height in feet which would produce the veloc- ity v. rm, ^ WV _ WV 2 Wh gt 2 g R R Example.— A steam hammer weighing 1,000 lb. (with its piston) falls from a height of 8 ft., and compresses a piece of iron £ in.; what is its striking force? Solution.— If gravity be considered as the only force acting, the steam on top of the piston being used to prevent a rebound of the hammer, „ Wh 1,000 X 8 f= -r- = — ■ = 1,000 x 8 x 8 x 12 = 768,000 lb. (* + 12 ) Divide | in. by 12, to obtain the amount of compression in feet or parts of a foot. BELTING. D = diameter of larger pulley in inches; d = diameter of smaller pulley in inches; N = revolutions per minute of larger pulley; n = revolutions per minute of smaller pulley; W = width of double belt in inches; w = width of single belt in inches; H = horsepower that can be transmitted by the belt. BELTING. 141 Then, H = — for single belts. 2,750 I) N W II = , for double belts. 1,925 2,750 H 2,750. H w ~~ j)jsf — dn 1,925 H 1,925 H W ~ DN ~ dn ' D = for single belt. wN D = for double belt. N = 2,/50 ^ for single belt. wD 2VT = for double belt. W D The above rules are for open belts and pulleys having the same diameter, the arc of contact being, in this case, half the circumference, or 180°. For open belts and pulleys of different diameters, the arc of contact is less than 180° on the smaller pulley, and a different constant, to be taken from the fol- lowing table, must be substituted in the formulas. To find the arc of contact, let l be the distance in inches between the centers of the pulleys. Then, = cosine of half the angle Find this half angle from a table of natural cosines, and Degrees. Fraction of Circumference. Single Belt Constant. Double Belt Constant. 90 V± = .25 6,080 4,250 11234 A - .3125 4,730 3,310 120 y = .3333 4,400 3,080 135 % = .375 3,850 2,700 150 A = .4167 3,410 2,390 15734 * = .4375 3,220 2,250 180 to 270 34 to % = .5 to .75 2,750 1,925 multiply by 2. The result is the arc of contact in degrees. Find the number in the first column of the table, w T hich is nearest to this result, and use the constant corresponding to 142 MECHANICS. that number. If a table of natural cosines is not at hand, measure the length of the arc of contact on the smaller pulley and divide it by the circumference of the pulley. Find the fraction in the Second column that corresponds nearest to this result, and opposite this its corresponding constant. Example.— What must be the width of a single belt to transmit 12 horsepower, when the diameter of the larger pulley is 42 in., of the smaller pulley 20 in., distance between their centers 14 ft. = 168 in., and R.P.M.of smaller pulley 150? 42 20 Solution.— 2 ^ — = .06548 = cosine of half the arc of contact, which thus = 86° 15', nearly; 86° 15' X 2 = 172£° = arc of contact; the nearest number in the table is 180°, and the corresponding constant is 2,750; hence, w = = 11 in. Oak-tanned leather makes the best belts. When belts are run with the hair side over the pulley, they have greater adhesion. The ordinary thickness of leather belts is ^ in., and their weight is about 60 lb. per cu. ft. Ordinarily, four-ply cotton belting is considered equivalent to single-leather belting. RULES FOR CALCULATING THE SPEED OF GEARS OR PULLEYS. In calculating for gears, multiply or divide by the diameter or the number of teeth, as may be required. In calculating for pulleys, multiply or divide by their diameters in inches. The driving wheel is called the driver , and the driven wheel the driven or follower. Problem I. The revolutions of driver and driven, and the diameter of the driven, being given, required the diameter of the driver. Rule .— Multiply the diameter of the driven by its number of revolutions , and divide by the number of revolutions of the driver. Problem II. The diameter and revolutions of the driver being given, required the diameter of the driven to make a given number of revolutions in the same time. PUMPS. 143 Rule . — Multiply the diameter of the driver by its number of revolutions, and divide the product by the required number of revolutions. Problem III. The diameter or number of teeth, and number of revolu- tions of the driver, with the diameter or number of teeth of the driven, being given, required the revolutions of the driven. Rule . — Multiply the diameter or number of teeth of the driver by its number of revolutions , and divide by the diameter or num- ber of teeth of the driven. Problem IV. The diameter of driver and drwen, and the number of revolutions of the driven, being given, required the number of revolutions of the driver. Rule . — Multiply the diameter of the driven by its number of revolutions, and divide by the diameter of the driver. PUMPS. In all pumps, whether lifting, force, steam, single-acting, double-acting, or centrifugal, the number of foot-pounds of work performed by the pump is equal to the weight of the water discharged in pounds, multiplied by the vertical dis- tance in feet between the level of the water in the well or source and the point of discharge, plus the work done in overcoming the friction and other resistances. (It is assumed that the water is delivered with practically no velocity.) To find the discharge of a pump in gallons per minute: Let T = piston travel in feet per minute; d = diameter of cylinder in inches; G = number of gallons discharged per minute. Then, G = .03264 Td *. To find the horsepower of a pump, use the following formula, in which Tand d are the same as above, and h is the vertical distance in feet between the level of the water at the source and the point of discharge: H. P. = .00033724 Gh = .00001238 Td* h. Jn both the above formulas, allowance has been made ior friction, leakage, etc. 144 MECHANICS. DUTY. The duty of a pump is the number of foot-pounds of work actually done for 100 lb. of coal burned. Duty = 835.53 w where W = weight of coal burned, in pounds. HYDROMECHANICS. HYDROSTATICS. Hydrostatics treats of liquids at rest under the action of forces. If a liquid is acted on by a pressure, the pressure per unit of area exerted anywhere on the mass of liquid is trans- mitted undiminished in all directions, and acts with the same force on all surfaces, in a direction at right angles to those surfaces. General Law for the Downward Pressure on the Bottom of Any Vessel.— The pressure on the bottom of a vessel containing a liquid is independent of the shape of the vessel, and is equal to the weight of a prism of the liquid whose base is the same as the bottom of the vessel, and whose altitude is the distance between the bottom and the upper surface of the liquid, plus the pressure per unit of area upon the upper surface of the liquid multiplied by the area of the bottom of the vessel. General Law for Upward Pressure.— The upward pressure on any submerged horizontal surface equals the weight of a prism of the liquid whose base has an area equal to the area of the submerged surface, and whose altitude is the distance between the submerged surface and the upper surface of the liquid, plus the pressure per unit of area on the upper surface of the liquid multiplied by the area of the submerged surface. General Law for Lateral P ressu re. -^The pressure on any ver- tical surface due to the weight of the liquid is equal to the weight of a prism of the liquid whose base has the same area as the vertical surface, and whose altitude is the depth of the center of gravity of the vertical surface below the level of the liquid. Any additional pressure is to be added, as4n the previous cases. HYDROMECHANICS. 145 Pressure on Oblique Surfaces.— The pressure exerted by a liquid in any direction on a plane surface is equal to the weight of a prism of the liquid whose base is the projection of the surface at right angles to the given direction, and whose height is the depth of the center of gravity of the surface below the level of the liquid. If a cylinder is filled with water, and a pressure applied, the total pressure on any half section of the cylinder is equal to the projected area of the half cylinder (or the diameter multiplied by the length of the cylinder) multiplied by the depth of the center of gravity of the half cylinder, multiplied by the weight of a cubic inch of water, plus the diameter of the shell, multiplied by the pressure per square inch, multi- plied by the length of the cylinder. If d = the diameter, and l = the length of the cylinder, the pressure due to the weight of the water when the cylinder is vertical upon the half cylinder = d X l X ^ X the 1 2 weight of a cubic inch of water = d X X the weight of a cubic inch of water; d and l are to be measured in inches. The pressure in pounds per square inch due to a head of water is equal to the head in feet multiplied by .434. The head equals the pressure in pounds per square inch multiplied by 2.304. Example.— (a) What is the pressure per square inch cor- responding to a head of water of 175 ft. ? (6) If the pressure had been 90 lb. per sq. in., what would the head have been? Solution.— (a) 175 x .434 = 75.95 lb. per sq. in. (5) 90 X 2.304 = 207.36 ft. HYDROKINETICS. Hydrokinetics , also called hydrodynamics and hydraulics, treats of water in motion. When water flows in a pipe, con- duit, or channel of any kind, the velocity is not the same at all points of the flow, unless all cross-sections of the pipe or channel are equal. That velocity which, being multiplied by the area of the cross-section of the stream, will equal the total quantity discharged, is called the mean velocity. 146 MECHANICS. Let Q = quantity that passes any section in 1 second; A = area of the section; v = mean velocity in feet per second. Then, Q = Av, and v = The vertical distance between the level surface of the water and the center of the aperture through which it flows, is called the head. Let V = mean velocity of efflux through a small aperture; h = head in feet at the center of the aperture; tv = weight of water flowing through the aperture per second. Then, V = 1/2 g h; that is, the, velocity of efflux is the same as if the water had fallen through a height equal to the head. Let Q = theoretical number of cubic feet discharged per second; V m = mean velocity through orifice in feet per second; A = area of orifice; h = theoretical head necessary to give a mean velocity V m ; Q a = actual quantity/discharged in cubic feet per second. Then, for an orifice in a thin plate, or a square-edged orifice (the hole itself may be of any shape, triangular, square, circular, etc., but the edges must not be rounded), the actual quantity discharged is Q a = .615 Q = .615 A V m . The weir is a device used for measuring the discharge of water. It is a retangular orifice through which the water flows. If d = the depth of the opening in feet, and b its breadth in feet, the area of the opening is A = d X b, and the theo- retical discharge is Q = dxb X V m = db X \ V % 9 d, the nead for this case being taken as d. The actual discharge when the top of the weir lies at the surface of the water is . Q a = .615 Q =. 615 Xd5Xf l/27d = .615 XI & l/ 2gd 3 = 3.2885 1 / d 3 . HYDROMECHANICS. 147 If h\ is the depth in feet of the top of a weir below the surface of the water, and h is the depth in feet of the bottom of the weir below the surface of the water, the actual dis- charge Q ay in cubic feet per second, is Q a = .615X1 b V^9WW-VW) = 3.288 b FLOW OF WATER IN PIPES. Let V m = mean velocity of discharge in feet per second; h — total head in feet = vertical distance between the level of water in reservoir and the point of discharge; l = length oDpipe in feet; d = diameter or pipe in inches; f = coefficient of friction. Then, for straight cylindrical pipes of uniform diameter, the mean velocity of efflux may be calculated by the formula, V m = 2.315 - ' Vj hd ' fl +.125 d‘ (a) Note.— The head is always taken as the vertical distance between the point of discharge and the level of the water at the source, or point from which it is taken, and is always measured in feet. It matters not how long the pipe is— whether vertical or inclined, whether straight or curved, nor whether any part of the pipe goes below the level of the point of discharge or not— the head is always measured as stated above. Example.— What is the mean velocity of efflux from a 6" pipe, 5,780 ft. long, if the head is 170 ft. ? Take / = .021. Solution — V m = 2.315 yj fl + md = 2.315 ^ 021 x 5,730 + (.125 X =^= 6.69 ft. per sec. When the pipe is very long compared with the diameter, as in the above example, the following formula may be used: y V m = 2.315 "y jy’ Jb) in which the letters have the same meaning as in the prece- ding formula. This formula may be used -when the length of the pipe exceeds 10,000 times its diameter. 148 MECHANICS. The actual head necessary to produce a certain velocity V m may be calculated by the formula + - 0233 v * (0 If the head, the length of the pipe, and the diameter of the pipe are given, to find the discharge, use the formula « = ' OS 445 <* 2 V/ITl25d : W that is, the discharge in gallons per second equals .09445 times the square of the diameter of the pipe in inches, multi- plied by the square root of the head in feet, multiplied by the diameter of the pipe in inches, divided by the coefficient of friction times the length of the pipe in feet, plus .125 times the diameter of the pipe in inches. To find the value of /, calculate V m by formula ( b ) assu- ming that / = .025, and get the final value of * from the following table: V m / V m f V m / .1 .0686 .7 .0349 2 .0265 .2 .0527 .8 .0336 3 .0243 .3 .0457 .9 .0325 4 .0230 .4 .0415 1 .0315 6 .0214 .5 .0387 IK .0297 8 .0205 .6 .0365 i % .0284 12 .0193 Example.— The length of a pipe is 6,270 ft., its diameter is 8 in., and the total head at the point of discharge is 215 ft. How many gallons are discharged per minute ? Solution.— V m = 3-315 ^7 025X^,270 = 7 ' 67 ft ' PCT SeC " nearly ' Using the value of / = .0205 for V m = 8 (see table), Q = 21 5 V 8 ^05 X 6,270 +(.125 X 8) = 22 -° 3 gal ' P6r ^ = 22.03 X 60 = 1,321.8 gal. per min. If it is desired to find the head necessary to give a discharge of a certain number of gallons per second through a pipe HYDROMECHANICS. 149 whose length and diameter are known, calculate the mean velocity of efflux by using the formula V m = 24.51 Q . d? («) findthe value of /from the table, corresponding to this value of V mi and substitute these values of / and V m in the formula for the head. Example.— A 4" pipe, 2,000 ft. long, is to discharge 24,000 gal. of water per hr.; what head is necessary? 24,000 _ _ TT _ 24.51X61 Solution.— 60X60 = 6f gal. per sec. V m 42 = 10.2 ft. per sec. From the table, / = .0205 for V m = 8, and .0193 for V m = 12; assume that / = .02 for V m — 10.2. Then, h = - 2 - X ^^?- 22 + .0233 X 10.22 = 196.53 ft. To find the diameter of a pipe that -will give any required discharge in gallons per second, the total length of the pipe and the head being known, find the value of d by formula (/); substitute this value in formula (e), and find the value of V m . Then find from the table the value of f corresponding to this value of V m . Substitute the values of d and f just found in the right- hand member of formula (g) and solve for d; the result will be the diameter of the pipe , accurate enough for all practical purposes. d -1.229^f. 00 d = 2.57^p±M^. (,) Example.— A pipe 2,000 ft. long is required to discharge 24,000 gal. of water per hr. The head being 195 ft., what should be the diameter of the pipe? Solution. — Q = = 6 § S al * P er sec - Substitu- ting in formula (/), d = 1.229^j 2,000 ^ - - — = 4.18 + in. Substituting this value in formula (e), V m = — = 9.352 ft. per. sec. From the table, the value of/ for V m = 9.352 is .0201. Substituting this value of / and the value of d, found above, in formula (g), S _ O /( 0201 x 2,000 -fix 4.18) X (6f )2 150 STRENGTH OF MATERIALS. STRENGTH OF MATERIALS. The ultimate strengths of different materials vary greatly from the average values given in the following tables. In actual practice, the safest procedure would be to make a test of the material for its ultimate strength and coefficient of elasticity, or else specify in the contract that it shall not fall below certain prescribed limits. In the following formulas, A = area of cross-section of material in square inches; E = coefficient of elasticity in pounds per square inch; O 2 = square of least radius of gyration; I — moment of inertia about an axis passing through the center of gravity of the cross-section; M = maximum bending moment in inch-pounds; P = total stress in pounds; P = moment of resistance; S = ultimate stress in lb. per sq. in. of area of section; W = weight placed on a beam in pounds; b = breadth of cross-section of beam in inches; d = depth of beam (in.) = diam. of circ. section = alti- tude of triangular section = length of vertical side; e = amount of elongation or shortening in inches; / = factor of safety; l = length in inches; p = pressure in pounds per square inch; 7 r == ratio of circumference to diameter = 3.1416, nearly; q = a constant used in formula for columns; r = radius of a circular section; s = elastic set or deflection in inches of a beam under a . transverse (bending) stress; t — thickness of a shell or hollow section. For tension, compression (where the piece does not exceed 10 times its least diameter), and shear, To find the breaking stress (P), make/ = 1. For safe load, take / from Table I, and S from Table II, according to the nature and character of stress. STKENGTH TABLES. 151 TABLE I. Factors of Safety (/). Name of Material. Steady Stress. Varying Stress. Shocks (Ma- chines). Cast iron 6 15 20 Wrought iron 4 6 10 Steel 5 7 15 Wood 8 10 15 Brick and stone 15 25 30 TABLE II. Ultimate Strengths (S). Name of Material. Tension. Com- pression. Shear. Flexure. Cast iron Wrought iron Steel Wood Stone 20,000 50.000 100,000 10.000 90.000 50.000 150,000 8,000 6,000 2,500 20,000 47.000 70.000 600 to 3,000 36.000 50.000 120,000 9.000 2.000 Brick 200 Example— A square cast-iron pillar 18 in. long is required to sustain a steady load of 75,000 lb.; what must be the length of a side ? Solution.— From the table, / = 6, and 5 = 90,000. By formula (1), AS f ' Pf S 75,000 X 6 90,000 5 sq. in. Length of side = j/ 5 = 2.236 in., say 2| in. The amount of elongation or of shortening of a piece under a stress is given by the formula e PI AE' ( 2 ) The coefficient of elasticity ( E ) must be taken from the following table: 152 STRENGTH OF MATERIALS. TABLE III. Name of Material. Coefficient of Elasticity. Elastic Limit for Tension. Cast iron 15,000,000 6,000 Wrought iron 25,000,000 25,000 Steel 30,000,000 50,000 Wood 1,500,000 3,000 A wrought-iron bar 24 ft. long, 1£ in. in diameter, would elongate, under a tensile stress of 15 tons, (15 X 2,000) X (24 X 12) _ i 77 (H)2 X 25,000,000 To find the breaking strength of a beam , use the formula M = SR. (3) Obtain M and R from the two following tables, according to the kind of beam and nature of cross-section. A simple beam is one merely supported at its ends. In the expression for R, d is always understood to be the vertical side or depth; hence, that beam is the stronger which always has its greatest depth or longest side vertical. The moment of inertia I is taken about an axis perpendicular to d, and lying in the same plane. TABLE IV. Kind of Beam and Man- ner of Loading. Bending Moment. M Cantilever, load at end Wl Cantilever, uniformly loaded A Wl Simple beam, load at mid- dle A Wl Simple beam, uniformly loaded Vs Wl Beam fixed at both ends, load at middle Vs Wl Beam fixed at both ends, uniformly loaded A Wl Deflection. s STRENGTH OF MATERIALS. 153 TABLE V. Name of Section. I R £2 Solid circular 7i d 4 nd* 32 d 2 16 Hollow circular 7r (d4_^ 1 4) 64 n(d^-d^) 32 d d 2 + d x 2 16 Solid square Hollow square d 4 12 d 4 — d x 4 12 d 3 6 d 4 — d x 4 6d d 2 12 d 2 + d x 2 12 Solid rectangular Hollow rectangular 5d 3 12 fcd 2 6 b 2 12 ftdS-M ! 3 12 bd 3 —bidi 3 6d 6 3 d - &i 3 di 12 (6d — 6idi) Solid triangular fcd 3 36 fed 2 24 d 2 18 Solid elliptical nbd 3 64 _jr6d2 32 16 Hollow elliptical I-beam Cross with equal arms (approxi- mate) Angle with equal arms (approxi- mate) ^(&d 3 -Mi 3 ) 7r(6d 3 -6idi 32d 3 6 3 d - b^di 16(bd — 6idi) 6d 3 -6 1 d 1 3 12 6d 3 -6 x di 3 6d 6 3 d - &! 3 d! 12 (6d — Mi) d 2 22.5 d 2 25 154 STRENGTH OF MATERIALS. Thus, the breaking strength of a cast-iron simple beam uniformly loaded and 20 ft. long between the supports, hav- ing a hollow rectangular cross-section 8 in. by 6 in. outside and 6 in. by 4 in. inside, is given by the formula M = SR, or \Wl = 36,000 X — . &1 ^ o CL Using a factor of safety of 6, the beam should support 55,200 - = 9,200 lb. with perfect safety. The value of S for beams should be taken from the flexure column of Table II. To find the amount of deflection in a beam due to a load, sub- stitute the values of W f l, E, and I in the different expres- sions for the deflection s in Table IV. The value of I is to be taken from Table V. Example.— What is the deflection of a wrought-iron beam fixed at both ends, 7 ft. long between the supports, having a solid rectangular cross-section 6 in. wide and 2£ in. deep, carrying a load of 21,000 lb. in the middle ? Solution.— F rom the table, WP WP 21,000 X (7 X 12) 3 X 12 _ S 192 El 6d3 192 X 25,000,000 X 6 X (2£) 3 * 192 Ji. X 12 Example.— It is desired to calculate the depth (d) of a cast-iron cantilever 36 in. in length (= l) that will sustain at its end a weight of 4,000 lb. (= W), the lever to be of rect- angular section and 2 in. in width. Solution.— The ultimate stress per square inch for cast iron in flexure is given in Table II as 36,000 lb. (=5). The weight will be a steady load, and therefore, according to Table I, a factor of safety of 6 should be used. By for- mula (3), M = SR. For a cantilever beam carrying a load at the end, M = Wl (Table IV); and for a rectangular sec- tion, R = (Table V). D Then, as W = 4,000, l = 36, b = 2, / = 6, we have STRENGTH OF BEAMS. 155 The value of d is found by substituting in this equation the known values of S, b , W, l, and/, as follows: v d = 4,000 X 36; whence, d = 8.49 in. o X o At the point where the beam is supported, the required depth is found to be 8.49, or, practically, 8£ in. At a point 6 in. from the support, the depth may again be calculated by substituting in the equation the value of l (the overhanging length beyond this point); l = 30, and the equation becomes a xg . 4,000 X 80. b X o d = 7.75 in. At a point 12 in. from the support, l = 24, and 36,000 X 2 X d? 6X6 = 4,000 X 24; whence, d = 6.93 in. At a point 18 in. from the support, l = 18; and from the equation, d = 6 in.; at 24 in. from the support, l = 12 and d = 4.9 in.; at 30 in. from the support, 1=6 and d = 3.46 in.; at 36 in. from the support, or at the end of the beam, l = 0 and d = 0. The depths required to be given to the lever or beam at the point of support and at intervals of 6 inches along its length, are found to be 8.49, 7.75, 6.93, 6, 4.90, and 3.46 inches, respectively. The lever is shown in the figure; theoretically, it would taper to nothing at the end, as indicated by dotted lines, but practically sufficient metal must be added at that point to provide means of attaching the weight. 156 STRENGTH OF MATERIALS. Note.— I n the preceding examples the weight of the beam has been neglected. If, however, this weight is large in com- parison with the weight or weights carried by the beam, it should be taken into account, considering it (when the cross- section of the beam is the same throughout) as a load uni- formly distributed over the whole length of the beam. COLUMNS. To find the breaking strength of a column, use the follow- ing formula: P = SA V * 1 + q GP ( 4 ) S is taken from Table II, in the column for compression, G 2 from Table V, and q from the following table, according to the character of the ends. TABLE VI. Material. Both Ends One End Both Ends Flat or Fixed. Round. Round. Cast iron 1 1.78 4 5,000 5,000 5,000 Wrought iron 1 1.78 4 36,000 36,000 36^000 Steel 1 1.78 4 25,000 25,000 25,000 Wood 1 1.78 4 3,000 3,000 3,000 The breaking load of an elliptical wooden column 18 ft. long, having rounded ends, the diameters of the cross-section being 12 in. and 8 in., is SA _ 8,000 X (i it X 12 X 8) P = 4 (18 X 12) 2 3,000 8 2 = 36,442 lb. 16 Using a factor of safety of 8, the column should support 36,44 2 8 ! = 4,565 lb. with perfect safety. SHAFTING. 157 SHAFTING. The diameter of a shaft may be found by the following formulas. The first is used when great stiffness is required, and the shafts are very long; the second when strength only is required to be considered. d = diameter of shaft in inches; H = horsepower transmitted; N = number of revolutions per minute; c = constant in formula (5); k = constant in formula (6). c = 5.26 for cast iron; 4.75 for wrought iron; 3.96 for steel; k = 4.02 for cast iron; 3.63 for wrought iron; 3.03 for steel. Note.— T o extract the fourth root, extract the square root twice. p = pressure in pounds per square inch; d = diameter of pipe or cylinder in inches; t = thickness in inches; S = ultimate tensile strength taken from Table II; r = inside radius in inches; / = factor of safety, usually taken as 6 for wrought iron and 12 for cast iron. For thin pipes, pdf = 2 tJS. (7) For thick pipes or cylinders, D = diameter of the rope in inches = diameter of iron from which the link in chain is made; W = safe load in tons of 2,000 lb. For common hemp rope, W = £ D 2 . For iron-wire rope, W = § D 2 . For steel-wire rope, W = D 2 . For close-link wrought-iron chain, W — 6 D 2 . For stud-link wrought-iron chain, W = 9 D 2 . (5) d = k^j§. (6) PIPES AND CYLINDERS. ROPES AND CHAINS. 158 BOILERS. BOILERS. BOILER DESIGN. TO DEVELOP THE DOME OF A BOILER. A side view of the dome, together with a section of the boiler, is shown in Fig. A. Draw Fig. B, the end view of the dome and of the boiler. Above the dome draw a circle ine" m of the same diameter as the dome. Divide the lower half of this circle, as n e" m, into any number of equal parts, as me", c" d", d" e", e"f", and/" g". The greater the num- ber of these divisions, the more accurate will be the results. From the points of division c", d ", e",/", and g", draw lines parallel to the vertical center line of the boiler, as c" c', d" d\ /"/', and g" g'. We are now ready to draw the templet of the dome, as shown in Fig. C. Draw a straight line of indefinite length, and on it lay off a distance h i equal to the circumference of BOILER DESIGN. 159 the dome. (The circumference of the dome is found by multiplying the diameter a b of the dome by 3.1416.) Divide the distance h i into twice the number of equal parts that the semicircle above the dome in Fig. B has. In the figure it has been divided into 6 equal parts; therefore, divide this line into 2 X 6 = 12 equal parts, as bg, gf, fe , ed , etc., and through these points of division draw lines at right angles to the line hi, as shown; make the length of each of these lines the same as the length of the line that corresponds to it in Fig. B. Thus, e e' is equal to e e' in Fig. B, d d ' is equal to d d' in Fig. B, a a' is equal to a a' in Fig. B, etc. After hav- ing laid off the lengths of these lines, draw the curved line i' c' h'. This being done, we have the templet of the dome on the seam. The lap for riveting must be allowed, as shown by the dotted lines around the templet. TO DEVELOP THE SLOPE SHEET abed OF A BOILER, SHOWN AT A IN THE FIGURE BELOW. Draw a straight line a b, as shown in Fig. B, and on it lay off the distance ad, equal to be, Fig. A. At a and d, erect perpendiculars a c and d e , respectively, making a c equal to b a, and d e equal to c d, of Fig. A. With a point 6 on a 6 as a center, and a radius d e, describe the quadrant/ g. Divide this quadrant into any number of parts; the greater the number, 160 BOILERS. the more accurate will he the results. Here it is divided into three, as g-i, 1-2, and 2-f. Through the points g, i, and 2, draw lines parallel to a o, intersecting the perpendicular d e in e, 1', and 2', and the perpendicular bginh and i. Througn the points V, 2 ', and d, draw lines parallel to c e. Through any point, as J , on the line ce, draw JK perpendicular to ce, cut- ting the lines 2"-2', and 3"-d in the points i, n , and K , respectively. From the line J K lay off the distances i m, n o, and Kp, equal to the distances hi, i 2, and bf, respectively, and pass the dotted curve Jmop through the points. Now draw Fig. C. Draw the straight line kq, and through the point J draw ec perpendicular to it. Lay off on the line kq, on each side of the line c e, points ra' and m' at distances from it equal to the length of Jm in Fig. B. Lay off, also, points o' and o' at distances from m' and m' equal to m o in Fig. B\ also, points p' and p' at distances from o' and o' equal to op of Fig. B. Through the points thus laid off, draw lines parallel to c e. Lay off the distances J c and J e from J, in Fig. C, equal to Jc and Je , respectively, in Fig. B\ the dis- tances m' V" and m' 1" from m' equal to i 1 " and i V in Fig. B; o' 2"' and o' 2" from o' equal to n2" and n2 '; and p f 3'" and p' 3" from p' equal to K3" and Kd of Fig. B. Through the points thus laid off draw the curved lines S'" c3'" and 3" e 3". With the points 3" as centers and a radius a d, Fig, B, describe the arcs r and r. With the points S'" as centers and a radius 3" a, Fig. B, describe the arcs s and s. From the points of intersection of these arcs, draw lines to the points 3'" and 3". This being done, we have the templet of the slope sheet on the seams. The laps for rivet- ing must be allowed as shown by the dotted lines around the templet. _____ TO DEVELOP THE SLOPE SHEET ImnO OF A BOILER, SHOWN AT A IN THE FIGURE ON THE FOLLOWING PAGE. Draw the two views of the sheet as shown in Figs. B and C. Suppose the seam to be at o n, Fig. A, and the sheet to be made in one piece. Divide the semicircles a dg and a' d'g', Fig. C, into any number of equal parts; the greater the number BOILER DESIGN. 161 of these divisions, the more accurate will be the results. Join the points b and b', c and c', d and d f , e and e', and / and f by full lines, and join the points b and a', c and b ', d and c', e and d', / and e', and g and f by dotted lines, as shown. Then draw Figs. D and E. Draw at right angles to one another the lines wa and wx , also the lines za' and zy. Make the length of the line wx equal to r, Fig. B, and the length of the line w a equal to a a', Fig. C. From w lay off on the line wa, Fig. D, distances wb, w c, w d, w e, w f, and wg, respectively, equal to the lengths of the full lines bb', cc\ etc. of Fig. C, and draw the lines ax, bx, cx, dx, ex, fx, and gx, as shown. Make the length of the line zy, Fig. E, the same as that of w x, Fig. D. From z lay off on the line z a\ 162 BOILERS. Fig. E, distances za zb zc\ zd', ze ', and zf, respectively, equal to the lengths of the dotted lines ba', cb', etc., in Fig. C, and draw the lines a' y , b ’ y, c' y, f y, d ' y, and e' y. We are now ready to draw the templet of the slope sheet. Instead of drawing the whole templet, we will draw only one-half of it, as is shown in Fig. F, since the other half is exactly the same. Draw the line a a', and make it equal in length to the distance ax, Fig. D. With a' as a center, and a radius ya', Fig. E, describe an arc at b. With a as a center and a radius = arc a b, Fig. C, describe another arc inter- secting the first arc in b. With a' as a center, and a radius = arc a' b', Fig. C, describe an arc at b'. With b as a center, and a radius x b, Fig. D, describe an arc, intersecting the arc already drawn, at b'; draw the full line bb ' and dotted line b a'. With b' as a center, and a radius y b', Fig. E, describe an arc at c. With & as a center, and a radius = arc c b, Fig. C, describe an arc cutting the last arc at c. With b ' as a center, and a raditis = arc c'b ', Fig. C, describe an arc at c'. With c as a center, and a radius x c, Fig. D, describe an arc cutting the last arc at c'; draw the full line cc’ and dotted line cb'. Continue to construct the remaining portion of the half templet in a similar manner, taking the distances for the full lines from Fig. D, and those for the dotted lines from Fig. E. Through the points a, b, c, d , e, /, and g, and through the points a', b', c', d', e', f, and g', draw the curved lines shown. Since this is the development of the slope sheet at the seam, the laps for riveting must be allowed; they are shown by the dotted lines around the templet in Fig. F. CARE AND INSPECTION OF BOILERS. POINTS TO BE OBSERVED. Preliminary to a boiler inspection, the boiler, flues, mud- drum, ash-pit, and all connections should be thoroughly cleaned, to facilitate a careful examination. Blisters may occur in the best iron or steel, and their presence, and also that of thin places, is ascertained by going over all parts ot the boiler with a hammer. When blisters are discovered, the plates should be repaired or replaced. Repairing a blister CARE OF BOILERS. 163 consists in cutting out the blistered space and riveting a “hard patch” over the hole on the inside of the boiler, if possible, to avoid forming a pocket for sediment. All seams, heads, and tube ends should be examined for leaks, cracks, corrosions, pitting, and grooving, detection of the latter possibly requiring the use of a magnifying glass. Uniform corrosion is a wasting away of the plates, and its depth can be determined only by drilling through the plate and measuring the thickness, afterwards plugging the hole. Pitting is due to a local chemical action, and is readily perceived. Grooving is usually due to buckling of the plates when under pressure, and frequently to the careless use of the sharp calking tool. Seam leaks are generally caused by overheating, and demand careful examination, as there may be cracks under the rivet heads. If such cracks are discovered, the seam should be cut out, and a patch riveted on. Loose rivets should be carefully looked for, and should be cut out and replaced, if found. Pockets, or bulging, and burns should be looked for in the firebox. The former are not necessarily dangerous, but if there are indications of their increasing, they should be heated and forced back into place or cut out and a patch put on. Burns are due to low water, the presence of scales, or to the continuous action of flames formed on account of air leaking through the brickwork. The burned spots should be cut out and patched as previously described. The conditions of all stays, braces, and their fastenings should be examined and defective ones replaced. The shell of the boiler should be thoroughly examined externally for evidences of corrosion, which is liable to set in on account of dampness, exposure to weather, leakage, etc., and may be serious. The boiler should be so set that joints and seams are accessible for inspection, and should have as little brickwork in contact with it as possible. The brickwork should be in good condition, and not have air holes in it, since they decrease the efficiency of the boiler and are liable to cause injury to the plates by burning, as above explained, and also by unevenly heating and distorting them. The mud-drum and its connections are liable to corrosion, pitting, and grooving, and should be examined as carefully as the boiler. 164 BOILERS. All valves about a boiler should be easy of access, and should be kept clean and working freely. Each boiler should have at least three gauge-cocks, properly located, and it is of the utmost importance that they be kept clean and in order, and the same may be said of the glass water gauge. The middle gauge-cock should be at the water level of the boiler, and the other two should be placed one above and one below it, at a distance of about 6 in. The condition of the pumps or injectors should be looked into to make sure that they are in the best working order. The steam gauge should be tested to ascertain that it indicates correctly, and if it does not, it should be corrected. If the hydraulic test is to be used, the boiler should be tested to a pressure of 50fo higher than that at which the safety valve will be set. External Inspection When Boiler Is Under Steam.— The gauge- cocks, and also the gauge glass, should be tried, to make sure that they are not choked. The steam gauge should be taken down, if permissible, and tested, and corrected if necessary. The gauge pointer should move freely. Blowing out the gauge connection will show whether it is clear or not. The boiler connections should be examined for leaks. The safety valve should be lifted from its seat, to make sure that it does not stick from any cause, and it should be seen that the weight is in the right place. Observe from the steam gauge if the valve blows off at the pressure it is set for. See that all pumps and feed-apparatus are working properly, and that the blow-off and check-valves are in order. Blisters and bagging may sometimes be detected in the furnace. The condition of the brickwork is of considerable importance, since the existence of air holes is a source of trouble, as' already explained. Incrustation. — One of the chief sources of trouble to the boiler user is that of incrustation. All w T ater is more or less impure; and as the water in the boiler is continuously evapo- rated, the impurities are left behind as powder or sediment. This collects on the plates, forming a scaly deposit, varying in nature from a spongy, friable texture to a hard, stony one. This deposit impedes the transmission of heat from the plates CARE OF BOILERS. 165 to the water and often causes overheating and injury to the plates. It is probable that ^ in. of scale necessitates the con- sumption of 12$ to 20$ more fuel. The various impurities in the water may be either in suspension or solution. If the former, the water can be purified by filtration before going into the boiler. If the latter, the substances must first be precipitated and then filtered. Many impurities (sulphate and carbonate of lime, etc.) may be removed by heating the water before feeding it into the boiler. The first thing to do, when dealing with a water supply, is to have an analysis of it made by a competent chemist. The fact that a water contains a certain amount of solid matter is no criterion as to its unfitness for boiler use. The presence of certain salts, as carbonate or chloride of sodium, even in large quantities (say 40 to 50 gr. per gal.), would not be serious if due attention were given to the blowing off. On the other hand, salts of lime in the above proportion would be very objectionable, requiring greatly increased attention in the matter of purification and blowing off or else cleaning out. The various methods of dealing with impure water may be classed as follows: 1. Filtration .— Where the matter (sand, mud, etc.) is held in suspension, it can be removed, before the water enters the boiler, by the aid of settling tanks or by filtering, or by forcing the water up through layers of sand, broken brick, etc., or by using filtering cloths in a proper machine. 2. Chemical Treatment. — Clark’s process, combined with a subsequent filtration (the joint process being known as the Atkins system), has been successfully applied on both small and large scales in the chalk districts of England. Lime water is mixed with the water to be purified, the amount used depending on the composition of the water, as deter- mined by a careful analysis. The lime is thus precipitated, and the water is then filtered in a machine containing travel- ing cotton cloths. Not only is the carbonate of lime entirely removed, but it has been proved that any sulphate of lime that may be present is also prevented from incrusting. This is important, as the latter impurity forms, perhaps, the "worst scale one has to contend with. 166 BOILERS. Various chemical compounds are in use for boilers. Car- bonate of soda is perhaps the best general remedy. It forms the basis, in fact, of nearly all boiler compounds, whatever their name or appearance. This soda deals efficaciously both with the carbonate and the sulphate of lime. The precipi- tates thus thrown down do not form a hard crust; they can be washed out in the form of sludge or mud. Carbonate of soda is also useful where condensers are employed, as it counteracts the effect of the grease, which is brought over with the exhaust steam. If used in too large quantities, it will cause priming. The best way to use it is to make a solution of it and connect with the feed, fixing a cock so as to regulate the amount fed in. Soda ash is cheaper, but more of it is required, and, besides, it is generally impure. Caustic soda removes lime scale quicker than ordinary soda does, but it is much stronger and liable to attack the plates. It should be used in smaller quantities than the ordinary kind. Barks, molasses, vinegar, etc. develop acids that attack the plates. Animal and vegetable oils do the same, and also harden the deposits and make their removal more difficult. It is a good rule to keep all animal and vegetable matter out of boilers altogether. Feed-Water Heaters .— Carbonates and sulphates of lime are precipitated by high temperatures. The heaters should be arranged so that the deposit forms chiefly on a series of plates that can be easily removed for cleaning. If -the deposit gathers in pipes, however, it is simply transferring the evil from one vessel to another. A double advantage is gained by these heaters, for the feedwater is put into the boiler already heated, and so fuel is saved. Mechanical Aids.— Deposits take place chiefly in sluggish places. Various devices to aid circulation have been brought out. With good attention and a not too impure water, they give satisfactory results. Potatoes, linseed oil, molasses, etc. are sometimes put into the boiler with the idea of lessening scale formation, by form- ing a kind of coating round the particles of solid matter and so preventing their adhering together. This certainly takes place, but the substances are injurious, as already pointed out. CARE OF BOILERS. 167 Whenever a boiler has been cleaned out, we may with advan- tage give the inside a thin coating of oil, or tallow and black lead; this arrests the incrustation to a great extent. Sand, sawdust, etc. are often used, the idea being that their grains act as centers for the gathering together of the solid matter in the water, the resulting small masses not readily collecting together themselves and therefore being easily washed out. This may be so, but the cocks, valves, etc. are liable to suffer from the practice. Kerosene is strongly recommended by some boiler users. There is no doubt that in many cases its use has given good results. It prevents incrustation, by coating the particles of matter with a thin covering of oil, the deposit thus formed being easily blown out. The oil also seems to act on the scale already formed, breaking it up and thus facilitating its removal. As already remarked, it is a good plan, when the boiler is empty, to give the inside a good coating of this oil, afterwards putting it in with the feed, the supply being regu- lated automatically. As to the quantity required, this will be found to vary in different cases, according to the nature of the water; an average of 1 qt. per day for every 100 horse- power will give good results in most cases. In marine boilers, strips of zinc are often suspended; the deposit largely settles on them instead of on the boiler plates. Also, any scale that may be formed on the latter is less hard and compact and more easily broken up. Further, any acids formed by the oil and grease brought over from the con- denser attack this zinc instead of the boiler plates. Miscellaneous Acids are often introduced into boilers to dissolve the scale already formed, the solid matter then being washed out. This treatment should be adopted with great care, if at all, as the plates are likely to be affected. Scale is often loosened and broken up by deliberately inducing sudden expansion or contraction in the boiler. In the former case, the expansion is brought about by blowing off the boiler, and then, when it is quite cooled down, turn- ing on steam at as high a temperature as obtainable, thus causing the scale to expand more quickly than the plates and thus become loose. 168 BOILERS. In the second method, the boiler is blown off when the steam (and therefore the temperature) is at its highest and a stream of cold water then turned in. The fires are then drawn and the fire-hole doors, dampers, etc. opened, letting in a rush of cold air. All this cools the plates and, by the contraction thus brought about, loosens the scale. These two practices should be guarded against. Foaming or priming is usually due either to forcing a boiler beyond its capacity for furnishing dry steam, or to the presence of foreign matter. It is dangerous if occurring to any great extent, since water may be carried along with steam into the engine, and a cylinder head knocked out. Foaming, when it cannot be checked by the use of the sur- face blow-out apparatus, may necessitate the emptying of the boiler, which must then be filled with fresh water; this rids the boiler of the impurities thdt have collected during the operation of the boiler. HORSEPOWER OF BOILERS. In actual practice, the result of a great many tests has shown that an evaporation of 30 lb. of water per hr. from a feedwater temperature of 100° F. into steam at 70 lb. gauge pressure is the equivalent of 1 horsepower, or that this steam, in a properly designed engine, will do the equivalent of 33,000 X 60 = 1,980,000 ft.-lb. of work per hr. In order, how- ever, to have a more ready standard of comparison, the above evaporation has been reduced to another standard, and is found to be equal to the evaporation of 34.5 lb. of water from and at a temperature of 212° F. under atmospheric pressure, and it is on this latter quantity that the calculations of the horsepower of boilers are usually based. In making an approximation of the horsepower of a given boiler, the square feet of water-heating surface of the boiler should first be determined, and in doing this the area of all the surfaces exposed to the fire and hot gases, which, on their opposite sides come in contact with the water in the boiler, should be taken into account. Example.— A n externally-fired flue boiler, having a shell 38 in. in diameter, and containing two flue pipes 10 in. in HORSEPOWER OF BOILERS. 169 / diameter, is 22 ft. long without the smokebox. If the greatest depth of the water in the boiler is f X 38 = 25.33 in., what is the total water-heating area of the boiler ? Solution. — Six feet of the circumference of the boiler shell lies below the water-line, as could be found by actual measurement, and the circumference of the two flues is equal to ( 1^1416 ) x2 = 5 . 24ft . Therefore, the water-heating surface of the shell is 6 X 22 = 132 sq. ft., and that of the flues is 5.24 X 22 = 115.28 sq. ft. The w’ater-heating surface of the heads of the shell (that is, the area below the water-line, minus the area of the flues, which could be obtained by direct measurement) is 4.5 X 2 = 9 sq. ft. Therefore, the total water-heating surface of the boiler is the sum of all these, or 256.28 sq. ft. Having determined the water-heating surface of a boiler, to approximate its horsepower: Rule . — Divide the total water-heating surface in square feet by the number of square feet of heating area, as given in the table below, required to produce an evaporation equivalent to 1 horsepower in boilei'S of the given type. Example.— The total water-heating surface of the above externally-fired flue boiler is 256.28 sq. ft. What is the horse- power of the boiler? Solution.— B y referring to the table, we find that it takes about 10 sq. ft. of heating surface to produce 1 horsepower; therefore, the above boiler would be rated at about = 25.63 H. P. Water-Heating Ratio of Water- Type of Boiler. Surface for 1 Horsepower. Heating Area to Grate Area Square Feet. Required. Cylindrical Flue 9 From 12 to 15 : 1 10 From 20 to 25 : 1 Firebox tubular 12 From 25 to 35 : 1 Return tubular 15 From 25 to 35 : 1 Vertical 15 From 25 to 30 : 1 Water tube 11 From 35 to 40 : 1 170 BOILERS. The above rule must not be taken as furnishing anything but an approximate method, since the same boiler will give a different horsepower whenever the conditions under which it is operated are changed; or, in other words, the horsepower developed depends largely on the amount of coal burned per square foot of grate area per hour, the velocity and character of the furnace draft, and the quality of the coal used. In ordinary practice, however, we may expect an evaporation of from 8 to 11 lb. of water from and at 212° F. for each pound of good coal burned, where from 11 to 13 lb. of coal are consumed per sq. ft. of grate surface per hr., or about from 3 to 4 lb. per H. P. per hr. CHIMNEYS. The chimney serves the double purpose of creating a draft and carrying away obnoxious gases. The production of the draft depends on the fact that the furnace gases (the products of combustion) passing up the chimney have a high tempera- ture, and are, consequently, lighter than an equal volume of outside air at the ordinary temperature; that is, the pressure within the chimney is slightly less than the pressure of the outside air. Consequently, the air will flow from the place of higher pressure to the place of lower pressure, that is, into the chimney through the furnace. Suppose, for example, the average temperature of the gases in a chimney 150 ft. high is 500° F. A pound of the gases at 62° F. has a volume of 12.5 cu. ft.; its volume at 500° is, then, ~ 2 ' - 5 + 46 °' ) = 23 cu. ft. Therefore, a column of the 62 + 460 150 gases 1 ft. square and 150 ft. long would weigh — = 6.52 lb. 150 A similar column of air at 62° F. would weigh = 11.42 lb., nearly. Hence, the pressure of the draft is 11.42 — 6.52 = 4.9 lb. per sq. ft. = .941 in. of water. It is evident that the pres- sure of the draft depends on the temperature of the furnace gases and the height of the chimney. The higher the chim- ney, the lower may be the temperature of the gases to produce CHIMNEYS. 171 the same draft, and the greater will be the economy of the furnace. In general, chimneys are not built much less than 100 ft. in height. The relation between the height of the chimney and the pressure of the draft in inches of water is given by the follow- ing formula: rr/7.6 7.9\ P ~ H \ T a T')' where p = draft in inches of water; H = height of chimney in feet; T a = absolute temperature of outside air; T c = absolute temperature of chimney gases. Absolute temperatures are found by adding 460° F. to the ordinary temperatures. Example.— What draft pressure will be produced by a chimney 120 ft. high, the temperature of the chimney gases being 600° F. and the external air 60° F.? Solution.— B y the formula we find 7.6 7.9 j) = - 8 in. of V 460 + 60 460 + 600,/ water. The draft pressures ordinarily produced by chimneys vary from 0 to 2 in. of water. A water-gauge pressure of 1 in. is equivalent to .03617 lb. per sq. in. Wood requires least draft, and the small sizes of anthracite coal the greatest draft. To successfully burn anthracite, slack, or culm, a draft of 1£ in. is necessary. To find the height of chimney to give a specified draft pressure, the formula may be transformed: H = 7 1 6_7 1 9* T a T c Example.— Required the height of the chimney to produce a draft of 1£ in. of water, the temperature of the gases and of the external air being, respectively, 550° and 62° F. Solution.— By the formula we find p _ 1.125 II = 7^6 T M a 1A T e 7.6 522 ~ 7.9 1,010 = 167 ft. The sizes of chimneys for boilers of various horsepowers are given in the following table: 172 BOILERS. Sizes of Chimneys and Horsepowers of Boilers. Height of Chimney in Feet. Actual Area in Sq. Ft. Side of Sq. in In. Diameter in In. | 50 60 70 80 90 100 110 125 150 175 200 Commercial Horsepower. 23 25 27 1.77 16 18 35 38 41 2.41 19 21 49 54 58 62 3.14 22 24 65 72 78 83 3.98 24 27 84 92 100 107 113 4.91 27 30 115 125 133 141 5.94 30 33 141 152 163 173 182 7.07 32 36 183 196 208 219 8.30 35 39 216 231 245 258 271 9.62 38 42 311 330 348 365 389 12.57 43 48 363 4271 449 472 503 551 15.90 48 54 505 539 565 593 632 692 748 19.64 54 60 658 694 728 776 849 918 981 23.76 59 66 792 835 876 934 1,023 1,105 1,181 28.27 64 72 995 1,038 1,107 1,212 1,310 1,400 33.18 70 78 1,163 1,214 1,294 1,418 1,531 1,637 38.48 75 84 1,344 1,415 1,496 1,639 1,770 1,893 44.18 80 90 1,537 1,616 1,720 1,876 2,027 2,167 50.27 86 96 Example.— A round chimney 100 ft. high is to he used for a battery of boilers of 550 H. P. What should be the internal diameter? Solution.— Looking under column 100 in “Height of Chimney in Feet” the nearest horsepower is 565, and the diameter corresponding is 60 in., which should be the inter- nal diameter of the chimney. Chimneys are usually built of brick, though in some cases iron stacks are preferred. The external diameter of the base should be T V of the height, in order to provide stability. The taper of a chimney is from ^ to £ in. to the foot on each side. The thickness of brickwork is usually 1 brick (8 or 9 in.) for 25 ft. from the top, increasing k brick for each 25 ft. from the top downward. If the inside diameter is greater than 5 ft., the top length should be 1£ bricks, and if under 3 ft., it may be EXHAUST HEATING. 173 & brick in thickness for the first 10 ft. A round chimney is better than a square one, and a straight flue better than a tapering one. If the flue is tapering the area for calculation is measured at the top. The flue through which the gases pass from the furnaces to the chimney should have an area equal to, or a little larger than, the area of the chimney. Abrupt turns in the flue or contractions of its area should be carefully avoided, as they greatly retard the flow of the gases. Where one chimney serves several boilers, the branch flue from each furnace to the main flue must be somewhat larger than its proportionate part of the area of the main flue. SAFETY VALVES. Balance the valve and lever over a sharp, knife-like edge, and measure the distance from the point of suspension to the fulcrum (center of pin on which the lever turns). Let a = distance thus measured in inches; b = distance from center of valve to fulcrum in inches; x = distance of weight from fulcrum in inches; W = weight in pounds hung on lever; Q = weight of lever and valve in pounds; A = area of safety valve in square inches; p = pressure per square inch in the boiler. Apb—Qa JTr Apb—Qa Wx + Qa Then, x = - Tr . - ; W = — - — ; p = W x * Ab EXHAUST HEATING. Exhaust steam from non-condensing engines usually con- tains from 20$ to 25$ of water and oil, the latter being employed to lubricate the engine cylinders. Before exhaust steam is allowed to enter a heating system, the water and oil should be separated from it. The effect of turning exhaust steam into a heating system is to form a back pressure on engine, which must be avoided as far as possible by using large steam-distributing pipes. A direct connection to the steam boilers through a pressure- reducing valve must be employed, to automatically furnish 174 BOILERS. steam to the heating system when the exhaust fails. A relief valve, also, should be placed upon the system, so that surplus exhaust steam may escape to the atmosphere. To proportion an exhaust-heating system, it is necessary to know about how many square feet of radiating surface we should employ to properly condense the exhaust steam from the non-condensing engines. To do this we must first know the weight of steam that would be discharged from the engine. Class of Non-Condensing Engine. Water Used per Hour for Indicated Horsepower. Compound automatic Simple Corliss Simple automatic Simple throttling 25 lb. 30 lb. 35 lb. 40 lb. From this must be deducted about 10$ for condensation in the cylinders, etc., in order to obtain the real available weight of steam for heating purposes. Approximate Ratio Between Cubic Contents and Radi- ator Surface for Exhaust Heating. Class of Building. Direct Radiation. Indirect Radiation. Blower System. Dwellings sq.ft, cu.ft. 1 to 50 1 to 70 1 to 100 1 to 200 sq.ft, cu.ft. 1 to 40 1 to 60 1 to 80 1 to 150 sq.ft, cu.ft. 1 to 300 1 to 365 1 to 500 1 to 900 Offices Stores and shops Churches, etc The figures in the foregoing tables simply form a reason- able average, and allowance must be made for exposure, etc. Each square foot of direct radiating surface gives off to the air around it about 1£ thermal units per hour per degree of difference between the temperature of the steam and that of the surrounding air. This is equivalent to about £ lb. of steam per hr., or, in other words, about 4 to 4£ sq. ft. of surface to each pound of steam to be condensed. BLUEPRINTS. 175 MACHINE DESIGN. BLUEPRINTS. Blueprint paper for copying tracings of plans and other drawings may be prepared as follows: Dissolve 1 oz., avoir- dupois, of ammonia citrate of iron in 6 oz. of water, and in a separate bottle dissolve the same quantity of potassium ferri- cyanide in 6 oz. of water. Keep these solutions separate, and in a dark place, or in opaque bottles. To prepare the paper, mix equal quantities of the two solutions, and with a sponge spread it evenly over the sur- face. Let the paper remain in a horizontal position until the chemical has set on the surface, which will take but a few minutes; then hang the paper up to dry. In preparing the paper darken the room by pulling down the shades, as direct rays of light aifect sensitized surfaces. The prepared paper should be kept in a closed drawer, well covered with heavy paper, so that no light can come in contact with the sensitized surface; otherwise it will lose much of its value. To make a blueprint from a tracing, lay the tracing with ink side down against the glass of the printing frame, then take the prepared paper, and place the sensitized surface down on the tracing. On the top of the paper place the felt cushion, on top of which place the hinged back of the printing frame, after which expose to the sunlight. The exposure will vary in sunlight from about 3 to 10 minutes. After the exposure, wash the paper thoroughly in a trough of cold water for about 10 minutes, and hang it up to dry. The print after washing should be of a deep-blue color, with clear white lines. If the color is a pale blue, this indi- cates that the print has not had sufficient exposure, and if the lines of the drawing are not perfectly clear and white, that the exposure has been too long. Corrections may be made on the print with an ordinary writing or ruling pen and a solution of washing soda, caustic" potash, strong ammonia, or any other alkali. When any of these are mixed with carmine ink, the marks on the print will be red, thus making the corrections clear. 176 MACHINE DESIGN. MACHINE TOOLS. SPEED OF EMERY WHEELS. The speed most strongly recommended by their manufac- turers is a peripheral velocity of 5,500 ft. per min. for all sizes. All things being considered, it is stated that no advantage is gained by exceeding this speed. If run much slower than this, the wear on the wheels is much greater in pro- portion to the work accomplished, and if run much faster, the wheel is likely to burst. SPEED OF GRINDSTONES. Grindstones used for grinding machinists’ tools are usu- ally run so as to have a peripheral speed of about 900 ft. per min., and those used for grinding carpenters’ tools at about 600 ft. per min. With regard to safety, it may be stated in general that with any size of grindstone having a compact and strong grain, a peripheral velocity of 2,800 ft. per min. should not be exceeded. . SPEED OF POLISHING WHEELS. Polishing wheels are run at about the following peripheral speeds: Leather-covered wooden wheels 7,000 ft. per min. Walrus-hide wheels. 8,000 ft. per min. Rag wheels 7,000 ft. per min. SPEED OF CUTS FOR MACHINE TOOLS. Brass: Use high speeds, about the same as for wood. Bronze : 6 to 18 ft. per min., according to alloy used. Cast or wrought iron: 20 ft. per min. is a good average for all machines, except millers. 30 is about the maximum. Machinery steel: 15 ft. on shapers, planers, and slotters. 20 to 45 on turret lathes, according to cut. Tool steel: 8 to 10 ft. Milling Cutters .— Gun metal , 80 ft. per min.; cast iron , 30; wrought iron , 35 to 40; machinery steel , 30. These are good speeds to adopt, with a view to economy, time required for regrinding, etc. MACHINE TOOLS. 177 Twist Drills.— The best results are obtained when the rates of speed of twist drills are as given in the following table: Revolutions of Drills per Minute. of Drills. Steel. Iron. Brass. Ps 940 1,280 1,560 460 660 785 8 310 420 540 230 320 400 Ps 190 260 320 150 220 260 p 130 185 230 115 160 200 Ps 100 140 180 95 130 160 I a 85 115 145 75 105 130 70 100 120 Vs 65 90 115 It 62 85 110 1 58 80 100 IPs 54 75 95 52 70 90 i % 49 66 85 46 62 80 IPs 44 60 75 42 58 72 Ip 40 56 69 39 54 66 IPs 37 51 63 36 49 60 ill 34 47 58 33 45 56 ill 32 43 54 31 41 52 111 30 40 51 2 29 39 49 The following are recommended as the best rates of feed for twist drills: Diameter of drill in inches Number of revolu- K % K % 1 IK tions per inch depth of hole 125 125 120 to 140 1 in. feed per min. 178 MACHINE DESIGN. CHANGE GEARS REQUIRED FOR CUTTING SCREW THREADS. The pitch of a single-threaded screw is the distance between two adjacent threads, measured on a line parallel to the axis of the screw; or, in any screw, whether single- or multiple-threaded, it is the distance the nut is moved by 1 revolution of the screw. Usually, a screw is spoken of as having a certain number of threads to the inch, and this is equal to the number of revolutions the screw must make in order to move the nut a distance of 1 inch; so, whether the screw is single- or multiple-threaded, the pitch is always equal to 1 divided by the number of revolutions that the screw must make in order to move the nut 1 inch. The Simple-Geared Lathe.— In Fig. 1 is shown the usual arrangement of the change gears of a simple-geared screw- cutting lathe. By a simple-geared lathe is meant a lathe in which the change gears are so arranged that the circum- ferential velocity of the change gear on the stud is the same as that of the change gear on the lead screw, which means that, when the change gear on the stud has rotated, say, 5 teeth, the change gear on the lead screw has also rotated 5 teeth, whatever the diameter of these gears, or of any intermediate gears between them, may be. Referring to Fig. 1, the gear a is fastened to the spindle b and drives another gear c by means of either one of the MACHINE TOOLS. 179 reversing gears d, d' . The gear c is keyed to one end of the spindle e; this spindle is called the stud , and carries on its outer end a change gear /. The lead screw g carries a change gear h\ and these two change gears / and h are connected by means of the idler gear i, so that gear/ drives gear h, and with it, the lead screw g. In making calculations for the change gears of a simple- geared screw-cutting lathe, the idler gear i is ignored, as it is only introduced to connect gears / and h. The gears d and d' are also ignored, since they are only used to change the direction of rotation of the gear c, their duty being to facilitate the cutting of either right-hand or left-hand threads; when d meshes with gear a , as shown in Fig. 1, a a right-hand thread is cut, and when d f meshes with gear o, a left-hand thread is cut. The number of teeth in the gear a is not always the same as the number of teeth in the gear c; it is so in some lathes, but in others it is not; hence, in calculating the change gears for any lathe, the number of teeth in the gears a and c must be taken into account. By the following formulas and rules, the number of teeth required in each change gear in order to cut a given number of threads to the inch, or the number of threads to the inch that given change gears will produce may be found. Let a = number of teeth in the spindle gear a ; c = number of teeth in the gear c; / — number of teeth in the change gear on stud; h = number of teeth in the change gear on lead screw; g = number of threads to the inch in the lead screw; n = number of threads to the inch to be cut. Then, n = _ gch a/ ' (1) h = ^f. gc (3) n a (2) f _ & ch J na' (4) Now, of the gears /*,/, c, a, a and /are the drivers , and c and h being driven by a and /, are called the driven gears; remembering this, we deduce, from formula (1), the following rule for simple-geared screw-cutting lathes: 180 MACHINE DESIGN. Rule .— The number of threads to the inch to be cut is equal to the number of threads to the inch in the lead screw , multiplied by the product of the number of teeth in each driven gear , and divided by the product of the number of teeth in each driving gear. Example.— If the lead screw g of a simple-geared lathe has 5 threads to the inch, and the gear a has 21 teeth, the gear c 42 teeth, the change gear / 60 teeth, and the change gear h 72 teeth, how many threads to the inch will be cut? Solution. — U sing formula (1), we have gch __ 5 X 42 X 72 = 12 teeth. af “ 21X60 From formula (2) we deduce the following rule for simple- geared screw-cutting lathes: Rule. — The number of teeth in the change gear on the lead screw, divided by the number of teeth in the change gear on the stud, is equal to the product of the number of threads to the inch to be cut and the number of teeth in the driving spindle gear , divided by the product of the number of threads to the inch in lead screw and the number of teeth in the fixed gear on the stud. Example.— If the lead screw g of a simple-geared lathe has 8 threads to the inch, and the gear a has 16 teeth, and the gear c 32 teeth, how many teeth must there be in each of the gears / and h in order that the lathe may cut 10 threads to the inch? Solution.— Using formula (2), h na 10 X 16 5 f ~ gc~ 8X32 ~ 8’ and, if it were possible to have gears with 5 and 8 teeth, respectively, then a solution of the problem would be, h = 5,/ = 8. It is evident that such gears are impracticable; but, as it does not change the value of a fraction to multiply both numerator and denominator by the same number, we may multiply 5 and 8, each by such a number that the result- ing numbers of teeth in the gears are satisfactory. There is evidently, therefore, more than one solution to the problem— for if we multiply by 10 we, shall have h = 50, / = 80, which would give 12 threads to the inch; and if we multiply by 13, we shall have, as another solution, h = 65, / = 104, which would also give 12 threads to the inch, because fife = |. MACHINE TOOLS. 181 Having found that j = |, it is customary in practice to choose the change gears in the following manner: From the assortment of gears belonging to the lathe, choose one of convenient diameter, the number of whose teeth is divisible by either the numerator 5 or the denominator 8, and, after dividing by one of these numbers, multiply both numerator and denominator by the quotient. Example.— G iven, - = •§, to find the number of teeth in the two change gears h and/, respectively. Solution.— Choose a gear of convenient diameter, the number of whose teeth, say 60, is divisible by either 5 or 8, in this case by 5; divide 60 by 5, and the answer is 12. Then, • 5X12 = 60 8 X 12 96’ that is, h has 60 teeth, and / 96 teeth. If one of the change gears is given, and it is desired to find the number of teeth in the other change gear in order to cut a given number of threads to the inch, use either formula (3) or formula (4) according as the number of teeth in gear h or in gear /is required. After the examples given, these formu- las will not need explanation. In a simple-geared screw-cutting lathe, it is often possible to cut a fractional number of threads to the inch, as is the case in the following example: Example. — If the lead screw g has 2 threads per inch, and the gear a has 20 teeth, and the gear c has 20 teeth, how many teeth must there be in each of the change gears / and h , in order to cut 5£ threads to the inch ? Solution.— Using formula (2), h na 5i X 20 5£ 7 “ Jc~ 2X20'“ IT* Then, choosing a gear whose number of teeth, say 32, is divisible by 2, divide 32 by 2 and the quotient is 16. Then, Slvifi «4 * - — = j that is, h has 84 teeth, and / 32 teeth. In many cases, however, it is impossible, out of the assortment of gears supplied with a simple-geared screw-cutting lathe, to 182 MACHINE DESIGN. find gears to cut a screw of the required number of threads to the inch. In such cases, it becomes necessary either to make suitable gears or to resort to a compound-geared lathe. The Compound-Geared Lathe. — In Fig. 2 is shown the usual arrangement of the change gears of a compound-geared screw-cutting lathe. The difference between this and the simple-geared lathe lies in putting two change gears of differ- ent sizes on one spindle, in place of the idler between the gear on the stud and the gear on the lead screw. These two gears on one spindle are shown at i and j in Fig. 2, gear j meshing with gear h on the lead screw, and gear i meshing with gear/ on the stud. From the following formulas, the number of teeth in each change gear, or the number of threads per inch that can be cut with given change gears, can be found. Let a = number of teeth in the spindle gear a; c = number of teeth in the gear c; / = number of teeth in the change gear/; h = number of teeth in the change gear h ; i == number of teeth in the change gear i , which meshes with the change gear/; j = number of teeth in the change gear j, which meshes with the change gear h ; g = number of threads to the inch in the lead screw; n = number of threads to the inch to be cut. gXchi /rx Then, n = --jt-. (o) MACHINE TOOLS. 183 Now, remembering that gears a, f and j are the drivers, and gears c, h, and i are the driven gears, and also that the idlers are ignored in all calculations, we can, from formula (5), deduce the following rule for compound-geared screw- cutting lathes: Rule . — The number of threads to the inch to be cut is equal to the number of threads to the inch in the lead screw , multiplied by the product of the number of the teeth in each of the driven gears , and divided by the product of the number of teeth in each of the dnving gears. Example.— I f the lead screw (7 of a compound-geared lathe has 2 threads to the inch, and the gear a has 20 teeth, gear c 40 teeth, change gear / 48 teeth, change gear i 72 teeth, change gear j 36 teeth, and change gear h 96 teeth, how many threads to the inch will be cut ? Solution— U sing formula (5), we have gX chi _ 2X40X96X72 afj = -20 X 48 X 36" = 16 threadS t0 the iQCh - If it is desired to find what combination of change gears will enable us to cut a given number of threads to the inch, the following formula may be used: ( 6 ) i_ _ naf J ~ gch' From this formula the following rule is deduced: Rule . — Of the change gears of a lathe , any driven gear divided by any driver gear is equal to the product of the numbers of teeth in each of the other driver gears and the number of threads to the inch to be cut , divided by the product of the numbers of teeth in each of the other driven gears and the number of threads to the inch in the lead screw. Example.— I n a compound-geared lathe, in which the lead screw has 5 threads to the inch, gear a 20 teeth, gear c 40 teeth, and the number of threads per inch to be cut is 3i, what must be the number of teeth in each of the change gears h t i t j,f? Solution.— U sing formula ( 6 ), we have i _ naf j ~ g~cK 184 MACHINE DESIGN. From the assortment of gears belonging to the lathe, choose, for the driven gear h, one whose number of teeth, say 28, can be divided by the number of threads per inch to be cut, in this case 3£; 28 is a multiple of 3£, because it is obtained by multiplying 3£ by 8. Substitute this value in place of h; then choose any gear of convenient size, say one having 40 teeth, and substitute 40 in place of /; we shall then have, i _ n a X 40. j ” g c X 28’ or, substituting the given values of n, a, g, and c, i _ 3£ X 20 X 40 1 j 5X40X 28 “2* Choose, for j, a gear whose number of teeth, say 60, is divisible by 2; then, dividing the number of teeth in,; by 2, we have 60 2 = 30. Now multiplying both terms of the frac- tion £ by 30, £ _ 1 X 30 _ 30, j ~ 2 X 30 “ 60 ’ that is, i — 30, and j = 60. Hence, one solution of the prob- lem is, h = 28; i = 30; j = 60; / = 40. HORSEPOWER OF ENGINES, BOILERS, AND PUMPS. THEORETICAL HORSEPOWER. The theoretical horsepower of any machine that uses a fluid (steam, gas, water, etc.) as a motive power, or that dis- charges a fluid (i. e., a pump or a fan), may be readily com- puted by the following formula, in which v is the volume of the fluid used or discharged in cubic feet per minute, and p is the average pressure in pounds per square inch: tt p _ 144 vv ' ' 33,000 ' If, in the above formula, allowance for friction, etc. is made, the final result will be the actual horsepower. Example.— A ventilating fan delivers 5,000 cu. ft. of air per min. at a pressure of .56 lb. above the atmospheric pressure; what is the theoretical horsepower required to drive the fan ? HORSEPOWER. 185 Solution.— H. P. = 144 v p 33,000 144 X 5,000 X .56 = 12.218. 33,000 If all hurtful resistances are taken in this case as 20$ of the total horsepower, the actual horsepower will be 12.218 (1 — .20) = 12.218 -r- .80 = 15.27 H. P. Example.— T he mean effective pressure computed from an indicator card taken from the air cylinder of an air com- pressor is 30.6 lb. per sq. in.; diameter of cylinder, 28 in.; stroke, 48 in.; number of strokes per minute, 108; what is the horsepower? Solution.— I n this case v Hence, 144 vp __ 3p00 — 28 2 X .7854 X 48 X 108 1,728 cu. ft. per min. 144 X 282 X .7854 X 48 X 108 X 30.6 1,728 X 33,000 246.66 H. P. HORSEPOWER OF AN ENGINE. Let P = mean effective pressure in pounds per square inch on the piston during one stroke; L = length of stroke in feet; A = area of piston in square inches; N = number of strokes per minute; D — diameter of piston in inches. Then, to find the indicated horsepower, PLAN _ 238 P L D 2 N * * * 33,000 ~ 10,000,000 ’ The actual horsepower may be taken as three-fourths of the indicated horsepower. The mean effective pressure may be found exactly by taking some indicator cards, finding the areas by means of a planimeter, and dividing the area by the length of the card. Multiply the result by the scale of the indicator spring, and the product will be the mean effective pressure, or M. E. P. If no planimeter is at hand, divide the card into 10 equal parts and measure each part in the middle, as shown by the dotted lines in the following figure. Add, all the dotted ordinates together, and divide by 10; this result, multiplied by the scale of the indicator spring, gives the M. E. P. 186 MACHINE DESIGN. Thus, suppose a double-acting engine 26" X 30", making 80 rev. per min. (80 R. P. M.), gives an indicator card that, being divided up as shown in the figure and measured, gives, for the total length of the ordinates, 21.4 in. This divided by 10 = 2.14 in. for the length of the mean ordinate. If a No. 40 spring is used in the indicator, every inch measured ver- tically on the diagram = 40 lb. per sq. in., and 2.14 X 40 = 85.6 lb. per sq. in. for the M. E. P. on the piston. Then the indicated horsepower, or I. H. P., equals PLAN 85.6 X X (.7854 X 262) X (2X80) _ 83,000 33,000 U,88 ‘ The calculation is rendered much easier by using the sec- ond formula. Thus, T tt t> 238 X 85.6 X f § X 262 X (2 X 80) ecn 00 L - 16,000,600 “ 550 - 88 ' If an indicator card cannot be obtained, a fair approxima- tion to the M. E. P. may be obtained by adding 14.7 to the gauge pressure, and multiplying the number opposite the fraction indicating the point of cut-off in the following table by the boiler pressure. Subtract 17 from the product, and multiply by .9. The result is the M. E. P. for good simple non-condensing engines. If the engine is a simple con- densing engine, subtract the pressure in the condenser instead of 17. The fraction indicating the point of cut-off is obtained by dividing the distance that the piston has traveled when the steam is cut off by the whole length of the stroke. Thus, if the stroke is 30 in., and the steam is cut off when the piston THE SLIDE VALVE. 187 has traveled 20 in., the engine cuts off at = | stroke. For a f cut-off, and 92-lb. gauge pressure in the boiler, the M. E. P. is [(92 + 14.7) X .917 — 17] X .9 = 72.76 lb. per sq. in. Cut-off. Constant. Cut-off. Constant. Cut-off. Constant. % .566 % .771 % .917 * .603 .4 .789 .7 .926 .659 A .847 % .937 .3 .708 .6 .895 .8 .944 Yz .743 % .904 Vs .951 THE SLIDE VALVE. Figs. A, B, C , and D show sections of an ordinary D slide valve at different points of its travel. Fig. A shows the valve in its central position, with the center of the valve in line with the center line of the exhaust port. The names of the various parts are as follows: p and p are the steam ports; e is the exhaust port; s, s is the naive seat; the amount o by which the valve overlaps the outer edges of the steam ports is the outside lap; the amount i by which the valve overlaps the inside edges of the steam port is called the inside lap; the amount l (Fig. C) that the port is open when the piston is at the end of the stroke is called the lead. The valve travel is the total distance in one direction that the valve can be moved by the eccentric; it is the total distance between two extreme positions of the valve. The displacement of the valve is the distance that the valve has moved (in either direction) from its central position. The line joining the center of the eccentric with the center of the crank-shaft is called the eccentric radius. When the eccentric radius makes a right angle with the center line of the crank, that is, when the eccentric radius is vertical (see oe, Fig. E), the valve is in its central position, provided the valve seat is horizontal, as is usually the case. When the crank is on a dead center, say a, Fig. E , the valve must be in the position shown in Fig. C; that is to say, the crank must 188 MACHINE DESIGN. have moved from its central position an amount equal to the outside lap plus the lead. In order that this may happen, the eccentric must be at c, Fig. E. The angle eoc, through which the eccentric must be moved from its vertical position when the crank is on a dead center, is called the angle of advance. THE SLIDE VALVE. 189 In Fig. B, the valve is shown in its extreme position at the right. The distance marked m is the maximum port opening. It matters not whether the outer edge of the valve travels beyond the inner edge of the port or falls short of it, as in the figure, the distance m between the edge of the valve and the edge of the port when the valve is in its extreme position is the maximum port opening. If, in Fig. C, the valve were shown moving to the left, a little farther movement would bring the left outer edge just even with the outer edge of the left steam port, and from here on to the end of the stroke no more steam could enter the left end of the cylinder; in other words, the valve cuts off at this point. A little farther move- ment of the valve to the left brings the valve to the position shown in Fig. D, with the right inner edge opposite the inner edge of the right steam port; it is at this point that compres- sion begins. When designing a valve for an engine, some of the above quantities are assumed and the remaining ones are required; these may be found by means of the diagram shown in Fig. E. Let a b, Fig. E, drawn to any convenient scale, represent the stroke of the engine; then a db will represent the crank- pin circle. About o, the center of the crankpin circle, describe a circle a' eb whose diameter a'b' is equal to the actual travel of the valve. Draw the line gh parallel to ab and at a distance from it equal to the lead of the valve. Then, with a radius o' j equal to the outside lap of the valve, describe a circle, called the outside lap circle , tangent to the line gh, and having its center o' on the circle a'eb'. Draw the line oo', and produce it to/; then fob = eoc = angle of advance. Now, draw any position of the crank center line, such as a o, and drop upon it, from the point o', a perpendicular; the length of this perpendicular (marked r in Fig. E) is the dis- placement of the valve for that position of crank center line. About the center o' with a radius equal to the inside lap of the valve, describe a circle; this is called the inside lap circle. The radius od, drawn from the point o tangent to the outside lap circle, is the position of the center line of crank at the point of cut-off. Drop a perpendicular from point d , 190 MACHINE DESIGN. meeting the line ab at then ak is the distance moved by piston before cut-off, and the fraction of the stroke at which cl k the valve cuts off is represented by the fraction — . Draw the radius o l tangent to the upper side of the inside lap circle, and it will be the position of the center line of the crank when compression commences; if a perpendicular is dropped from point l , meeting the line ab at p, the fraction of the stroke of piston at which compression begins will be represented by the fraction In like manner, the radius ora, drawn tangent to the lower side of the inside lap circle, is the position of the center line of the crank at the moment of release; and — ? is the a b fractional part of the stroke at which the expanding steam is released. The maximum steam-port opening is equal to on, n being the point of intersection of the outside lap circle with the angle of advance line o /. The essential features of the valve diagram having been given, the following examples will make clear its application in practice: Example 1 . — Given, the point of cut-off, the point of release, the lead, and the maximum port opening, to find the valve travel, the outside and inside lap, the angle of advance, and the point of compression. Solution. — Draw to a convenient scale the crankpin circle ad b, Fig. E, having its center at o , and its diameter ab equal to the stroke of the piston. From the point a, lay off, on the line a b, the distances a k and ay, so that and are equal, respectively, to the fractions of the stroke at which cut-off and release are to occur. At k and y draw perpendiculars to the line a b, inter- secting the crankpin circle at d and ra, respectively; the radii o d and ora will represent the positions of the crank at cut-off and release, respectively. Now draw gh parallel to ab, and at a distance above it equal to the lead; then, about o as THE SLIDE VALVE. 191 a center, and with a radius equal to the given maximum port opening, describe an arc. Find by trial a center o', from which a circle can be drawn tangent to this arc, and also to the radius o d, and to the line g h. The radius of this circle will be the required outside lap; and its center o' will be a point in the valve circle whose center is at o; this circle can now be drawn, since the radius o o' is known. The diameter a'b' is equal to the required valve travel. Now, with o' as a center, draw’ a circle tangent to o m, and the radius of this circle will be the required inside lap. Draw 0 / through 0 ' and the angle fob is the required angle of advance. Draw the radius ol tangent to the inside lap circle on its upper side, and Ip perpendicular to a b. Then, ^ represents the fraction of the stroke at which compression begins. Example 2. — Given, the valve travel, the angle of advance, the cut-off, and the point of compression, to find the lead and the outside and inside lap. Solution.— Draw the crankpin circle, as before, and the valve circle a' eb'\ construct the angle /06 equal to the angle of advance. By the same method as employed in the last example, locate the radii od and ol, representing the posi- tions of the crank at the points of cut-off and compression, respectively. About the point o', at which 0 f intersects the valve circle, describe a circle tangent to od, and the radius o'j of this circle will be the required outside lap. Now draw the line gh parallel to a 6 and tangent to the outside lap circle; then, the perpendicular distance between gh and a 6 is the required lead. The radius of a circle drawn from o' tangent to o l will be the inside, lap. Example 3.— Given, the valve travel, outside lap, and the lead, to find the point of cut-off and angle of advance. Solution. — Draw the crankpin circle and the valve circle a' e b' as before; draw a line parallel to a 6 , at a distance above it equal to the outside lap r plus the lead, intersecting the valve circle at the point o'. About o' as center, and with a radius equal to the given lap, describe a circle; draw od 192 MACHINE DESIGN. tangent to this circle, and drop a perpendicular from c2, meet- ing line ab at a point k; then the required cut-off is represented CL Jc by the fraction Draw the radius of through the point o' and the angle /o 6 is the required angle of advance. Example 4.— Given, the outside lap, the lead, and the point of cut-off, to find the valve travel and the angle of advance. Solution.— Draw the crankpin circle as before, and by the same method as employed in Example 1 locate the radius o <2, the position of the crank at the point of cut-off. Draw g h parallel to a b , and at a distance above it equal to the lead. At a distance above the line ab equal to the lap plus the lead, draw another line parallel to ab ; about a center o' on this line, and with a radius o' j equal to the outside lap, describe a circle tangent to o d and g h. Draw the radius of through o', then/o b will be the required angle of advance. About o as a center, and with a radius o o', describe the valve circle a f e b', and a' b' will be the required valve travel. LOCKNUTS. A good method of locking a nut is shown in the figure. The lower portion of the nut is turned down, and in the center of the circular portion a groove is cut. A collar is fastened by means of a pin to one of the pieces to be con- nected, and into this collar is fitted the circular part of the nut. The nut is then bound to the collar by a setscrew passing through the latter, the point of the setscrew engaging into the groove turned in the nut. The following proportions have proved very satisfactory, in which <2, the diameter of the bolt, is taken as the unit. All dimensions are in inches: a = l£ <2 — T V'; /=£<* + £"; b = l£<2 + £"; . g = £<2 + ^"; c = £<2 + £"; h = £<2 + £". e = £c2; LINE SHAFTING. 193 PROPORTION OF KEYS. In common designing, the sizes of keys are determined by empirical formulas, which give an excess of strength. For an ordinary sunk key, these proportions may be adopted: t — thickness of key in inches; b = breadth of key in inches; d = diameter of shaft in inches; b = id; t = *6 = id. LINE SHAFTING. The speed of a shaft is fixed largely by the speed of the driving belt or the diameters of the pulleys upon it. In general, machine-shop shafts run about 120 to 150 rev. per min.; shafts driving wood-wor king machinery, about 200 to 250 rev. per min.; in cotton mills, the practice is to make the shaft diameter smaller and run at a higher speed. Line shafts should generally not be less than li in. in diameter. The distance between the bearings should not be great enough to permit a deflection of more than yfoy in. per foot of length; hence, the bearings must be closer when the shaft is heavily loaded with pulleys. The maximum distances between bearings of different sizes of continuous shafts used for transmitting power are: Distances Between Bearings. Diameter of Shaft. Inches. Distance Between Bearings in Feet. Wrought-Iron Shaft. Steel Shaft. 2 11 11.50 3 13 13.75 4 15 15.75 5 17 18.25 6 19 20.00 7 21 22.25 8 23 24.00 9 25 26.00 Pulleys that give out 'a large amount of power should be placed as near a hanger as possible. 194 MACHINE DESIGN. SHAFT COUPLINGS. A box, or muff, coupling is shown in the figure. It consists of a cast-iron cylinder that fits over the ends of the shaft. The two ends are prevented from moving relatively to each other by the sunk key. The key way is cut half into the box and half into the shaft ends. Quite commonly the ends of the shafts are enlarged to allow the key way to be cut without weakening the shaft. The key may be proportioned by the formula already given. For the other dimensions, take l = 2£d + 2" t = .4d + .5" Example. — Find the dimensions of a muff coupling for a shaft 2b in. in diameter. Solution. — For the key we use the formula previously t = Ad + .5" = .4 X 2J + .5" = 1 A flange coupling is shown in the following figure. Cast- iron flanges are keyed to the ends of the shafts. To insure a given, For the muff, PEDESTALS. 195 perfect joint the flange is usually faced in the lathe after being keyed to the shaft. The two flanges are then brought face to face and bolted together. Sometimes the ends of the shafts are enlarged to allow for the keyway. To prevent the possibility of the shafts getting out of line, the end of one may enter the flange of the other. The following proportions may be used for this form of flange coupling: d = diameter of shaft; n = number of bolts. D = lfd +1" A = 2£d + 2" l = l*d + l" n= 3 + | (Take the nearest whole number for n.) A = 1.4 A b = %d + f" e = 2b t = id The proportions for the key have already been given. In the accompanying figure is shown a flexible coupling, or uni- versal joint. These joints, when constructed of wrought iron, may have the following proportions in terms of the diameter d of the shaft: a = 1.8 <2 g = .6 d b = 2d h— .5 d c = d k = .6d e = 1.6 d PEDESTALS. The names pedestal , pillow-block , bearing , and journal-box are used indiscriminately. They are all a form of bearing, and indicate a support for a rotating piece. 196 MACHINE DESIGN. A form of journal-box frequently used for small shafts is shown in Fig. 1. It consists of two parts: (1) the box that supports the journal, and (2) the cap that is screwed down to the box. In this journal-box the seats are of babbitt, or, as it is commonly expressed, the box is babbitted. The cap is held in place by what are called capscrews. This is invariably done in small pedestals. The proportioning of a pedestal is largely a matter of experience. Few or none of the parts are calculated for strength. All the proportions of the pedestals that follow are based on the diameter of the journal d as the unit; the length of the seats is the same as that of the journal. For the journal-box shown in Fig. 1, the following propor- tions may be used for sizes of journals from £ in. to 2 in. diam- eter, inclusive. The diameter of shaft d is the unit. PEDESTALS. 197 Fig. 2. 198 MACHINE DESIGN. m = .25 d + .1875"; n = .5 d; o = .625" (constant); p = 1.5 d ; q = 1.333 <2; y = .08 ( 2 ; s = .125" (constant); * = .16 d; u * * 1.333 d; v .125(2. a = 2.25 (2; 5 = 1.75 d\ c — d; c = .375 d; /= .08 d -t- .0625"; g = 1.75 (2; h = 2.45 d; t = .3 d ; J = .33 d; = .25 d + .125"; Z = .08 d; In Fig. 2 is shown a common form of pedestal that is used for somewhat larger journals than the one shown in Fig. 1 . It consists of (1) a foundation plate that is bolted to the foundation on which the pedestal rests; the plate is essential when the pedestal rests on brickwork or masonry, but may be dispensed with when the pedestal rests on the frame of the machine; ( 2 ) the block that carries the seats and supports the journal; (3) the cap that is screwed down over the seats. The bolt holes in both foundation plate and block are oblong, so that the pedestal may be readily adjusted. The following proportions may be used for this kind of. pedestal, having journals from 2 in. to 6 in., inclusive. An oil cup having a £ in. pipe-tap shank may be used on pedestals for journals having diameters from 3 in. to 4 in., and f in. pipe-tap shank for larger sizes up to 6 in. diameter. Note.— The shanks of oil cups and grease cups bought in the market are made with a £", f", or £" pipe thread. The amount of oil or grease the cup holds when filled is usually expressed in ounces. The diameter of journal d is the unit. a = 3.25d; j = .375(2; b = 1.75<2; k = 1.0625(2; c — c 2 ; 2 = .875(2; e = .5(2; m = 1.75(2; / = .4375(2; n = 1.25(2; g = .09(2; o = .125" (constant); h = .3125(2; p = .875" (constant); i = .25(2; q = .625 <2; r = .25 d; s = .1875d; t = .65(2; u = .75 cf; v — 1.375(2; x = .25(2; y = .5<2; z = .0625d. PEDESTALS. 199 Fig. 3. 200 MACHINE DESIGN. Fig. 3 shows a pedestal suitable for the crank-shaft of a horizontal engine with journals from 8 in. to 20 in. in diameter. The block may be complete in itself, as shown in the figure, but more often it forms part of the engine bed. The seats are in three parts, and may be adjusted hori- zontally by means of the wedges W. The lower seat may be raised by placing packing pieces under it. To obtain its dimensions, use the following proportions, which are based on the unit d = the diameter of the crank-shaft journal. a = d + 1"; b = .5 (2 + 1"; c = .66 (2; e = .825(2 — .25"; / = .6(2; g = .1 d + .5625"; h = .1(2 + .25"; h' = .08(2; i = .11(2; j = .625" (constant); Jc = .5(2 + 1.25"; l = .375" (constant); m = .175(2 + .31.25"; n = .25(2 + 25"; n' = .1(2 + .375"; q' = 1.5(2; r = .15(2; r' = .1(2; ri — <2; s = .9(2; 2 = 15 (2 + .375"; 2' = .9(2; u = 1.5(2; v = .25(2 + .375"; w = 1.45(2; u>' = 1.47(2 Wi = 1.75(2; x =3 .1(2; y = .3 (2 + .75"; y' = .2 (2 + .5"; z = .09 (2; ^ = 2.5" (constant). o = 1" (constant); p = .25(2 + .625"; (? = 1.75(2; Taper of adjusting wedge, 1 : 10. Further details of the bottom seat and the cap are shown in Fig. 4, in which the unit is the same as in Fig. 3, and the proportions are as follows: a = 1" (constant); c = .08(2; b = 1.65(2— .5"; d = .1(2. The foundation casting, or the bed casting, is shown in Fig. 5, and has dimensions to suit the pedestal that is shown in Fig. 3. The proportions of the casting are given in con- nection with Fig. 5, on page 201. The diameter d of the crank-shaft journal is taken as the unit. PEDESTALS. 201 b = o' = t = 2.45 (2 +7.25"; 2.3 d + 5.25"; .5(2 + 3.5"; 3.5(2 + 2"; .25(2 + .5"; .25 (2 + 1.75"; .25 d + 2.25"; .05 d + .5"; .05 c? + 1.125"; .05 d + .75"; .25 <2 + .75"; Ad) .6 d) 1.55 d + 2.5"; .25 d + 2"; .25(2 + .5"; .5 4 34 1 ! p~ 10 34 T 3 6 1 5 T6 T5 3 ‘ >4 34 1 34 6 8 31 34 1* gf 10 Its 34 5% % 134 12 12 4 31 34 1 334' "34" "34" i 34 6 1/4 4 8 5 "34 134 34 14 10 12 4 6 A+ *+' T3 34 1/4 134 14 "ST 534 6 /| 334 434 I I i 134 /! 8 1 3 6 1 5 6 lie 5 10 6 16 12 4 31+ 34 1H 1% 13 s fi ... . T" 34 6 % * % 134 % 8 T%+ T B 6 l*’ % 5 10 6 ....... 12 16 7 "ii" ti 37 3l 1% if 34 5 4 I 18 4 TS TS 1* A 4 % % 34 6 434 8 31 51 134' "if 534 "54" 10 6 12 734 % % 134 16 34 34 234 134 8 20 9 20 4 1S + TS 1% "34 4 % "34" 134 34 6 434 8 5 10 31 '"54" 6 12 7 134' 16 A T5 234' 134" 8 % "34 20 10 1 PROPORTIONS OF PULLEYS. 207 Table— ( Continued). a cC 0> o Rim. Arm. Hub. Boss. A A B C x> E F G H J 22" 4 IS IK K 4 K K iK K 6 4K k 8 5 10 *+ ii 1% If 6K "K .. 12 iK 16 "s% i 20 +3 2K IK 11 IK iK 24 4 32 4* 1t 9 3 ll 4 K K iK K 6 8 10 % K~ IT "iK - 0/2 7 "Vs iK 12 16 ’9K i " 20 24 IS 32 2% 1% IK K iK 26 4 6 32 hi 1U “k f* “k iK K 8 6 Va iK 10 K 2*"" 7 12 7K 16 10 IK "Va 'i K 20 24 T S 3 + 32 211 iii S* 4^ 5K 28 4 6 3 7 2+ ii" I "I iK iK K 8 7 10 7K 12 16 20 K+ K 2K 11 8 10 11 i T S 3 I 32 IK If iK "I K iK 30 24 4 6 8 10 12 32+ f "§ f "k K i'K iK K 13 2K T ' 8 T" 16 20 8K UK iK Va iK 32 24 4 1 iK if 13 4K iK K 6 8 5M 6% l 10 7^ 208 MACHINE DESIGN. Table— ( Continued). a c3 6 o Rim. Arm. Hub. Boss. s A B C D E F 0 H 7 12 TS 33 2^ 1* 8 134 % 134 16 934 20 11 '134' 24 13 34 4 34+ 34 234 15 TS 434 6 1 / Vs % 134 34 8 °/2 634 i ’ 10 12 16 TS 'W 2/s Its 1 934 134 134' 20 12 "134' 24 13 36 4 g 34+ 34 2t 3 s TS 434 534 "Vs "'34 134' 8 m ....... 10 734 12 TS 33 10 i 134' "Vs 134 16 2 t 9 s 134 •134 20 12 40 24 8 T6 M ‘2*’ 1 ' f 134 1 1 34 1 3l 12 434 % 134 16 M 34 2% 'l34' 10 $ 20 1134 i i% 34 24 1534 A 44 8 12 16 S3 TS iy* 134 6/1 g % 134 34 33 % 3 ‘ 'i* 10 "134' rj 20 12 1 134' "34" 24 3/4 234 134 15 48 8 12 3 9 3 + TS 1 '134 34 16 % TS 334 l/s 10 134 1 "34“ 20 12 54 24 12 16 TS+ 1 6 33 3 " 'I* 15 9 34 II34 I 134 i ” 134 '34" 20 H M 334 134 134 24 15 ’ '1% 2 " 60 12 33 34 3ys Its 10 m 1 134 34 16 20 1/5 34 m 134 1134 12/4 i 134 " 2 " 24 15 ROPE BELTING. 209 Table — ( Continued). a ci o w Rim. Arm. Hub. Boss. 5 A B g D E F G H I O) q> 12 u X 3ft 1* 10 11 4 1 y* 16 20 X % 4 X lit 13^ 1 X 2 24] 15 72 12 16 % t 9 s We IB 10K ill I "ix 2 X 20 re f§ 2* 4 24 15 2 ROPE BELTING. There is a growing tendency toward the substitution of hemp and cotton ropes for belting and line shafting as a means of transmitting power in large factories and shops. The advantages claimed for the rope-driving system are: 1. Economy; for a rope system is cheaper to install than either leather belting or shafting. 2. In the rope system there is less loss of power by slipping. 3. Flexibility; that is, the ease with which the power is transmitted to any distance and in any direction. In this country, a single rope is carried round the pulley as many times as is necessary to produce the required power, and the necessary tension is obtained by passing the rope round a tension pulley weighted to give the desired tension. The ropes used in rope transmission are either of hemp, manila, or cotton. Manila ropes are mostly used in this country. They are of three strands, hawser laid, and may be from | in. to 2 in. in diameter. The weight of ordinary manila or cotton rope is about .3 D 2 lb. per ft. of length, where D represents the diameter of the rope in inches. Letting w = the weight per foot of length, w — .3 Z) 2 . The breaking strength of the rope varies from 7,000 to 12,000 lb. per sq. in. of cross-section. The average value may be taken as 7,000 Z) 2 . when D is the diameter of rope. 210 MACHINE DESIGN. II - For a continuous transmission, it has been determined by- experiment that the best results are obtained when the tension in the driving side of the rope is about ^ of the breaking strength. That is, 7 000 T ) 2 Ti = tension in tight side = -h — — = 200 D 2 . The ropes run in V-shaped grooves, and the coefficient of friction is, of course, greater than on a smooth surface. The coefficient for grooves with sides at an angle of 45° may be taken at from .25 to .33. The horsepower that can be transmitted by a single rope running under favorable conditions is given by the formula ' = ^( 20 0--^-V 825 V 107.2/ in which H = horsepower transmitted; D = diameter of rope in inches; v = velocity of rope in feet per second. The maximum power is obtained at a speed of about 84 ft. per sec. For higher velocities, the centrifugal force becomes so great that the power is decreased, and when the speed reaches 145 ft. per sec. the centrifugal force just balances the tension, so that no power at all is transmitted. Consequently, a rope should not run faster than about 5,000 ft. per min., and it is preferable on the score of durability to limit the velocity to 3,500 ft. per min. Example.— A rope flywheel is 26 ft. in diameter, and makes 55 rev. per min. The wheel is grooved for 35 turns of 1¥' rope. What horsepower may be transmitted ? Solution.— Velocity in feet per second = 26 X 7T X 55 v = : 60 Applying the formula, v D 2 1 H = 2 / V 2 \ 825 I 200 " 107.2)- the horsepower transmitted by one rope or turn is 74.9 X (H) 2 / 90n _ (74.9) 2 \ 825 \ 200 107.2 Then, 30.16 XS5 = 1,055.6 = horsepower transmitted by the 35 ropes. ')= 30.16. ROPE BELTING. 211 Example.— H ow many times should a 1" rope be wrapped around a grooved wheel in order to transmit 200 horsepower, the speed being 3,500 ft. per min.? Solution.— 3,500 ft. per min. = = 58| ft. per sec. Applying the formula, the horsepower transmitted with one turn is, Hence, 200 -r- 11.9 = 16.8, say 17 turns. Rope pulleys differ from belt pulleys only in their rims. The inclination of the sides of the grooves may vary from 30° to 60°. The more acute the angle, the greater the coefficient and, consequently, the wear on the rope. A section of a grooved rim in which the sides of the grooves are formed with circular arcs is shown in the figure. The proportions for this rim are as follows, using the diameter D of the rope as a unit: a = ££>; b = ZD + rV'l c = D; d = 1.6 D; e = i 2) + T V'; /= i2> + iV'; g = h = \D + The radii ri and r 2 are to be found by trial; they should be of such lengths as to make the curves drawn by them tangent to the required lines. 212 MACHINE DESIGN. The long radius R is determined by drawing a line through the center of the rope at an angle of 22£° with the horizontal, and producing it until it intersects a line drawn through the tops of the dividing ribs; then, with this point of intersection as a center, draw the curve forming the side of the groove tangent to the circumference of the rope. The advantage claimed for this groove is that the rope will turn more freely in it, thus presenting new sets of fibers to the sides of the grooves and increasing the life of the rope. The diameter of a rope pulley should be at least 30 times the diameter of the rope. Good results are obtained when the diameters of pulleys and idlers on the driving side are 40 times, and those on the driven side 30 times, the rope diameter. Idlers used simply to support a long span may have diameters as small as 18 rope diameters, without injuring the rope. When possible, the lower side of the rope should be the driving side, for in that case the rope embraces a greater portion of the circumference of the pulley, and increases the arc of contact. When the continuous system of rope transmission is used, the tension pulley should act on as large an amount of rope as possible. It is good practice to use a tension pulley and carriage for every 1,200 ft. of rope, and have at least 10$ of the rope subjected directly to the tension. Aside from the grooved rim, rope pulleys are constructed the same as other pulleys. They may be cast solid, in halves, or in sections. The pulley grooves must be turned to exactly the same diameter; otherwise, the rope will be severely strained. TRANSMISSION OF POWER BY WIRE ROPE. Wire rope for transmitting power is made up of 6 strands twisted about a hemp core, each strand being composed of either 7 or 19 wires, according to the size of the sheaves, the 19-wire rope being employed in cases where it is impracticable to use the larger sheaves required by the 7-wire rope. Where the conditions, however, do not preclude the use of the WIRE ROPE. 213 proper size of sheaves, the 7-wire rope is to be recommended in preference to the other, except sometimes on very- short spans, where 19-wire rope is to be preferred, composed of the same size of wires as the smaller 7-wire rope, such as would ordinarily be used to transmit the power, and run under a tension corresponding to the smaller rope, or con- siderably below the maximum safe tension of the rope used. This is done in order to avoid stretching, which would other- wise occur, and the consequent use of mechanical appliances for preserving the necessary tension. In flying transmission, where the rope makes a single half lap at each end, the sheaves are usually made of cast iron, with rims having grooves lined with segments of rubber and leather, dipped in tar, and laid in alternately, upon which the rope tracks. The diameters of the minimum sheaves, corresponding to a maximum efficiency, are as follows, according to a prominent manufacturer: Diam. of sheave for 7-wire steel rope, 77 times diam. of rope. Diam. of sheave for 19-wire steel rope, 46 times diam. of rope. Diam. of sheave for 7-wire iron rope, 160 times diam. of rope. Diam. of sheave for 19- wire iron rope, 96 times diam. of rope. In long-distance transmissions, where the rope makes 2 or more half laps at each end about a pair of drums or several sheaves, the rims may be lined with wood or the rope may be run in plain turned grooves. The horsepower capable of being transmitted is deter- mined by the general formula: N = [c D 2 — .000006 (w + g\ + g^)]v , in which D — diameter of rope in inches; v = velocity of rope in feet per second; w = weight of rope in pounds; < 7 i = weight of terminal sheaves and shafts; g>z = weight of intermediate sheaves and shafts; c = constant depending on the material of which rope is made, the character of the filling or sur- face material in the sheaves or drums upon which the rope tracks, and the number of half laps at each end. 214 MACHINE DESIGN. The values of c for from 1 up to 6 half laps for steel rope are given in the following table: c for Steel Number of Half Laps at Each End. Rope on 1 2 3 4 5 6 Iron. Wood. Rubber and Leather. 5.61 6.70 9.29 8.81 9.93 11.95 10.62 11.51 12.70 11.65 12.26 12.91 12.16 12.66 12.97 12.56 12.83 13.00 The values of c for iron ropes are one-half the above. It is apparent from this table that, when more than 3 half laps are made, the character of filling or surface in contact is immaterial so far as slipping is concerned. Where the distance is comparatively short, as in most flying transmissions, the effect of the weight of the rope and sheaves is so slight that it may be neglected, and we have the general rule, that the actual horsepower capable of being transmitted by a wire rope approximately equals c times the square of the diameter of the rope in inches, multiplied by the speed of the rope in feet per second. The tension of the rope is measured by the amount of sag or deflection at the center of the span, and the deflection corresponding to the maximum safe working tension is deter- mined by the following formulas, in which s represents the span in feet: Deflection. Steel Rope. Iron Rope. Still rope at center, in ft h = .00004 s 2 h = .00008 s 2 Driving portion, running, in ft... hi = .000025s 2 hi = .00005s 2 Slack portion, running, in ft .../^ = .0000875s 2 h ? = .000175s 2 In very long transmissions it often happens that the con- ditions will not allow of the required amount of tension to drive properly with but a single half lap on the pulley. In such cases it is customary to give the rope a sufficient num- ber of half turns around successive grooves in the driving pulley and a series of guide pulleys that serve to lead the rope from one groove on the driving pulley to the next. With this arrangement a guide pulley at one end of the PIPE FLANGES. 215 line is usually made to serve the purpose of a tension pulley by being mounted in a movable frame that can be drawn by means of a screw or a weight so as to give the rope the desired tension. PIPE FLANGES. The figure shows the method of flanging and bolting the ends of two cast- iron pipes. The dimensions of the flanges for the various sizes of pipes are given in the following table: Standard Pipe Flanges. n = number of bolts. a b c d n e / 9 2.0 .409 % 2.000 4 Vs 4.75 6.00 2.5 .429 % 2.250 4 ji 5.50 7.00 3.0 .448 VS 2.500 4 k 6.00 7.50 3.5 .466 vs 2.500 4 Ji 7.00 8.50 4.0 .486 k 2.750 4 it 7.50 9.00 4.5 .498 ft 3.000 8 if 7.75 9.25 5 .525 % 3.000 8 it 8.50 10.00 6 .563 3 / 3.000 8 1 9.50 11.00 7 .600 74 3.250 8 Its 10.75 12.50 8 .639 ft 3.500 8 ix 11.75 13.50 9 .678 % 3.500 12 IX 13.25 15.00 10 .713 3.625 12 l T s 14.25 16.00 12 .790 vs 3.750 12 l|f 17.00 19.00 14 .864 1 4.250 12 1% 18.75 21.00 15 .904 1 4.250 16 1% 20.00 22.25 16 .946 1 4.250 16 Its 21.25 23.50 18 1.020 Ws 4.750 16 1t 9 s 22.75 25.00 20 1.090 VA 4.750 20 lit 25.00 27.50 22 1.180 134 5.500 20 lit 27.25 29.50 24 1.250 1X 5.500 20 lfl 29.50 32.00 26 1.300 IX 5.750 24 2 31.75 34.25 28 1.380 134 6.000 28 2t x s 34.00 36.50 30 1.480 13Z 6.250 28 2ps 36.00 38.75 36 1.710 1/i 6.500 32 2% 42.75 45.75 42 1.870 IX 7.250 36 2/4 49.50 52.75 48 2.170 i/t 7.750 44 56.00 59.50 216 MACHINE DESIGN. LINING FOR SEATS. Seats for large bearings are often lined with Babbitt metal, or anti-friction metal. It has been found by experience that a bearing will run cooler when so lined, probably because the Babbitt metal, being softer, accommodates itself to the journal more readily than the more rigid gun metal. Some of the common methods of lining the seats are shown in the figure. At (a) the Babbitt metal is shown cast into shallow helical grooves; at (&), into a series of round holes; and at (c), into shallow rectangular grooves. Conse- quently, the journal rests partly on the brass and partly on the Babbitt metal. In cheap work, very frequently the seats are made entirely of Babbitt metal. A mandrel the exact size of the journal is placed inside the bearing, and the melted Babbitt metal is poured around it. In better work a smaller mandrel is used, and the metal is hammered in, the bearing being then bored out to the exact size of the journal. CYLINDERS AND STEAM CHESTS. Fig. 1 shows a cylinder designed for a simple slide-valve engine. The front head A is cast solid with the cylinder. The method of fastening to the frame B is clearly shown. The principal dimensions of this cylinder may be deter- mined from the following proportions: D = diameter of cylinder; L = length of stroke + thickness of piston + twice the '•piston clearance; C = length of stroke + distance from outer edge to outer edge of piston rings — (.01 D + .125"); a = 5.5 i; CYLINDERS AND STEAM CHESTS. 217 b = 4.2 i; c = i ; d = i; e' = net area of a single cylinder-head bolt whose nominal A P diameter is e = . , 4,000 n where A = area of cylinder head in square inches; P = steam pressure; n = number of bolts. The pitch of the bolts may be from 4.5 to 5.5 in., but should never be more than 5 /. / = 1.5 v, g = .04 D + .125". Take the nearest nominal size pipe tap. h = twice the outside diameter of drain pipe. i ■— .0003 PD + .375", where P is the steam pressure. If the steam pressure is less than 100 lb., make P = 100. 3 = .85 i; k = 4i; l = .75 i; ' m = 1.01 D + .125"; n = m + 6e, never less. Here, e is the nominal diameter of the bolt. o = the nominal diameter of steam-chest bolts. The net A'P area of a single steam-chest bolt = — — r — -» & 4,000 n' where A ' = area of steam chest; n' = number of bolts*ln steam chest. p = 2.75 o; q = 1.5 r; r = 1.25 i; s = i. This is required only when the length of the port is greater than 12 in. t = 1.25 i. When D is greater than 24 in., use 4 bolts in the standard and make t = 1.1 i. u= 1.5 i; v — .25" (constant). The dimensions of the steam ports , exhaust ports , and other steam passages depend on the velocity of the flow of steam. The ports and passages must be large enough to allow the steam to follow up the advancing piston without loss of 218 MACHINE DESIGN. Fig. CYLINDERS AND STEAM CHESTS. 219 pressure. The maximum allowable velocity of the steam in the passages, when they are short, is about 160 ft. per sec. But, with the ordinary ratio between the length of connect- ing-rod and length of crank, the average velocity is about five-eighths of the maximum. Hence, the allowable average velocities are 100 to 125 ft. per sec. for long and short passages, respectively. Let l — length of port in inches; 6 = breadth of port in inches; A — area of cylinder; S = average piston speed in feet per second; v — average velocity of steam in feet per second. Then, area of port X velocity of steam = area of piston X velocity of piston, or Ibv = AS; whence, ib-±* v For long indirect passages, take v = 100; and for short direct passages, take v = 125. The constant 100 may be used for v, when designing plain slide-valve engines of the ordinary type, which cut off late in the stroke, and 125 may be used for high-speed engines with early cut-off, and for the Corliss type. The area of the exhaust port or ports may be from If to 2* times the area of a steam port. The area of the cross-section of the steam pipe is approxi- mately equal to the area of the steam port; likewise, the area of the exhaust pipe should be equal to that of the exhaust port. The length l of the port may be .6 D to .9 D for slide-valve engines, and about .9 D to D for the Corliss type. The height w, Fig. 1, of the valve seat must be such that the area of the most contracted part of the exhaust port is not less than 75$ of the area of the steam port. THE STEAM CHEST. Fig. 2 shows a steam chest for the cylinder illustrated in Fig. 1. The principal dimensions are to be determined by the following proportions, which are based on the thickness i of the cylinder walls, and on the travel and dimensions of the valve: 220 MACHINE DESIGN. a = length of valve + travel of valve -f twice the clear- ance between the valve and the steam chest at ends of valve travel; b = breadth of valve + twice the clearance between one valve and steam chest; c = .75 i; d = 2.75 o, where o is the nominal diameter of the steam- chest bolts, as in Fig. 1; e = .04 j/ A' H- .125" for all areas above 100 sq. in. A' = area of steam chest, outside measurement, in square inches; / = 1.3 e; g = .85 i ; h — height of valve + necessary clearance; t = .85 i ; j = 2.5 i. Note.— W hen the area of the steam-chest cover is less than 100 sq. in., its thickness e may be made equal to i. If the area of the steam-chest cover exceeds 600 sq. in., the height of the ribs should be 3.5 i, and their number should be increased. CYLINDERS AND STEAM CHESTS. 221 Fig. 3 shows a design for a steam-chest cover when the steam-pipe flange is on one side of the steam chest. Deter- mine the thickness e-by the same formula and rules as for the cover in Fig. 2. The other dimensions are found as follows: c = .75 e; j = 2.6 c; / = 1.3 c; r = 6 c. p should never exce ed the distance in inches given by the /40 C1 2 formula p tion expressing the thickness of the cover in sixteenths of an inch, and p g is the gauge boiler pressure in pounds per square inch. Example.— Find the maximum pitch of the ribs for a cover £§ in. thick, subjected to a steam pressure of 160 lb. per sq. in. Solution. where Ci is the numerator of the frac- Substituting in the formula for p, we have 140 X J f = v-5- = V ;40 X 15 2 = 7.5 in. p g \ 160 Fig. 4 shows a Corliss engine cylinder that may be designed according to the following proportions: D — diameter of cylinder. a = 1.21 D + 2c + 1.22"; b = 2D + 1.125"; c = .048 D; & = .079 D; d = .17 D; c = .0003 P D + .375", if boiler pressure is above 100 lb.; otherwise, c = .03 D + .375"; / = .82 c; g = -9e; h = b +2(c + p); h' = h; i = 1.8 c; j = e; k = 1.2c; l = 1.7 x + 2" — 1.2 c, where x = diameter of piston rod; V = .32 D, about; 222 MACHINE DESIGN. m — .25 D; u n = .32 D; 0 = 1.25 e; V P = 1.3 e; Q = .252); w Q' .32 D; r = 1.2 e; y s = 1.5 c; z = e; take diameter nearest standard size bolt; = 1.2 e; take diameter nearest standard size bolt; = 1.7 x + 2.25", where x =* diameter of piston rod; = D) = 1.5 c. -Mi T ex 1 /f k\ 1 i ^ )mtu pan PISTONS. 223 A is to be made according to proportions given on page 215. Bolts to be made according to the same table. Note. — The bolts for cylinder heads are to be calculated from the formula given for cylinder-head bolts in connec- tion with Fig. 1. In this cylinder the stuffingbox £ is a separate piece that is to be bolted to the cylinder head. CRANK-SHAFTS. For high-speed, automatic short-stroke engines, the follow- ing formula corresponds with good practice: d = .44 D + where d is the diameter of shaft and D is the diameter of cylinders. For the Corliss type, in which the stroke is equal to or greater than twice the diameter, d = .34 D + 2£", when D is equal to or greater than 16 in. When D is less than 16 in., d = %D. PISTONS. A form of piston that is much used is shown in the follow- ing figure. It consists simply of a hollow circular disk of 224 MACHINE DESIGN. The packing rings $, s are made of cast iron, and are split and sprung into place. Their elasticity causes them to press against the cylinder walls and thus prevent the leakage of steam. The following proportions will give dimensions suitable for this piston: D = diameter of cylinder in inches; a =* .2D + 1.5"; e = .75c; 6 = diameter of piston rod; r = .5 c; 6' — 26; p = coreplug; c = .18/2 D -.1875"; number of ribs = .08(2) +34). CONNECTING-RODS. The figure shows a strap-end connecting-rod. The straps Ci and c 2 are fastened to the ends of the rods by means of the gibs % and a 2 and the cotters 6i and 6 2 . The cotters are held in place by the setscrews Si and s 2 . Small steel blocks shown between the ends of the setscrews and the cotters are used to prevent injury of the cotter by the setscrews. The rod, cotters, gibs, and straps may be made of either wrought iron or steel. The crankpin brasses are shown babbitted and wristpin brasses without babbitt. The brasses are adjusted by means of the cotters, which draw the straps farther on to the rod when they are driven in. The dimensions for the rod are given by the following proportions: For wristpin end: D = diameter of cylinder; d = .2 D = diameter of wristpin; n = .155 2) + .0625"; x = - w 2 = a factor for use 4 in finding proportions below; a = .75 d + .125"; a' = .75 d + .125"; 6 = j/z5x; c = .256; e = .125 d; f = .26 D + .5" for cylinders to 26" in diameter, and / = .28 D for cylinders above 26" in diameter; g = 1.3 n; .5 s < g-c' .32 s h ’ h = CONNECTING-RODS. 225 k - — • 1.8 d’ l = .375 b ; o = .25 5; m = 1.35 d for wristpins up to 3.5" in diameter, andm = 1.48 n for pins above 3.5" in diameter; 226 MACHINE DESIGN. p = .33 6; Q = 1.125 d for wristpins up to 3.5" in diameter, and q = 4", constant, for pins above 3.5" in diameter; The taper of the cotter is | in. per foot. Proportions for the crankpin end: D = diameter of cylinder in inches; d' = .28 D = diameter of r = n; s = .125 d; t = 1.35 d; u = .02 D + .25"; v = .125 d. .32jr' h crankpin; n f = 1.1 n\ (n = .155 D + .0625"); x ’ = ~ n ' 2 = a factor used 4 below; a = .75 d'; a' = .75 d'; 6 = j/ 2.5 x'; c' = .25 6; e = .125 d'; f = .26 D for cylinder diam- eters up to 26", and / = .28 D for cylinders above 26" in diameter; g = 1.3 n = same as wrist- pin end; The taper of the cotter is $ i: , = x same as wristpin 1.8 d end; l = .375 6; m = 1.3 d'\ o = .25 6; p = .33 6; q = same values as for wristpin end; r = 1.1 n; s = .125 d; * = 1.35 d'; v = .125' (constant); w = .02 Z> + .0625"; where D = length of rod, and >8 = stroke, both in inches. l. per foot. ECCENTRIC AND STRAP. The figure shows an eccentric sheave and strap, both of cast iron. The eccentric sheave is cast solid, and must be slipped over end of shaft. The eccentric rod is held in a boss on the strap by a cotter. For eccentrics used with valve stems £ in. in diameter or less, holes for bolts j are not to be cored. A = boss for oil cup; B = cross-section of rib r. ECCENTRIC AND STRAP. 227 The proportions are as follows: D = diameter of valve stem; d = diameter of shaft; Qi = d -f- 2 q -}- 2 ft; b = 2D + .125"; V = 2.25 D + . 125"; c = 1.5 D; e = .75 D; e' = .75 D; / = .7 D; g = 1.25 d; ft = D + .125"; i = .25 D + . 0625"; j = area of bolt at root of thread = .38 D 2 ; use the nearest standard l =3\ d-\-2q-\-2h-\~ 2 f m = g : m'= m; = D + .125"; n' = D + .125"; o = .75 j; P — D; q — eccentricity; r = D; s = 1.25 D; * = 2.25 D + 1.25"; u = D; v = 2.25 D; v' = 1.125 D; size bolt; f =j + - 1875; ft = 4D; w = 2.5 D; x = 2.25./. 228 MACHINE DESIGN. STUFFINGBOXES. The stuffingbox ol the form shown in the figure is generally used for small work, such as the spindles of valves, etc. The outside of the stuffingbox is threaded to receive a hexagonal nut that fits over the gland. As the nut is screwed down, the gland is pressed down- wards and compresses the packing. The proportions used are: d — diameter of rod; a = 2.5 d + .5"; b = 1.5<2 + .125"; c = 3d + .25"; e = 3.5(2 + 625"; / = d + .125"; g = 2d + .25"; h = 1.5 <2 + .25"; i = .25(2 + .0625"; k = .5 d. This design may be used for rods up to H in. in diameter. Make the number of threads per inch the same as for a bolt whose diameter is equal to the diameter of the rod. GEARING. The circular pitch of a gear-wheel is the distance in inches measured on the pitch circle from the center of one tooth to the center of the next tooth. If the distance of the teeth of a gear thus measured were 2£ in., we would say that the circular pitch was 2£ in. Let P = circular pitch; D = diameter of pitch circle, in inches: C = circumference of pitch circle, in inches; N = number of teeth; 7t = 3.1416. GEARING. 229 Then, P = C ttD C n D n oi ^t n =p oi -f- C = PA or 7T D. Addendum = .3 P. Root = .4 P. The thickness of the teeth for a cut gear is equal to .5 P, and for a cast gear .48 P. The diametral pitch of a gear-wheel is the name given to the quotient that is obtained by dividing the number of teeth in the wheel by the diameter of the pitch circle in inches; or, the diametral pitch may be defined as the number of teeth on the circumference of the gear-wheel for 1 in. diameter of pitch circle. A gear -with a pitch diameter of 5 in., and having 40 teeth is 8 pitch; one with the same pitch diameter and having 70 teeth is 14 pitch. In the gear of 8 pitch there are 8 teeth on the circumfer- ence for each inch of the diameter of the pitch circle; and in one of 14 pitch there are 14 teeth on the circumference for each inch of the diameter of the pitch circle. Let P = diametral pitch; D = diameter of pitch circle, in inches; N = number of teeth; d = outside diameter; l = length of tooth; t = thickness of tboth; i>.= * D N jr n=pd. N + 2 2.157 a ~ P • P ‘ _ i - 57 _ P* The circular pitch corresponding to any diametral pitch may be found by dividing 3.1416 by the diametral pitch; and the diametral pitch corresponding to any circular pitch may be found by dividing 3.1416 by the circular pitch. (а) If the diametral pitch of a gear is 6, what is the cor- responding circular pitch? (б) If the circular pitch is 1.5708 in., what is the corre- sponding diametral pitch? , . 3.1416 coo ^ . ... 3.1416 <°> "IT “ - 5236m - (6) 1T5708 - 2 ' 230 ELECTRICITY. Diametral Pitches With Their Corresponding Circular Pitches. Diametral Pitch, or Teeth, per Inch in Diameter. Correspond- ing Circular Pitch. Diametral Pitch, or Teeth, per Inch in Diameter. Correspond- ing Circular Pitch. 1 3.1416 8 .3927 2 1.5708 9 .3491 3 1.0472 10 .3142 4 .7854 12 .2618 5 .6283 14 .2244 6 .5236 16 .1963 7 .4488 20 .1571 ELECTRICITY, PRACTICAL UNITS. The volt is the practical unit of electromotive force or elec- trical pressure. It is that electromotive force which will main- tain a current of 1 ampere in a circuit whose resistance is 1 ohm. The electromotive force of a Daniell’s cell is 1.072 volts. The ampere is the practical unit denoting the strength of an electric current, or the rate of flow of electricity. It is that strength of current or rate of flow which would be maintained in a circuit whose resistance is 1 ohm by an electromotive force of 1 volt. One ampere decomposes 00009342 gram of water (H 2 0) per second; or deposits .001118 gram of silver per second. The ohm is the practical unit of resistance. It is that resist- ance which will limit the flow of an electric current under an electromotive force of 1 volt to 1 ampere. The legal ohm is the resistance of a column of mercury 106 centimeters long and 1 square millimeter sectional area at 0° C. One mile of pure copper wire in. in diameter has a resistance of 13.59 ohms at a temperature of 59.9° F, PRACTICAL UNITS. 231 To makeHhe significance of these units clearer, take the analogous case of water flowing through a pipe under a pres- sure of a column of water. The force that causes the water to flow is due to the pressure or head; the flow or current of water is measured in gallons per minute; and the resistance that opposes or resists the flow of water is caused by the fric- tion of the water against the inside of the pipe. In electrotechnics, the electromotive force or electrical potential expressed in volts corresponds to the pressure or head of water; and the resistance in ohms to the friction in the pipe. The unit that expresses the rate of transmission of electricity per second is called the ampere , while the flow of water is ex- pressed in gallons per minute. In either case the strength of current or rate of flow depends on the ratio between the pressure and the resistance; for, as the pressure increases, the current increases proportionately; and as the resistance increases, the current diminishes. This relation, as applied to electricity, was discovered by Dr. G. S. Ohm, and has since been called Ohm's law. Ohm’s Law. — The strength of the current in any circuit is directly proportional to the electromotive force in that circuit and inversely proportional to the resistance of that circuit , i. e. t is equal to the quotient arising from dividing the electromotive force by the resistance. Let E = electromotive force in volts; R = resistance in ohms; C = strength of current in amperes. Then C = f . R = E — CR. 1C (J Example.— The electromotive force of a circuit is 110 volts, and its resistance is 55 ohms; what is the strength of current? Solution.— E = 110 volts. R = 55 ohms. C = ^ ~ xl 55 = 2 amperes. The unit by which electrical power is expressed is called the watt. It is that rate of doing work when a current of 1 ampere is passing through a conductor under an electro* motive force of 1 volt, and is equal to ^ of a horsepower. 232 ELECTRICITY. Let E = electromotive force in volts; C = strength of current in amperes; E = resistance in ohms; W = power in watts; H. P. = horsepower. E 2 W = EX C= C 2 X E = -5-. K H. P. = EX 0 C 2 X E E 2 W 746 ~ 746 ~ EX 746 ~ 746' One kilowatt is equal to 1,000 watts: sometimes abbrevi- ated to K. W. Watt hour is a unit of work. It is used to indicate the expenditure of an electrical power of 1 watt for 1 hour. Example. — The resistance of a lighting circuit is 5 ohms and the electromotive force is 110 volts, (a) What is the amount of electrical power in watts required for this current? (6) What is the equivalent horsepower? Solution.— E = 110. E — 5. E 2 ~E E 2 E X 746 110 2 = 2,420 watts. 5X 746 Conductivity is the name given to the reciprocal of the resistance of any conductor. There is no unit by which to express conductivity. Note. — The reciprocal of any number is unity divided by that number. Thus, the reciprocal of 2 is £ or .5. CURRENTS. RULES FOR DIRECTION OF CURRENT, ETC. To determine the direction of a current in a conductor by the aid of a compass: Rule . — If the current flows from the south pole over the needle to the north , the north end of the needle will point towards the west , as in Fig. 1. If the compass is placed over the conductor so that the current will flow from the south under the needle to the north , the north end of the needle will point towards the east , as in Fig. 2. CURRENTS. 233 To determine the polarity of an electromagnet: Rule.— In looking at the face of a pole (Fig. 3), if the current Fig. 3. ^rRBCTTOS or UmSB OF FO&CLE. flows in the direction a , of the hands of a watch , it will he a south pole, and if in the opposite direction h, it will he a north pole. To determine the direction of an^induced current in a conductor that is moving in a magnetic field: Rule.— Place thumb , forefinger , and middle finger of right hand, each at a right angle to the other two, as shown in Fig. h; if the forefinger shows di- rection of lines of force and the thumb the direc- tion of motion of conduc- tor, then the middle finger will show the direction of the induced current. Fig. 4. Note.— The above rule will give the polarity of a dynamo. To determine the di- rection of motion of a conductor carrying a cur- rent when placed in a magnetic field: 234 ELECTRICITY. Rule . — Place thumb , forefinger , and middle finger of the left hand , each at a right angle to the other two , as shown in Fig. 5; if the forefinger shows the direction of the lines of force and the middle finger shows the direction of the current , then the thumb will show the direction of motion of the conductor . Note.— T he above rule will give the polarity of a motor. DERIVED OR SHUNT CIRCUITS. A circuit divided into two or more branches, each branch transmitting part of the current, is said to be a derived circuit; the individual branches are in multiple-arc, or parallel with each other. To find the joint resistance of a derived circuit: Rule. — As the conductivity of any conductor is equal to the reciprocal of its resistance , then the joint conductivity of two or more circuits in parallel is equal to the sum of the reciprocals of their separate resistances. The joint resistance of two or more circuits in parallel is equal to the reciprocal of their joint conductivity. In a derived circuit of three branches, let r\, r 2 , and r 3 be the resistances of the three branches, respectively. Their joint conductivity, or the sum of the reciprocals of their resistances, is 1,1 ,1 or r 2 r 3 -f r x r 3 + r x r 2 r\ r % r 3 n r 2 r 3 Their joint resistance is, therefore, L__ or r ir 2 r 3 r 2 r 3 + n r 3 + r x r 2 * r 2 r 3 + r x r 3 + n r 2 * n r 2 r 3 The joint resistance of a derived circuit with but two branches in parallel may be thus expressed: product of their resistances sum of their resistances Example.— T he resistances of two branches of a derived circuit are 20 and 30 ohms, respectively. Find their joint resistance. Solution. — product of their resistances _ 600 _ 12 o ^ mg sum of their resistances ~ 50 WIRING, 235 To find the strength of current in the separate branches of a derived circuit: Rule. — A current is divided among the branches of a derived circuit in 'proportion to their conductivities — i. e., to the reciprocal of their resistances. Example.— If the resistances of the two branches A and B of a derived circuit are 20 and 30 ohms, respectively, and the total current in the main circuit is 60 amperes, what is the current in each ? The conductivity of A is ^ and of B ^ . Solution.— If Ci represents the current in A, and C 2 represents the current in B, then, Ci : C 2 = ■ aV Hence £ _ * _ » _ « C6 ’ C 2 “ *’° r C 2 “ 20 “ 2- Now. Ci + C 2 = 60, or C 2 = 60 - Ci. Ci _ 3. 60— Ci 2’ Ci = 36, and C 2 = 24. Substituting, WIRING. INTERIOR WIRING. A mil is a unit of length used in measuring the diameters of wires, and is equal to .001 in. A circular mil is a unit of area used in measuring the cross-sections of wires, and is equal to S( l- in - The sectional area of a wire expressed in circular mils is equal to the square of its diameter in mils. Let c. m. = circular mils; C = total current in amperes; c = current in amperes to each lamp; n = number of lamps in multiple; v — volts lost in line; r = resistance per foot of wire; d = distance from dynamo to lamps. The resistance of 1 ft. of commercial copper wire, 1 mil in diameter, at a temperature of 75° F., is 10.8 ohms. 236 ELECTRICITY. A 16 c. p. (candlepower) 110- volt lamp takes about .5 ampere; a 16 c. p. 55-volt lamp takes about 1 ampere. All calculations for size of wire must be checked by com- paring with a table of safe carrying capacity (see table on pages 238 and 239), and the current value there given must not be exceeded. To find the size of wire for 110-volt circuit with 16 c. p. lamps: v r = — v. nd „ . 10.8 nd For large cables, c. m. = • — - — . Example. — Find the size of wire necessary for a circuit supplying current to 50 110-volt 16 c. p. lamps, 300 ft. from the dynamo, allowing a loss of 5 J in line. Solution.— Volts at dynamo = i?? = 115.8. .9o Volts lost in line = 115.8 — 110 = 5.8 = v. Then ’ r = JT3 = 5oHoO = - 000386 0hm PCT ft " = .386 ohm per 1,000 ft. The nearest size of wire, as given in the table on page 238, is No. 6 B. & S., and its current capacity is 35 amperes; there- fore it is safe. To find the size of wire for a 55-volt circuit with 16 c. p. lamps: For large cables, c. m. = 2 nd' 21.6 nd Example.— What size of wire should be used for supplying current to 75 16 c. p. lamps on a 55-volt circuit, the distance from dynamo being 230 ft., and line loss, 4 volts? Solution.— r = 2 h = 2 X 75 X 230 “ - 000116 ° hm P6r ft " = .116 ohm per 1,000 ft. By referring to the table, (page 238) the nearest wire is found to be No. IB. & S., and its carrying capacity is greater than the current (75 amperes) that it is to conduct. INTERIOR WIRING. 237 To find the size of wire for any circuit on a 2-wire system: In general, r = c^Td’ or, c. m. = 10 ' 8 x 2 ^ X C V Example.— What wire should he used to carry 450 amperes a distance of 600 ft., the allowable drop being 60, and the E. M. F. at the end of the circuit 115 volts? 115 Solution.— Volts at dynamo = — = 122.3. Volts lost in line = 7.3. Then, c. m. = 10.8 X 2 X 600 X 450 7.3 798,900. Comparing this number with the table on page 239, giving current capacity of cables, it will be seen that it is within the prescribed limits. These formulas may be used for feeders, mains, branch mains, service mains, and inside wiring on continuous-current circuits, and for secondary wiring on alternating systems. To find the size of wire for a 110-volt circuit, 3-wire system, 16 c. p. lamps: 4 v r — — = for each wire. n d For large cables, 2.7 n d . , c. m. = for each wire. v In checking for carrying capacity, remember that the wire carries only one-half the current that would be used on a 2-wire system, as the voltage between the outside conductors is double the voltage at the terminal of 1 lamp. Example.— What should be the size of the conductors for a 3-wire system, when 132 110-volt, 16 c. p. lamps are installed at a distance of 210 ft. from the source of supply, the loss being 4 volts? Solution.— r = isl$lio = • 000B77ohmperft -’ = .577 ohm per 1,000 ft. This would call for a wire between Nos. 7 and 8. The 238 ELECTRICITY. 132 X 5 current will be = 33 amperes; but this is too much for the wire to carry, and No. 6 B. & S. wire should be used, notwithstanding the somewhat less drop in volts that will result. For continuous-current circuits, 5$ loss is usually allowed, with full current - from the dynamo to the lamps. For long distances a larger line loss may be allowed, if the dynamo is wound for that loss. Dimensions, Weight, and Resistance of Copper Wire. w Diameter in Mils (d). lmil = .001 in. Area. Weight and Length. Resistance at 75° F. Ohms per 1,000 ft. Current. Amperes. B. & S. Gauge. Circular Mils (d 2 ). Lb. per 1,000 Ft. Ft. per Lb. Exposed. 1 Concealed. 0000 460.000 211,600.0 639.33 1.56 .049 300 175 0000 000 409.640 167,805.0 507.01 1.97 .062 245 145 000 00 364.800 133,079.0 402.09 2.49 .078 215 120 00 0 324.950 105,592.0 319.04 3.13 .098 190 100 0 1 289.300 83,694.0 252.88 3.95 .124 160 95 1 2 257.630 66,373.0 200.54 4.99 .156 135 70 2 3 229.420 52,634.0 159.03 6.29 .197 115 60 3 4 204.310 41,742.0 126.12 7.93 .248 100 50 4 5 181.940 33,102.0 100.01 10.00 .313 90 45 5 6 162.020 26,250.0 79.32 12.61 .395 80 35 6 7 144.280 20,817.0 62.90 15.90 .498 67 30 7 8 128.490 16,509.0 49.88 20.05 .628 60 25 8 9 114.430 13,094.0 39.56 25.28 .792 9 10 101.890 10,381.0 31.37 31.88 .999 40 20 10 11 90.742 8,234.1 24.88 40.20 1.260 11 12 80.808 6,529.9 19.73 50.69 1.589 30 15 12 13 71.961 5,178.4 15.65 63.91 2.003 13 14 64.084 4,106.8 12.41 80.59 2.526 22 10 14 15 57.068 3,256.7 9.83 101.65 3.186 15 16 50.820 2,582.9 7.80 128.17 4.017 15 5 16 17 45.257 2,048.2 6.19 161.59 5.066 17 18 40.303 1,624.3 4.91 203.76 6.388 10 18 19 35.890 1,288.1 3.89 257.42 8.055 19 20 31.961 1,021.5 3.08 324.12 10.158 5 20 INTERIOR WIRING. 239 Carrying Capacity of Cables. Area. Circular Mils. Current. Amperes. Area. Circular Mils. Current. Amperes. Exposed. Concealed. Exposed. Concealed. 200,000 299 200 1,200.000 1,147 715 300.000 405 272 1,300,000 1,217 756 400.000 503 336 1,400,000 1,287 796 500.000 595 393 1,500,000 1,356 835 600.000 682 445 1,600.000 1,423 873 700.000 765 494 1,700,000 1,489 910 800.000 846 541 1,800,000 1,554 946 900.000 924 586 1,900.000 1,618 981 1.000.000 1,000 630 2,000,000 1,681 1,015 1,100,000 1,075 673 To find the size of wire on primary circuits for alternating svstem: 10.8 X 2 d X C l v ~ v ’ ! ~ C 1 X 2 & S. gauge . DYNAMO DESIGN. The fundamental principle of dynamo design is expressed by the formula _ NCn * 10 8 X 60’ in which E = electromotive force in volts given by the dynamo; N = number of lines of force used to magnetize the armature; C = number of conductors in a bipolar machine, measured all round the outside of the armature (whether in one DYNAMO DESIGN. 255 or more layers), or in a multipolar machine, as measured from a point opposite one north pole to a corresponding point opposite the next succeeding north pole; n = number of revolutions per minute of the armature. For example, a 2-pole dynamo has 2,000,000 lines of force passing from the north pole through the armature to the south pole; there are 200 conductors on the surface of the armature, and the speed is 1,500 rev. per min. The electro- motive force generated will then be E = 2,000,000 X 200 X 1,500 100,000,000 X 60 = 100 volts. If a 4-pole dynamo were used, having a 4-circuit armature and 4 sets of brushes, with 1,000,000 lines of force passing through any one pole piece, then the total number would be 2,000,000, because the same lines of force pass into a south pole that emerge from a north pole. With the same arma- ture as above, the number of conductors to be counted is only 100, as taken from one north pole to the next, and the electromotive force is E = 2,000,000 X 100 X 1,500 100,000,000 X 60 = 50 volts. For determining the number of lines of force required in a specific case, the above formula may be reversed, and we have AT EX 10 8 X 60 These lines of force have a cir- cuit to traverse composed of three different paths. One of these is through the field magnet and yoke M, Fig. 1; next, through a double air gap (?; and, lastly, through the armature core A. A given density of lines of force may not be ex- ceeded, this limit being for ordi- nary cast iron about 50,000 lines per square inch; for wrought-iron forgings or cast steel, about 90,000; and for soft sheet iron, 110,000. The ratio of magnetization to magnetizing force is called Ones of Force per Square lnok» 256 ELECTRICITY. Fig. 2. DYNAMO DESIGN. 257 the 'permeability. The permeability of air is very low, the intensity of magnetization being a direct measure of the magnetizing force required; therefore, the air gap is usually made short. In order to drive the lines of force through the magnetic circuit, magnetizing coils are wound on the cores at M, M. A certain number of ampere-turns will be required, depending on the density of the lines of force and the permeability of the different portions of the circuit. The number of turns may be found by taking a convenient current value, and dividing the ampere-turns by this. Reference to a wire table will then determine whether the resistance of the wi*e will be such that the terminal E. M. F. of the machine will supply the proper current. A margin should be allowed for regulating, and for the increase in resistance due to rise in temperature, which is about A +5 I> : °° T- t r- ! o rH c4 05 w -4 r4 t3. : •l-t e «-H P o : p^ 1—1 p3 • o> *.3 o-g : 02 Q • . N p £ „,P : § a S O d o c3 II <12 Q. • dpS a S'* g wco 0) « O d 2 Co lai £ i 5? !z i W s H « S a -rCj CO o3 ; o „ -C :S| pi-— Pi — .3 3 § ,S d ^ « ^r-H^^H Q-— cc cc co o3^° <3J .2 S« ' .2 2 CC O L rn *. 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"oS P ^ ^ ^ V, ft k >3 p • 'drd 5 m ±: E _ rH V C P rQ 'd « c jS P O Sl^St ft o ft 266 ELECTRICITY. STORAGE BATTERIES. 267 STORAGE BATTERIES. Storage batteries or accumulators are composed of plates of prepared lead, placed side by side in glass cells or wooden boxes lined with rubber or lead, alternate plates being con- nected together, thus forming two sets, which constitute the positive and negative elements. The plates are entirely sub- merged in dilute sulphuric acid, specific gravity 1.17. The charging E. M. F. is about 2.5 volts per cell, so that, if 10 cells are connected in series, the required E. M. F. will be 25 volts. The discharging E. M. F. is usually taken as 1.9 volts, so that an installation to supply current at 115 volts should consist of 115 — = 61 cells, with a few added to replace any that are out of order or to serve as regulators to vary the E. M. F. As soon as the battery is set up and the electrolyte added, the charging should commence, the first charge being continued a long while at a comparatively slow rate. Observe that the direc- tion of current through the cell in charging is from the positive or brown plate to the negative or gray one. Discharging should be at a low rate, as rapid discharge leads to deteriora- tion of the positive plates. The rating of the capacity of accumulators is usually made on the basis of a discharge current that will cause the E. M. F. to fall to 1.8 volts in 10 hours, but it is well to stop discharging when the E. M. F. falls to 1.9 volts. Storage-Battery Regulation.— -In electric-lighting plants, an equalization of load on the dynamos is sometimes obtained by installing accumulators or storage batteries. Automatic or hand regulation may be employed, the usual method being to cut out one or more cells when the load is light and change the remainder, these cells being connected in again when the load rises. The following method obviates the many disadvantages of this system. A shunt dynamo d, Fig. 1, supplies current to the lighting mains m, n, this current passing through the fields c of a low- voltage dynamo or booster 6, driven by a shunt motor and con- nected across the mains in series with the battery B. The E. M. F. of the dynamo d is a little greater than that of the battery, so that it will charge the battery when there is no 268 ELECTRICITY. Sa. external load. When all the lights are turned on, the booster field will be fully energized, and the E. M. F. of the booster will be added to that of the bat- tery, thereby causing the battery to discharge and assist the dy- namo. At a medium load, the battery will be neutral, neither taking current nor discharging, while the dynamo is running at full load. Any increase that may be made in the load will then be taken up by the battery. In electric-railway plants the dynamos are usually over- compounded, thus giving a higher E. M. F. at the brushes at full load than at light load. In a case of this kind, a differential winding is employed, as shown in Fig. 2, which Fig. 1. causes the booster to work both ways. On light loads a differential winding will assist the dynamos d' and d" to charge the battery, raising the E. M. F. to the required value; but on heavy loads the series winding c will over- power the shunt s, and the battery will discharge into the outer circuit. The shunt field must be regulated so that the total charging and discharging that is done within a given time will balance each other, as the battery will otherwise tend either to overcharge or to undercharge. If the shunt field is strengthened, it will cause the batteries to charge, while if the field is weakened, it will cause the batteries to discharge at a lower value of the external load than before. ELECTRIC GAS LIGHTING 269 ELECTRODEPOSITION. For electrodeposition of metals, low-resistance primary bat- teries giving from 2 to 10 volts may be used when the work is on a small scale. For larger work, accumulators may be employed, or the current may be taken directly from a low- voltage dynamo. The electroplating bath consists of a solu- tion that has little or no chemical action on the objects to be plated, and that are suspended in it and electrically connected to the negative pole of the battery. The anode is a plate of the metal that it is desired to transfer; it is also submerged in the solution and connected through a resistance, if necessary, to the positive pole of the battery. For deposition of copper, the bath is made by taking 4 parts saturated solution of sul- phate of copper mixed with 1 part of water containing one- tenth its volume of sulphuric acid. The current used must not exceed 18 amperes per square foot of surface of cathode. For nickel, use the double sulphate of nickel and ammonia, specific gravity 1.03; the current density must be low, and the solution should be neutral or slightly alkaline, as an acid bath will cause the nickel to peel off. For silver, the bath is a solution of cyanide of silver dissolved in cyanide of potas- sium. For gold, use cyanide of gold dissolved in cyanide of potassium. This solution is kept at 150° F. while in use. ELECTRIC GAS LIGHTING. The arrangement of the apparatus required for electric gas lighting is shown in the figure, A battery of about 6 Leclanch6 cells c, c, etc., joined up in series, is connected to one terminal of a spark coil k, the other terminal of which is soldered to a gas pipe p. The wire from the free end of the 270 ELECTRICITY. battery is carried up through the house, and branches are run to the burners as at 5, wherever needed. The insulation of this wire must be very thorough, special precautions being taken when it is carried through or along the fixtures. The burners are provided with a chain a attached to a movable contact spring, which is drawn past the burner, producing a spark of sufficient intensity to ignite the gas if it is previously turned on. In multiple gas lighting, a fine wire is run from one burner to another of a group, as on a chandelier, leaving a small air gap at each one, and a current of very high tension is used, generated by a small frictional machine, causing a spark at each burner. The last contact in a series of burners is con- nected to the gas pipe. THE WHEATSTONE BRIDGE. A diagrammatic sketch of the Wheatstone bridge is shown in Fig. 1. This instrument is widely used for the determina- tion of unknown resistances, and consists of such an arrange- ment of three circuits, M, N, P, of variable resistance, that the value of a fourth may be found from their relation. This unknown resistance is connected between the points b and c, and the battery B be- tween a and b. The variable resistances are then so adjusted that there shall be no difference of potential between c and d , which form the termi- nals of the galvanometer G. The drop in potential from a to c will then be the same as from a to d, and a c bears the same proportion to a c b as a d bears to adb. From this it follows MP Fig. 1. that ac: ad = cb : db, or the unknown resistance X — M 10 For a certain test, the ratio of the arms, — = — — . N lOO N ' On CABLE TESTING. 271 adjusting the resistance P, a balance is obtained when it is equal to 7,800 ohms. Then, „ 10 X 7,800 _ OA , X = — • - = 780 ohms. 100 A commercial form of bridge is shown in Fig. 2. The same letters of reference are used as in the preceding diagram. Two keys, K and K', are added, to be used in closing the r u K’ \ jOC K >_j A) I “ • <1 10 too 1000 ]CDCjt lOOOlOO 10 )OGDC 1 2 2 J 5 10 20 20 > -OOGDaODCX 3000 2000 1000 5 OO 200 200 LOO 50 Fig. 2. <1 .P g si circuits. Resistances are put in by withdrawing the plugs. In the arm JV there is a resistance of 10 ohms; in M, 1,000 ohms; in P, 5,838 ohms. If the galvanometer G indicates a balance, the value of the unknown resistance X = 1,000 x 5,838 coo OAA , — Jq-- — = 583,800 ohms. CABLE TESTING. Test for Capacity.— A condenser of known capacity k is charged by a battery and discharged through a galvanometer, producing a deflection c^. The cable, having an unknown capacity fc 2 , is charged and discharged in similar manner, giving a deflection d 2 . Then k 2 = ki The connections oh for the test are shown in Fig. 1. A' plug commutator p may be used to make connection with the insulated line wire L or 272 ELECTRICITY. with one side of the condenser c, by putting a plug in 1 or 2. On depressing the key k, contact is made with one pole of the battery B, having about 100 cells; on releasing the key, the discharge from the line or the condenser passes through the galvanom- eter to the ground at O. Example.— T he d e- fiection through a con- denser of 1.5 microfarads (mfds.) was 82 divisions, and through a cable, 154 divisions. Find the capacity of cable. Solution.— From the formula given, 154 k 2 = 1.5 X “oo" == 2 - 8 microfarads. Voltmeter Method of Testing Insulation.— An ordinary Weston voltmeter with a range of 150 volts has a resistance of about 19.000 ohms. If, then, this instrument is connected across a 110-volt circuit, it will indicate the resistance of the circuit, that is, of itself, since the resistance of the armature and leads is very low. If v is the voltage across the mains, r the resistance of the voltmeter, and x the voltmeter reading, V T then the resistance to be determined, R = — . When the x voltmeter is put across the mains, v = 110, r = 19,000, and x = 110. The only resistance in the circuit is the voltmeter 11 0 V 1 9 OftO itself, for R = — - = 19,000 ohms. If we now put in series with the voltmeter a high resistance, thereby reducing 110 X 19 000 the reading to 2 divisions, the total resistance R = 2 ^ = 1,045,000 ohms. From this we must subtract the voltmeter resistance in order to find the added resistance, which is 1.045.000 — 19,000 = 1,026,000 ohms. A deflection of one division gives 2,071,000 ohms. To obtain higher readings, a special high-resistance voltmeter should be used. The con- nections are made as shown in Fig. 2, where V is the CABLE TESTING. 273 voltmeter, F the feeder, and D the source of current. If I is the insulation resist- ance of a feeder, the corrected for- mula becomes .. v r I = r. r Fig. 2. When a voltmeter is used having a resistance of 1 megohm (1,000,000 ohms), then a deflection of 1 division, when con- nected up as shown, would give an insulation resistance 7 = 110 x 1,000,000 — 1,000,000 = 109 megohms. Loss-of-Charge Method of Cable Testing.— The core of the cable must first be put to earth a sufficient length of time to be thoroughly clear from any charge due to previous elec- trification; then the far end is freed, and connections are made as shown in Fig. 3. On depressing the key k, the cable is put to earth through the condenser c, which should be of very small capacity, say one-fiftieth of a microfarad. Both the cable L and the condenser c are then charged from the battery B by depressing the key k', and on releasing k, the condenser is discharged through the bal- listic galvanometer g , a mo- ment being chosen when the galvanometer is at zero, show- ing that the charge is steady. The deflection produced (di) represents the full charge held by the cable. The key k is then again depressed, and cable and condenser are charged for, say, half a minute, after which the battery is discon- nected at k', and leakage of the charge is allowed to take place for perhaps 5 minutes. Selecting a moment when the charge is steady, indicating an even distribution, the key k is raised, and the condenser discharged through the 274 ELECTRICITY. galvanometer. The deflection (d 2 ) obtained will be less than the first one, owing to the leakage of charge during the 5 minutes, and will therefore be a measure of the conducting power of the cable covering, or its insulation resistance. The ratio of these two deflections, di and d 2 , will ordinarily be sufficient to indicate the condition of the cable without further calculation; the exact insulation resistance may be found by the following formula, x _ 26.06 1 ~ Klos % where I = insulation resistance of the cable in megohms; t = time in minutes during which the charge is allowed to leak; K = capacity of the cable in microfarads; di = initial discharge deflection; d 2 — final deflection after t minutes. Example. — In a loss-of-charge insulation test, the initial deflection was 238 divisions, and the deflection after 5 min- utes’ leakage was 137 divisions. The capacity of the cable being 1.8 microfarads, what was the insulation resistance? Solution.— I 26 ‘°— b = 301.8 megohms. 1.8 X log ~ The battery used in this test may be about 100 chloride-of- silver cells, or the same number of Leclanch6 cells. In the latter case it will be better to make the electrolyte of only about one-fifth the usual strength, to prevent creeping of the salts, as only very small currents are required for these tests. The battery must be very thoroughly insulated. Location of Faults. A fault in a cable usually develops slowly; and there is \ considerable resist- ance at that point; therefore, in determining the location of the fault, its resist- ance must be taken into account. Let A B, Fig. 4, be the cable, and let a fault F connect to the ground at G through CABLE TESTING. 275 a resistance R. When the end B of the cable is insulated, the resistance >is measured at the station A, and is equal to the resistance of that portion of the cable between the station and the fault plus the resistance of the fault, that is, x + R. B is then grounded at G', and the resistance is + y + R' x-\- E = r. yr Let Let Let Then, x + y = r". x = r' — j/ (r — r')(r" — r'); y = r" — r' + |/ (r -- r f ){r" — r'). If L = length of cable in feet, the distance from A to the fault is Lx z~-fy‘ Example. — The resistance of a cable in good condition is 3 ohms. A fault develops, and, on testing, the resistance through it is 160 ohms, the far end of the cable being insu- lated. When the far end is grounded, the resistance is 2.95 ohms. What is the distance to the fault, the length of cable being 5,180 ft.? Solution.— r = 160, r* =±= 2.95, r" = 3. Then, x = 2.95 - 1 / 157.05 X .05 - .15 ohm. y ■= 3 — 2.95 + j/ 157.05 X -05 = 2.85 ohms. The distance to the fault — 5.180 X - 15 — 259 ft. 3 276 SURVEYING. SURVEYING. COMPASS SURVEYING. The magnetic bearing of a line is the angle that the line makes with the magnetic needle. The length of a line, together with its bearing, is termed a course. To take the bearing of a line, set the compass directly over a point in it, at one extremity, if possible. This may be done by means of a plumb-bob suspended from the compass. Bring the compass to a perfectly level position. Let a flag- man hold a rod carefully plumbed at another point of the line, preferably the other extremity, if he can be distinctly seen. Direct the sights upon this rod and as near the bottom of it as possible. Always keep the same end of the compass ahead— the north end is preferable, as it is readily distin- guished by some conspicuous mark, usually a fleur-de-lis— and. always read the same end of the needle, that is, the north end of the needle if the north point of the compass is ahead, and vice versa. Before reading the angle, see that the eye is in the direct line of the needle, so ac to avoid the error that would otherwise result from parallax , or apparent change of the position of the needle, due to looking at it obliquely. The angle is read and recorded by noting, first , whether the N or S point of the compass is nearest the end of the needle being read; second , the number of degrees to which it points; and third , the letter E or W nearest the end of the needle being read. Let A B in Fig. 1 be the direction of the magnetic needle, B being at the north end. Let the sights of the compass be directed along the line CD. The north point of the compass will be seen to be nearest the north end of the needle which is to be read. The needle, which has remained stationary while the sights were being turned to CD, now points to 45° between the N and E points, and the angle is read north forty- five degrees east (N 45° E). A sure test of the accuracy of a bearing is to set up the compass at the other end of the line, i. e., the end first sighted COMPASS SURVEYING. 277 Fig. 1. to, and sight to a rod set up at the starting point. This proc- ess is called backsighting. If the second bearing is the same as the first, the reading is correct. If it is not the same, it shows that there is some disturbing influence at either one or the other end of the line. To determine which of these two bearings is the true one, the compass must be set up at one or more intermediate points, when two or more similar bearings will prove the true one. The magnetic meridian is the direction of the magnetic needle. The true meridian is a true north t n,4T¥ro and south line, which, if produced, would | jj L pass through the poles of the earth. The 0 declination of the needle is the angle that the magnetic meridian and the true meridian make with each other. Example of the Use of the Compass in Rail' road Work.— Suppose CAD in Fig. 2 to be a railroad in operation, and that it has been decided to run a compass line from the point A along the valley of the stream X to the point B. The bearing of the tangent A D cannot be determined by set- ting up the compass at A on account of the attraction of the rails. The direction of this tangent, however, can be obtained by setting up at A and sighting to a flag held at D. The point A, which is the starting point of the line to be run, is marked 0. Producing the line A D 440 ft., the point E is reached, which has been previously de- cided on as a proper place for changing the r> direction of the line. The compass having Fig. 2. been set up at E , the bearing of the line A E , which is the 278 SURVEYING line A D produced, is found by sighting to A, or, what is still better, to the point D, if that point can be seen. The number of Sta. (Station) E , namely, 4 + 40, and the bearing of A E are then recorded by the compassman. By this time the chief of party has located the point F y and the flag is in place for sighting. The axmen, if there is work for them to do, are now put in line by the head chainman; the axmen clear only so much as would interfere with rapid chaining. The bearing of the line EF haying been recorded, the compass is moved quickly to F, replacing the target left by the flagman, leveled up, and directed toward the point G , which is already located. The chainmen reaching F, its number 11 + 20 is recorded by the compassman and the instrument sighted to G and the work continued as before. FORM FOR KEEPING NOTES. A plain and convenient form for keeping compass notes is the form given on page 279, which is a record of the survey platted in Fig. 2. The first column of the table contains the station numbers, the notation running from the bottom to the top of the page. By means of this arrangement, the lengths of the courses are found by subtracting the number of the station of one compass point from the number of the station of the next succeeding compass point. Before work has commenced on the plat, the subtractions are made and the lengths of the courses are written in red ink between the station numbers. The second column contains the bearings of the lines. The bearing recorded opposite to a station is the bearing at the course between the given station and the one next above. Thus, the bearing recorded opposite Sta. 0 is 75° 00' W, and is the bearing of the line extending from Sta. 0 to Sta. 4 + 40 next above. The length of the course is the difference between 0 and 4 + 40 equal to 440 ft. The bearing recorded opposite to 4 + 40 is N 25° 00' W. It is the bearing of the line extending from Sta. 4 + 40 to Sta. 11 + 20 next above. Its length is found by subtracting 4 + 40 from 11 + 20 equal to 680 ft., and so on. TRANSIT SURVEYING. 279 In the third column, under the head of remarks, are recorded notes of reference, topography, and any informa- tion that may aid in platting or subsequent location. Station, Bearing. Remarks. 47 + 75 End of line. 35 + 75 N 25° 40' E 27 + 50 N 14° 10' E 20 + 35 N 2° 30' W Woodland. 11 + 20 N 15° 10' W 4 + 40 N 25° 00' W 0 N 75° 00' W Sta. 0 is at P. C, of 14° curvejo left at Bellford Sta. O. & P. R. R. TRANSIT SURVEYING. The Vernier. — A vernier is a contrivance for measuring smaller portions of space than those into which a line is actually divided. The divided circle of the transit is gradu- ated to half degrees, or 30'. The graduations on the verniers run in both directions from its zero mark, making two dis- tinct verniers, one for reading angles turned to the right and the other for reading those turned to the left. In reading the vernier, the observer should first note in which direction the gradu- p IG ^ ations of the divided circle run. In Fig. 1 the graduations increase from left to right and extend from 57° to 91°. Next, he should note the point where the zero mark of the vernier comes on the divided circle. In Fig. 1 the zero mark cOmes between 74° and 74£°. Now, as the circle graduations read from left to right, we read the right-hand vernier and find that the 23d graduation on the vernier coincides with a graduation on the 280 SURVEYING. divided circle and the vernier reads 23', which we add to 74°, making a reading of 74° 23', an angle to the left. In Fig. 2 the graduations on the circle increase from right to left, and we accordingly- read the left-hand ver- nier. The zero mark of the vernier comes be- tween 67£° and 68°. Reading the vernier, we find that the 13th graduation on the vernier coincides with a graduation on the circle and the vernier reads 13'. Accordingly, we add to 67£°, the reading = 13', making a total reading of 67° 43', an angle to the right. Setting Up the Instrument.— In setting up a transit, three preliminary conditions should be met as nearly as possible: 1. The tripod feet should be firmly planted. 2. The plate on which the leveling screws rest should be level. Fig. 2. 3. The plumb-bob should be directly over the given point. When these three conditions are met, the completion of the operation is quickly performed with the leveling screws. How to Prolong a Straight Line.— Let A B, in Fig. 3, be a straight line which it is required to prolong or “produce. ,, Fig. 3. 400 ' The line can be prolonged in two ways: by means of foresight or by means of backsight. 1. By foresight, set up the transit at A and sight to B; measure 400 ft. from B in the opposite direction from A. Then, by means of signals, move the flag to the right or left until the vertical cross-hair shall exactly bisect the flag held at C. Then, the line B C will be the prolongation of the line AB. 2. By backsight, set the transit at B and sight to A. Reverse the telescope, and having measured 400 ft. from B in the opposite direction from A, set the flag at C ; then will the line B C be the line A B produced. TRANSIT SURVEYING. 281 Horizontal Angles and Their Measurement.— A horizontal angle is one the boundary lines of which lie in the same horizontal plane. Let A, B , and C , in Fig. 4, be three points, and let it be required to find the horizontal angle formed by the lines A B and AC joining these # 0 points. Set up the instrument v * 3 ° 3<> precisely over the point A, and / carefully level it. Set the ver- _ / nier at zero, and place flags at B B and C. Sight to the flag at FlG - 4 * B and set the lower clamp. Then, by means of the lower tangent screw, cause the vertical cross-hair to exactly bisect the flag at B. Loosen the upper clamp. With a hand on & either standard, turn the telescope in the same direction as that of the hands of a watch until the flag at C is covered or nearly covered by the vertical cross-hair. Clamp the upper plate, and with the upper tangent screw bring the line of sight exactly on the flag at C. The arc of the graduated circle traversed by the zero point of the vernier will be the measure of the angle BAC , as 143° 30'. The points A, B, and Care not necessarily in the same hori- zontal plane, but the level plate of the instru- ment projects them into the horizontal plane .© in which it revolves. A Deflected Line.— A deflected line, or “ angle line,” is a consecutive series of lines and angles. The direction of each line is referred to the line immediately preceding it, the latter being, in imagination, produced, and the angle measured between it and the next line actually run. The angles are recorded R T or L T , according as they are turned to the right or left of the prolongation of the immediately preceding line. An example of a deflected line is shown in Fig. 5; it starts from the head block of switch at Benton Station, O. & P. R. R. Set up the transit at A with vernier at zero. Sight to a flag 282 SURVEYING. held at F on the center line of the track, O. & P. R. R. Loosen the vernier clamp, the point B being determined, and turn the telescope until the point B is distinctly seen; clamp the vernier, and accurately sight to flag held at B\ the angle reads 32° 30' and is recorded R T 32° 30', with a sketch showing the connection. The bearing of the line A B cannot be taken at A on account of the attraction of the rails. The point A is in the head block of the switch (which is designated by the abbreviation H. B.) at Benton Station, O. & P. R. R. The instrument is now moved to B , the vernier set at zero and backsighted to A; the bearing of A B , viz., N 75° 00 / E, is taken, and the number of station B , viz., 2 + 90, together with the bearing of A B recorded. The telescope is then reversed, pointing in the direction BB'. The point C being determined, the upper clamp is loosened and the telescope turned to the right and sighted to C. The. reading is found to be 14° 30' and recorded R T 14° 30'. It measures the angle B' B C. The bearing N 89° 20' E is then recorded. The instrument is next set up at C, the vernier set at zero, back- sighted to B, and then reversed; the deflection to D, viz., R T 10° 00' read and recorded, together with the number of the station at (7, viz., 6 + 85. This deflection measures the angle C' CD and gives the direction of the line CD. A good form of notes for such a survey is the following: Station Deflection. Mag. Bearing. Ded. Bearing. Remarks. 13+63 End of Lin ie. 10+31 L T 30°00 f N. 69°Z5’E. E. 69°30'E. 6+85 R r 10W S. 80°30'E. S. 80°30'E- 9+90 R*14°3 6. recorded N 60° 30' E. The magnetic bearing being N 89° 20' E, would have at once revealed the error. The confusion of the directions R T and L T is the commonest source of error in recording deflections, though sometimes a mistake of 10 degrees is made in reading the vernier. Both angle and bearing should be read after they are recorded, and compared with the recorded readings. TRIANGULATION. Triangulation is an application of the principles of trigo- nometry to the calculation of inaccessible lines and angles. A common occasion for its use is illustrated in Fig. 1, where the line of survey crosses a stream too wide and deep for actual measurement. Set two points A and B on line, one on each side of the stream. Estimate roughly the distance A B. Suppose the estimate is 425 ft. Set another point C, making the dis- tance A C equal to the estimated Set the transit at A and measure the Next set up at the point C and distance A B = 425 ft. angle BA C = say, 79° 00'. 284 SURVEYING. measure the angle A CB = say, 56° 20'. The angle ABC is then determined by subtracting the sum of the angles A and C from 180°; thus, 79° 00' + 56° 20' = 135° 20 '; 180° 00' - 135° 20 ' _ 440 4 (y = the angle ABC. We now have a side and three angles of a triangle given, to find the other two sides A B and CB. In trigonometry, it is demonstrated that, in any triangle the sines of the angles are proportional to the lengths of the sides opposite to them. In other words, sin A : sin B = B C: AC; or, sin A : sin C = B C : A B, and sin B : sin C = A C : A B. Hence, we have sin 44° 40' : sin 56° 20' = 425 : side A B\ sin56°20 / = .83228; .83228 X 425 = 353.719; sin 44° 40' = .70298; 353.719 -r- .70298 = 503.17 ft. = side A B. Adding this distance to 76 + 15, the station of the point A , we have 81 + 18.17, the station at B. Another case is the following: Two tangents, A B and C D (see Fig. 2 ), which are to be united by a curve, meet at some inaccessible point E. Tangents are the straight portions of a A B, and two points Uand D of the tangent CD , being care- fully located, set the transit at B, and backsighting to A , measure the angle EB C = 21° 45'; set up at C, and, back- sighting to D, measure the angle ECB = 21° 25'. Measure the side B C — 304.2 ft. Angle C EF being an exterior angle of triangle EB C equals sum of EB C and ECB — 21° 45' + 21° 25' = 43° 10'; angle B E C — 180° — CEF — 136° 50'. From trigonometry, we have sin 136° 50' : sin 21° 45' = 304.2 ft. : CE) sin 21° 45' = .37056; .37056 X 304.2 = 112.724352; sin 136° 50' = .68412; side CE « 112.724352 - 4 - .68412 = 164.77 ft. Fig. 2. r line of railroad. The angle CEF, which the tangents make with each other, and the dis- tances BE and CE are required. Two points A and B of the tangent TRI ANGULATION. 285 Again, we find B E by the following proportion: sin 136° 50' : sin 21 ° 25' = 304.2 : side B E; sin 21° 25' = .36515; -J50L- -f- i~ / .36515 X 304.2 = 111.07863; sin 136° 50' = .68412; side B E = 111.07863 .68412 = 162.36 ft. A building H, Fig. 3, lies directly 'in the path of the line A B f which must be produced beyond H. Set a plug at B, and then turn an angle BBC = 60°. Set a plug at C in the 4 „ line B C , at a suitable distance \ from B, say, 150 ft. Set up at C, and turn an angle B CD = 60°, and set a plug at D, 150 ft. from \ / C. The point D will be in the # prolongation of A B. Then, set Fig. 3. up at Dy and backsighting to Cy turn the angle C D D' — 120°. D D' will be the line required, and the distance B D will be 150 ft., since BCD is an equilateral triangle. A B and CD, Fig. 4, are tan- gents intersecting at some in- accessible point H. The line A B crosses a dock O P, too wide for direct measurement, and the wharf LM. F is a point on the line A B at the wharf crossing. It is required to find the distance B H and the angle FHG. At B , an angle of 103° 30' is turned to the left and the point E set 217' from B = to the estimated dis- tance BF. Setting up at E , the angle B E F is found to be 39° 00'. Whence, we find the angle (103° 30' + 39°) = 37° 30', BFE — 180° - 286 SURVEYING. From trigonometry, we have sin 37° 30' : sin 39° 00' = 217 ft. : side BF\ sin 39° 00' = .62932; .62932 X 217 = 136.56244; sin 37° 30' = .60876; side B F = 136.56244 -f- .60876 = 224.33 ft. Whence, we find station F to be 20 4- 17 4- 224.33 = 22 + 41.33. Set up at F and turn an angle HFG = 71° 00' and set up at a point G where the line CD prolonged intersects FGo Measure the angle FGH = 57° 50', and the side FG = 180.3. The angle FHG = 180° - (71° + 57° 50') = 51° 10'. From trigonometry we have sin 51° 10' : sin 57° 50' = 180.3 : side FH. Sin 57° 50' = .84650; .84650 X 180.3 = 152.62395; sin 51° W = .77897; side FH = 152.62395 -f- .77897 = 195.93 ft.; whence we find station H to be 24 4- 37.26. CURVES. Two lines forming an angle of 1° with each other will, at a distance of 100 ft. from the angular point, diverge by 1.745 ft. The degree of a curve is deter- mined by that central angle which is subtended by a chord of 100 ft. Thus, if BOG (Fig. 1) is 10° and B G is 100 ft., BGHKC is a 10° curve. The deflection angle of a curve is the angle formed at any point of the curve between a tangent and a chord of 100 ft. The deflection angle is there- fore half the degree of the curve. Thus, if the chord B G is 100 ft., the angle EBG is the deflec- tion angle of curve BGHKC, and is half the angle BOG. Example— Given, the deflection angle EBG = D (Fig. 1), to find the radius B 0 = R. CURVES. 287 ' Solution.— Draw 0 L perpendicular to B G. In the right- JB L angled triangle BOL, we have sin BOL = but BOL = E B G — D, since 0 L , being perpendicular to the chord B G, bisects the arc B LG. But the angle D = \ B 0 G; hence, angle BOL = D. BL = 50 ft., and the radius B O = R. Substituting these values in the given equation, we have sin 1) — 4?; whence, R sin D — 50, and R = — — — . R an L For curves of from 1° to 10°, the radius may be found by dividing 5,780 ft. (the radius of a 1° curve) by the degree of the curve. The results obtained are sufficiently accurate for all practical purposes. For sharp curves, i. e., for those 50 exceeding 10°, the above formula, viz., R = — — should be sin D used, especially if the radius is to be used as a basis for further calculation. Tangent Distances.— When an intersection of tangents has been made and the intersection angle measured, the next question is the degree of curve that is to unite them, which being decided, the next step in order is the location of the points on the tangents where the curve begins and ends. These two points are equally distant from the point of intersection of the tangents, which is called the P. I. The point where the curve begins is called the point of curve , or the P. C., the point where the curve terminates is called th^ point of tangent , or the P. T. The distance of the P. C. and P. T. from the P. I. is called the tangent distance. In Fig. 1, let A B and CD be tangents intersecting at the point E and forming an angle CEF = 40° 00' with each other. It is decided to unite these tangents by a 10° curve, whose radius is 573.7 ft. Call the angle of intersection J, the radius B O, R , and the tangent distance B E, T. From geom- etry we know that B 0 C = CEF, hence the angle B O E = \CEF. From the right triangle E B 0, we have tan £0£ =§l- Substituting the above equivalents, we have tan 1 1 = ot T = R tan *1; R = 573.7; * / = 20°; tan 20° = .36397; xC 288 SURVEYING. 573.7 X .36397 = 208.81 ft. Measure back from the point E on both tangents the distance 208.81 ft. to the points B and C. Drive plug flush with the ground at both points and set accurate center points, marked by tacks, in both. Directly opposite each of these plugs drive a stake, called a guard stake because it guards or rather indicates where the plug is. The stake at B, if the numbering of the stations runs from B toward C, will be marked P. C., and the stake at C will be marked P. T. To Lay Out a Curve With a Transit. — Having set the tangent points B and C, Fig. 1, set up the transit at B, the P. C. Set the vernier at zero and sight to E , the intersection point. Suppose B to be an even or “ full station,” say 18, and that it has been decided to set stakes at each hundred feet. Let the central angle BOG , measured by the 100-ft. chord B G, be 10°; then, the deflection angle E B G, whose vertex B is in the circumference and subtended by the same chord B G, will be £ B 0 G, or 5°. Turn an angle of 5° from B, which in this case will be to the right, measure a full chain 100 ft. from B and line in the flag at G; drive a stake at G, which will be marked 19. Turn off an additional 5° making 10° from zero, and at the end of another chain from G, at H, set at a stake marked 20. Continue turning deflections of 5° until 20° or one-half of the intersection angle is reached. This last deflection, if the work has been correctly done, will bring the head chainman to the point of tangent C. It is but rarely that the P. C. comes at a full station. When the P. C. comes between full stations it is called a substation , and the chord between it and the next full station is called a sub- chord. Had the P. C. come at a substation, say 17 + 32, the deflection for the subchord of 100 — 32, or 68 ft., the distance to the next station, is found as follows: The deflection for a full station, i. e., 100 ft., is 5° = 300', and the deflection for 1 ft. is ^5^. = and for 68 ft. the deflection will be 68 X 3 = 204' = 3° 24', which is turned off from zero and a stake set on line, 68 ft. from the transit, at station 18. The length of a curve uniting two given tangents whose intersection is determined, is found as follows: CURVES. 289 * Suppose I = 32° 40' and that the tangents are to be united by a 6° curve. 32° 40' reduced to the decimal form is 32.667°; as each central angle of 6° will subtend a 100-ft. chord or one chain, there will be as many such chords or chains as the number of times 6 is contained in 32.667, which is 5.444, that is, there will be 5.444 chains in the curve, or 544.4 ft., which is the required length of the curve. The P. C. and P. T. having been set and the station of the P. C. determined by actual measurement, say 58 + 71, the station number of the P. T. is found by adding to 58 + 71, the station number of the P. C., the calculated length of the curve 544.4 ft. 58 + 71 -f 544.4 = 64 + 15.4, the station of the P. T. Tangent and Chord Deflections. — Let A B in Fig. 2 be a tan- gent, and B C EH a curve commencing at B. Produce the tangent A B to the point JD. The line CD is a tangent deflec- tion , and is the perpendicular distance from the tangent to the curve. If the chord B C is produced to the point G , making CG = BC = CE, the distance G E is a chord deflection and is double the tan- gent deflection D C. Given, the radius BO — R, Fig. 2, to find the chord deflection EG and the tangent deflection CD = FE. The triangles 0 CE and CE G are similar, since both are isosceles, and the angle G CE = angle COE. Hence, we have 0 C: CE = CE: EG. Denoting the chord CE by c and the chord deflection E G by d, we have, from the above propor- c 2 tion, R : c = c : d. Therefore, d — To find the tangent R deflection, draw C F to the middle point of EG. Then FE is equal to the tangent deflection, or D C. Hence, the tan- gent deflection is equal to one-half the chord deflection, or c 2 the tangent deflection = — z n 290 SURVEYING If the P. C. does not fall at a full station (and this is usually the case), compute the chord deflection by substituting for c in the formula for chord deflection £ c (c + c'). Where c' is the length of the chord from the P. C. to the full station; or if the tangent deflection / for a chord of 100 feet has been previously found, the chord deflection for the second station beyond the P. C. is d 0 =/ (l -f . Laying Out Curves Without a Transit.— During construction, the engineer is often called upon to restore center stakes on a curve when the transit is not at hand. This can be accom- plished reasonably well with a tape, as follows: In Fig. 3, A B is a tangent and P, at Sta. 8 -f 25, is the P. C. of a 4° curve; a stake is required at each full station. The stakes at A and B are restored, determining the P. C. and the direction of the tangent. For a 4° curve the regular chord deflection for 100 feet is 6.98 ft., and the tangent deflection is 6.98 -4- 2 = 3.49 ft. The distance from the P. C. to the next station C is 75 ft.; hence, the tangent deflection C F = 75 2 -i- (2 X 5,730 -4- 4) = 1 96 ft. The point F is found by first measuring 75 feet from B, thus locating the point C ) in the line with A B, then from C measuring C F = 1.96 feet, at right angles to B C\ the point F thus determined will be Station 9. Next, the chord B F is prolonged 100 feet to D; as B F is only 75 feet, D G = d 0 = 3.49 X (1 + t 7 t&) = 6.11 feet. This distance is measured at right angles to B D; the point G thus determined will be Station 10. The position of Station 11, the P. T., is determined in the same manner, except that, as the chords F G and G H are each 100 feet long, the regular chord deflection of 6.98 feet is used for EH. A stake is driven at each station thus located. To Determine Degree of Curve by Measuring a Middle Ordinate. In track work, it is often necessary to know the degree of a curve when no transit is available for measuring it. The degree can be found by measuring the middle ordinate of any CURVES. 291 convenient chord, and multiplying its length by 8, which will give the chord deflection for that curve. Let A B, in Fig. 4, be a 50-ft. chord, measured on the track, and let the middle ordinate ab be .44 ft. .44 X 8 = 3.52 = chord deflection for 50 ft., which, expressed in decimal parts of a full station, is ".5; .5 2 = .25. The chord deflection for 100 ft. ^ multiplied by .25 = the chord de- flection for 50 ft., which we know by calculation to be 3.52 ft. Hence, ^ IG * 3.52 h-. 25 = 14.08 ft., the chord deflection for 100 ft., which, if divided by 1.745, the chord deflection for a 1° curve, gives a quotient of 8.07, nearly. The inference is that the curve is 8°. How to Keep Transit Notes.— A good form for location notes is the following: DtflocUem. Tot- Angle. tfaff. Bearing Dm*. Bearing Bern June JO 1894 irks. 9 8 1 8+96 4*54’ P.T. 16*00 ‘ ~ S- 3*8* B. 2r.36°l*£. 6+60 4*0* 6 3*0* 6+60 3*0* 6+80 6 I a OO' 6+60 6+60 8*3* 6*1 r 4 1*3* Ini. Angle- 15*00' 4*Cnrvt B T 3+60 0*3* T- 188.81 ft. fief. Angle for 60 flrl'OO 3+30 P.C4*B r PC- 3+30 Def. Angle for 1 fl - 1-8“ 3 Length of Ourve-376fi » PJP-C+96 1 O y. zone's. B 80*1** In the first column the station numbers are recorded. In the second column are recorded the deflections with the abbreviations P. C. and P. T., together with the degree of curve and the abbreviation R T or L T , according as the line curves to the right or left. At each transit point on the curve, the total or central angle from the P. C. to that point is calculated and recorded in the third column. This total angle is double the deflection angle between the P. C. and the transit point. In the above notes there is but one inter- mediate transit point between the P. C. and P. T. The 292 SURVEYING. deflection from P. C. at Sta. 3 + 20 to the intermediate transit point at Sta. 4 -f 50 is 2° 36'. The total angle is double this deflection, or 5° 12', which is recorded on the same line in the third column. The record of total angles at once indi- cates the stations at which transit points are placed. The total angle at the P. T. will be the same as the angle Of inter- section, if the work is correct. When the curve is finished, the transit is set up at the P. T., and the bearing at the for- ward tangent taken, which affords an additional check upon the previous calculations. The magnetic bearing is recorded in the fourth column, and the deduced or calculated bearing is recorded in the fifth column. LEVELING. Examples in Direct Leveling.— The principles of direct level- ing are illustrated in the figure. Let A be the starting point, which has a known elevation of 20 ft. The instrument is set at B, leveled up and sighted to a rod held at A. The target being set, the reading, 8.42 ft., called a backsight , is the distance that the point where the line of sight cuts the rod is above the point A , and is to be added to the elevation of the point A. 20.00 -f 8.42 — 28.42 is called the height of instrument and is designated by H. I. The instrument being turned in the opposite direc- tion, a point C is chosen, which must be below the line of sight. This point is called a turning point , and is designated by the abbreviation T. P. Drive a peg at (7, or take for a turn- ing point a point of rock or some other permanent object upon which the rod is held. The reading at this point is a foresight, and is to be subtracted from the height of the instrument at B to find the elevation of the point at C. Let the rod reading be 1.20 ft. As this reading is a fore- sight, it must be subtracted from 28.42, the height of instru- ment at B ; 28.42 — 1.20 = 27.22 ft., the elevation of the point C. The leveler carries the instrument to B, which should be of such a height above C that, when leveled up, the line of sight will cut the rod near the top. The backsight to C gives a reading of 11.56 ft., which, added to 27.22 ft., the elevation LEVELING. 293 of C , gives 38.78 ft., the height of the instrument at D, The rodman then goes to E, a point where a foresight reading is 1.35, which, subtracted from 38.78, the H. I. at D, gives 37.43 ft., the elevation of E. The level is then set up at F \ being careful that line of sight shall clear the hill at L. The back- sight, 6.15 ft., added to 37.43 ft., the elevation of E, gives 43.58 ft., the H. I. at F. The rod held at G gives a foresight of 10.90 ft., which, sub- tracted from 43.58 ft., the H. I. at F, gives 32.68 ft., the elevation at G. Again moving the level to H, the backsight to G of 4.39 ft. added to 32.68 ft., the elevation of G, gives 37.07 ft., the H. I. at H. Hold- ing the rod at K , a foresight of 5.94, subtracted from 37.07, gives 31.13, the elevation of the point K. The elevation of the starting point A is 20.00 ft.; the elevation of the point K is found by direct leveling to be 31.13 ft., and the difference i n the elevations of A and K is 31.13 — 20.00 = 11.13 ft.; that is, the point K is 11.13 ft. higher than the point A. Turning points previously men- tioned are the points where back- si ghts and fore- sights are taken. The backsights are plus ( + ) readings, and 294 SURVEYING. are to be added; the foresights are minus (— ) readings, and are to be subtracted. A point for a foresight having been determined, the rodman drives a peg firmly in the ground and holds the rod upon it. After the instrument is moved, set up, and a backsight taken, the peg is pulled up and carried in the pocket until another turning point is called for. Turn- ing points should be taken at about equal distances from the instrument, in order to equalize any small errors in adjust- ment. In smooth country an ordinary level will permit of sights of from 300 to ,500 ft. To Keep Level Notes.— Many forms are used. The distin- guishing feature of one of the best (see page 295) is a single column for all rod readings. The backsights being additive and the foresights subtractive readings, they are distinguished from other rod readings by the characteristic signs + (plus) and — (minus). The turning points, whose foresight read- ings are — , are further abbreviated T. P. To Check Level Notes.— A well-known method of checking level notes provides for checking the elevations of turning points and heights of instrument only, which is sufficient, as all other elevations are deduced from them. The method depends on the fact that all backsights are additive (i. e. +) quantities, and all foresights are subtractive (i.e.—) quantities. The notes given on page 295 are checked as follows: The ele- vation of the bench mark at station 0 is 100.00 ft., to which all backsights, or + readings, are to be added and from this sum all foresights, or — readings, are to be subtracted. The sum of the backsights, with elevation of bench mark at 0, is 122.59. Sum of foresights is 24.27, and difference is 98.32 ft., the eleva- Thus, + 100.00 5.61 5.41 11.57 122.59 ' 24.27 ”~9&32 10.22 2.52 11.53 24.27 tion of the turning point last taken. As soon as a page of level notes is filled, the notes should be checked and a check mark y' placed at the last height of instrument or elevation checked. When the work of staking out or cross- sectioning is being done, the levels should be checked at each bench mark on the line. After each day’s work, the leveler must check on the nearest bench mark. LEVELING. 295 2. Date. o c3 CO t-4 © p< a p CO Remarks. On root of white oak Spring Brook. Fill. Cut. Grade. Eleva- tion. o o o o 99.5 1 CO CO 05 d 05 1 94.5 1 96.6 CO 05 CO s $ q CO o CO o I> 05 05 98.32 Ht. Instru- ment. 105.61 SB © o rH 1 iO 00 05 o Rod Read- ing. + 5.61 «d CO 00 9.2 1 1 —10.22 id + CO © +3 fcoo p a> h c3 o U _d © Soy a © S> *d o o Jh © a> ft ft ^ 0) ft S«S r_| © H ft a> ft ft w © ft £_j making the angle nPh — 33° 41'. Draw n h intersecting h P in h ; then will n h to the same scale equal the friction of the earth against the back of the wall. Completing this parallelo- gram, nhkP, the diagonal hP = 1,139 in., which, to a scale of 2,000 lb. = 1 in., amounts to 2,278 lb., and is the resultant of the pressure and the friction. Produce the resultant h P to u. We next find the center of gravity g of the wall abdc. The section of the wall is a trapezoid, and the center of gravity g is readily found as follows: Produce the upper base of the section to x , and make ax = cd = 4.5 ft. Then produce the lower base in the opposite direction to y, and make d y — a b = 2 ft. Join x and y. Find the middle points x' and y' of the upper and lower bases of the section. Join these points. The inter- section g of the lines x y and x' y' is the center of gravity of the trapezoid abdc. The volume of the section of wall abdc is readily found. The sum of top and bottom widths = 2.0 + 4.5 = 6.5 ft. 6.5 -r- 2 = 3.25 ft. 3.25 X 9 = 29.25 cu. ft. 29.25 X 154 = 4,504 lb. (the weight per cubic foot of good mortar rubble = 154 lb.) = the weight of the section abdc. Draw through g a verti- cal line g i, and lay off on it, to a scale of 2,000 lb. to the inch, from the point l, where the line of gravity intersects the pro- longation of the line of pressure h P, the length 1 1 equal to 4,504 lb., the weight of the wall. Lay off from l on the pro- longation of hP, Im equal to 2,278 lb. to the same scale. Complete the parallelogram Imst. The diagonal Is represents the resultant of the pressure and of the weight of the wall. The distance c r from the toe c to the intersection of the resultant l s with the base c d is more than one-third of the width of the base, which insures ample stability. TUNNELS. 305 Fig. 4. Pressure of the Backing on Surcharged Walls.— In Fig. 4 the surcharge of backing mbo slopes from b at its natural slope, and attains its maximum pressure where the slope of maximum pressure d k intersects the natural slope b m at /. Any addi- tional height of sur- charge does not increase this pressure. If the sur- charge slopes from a, as shown by the line ap , or from any point between a and b , then the slope of maximum pressure must be extended, intersecting the slope from a in the point k. The prism of maximum pressure will then be dik. The triangle of earth abi on the top of the wall exerts no pressure against the back of the wall, but adds to its stability. Having found the weight of the triangle bdf we have approximate pressure — weight of triangle bdfX .643, which includes the pressure of the backing and the friction of the earth against the back of the wall. Draw Pn perpendicular to the back of the wall and draw hP making the angle nPh = 33° 41, the angle of wall friction. Then, hP will be the direction of the pressure. The point of application of this pressure will not always be at P, one-third of the height of bd measured from d , but above P, as at r, where a line drawn from the center of gravity g of the prism of maximum pressure dik (omitting any earth resting directly upon the top of the wall), and parallel to the line d k of maximum pressure, cuts the back b d of the wall. The center of pressure P will be at one-third the height of the wall when the sustained earth dbs or dbf forms a complete triangle , one of whose angles is at b, the inner top edge of the wall. For all other surcharges, the point of pressure will be above P. 306 SURVEYING. TUNNEL SECTIONS. Tunnel sections vary somewhat, according to the material to be excavated, but the general form and dimensions are much the same. Section of Section of Double-Track Tunnel . Single-Track Tunnel. Fig. 1. Fig. 2. The general dimensions are as follows: For double track, from 22 to 27 ft. wide and from 21 to 24 ft. high, and for single track, from 14 to 16 ft. wide and from 17 to 20 ft. high (see Figs. 1 and 2). In seamy or rotten rock the section is sufficiently enlarged to receive a lining of substantial rubble or brick masonry laid in good cement mortar. When the material has not sufficient consistency to sustain itself until the masonry lining is built, resort is had to timbering, which furnishes the necessary support. CALCULATION OF EARTHWORK. In calculating the quantity of material in excavation and embankment, two general methods are used, namely, the end-area formula and the prismoidal formula. Calculation by the end-area method consists in multiply- ing the mean, or average, area in square feet of two consecu- tive sections by the distance in feet between them. Thus, EARTHWORK. 307 let A represent the area in square feet of one section; B, the area in square feet of the next section; C, the number of feet between the sections; and D, the total number of cubic feet in the prismoid lying between these sections. Then, A + B D = — - — X C , approximately. The distance between sections should not be more than 100 ft., and should be less if the surface of the ground is irregular. A more accurate result is obtained by the use of the pris- moidal formula. In applying the prismoidal formula to the calculation of cubic contents, it is requisite to know the middle cross-section between each two that are measured on the ground. The dimensions of this middle section are the means of the dimensions of the end sections. Calling one of the given sections A, the other B, the mid- dle (not the mean) section M, the distance between the sections L, and the required contents S t we have, by the prismoidal formula, S=§M + 4M+B). Example.— Two sections are repre- sented by Figs. 1 and 2, and are de- noted by the letters A and B. The per- pendicular dis- tance between them is 50 ft. It is required to find the cubical contents of the prismoid. Solution.— The sec- tion given in Fig. 1 is composed of the four triangles a, b, c , and d. The triangles a and b have equal bases of 9 ft. , the half width of the Fig. 2. roadway; hence, if we J— i ZL& £5 * \ « ^ 1 ' — i. " j t“ ' ^ Fig. 1. 308 SURVEYING. take half the sum of their altitudes and multiply it by the common base we shall have the sum of the areas of the triangles a and b. The triangles c and d have a common base 8 ft., the center cut of the section, and if we take the half sum of the side distances and multiply it by 8 ft., we shall obtain the areas of the triangles c and d. Taking the dimensions of section A given in Fig. 1, we have 12 g _i_ 5 Areas of triangles a + b = — — X 9 = 80.1 sq. ft. Areas of triangles c + d = X 8 = 143.2 sq. ft. Total area of section A — 223.3 sq. ft. Taking the dimensions of the section B given in Fig. 2, we have 9 7 + 22 Areas of triangles a' + b’ = — 1 — - — — X 9 = 53.55 sq. ft. 18 7 4- ii 2 Areas of triangles c' + d' = — : — — — - X 5 = 74.75 sq. ft. Total area of section B = 128.3 sq. ft. In applying the prismoidal formula we calculate the area of a section midway between the given sections, and for its dimensions we take the mean of the dimensions of the given sections. These dimensions will be as follows : Center cut, - ^ 5 = 6.5 ft. Right-side distance, — = 12.6 ft. Left-side distance, = 20.25 ft. 2 TRACKWORK. 309 With dimensions thus found, construct the section M shown in Fig. 3. The area of section M is computed by the same method as that used with sections A and B in Figs. 1 and 2, and is as follows: Area of triangles a"+ 6" = 11-2 + X 9 = 66.6 sq. ft. on o _l i o fi Area of triangles c"+ d," — — — - — - X 6.5 = 106.6 sq. ft. Total area of section M = 173.2 sq. ft. Denoting the distance between the sections by L and the cubical contents of the prismoid by S , we have, by substi- tuting in the prismoidal formula, S = |u+4Jf + £). 50 S = -g (223.3 + 4 X 173.2 + 128.3) = 8.703 cu. ft. = 322.3 cu. yd. TRACKWORK. Curving Rails.— When laying track on curves, in order to have a smooth line, the rails themselves must conform to the curve of the center line. To accomplish this, the rails must be curved. The curving should be done with a rail bender or with a lever, preferably with the former. To guide those in charge of this work, a table of middle and quarter ordinates for a 30-ft. rail for all degrees of curve should be prepared. The following table of middle ordinates for curving rails is calculated by using the formula in which m = middle ordinate; c = chord, assumed to be of the same length as the rail; R = radius of the curve. The results obtained by this formula are not theoretically correct, yet the error is so small that it may be ignored in practical work. 310 SURVEYING. In curving rails, the ordinate is measured by stretching a cord from end to end of the rail against the gauge side, as shown in Fig. 1. Suppose the rail A B is 30 ft. in length, and the curve 8°. Then, ^ ”T — ^ by the previous prob- 7 lem > the middle K * * ordinate at a should FlG> be 1§ in. To insure a uniform curve to the rails, the ordinates at the quarters b and b' should be tested. In all cases the quarter ordi- nates should be three-quarters of the middle ordinate. In Fig. 1, if the rail has been properly curved, the quarter ordinates at b and b' will be £ X If in. = Iff, say 1§ in. Middle Ordinates for Curving Rails. TRACKWORK. 311 In trackwork it is often necessary to ascertain the degree of a curve, though no transit is available for measuring it. The following table contains the middle ordinates of a 1° curve for chords of various lengths: The lengths of the chords are varied, so that a longer or shorter chord may be used, according as the curve is regular or not. The table is ap- plied as follows: Suppose the middle ordinate of a 44-ft. chord is 3 in. We find in the table that the middle ordi- nate of a 44-ft. chord of a 1° curve is £ in. Hence, the degree of the given curve is equal to the quotient of 3 £ = 6° curve. Elevation of Curves.— To counteract the centrifugal force developed when a car passes around a curve, the outer rail is elevated. The amount of elevation will depend on the radius of the curve and the speed at which trains are to be run. There is, however, a limit in track elevation as there is a limit in widening gauge, beyond which it is not safe to pass. The best authorities on this subject place the maximum elevation at one-seventh the gauge, or about 8 in. for standard gauge of 4 ft. 8£ in. The gauge on a 10° curve elevated for a speed of 40 miles an hour should be widened to 4 ft. 9i in. All curves, when possible, should have an elevated approach on the straight main track, of such length that trains may pass on and off the curve without any sudden or disagreeable lurch. A good rule for curve approaches is the following: For each half inch or fraction thereof of curve elevation, add 30 ft., for 1 rail length, to the approach; that is, if a curve has an elevation of 2 in., the approach will have as many rail lengths as the number of timqp \ is contained in 2, or 4. The approach will, therefore, have a length of 4 rails of 30 ft. each, or 120 ft. Length of Chord. Feet. Middle Ordinate of a 1° Curve. Inches. 20 Va 30 M 44 y% 50 V* 62 l 100 2/^ 120 3% 312 SURVEYING. The following table for elevation of curves is a compromise between the extremes recommended by different engineers. It is a striking fact that experienced trackmen never elevate track above 6 in. and many of them place the limit at 5 in. Degree of Curve. Length of Approach. Feet. Elevation. Inches. Width of Gauge. Speed of Train. Miles per Hour. 1 60 1 4' 834" 60 2 120 2 4' 834 " 60 3 4 150 180 1 4' 8%" 4' 8%" 60 55 5 180 3 4' 8%" 50 6 210 3)4 4' 8%" 45 7 210 3)1 4' 9" 40 8 240 3 % 4' 9" 35 9 240 4 4' 9" 30 10 270 4 )4 4' 9" 25 11 270 4)| 4' 9)4" 20 12 270 4 % 4' 934" 15 13 240 4)| 4' 934" 10 14 240 434 4' 9)4" 10 15 240 4 4' 9)|" 10 16 240 4 4' 934 " 10 The Elevation of Turnout Curves.— The speed of all trains in passing over turnout curves and crossovers is greatly reduced, so that an elevation of ) in. per degree is amply sufficient for all curves under 16°. On curves exceeding 16°, the elevation may be held at 4 in. until 20° is reached, and on curves extending 20°, t 3 «j in. of elevation per degree may be allowed until the total elevation amounts to 5 in., which is sufficient for the shortest curves. The Frog.— The frog is a device by means of which the rail at the turnout curve crosses the rail of the main track. The frog shown in Fig. 2 is made of rails having the same cross- section as those used in the track. The wedge-shaped part A is the tongue , of which the extreme end a is the point. The space 6, between the ends c and d of the rails, is the mouth , and the channel that they form at its narrowest point e is the throat. The curved ends / and g are the wings. TRACKWORK. 313 That part of the frog between A and A' is called the heel. The width h of the frog is called its spread. Holes are drilled Fig. 2. in the ends of the rails c, d , k, and l to receive the holts used in fastening the rail splices, so that the rails of which the frog is composed form a part of the continuous track. The Frog Number.— The number of a frog is the ratio of its length to its breadth; i. e., the quotient of its length divided by its breadth. Thus, in Fig. 2, if the length a' l, from point to heel of frog is 5 ft., or 60 in., and the breadth h of the heel is 15 in., the number of the frog is the quotient of 60 -s- 15 = 4. Theoret- ically, the length of the frog is the distance from a to the middle point of a line drawn from k to l\ practically, we take from a to l as the distance. As it is often difficult to deter- mine the exact point a of the frog, a more accurate method of determining the frog number is to measure the entire length dl of the frog from mouth to heel , and divide this length by the sum of the mouth width b and the heel width h. The quotient will be the exact number of the frog. For example, if, in Fig. 2, the total length dl of the frog is 7 ft. 4 in., or 88 in., and the width h is 15 in., and the width b of the mouth is 7 in., then the frog number is 88 -4- (15 + 7) = 4. Frogs are known by their numbers. That in Fig. 2 is a No. 4 frog. The Frog Angle.— The frog angle is the angle formed by the gauge lines of the rails, which form its tongue. Thus, in Fig. 2, the frog angle is the angle l a' k. The amount Fig. 3. of the angle may be found as follows: The tongue and heel of 314 SURVEYING. the frog form an isosceles triangle (see Fig. 3). By drawing a line from the point a of the frog to the middle point b of the heel c d, we form a right-angled triangle, right-angled at b. The perpendicular line a b bisects the angle a, and, by b c trigonometry, we have tan £ a = The dimensions of the frog point given in Fig. 3 are not the same as those given in Fig. 2, but their relative proportions are the same, viz., the length is four times the breadth. The length ab = 4 and the width cd — 1; hence, be = £. Substituting these values, we have tan £ a = ^ = £ = .125. Whence, £ a = 7° 7£' and a — 14° 15'; that is, the angle of a No. 4 frog is 14° 15'. Frog numbers run from 4 to 12, inclu- ding half numbers, the spread of the frog increasing as the number decreases. The Parts of a Turnout. — The several parts of a turnout are \ represented in Fig. 4. The dis- tance pf from the P. C. of the turnout curve to the point -of frog is called the frog distance. Fig. 4. The radius c o of the turnout curve, the frog distance, the frog angle, and the frog number bear certain relations to one another, -which are expressed by the following formulas: Tangent of half fr og angle = gauge -4- frog dista nce. Frog number = |/ radius c o -f twice the gauge. Frog number = 1 -r- £ the tangent of £ the frog angle. Radius co = twice the gauge X square of the frog number. Radius co — (frog distance pf + sine of frog angle) — i the gauge. Radius co — gauge -j- (1 — cosine of frog angle) — £ the gauge. Frog distance pf — frog number X twice the gauge. Frog distance pf = gauge p q tangent of £ the frog angle. Frog distance pf = (radius co + half the gauge) X sine of frog angle. TRACKWORK. 315 Middle ordinate (approximate) = £ the gauge. Each side ordinate (approximate) = £ the middle ordinato = 1 % (or .188) of the gauge. Switch length (approximate) = V throw in feet X 10,000 tan deflection for chords of 100 ft. for radius co of turnout curve* The tangent deflection may be obtained from the table on pages 298-300. Turnouts From a Straight Track. Gauge , 4 ft. in. Throw of switch , 5 in. Frog Number. , Frog «5 £ Turnout Radius. Degree of I Turnout Curve. Frog Distance. Middle Ordinate. Side Ordinate. Stub Switch Length. o r Feet. o / Feet. Feet. Feet. Feet. 12 4 46 1,356 4 14 113.0 1.177 .883 34 11% 4 58 1,245 4 36 108.3 1.177 .883 32 11 5 12 1,139 5 02 103.6 1.177 .883 31 10% 5 28 1,038 5 31. 98.9 1.177 .883 29 10 5 44 942 6 05 94.2 1.177 .883 28 9% 6 02 850 6 45 89.5 1.177 .883 27 9 6 22 763 7 31 84.7 1.177 .883 25 8% 6 44 680 8 26 80.0 1.177 .883 24 8 7 10 603 9 31 75.3 1.177 .883 22 7% 7 38 530 10 50 70.6 1.177 .883 21 7 8 10 461 12 27 65.9 1.177 .883 20 6%' 8 48 398 14 26- 61.2 1.177 .883 18 6 9 32 339 16 58 56.5 1.177 .883 17 5% 10 24 285 20 13 51.8 1.177 .883 15 5 11 26 235 24 32 47.1 1.177 .883 14 4% 12 40 191 30 24 42.4 1.177 .883 13 4 14 14 151 38 46 37.7 1.177 .883 11 The switch lengths in the above table merely denote the shortest length of stub switch that will at the same time form part of the turnout curve, and give .5 in. throw. Point or split switches require a throw of not more than 3£ in., though many have a throw of 5 in., with an equal space between the gauge lines at the heel. The heels of a split switch, which occupy the same position as the toes of a stub switch, should 316 SURVEYING. be placed at the point where the tangent deflection or offset is 5 in. The point where the tangent deflection is but 4£ in. will answer for many rail sections, but for those above 65 lb. per yd., 5 in. should be taken. In the table on pages 298-300, tangent deflections for chords of 100* ft. are given for all curves up to 20°; and for a curve of higher degree, the tangent deflection may be found by apply- C 2 ing the formula tan deflection = — 2 K In complicated trackwork, where space is limited, curves must be chosen to meet the existing conditions, and not with reference to particular frog angles, in which case the frogs are called special frogs and are made to fit the particular curve used. The determi- nation of the frog distance, switch length, and frog angle may be understood by referring to Fig. 5. Let the main track ab be a straight line; the gauge p q = 4 ft. 8£ in. (= 4.71 ft.); the de- gree of the turnout curve = 13°; the chord qd = 100 ft.; cd = the tangent deflection of the chord q d; and pf = the frog distance. From the table on page 299, we find the tangent deflection for a chord 100 ft. long of a 13° curve is 11.32 ft. Then, from Fig. 5, we have the proportion cd:ef = qc 2 : qe 2 . Now, in curves of large radius, qc and qd are assumed to be equal. Also, qe = pf , the frog distance, and substituting these equivalents w r e have the proportion Substituting the above given quantities in the proportion, we have 11.32 : 4.71 = 100 2 : pf 2 ; whence, P/ 3 = and the frog distance, pf = 64.5 ft. TRACKWORK. 317 If the space between the gauge lines at the heels of a split switch be taken at 5 in. = .42 ft., the distance from the P. C. of the turnout curve to the heel of the switch may be found as follows: In Fig. 5, let h, the tangent offset at the heel of the switch = .42 ft., we have the proportion cd : h = qd 2 : qh 2 , and substituting known values, we have 11.32 : .42 = 100 2 : qh 2 , whence, qh = 10,000 X .42 11.32 = 371.02, and q h = 19.26 ft. This locates the heel of a split switch and the toe of a stub switch. The frog angle is the angle kfl (see Fig. 5) formed by the gauge line of the main rail / k and the tangent to the outer rail qf of the turnout curve at the point where the two rails intersect. This angle is equal to the central angle qof. The arcs qf and r s are assumed to be of the same length. The turnout curve being 13°, the central angle for a chord of 1 ft. 13 X 60 is — = 7.8', and the central angle for 64.5 ft. the frog distance , is 7.8' X 64.5 = 8° 23', the frog angle for a 13° curve. By this process the frog distance, switch length, and frog angle may be calculated for curves of any radius. To Lay Out a Turnout From a Curved Main Track. — There are two cases: Case I.— When the two curves deflect in opposite direc- tions, illustrated in Fig. 6. Case II. — When the two curves deflect in the same direc- tion, illustrated in Fig. 7. In Fig. 6, the curve ab is 3° 30', and it is proposed to use a No. 8 frog. By reference to the table on page 315, we find that the degree of curve corresponding to a No. 8 frog is 9° 31'. Accordingly, we use a turnout curve a e, whose degree when added to the degree of curve of the main track shall equal the degree required for a No. 8 frog; i. e., we use a 6° turnout curve, which is within 1 minute of the required degree, and close enough for practical purposes. We know that for 318 SURVEYING. curves of moderate radii, i. e., from 1° up to 12°, the tangent deflections or offsets increase as the degree of the curve. That is, the tangent deflection of a 2°, 4°, and 6° is two, four, and six times, respectively, that of a 1° curve. In the accom- panying cuts illustrating the location of frogs and switches, each curve is represented by two lines indicating the rails, whereas only the center lines of the curves are run in on the ground t In Fig. 6, the line c d is tangent to the center lines of the curves. These center lines do not appear in the cut. Again referring to Fig. 6, if a tangent cd be drawn at c, the point common to the center lines of the curves, the sum of the deflections of both curves from the com- mon tangent will be equal, in this case, to the tangent deflec- Fig. 6. tion of a 9° 30' curve from a straight line. Accordingly, to find the frog distance for a 6° turnout curve from a 3° 30' curve, the curves being in opposite direc- tions, as shown in Fig. 6, we find the tangent ^deflection of a 9° 30' curve for a chord of 100 ft. This deflection is 8.28 ft., as given in the table on page 299. Assuming the gauge of track to be standard, viz., 4 ft. 8£ in. = 4.71 ft., and denoting the required frog distance by*, we have the following proportion: 8.28 : 4.71 = 1002 ; „ 10,000 X 4.71 whence, 8.28 = 5,688.4, and the frog distance, * = 75.42 ft. We use the tangent deflection for a 9° 30' curve, which very nearly equals the tangent deflection for a 9° 31' curve, thus saving the labor of a calculation; this will not appreci- ably affect the result. We locate the heel of the switch in the same way, using for the second term of the proportion, .42 ft., the distance between the gauge lines at the heel, instead of 4.71 ft., the gauge of the track. TRACKWORK. 319 In Fig. 7, which comes under Case II, both curves deflect in the same direction, and the rate of their deflection from each other is equal to the rate of the deflection' of a curve whose degree is equal to the difference of the degrees of the two curves from a tangent. Let the main-track curve a b be 5°, and the turnout curve ac be 10°. Then, the rate of deflection or divergence of the 10° curve from the 5° curve equals the divergence of a (10° — 5°) = 5° curve from a straight track or tangent. Accordingly, we find, in the table on page 298, the tangent deflection for a 5° curve for a chord of 100 ft. = 4.36 ft. Denoting the required frog distance by x , we have the fol- lowing proportion: 4.36 : 4.71 = 100 2 : x 2 , Fig. ' whence, x 2 — 10,000 x 4.71 4.36 = 10,802.8, and the frog distance, x = 103.9 ft. Distances are not calculated nearer than to tenths of a foot. How to Lay Out a Switch.— In laying out a switch, locate the frog so as to cut the least possible number of rails. Where there is some latitude in the choice of location, the P. C. of the turnout curve can be located so as to bring the frog near the end of a rail. To do this, take from the table on page 315 the frog dis- tance corresponding to the number of the frog to be used. Locate approximately the P. C. of the turnout curve, and measure from it, along the main-track rail, the tabular frog dis- tance. If this brings the frog point near the end of the rail, the P. C. of the turnout curve may be moved so as to require the cutting of but one main-track rail. Measure the total length of the frog, and deduct it from the length of the rail to be cut, marking with red chalk on the flange of the rail the point at which the rail is to be cut. Measure the width of the frog at the heel, and calculate the distance from the heel 320 SURVEYING. to the theoretical point of frog. For example, if the width of the frog at the heel is 8£ in., and a No. 8 frog is to be used, the theoretical distance from the heel to the point of frog is 8.5 X 8 = 68 in. = 5 ft. 8 in. Measure off this distance from the point, marking the heel of the frog. This will locate the point of the frog, which should be distinctly marked with red chalk on the flange of the rail. It is a common practice to make a distinct mark on the web of the main-track rail, directly opposite the point of frog. This point being under the head of the rail, it is protected from wear and the weather. The P. C. of the turnout curve is then located by measuring the frog distance from the point of frog. .From the table on page 315, we find the frog distance for a No. 8 frog is 75.3 ft., and the switch length, i. e., distance from P. C. of turnout curve to heel of split switch or toe of stub switch, is 22 ft. If a stub switch is to be laid, make a chalk mark on both main-track rails on a line, marking the center of the head- block. A more permanent mark is made with a center punch. Stretch a cord touching these marks, and drive a stake on each side of the track, with a tack in each. This line should be at right angles, to the center line of the track, and the stakes should be far enough from the track not to be dis- turbed when putting in switch ties. Next, cut the switch ties of proper length; draw the spikes from the track ties, three or four at a time, and remove them from the track, replacing them with switch' ties, and tamping them securely in place. When all the long ties are bedded, cut the main- track rail for the frog, being careful that the amount cut off is just equal to the length of the frog. If, by increasing or decreasing the length of the lead 5}, it is possible to avoid cutting a rail, do not hesitate to do so, especially for frogs above No. 8. Use full-length rails (30 ft.) for moving, or switch, rails, and be careful to leave a joint of proper width at the head- chair. Spike the head-chairs to the head-block so that the main-track rails will be in perfect line. Spike from 8 to 11 ft. of the switch rails to the ties, and slide the cross-rods on to the rail flanges, spacing them at equal intervals. The cross- rods are placed between the switch ties, which should not TRACKWORK. 321 be more than 15 in. from center to center of tie. The switch ties, especially those under the moving rails, should he of sawed oak timber. Southern pine is a good second choice. Attach the connection-rod to the head-rod and to the switch stand. With these connections made, it is an easy matter to place the switch stand so as to give the proper throw of the switch. It is common practice to fasten the switch stand to the head-block with track spikes, but a better fastening is made with bolts. The stand is first properly placed, and the holes marked and bored, and the bolts passed through from the under side of the head-block. This obviates all danger of movement of the switch stand in fastening, which is liable to occur when spikes are used, and insures a perfect throw. The use of track spikes is quite admissible when holes are bored to receive them, in which case a half-inch auger should be used for standard track spikes. The switch stand should, when possible, be placed facing the switch, so as to be seen from the engineer’s side of the engine— the right-hand side. Next stretch a cord from a, Fig. 8, a point on the outer main-track rail opposite the P. C. of the turnout curve to b r , the point of the frog. This cord will take the position of the chord of the arc of the outer rail of the turn- out curve. Mark the middle point c and the quarter points d and e. Whatever the degree of the turnout curve, the distance from the middle point c of the chord to the arc a b' is 1.18 ft., and the distances from the quarter points d and e are .88 ft.; hence, at c lay off the ordinate 1.18 ft., and at both d and e the ordinate .88 ft., three-quarters of the middle ordinate. These offsets will mark the gauge line of the rail a b' . Add to these offsets the distance from the gauge line to outside of the rail flange, and mark the points on the switch ties. Spike a lead rail to these marks, and place the other at easy track gauge from it. Spike the rails of the turnout as far as the point of frog to exact gauge, unless the gauge has been widened owing to the sharpness of the curve. Beyond the Fig. 8. 322 SURVEYING. point of frog the curve may be allowed to vary a little in gauge to prevent a kink showing opposite the frog. In case the gauge is widened at the frog, increase the guard-rail dis- tance an equal amount. For a gauge of 4 ft. 8£ in., place the side of the guard rail that comes in contact with the car wheels at 4 ft. 6£ in. from the gauge line of the frog. This gives a space of 1 $ in. between the main and guard rails. In case the gauge is widened i or | in., increase the guard- rail distance an equal amount. When the turnout curve is very sharp, it will be necessary to curve the switch rails, to avoid an angle at the head-block. The lead rails should be carefully curved before being laid, and great pains should be taken to secure a perfect line. If a point , or split, switch is to be laid, the order of work is nearly the same. The same precautions must be taken to avoid the unnecessary cutting of rails, with the additional precaution of keeping the switch points clear of rail joints, as the bolts and angle splices will prevent the switch points from lying close to the stock rails. As already stated, these conditions can usually be met where there is some range in the choice of the location of the switch. Where there is none, the main-track rails must be cut to fit the switch. Having located the point of frog, the P. C. of the turnout curve, and the heel line of the switch, measure back from the heel line a distance equal to the length of the switch rails, and place on the flange of each rail a chalk mark to locate the ends of the switch points. This will also locate the head-block. Prepare switch ties of the requisite number and length, and place them in the track in proper order. As in the case of stub switches, see to it that all long switch ties are in place before cutting the rail for placing the frog; also, that the ends of the lead rails, with which the switch points connect, are exactly even; otherwise, the switch rods will be skewed, and the switch will not work or fit well. Fasten the switch rods in place, being careful to place them in their proper order, the head-rod being No. 1. Each rod is marked with a center punch, the number of the punch marks corresponding to the number of the rod. Couple the switch points with the lead rails, and place the TRACKWORK. 323 sliding plates in position, securely spiking them to the ties. Connect the head-rod with the switch stand, and close the switch, giving a clear main track. Adjust the stand for this position of the switch, and bolt it fast to the head-block. Next, crowd the stock rail against the switch point so as to insure a close fit, and secure it in place with a rail brace at each tie; then continue the laying of the rails of the turnout. If there is no engineer to lay out the center line of the turnout, the section foreman can put in the lead from ordinates, as explained in Fig. 8. In modern railroad practice, how- ever, most trackwork is done under the direction of an engineer, in which case the center line of the turnout is located with a transit. This insures a correct line and expedites work. For ordinary curves, center stakes at intervals of 50 ft. are sufficient, excepting between the P. C. of the turnout and the point of frog, where there should be a center stake at each interval of 25 ft. Place a guard rail oppo- site the point of frog on both main track and turnout. The guard rail should be 10 ft. in length; this is an economical length for cutting rails, as each full-length rail makes three guard rails. Two styles of guard rails are shown in Fig. 9. That shown at B is in general use, but the style shown at A is growing in favor. The latter is curved throughout its entire length. At its middle point a, directly opposite the point of frog, the guard rail is spaced If in. from the gauge line of the turnout rail be. From this point the guard rail diverges in both directions, giving at each end a flangeway of 4 in. This allow;s the wheels full play, excepting at the point of frog, where the guard rail is exactly adjusted to the track gauge, and holds the wheels in true line, preventing them from climbing , or mounting , the frog. The style of guard rail shown at B, though still much used, has two objectionable features; Fig 324 SURVEYING. viz., first, the abruptly curved ends d and e often receive an almost direct blow from the wheel flanges, which causes a car to lurch violently; and second, the flangeway of uniform width, though proper for the main track when straight, as in Fig. 9, is unsuited for sharp curves on either a main track or a turnout, as it compels the wheels to follow a curved line; whereas the normal position of the wheel base of each truck is that of a chord of, or a tangent to, the curve. These two defects alone produce what is known as a rough-riding frog, even though the frog is well lined and ballasted. Location of Crotch Frog.— A crotch , or middle , frog is a frog placed at the point where the outer rails of both turnouts of a three-throw switch cross each other. When both turnouts are of the same de- gree, the crotch frog comes midway be tween the main-track rails. Its location and angle may be determined as follows: Let the turnout curves A and B, Fig. 10, be each 9° 30', uniting with the main track C by a three-throw switch. Let a be the P. C. common to both curves, and b, the location of crotch, or middle, frog. It is evident that the point of the crotch frog should be exactly midway between the gauge lines of the main-track rails, and if the gauge is 4 ft. 8& in. = 4.71 ft., the point of the crotch of the frog will be 4.71 = 2.35 ft. from each rail. Now, the problem is to find the frog distance from a, the P. C., to the point c, where the tangent deflection will equal 2.35, or half the gauge. From the table on page 299, we find the tangent deflection of a 9° 30' curve is 8.28 ft. Applying the principle explained in connection with Fig. 5, and letting x represent the required frog distance, we have the following proportion: 8.28:2.35 = 100 2 : x-; whence, x 2 = 100 2 x 2.35, 8.28 2,838.2 ft., and the required frog distance x = 53.3 feet, nearly. TRACKWORK. 325 Now, there are two curves starting at the common point a; the outer rails intersect at 5, and the angle d b e, formed by tangents drawn at the point of intersection, is the angle of the crotch, or middle frog. The angle is equal to the sum of the angles afb and af' b; that is, equal to double the central angle of either curve between the P. C. and the point of intersection b. The degree of the curve is 9° 30' = 570', and 570 ' the central angle or total deflection for each foot is — = 5.7'; and for the frog distance of 53.3 ft., the central angle is 53.3 X 5.7' = 303.8' = 5° 03.8'. The angle of the crotch frog is double this angle; i. e., 5° 03.8' X 2 = 10° 07.6'. The crotch frog should be accurately located and spiked in place before the lead rails are placed. The one objection to the three-throw switch is the open joint at the head-block, the inevitable attendant of the stub switch, but its advantages are so great that it will continue to be used, especially in yard service. Crossover Tracks.— A crossover is a track by means of which a train passes from one track to another. The tracks united are usually parallel, as are the tracks of a double-track road. Such a crossover is shown in Fig. 11. The tracks a b and c d are 13 ft. apart from center to center, which is the standard distance for double tracks. The crossover consists of two Fig. 11. turnout curves, ef and g h. These curves are usually, though not necessarily, of the same degree. The curves terminate at the points of frog / and h, between which the track fh is a tangent. The essential point in laying out a crossover is to so place the frogs that the connecting track shall be tangent to both curves. In Fig. 11, suppose the frogs are No. 9, requiring 7° 31' turnout curves. From the table on page 315, we find the required frog dis- tance is 84.7 ft., and the switch length 25 ft. As previously 326 SURVEYING. noted, if there is considerable range in choice of location, the frogs can be so placed as to largely avoid the cutting of rails; but usually crossovers are required at certain precise places, and the rails must be cut as occasion demands. Hav- ing located the point of frog at /, we determine the point of the next frog at A, as follows: A No. 9 frog is one that spreads 1 in. in width to every 9 in. in length; and, as the track between the frog points is straight, the distance /A between these points will be as many times 9 in. as is the space k between the tracks at the frog point /. The main-track centers are 13 ft. apart, making the space between the gauge lines of the inside rails 8 ft. in. As it is the rail l of the turnout that joins the second frog at A, we subtract the gauge, 4 ft. 8* in. from 8 ft. 3£ in., leaving 3 ft. 7 in., the distance k, between the gauge line of the rail l , opposite the frog point /, and the gauge line of the nearest rail of the track c d. This distance multiplied by 9 in. will give the distance from the frog point / to the frog point A; 3 ft. 7 in. = 43 in.; 43 X 9 = 387 in. = 32 ft. 3 in. Accordingly, having located the point or frog /, we mark a corresponding point on the nearest rail of the opposite track. From this point we meas- ure along the rail the distance 32 ft. 3 in., locating the second frog point A, and again the frog distance 84.7 ft. to the P. C. of the second turnout curve at g. If frogs of different numbers, say 7 and 9, were to be used, the distance between the frogs is found as follows: As the No. 7 frog spreads 1 in. in 7 in., and the No. 9 frog 1 in. in 9 in., the two will together spread 2 in. in 7 4- 9 = 16 in., or 1 in. in 8 in. Now, if the rails to be united are 3 ft. 7 in., or 43 in., apart, as in the previous problem, the distance between the frog points will be 43 X 8 = 344 in. = 28 ft. 8 in. In locating crossover tracks, regard should be paid to the direction in which the bulk of the traffic moves, and the crossover tracks should be so placed that loaded cars will be backed, not pushed, from one track to the other. At all stations on double-track roads there should be a crossover to facilitate the exchange of cars and the making up of trains. PERPETUAL CALENDAR. 327 PERPETUAL CALEN DA R— 1797-1904. 3 4 5 6 0 1 2 June. Sept. Dec. April. July. Jan. Oct. May. Aug. Feb. Mar. Nov. 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 . 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 Sun. Mon. Tues. Wed. Thur. Fri. Sat. 1 2 3 4 5 6 <7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 .34 35 36 37 38 39 40 41 / 42 43 44 328 PERPETUAL CALENDAR. By means of the table given on the preceding page, the day of the week corresponding to any date between 1797 and 1904, inclusive, may be readily found. Before every leap year there is a blank space. To find the day of the week on which January 1 of any year fell, find that year in the table; glance down the column containing that year, and the day of the week at the foot of the column will be the day of the week required. Thus, to find on what day of the week January 1 , 1895, fell, we find under 1895 in the table, Tuesday. For leap years, we look for day of week under the blank space before the year. Thus, January 1 , 1896, fell on Wednes- day, Wednesday being in the column containing the blank space before 1896. To find the day of the week for any other date, add (mentally) to the day of the month the first number under the day of the week that is contained in the column containing the year of the century; to this sum, add the number above the month at the top of the table. Find the number thus obtained in the columns of figures under the days of the week; the day of the week at the head of the column containing this number will be the day required. Thus, to find on what day of the week September 10, 1813, fell, we find 1813 in the table. The number under the day of the week in the column containing 1813 is 6, and the number above September at the top of the table is 4. Hence, 10 + 6 + 4 = 20. The day of the week above 20, in the lower part of the table, is Friday. For dates in January and February of leap years, take one day less, or add the number beneath the day of the w T eek under the blank space preceding the year. Thus, for Feb- ruary 12, 1896, we have 12 + 4 + 2 = 18, and the day of the week above 18 is Wednesday. The table may also be used for fixing dates. Thus, Thanksgiving Day is the last Thursday in November; on what day of the month did it fall in 1897 ? Since the earliest day on which it can fall is the 24th, we find on what day of the week November 24 falls, and then count ahead to Thurs- day. Referring to the table, 24 + 6 + 2 = 32; the day of the week above 32 is Wednesday, and since Thursday is one day late*, it follows that Thanksgiving Day in 1897 fell on the 25th.