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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I 11ilUl'l'i'JllJ'Jl'Jl'!IJl'!IJ'.liliUi: iHiiiii i iniiummmm GaniiHii mmmiminiiHnumminiiii ^iii>iiiiiluilliiiMiii'rtMii»<riil<LHuiiutalMiH<HrmWt«Minil>i>aiuiiiirtiiiiiin<rfMiiiii'r:l fiiiiiiHiiiiiiiiHiiiiiiiiiiuuiuittiiuyiiiuiiiiiuiiiuiiiuiuiiJiBfliuiiiuuiUinniDiM THE GIFT OF lilllllllillilillllllillllilillillilllMII.111111111 lilllkllllillllflUMIIIIII OB l*- 1^ r THE CIVIL ENGINEER'S POCKET-BOOK JOHN 0. TRAUTWINE CIVIL ENGINEER EKVISBD BY JOHN C. TKAUTWINE, Jb. AND JOHN C. TRAUTWINE, 3d. CIVIL ENGINEERS EIGHTEENTH EDITION, NINETIETH THOUSAND NEW YORK JOHN WILEY A SONS LovDoir: CHAPMAN & HALL, Limited 1907 \. Entered, according to Act of Congress, in the year 1882, hj JOHN C. TRAUTWINE, in the Office of the Librarian of Congress at Washingron. Copyright by John C. Trautwine, Jr., 1902. > WM. F. FELL COMPANY A. REED & CO. ELECTROTYPERS AND MIINTKRS BINDERS PHILADELPHIA PHILADELPHIA THE AUTHOR DEDICATES THIS BOOK TO THE MEMORY OF HIS FRIEND, THE LATS BENJAMIN H. LATROBE, Esq., CITIL ENQINEXK. No pains have been spared to maintain the position of this as the foremost Civil Engineer's Pocket-book, not only in the United States, but in the EngUsh language. JOHN 'WILEY & SONS, Scientific Publishers, IS East Nineteenth Street, New Yor^ City. PREFACE TO FIRST EDITION, 1872. QHOULD experts in engineering complain that they do not find ^ anything of interest in this volume, the writer would merely remind them that it was not his intention that they should. The book has been prepared for young members of the profession ; and one of the leading objects has been to elucidate, in plain En^ish, a few important elementary principles which the savants have envel- oped in such a haae of mysteiy as to render pursuit hopeless to any but a confirmed mathematician. Comparatively few engineers are good mathematicians ; and in the writer's opinion. It is fortunate that such is the case ; for nature rarely combines high mathematical talent, with that practical tact, and observation of outward things, so essential to a successful engineer. There have been, it is true, brilliant exceptions ; but they are very rare. But few even of those who have been tolerable mathe- matidana when young, can, as they advance in years, and become engaged in business, spare the time necessary for retaining such accomplishments. Nearly all the scientific principles which constitute the founda- tion of civil engineering are susceptible of complete and satis- factory explanation to any person who reaUy possesses only so much elementary knowledge of arithmetic and natural philosophy as is Bupposed to be taught to boys of twelve or fourteen in our public schools.* * Let two little boys weigh each other on a platform scale. Then when thej iMdanoe each other on their board see-eaw, let them see (and measure for them- •elTbs) that the lighter one is farther from the fence-rail on which their boaid is placed, in the same proportion as the heavier boy outweighs the lighter one. Tfaey will then have learned the grand principle of the iever. Then let them measure and see that the light one see-saws farther than the heavy one, in the same proportion ; and they will have acquired the principle of virtual veloeiUa^L.^ Explain to them that eqwUUy qf moments means nothing more than that V VI PREFACE. ^^^ • The little tbat is beyond this, might safely be intrusted to the savants. Let them work out the results, and give them to the engi- neer in intelligible language. We could afford to take their words for it, because such things are their specialty ; and because we know that they are the best qualified to investigate them. On the same principle we intrust our lives to our physician, or to the captain of the vessel at sea. Medicine and seamanship are their respective specialties. If there is any point in which the writer may hope to meet the approbation of proficients, it is in the accuracy of the tables. The pains taken in this respect have been very great. Most of the tables have been entirely recalculated expressly for this book ; and one of the results has been the detection of a great many errors in those in common use. He trusts that none will be found exceed- ing one, or sometimes two, in the last figure of any table in which great accuracy is required. There are many errors to that amount, they seat themselves at their measured distances on their see-saw, ikey balance each other. Let them see that the weight of the heavy hoy, when multiplied hy his distance in feet from the fence-rail amounts to just as inuch as the weight of the light one when multiplied by his distance. Explain to them that each of the amounts is in foot-pounds. Tell them that the lightest one, because he see- saws so much faster than the other, will bump against the ground Just as hard as the heavy one ; and that this means that their momentums are equal. The boys may then go in to dinner, and probably puzzle their big lout of a brother who has just passed through college with high honors. They will not forget what they have learned, for they learned it as play, without any ear-pulling, spanking, or keeping in. Let their bats and balls, their marbles, their swings, Ac, once become their philosophical apparatus, and children may be taught {really taught) many of the most important principles of engineering before they can read or write. It is the ignorance of these principles, so easily taught even to children, that constitutes what is popularly called " The Practical Enginkeb ; " which, in the great majority of cases, means simply an ignoramus, who blunders along without knowing any other reason for what he does, than that he has seen it done BO before. And it is this same ignorance that causes employers to prefer this practical man to one who is conversant with principles. They, themselves, were spanked, kept in, &jc, when boys, because they could not master leverage, equality of moments, and virtual velocities, enveloped in x's, p's, Greek letters, square- roots, cube-roots, &c, and they naturally set down any man as a fool who could. They turn up their noses at science, not dreaming that the word means simply, Juwwing why. And it must be confessed that they are not altogether without reason ; for the savants appear to prepare their books with the express object of preventing purchasers, (they have but few readers,) from learning why. PREFACE. Vll especially where the recalcalation was very tedious, and where, oousequently, interpolation was resorted to. They are too small to be of practical importance. He knows, however, the almost impos- sibility of avoiding larger errors entirely; and will be glad to be informed of any that may be detected, except the final ones alluded to, that they may be corrected in case another edition should be called for. Tables which are absolutely reliable, possess an in- trinsic value that is not to be measured by money alone. With this consideration the volume has been made a trifle larger than would otherwise have been necessary, in order to admit the stereotyped sines and tangents from his book on railroad curves. These have been so thoroughly compared with standards prepared independ- ently of each other, that the writer believes them to be absolutely correct. In order to reduce the volume to pocket-size, smaller type hat been used than would otherwise have been desirable. Many abbreviations of common words in frequent use have been introduced, such as abut, oen, diag, hor, vert, pres, &c, instead of abutment, center, diagonal, horizontal, vertical, pressure, &c. They can in no case lead to doubt ; while they appreciably reduce the thickness of the volume. Where prices have been added, they are placed in footnotes. They are intended merely to give an approximate or comparative idea of value ; for constant fluctuations prevent anything farther. The addresses of a few manufacturing establishments have also been inserted in notes, in the belief that they might at times be found convenient. They have been given without the knowledge of the proprietors. The writer is frequently asked to name good elementary books on civil engineering ; but regrets to say that there are very few such in our language. "Civil Engineering," by Prof. Mahan of West Point ; " Roads and Railroads," by the late Prof. Gillespie ; and the '* Handbook of Railroad Construction," by Mr. George L. Vose, Civ. Eng. of Boston, are the best. The writer has reason to know that a new edition of the last, now in press, will be far Viii PREFACE. superior to all predecessors ; and better adapted to the wants of the young engineer than any book that has appeared. Many of Weale's series are excellent. Some few of them are behind the times ; bat it is to be hoped that this may be rectified in iiitare editions. Among pocket-books, Haswell, Hamilton's Usefhl Information, Henck, Molesworth, Nystrom, W^^^®) ^f abound in valuable matter. The writer does not include Rankine, Moseley, and Weisbach, because, although their books are the productions of master-minds, and exhibit a profundity of knowledge beyond the reach of ordi- nary men, yet their language also is so profound that very few engineers can read them. The writer himself, having long since foigotten the little higher mathematics he once knew, cannot. To him they are but little more than striking instances of how com- pletely the most simple &cts may be buried out of sight under heaps of mathematical rubbish. Where the word *'ton '' is used in this volume, it always means 2240 lbs. There is no table of errata, because no errors are known to exist except two or three of a single letter in spelling ; and which will probably escape notice. John C. Tbautwhi*. Philadelphia, November 13th, 1871. PREFACE TO NINTH EDITION. TWENTY-SECOND THOUSAND, 1885. CI INCE the appearance of its last edition (ihe twentieth thousand) '^ in 1883, the " Ppcket-Boo]c " has been thoroughly revised, and many important additions and other alterations have been made. These necessitated considerable change in the places of the former matter, and it veas deemed best to turn this necessity to advantage, and to make a thorough re-arrangement, putting all of the articles, as far as possible, in a rational order. The list of new matter and of revisions and extensions is condensed as foUows, 1902 : New articles on the steam-bammer pile driver, machine rock drills, air com- pressors, high explosives, cost of earthwork by drag and wheel scrapers and by steam excavators, iron trestles, track tanks, artesian well-boring and standard time, and new tables of railroad curves in metric measure, circumferences and areas of circles, thermometric scales, and fractions with their decimal equivalents. Articles revised and extended, on circular arcs, thermometers, flotation, flow in- pipes, waterworks appliances, velocities, d;c, of falling bodies, centrifugal force, strength of timber, strength of beams, riveting, riveted girders, trusses, Bospension bridges, rail joints, turnouts, turntables, locomotives, cars, railroad statistics and manufactured articles, including columns, beams, channels, angles and tees. Most of the new matter is in nonpareil, the larger of the two types heretofore used. Boldfoced type has been freely used ; but only for the purpose of guiding the reader rapidly to a desired division of a subject. For emphasis, italics have been employed. Illustrations which were lacking in clearness or neatness have been re-touched and re-lettered, or replaced with new and better cuts. The new matter is very freely illustrated. New rules have been put in the shape of formulae, and many of the old rules have been re-cast into the same form. ix X FB£:fAC£. The addition of new matter, and a number of blank spaces necessarily left in making the re-arrangement, have increased the number of pages about one-fifth. The new index is in stricter alphabetical order than that of former editions, and contains more than twice as many entries, although much repetition has been avoided by the free use of cross- references, without which this part of the work might have been indefinitely extended. The selection of articles of manufacture or merchandise for illus- tration, has been guided by no other consideration than their fitne^ for the purpose, and the courtesy of the parties representing them, in supplying information. The writer gratefully acknowledges the kindness of those who have assisted in furnishing and arranging data. Philadelphia, January, 1886. J. C. T., Jb. PREFACE TO EIGHTEENTH EDITION. (SEVENTIETH THOUSAND, 1902.) IN preparation for its eighteenth edition, The Civil Engineer's Pocket Book, the first edition of which appeared thirty years ago, has undergone a far more extensive revision than at any other time. More than 370 pages of new matter have been added ; and the new edition is larger, by about 100 pages, than its recent predecessors. Among the new matter in this edition will be found : Pages 43- 46 Annuities, Depreciation, etc. 70- 72 Logarithms. 73- 77 Logarithmic Chart and Slide Kule. 80- 91 New Table of Logarithms. 228- 253 Conversion Table of Units of Meaaurement. 300- 301 Isogonic Chart. 532- 635 Venturi Meter. 536 Ferris-Pitot Meter. 546 Miner's Inch. * 649 Water Consumption in Cities. 658- 659 Cost of Water Pipe and Laying. 745- 764 Digests of Specifications for Bridges and Buildings. 816 Tie Plates. 870- 873 Digest of Specification for Iron and Steel. 905- 906 Gray Column. 914 Trough Floor Sections. 983- 995 Price List of Manufactured Articles. 996-1007 Business Directory. 1008-1023 BibUography. The following articles have been almost or entirely rewritten: Nkw Pages Old Pages 35- 47 Arithmetic 33-37 210-211 Specific Gravity 380-381 265-266 Time 395 282-283 Chains and Chaining 176 284-290 Location of the Meridian 177-179 322-325 Rain and Snow 220-221 358-453 Statics 318 f-361, 370-375 xi • • XU PREFACE. New Pages Old Pages 466-494 Strength of Beams 478-520, 528-536 499 Shearing Strength 476 499-500 Torsional Strength 476-477 501-503 Opening Remarks on Hydrostatics 222-224 537-538 Effect of Curves and Bends on Flow in Pipes 255-256 689-744 Trusses 647-614 856-864 Locomotives 805-810 865-866 Cars 811-813 867-869 Railroad Statistics 814-818 892-899 I Beams, Channels, Angles and T Shapes 521-527 930-942 Cement 673-678 943-947 Concrete 678-682 954-956 Timber Preservation 425-425 a The articles on arithmetic are considerably extended, notably by the addition of new matter relating to interest, annuities, depreciation, etc., including several tables. The new and greatly enlarged table of five-place logarithms is arranged in a somewhat novel form. In constructing this table, the effort has been to obviate the difficulty, present in all tables where the difference between successive numbers is constant throughout, that the differences between successive logarithnas of the lower numbers are relatively very great. In the new table the differences between logarithms are much more nearly con- stant. For convenience in rough calculations, the old table of five-place logarithms, on two facing pages, is retained. The Conversion Tables contain the equivaleilts of both English and metric units, and of each of these in terms of the other; but, owing to the extreme ease with which one metric imit may be converted into others of the same system, it has been unnecessary to burden the table with many of the metric units. The tables have been separately calculated by at least two persons, and their results compared and corrected. One of these results has then been used by the compositor in setting the type, and the proofs have been compared with the other. The new article on the location of the meridian is much more complete than its predecessors, and a new table of azimuths of Polaris, corresponding to different hour-angles, has been added. Perhaps the most radical and extensive of all the changes in this edition are those in the articles on Statics, on Beams and on Trusses These have been almost entirely rewritten and com- pletely modernized. Under Trusses, modern methods of cal- culating the stresses in and the dimensions of the several FBEFAOS. xiii members, and modern methods of construction, are explained, and several modern roofs and bridges are described and illus- trated. One of the most notable features in the new article is the digest of prominent modem specifications for bridges for steam and electric railroads and for highways. The articles on the strength of beams are greatly simplified and brought into harmony with modern. methods of dealing with that subject. In preparing the digests of specifications for iron and steel, use has been made of the specifications recently adopted by the American Section of the International Association for Testing Materials; while those of the American Society of Civil Engineers and of the recent report of a Board of United States Army engineer officers have been similarly used in con- nection with cement. The price list of engineering materials and appliances has been prepared merely as a useful guide in roughly estimating the ap- proximate costs of work, and it is not to be supposed that it can, in any important case, take the place of personal inquiry and correspondence with manufacturers or their agents, nearly 700 of whom are named in the accompanying list of names and addresses of manufacturers, etc. From its first appearance, the Pocket Book has undertaken to give prices of certain manufac- tured articles, and addresses of those from whom they may be obtained; but these, scattered as they were throughout the voliune, were necessarily desultory, and limited in their extent and usefulness. It is hoped that the present articles will be found at least an acceptable substitute for them. As in preceding editions, all new work and all revisions have been the subject of our personal attention, and " scissors-and- paste" methods have been scrupulously avoided. Even in using lists of manufactured articles, etc., although their statements have in general been left unchanged, the matter has in most or all cases been rearranged and classified, to suit the requirements of this work. For instance, the ''digests" of specifications for Cement, for Steel and Iron, for Railroad and Highway Bridges and for Steel Buildings, are by no means mere quotations from the originals; but, as their name implies, the result of careful digesting of the contents of the specifications selected for the purpose; their several provisions being carefully studied, in nearly all cases re- worded or reduced to figures, and tabulated in form convenient XIV PREFACE. for reference, the whole being arranged in such logical order as to facilitate reference. As in all cases heretofore, every rule or formula and every description of methods, etc., can be readily understood and ap- plied by any one, engineer or layman, understanding the use of common and decimal fractions, of roots and powers, of loga- rithms, and of sines, tangents, etc., of angles. On the other hand, one who is not possessed of this very meager stock of mathemati- cal knowledge will hardly approach engineering problems, even as an amateur; -and we have therefore followed the precedent, established seventeen years ago, of putting rules in the shape of formulas, which have " the great advantage of showing the whole operation at a glance, of making its principle more apparent, and of being much more convenient for reference" (From Preface to ninth edition, 1885). The new matter is very fully illustrated. As heretofore, all cuts have been engraved expressly for this work. As in preparing for the ninth edition (1885), all the matter of the book has been rearranged. This has necessitated a new paging; and, in making this, the lettering of pages, introduced from time to time as new editions have appeared in the past, has been eliminated. The rearrangement and the addition of so much new matter have of course necessitated the preparation of a new table of contents and a new index. In this, as in all previous editions since the eighth (1883), practically all new matter has been set in nonpareil, the larger of the two types hitherto used, and much of the old matter retained has been reset in the larger type. We take pleasure in acknowledging our indebtedness to many who have kindly assisted us in our work, notably to Messrs. Otis E. Hovey and Wm. M. White, of the American Bridge Co., for painstaking examination of the article on Trusses; to Mr. C. Robert Grimm and Professor E. J. McCaustland for similar as- sistance in connection with the article on Statics; to Misses Laura Agnes Whyte and Louise C. Hazen for suggestions respecting mathematics and astronomy ; and to the following gentlemen for valuable information respecting the subjects named : Isogonic Chart, Mr. O. H. Tittmann, Sup't, U. S. Coast and Geodetic Survey. Trusses, Messrs. Wm. A. Pratt, Engineer of Bridges, Pennsyl- vania Railroad; W. B. Riegner, Engineer of Bridges, Philadel- PREFACE. XV phia and Reading Railway; Paul L. Wolfel, Chief Engineer, American Bridge Co.; J. Sterling Deans, Chief Engineer, and Moritz G. Lippert, Assistant Engineer, Phoenix Bridge Co. ; Ralph Modjeski, Northern Pacific Railway; D. J. Whittemore, Chief Engineer, and C. F. Loweth, Engineer and Superintendent of Bridges and Buildings, Chicago, Milwaukee and St. Paul Railway. Specifications for Bridges and Buildings, Messrs. C. C. Schnei- der, Vice President, American Bridge Company; J. E. Greiner, Engineer of Bridges and Buildings, Baltimore and Ohio Railroad ; Theodore Cooper; W. K. McFarlin, Chief Engineer, Delaware, Lackawanna and Western Railway; Mason B. Strong, Bridge Engineer, Erie Railroad; F. C. Osborn, President, Osborn En- gineering Co. ; Wm. A. Pratt, Engineer of Bridges, Pennsylvania Railroad ; W. B. Riegner, Engineer of Bridges, Philadelphia and Reading Railway; W. J. Wilgus, Chief Engineer, New York Central Railroad. Locomotives, Baldwin Locomotive Works; Messrs. Wilson Miller, President, Pittsburgh Locomotive and Car Works ; Theo. N. Ely, Chief of Motive Power, Pennsylvania Railroad; A. E. Mitchell, C. W. Buchholz and A. Mordecai, of the Erie Rail- road; Edwin F. Smith, Wm. Hunter, A. T. Dice and Samuel F. Prince, Jr., of the Philadelphia and Reading Railway; and Thomas Tait, Manager, Canadian Pacific Railway; and Major E. T. D. Myers, of the Richmond, Fredericksburg and Potomac Railroad. Cars, Allison Manufacturing Co., Harlan & HoUingsworth Co., and Mr. Jos. W. Taylor, Secretary, Master Car Builders* Associa- tion. Railroad Statistics, Mr. Edward A. Moseley, Secretary, Inter- state Commerce Commission. Iron and Steel, Mr. Wm. R. Webster. Cement, Mr. Richard L. Humphrey. Concrete Beams, Mr. Howard A. Carson, Chief Engineer, Bos- ton Transit Commission. Preservation of Timber, Mr. O. Chanute. Building Material, Mr. John T. Willis. John C. Trautwine, Jr., John C. Trautwine, 3d. Philadelphia, October, 1902, Folios xvi to xxiv inclusive are left blank, to provide for future additions to prefaces. XTi CONTENTS. MATHEMATICS, paob Mathematical Ssnnbote 33 Greek Alphabet 34 Aritliinetie. Factors and Multiples 35 Fractions 35 Decimals 37 Ratio and Proportion 38 Progression 39 Permutation, Combination, Al- ligation 40 Percentage, Interest, Annuities 40 Simple Interest 41 Equation of Payments 42 Compound Interest 42 Annuity^ Sinking Fund, De- preciation, etc 43 Equations and Tables. . .44r-46 Duodenal Notation 47 Reciprocals 48-52 Roots and Powers. Square and cube. Tables ; 64 Rules 66 Fifth Roots and Powers .... 67 LoKarithms 70 Rules 70 Logarithmic Chart and Slide Rule 73 Two-page Table 78 Twelve-page Table 80 Geometry. Alensiiration, and Tnyonometrjr. liines. Definitions 02 Angles- Definitions 92 Construction 93 Bisection 94 Inscribed 94 Complement and Supplement . 94 In a Parallelogram 95 Minutes and Seconds in Deci- mals of a Degree, Table of — 95 Approximate Measurement of Angles 96 Sine, Tangent, etc 97 Definitions 97 Table 98 Ohonk. Table' d!-^ '.'.'.! !!'.'. 143 PAOB Polygons. R^^ular — , Tables, etc.. of — 148 Triangles. Dennitions. Properties 148 Right-angled — 150 Trigonometrical Problems . . 150 Parallelogram '. 157 Trapezoid. Trapezium 158 Polygons 159 Regular 159 Reduction of Figures. . .159, 160 Circle 161 Radius. Diameter 161 Area, Center, to Find — ... 161 Problems 161. 162 Tables of — . Diameter in Units, Eighths, etc 163 Diameters in Units and Tenths 166 Diameters in Units and Twelfths 172 Arc. Circular. Chord, Length 179 Radius, Rise, and Ordinates. 180 Of Large Radius, to Draw — 181 Tables of — 182-185 Circular Sector, Ring, Zone, and Lune 186 Circular Segment. Area of — ; to Find 186 Area of — ; Table 187 Ellipse. Properties of ^ 189 Ordinates and Circumference of —; to Find — 189 Elliptic Arc 189 Tables of Lengths of — ... 190 Area of; to Find — 190 Construction. Tangents. . . 190 Oval or False — 191 C^ma Recta, Cyma Reversa, Ogee 191 Parabola. Properties of — 192 Parabolic Curve. Length of-^- 192 Area 192 Parabolic Zone or Frustum . 192 Construction 193 Cycloid 194 Solids. Regular Bodies. Tetiahedron, Hexahedron, etc 194 Guldinus Theorem 194 Parallelopiped, Properties 105 XXV XXVI CONTENTS. PAGE Priam .- 195 Frustum 195 Cylinder. Volume and Surface of — . . 196 Volume. Table of — , in Cu. Ft. and U. S. Gala 197 Wella; Contenta of — and Masonry in Walla of — ... 198 Cylindrio Ungula 199 Pyramid and Cone 200 Frustums of 201 Prismoid 202 Wedge 203 Sphere. Properties 204 Volume, Surface, etc. Formulas for — 204 Tables of — 205-207 Segment and Zone of — . . . . 208 Spherical Shell 208 Spheroid or Ellipsoid 208 Paraboloid 209 Frustum of — 209 Circular Spindle 209 Circular Ring 209 Specific OraTity. Principles 210 Table 212-216 Welgrbts and Measures. U. S., British and Metric — , Units of — 216 Coins; Foreign and U.S. — 218 Gold and Silver 219 Weights; Troy, Apothecaries' and Avoirdupois — 220 Long Measure 220 Degrees of Longitude. Length. 221 Inches Reduced to Decimals of a Foot. Table 221 Square or Land Measure 222 Cubic or Solid Measure 222 Liquid Measures 223 Diy Meaaure 223 British Imperial Measures 224 Volumes and Weights of Water 224 Metric Units 226 Systfeme Usuel, — Ancien 226 Russian 227 Spanish 227 Conversion Tables 228 Introduction and Explana- tion 228 List of Tables 229 Fundamental Equivalents . . 230 Abbreviations 230 Equivalents and Numbers in Common Use 231 Metric Prefixes 231 Tables 232 Aorea per Mile and per 100 feet. Table 254 PAGE Grades, Tables of — 255-257 Heads and Pressures of Water; Tables of — 258-260 Discharges in Gals, per Day and Cu. Ft. per Second; Tables 261-265 Time. Definitions, etc 265 Standard Railway — 267 Dialing 268 Board Measure. Table 269 Survey infT. Testa of Accuracy, Distribution of Error, etc 274 Chaining 282 Location of Meridian 284 By Circumpolar Stars 284 Definitiona 284 By Meana of Polaris 285 By Means of Any Star at Equal Altitudes 287 Times of Elonflnition and Cul- mination of Polaris 288 Azimuths of Polaris, Table. . 289 Polar Distances and Azi- muths of Polaris, Table. . 290 Engineer's Transit 291 Adjustment and Repairs. . . . 294 Vernier 296 Croas-hairs; to Replace 296 Bubble Glasa; to Replace. . . 296 Theodolite . . ; 296 Pocket Sextant 297 Compaaa. Adjustment 298 Magnetic Declination and Variation. Isogenic Chart of U. S 300 Declination 301 Variation 301 Demagnetization 302 Leveling. Contour Lines 302 Y Level 306 Adjustment 307 Forms for Notes 309 Hand Level, Adjustment . . . 310 Builder's Plumb Level 311 Clinometer or Slope Inst .... 311 Leveling by the Barometer or Boiling Point 312 Table 316 NATrRAI. PHENOMENA. Sound. Volocity of 316 Heat. Expansion and Melting Points. Table 317 Thermometer. Conversion of Scales 318 Tables 318, 319 CJ0NTENT8. XXVll Air. Atmospliere. page Properties 320 Pressure in Diving Bells, etc. . . 321 Dew Point 321 Heat and Cold, Records of ... . 321 Wind. Velocity and Pressure. Table. 321 Bain »nd Snow. Precipitation. Average 322 Effect of Climate on — 322 and Stream-flow 323 Maximum Rates of — 323 Weight of Snow 323 Rain Gau^ 324 Precipitation, Details of — in U.S., Table 325 Water. Composition, Properties 326 Ice 326 Effects of Water on Metals, etc. 327 Tides 328 KTaporatlon, ratration, lieakai^e 329 MECHANICS, FOBCE IN RieiD BOBIES. Definitions 330 Matter; Body 330 Djmaiiiies. Motion, Velocity 331 Force 332 Action and Reactioti 333 Acceleration 334 Mass 336 Impulse 337 Density; Inertia 338 Opposite Forces 339 Work :. 341 Power 842 Kinetic Energy 343 Momentum 345 Potential Energy 346 Impact 347 Gravity, Falling Bodies 34$ Descent on Inouned Planes . . . 349 Pendulums 350 Center of Oscillation 351 Center of Percussion 351 Angular Velocity 351 Moment of Inertia 351 Radius of Gyration 352 OnthfuffBd Force 354 StatlctB. PAoa Forces .• 358 Line of Action 359 Stress 359 Moments 360 Classification of Forces 361 Composition and Resolution of Forces 362 Force Parallelogram 364 Foi-ce Triangle 367 Rectangular Components 369 Inclined Plane 369 Stress Components 371 Applied and Imparted Forces . . 372 Resolution, etc., by means of Co-ordinates 372 Force Polygon 374 Non-coneurrentCopUnarForoes 375 Equilibrium of Moments 376 Cord Polygon 377 Concurrent Non - coplanar Forces 380 Non-concurrent Non-coplanar Forces 381 Parallel Forces 382 Coplanar 382 Non-coplanar 385 Center of Gravitv 386 Stable, Unstable, and Indif- ferent Equilibrium 387 General Rules 387 Special Rules 391 Line of Pressure. Center of Force or of Pressure 399 Position of Resultant 399 Distribution of Pressure .... 400 "Middle Third" 402 Couples 404 Friction 407 Coefficient ' 408 Morin's Laws 410 Table of Coefficients 411 Other Experiments 412 Rolling Friction 414 Lubricated Surfaces 415 Friction Rollers 417 Resistance of Trains 417 Workof Overcoming Friction 418 Natural Slope 419 Friction of Revolving Shaft 419 Levers 419 StabUity 422 Work of Overturning 422 On Inclined Planes 424 The Cord 425 Funicular Machine 427 Toggle Joint 427 PuHey , . . . 428 Loaded Cord or Chain 428 Arches, Dams, etc. Thrust and Resistance Linec .... 430 Arches 430 Graphic Method 430 Practical Considerations. . 432 Masonrv Dam 433 Graphic Method 435 Practical Considerations. . 436 The Rcrew 436 zxviii OONTBKTB. PAOB Forces Acting upon Beams and Trusses 437 Conditions of Equilibrium. . 437 End Reactions 439 Moments 440 In Cantilevers 442 In Beams 443 Inclined Beams 445 Curved Beams 446 Shear 446 Influence Diagrams 449 For Moments 449 For Shear 460 Relation between Moment and Shear 452 STREHGTS OF HATE- 1IIAI.S. Ctoneral Principles. 454 Stretch, Stress and Strain .... 455 Modulus of Elasticity 456 Limit of Elasticity 458 Yield Point 459 Resilience 460 Suddenly Applied Loads 460 Elastic Ratio 461 Strengths of Sections 462 Fatigue of Materials 465 TransTerae Streng^tb Conditions of Equilibrium .... 466 Neutral Axis 466 Resisting Moment 467 Modulus of Rupture 468 Moment M Inertia. 468 Table 469 Section Modulus 473 Loading. Strength 473 Table 474 Beam of Unit Dimensions .... 475 Coefficients, Table 476 Weight of Beam as Load 477 Comparison of Similar Beams. 478 Horizontal Shear 478 Deflections 480 Elastic Limit 482 Elastic Curve •. . 482 Deflection Coeffioi^it 483 Eccentric Loads 484 Uniform Loads • 486 Inclined Beams ....'. 485 Sirlindrical Beams 485 aximum Permissible — . . . . 485 Suddenly Applied Loads . . . 486 Uniform Strength 486 Cantilevers. Table 487 Beams. Table 488 Continuous Beams 489 Table 490 Cross-shaped Beam 492 Plates 492 Transverse and Longitudinal Stresses Combined 493 PAoa Strengrtb of Piilam. 496 Radius of Gyration 496 Table 496 Remarks 40S Slieariiiff Strentrtli . 499 ToMtanal 8ir«iivtli. 490 HTDBOSTATICfiL Principles 601 Center of Pressure 601 Air Pressure 602 Horisontal and Vertical Components 603 Pressure in Vessels 503 Opposite Pressures 503 Rules 604 Transmission of Pressure 606 Center of Pressure 609 Walls to Resist Pressure 608 Thickness at Base 609 Stability 510 Contents 510 Liability to Crush 51Q Thickness for Cylinders 511 Iron Pipes 512 Lead Pipes 513 Buovancy 513 dotation. Metaeenter 614 Draught of Vessels 515 HTDRAUI«ICS. Flow Of W«ter tbrouffb Pipes 610 Head of Water 616 Velocity Head 616 Entry Head 616 Friction Head 616 Pressure Head 618 Piezometers 618 Hydraulic Grade Line 519 Siphon 620 Velocity Formulae 622 Kutter's Formukk 523 Weight of Water in Pipes 526 Areas and Contents of Pipes . . . 526 Total Head Required 627 Table of Velocity and Friction Heads and Discharge 628 Compound Pipe 631 Venturi Meter. Theory 632 Tube 634 Register 536 Ferris-Pitot Meter 53ft Curves and Bends 637 OONnsStB, PAOK Flow thronff li Ortflees Tbeoretical Velocities £39 With Short Tubes 640 Through Thin Partition 641 Discharge from One Reservoir to Another 643 Rectangular Openings 644 Time of Emptying Pond. . . . 646 Miner's Inch 646 Flow OTor Wolrs End Contractions 647 Measiu«ment of Head 648 Formulae , 649 Francis 660 Table of Discharges 561 Basin 662 Values of m 663 Submerged Weirs 664 Velocity of Approach 666 Iztelined Weirs 668 Broad-crested Orerf all 669 Triangular Notch 669 Trapezoidal Notch 669 Flow In Open Channels ligations of Velocities 660 Steam Gauging 660 Pitot Tube, etc 661 Wheel Meter 662 Abrasion of Channel 663 Theory of Flow 663 Kutter's Formula 664 Coefficient of Roughness 664 Coeffs of Roughness. Table 666 Coefficient, e. Table 666 To Draw Kutter Diagram. 670 Flow in Sewers 674 Flow to Sewers 676 Flow in Drain-pipes 676 Constriction of Channel 676 Scour 677 Obstruction's in Streams 677 Power of Falling Water 678 Water Wheels. 678 Hydraulic Ram 678 Power of Running Stream .... 678 COVSTBVCnONS, ETC. "Dredging* Cost of Dredging 680 Horse Dredges 681 Weight of Material 681 Foundations. Foundations 682 Borings in Common Soils 682 Unreliable Soils 683 Resistanoe of Soils. . , 688 PAOB Rip-rap 583 Protection from Scour 683 Timber Cribs 684 Caissons 685 Coffer-dams 686 Earth Banks 686 Crib Coffer-dams 687 Mooring Caissons or Cribs 689 Sinking through Soft Soil 689 PUes 689 Sheet Piles 690 Grillage 690 Pile Drivers 690 Resistance of Piles 592 Penetrability of Soils 693 Driving 693 Screw Piles 694 Drivin/s by Water Jet 695 Hollow Iron Cylinders 696 Pneumatic Process 696 Timber Caisson 598 Masonry Cylinders 699 Fascines 699 Sand-Piles 699 Stonework. Cost, etc 600 Retaining Walls. General Remarks 603 Theory 606 Surcharged Walls 609 Wharf Wails 611 Transformation of profile 611 Sliding, etc 612 Stone Bridg^es. Definitions 613 Depth of Keystone 613 Pressures on Arch-stones 614 Table of Arches 615 Abutments 617 Abutment Pi^s 619 Inclination of Courses 620 Culverts 622 Wing Walls s, 624 Foundations 627 Drains 627 Drainage of Roadway 62S Contents of Piers 62$ Brick Arches 62P Centers 631 Timber Bams. Primary Requisites 642 Examples 642 Abutments. Sluices, Ground Plan, Cost 645 Measuring Weirs 64i Trembling 648 Thickness of Planking Re- quired 648 CONTENTS. WATER SUPPI<T. PAGE Consumption, Use and Waste. 649 Waste Restriction ; Water Meters 649 Water for Fire Protection . . . 650 Reservoirs 650 Leakage through — , Mud in— 651 Storage Reservoirs 652 Valve Towers, etc 652 Comj^ensation 653 Distributing Reservoirs .... 653 Water Pipes 653 Concretions in — , preven- tion of — 655 Weights of Cast Iron Pipes . . 666 Wrought Iron Pipes 656 Wooden and Other Pipes . . . 657 Costs of Pipes and Laying . . 658 Pipe Joints 660 Pipe Jointer 660 Flexible Joints.. 661 Special Castings 661 Repairs and Connections. . . 662 Air Valves 662 Air Vessels, Stand-pipes 663 Service Pipes 664 Tapping Machines 664 Anti-bursting Device 665 Valves, Gates 666 Fire Hydrants 668 TEST AND WEI^Ii BORING. Test Boring Tools 670 Artesian Well Drilling 671 ROCK DRII4I1S. Diamond . Drills 675 Percussion DrjUs 676 Hand Drills 681 Channeling 681 Air Compressors 681 TRACTION, ANIMAIi POWER. On Roads, Canals, etc 683 TRUSSES. Introdnetion. General Principles 689 Loading, Counterbraoing 690 Cross bracing 691 Types of Trusses 691 Camber 696 Cantilevers 696 Movable Bridges 696 Skew Bridges 697 Koof Trusses 698 Stresses in Trnss Mem- bers €(eneral Principles 698 Method by Sections 700 Chord Stresses, Moments, Chord Increments 701 FACIB Shear 702 Influence Diagram 702 Dead Load Stresses 703 Live Load Stresses 705 Typical Wheel Loads 706 Cooper's 706 Live Load Web Stresses 706 Live Load Chord Stresses. . . 709 Wind Loads 710 Impact, etc 711 Maximum and Minimum Stresses 712 Effect of Curves 712 Counterbracing 713 Stresses in Roof Trusses 713 Weights and Loads 713 Wind Pressures 714 Graphic Method 715 Timber Roof Trusses 716 Deflections 718 Redundant Members 720 Brtdg^e I>etalls and Con- struction General Principles 720 Floor System and Bearings. . 720 Design 721 Flexible and Rigid Tension Members 721 Compression Members 721 Pin and Riveted Connec- tions 721 Floor Beam Connections 721 Tension Members, Detail . . . 722 Compression Members, De- tail 722 End Post and Portal Bracing 723 Joints 724 Pin Plates 724 Pins 725 Expansion Bearings 725 Loads, Clearance, etc., for Highway Bridges 726 Camber 726 Examples 726 Weights of Steel Railroad Bridges 731 List of Large Bridges 732 Timber Trusses 732 Joints 733 Howe Truss Bridges 736 Examples 738 Metal Roof Trusses 740 Broad Street Station, Phila. . 740 List of Large Arched Roofs. 742 Timber Roof Trusses 742 Transportation and Erection . . 743 Digests of Speelfleations for Brldgres and Buildings. For Steel Railroad and Highway Bridges. General Design 745 Material 751 i Loads 755 C0NTEKT8. PAOB Btreeses and Dimensioos 759 Protection 763 Erection 763 For Combination Railroad Bridyes. General Design 763 Material 763 Loads 764 Stresses and Dimensions 764 Protection 764 For Roofli, Bulldlngns* etc. General Design, Material, etc.. 764 Sl^SPENSIOM BRIDOIS. Data Required 765 - Formulas 766 Anchorages 770 RITETS AND RITETINe. Rules and Tables 772 RAIIiROADS. Carves. Definitions 780 Tables, etc 784 EartliworlK. Table of Level Cuttings 790 Shrinka^ of Embankment .... 799 Cost of Earthwork 800 Tunnels. Coostruction 812 Trestles. Construction 813 Track. Ballast 815 Ties 816 Tie Plates 816 Rails 817 Spikes 818 Rail Joints 819 Turnouts 824 Eqnlpment. Turntables I 845 Water Stations 851 Track Tanks 853 Track Scales, Fences, etc 854 Cost of Mile of Track 855 Rolling Stoe J? XXXI PASS Locomotives. Dimensions. Weights, etc. . . 856 Performance 860 Tonnage Rating 862 Fast Runs 863 Running Expenses 864 Cars 865 Statistics. Earnings, Expenses, etc. 867 MATERIAUS). Metals. Iron and Steel. 'Requirements. International Ass'n for Testing Materials. 870 Cast Iron 874 Weight 875 Weight of Cast Iron Pipes. . 876 Weight of Wrought Iron and Steel 877 Roofing Iron 880 Corrugated Iron 881 Wrought Iron Pipes and Fit- tings 882 Screw Threads, Bolts, Nuts and Washers 883 Lock-nut Washers 885 Buckle Plates 885 Bolts. Weight and Strength, Table 886 Wire Gauges 887 Circular Measure 889 Wire, Table , 891 Structural Shapes. I Beams 892 Channels 894 Angles and T Shapes 896 Separators for I Beams 900 Z-Bar Columns 901 Phcenix Segment Columns . . 904 Gray Column 906 Strengths of Iron Pillars, Tables 907 Floor Sections 914 Chains 915 0kber Metals. Tin and Zinc 916 Copper,' Lead, etc 918 Tensile Strengths, Table 920 Compressive Strengths, Table. 921 Stone, etc. Tensile Strengths. Table 922 Compressive Strengths, Table 923 Transverse Strength, ^^able. . . 924 XXXll CONTENTS. lIortov,Briclu»efe. page Lime Mortar 926 Bricks 927 Cement 930 Cement Mortar 931 Sand 935 Effects on Metab 936 Efflorescence 936 Silica Cement 937 Recommendations, Am. Soc. C. E 937 Tests 938 Report of Board of U. S A. Engineer Officers 940 ' Tests 941 Requirements 942 Concrete 943 Properties 943 Handling 946 Explosives. Nitro^ycerine and Dynamite. 948 Blasting Powders 951 Firing 962 Gunpowder 963 Timber. Decay and Preservation 954 Tensile Strength 957 Compressive Strength 958 Transverse Strength 959 Strength as Pillars 963 B«lldlii|r Materials and Op^V^^OXS. PAOS Plastering 966 Slating 969 Shingtes 971 Painting 971 Glass and Glasing 973 Sundry Materials. Rope 976 Wire Ropes 976 Paper 978 Blue Prints, etc 979 Price lilst and Business Bt- rectoiry. Prieelist 984 Business Directory 996 Biblioffrapiiy. List of Engineering Books 1008 GLOSSARY 1026 INDEX 10» KATHEMATIGS. MATHEMATICAI. STMBOIA. •f Pins, positive, add. 1.414+ means 1.414 -f other decimala. — Minas, nejg^ative, subtract. ± Plus or minus, positive or negative. Thus, y^a* — ±a. 7 Minus or plus. X Multiplied by, times. Thus, x'Xy = x:.y=x7;3X4 = 12, : vDivided by. Thus, a -4- b = a : b = a/b = -r-- y) ^ : : : Proportion. Thus, a : b : : c : <2, as a is to 6, so is to <<. -= Equals, is equal to. > Is ffreater than. Thus, 6 > 5. < Is less than. Thus, 5 < 6. '^ Is not equal to. :^ Is greater or less than. j^ Is not greater than. ^ Is not less than. ;^ Is equal to or greater than. ^ Is equal to or less than. oc la proportional to, varies with. 00 Innnity. J. Is perpendicular to. ^ \ Angla 'v Is similar ta I la parallel to. V l^~Root of. Thus, "i/oor r/o^ square root of o, i/ o =* 8d or cube root of a, ** J a s— nth root of a. Parenthesis. 11 Braclcets. I Quantities enclosed or covered by the symbol are to be I taken tpgether. -Vinculum. J *.* Since, because. .*. Hence, therefore. o Degrees. ' Minutes of arc,* feet. " Seconds of arc,* inches. * / ff /// gtc^ Prime, second, third, etc Distinguishing accents. Thus, a', a prime ; of', a second, etc. Circumference „-..,„„««.. r • • 1 <«»«« n — y- 7 = 8. 14159265 +, arc of semicircle, or 180°. Diameter ' E, Modulus of elasticity. e c, Base of Napierian, natural or hyperbolic logarithms = 2.718281828. g, Acceleration of gravity = approximately 32.2 feet per second per second » approximately 9.81 meters per second per second. * Minutes and seconds of time, formerly also denoted by ' and '', are now de- noted by m aud «, or by min and sec, respectively. 3 33 34 OBEEK ALPHABET. THE eREEK AI.PHABET. This alphabet is inserted for the benefit of those who have occasion to consult scientific works in which Greek letters are used, and who find it inconvenient to memorize the letters. Greek letters. Name. Approximate equivalent. Commonly used to designate Capital. Small. * A a Alpha a Angles, Coefficients. B ^ Beta b it u r y Gamma g " " Specific gravity. A i Delta d « " Density, Variation. /Base ot hyperbolic logarithms » s « Epsilon e (short) -j 2.7182818. V Eccentricity in conic sections. z < Zeta * Co-o'rdinates, Coefficients. H n Eta e (long) ii (I e 9& Theta th Angles. I I loU i K iC Kappa k A A Lambda 1 Angles, Coefficients, Latitude. M Jtt Mu m tt t< N V Nn B t( B f Xi X Co-ordinates. O o Omicron (short) n w Pi P Circumference -i- radios.* p p Bho r Badius, Batio. 2 o-« Sigma • Distance (space).t T T Tau t Temperature, Time. Y V Upsilon u or y « * Phi ph Angles, Coefficients. X X Chi ch ♦ ^ Psi P8 Angles. o w Omega o (lon«) Angular velocities. * The small letter fr (pt) is universally employed to designate the number of times (= 3.14159265 . . .) the diameter of a circle is oootained in the circum- ference, or the radius in the semi-circumference. In the circular measure of angles, an angle is designated by the number of times the radius of any circle is <k>ntained iu an arc of the same circle subtending that angle. ir then stands for an angle of 180° (= two right anglesX because, in any circle, ir X radius = the semi-clrcumferenoe. The capital letter n (;>i) is used by some mathematical writers to indicate the product obtained by multiplying together the numbers 1, 2, 3, 4, 5 . . . etc., up to any given point. Thus, n 4 = 1 X2 X 3X4 = 24. t The capital letter 2 (sigma) is used to designate a mm. Thus, in a system of pandlel forces, if we calf each of the forces (irrespective of their amounts) F, then their resultant, which is equal to the (algebraic) sum of the forces, may he written B = 2 F. AssTButata. ' 35 ABITHMETIO. FACTORS AND MVI4TIPI1ES. (1) Factors of any number, n, are numbers whose product is = n. Thus, 17 and 4 are factors of 68 ; so also are 34 and 2 ; also 17, 2, and 2. <3) A prime number, or prime, is a number which has no factors, except itself and 1 ; as 2, 3, 5, 19, 2S&. (8) A common HicAor, common diwiflor or common meaanre, of two or more numbers, is a number which exactly divides each of them. Thus, 8 is a common dirisor of 6, 12, and 18. (4) Tlie hiipiieBt common fiictor or nreatest common diwiaor, of two or more numbers, is called their H. C. F. or their O. C I>. Thus, 6 is the H. C. F. of 6, 12, and 18. (5) To find (lie H. €• F. of two or more numbers ; find the prime factors of each, and multiply together those factors which are common to all, taking ••di factor only once. Thus, required the H. C. F. of 78, 126, and 284 78 = 2 X 8 X 13 126 = 2X3X3X7 284 = 2X3X3X13 and H. G. F. * 2 X 8 — 6. (6) To find tlie H. C. F* of two large numbers ; divide the greater by the less ; then the less by the remainder, A : A by the second remainder, B ; B by the third remainder, G ; and so on until there is no remainder. The last divisor Is the H. G. F. Thua, required the H. a F. of 575 and 782. 675)782(1 575 A 207)575(2 414 B 161)207(1 161 G 46)161(8 188 D 28)46(2 H. G. F. =- D » 2& 46 (7) A comnMMi maltiple of two or more numbers is a number which is exactly divisible by eaoh of tn^m. (8) Tbe least common maltiple of two or more numbers is called iheir li. €. M. (9) To find the !<• C. M. of two or more numbers ; find the prime factors of each. Multiply the factors together, taking each as many times as it is con- tained in that number in which it is oftenest repeated. Thus, required the L. G. M. of 7, 80, and 48. 7 = 7 30 = 2X3X5 48 = 2 X2X 2X2X3 L. C. M. = 7X2X2X2X2X8X5 = 1680. (10) To find the !<• C M. of two large numbers; find the H. C. F., as above ; and, by means of it, find the other factors. Then find the product of the fKtors, as before. Thus, required the L. G. M. of 575 and 782. As above, H. G. F. =23; ^ = 25; and^ = 34. Hence, 575 = 23 X 25 782 = 23 X 34 and L. G. M. = 28 X 25 X 84 = 19,660. FRACnOBTS. CI) A conuBfMi denominator of two or more fhictions is a common moltiple of their denominators. (2) The least common denominator, or !<• €• D«, of two or more firactions is the L. G. M. of their denominators. 36 ARITHMETia (8) To rednce to a oommoii denominfttor. Let N °. the new numerator of any fraction n = its old numerator d a its old denominator C » the common denominator Then _, C Thus, ^t -f-> j-* C *" L* C* ^' o^ denominators «« 24. S ^ ^^ 4 8X6 18. 5 5X4 20. 7 ^ 7X8 ^ 21 4~^,,24"4X6''24* 6"'6X4''a4' 8 8X8~24* 4X-4 If none of the denominators have a common factor, then C^the product of all th« denominators, -: = the product, P, of all the otAer denominators, and N » P n. Thu8,|,l^,f c = 84 2 _ 2X4X7 se. 1 _ 1X3X7 _ 21. 5 _ gX3X4 eo t ~ 84 T¥' T ~ 84 ^^' 7" 84 TT* (4) Addition and Subtraction. If necessary, reduce the fractions to a common denominator, the lower the better. Add or subtract the numerators. Thus, 1 4.1 _2_i.8 4.1_4_i.8 4.5_27 .20_47_i 11. 3_l7_64.7_13_,6 f_l— 2_1.8_5_2_7_20_jr.7_3_7 6_1 (5) Multiplication. Multiply together the numerators, also the denomi- nators, cancelling where possible. Thus, lvl_l. 8vl S_« 3 V 5 v^ 2 _ 6 . 84 X i| = ^ X I = Jjft^ = 5|f ; I X f = |; |of|of|of^ = f X^Xf X| = |. (6) IMvision. Invert the divisor and multiply. Thus, l^l=,lv2_2_,. 3^1_8v4_8„-. i;^7_Bv8_40_e5 o-7--g- — oXir — 7V- — 5-S-. (7) A fraction is said to be in its lowest terms, or to be simplified* when its numerator and denominator have no common factor. Thus, 1^ simplified = |-. (8) To reduce to low<$st terms. Divide numerator and denominatox 34 by their H. C. F. Thus, required the lowest terms of ^. H. C. F. Of 34 and 85=^17; and ?^ « ?i:ti? = ?. 85 85 + 17 S ARITHMETIO. 87 (9) Mnltlplleatlon. The prodnct has as many decimal places as th« factors combined. Thus, . Factors: 100X3X3.5X0.004X465.21 = 1953.882000 Number of decimal places: + 0+1+ 8+ 2= 6 (10) DiTisloii. The number of decimal places in the quotient = those in the dividend minus those in the divisor. Thus, 5.125 ,„_ 5 5.00 i„^.3 3.00 ■^. 0.42 _ 0.4200 _ ^^ = 1.25; -= — =1.25; 4 = "X = ^'^^ 00021 "" 0:0021 ^ ^' When the divisor is a fraction or a mixed number, we may multiply both divisor and dividend by the least power of 10 which will make the divisor a whole number. Thus, 2.679454 26,794.54 .^ ,_ 0.0062 62 (11) To rednee a common fraction to decimal form ; dividt the numerator by the denominator. Thus, ^ = 0.8 ; 1-|- = -|. =» 1.6. Table 1. Decimal eqniTalents Of common fractions. 8thB 16tha SMi 64t]u , 8ths lethg 82dB 64tlis 1 :015625 S3 .515625 1 2 3 .03125 .046875 17 34 . 35 .53125 .546875 1 2 *4 5 .0625 .078125 9 18 36 ,37 .5625 .578125 8 6 7 .09375 .109375 19 38 39 .59375 .609875 1 2 4 8 9 .126 .140625 5 10 20 40 41 .625 .640625 5 10 11 .15625 .171876 21 42 48 .65625 .671875 ' 8 6 12 13 .1875 .203125 11 22 44 45 .6875 .708125 7 14 15 .21875 .234375 23 46 47 .71875 .734375 2 4 8 16 17 .25 .265625 6 12 24 48 49 .75 .765625 9 18 19 .28125 .296875 25 50 51 .78126 .796875 5 10. 20 21 .3125 .328126 • 13 26 52 63 .8126- .828125 11 22 23 .34375 .359375 27 54 55 .84376 .859375 8 6 12 24 25 .375 .390625 7 14 28 56 57 .875 .890625 13 26 27 .40625 .421875 29 58 59 .90625 .921875 7 14 28 29 .4375 .453125 15 30 60 61 .9375 .958125 15 80 31 .46875 .484375 31 62 63 .96875 .984375 4 8 16 82 .5 8 16 82 64 1. (12) To reduce a decimal fraction to common form. Supply the denominator (1), and reduce the resulting fraction to its lowest terms. Thus : 0.25 0.25 1.00 25 100 1 4' = . ; 0.75 = To 100 3 4' ^ : 0.800626 = 890626 1000000 57 64* 38 ABITHMETIO, (IS) Becnriinff, etrealattny, or repeattny decimals are those in which certain digits, or series of digits, recur indefinitely. Thus, ^ =» 0.8338...., and so on ; ^^ ^ 1.428571428571 and so on. Becurring decimals may be in« dicated thus : 0.3, 1.428571 ; or thus : 0.*3, l.*428571. RATIO AND PlU^PORTIOir* (1) Batio. The ratio of two quantities, as A and B, is expressed by their qaotient, ^ or •-. Thus, the ratio of 10 to 5 is =» - =a 2 : the ratio of 5 to 10 A* (2) Dapllcate ratio is the ratio of the tquares of numbers. Thus, ^-s is the duplicate ratio of A and B. (S) Proportion is equality of ratios. Thus, ^ = -^. = ^A*? = 2. I9 the figure, which represents s^ments, A, B, C, and D, between parulel lines ; A : B : : C : D, or 5 = ^. (4) The first and fourth terms, A and D, are called the extremes, and the second and third, B and C, are called the means. The first term, A or C, of each ratio, is called the antecedent, and the second term, B or D, is called the consequent. D is called the fonrtli proportional of A, B, and C. (5) In a proportion, A : B = C : D, we have : Product of extremes = product of means. A D >= ..... A C A B Alternation. 3 = 5; c " D* _ , B D B A D C Inversion. ^ = ^; ^ - ^; 5 = ;^. ^ ... A + B C +D . A-f B Composition. — - — = — ^ — ; — g— -., ,, A-B C-D A — B Diyision. — ^ = -^ ; -^^ A 4- B 'Composition and division. =, = _ ... We have, also : mA ^ A ^ C ^ n^ ^ nC, mA^mC^ ^^^, */a ^ ^y/g mB B D nB nl)' nB ~nD' b*~D"' **|/B "" *i/D (6) If, in the proportion, A : B = C : D, we have B = C = m, then A : m « TO : D, or — = - or m * ■" A D, or m = 1/ A D. ml) (7) In such cases, m is called the mean proportional between A and D, mnd D is called the tbird proportional of A and m. A «M»ntinned proportion is a series of equal ratios, as A:B = C:D = E:F, etc. = R; or ^ = ~ = y, etc «- E In continued proportion, A + C -f E + etc. _AC_E _ B 4- D + F + etc. "^ B ~ D ~ F ^^^' ~ '^ „ A _ C A' C' A" _ C^' A A^ A» _ C C^ €<> B "■ D' b' ~ D'*' i3'' ~' iy' B B' B» - DDT)"®^ (8) Let A, B, and C be any three numbers. Then A_AB AAC C ' B • C' *°** B " C • B" ■"^ ♦ 0.*8, l.*428571, etc., sUnding for 0.3333...., 1.428571428671...., etc. ABITHMETIC. 39 (0) Reciprocal or inverae proportion. Two quantities are said to be redproeally or inversely proportional, when the ratio ^ of two values, A B' and B, of the one, is => the reciprocal, -j-,^ of the ratio of the two corresponding values of the other. Thus, let A = a velocity of 2 miles per hour, and B == 3 miles per hour. Then the hours required per mile are respectively. A' = — = i» andB' = | = -J-. HereA: B = B' : A', or | = ?^„ or | = | = i = l-s-^'. (10) If two variable numbers, A and B, are reciprocally proportional, so that A' : B' = B" : A", the product, A' A", of any two values of one of the numbers is equal to the product, B' B'' pf the two corresponding values of the other. (11) The application of proportion to practical problems is sometimes called the rale 01 three. Thus : sing^le rule of tbree : If 3 men lay 10,000 bricks in a certain time, how many could 6 men lay in the same time? As 3 men are to 6 men, so are 10,000 bricks to 20,000 bricks; or, 10,000 bricks X -g- = 20,000 bricks. If 3 men require 10 hours to lay a certain number of bricks, how many hours would 6 men require to lay the same number? As 6 men are to 3 men, so are 10 hours to 5 hours ; or, 10 hours X -|- = 5 hours. (12) Double rule of tbree. If 3 men can lay 4,000 bricks in 2 days, how many men can lay 12,000 bricks in 3 days? Here 4,000 bricks require 3 men 2 days, or 6 man-days, and 12,000 12 000 bricks will require 6 X XaSa = 6 X 3 = 18 man-days ; and, as the work is to be done in 3 days, -^ = 6 men will be required. PROGRESSION. (1) Aritbmetteal Prog^ression. A series of numbers is said to be in arithmetical progression when each number differs from the preceding one by the same amount. Thus, —2. —1, 0, 1, 2, 8, 4, etc., where diff'erence = 1 ; or 4, 3, 2, 1, 0, —1, —2, etc.. where diflTerence == —1 : or —4, —2, 0, 2, 4, 6, 8, 10, where dlffiapence = 2 ; or % 1%, 1, %, %, %, 0, -% —3^, etc., where diffference = —^ (2) In any such series the numbers are called terms. Let a be the first term, I the last term, d the common differdnce, n the number of terms, and s the sum of the terms. Then i = a + (n — 1) d Required I Given a d n I ads s a d n ? = — l.rf±|/2d* + (a — ^cf)S , = 1. n [2 a + (n — 1) d] dls o=»-|-(f± l/(/-|-^d)8 — 2d* d — 2 a ± ^(2 a — d)8 -I- 8 d * n ads n =a 2d n dls 21 + d ± >/(2/ + d)2— 8dj 2d (S) ISeometrieal Progression. A series of numbers is said to be in geometrical progression when each number stands to the preceding one in the same ratio. Thus: •^, -J-, 1, 8, 9, 27, 81, etc., where ratio => 8; or 48, 24, 12, 6, J, 1^, 4, f, etc., where ratio =- -J-; or ^, 1-J-^, 3|, 6|, 13^, 27, etc., where iatio = 2. 40 AKITHMETIO. (4) Let a be the flnt term, I the last term, r the constant ratio, n the numbet of terms, and 4 the sum of the terms. Then : Bequired I Given a r n I art 1 r H * ^^ g + (r- 1)* r ^ r" — 1 a n Z «=> r n I * = r«-.r*~* «»-i PiaKMVTATIOH, Ete. (1) Permatation shows in how many positions any namber of things oatt be arranged in a row. To do this, multiply together all the numbers used in. counting the things. Thus, in how many positions in a row can 9 things be placed? Here, 1X2X3X4X6X6X7X8X9 = 362880 positions. Ans. (2) Combinatton shows how many combinations of a few things can be made out of a greater number of things. To do this, first set down that number which indicates the greater number of things; and after it a series of numbers, diminishing by 1, until there are in all as many as the number of the few thinga that are to form each combination. Then beginoing under the last one, set down said number of few things \ and going backward, set down another series, also^ diminishing by 1, until arriving under the first of the upper numbers. Multiply together all the upper numbers to form one product; and all the lower ones to form another. Divide the upper product by the lower one. Ex. How many combinations oi 4 figures each, can be made from the 9 figure* 1, 2, 3, 4, 5, 6, 7, 8, 9, or from 9 any things? 9X8X7X6 3024 ,„^ ., ^, . r x 2 X 8 X 4 ^'2r^ combinations. Ans. (3) AlUg^tion shows the value of a mixture of different ingredients, When the quantity and value of each of these last is known. Ex. What is the value of a pound of a mixture of 20 fi>s of sugar worth 15 ots per lb ; with 80 lbs worth 25 cte per fi>? fts. cts. cts. 20 X 15 = 800 _, - 1050 „, , 80 X 25 = 750 Therefore, -^ = 21 cts. Ans. 60 lbs. 1050 cts. PEBCENTAOE, INTEREST, ANNUITIES. Percentagre* (1) Batio is often expressed by means of the word " per." Thus, we speak of a grade of 105.6 feet per mile, i. e., per 5280 feet. When the two numbers in the ratio refer to quantities of the same kind and denomination, the ratio is often expressed as a percentage (perAundredage). Thus, a grade of 105.6 feet per mile,. * Equations involving powers and roots are conveniently solved by means of logarithms. AMTtBUmiC. 41 or per 6280 feet, is equivalent to a grade of 0.02 foot per foot,* or 2 feet per 100 feet, or simply (since botli dimensions are in feet) 2 per 100, <» 2 per " cent.'* (2) One-fiftietli, or 1 per 50, is plainly equal to two hundredths, or 2 per Atm- dred, or 2 per cetU. Similarly, 3^ = 25 per cent, % =,3 X 26 per cent. = 75 per cent., etc Heace, to reduce a ratio to the form of percentage, divide 100 times* the first term by the second. Thus, in a concrete of 1 part cement to 2 of sand and 5 of broken stone, there are 8 parts in all, and we have, by weight— f Cement = X » 0.126 = 12.6 per cent, of the whole. Sand =2. = 0.260= 26.0 " " Stone =|. = 0.626= 62.6 " " Concrete = f = 1000 = 100.0 " " (3) Percentage is of very wide application in money matters, payment for service in such matters being often based upon the amount of money involved. Thus, a purcliasing or selling agent may be paid a brokerage or commission which forms a certain percentage of the money value of the goods bought or sold ; the premium paid for insurance is a percentage upon the value of the goods insured; etc. Interest. (4) Interest is hire or rental paid for the loan of money. The sum loaned is caDea the -prlneiiMftl, and the number of cents paid annually for the loan of each dollar, or of dollars per hundred dollars, is called the rate of interest* The rate is always stated as a percentage. (5^ If the interest is paid to the lender as it accrues, the money is said to be at siniple interest ; but if the interest is periodically added to the princi- pal, so that it also earns interest* the money is said to be at eomponncl Interest, and the interest is said to be compounded. Simple Interest. (6) At the end of a year, the interest on the principal, P, at the rate, r, is » P r, and the Amoant, A, or sum of principal and interest, is A =- P + P r = P (1 + r). (7) At the end of a number, n, of years, the interest is » P rn (see right- hand side of Fig. 1), and A = P + P rn =» P (1 + rn). Thus, let P = $866.32, r = 3 per cent., or 0.03, n=l year, 3 mouths and 10 days =» 1 year and 100 days = 1-J^ Y^axB =» 1.274 years. Then A — P (1 + rn) — S866.82 X (1 + 0.03 X 1.274) » $866.32 X 1.08822 => 8898.39. (8) For the present worth, principal, or eapltallEatlon, P, of the amount, A, we have p 1 + rn Thns, for the sum, P, which, in 1 year, 8 months, 10 days, at 8 per cent. 898 39 simple interest, will amount to S898.39, we have P «- , ^ no v^ i otA = ^^866.32. (9) In commercial business, interest is commonly ealenlatecl approxl* nuktely by taking the year as consisting of 12 months of 30 days each. Then, at 6 per cent., the interest for 2 months, or 60 days, = 1 per cent; 1 month, or 30 days, = Hp^ cent.; 6 days = 0.1 per cent. Thus, required the interest on $1264.35 for 6 months, 28 days, at 6 per cent. *A.Jraetianj as ^^ •^, etc., or its decimal equivalent, as 0.125, 0.3126, etc., is compared with unUy or one; but in percentage the first terra of the ratio is compared with one hwndred units of tue second term. Mistakes often occur through n^lect of this distinction. Thus, 0.06 (six per cent, or six per hundred) is sometimes mis-read six one-hundredths of one per cent, or six oue-hun- dredths per cent, t For proportions by volume, see pp 936 and 943. 42 ARITHMETIC. Principal .tl264.85 Interest, 2 mos, 1 per cent 12.64 2mo8, 1 " 12.64 " Imo, h " 6.82 " 20 days, I " 4.21 " 6 days, 0.1 " 1.26 " 2 days, ^ " 0.42 Interest at 6 per cent $37.^ Deduct one-sixth 6.25 Interest at 5 per cent $31.24 Equation of Paymente. (10) A owes B $1200 ; of which $400 are to be paid in 3 months ; $500 in 4 months; and $300 in 6 months; all bearing interest until paid; but it has been Agreed to pay all at onc& Now, at what time must this payment be made so that neither party shall lose any Interest? $ months. 400 X 3 = 1200 . _.. 6000 ^.. ., . 500 X 4 = 2000 Average time = T^ = ^ months. Ans. 300 X 6 = 1800 1200 5000 Compound Interest. (11) Interest is usually compounded annually, semi-annually, or quarterly. If it is compounded annually, then (see left side of Fig. 1) at the end of 1 year A = P (1 + r) 2 years A = P (1 + r) (1 + r) = P (1 + r)« n years A = P (1 + r)**; and ^=(T:n^n=A(i + r)- p = (l+r)« (12) If the int^est is compounded g times per year, we have (la) The principal, P, is sometimes called the |»i*esent worth or present Talue of the amount, A. Thus, iu the following table, $1.00 is the present worth of $2,191 ^ue iu 20 years at 4 per cent, compound interest, etc, etc <i M (( «( i k / z. 21 y x*- rTv <^ F(l + r)n ^ 5 ^ r J ?r i ». ^ ^ « ^ w^^ <M _r^^ at ^^ >; \ r^^ *— \ ' •8^' ^ 1 I ^L t J 3 o • J > > ' > ' > r < ' i > J \ A ; ii r 4 \ i F < :; 1 t r 1 9 » I Years Figr. 1. ABITHHETIC. 43 Ttible S« CompouiMl Interest. Amount of 81 at Compoand Interest. 8 »H 4 ^ 6 6H 6 «H Yean. per per per per per per per per cent. oent. cent. cent cent oent cent. cent 1 1.030 1.035 1.040 1.045 1.060 1.066 1.060 1.065 2 1.061 1.071 1.082 1.002 1.103 1.118 1.124 1.134 8 1.098 1.109 1.126 1.141 1.168 1.174 1.191 l.i08 4 1.126 1.148 1.170 1.193 1.216 1.239 1.262 l.f86 5 1.159 1.188 1.217 1.246 1.276 L807 1.338 1.870 6 1.194 1.229 1.265 1.302 1.340 1.379 1.419 1.459 7 1.230 1.272 1.316 1.361 1.407 1.455 1.504 1.654 8 1.267 1.817 1.869 1.422 1.477 1.635 1.594 1.655 9 1.805 1.863 1.423 1.486 1.651 1.619 1.689 1.763 10 1.844 1.411 1.480 1.553 1.629 1.708 1.791 1.877 11 1.384 1.460 1.539 1.623 1.710 1.802 1.898 1.999 18 1.426 1.511 1.601 1.696 1.796 1.901 2.012 2.129 18 1.469 1.564 1.665 1.772 1.886 2.006 2.133 2.267 14 1.518 1.619 1.732 1.852 1.980 2.116 2.261 2.415 15 1.558 1.675 1.801 1.935 2.079 2.282 2.397 2.672 16 1.606 1.734 1.878 2.022 2.188 2.355 2.540 2.739 17 1.653 1.795 1.948 2.113 2.292 2.486 2.693 2.917 18 1.702 1.868 2.026 2.208 2.407 2.621 2.854 3.107 19 1.754 1.923 2.107 2.308 2.527 2.766 3.026 3.309 98 1.806 1.990 2.191 2.412 2.653 2.918 3.207 3.524 91 1.860 2.069 •2279 2.520 2.786 8.078 8.400 3.753 92 1.916 2.132 2.370 2.634 2.925 3.248 3.604 3.997 98 1.974 2.206 2.465 2.752 3.072 3.426 8.820 4.256 94 2.033 2.283 2.563 2.876 3.225 3.615 4.049 4.533 95 2.004 2.863 2.666 3.005 3.386 a8i8 4.292 4.828 98 2.157 2.446 2.772 3.141 3.556 4.023 4.549 5.141 97 2.221 2.532 2.883 3.282 3.733 4.244 4.822 5.476 98 2.288 2.620 2.999 3.430 3.920 4.478 6.112 5.832 98 2.857 2.712 3.119 3.584 4.116 4.724 5.418 6.211 80 2.427 2.807 3.243 3.745 4.822 4.984 &743 6.614 81 2.500 2.905 3.373 3.914 4.538 6.268 6.088 7.044 89 2.575 3.007 3.508 4.090 4.765 6.547 6.453 7.502 88 2.652 8.112 3.648 4.274 5.008 5.852 6.841 7.990 84 2.732 8.221 3.794 4.466 5.253 6.174 7.251 8.509 85 2.814 8.834 3.946 4.667 5.516 6.514 7.686 9.062 88 2.898 3.450 4.104 4.877 6.792 6.872 8.147 9.651 87 2.985 3.671 4.268 5.097 6.081 7.250 8.636 10.279 88 8.075 3.696 4.439 5.826 6.385 7.649 9.154 10.947 89 3.167 3.825 4.616 6.566 6.706 8.069 9.704 11.658 40 8.262 3.959 4.801 6.816 7.040 8.613 10.286 12.416 Compoand interest on M dollars, at any rate r for n years =» M X compoand interest on $1 at same rate, r, and for n years. AnBnity, Sinkinir Fand, Amortisatloii, ]>epreeiaftloii. (14) Under "Interest" we deal with cases where a certain sum or "prin- cipal,** P, paid once for all, is allowed to accumulate either simple or compound interest ; but in many cases equal periodical payments or appropriations, called •mnaltiee, are allowed to accumulate, each earning its own interest, usually compoand. 44 ARITHMEnO. (15) Thua, a sum of money is set aside annually to accumulate oompoand interest and thus form a stiikliiil^ ftind, in order to extinguish a debt. In this way, the cost of engineering works is frequently paid virtually in instal- ments. This process is called amortlBatlon. (16) In estimating the operating expenses of engineering works, an allowance is made for depreelatlon. In calculating this allowance, we estimate or assume the life-time, n, of the plant, and find that annuity, p, which, at an assumed rate, r, of compound interest, will, in the time n, amount to the cost of the plant, and thus provide a fund by means of which the plant may be replaced when worn out or superseded. (17) The present wortb, present walae, or capltaliBation, W. Fig. 2, of an annuity, p, for a given number, n, of years, is that sum whidi. if now placed at compound interest at the assumed rate, r. will, at the end of that time, reach the same amount, A, as will be reached by tnat annuity. i > 1 1 z • 1 I (*+'>'■ ^ ^ ^ ^ ^ 1 1 i .^ ^ J V t 1 1 L^ a ,^ f > r > f J y f \ f 1 r \ r < > J [ J \ a ( 4 \ I S i i : r « r I i » % Years Flff.l. O X 2 3 4 s a Year* FlV. 2. 7 S 9 n (18) Equations for Compoand Interest and Annnltles. (See Figs, land 2.) P = principal ; r => rate of interest ; n = number of years ; A =■ amount ; p = annuity ; W = present worth. The interest is supposed to be compounded, and the annuities to be set aside, at the end of each year. Compound Interest. (1) The amount. A, of $1, at the end of n years, see (11), is A => (1 + r)". (2) Since the present worth of (1 + r)\ due in n years, is $1, see (1), it Uows, by proportion, that tlie present worth, W, of $1, due in n yean, fol isW = (1 + r)' = (1 + r) Annuities. (3) In n years, an annuity of $r will amount to (1 + r)** — 1.* Hence, the amount. A, of an annuity of $1, at the end of n years, is *In the case of compound interest on $1, the rate, r, may be regarded as an annuity, earning its interest; and, at the end of n years, the amount of the several annuities (each = the annual interest, r, on the $1 principal) with the interests earned by them, is = the amount, (1 + r)", of $1 in n years at rate, r, minus tiie $1 principal itself; or, amount of annuity = (l -f r)** •— 1. ARITUMETIG. 45 (4) For the present wortli, W, of an nnnnity of $1 for n years, we oave, trom Eqaations (1) and (3) : 1 i— (l + r)*:l = ^^^^:^^^: i-iW. Hence. W = )-f-f^i jr-^^ r (1 + r) r See Table 3. (6) Tlie annuity for n years, which $1 will purchase, is * 1* r P='W^ i — 1 — (6) Tlie annnl^ which, in n years, will amount to $1, is jf = p -T W ft 1 — (l + r)*-l See Table 4. (1 + r) * Table 8. Present Talne of Annuity of $1000. See Equation (4). Bate of Interest (Compound). 2^ 8 8H 4 4^ 6 6Ji 6 Tears. per per per per per per per per • cent. cent. cent • cent. cent. oent. cent. cent. 6 4,646 4,580 4,515 4,452 4,390 4,829 4,268 4,212 10 8,752 8,580 8,816 8,111 7,913 7,722 7,688 7,360 16 12,381 11,938 11,517 11,118 10,740 10,380 10,037 9,712 ao 15,589 14,877 14,212 13,590 18,008 12,462 11,950 11,470 26 18,424 17,413 16,482 15,622 14,828 14,094 13,414 12,783 80 20,930 19,600 18,392 17,292 16,289 15,372 14,534 13,765 S6 23,145 21,487 20,000 18,664 17,461 16,374 15,391 14,498 40 25,103 23,115 21,865 19,793 18,401 17,159 16,045 16,046 46 26,833 24,519 22,495 20,720 19,156 17,774 16,648 15,456 60 28,362 25,730 23,456 21,482 19,762 18,256 16,982 15,762 100 36,614 31,599 27,655 24,505 21,950 19,848 18,096 16,618 (19) In comparing the merits of proposed systems of improvement, it is usual to add, to the operating expenses and to the cost of ordinary repairs and nuUntenance, (1) the interest on the cost, (2) an allowance for depreciation, and sometimes (3) an annuity to form a sinking fund for the extinction of the debt incurred by construction. The cilpitalization of the total annual expense, thus obtained, is then regarded as the true first cost of the construction. Ail the elements of eost are thus reduced to a common basis, and the several propositions become properly comparable. (20) Thus, in estimating, in 1899,^ the cost of improving the water supply of Fliiladelphia, the rate, r, of interest was assumed at 3 per cent, and depreciation was assumed as below. Under "Life" is given the assumed life-time of each class of structure or apparatus, and under *' Annuity " the sum which must be set aside annually in order to replace, at the expiration of that life, $1,000 of the corresponding value. Present worth Annuity * Because, W $1.00 Equation (4) Present worth Annuity Sl.OO : p. Hence, j9 Equation (5) 1 Annuity .Amount Annuity Amount ^ tBecause, r : (1 + r) " — 1 : : p' : $1.00. Hence, p' = .^ , \ n — 7. Equation (8) Equation (6) (1 + r) " — 1 X Report by Rudolph Hering, Samuel M. Gray, and Joseph M. Wilson. 46 ▲BITHMEnO. BTBUCVDBm, Apparatus, etc. Lvb, Ahkoitt in years f Masonry conduits, filter beds, reservoirs ^..Indefinite 0.00 Permanent buildings 100 1.65 Cast iron pipe, railroad side-tracks 80 8.11 Steel pipe, valves, blow-o£b, and gates 85 16.M Engines and pumps 30 21.02 Boilers, electric light plants, tramways and equipment, iron 'fences 20 87.22 Telephone lines, sand-washer, and regulating apparatus.... 10 87.24 (21) Calculated upon this basis, two projects, each designed to fiimish 450 million gallons per day, compared as follows : BiVER Watkb, takkn within City Ldcixb and Filtbbkd. Unfiltbbed Watbb, by Aqubduct. First Out. 8toraffe leservoirs. 930,900,000 Aqueducts 47,730,000 Distribution 8,655,000 Distributing reservoir 1,000,000 Total $88,185,000 Annual. Interest on |68,185,00a 82,485,550 I%rstCbH. Filter plants 828,174,680 Mains ^ » 10,980,000 Depreciation Operation and Maintenance. Analyses and inspec- tion 841,620 Ordinary repairs ^,150 Pumping and wages 140,770 198,640 281,540 Total $84,154,68^ AninuaL Interest on 884,154,680 $1,024,840 Depreciation 206,540 Operaiion and MaMenanee. Pumping 81,216,021 Filtration 525,600 82,925,780 1,741,621 82,971,801 It will be noticed that, although the first cost of the filtration project was much less than half that of the aqueduct project, its large proportion of perishable parts made its <diarge for depreciation somewhat greater, while its cost for oper- ation and maintenance was more than seven times as great, and its total annual charge a little greater. Table 4. Anniilty required to redeem $1000. See Equation (6). Bate of Interest (Compound). 1 2 2^ t «K 4 5 6 Years. per per per per per per per per cent. cenL cent. cent cent. cent. cent. cent. 5 196.04 192.16 190.24 188.36 186.49 184.63 180.98 177.80 10 95.58 91.33 89.25 87.23 85.24 83.29 79.60 75.87 15 62.12 57.83 66.77 53.77 61.82 49.94 46.34 42.90 20 45.42 41.16 89.14 37.22 85.36 33.58 30.24 27.18 85 85.41 31.22 29.27 27.43 25.67 24.01 20.96 18.28 SO 28.75 24.65 22.78 21.02 19.37 17.83 15.05 12.65 S5 24.00 20.00 18.20 16.54 15.00 13.68 11.07 8.97 40 20.46 16.55 14.84 13.26 11.88 10.62 8.28 6.46 45 17.71 13.91 12.27 10.79 9.45 8.26 6.26 4.70 50 15.51 11.82 10.26 8.87 7.63 6.55 4.78 8.44 60 12.24 8.77 7.35 6.18 6.09 420 2.83 1.88 70 9.93 6.67 5.40 434 3.46 2."74 1.70 1.08 80 8.22 5.16 4.03 8.11 2:88 1.81 1.08 0.578 90 6.91 405 8.04 2.26 1.66 1.21 0.627 0.318 100 5.87 3.20 2.31 1.65 1.16 0.808 0.383 0.177 ARITHMETio. 47 I>rODENAI« OB BUOBBNART NOTATION.* (1) In the Arabic system of notation 10 is taken as the base, but in dnodenal notation 12, or " a dozen," is the base. While 10 is divisible only by 0, and (once only) by 2, 12^s divisible twice by 2, and ouce by 8, by 4, and by $. This accounts for tne popularity of the dozen as a basis of enumeration ; of weights, as in the Troy pound of 12 ounces ; of measures, as in the foot of 12 inches ; thoTear of 12 months, and the half day of 12 hours ; and of coinage, as in the British shilling of 12 pence. (S) The dnodenal notation uses the dozen (12), the gross (12^ = 144), and the great gross (12^ == 12 gross =» 1728), as the decimal system uses the ten (10), the hundred (10^ = 100), and the thousand (10^ =» 10 hundred => 1000). Two arbitrary single characters, such as T and E, represent ten and eleven respectively ; the symbol 10 represents a dozen ; 11 represents thirteen, and so on. Thus, the num- erals of the two systems compare as follows : Decimal 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 20 21 22 28 24 25 36 48 60 Duodenal 1 2 3 4 5 6 7 8 9 T E 10 11 12 ... 18 19 1T1E20 21 30 40 50 Decimal 72 84 96 99 100 108 109 110 111 112 113 117 118 119 120 121 122 Duodenal 60 70 80 83 84 90 91 92 93 94 95 99 9T 9E TO Tl T2 Decimal 129 130 131 182 133 138 140 141 142 143 144 288 1728 20736 etc. Dnodenal T9 TT T£ EO El E6 E8 E9 ET EE 100 200 1000 10000 etc. (8) IHiodeclmaUL Areas of rectangular figures, the sides of which are eadbressed in feet and inches, are still sometimes found by a method called *' Duodecimals," in which the products are in square feet, in twelfths of a square toot (each equal to 12 square inches) and in square inches ; but, by means of our table of *' Inches, reduced to decimals of a foot." page 221, the sides may be taken in feet and decimals of a foot, and the multiplication thus more conveniently performed, after which the decimal fraction of a foot in the product may, if oesired, be converted into square inches by multiplying by 144. •See Elements of Mechanics, by the late John W. Nystroa. 48 RECIPBOCALS OP NUMBERS. Table of Reetprocate of STuinbers. 8m p. 9S. No. Reciprocal. No. Reciprocal. No. fleciprocal. No. ReciprocaL 1 1.000000000 56 .017857148 Ill .009009009 166 .006024096 2 0.500000000 57 .017543860 112 .008928571 167 .005988024 3 .333333333 58 .017241379 113 .008849558 168 .005952381 4 .250000000 59 .016949153 114 .008771930 169 .005917160 5 .200000000 60 .016666667 115 .006695652 170 .005882353 6 .166666667 61 .016393443 116 .008620690 171 .005847953 7 .142857143 62 .016129032 117 .008547009 172 .005813953 8 .125000000 63 .015873016 118 .008474576 173 .005780347 9 .111111111 64 .015625000 119 .006403361 174 .005747126 10 .100000000 65 .015384615 120 ..008333333 175 .005714286 11 .090909091 66 .015151515 121 .008264463 176 .005681818 12 .083333333 67 .014925373 122 -.008196721 177 .006649718 18 .076923077 68 .014705882 123 .008130081 178 .005617978 14 .071428571 69 .014492754 124 .008064516 179 .005586592 15 .066666667 70 .014285714 125 .008000000 180 .005565656 16 .062500000 71 .014084507 126 .007936608 181 .005524862 17 .058828529 72 .013888889 127 .007874016 182 .005494505 18 .055555556 73 .013698630 128 .007812500 183 .005464481 19 .052631579 74 .013513514 129 .007751988 184 X)05434788 20 .050000000 75 .013333333 130 .007692308 185 .005405405 a .047619048 76 .013157895 181 .007633588 186 .005876844 •22 .045454545 77 .012987013 132 .007575758 187 .005347594 28 .043478261 78 .012820513 138 .007518797 188 . .005319149 24 .041666667 79 .012658228 134 .007462687 189 .005291005 25 .040000000 80 .012500000 135 .007407407 190 .005263158 26 .038461538 81 .012345679 136 .007352941 191 .005235602 : 27 .037037037 82 .012195122 137 .007299270 192 .005208333 28 .035714286 83 .012048193 138 .007246377 198 .005181847 29 .034482759 84 .011904762 139 .007194245 194 .005154639 30 .033333333 85 .011764706 140 .007142857 195 .005128205 31 .032258065 86 .011627907 141 .007092199 196 .005102041 32 .031250000 87 .011494253 142 .007042254 197 .005076142 33 .030303030 88 .011363636 143 .006993007 198 .005050505 34 .029411765 89 .011235955 144 •UUOU's^.'z 4fB 199 .005025126 35 .028571429 90 .011111111 145 .006896552 200 .005000000 36 .027777778 91 .010989011 146 .006849815 201 .004975124 37 .027027027 92 .010869565 147 .006802721 202 .004950495 38 .026315789 93 .010752688 148 .006756757 203 .004926108 39 .025641026 94 .010638298 149 .006711409 204 .004901961 40 .025000000 95 .010526316 150 .006666667 205 .004878049 41 .024390244 96 .010416667 151 .006622517 206 .004854369 42 .023809524 97 .010309278 152 .006578947. 207 .004830918 43 .023255814 98 .010204082 153 .006535948 208 004807692 44 .022727273 99 .010101010 154 .006493506 209 .004784689 45 .022222222 100 .010000000 155 .006451613 210 .004761905 46 .021739130 101 .009900990 156 .006410256 211 .004739336 47 .021276600 102 .009803922 157 .006369427 212 .004716981 48 .020833333 103 .009708738 158 .006329114 213 .004694836 49 .020408163 104 .009615385 159 .006289308 214 .004672897 60 .020000000 105 .009523810 160 .006250000 215 .004651168 bl .019607843 106 .009433962 161 .006211180 216 .004629680 52 .019230769 107 .009345794 162 .006172840 217 .004608295 £3 .018867925 108 .009259259 163 .006134969 218 .004587156 £4 .01851&'>19 109 .009174312 164 .006097561 219- .004566210 Sb .018181818 110 .009090909 165 .006060606 220 .004545455 BECIPROCALS OF NUMBEBS. 49 Table of BeeiproMOa of Hnmbom.— {Cbn/imied.) See p. 62. Ka BedprooaL No. Reciprocal. Na Beciprooal. No. BeciprocaL 221 .004524887 276 .003623188 831 .008021148 886 .002590674 222 .004504505 277 .008610108 832 .008012048 887 .002588979 228 .0044848a'> 278 .003597122 888 .003008003 888 .002577320 224 .004464286 .004444444 279 .008584229 834 .002994012 889 .002570694 225 280 .008571429 83l> .002965075 890 .002564103 226 .004424779 281 .003558719 836 .002976190 901 .002557545 227 .004405286 282 .008546099 887 .002967859 892 .002551020 228 .0048a5965 283 .003533569 838 .002958580 893 .002544529 229 .004366812 284 .003521127 339 .002949853 394 .002538071 280 .004347826 285 .008508772 340 .002941176 895 .002531646 231 .004329004 286 .003496503 341 .002982551 396 .002525258 232 .004310345 287 .003484321 842 .002923977 897 .002518892 238 .004291845 288 .003472222 343 .002915452 896 .002512568 234 .004278504 289 .008460208 344 .002906977 399 .002506266 235 .004255819 290 .008448276 845 .002898551 400 .002600000 236 .004237288 291 .003436426 846 .002890173 401 .002493766 237 .004219409 292 .003424658 347 .002881844 402 .002487562 238 .004201681 293 .008412969 818 .002873563 408 .002481890 289 .004184100 294 .003401861 349 .002865330 404 .002475248 240 .004166667 295 .003888831 350 .002857143 405 .002469186 241 .004149878 296 .003378378 351 .002849008 406 .002463054 242 .004132231 297 .008367003 352 .002840909 407 .002457002 243 .004115226' 298 .003855705 858 .002832861 408 .002450960 244 .004098861 299 .008344482 354 .002824859 409 .002444988 245 .0040K1638 800 .008338833 355 .002816901 410 .002439024 246 .004065041 301 .003322259 856 .002808989 411 .002438090 247 .004048583 802 .008811258 857 .002801120 412 .002427184 a<8 .004082258 308 .003300830 358 .002798296 418 .002421808 249 .004016064 804 .008289474 3591 .002785515 414 .002415459 250 .004000000 805 .008278689 360 .002777778 415 .00240968t 251 .008984064 306 .003267974 361 .002770088 416 .002408846 252 .003968254 307 .003257829 362 .002762431 417 .002398062 258 .003952569 308 .003246753 363 .002754821 418 .002392344 254 .003987008 809 .003236246 364 .002747253 419 .002386685 255 .003921569 810 .003225806 365 .002739726 420 .002380962 256 .003906250 811 .008215434 866 .002782240 421 .002375297 267 .003891051 812 .003205128 867 .002724796 422 .002369668 258 .003875969 813 .003194888 868 .002717391 428 .002864066 259 .003861004 314 .008184718 869 .002710027 424 .002358491 260 X)08846154 81d .003174603 370 .002702703 425 .002352941 261 .008881418 816 .008164557 371 .002695418 426 .002347418 2G2 .003816794 817 .003154574 872 .002688172 427 .002341920 268 .008802281 818 .003144654 873 .002680965 428 .002336449 264 .003787879 319 .003134796 374 .00267^97 429 .002381002 265 .003778585 320 .003125000 375 .002666667 430 .002825561 266 .008759398 321 .003115266 376 .002659574 431 .002320186 267 .003745318 322 .008105590 377 .002652520 432 .002314^5 268 .003731348 323 .008095975 378 .002645503 433 .002309469 269 .003717472 324 .008086420 379 .002638522 484 .002304147 270 .003703704 325 .008076923 380 .002681579 485 .002298851 271 .003690037 826 .008067485 381 .002624672 436 .0022Sfe578 272 .003676471 327 .008058104 882 .002617801 437 .002288380 273 .003668004 328 .003048780 888 .002610966 438 .002283105 274 .000649685 329 .008039514 384 .002604167 489 .002277904 275 .003636864 830 .008080808 885 .002597408 440 .002272727 50 BEOIPROCALS OF mTHBXltS. TftM« of meetpra9M9 «ff Bfanibei«b--KOMiiMiin£> 996'^9i. Kc Recipi^ooal. N<y. Beeipvocal. No. Reoiprocul. No. Recipf^dal 441 44JJ 443 444 445 .002267574 .002262443 .002237836 .0022622B2 .002247191 496 m 496 499 500 .002016129 .002012072 .002000032 .002004008 .002600000 651 5S2 668 564 565 .001814882 .001811594 .001806818 .001806054 .001801802 606 697 668 609 610 .0016S(n6i .001643tt6 .00164087 .0016«aD86 .001639344 446 447 446 441> 450 .002242152 .002287136 .002282143 .002227171 .002222222 601 602 503 501 505 .001996008 .001992032 .001988072 .001984127 .001980198 556 657 558 559 660 .001798561 .001795332 .001792115 .001768909 .0Q198S714 611 612 618 614 615 .001686661 .001633887 .00169Uei .001628664 .001626016 m 492 458 464 469 .002217295 .002212889 .002207506 .002202643 .002197802 506 507 506 509 510 .001976285 .001972387 .001968504 .001964637 .001960784 561 562 568 564 665 .001782581 .001770859 .001776199 .001773050 .001769912 616 617 618 619 620 .001628877 .001620746 .001618128 .001615909 .00161f990S 46^ 467 4$^ «9 4m .002192982 ' .0021881S4 .002188406 .002178649 .002178918 511 512 618 514 515 .001966047 .001958125 .001949818 .001945629 .001941748 566 5<fr 568 569 570 .001766784 .001769661 .0O176056J .001767469 .00175488^ 621 622 m ess .001610806 .001607717 .00160006 .00160SN4 .001600000 m 4m .002169W7 .002164502 .002159827 .002155172 ' .002150638 516 517 518 519 620 .001937984- .001934236 .0019805021 .001926782 .001928077 571 572 573 574 676 .001761813 .001748252 .001745201 .001742160 .001789130 626 637 628 629 680 .001597444 .001594606 .001692067 .001588(325 .001587602 46$ 467 468 469, 470 .002145923 .002141828 .002136752 .002182i96 .002127660 521 522 528 524 525 .001919886 .0019157091 .001912046 .001908897 .001904762 676 577 578 579 680 .0017961111 .001788102 .001799104 .001727116 .001724138 681 682 688 694 685 .001j5847d6 .00158^8 .001679779 .001577B67 .00157^08 471 472 474 476 .002128142 .002118644 .002114165 .002109705 .002106263 526 527 528 529 530 .001901141 .001897533 .001898939 .001890859 .0018867921 681 582 588 584 885 .001721170 .001718213 .001716266 .001712329 .001709403 636 637' 638 639 640 .001572827 .001569869 .0015^7$98 .0015a«945 .001562500 476 477 478 479 480 .002100840 .002096436 .002092050 .002087683 .002088833 531 532 598 534 585 .001»^39l .001879699 .001876173! .001872659 .001869159 586 587 588 589 590 .00170648.1 .001703678 .001700680 .001697793 .001694913 641 642 643 644 645 .001566062 .001557602 .001558^0 .001592796 .001556808 461 482 408 464 48& .002079002 .002074689 .002070393 .002066116 .002061856 536 537 538 589 540 .001865672 .001862197 .001858736 ,001855288 .001851852 591 592 598 694 595 .001692047 .001689189 .001686841 .001683502 .001680672 646 647 648 649 650 .001547908 .001546695 .001548^0 .001546682 .001538^2 486 487 468 489 490 .002057613 .002053888 .002049180 .002044990 .002040816 541 542 •543 544 545 .001848429 .001846018 .001841621 .001838235 .001834862 596 597 598 599 600 .001677852 .001676042 .001672241 .001669449 .001666667 651 652 653 654 665 .0015360^ .001538742 .001531894 .001529062 .001526718 491 492 493 494 495 .002036660 .002032520 .002028398 .002024291 .002020202 546 547 548 549 550 .001831502 .001828154 .001824818 .001821494 .001818182 601 602 608 604 605 .001663894 ,001661130 .001658875 .001655629 .001662893 666 667 668 669 660 .001524890 .001622070 .001619787 .001517461 .001515162 RECIPROCALS OP NUMBERS. 51 Tfil»l« of kl««tpii#caf8 ^VKataiierik-^aMiMraMl.) Seep. S2. WtJL Beciprotat 668 66S 664 000- 667 668 668 «70 671 672 67B 674 «7fi? €fS ersf «S2 088 «M <I85 685 697 688 689 691 6M 695 696. 697 698 699 700 701 702 7€8 i 706 25§ 707 708 709 710 711 712 713 714 715 .001512869 .601510874 ;*0015e8996 .001506024 i0015aS759 ;OOi5erso2. .001499250 i001497Q06 .00149<?68' .00149e§37 .001490813 .001488095 .001488884: .001498660; .00148M81, .001479990 .001477105 • .001474926 .001472754. .001470088; .00146M29. .001466976. .00146«129 .001461988 .001459654 .00145V726 .0014S6604, .001488488; .00146*979' .001449075 .001447178 .001446087 .6014^19001 .0014409!» .001438849 .091436382 )1484720 .001426534 .0OT424501 .001422475 J001420455 .001418440 :06l4i643i .001414427 .001412429 .001410437 .001408451 .001406470 .001404494 .001402525 .001400560 .001398601 M«. 716 717 718 719 720 721 72» 72» 724 726 726 727 728 729 730 73t 732 738 734 786 796 7S7 788 789 740 741 742 748 744 745 746 747 74B 749 750 761 j2 754 766 756 757 758 759 760 7^ 762 763 764 765 766 767 7^8 769 770 Beciporocal. .001386648 .001394700 .001392758 .00139082]; .001388889 .001386963 .001385042 J0013a8126 .001381213 X)013?9810 .001377410 X)013755l6 X)01373626 X)0137174!| •.0Q1869863 .00186798* -.001366120 .001364250 .001862398 .001860644 i001358696 .001866852 .001865014 i001$58186 • .001351351 .00134952 .0019477C .00184689( .001844086 .00184228^ •iO0l84O488: .001388688 :001886e98 .001835113 .001333333, .001331558 .001329787 .001328021 .001326260 .001324508 .001322751 .001321004 .001819261 .001317523 .001315789 .0013l4d60 .001812386 .001310616 .001308901 .001307190 .001305483 .001303781 .001302083 .001300390 .001298701 No. 771 772 778 774 775 776 777 778 779 780 781 782 788 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 ^ 802 •«03 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 Becipracal. .001297617 .001296337 .001296661 .001291990 J0O129O323 J00128866Q J0O12870O1 J001283847 .001283697 .001282051 .001280410 .001278772 .00127713^ .001276610 .001278686 .001272265 .001270648 .001269036 .001267427 .001265828 .00126422$ .001269626 .00126)031 .001269446 .00126786^ .001266281 .001254705 .00126813$ .001261564 .001260000 .001248439 .001246883 .001146880' \ .001943781' J001242236 .001240695 .001239157 .001237624 .001286094 :00l2$4d68 .001233046 .001231527 .001230012 .001228501 .001226994 .001228990 .(X)1222494 .001221001 .001219512 .001218027 ".001216545 .001215067 .001213592 .001212121 No. 826 827 829 880 891 832 838 834 885 886 887 838 839 640 841 842 843 844 846 846 847 848 849 850 861 862 863 854 855 856 857 889 869 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 B^ciproOftL .001210664 .001200190 .001207729 .0012062^ .001204819 .001208869 .001201928 .001200480 .0011990a .001197605 .001196172 .001194743 .001193817 .001191885 .001190476 .001189061 .001187648 .001186240 .001184884 ,001183432 .001182083 .001180688 .001179245 .001177866 X)01176471 .001176088 .001178709 .001172883 .001170960 .00116S691 .001168224 .001166861 '.O0I168GO1 .001164144 .001162791 .001161440 .001160093 .001158749 .001157407 .001156060 .001154734 .0011584031 ,001152074 .001100748 .001149426 .00114fflOe .001146789 .001145475 .001144165 .001142857 876 .001141558 877 .001140251 878 .001138952 879 .001137656 880 t .001136864 62 BECIPROCALS OF NUMBEB8. Table of Reelproeals of If ambers.— {ObnMniMtf.) See below. No. Reciprocal. No. Reciprocal. No. 941 Reciprocal No. Redproesl 881 .001138074 911 .001097695 .001062699 971 .001029666 882 .001133787 912 .001096491 942 .001061571 972 .001028807 888 .001132nO3 913 .001005290 943 .001060445 973 .001027749 884 .001131222 914 .001094092 944 .001059322 974 .001026694 886 .001129944 915 .001092896 945 .001058201 975 .001025641 886 .001128668 916 .001091703 946 .001057062 976 .001024590 887 .001127396 917 .001090513 947 .001065966 977 .001023541 888 .001126126 918 .001089325 948 .001054852 978 .001022495 889 .001124859 919 .001088139 949 .001063741 979 .001021450 880 .001123596 920 .001086957 950 .001052632 960 .001020408 891 .001122334 921 .001065776 951 .001051526 961 .001019368 892 .001121076 922 .001084599 952 .001060420 982 .001018380 893 .001119821 923 .001068424 953 .001049318 983 .001017294 894 .001118568 924 .001062251 954 .001048218 964 .001016200 895 .001117318 925 .001061081 955 .001047120 966 .001015228 896 .001116071 926 .001079914 956 .001046026 986 .001014199 897 .001114827 927 .001078749 957 .001044932 987 .001018171 898 .001113586 928 .001077586 958 .001043841 988 .001012146 899 .001112347 929 .001076426 959 .001042753 989 .001011122 900 .001111111 930 .001075269 960 .001041667 990 .001010101 901 .001109878 931 .001074114 961 .001040588 991 .001009062 902 .001108647 932 .001072961 962 .001039501 992 .001008065 903 .001107420 933 .001071811 963 .001038422 998 .001007049 904 .001106195 934 .001070664 964 .001087344 994 .001006086 905 .001104972 935 .001069519 965 .001036269 995 .001005026 906 .001103753 936 .001068376 966 .001035197 996 .001004016 W7 .001102536 937 .001067236 967 .001034126 997 .001003009 908 .001101322 988 .001066098 968 .001083058 998 .001002004 909 .001100110 939 .001064963 969 .001031992 999 .001001001 «10 .001098901 940 .001063830 970 .001030928 1000 .001000000 BECIPBOCAIiS. (a) Tbe reeiproeal of a number Is the quantity obtained by divid- ing unity or 1 by that number. In other words, if n be any number, then Recip n = — . Thus, Redp 40 = — =s= 0.025 ; Recip 0.4 = — = 2.5, etc., etc Hence, Recip — = — , because Recip -- = l-»- — =«1X — =* — • Thus, since 1 yard = 36 inches, 1 inch = ^ yard = .027777778 yard, for Recip 36 = .027777778. Again, 1 foot head of water gives a pressure of .4336 lbs. per square inch. Hence a pressure of 1 lb. per square inch corresponds to a head of -^^ feet = 2.306805 feet, for Recip .4335 = 2.306805. (See b, below.) (b) It follows that if any number in the column headed *' No." be taken as the denominator of a common fraction whose numerator is 1, the corresponding reciprocal is the value of that fraction expressed in decimals.* Thus, ^ » .03126. Hence, to reduce a eommon fraction to decimal form, multiply the reciprocal of the denominator by the numerator. Thus, ^ sa .63125, because Recip 82 = .03125, and .03125 X 17 = .53125. (e; Conversely, if the reciprocal of a number n be taken as a number, then the number n itself becomes the reciprocal. In other words, Recip — =» n. Thus, Recip 0.025 = Recip -^ — 40 ; Recip 2.5 = Recip ~ = 0.4, etc., etc. * The numbers 2 and 5, and their powers and products, are the only ones whose reciprocalB can be exactly expressed in decimaUi BEGIPROOALS OF NI7MBEBS. 53 (d) The prodnet of any nmnber by its own redpioeal is equal to unity or 1 ; •r, n X — =r — = 1. * n n (e) Any number, a X Becip of a number, n = o x — = — . Hence, to aToid the labor of dlTiding, we may multiply by the redp- roctU of the divisor. Thus, 200 -+■ 48750 = 200 X Becip 48750 » 200 X.00002051282 (see ll, below)=.004102564 (f ) Any number, a -5- Becip of a number, n — a-i- — = an. fieiice, a -f- Becip a = a -*- — =»aX'7~ = a*. Thus. Eedp2 = 0.6.and-g^ = ^ = 4-2.. (:g) The numbers in the foregoing table extend from 1 to 1000 ; bat the recip- rocals of maltlples of these nmnbers by 10 may be taken from the table by adding one cipher to the left of the reciprocal (after the decimal point) for each cipher added to the number. Thus, Becip 390 = .002564103 ; Becip 3900 = .0002564103; Becip 39000 » .00002564108 ; and the reciprociJs of nambers eontaining' decimals may be taken firom the table by shifting the decimal point in the tabular reciprocal one place to the right for each decimal place in the number. Thus: Becip 227 = .004405286; Becip 22.7 = .04405286; Becip 2.27 » .4405286; Becip .227 = 4.405286; Becip .0227 = 44.05286. (k) The reciprocal of a number of more than three fkgnrea may be taken firom the table approxinvately by interpolation. Thus, to find Becip 236.4: Becip 236 =.004237288 Becip 237 =■ .004219409 Differences: 1, .000017879, 286.4—^ = 0.4. Then, 0.4 X .000017879 = .000007152, and Becip 236 =.004237288 minus .00 0007152 = Becip 236.4 =.004230136 by interpolation. The correct reciprocal is .004280118. (1) The reciprocals of numbers not in the table may be conveniently found bjr means of logarithms. Thus, to find the Becip 236.4 = : Log 1 =0.000000 Subtract Log 236.4 = 2.373647 '7.626353 = liOg 0.00428012 Becip 286.4 — 0.00423012. -, ^ J « , M24 286.4 To iUid Becip .^^^ =-3524", Log 236.4 = 2.378647 Subtract Log 8424 = 8.925518 "5.448129 = Log 0.0280627. fU24 Re«ip-^^ = 0.0280627. ' (J) Position of the decimal point. For the Nos. 10, 100, 1000, etc., the number of the decimal place occupied by the first significant figure in the teciprocal is equal to the number of ciphers in the No. ; but for all other Nos. it is equal to the number of the digrUs in the integral portion of the No. Thus : Becip 148.7 = .0069.., etc. Here the number of digits in the integral portion (143) of the No. is 8, and the first significant figure (6) of the reciprocal occupies tin Hdrd decimal place. BQU^KE AKD CUBE SOOTS. a Itoau awl Coke M*ot* vf BWnber* f) N 1 i It 8qui.be and cube boots. < and «!■»• Bwata t Ktemben fiKMH .1 WgjL 1 i Ml ■i" a S 1 1 1 i i i "1 i 1 i s i i i i J is ii Is IS i| Is IS i ii 9QjUAB3S&l>.CUBS9, AND BO0;r8. 5d TAMBUE of Sqinares. Cnbes, Square Roots, and Cube Boots. of Vumbers f^om 1 to lOOO. BuMAfF OH «Hi( I90LL0VIX* Tabuc. WbtToy^f \he «#eoi of a fifth 4(Bclmal in (he roots voQld lie M tad 1 td the fottrth aad flnel decimal tn the taole, (he addition has been made. 1ft errors. Bqpamf. Cpbe. 6<l. Rt. C. Rt. No. Sqnave. Cube. 6q. Kt. CBt. 1 1 1.0000 1.0000 61 .9721 220981 7.8102 3.9965 4 8 1.4i«3 1.3999 68 9844 238328 7.8740 3.9679 • 2T J.79»l l.(«29 J98 9999 250047 7.9873 8.0000 3.9791 U 64 9.0990 isooi 1.5874 «4 4096 262144 4. 15 1» I.TI90 65 4226 274625 8.0623 4.0207 96 316 ^4495 1.8171 66 4356 287496 8.1240 4.0411 1 S SM 9.M58 1.9129 67 4480 ■ 300789 . 8-1854 4.0615 # 51S 2.8384 2.9890 68 4624 31449S 8.2462 4.9617 (a 7fl» 8.0909 2.0691 69 4761 328909 8.3016 4.1016 10 IM 1600 9.1939 9.1544 70 4899 843090 8.3996 4.1218 B m 1331 9.S166 2.2240 71 5041 357911 8.4261 4.1406 144 * 1738 8.4941 2.2804 72 5184 373248 389017 8.4866 4.1602 19 3I6T 8.6966 2.3513 78 5476 8.5440 4.1798 n ^^ 3944 8.741t 2.4101 74 495224 8.6028 4.1989 ii 0n 8.8^90 2.4692 75 5625 421975 8.6696 4.2179 •» ^ 4096 40000 2.5198 76 5776 498076 466593 8.7178 4.2858 ^9 Svn *^& 1 2.5713 77 5999 8.7760 4.25a 19 32 56^ 2.9907 78 6094 474652 8.8318 : 4.9717 » Ml fMiO 2.9684 79 6341 498080 8.8883 4.2996 '» «D MM iiSsi i 2.7144 80 6400 512009 8.9448 4.8099 n m 1^107 2.7589 81 6561 591441 9. 4.3267 .» 2^ H.mNM ' fi.9920 «2 6794 551868 9.0554 4.8445 9 ^S 4.7^58 2.9489 88 6889 571787 9.1104 ; 4.3631 M . i 5!^ 9.9945 84 7066 592T04 9.1952 4.8796 » 2.9940 85 7395 614196 9.2195 4.8968 J7S7fi siSS- B.9625 8.9900 86 87 7396 7569 686056 658608 9.2736 9.3274 4.4140 4.4810 .19 M| 3H9I 5.'^ ] 8.0866 8.6783 88 7744 681472 9.3808 4.4480 » «g[ B4Mfr 80 7931 704999 9.4840 4.4647 » ^ MWJ ^!4772 8.1072 90 ' 8100 1 729000 9.4868 4.4814 n w bSos 5.5678 8.1414 ' 91 8381 758571 9.5394 4.4879 s> 'iSi 5.6599 • 8.1748 99 8464 778688 9.5017 4.6144 ft iohII piBST 6.7446 < 8.ao75 99 8649 804857 9.6487 , 4.5307 n ySH IN804 5.8610 B.S896' B.8711 94 8886 880584 9.6954 4.5480 •? in» «af5 5.9161 95 9025 857875 9.7466 44690 M 1U0 44556 6.0000 8.3019 96 9316 884796 9.7980 4.5780 n Xjg^ fi^^ 6.08eB 8J822' VJ 0499 912678 0.8488 4.6047 » 144ft oSSui 6.1644 3.8620 «8 9604 941192 9.8895 4.0104 SI isn 6961^ 6.3469 S.8S12 90 0891 970299 . 9.9499 4jKt61 40 1600 64000 6.3946 8.i«a00 100 10000 1000900 10. 4.6416 41 1Q61 tSS B.4081 8.4482 101 10201 1080301 10.0499 4.6570 tt 17M 8.4160 192 10404 1061206 10.0095 4.6728 48 1M» 'TttitfT Jgg 3.5034 169 10609 1002727 10.1480 4.0875 44 ^ . m 3.5903 104 10816 1124864 10.1980 4.7037' a 6!T98a BJ>5«9 105 U025 1157925 10.2470 4.7li77 « 2U6 iSSS 6.7828 3.5830 106 11236 1191016 10.2956 4]7336 4r a)()^ 6.885T S.9068 107 114l9f 1225048 10.3441 4.7475 48 11«SM 6.9989 8.0842 106 11604 1359712 10.3933 4.7023 II ^ 1350W 7.0000 3.6663 109 ]!l881 1295029 10.4408 4.7760 * 7.0^11 8.8840 110 12109 lasiooo 10.4881 4.7914 51 3Q01 1183661 7.1414 8.7084 111 12321 1867631 10.5357 4.8080 b 2T04 140986 7.aHii 8.7S25 112 12644 1404928 . 10..'i880 4.8988 tt aaoir 149B77 7.aB01 3.7B63 118 12789 1442897 10.9801 '4.8B46 & mn 167464 7.3485 3.n98 114 12908 1481544 10.6771 4.84B8 3035 166875 7.4162 3.8030 115 13225 1520875 10.7238 4.8639 M 3136 175616 7.4883 3.8259 116 13456 1560896 10.7703 4.8770 SM9 185198 7.5496 3.8485 117 18689 1601613 10.8187 4.8910 tt 8«64 19511S 7.6158 3.8709 118 13924 1643082 10.8628 4.9040 g S4S1 S600 305979 7.6811 S.88S0 110 14161 1686169 10.9087 4.9187 o %k9m 7.7460 8.0149 120 14400 1728000 10.9545 4.9896 56 SQUARES, CUBES, AND ROOTS. TABUE of Squares, Cabes, Square Boota, aud Cube oi^umbers firom 1 to lOOO— (Continued.) JTo. m m 123 134 i» IM 137 128 139 180 ISl 1S3 1S8 1S4 IM 1»6 18T in IM 140 141 148 148 144 146 14« 14T 148 149 IM ISl US 168 164 166 16« 16T 168 16» 100 181 108 188 164 166 166 167 168 160 170 m 173 178 174 176 176 177 178 170 180 181 183 188 184 186 Sqnmre. 14641 14864 15139 16876 16636 16876 16138 16884 16641 16800 17161 17434 17680 17866 18336 18486 18710 19044 19831 19600 19681 80164 80440 30780 81086 81816 31600 31904 33301 38600 33801 23104 23400 23716 34026 84886 84640 34864 86381 25600 35931 36344 36660 37226 37656 37880 28324 28561 S8900 892^ 29684 39939 30276 80626 80976 81839 81664 82041 83400 32761 33124 33488 33856 84226 Cube. 1771561 1815848 1860867 1906624 1868136 8000876 3048388 3007153 3146680 3197000 3348091 3399968 3853687 3406104 3460376 3616466 3671868 3638073 3886619 8744000 3808331 3034907 3886964 8048636 8113186 8176638 3341793 8307949 8875000 8442961 3511806 8581577 3663364 8733875 8796416 8869898 3944813 4019679 4096000 4173381 4351528 4830747 4410944 4483125 4674396 4657463 4741633 4836800 4913000 6000811 5068448 617ni7 5268034 5359375 5461776 5645233 6639752 5736339 6832000 5029741 6028668 6128487 6229604 6831625 8q. Bt. 1. 1.0464 1.0906 1.1366 1.1808 1.2260 1.2694 1.8187 1.8578 1.4018 1.4466 1.4891 1.5326 1.6758 1.6190 1.6619 1.7047 1.7478 1.7806 1.8833 1.8748 1.9164 1.9588 3. 3.0416 3.0680 3.1344 3.1666 3.3066 3.3474 3.2883 3.8388 3.8688 3.4007 3.4499 3.4900 3.6300 2.5696 2.6006 3.6491 2.6866 2.7279 3.7671 2.8062 3.8452 2.8841 2.9238 2.9616 8. 3.0884 3.0767 8.1148 3.1529 3.1909 8.2288 3.2666 8.3041 3.3417 8.3791 3.4164 3.4586 3.4907 3.5277 3.5647 3.6015 O.Bt. KO. 4.9461 186 4.9687 187 4.9783 188 4.9866 189 6. 190 6.0188 191 6.0365 193 6.0897 198 6.0638 194 6.0658 196 6.0788 196 6.0016 197 6.1045 196 6.1172 199 6.1399 300 6.1436 301 6.1551 303 6.1676 808 6.1801 304 6.1936 306 6.3048 306 6.3171 307 6.3398 306 6.3416 309 6.2636 310 6.3666 311 6.3776 313 6.3886 318 6.3015 814 6.3183 316 6.8351 316 6.8868 317 6.8486 318 6.8601 319 601717 220 6.8882 331 6.8947 333 6.4061 338 6.4176 324 6.4288 336 6.4401 236 6.4614 837 6.4626 338 6.4787 338 6.4848 330 6.4860 381 6.5068 233 6.5178 338 6.52R8 334 6.5397 286 5.5605 386 6.5613 287 6.5721 238 5.5828 280 6.5934 240 6.6041 341 6.6147 242 6.6252 248 5.6357 244 6.6462 246 5.6567 346 5.6671 247 5.6774 248 5.6877 249 5.6880 250 Sqiure. 84596 84869 86344 35721 86100 86481 86864 87249 87686 88026 88416 88809 89304 88601 40000 40401 40804 41300 41616 43036 43486 43848 43364 43681 44100 44621 45869 45796 46226 46666 47069 47524 47961 48400 48841 49384 49729 60176 60625 61076 61629 61984 63441 63800 68861 68824 64289 64756 65225 65696 56169 56644 57121 67600 bdOei 68564 69049 69686 60025 60516 61009 61604 62001 62500 Cube. 6434856 6539308 6644672 6751269 6659000 6867871 7077888 7189067 7301884 7414875 7629586 7646873 7762892 7880599 8000000 8120601 8842408 8366427 8489664 8615136 8741816 8869743 8998913 9129829 9961000 9893931 9628138 9663597 9800844 9938376 0077696 0318318 0360333 0503459 0648000 0793861 0941048 1089567 1339434 1890625 1543176 1697083 1852852 3008969 2167000 2826391 2487168 2649887 2813904 3877875 8144266 3313053 3481272 16651919 8824000 3997521 4172488 4348907 4526784 4706125 4886936 5069223 5252992 5438248 5625000 Bq. Ht. 13.6383 13.6748 13.7113 13.7477 18.7840 18.8308 18.8564 18.8934 13.9284 13.9643 14. 14.0067 14.0713 14.1067 14.1431 14.1774 14.3137 14.3478 14.3898 14.8178 14.8637 14.3876 14.4333 14.4568 14.4914 14.6258 14.5603 14.5046 14.6387 14.6628 14.6069 14.7800 14.7648 14.7866 14.8834 U.8661 14.8887 14.9832 14.9666 16. 15.0333 16.0665 16.0907 16.1327 16.1668 16.1987 15.2316 16.2643 16.2971 16.8297 16.3638 16.3948 16.4272 16.4686 16.4819 16.5243 15.5663 15.5885 16.6306 15.6626 15.6844 15.7162 15.7480 16.7797 15.8114 as*^ 6.7088 6.7186 6.7287 6.738a 6.7480 6.760(> 6.7680 6.7790 6.789a 6.7989 6.8e8» 6.8189 6.838S 6.8808 6.8486 6.8678 6.8676 6.8771 6.8ei» 6.8864 6.9089 6.9156 6.9360 6.9846 6.9489 6.9689 6.S6ST 6.9731 6.9614 6.990T 6. 6.0008 6.0186 6.037T •.0869 6.04i» 6.066« 6.0641 6.0782 6.0633 6.0912 6.1002 6.1091 6.1180 6.1369 6.1368 6.1440 6.1634 6.1633 6.1710 6.1797 6.1885 6.1972 6.3068 6.2145 6.2281 6.2817 6.3408 6.3488 6.367» 6.3668 8.2743 6.868» 6.3912 6.2996 SQUABES, CUBES, AND BOOTS. 67 TABliE of Sqimres, Onbes, Square Boots, and €al»e Roots* of A ambers troBa. 1 to 1000->(OoNTiNin£D.) No. BqxiAra. Cabe. Sq. Bt. C. Bt. No. Square. Cube. Sq. Bt. CBt. S61 68001 15818251 15.8480 6.3060 316 99856 31554496 17.7764 6.8113 3tU 68904 16003006 15.8745 6.3164 817 100489 31R55018 17.8045 6.8185 253 «4UW 16194277 15.9060 6.3847 818 101124 32157432 17.8886 6.8256 S64 64516 16387064 15.9374 &3880 819 101761 32461759 17.8606 6.8328 355 65035 16581875 15.9687 6.3413 830 102400 32768000 17.8886 6.8899 S66 665S6 16777216 16. 6.3486 821 103041 83076161 17.9165 6.8470 957 06048 16974598 16.0313 6.3579 822 103664 33386248 17.9444 6.8541 SS8 66564 17178512 16.0624 6.8661 828 104829 33698267 17.9722 6.8612 »e 67061 17378879 16.0885 6.3748 824 . 104976 34013824 18. 6.8683 160 67600 17576000 16.1245 6.8835 825 105625 34328185 18.0278 6.8763 381 68121 17779561 16.1555 6.3907 826 106276 84645976 18.0555 6.8824 363 6R644 17984728 16.1864 6.3968 827 106929 34965783 18.0831 6.8884 968 68169 18191447 16.2178 6.4070 828 107584 35287562 18.1108 6.8964 964 OTUlfD 18390744 16.2461 6.4151 829 106241 35611380 18.1884 6.9684 966 70325 18608625 16.2788 6.4282 880 108900 35987000 18.1659 6.91M 966 70766 18821096 16.3085 6.4312 881 109561 36264691 18.1934 6.9174 967 712H0 19084163 16.3401 6.4898 832 110224 36594868 18.2209 6.9944 968 71824 19248882 16.3707 6.4478 888 110889 36926087 18.2483 6.9813 960 72361 19465109 16.4012 6.4558 884 111556 37259704 18.2757 6.9382 970 72900 19688000 16.4317 6.4688 885 112225 37695375 18.3080 6.9461 871 73441 19902511 16.4621 6.4718 886 112886 37938056 18.8308 6.9531 973 73964 20129648 16.4824 6.4792 887 113669 88272758 18.3576 6.958» 973 74528 20346417 16.5227 6.4872 888 114244 38614472 18.3848 6»9668 974 76076 20570824 16.5529 6.4851 888 114921 38958219 18.4120 6.9727 375 75625 20796875 16.5881 6.5080 840 115600 89304000 18.4391 6.9796 976 76176 21024576 16.6132 6.5106 841 116881 39651821 18.4662 6.9664 977 76729 21258883 16.6488 6.5187 842 116864 40001686 18.4832 6.9088 378 77384 21484852 16.6788 6.5265 843 . 117649 40858607 18.5203 7. 379 77841 21717688 16.7063 6.5843 844 118886 40707584 18.5472 7.0068 360 78400 21952000 16.7832 6.5421 345 119035 41068635 18.5742 7.0136 981 78861 22188041 16.7681 6.5499 846 119716 41421786 18.6011 7.0206 383 79524 22435768 16.7829 6.5577 847 120409 41781928 18.6279 7.0271 3RS 80689 22665187 16.8226 6.5654 848 121104 4214492 18.6648 7.0838 984 80656 22906804 16.8528 6.5731 849 121801 42508549 18.6815 7.0406 885 81285 28146135 16.8819 6.5806 850 122500 42875000 18.7088 7.0478 3R6 81796 23398666 16.9116 6.9865 851 123801 43248651 18.7350 7.0540 987 82369 23688908 16.9411 6.5962 862 123904 43614206 18.7617 7.0607 «8 82944 23887872 16.9706 6.6089 858 124609 48966977 18.7888 7.0674 968 83521 24187568 17. 6.6115 854 125316 44361864 18.8149 7.0740 »0 saoo 24S8800O 17.0394 6.0191 855 126025 44788R75 18.8414 7.0807 sn 84681 24643171 17.0587 6.6267 856 126736 45118016 18.8680 7.0878 983 85264 24897068 17.0680 6.6348 857 127449 45499298 18.8944 7.0940 386 85848 25158757 17.1172 6.6419 858 128164 45882712 18.9209 7.1006 984 86488 25412184 17.1464 6.6494 859 128881 46268379 18.9473 7.1072 985 87025 25673875 17.1756 6.6569 860 129600 46656000 18.9737 7.1138 986 87616 25884886 17.2047 6.6644 861 130821 4704S881 19. 7.1804 987 88209 36198078 17.2887 6.6719 862 131044 47437828 19.0263 7.1260 986 88804 36468582 17.2627 6.6794 863 131760 47882147 19.0526 7.1386 988 88401 26780699 17.2916 6.6869 864 132496 48228544 19.0788 7.1400 800 90000 27000800 17.3305 6.6948 865 133225 48627125 19.1050 7.1466 801 90601 27276801 17.8484 6.7018 866 . 183956 48027896 19.1811 7.1531 803 91304 27548806 17.3781 6.7092 867 134689 49439863 19.1572 7.1586 806 91809 27818137 17.4068 6.7166 868 135424 49836082 19.1888 7.1661 804 93416 28084M4 17.4856 6.7240 369 136161 50248409 19.2094 7.1726 806 83025 28373635 17.4643 6.7818 870 136900 50658000 19.2354 7.1791 806 9S686 38659816 17.4929 6.7887 871 137641 51064811 19.2614 7.1866 «rr 94348 28884448 17.5214 6.7460 872 188884 51478848 19.2873 7.1920 806 94864 29316112 17.5499 6.7588 878 188129 51895117 19.3132 7.1984 808 85481 apfiowM 17.5784 6.7606 874 189676 53818624 19.3391 7.2048 AO 86100 29791000 17.6068 6.7678 875 140635 52784875 19.3649 7.2112 811 86731 80080881 17.6868 6.ni2 876 141876 58157876 19.8907 7.2177 813 87844 808n828 17.6685 6.7834 877 142129 53583688 19.4165 7.2240 818 87888 80684987 17.6818 6.7887 878 142884 54010152 19.4422 7.2804 814 86686 80860144 17.7300 6.7968 879 143641 54439969 19.4679 7.2368 816 96996 81366676 17.T48S 6.8041 880 144400 5487360(r 19.4986 7J4S2 58 SQUARES, •CUBES^ Ain> BOOTS. TABIiE off SqiiaveSy Cubes, flqvave Boots, oad Cube of srambem Drom 1 to 10O<^^*<OMrenriTXD.) STo. S81 M3 188 884 886 887 •qiuura. itfin 140M 147456 148»5 S»7 806 401 408 408 404 405 40T 408 400 •410 411 418 418 414 4U 43M 41T 419 4It 411 41t 4» 484 435 496 487 488 498 480 ■ai 4S2 488 434 436 486 487 488 480 440 441 448 '448 444 446 140768 150644 161S21 158100 153881 15S664 154440 155386 156035 156816 157600 158404 150301 160000 160601 161604 163400 168316 164036 164886 166640 166464 167881 168100 168031 160744 170660 171886 172335 173066 178889 174734 175661 176400 in341 178064 178038 179776 180686 181476 183880 18S184 184041 184900 185761 186624 187480 188S66 189386 190096 190060 191844 192721 193600 194481 195804 190349 1971S0 196026 Gab«. 8q. Bt. 65306841 65748068 66181887 66623104 67066616 67513«< 67960608 68411073 68868860 69318000 69770471 60236388 60696467 61169804 61639075 63099186 62570778 63044793 63531199 64000000 64481901 64864800 65460087 65880864 66480196 06038410 674101tt 1 67917818 68417929 68891000 09436681 68034638 70444007 7096V0a 71478875 71991386 72511718 73034603 73660660 74088000 74610101 75161440 75686007 76338094 76766096 7780en6 77854488 78403763 78058680 79507000 80003001 80631566 81183T87 81746604 83319876 82881866 83458458 84027673 84604619 85184000 85766131 86350888 869S8807 87528884 \ 881211<t5 19.5192 19.5448 19.5704 19.6950 19.6314 19.6468 19.6738 19.6977 19.7381 19.7484 19.7787 19.7990 19.8343 19.8494 19.8746 19.8887 18.9349 10.9499 19.8760 80. 30.0860 30.0480 30.0740 90.0908 a0.18«6 30.1494 30.1743 30.1990 20.2387 90.3485 90.2731 90.9078 90.3834 90.8470 90.8716 20.8881 30.4806 20.4460 20.4096 90.4088 90.5183 30.5496 20.5670 90.5918 90.6166 20.6396 20.6640 20.6882 20.7138 20.7864 20.7606 20.7846 20.8087 20.8387 20.8607 20.8806 20.9046 20.9884 20.9638 20.9708 21. 21.0838 21.0476 21.0713 21.0050 cut. 7.2495 7.2558 7.2028 7.2086 7.2748 7.2811 7.2874 7.J 7.3 7.8061 7.3134 7.8186 7.8946 7JS10 TJ87B 7.3484 7.3490 7.8660 7.3610 7.8681 7.8743 7.1 7.J 7. 7. 7.4047 7.4100 7.4100 7.4299 7.4990 7.4860 7.4410 7.4470 1.4 1.4 K«. 7.4660 7.4710 7.47T0 7.4899 1.4 T.4048 7.5007 7.5067 7.5190 7.6186 7.5944 7.5800 7.58a 7.5410 7.5478 7.5587 7.5585 7.5664 7.6711 7.5770 7.5898 7.5886 7.5044 7.6001 7.6060 7.6117 7.6174 7.6Sa2 7.6289 7.6846 446 447 448 460 451 469 468 454 466 466 457 468 460 460 401 463 468 484 486 SqtuuM. 467 468 460 470 471 473 473 474 475 476 477 478 479 480 481 483 488 484 485 486 487 488 480 480 491 493 483 494 485 496 tf7 498 499 fiOO 601 50S SOS 804 606 6DB 607 SOB 609 610 196916 190800 300704 301601 tM500 S0B4O1 304904 3a30O 306110 307085 307866 308840 300764 310681 311600 313531 318444 314368 315296 316396 117166 318080 310034 310061 330900 221841 222784 223720 224676 235636 nssfm 237590 928484 2204^ 280400 231361 232304 233280 234256 235235 236106 237109 238144 239121 240100 241081 242064 249040 244036 245035 346016 347000 348004 249001 260000 261001 252004 253600 254016 255026 256086 257040 258064 259081 860100 Gubo. Bq.Bt. 88716586 89614688 88015803 90516649 911360004 91788861 93845406 93968077 9S57W64 94186675 94816616 95448088 9a(moi8 96703570 21.1107 81.1404 21.1080 81.1880 11.2183 97071181 98611118 99361847 9988T844 100644096 101194880 103608883 103161760 10380000 104487111 105164048 105818817 106486434 107in676 107850176 106581888 108315861 108001180 llOSOMOO 111384041 1I1980168 113078567 113370804 114OOC10 114701360 115601808 116214372 116880160 117040800 iioswrri 119006488 iioeiffcr 120651104 121381tf» 122098806 123708478 123506082 124211408 125000000 123761601 126506089 127288617 128014804 128787086 128564116 13QS13M8 131006618 isisfrsaso 133651000 CL&U 11. 11. 31.! 91.3078 81.8987 31.3543 21.3776 21.4000 21.4248 31.4476 11.4709 11.4043 31.5174 31.5407 31.5680 11.5890 21.6103 31.6888 21.6664 21.6705 31.7025 31.7366 31.7486 31.7n6 31.7045 81.8174 31.8408 31.8681 21.8061 11.0000 11.9317 11.9545 31.9778 32. 33.0337 32.0454 32.0661 83.0907 22.1188 22.1860 32.1585 32.1811 22.3080 32.2381 12.3486 22.2711 23.3085 33.3160 32.8883 32.3607 23.3880 22.4054 23.4877 22.4480 28.4732 23.4844 4 22.5181 < 22.5680 ' 23.5610 28.6881^ 7.6400 7.0480 7.6517 7.6574 7j 7. 7.6744 7.6801 7.6867 7.6014 7.6870 7.74 7.71 7.718B 7.7U6 7.7860 7.7800 7.7801 7.7410 7.747t 7.7610 7.7604 7.7030 7.708t 7.7750 7.780i 7.7800 7.7015 7.7070 7.8005 7.80IO 7.8184 7.8180 7.8140 7. 7. 7.8100 7.8400 7.8U4 7.J 7.f 7.80M 7.8110 7.8784 7.8 7.8801 l.t 1.1 7.0051 7.M06 7.0108 7.0811 7.0184 7.90lt 7.9870 i: 7.9tTi l.i 1.1 7.S l.i 7.9700 7.9701 7.1 1.1 BqUAK£8» CUBES, AKD BOOX8. 5ft HI •18 •18 «n 618 St8 Mi 8ifr 584 •8T Bfl i4« Mi ««r C«8 6tT » WO m IT8 Sqoas*. Oab*. Sq. Bk CBt. K«. 961111 983144 388168 364188 1SS4S38S1 184317198 185006687 M6788744 186B86B26 83.6068 82.6374 33.6485 83.6710 33.6660 7.9048 8. &0062 &01O4 8.0166 670 677 678 670 600 387388 36BS84 3m4B8 1S78B6B88 138168fi8 isaoHoao 13910686* 140686888 83.7160 32.7870 32.7908 n.7810 33.6006 8.0060 8.QBtl 8.0008 8.0416 6»3 6tt4 666 sn4a 373184 318S88 374898 375895 1414MI61 143g664B 1438TSaM 144761185 33.8354 32.8418 32.8003 33.8810 22.9130 8.04(86 8.0617 8.0609 8.00aD 8.0671 688 667 680 600 600 3166n 37770 378784 378641 1456SIBK 146860188 147181668 148085668 148811088 22.9015 23.0780 33. 33.0817 8.0723 8.0774 8.6036 8.0876 8.0037 001 608 608 604 605 iiil! 148mS81 15050aM8 16141M87 163818684 168188875 SS.0404 23.0651 23.00n 23.1004 3S.I801 8.6878 8.1038 8.1019 8.1180 8.U80 686 607 100 660 000 387n8 388868 388444 380131 381680 ISItWHB 164854158 155780018 166688018 167461680 ss.iosr 23.1738 23.1048 23.2M4 2S.281t 8.1331 8.1381 8.U32 8.M82 8.1683 001 003 008 004 085 383681 98784 384648 386886 387895- 168840481 15|66066B lflM06688 166668184 161M60a5 23.3804 28.3010 28.S0U 28.3388 28.8458 8.1083 8.1533 8.1568 8.U8S 8.1)603 000 007 006 000 •10 386118 388988 800684 801401 809500 103191006 10808038 164806068 1666681188 166891000 28.8666 28.3688 23.4004 28.4807 2S.4ft8t 8.1783 8.1783 8.1883 8.1883 6.1983 Oil 013 018 014 015 808661 904704 806608 800818 806686 10f80tl51 168080868 16BUlBn 170681464 170666835 28.48U 28.4061 28.6100 23.6888 23.5604 8.8862 8.3881 8.8861 8.8180 8.0180 OM 017 010 a.* 808186 810848 811864 813481 818680 171878818 1787I1UI 1740M8IO 28.6709 28.0808 23.0080 28.8888 28.0840 8.3839 8.8378 8.3837 8.3377 8.3436 031 688 038 034 OK 814181 816844 816888 818086 818935 178506681 171004838 178U0649 17MI0lt4 180069186 2S.6854 28.7066 28.7376 28.7487 28.7607 6.9475 6.3684 6.3673 6.8681 8.«70 686 687 638 680 680 890658 831488 823634 838701 894800 181831406 183984868 188880483 1841890000 1861800QO 2B.780O 28.8110 2S.8080 23.8687 2B.87«r 8.8719 8.W66 84il6 8.0106 84^3 mt 088 884 036 ilii! 186100411 187140948 laouasu 188110894 ll8M60875 23.8066 33.0166 33.8074 33.9688 88.8798 8.89B2 6.aoao 6JW59 8.8107 8.U66 080 087 688 089 040 3S1770 333900 884004 886341 888714 34238( 843300 S446a 346744 3488101 S4810O 840981 860404 85104O 862880 864035 866310 860400 S6700i 868001 870681 8T3M» 873331 874644 876700 876600 8783)16 879450 880680 381984 883101 884400 8918X0 8081^ 884884 385041 886000 S86101 309444 400660 4OUIS1O 403836 404480 406700 407044 400881 409600 19U0887O 1931000SS 193100663 10a04680 196118000 311708786 8137U178 218849|M 2140U7W 210000080 237U066O 338188000 389488001 240|B1S8 241004107 2430^0^4 244Um 247Mp 260O430QO 267360466 266474868 2&80940;3 200017110 sraSior 24. 34.0306 34.0410 34.0034 34.0688 84.1060 34.1847 34.104 34.1081 34.1868 94.SM5 34.88U 34.S5U 34.Sm 34.4181 34.4836 24.4640 24.47I& 24.4040 34.0171 34!8n 34.0t70 34.0068 C. 34.8tt6 S4.8m 34.8000 35.2190 25.2380 35.3687 35.3384 36.3M8 8.3208 8.sasi 8.3900 8.SS48 0.S386 0.8448 8.3401 8.3^ 8.3607 8.8084 8.4 8.: m 8.4104 tt& 8.i 8. 8.^ 8. 8.4 8.4 8.4 8.4 8.4 0.^ 8.4688 1:1 0.400 ImIO 8.1 ei 8.r 8.1 8-H?o 6.6816 8.BV3 8J 8-i oj 8.C 8J 8^ 8.1 8.1 84|773 eSts 6.6807 6.6048 8.00B8 60 SQUARES, CUBES, AND BOOTS. TABI4E ofBonarea, Cabes, Sqnar« Root*, and Cube of iVambers from 1 to lOOO — (CoimiruBD.) No. SQnmre. Cube. Bq. Bt. O.Bt. No. Square. Cube. Bq.&t. CBU Ml 410881 363374721 25.3180 8.6233 706 498436 861886816 86.5707 8.90tt 643 412164 264609388 35.3377 6.6267 707 498848 353996343 86.6896 8.9686 643 413449 265847707 25.3574 8.6818 708 601364 854894818 96.6068 8.9187 644 414TS6 267089984 25.3773 8.6867 709 602681 856400838 66.6371 8.9168 646 416026 368336135 25.3969 8.6401 710 604100 867911000 96.6468 8.99U 646 417316 269586136 35.4165 8.6446 711 606681 859436481 96.6646 8.996S 647 418608 270840033 35.4363 8.6480 713 606844 860844138 86.6889 8.9386 648 419904 372097793 85.4558 8.66S6 m 608969 362467097 86.7081 8.9897 649 431201 273359449 86.4766 8.6679 714 6097W 863884844 86i7a06 8.9878 6&0 432500 274625000 85.4961 8.6684 716 611336 366686876 96.7996 8.8498 661 483801 375894451 85.5147 8.6668 716 61366« 967061686 96.7689 8.8489 652 48S104 377167808 35.5343 8.6718 717 614080 368601813 36.7769 8.9609 65S 426409 378445077 25.5539 8.6767 718 615584 370146888 36.7966 8.9646 «4 427716 279736364 35.5734 8.6801 719 616961 371684869 36.8148 8.9687 •66 439025 381011375 25.5990 . 8.6846 730 618400 973348000 36.8838 8.9899 666 430336 383900416 35.6135 8.6890 721 519641 374806961 36.8514 8.9nt <6T 431649 283593893 35.6330 8.6834 733 631884 376967048 36.8701 8.8711 668 432964 284890313 35.6515 8.6878 733 633789 377999067 36.8887 8.876? 66» 434281 286191179 36.6710 8.7033 724 634176 379609434 36.9073 8.8T84 660 436600 387496000 86.6906 8.7066 736 635686 981078136 36.9868 8.8696 661 436931 388804781 35.7099 8.7110 786 637076 883667176 ^m*w%Aw 8.987« 662 438244 390117538 35.7394 8.7164 737 638639 884840689 26.9689 8.981S «3 439569 391494347 85.7488 8.7196 738 689984 386888963 36.9816 8.986» 664 440896 393754944 25.7683 8.7341 739 631441 387480489 37. 8. 665 443326 394079636 85.7876 8.7886 730 633800 389017000 37.0186 8.00a 666 4436S6 395406396 85.8070 8.7389 731 6S496I 890617891 37.0970 8.6089 667 444889 396740968 85.8363 8.7373 733 636884 893888168 37.0666 8.0199 668 446324 298077633 25.8457 8.7416 733 637888 393833897 37.0740 8.01«4 668 447661 399418309 25.8650 8.7460 784 638766 395448804 37.0884 8.8906 mo 448900 600763000 36.8844 8.7606 736 640886 397066976 87.1108 8.6a4S 671 460241 303111711 36.9037 8.7647 736 641686 99e688SS« 400816669 87.1896 8.09W 673 461684 303464448 26.9830 8.7690 787 6491«8 87.1477 8.689B 67S 463939 304831317 35.9433 8.7634 738 644644 401947878 87.1668 8.Q868 674 464376 306183034 35.9616 8.7677 739 646181 409689418 87.1846 9.0410 676 465626 30764687S 86.9806 8.7791 740 647600 406884000 37.9088 9.04SO 676 466876 308915776 26. 8.7764 741 649081 406868081 87.8819 9.04n 677 468339 310288733 26.0198 8.7807 743 650664 4086IS488 87.3997 9.0699 678 469684 311665752 36.0384 8.7860 743 668048 410179407 87.8680 8.06T9 679 461041 313046839 36.0676 8.789S 744 663696 411890784 87.8764 9.90M 680 463400 314433000 36.0768 8.7987 746 666086 418486696 37.8847 8.06M 661 463761 815831341 36.0960 8.7960 746 666616 416160886 37.9190 8.0604 663 465134 317314568 26.1151 8.8083 747 668008 416689789 37.9913 8.0796 683 466489 318611967 36.1343 8.8066 748 669604 418608893 37.3486 8.0n6 684 467856 330013504 36.1534 8.8109 749 (^1001 563600 420189748 37.8679 8.06U 686 469335 331419136 26.1735 8.8163 750 421876000 37.9861 8.QM6 686 470596 333818866 36.1916 8.8194 751 564001 488664761 37.4044 8.068S 687 471969 S3434370S 36.3107 8.8887 763 665604 ITSXMK 87.4886 8.0987 688 473344 336660673 36.3896 8.8380 753 667008 430967777 37.4406 8.0877 689 474721 337083769 26.3488 8.8S2i 764 668616 428661664 37.4691 8.1017 600 476100 338609000 86.3679 8.8866 766 670036 430968876 37.4773 8.1067 691 477481 339939S71 36.3869 8.8406 756 671686 432061816 37.4966 8.1086 692 478864 SS1373888 36.3069 8.8461 757 573048 433796098 37.5136 8.118ft 698 480249 333813657 86.3349 8.8483 758 574664 436619619 37.6318 8.117ft 694 481636 334365384 26.3439 8.8686 758 676081 43784647* 37.6600 8.191ft •96 483035 S36703S75 26.3639 8.8678 760 577600 438976000 37.6681 8.196ft 696 484416 SS7158586 26.3818 8.8621 761 579181 440711061 37.5868 8.138ft •97 4A5809 338608873 26.4008 8.8663 763 580644 449460788 37.6043 8.199ft •98 4B7204 340068392 88.4197 8.8706 768 583168 444194847 37.6326 8.197S •99 488601 841533099 26.4886 8.8748 764 688686 446948744 27.6405 8.141ft 700 490000 848000000 26.4575 8.8790 766 685885 447697136 37.6686 8.146ft 701 491401 344472101 26.4764 8.8883 766 686756 448466006 37.6767 9.1488 702 492804 346948408 26.4963 8.8875 767 688989 461817668 37.6848 8.16ST 70S 494309 347438837 26.5141 8.8817 768 689634 468864888 37.7188 8.1677 T04 495616 348913664 36.5380 8.8868 769 681361 4547&6B08 37.7806 8.1ttT t06 497026 850403635 36.6618 8.8001 770 683800 466539000 37.7489 8.1861 SQUARES, CUBES, AKD BOOTS. 61 TABUB of Sqiiares, Cubes, S4|nare Roots, and Cabe Roots, of Nnmbers from 1 to lOOO— (Continued./ No. 807 810 811 812 818 814 815 810 817 818 810 830 821 814 836 888 827 838 880 881 888 884 fl86 Square. 50M41 S9S884 &075W 599076 600625 602176 603720 606284 606841 608400 609861 611534 613060 614666 616225 617796 619369 620944 622521 624100 625681 627264 628848 630436 632025 688616 635209 636804 638401 640000 641601 643204 644809 646416 648035 648636 651248 662864 664481 6S6100 657731 668344 662586 664235 665856 667480 669134 670761 672400 674041 676684 677339 678976 680625 682276 685584 687241 688800 600561 602234 MUUUM OMKKIO 007225 Onbe. 458314011 460090648 461889917 463684824 465484375 467388576 469007433 470010962 473730139 474653000 476870541 478311768 480048687 481800304 483736635 485687666 487443403 488803873 491168060 493039000 494913671 496798088 4086n257 600606184 602459675 604858336 506361573 608160603 510063399 512000000 613822401 515849608 617781637 619718464 521660135 523606616 535667948 637514113 529476130 681441000 533411731 635387838 637367797 538353144 541343375 643338486 545338513 647343433 649358259 661368000 653387661 666412248 657441767 550476224 561616625 563559076 565600388 567663552 568723780 571787000 573866191 675930868 678009537 580093704 683183875 8q. Bt. 27.7660 27.7848 27.8029 27.8300 27.8388 27.8568 27.8747 27.8927 27.9106 27.9285 27.9464 27.9643 27.9831 28. 38.0179 28.0367 28.0535 28.0713 28.0691 28.1069 28.1847 28.1425 28.1608 28.1780 28.1957 38.2185 28.2312 38.2488 28.2666 28.2848 28.8019 28.3106 38.3378 38.3540 28.3735 38.3901 28.4077 28.4353 38.4439 38.4606 38.4781 28.4956 38.5182 38.5307 38.5482 88.5667 38.5832 28.6007 38.6183 28.6366 38.6581 38.6705 88.6880 38.7054 38.7338 38.7402 28.7576 28.7760 28.7924 28.8097 28.8271 28.8444 28.8617 38.8791 38.8864 C. Bt. No. 9.1696 886 9.1736 837 9.1775 838 9.1815 839 9.1855 840 9.1894 841 9.1933 843 9.1973 843 9.2013 844 9.3063 845 8.2091 846 9.3130 847 9.3170 848 9.3300 849 0.3348 850 9.3287 851 9.2326 863 9.2365 853 9.2404 854 9.2443 855 9.2482 856 8.2521 867 9.2560 858 9.3599 850 9.3638 860 9.2677 861 9.27ie 862 9.2754 863 9.2793 864 9.3832 865 9.3870 866 9.3900 867 9.3948 868 9.2986 868 9.3025 870 0.3063 871 9.3102 872 9.3140 873 9.3179 874 9.8217 875 9.3355 876 9.3394 877 9.8332 878 9.3370 879 9.3406 880 9.3447 881 9.8486 882 9.3533 883 9.8561 884 9.8599 885 9.8637 886 9.3675 887 9.3713 888 9.3751 889 9.8789 890 9.3827 891 9.3865 893 9.S902 893 9.3940 894 9.3978 895 9.4016 896 9.4053 897 9.4091 898 8.4129 899 9.4166 900 Square. 700569 702344 703921 705600 707381 706964 710649 .712336 714025 716716 717409 719104 730801 732500 724201 735904 737609 739316 731025 782736 734449 736164 737881 738600 741321 743044 744760 746486 748225 748966 751689 758434 755161 756900 758641 760384 762129 763876 766625 767376 769139 770884 772641 774400 776161 777924 779689 781456 783225 784996 786760 788544 790S2I 792100 793881 796664 797449 799236 801025 802816 804609 806404 808201 810000 Cube. 8q. Bt. 584277056 586376258 588480(72 590688719 592704000 594828321 596947688 599077107 601311584 603851135 606496786 607646423 600800193 611900049 614135000 616395061 618470308 6206504n 622835864 626026875 637233016 629432793 631628713 633839779 636056000 638377381 640608938 642735647 644872644 647214626 648461896 651714368 658972032 656284809 658608000 660n6311 668064848 665838617 667627624 660921875 678321376 674636183 676836152 679151439 681473000 683797841 686128868 688465387 690807104 698154125 695506456 697864106 700227072 702595369 704969000 707847971 709732288 712121957 714516984 716817375 719323136 721734273 724150792 726673699 739000000 28.9137 28.9310 28.9483 28.9655 28.9828 29. 29.0172 29.0345 29.0517 29.0689 19.0861 29.1083 29.1204 29.1376 28.1648 29.in9 28.1890 29.2062 29.2283 29.2404 29.2675 29.2746 39.2916 29.3087 29.3258 89.8488 29.3598 29.8769 88.8939 29.4109 89.4279 29.4448 29.4618 29.4788 29.4958 29.5137 29.5296 29.5466 29.5635 29.5804 29.5973 29.6142 29.6311 29.6479 29.6648 29.6816 29.6985 29.7153 29.7321 29.7489 29.7668 29.7825 29.7993 29.8161 29.8329 39JB496 29.8664 29.8881 29.8998 29.9166 29.9383 29.9500 29.9666 29.9833 SO. O.BK 8.4204 8.4241 9.4279 9.4316 8.4854 8.4391 9.4429 9.4466 8.4503 9.4541 8.4578 9.4615 8.4652 9.4690 9.4727 8.4764 0.4801 9.4888 9.4875 9.4918 9.4948 9.4968 9.5028 9.5000 9.508T 9.5184 9.5171 9.5207 9.5244 8.5281 9.5817 9.6864 8.5S9I 8.5427 9.6464 8.6601 9.6537 9.5574 9.5610 9.5647 9.5688 9.5719 9.5756 9.5792 9.5828 9.5865 9.5901 9Ji937 9.5973 9.6010 9.6046 9.6062 9.6118 8.6154 8.6190 9.6226 9.6262 9.6298 9.6S34 9.6370 9.6406 9.6442 9.6477 9.6513 9.6549 62 8QUABEB, OUBB8, ANI> ROOXfiL VAMMmE of Stt«Mr«i» €«1>es« tenape Boots, mmA CqIm of N ambers from 1 to 14l0O--(Oo)(TunjEi>.) ITa Sqiuun. 901 m 903 904 905 906 *W , 908 * 900 910 911 912 9IS 9U 916 tie 917 •18 m M0 Ml &7 931^ 999 Mo 961 913 913 9M m 966, 9«r' 9S8 941 94S 94S 944 943 946 W 948 948 950 811801 813604 815409 817316 816036 830836 833648 834464 838381 838100 839931 831744 833569 835386 837335 839Q6ft 840889 84273 8464d0 84B3a 8500M 851939 85B776 856625 857476 859339 861184 868041 664900 866761 868624 870489 87235i 874335 876086 877968 8798U 881731 383600 885481 887364 889249 891136 893025 894916 896800 898704 900601 903500 Cul>«. , 8q. &t. 731433701 733870808 736314337 738763364 741217635 743677416 746143643 7486LS312 751089439 753571000 75606808] 758550638 761048497 763661944 766060875 768676386 7710063X3 773^32 77616I56» 778688000 781338861 78S777448 78633(Mff7 788888034 791^25 79402976 796597983 799178762 801765089 8O436710OO 806964481 8O95&7608 8131607 814780604 817400876 8200; 82 825283612 8279S60I9 830584000 833337621 8S5886tt8 838561 W7 Ml 233384 843908625 846590536 849278123 851971392 854670349 857375000 30.0167 30.0333 30.0500 80.0666 30.0832 30.0998 80.1164 30.1330 30.1486 30.1663 30.1838 so.iwi 30.2159 30.2334 30.3490 30.2666 30.2830 30.2986 30.3160 30.3316 30.3480 30.3«I5 3a3809 30.3974 30.4138 3O.4S0A 30.4467 30.4631 30.4796 30.4959 30.5133 30.5287 30.5460 30.5614 30.5778 30.5941 30.610$ 30.62i8 30.6431 30.6594 30.70< 30.7246 30.7409 S0.75T1 80,7734 30.7896 30.8058 30.8221 cut. Ko. fkiaave. 9.6586 9.6630 9.6666 9.6693 9.6737 9.6763 9.6799 9.6834 9.6970 9.6906 9.^1 9.7013 9.7047 9.7083 9.7118 9.7153 9.7188 9.7334 9.7359 9.738i 9.7338 9.73W 9.7400 9.7436 9.7470 9.7505 9.7540 9.7575 9.7610 9.7645 9.7680 9.7716 9.7750 9.7785 9.7819 9.7864 9.7889 9.7934 9.7959 9.7983 9.8038 9.8063 9.8087 9.8132 9.8167 9.8201 9.8236 9.8270 9.8305 951 962 963 954 956 956 967 968 969 960 % 9B3 964 966 % 966 909 970 971 973 973 974 976 976 977 rf78 9T9 980 981 d83 984 965 986 987 9e9 990 991 993 993 991 996 •fvQ 998 999 1000 904401 906304 908309 9L0U6 913036 918986 91689 917784 933LeW 938166 936088 83T034 838861 840800 943841 944784 946739 94867« 956636 953576 954539 956484 958441 968400 963361 964334 96628» 968256 9702^ 972196 974169 976144 978121 980100 9830^ 984094 986049 988036 990036 993016 994009 99000% 998001 1000000 CulM. Sq.su. 860085351 862801408 865633177 868360664 870083876 873733816 876467^ 87921^913 881874579 884736000 887603681 89037t(28 dKOBHsm 89684IS64 898683135 901438686 9O4S3t06S 907089333 909863309 913679000 91 91 931 t2401«i434 9368S98T6 939714176 9336Y4883 9S5«US63j 9383; 941 ll 944076141 946966168 949863087 952763904 966671635 958686356 961604803< 964430373 9678616m 970299000 973248371 976191488' 979146667 982107784 985074875 968047936 991030*73 994011992 997008999 1000000000 30.8383 30.8545 80.8707 30.8869 30.9031 30.9192 sasaN 8O.8$0 80.8877 80.8838 31. 81.0161 81.0333 81.04B 31.0644 31.0806 31.0893 81.113T 31.1288 81.1448 31.1608 31.176» 31.183» 31.2090 31.3360 31.3410 31.36T0 31.3730 81.3890 31.3060 31.3208 31.3369 81.3538 31.3688 31.8847 31.4006 31.41« 31.4335 31.4484 31.4643 31.4803 31.4966 31.6lf9 S1.627« 31.5438 31.5595 31.5753 31 .5911 81.6070 31.6228 CS^. 9.8339 9.8374 9.8408 9.8443 9.84!t 9.86U 9.8M« 9.8660 9.8614 9.8848 •••^ 9.9631 9.96M ISZ 9.9698 9.9738 9.97W 9.98» 9«V^Bo .9666 .9m 9. 9.1 9.9988 9.99Wr 10. To find tbe sonaro or eabo of any whole nnmber endlMP wltb cipbers. First, omit all the final ciphers. Take from the table w sqiMire or oub« (as the oaae maj be) of the rest of tbe number. To tbU tquare add twice M mt.nf ciphers as there were final ciphers in the original number. To the cube add three times as many at m the orlgioal number. Thus, for 905003; 9053 = 819025. Add twice 3 cipher*, obtaiuiog 8190250000. For iH)5803, go&3 = 741217625. Add 3 times 2 ciphers, obtaining 741217625000000. SQUABi: AND GITBB BOOTS. 63 No CTTora. Num. Sq. Rt. Ca. Rt. Num. Sq. Rt. Ca. Rt. 11.20 Nam. Sq. Rt. Cu. Rt. Nam. Sq. Rt. Cu.Rt. ido& 81.70 10.02 1405 87.48 1805 42.49 12.18 2205 46.96 1102 XOlO 31.78 10.03 1410 87.56 11.21 1810 42.54 12.19 2210 47.01 1?« 1015. 91.86 10.05 1416 87.62 U.23 1815 42.60 12.20 2216 47.00 19.04 low 31 .04 10.07 1420 87.68 11.24 1820 42.66 12.21 2220 47.12 1«.05 10» 82.0S ' 10.06 1426 87.76 11.26 1826 42.72 12.22 2226 47.17 I9.0ft U»0 82.oe 10.10 1430 87.82 11.27 1830 42.78 12.23 2230 47,22 47.28 i$.oe 1036. 32.17 10.12 1436 87.88 11.28 1836 42.84 12.24 2236 19.07 19.08 1040 82.25 10.13 1440 87.96 11.29 1840 42.90 12.25 2240 47.99 lOtf 106O 38.88 10.15 1446 88.01 lUl 1845 42.96 12.20 2246 47.98 19.00 82.40 10.16 1450 88.08 11.32 1850 43.01 12.28 2250 47.43 13.10 iioo 32.48 10.18 1456 38.14 U.33 1856 43.07 12.29 2256 47.^ 19.11 82.56 10.20 1460 88.21 88.21 11.34 1860 43.13 12.30 12.81 8260 47.64 14.12 106& $2.68 10.21 1466 11.36 1866 43.19 2266 47.89 1^13 I074» 82.71 10,23 1470 38.34 88.41 14.37 1870 1876 4S.2i 12.32 2270 47.64^ lil4 ^ 82.70 10.24 . 1476 11.38 43.30 12.33 2876 47.70^ l£lS $2.86 10.26 1480 38.47 U.40 1860 43.36 12.34 2280 47.75 19.10 1066 82.04 83.08 10.28 ' 1486 98.60 88.6t U.41 11.42 1886 43.42 12J5 2286 47,80 47.86 i9.n 109V 10.29 1490 1890 .43.47 18.36 2290 19!S 1 06 83.00 10.31 1496 11.43 1896 43.53 12.37 2296 47.91 l<N>^ 83.17 10.82 1500 38.73 U.46 1900 43.50 12.30 2300 47.0^ 19.20 101^ 89.34 10.84 1506 38.79 U.46 1906 43.3 12.40 2906 48.01 19.21 Ul« 33.8S 10,36 1510 88.86 98.99 11.47 1910 43.7)1 12.41 3310 48.00 19.22 uw 88.30 is.47 10.87 1516 U.49 1916 43.71 12.42 ' 2315 49.11 li29 UM 10.38 10.40 10.42 1520 89.12 11.50 19« 43.8! 12.43 zS20 49.17 isjit 88.54 88.68 1526 ■ 1530 11.51 li.63 1926 1930 49.8! 43.9: 12.44 12.46 2330 49.22 49.92 19:25 19.26 1 sfr 3^.60 10.43 1536 98.18 U.54 1936 43.9) 12.40 . 2336 19.27 1 40 83.76 UL46 1540 38.24 U.56 1940 44.06 lt47 2940 48-97 19.28 83.84 10.46 1646 S.'S 11.66 1946 44.10 12.48 2945 48.43 19.29 liso 83.01 10.48 1550 11.57 1950 44.16 12.19 2950 48.48 19.90 1^6 83.00 10.40 . 1656 89.49 U.59 1956 44.23 12.60 2856 48.63 19.90 ifiS 84.06 10.51 1560 S9.g 11.60 1960 44.27 12.51 2360 48.58 19*91 fj/i^ 84.18 10.63 1566 99!62 11.61 1966 US 44.U 12.63 2366 48.69 19.92 ^ 84.21 84.26 10.64 1(^65 1570 1575 11.62 11.69 1970 1976 ll54 12.66 2970 2376 48.68 48.79 19.98 19.94 UJBO 84.36 10.57 1680 16^ S9.7& ll.((5 1980 44.50 44.56 12,80 2380 48.70 li.3S n^K 84.43 10.58 ov.u 11.66 1986 12-§T 3986 48.84 13.98 iSo «4.5<) 10.60 1690 ^.87 11.67 1990 ^•^ lite liM 12.00 2S9D 48.89 iljst nj6 84.57 10.61 10.63 1696 g.9i 11.66 1996 44.fl» 2995 48.94 iSJiS 13.89 Qoo U.U 1600 1606 4o!m 11.70 2000 44.72 MOO 48.99 U06 84.71 34.70 10.04 11.71 11.72 2006 44.78 12.61 2106 49.04 13.40 uso 10.60 1610 40.12 2010 44.83 12.62 »10 49L<[» 18.41 ♦jll^ 94.80 10.67 1616 40.li 40.25 11.19 2016 44.n 12.09 i&5 4a.u 19.42 y<£3i 84.08 10,69 1620 11.74 2020 44.94 12.64 2480 tt.24 19.48 196 35.00 10.70 1626 40.31 40.St 11.76 2025 45.0D 12.66 2485 19.« y^ 36.21 10.71 1630 11.77 2030 45.0B 12.60 2430 40.ao 18.44 S£ 10.73 10.74 1636 1640 40.44 40.60 11.78 i;.7d 2036 2040 45.11 45.17 12.67 12.68 12.« 2436 2440 4».& 1I45 ll4ft 15.20 10.76 1646 40.59 11.80 2046 45.22 2445 4^*45 19.47 ;Ei6d 85.30 10.77 1650 40.62 11.82 2050 45.28 12.70 2460 4S.8O 19.48 466 95.43 10.79 1656 40.68 11.83 2055 45.33 12.71 2460 «^.60 19J2 85.50 10.80 1660 40.7i 11.84 2060 45.39 12.72 2470 49.70 !M6 35.67 10.82 1066 10.80 11.83 2066 45.44 12.73 2480 48.80 19.64 S9» 86.64 10.83 1670 40.87 11.86 2070 45.50 45.55 12.74 2490 49.90 19.66 U76 86.71 10.84 1675 40.99 11.88 2075 12.75 2500 60.00 19.67 85.78 10.86 1680 40.99 11.89 208O 45.61 12.77 2610 90.10 19.59 3B6 35.86 10.87 1686 41.06 11.90 11.91 2086 45.66 12.78 2520 60.20 13.61 aoo 85.92 10.89 1690 41.11 2090 46.72 12.79 26SO 2540 80.30 19.63 85.90 10.90 1695 41.17 11.92 2095 45.77 12.M 50.40 19.64 s 36.06 10.91 1700 41.23 11.93 2100 45.89 12.8T 2650 60.30 1166 80.13 10.99 1705 41.29 11.93 2105 43.88 12.82 2560 60.60 19.68 ^DO 86.10 10.94 1710 41.36 11.96 2110 45.93 12.83 2570 50.70 1170 lljiy ioiS 10.96 1715 41.41 11.97 2116 45.99 12.84 2580 50.79 1172 ICW 58S 1720 41.47 11.98 2120 46.04 12.83 2590 50.89 19.79 S5 96.40 1726 41.63 11.99 2125 46.10 12.86 2600 60.99 1175 S5o 36.47 11.00 1730 42.59 12.00 2130 46.15 12.87 2610 61.09 19.7T x56 96.54 U-Ol 1736 41.65 12.02 2135 46.21 .12.88 2620 51.19 1179 1|M# 90.61 11.02 1740 41.71 12.03 2140 46.26 12.89 2630 51.28 19.80 iMft 36.67 11.04 1746 41.77 12.04 2145 46.31 12.90 2640 51.38 1182 itso 96.74 11.06 1750 41.83 12.05 2150 48.37 12.91 2650 61.48 1184 S{ 96.81 11.07 1755 41.89 12.06 2155 46.42 12.92 2660 61.58 1I86 SS 90.88 11.08 1760 41.96 12.07 2160 46.48 12.93 2670 51.67 1187 Mt 90.96 11.09 1765 42.01 12.09 2165 46.53 12.94 2680 61.77 ll89 SM 97.01 11.11 1770 42.07 12.10 2170 46.58 12.95 2690 51.87 ll91 Bo 97.08 11.12 1776 42.13 12.11 2175 46.64 12.96 2700 61.96 18.92 97.U 11.13 1780 42.19 12.12 2180 46.69 12.97 2710 52.06 18.94 IW 97.82 1U6 lOo 1786 42.23 12.13 2185 46.74 12.98 2720 52.15 18.90 m 97.28 1790 4i.U 12.14 2190 46.80 12.99 2730 52.25 19.98 m 97.86 11.17 11.10 1795 42.37 12.15 2195 46.85 13.00 2740 52..35 19.99 um 87.42 1800 42.43 12.10 2200 46.90 13.01 2730 62.44 14.01 8QUAKE A.ND CUBE £ SQUABB AND CUBB BOOTS. 66 SQUARE AND CUBE ROOTS. Square Boots and Cube Roots oflf nmbem fWmi 1000 to lOOM — (GONTIirUXD.) Hun. Sq.Bt. Co. Bt. Nora. Sq.Bt. Od. Bt. Nam. 8q. Bt. Ca.Bt. Num. Bq.Bi. 01I.B4 tow W.29 ».M 0990 M.64 21.04 9660 97.79 21.22 97M 96.M I1J» MM 96.S4 ao.87 OSM M.6e 91.06 96W 97.78 21.22 97M .96.94 S1.8t 91M 96.89 ao.ae 9S40 M.04 91.M 9670 97.88 21.28 9eM W.M 31.M 9110 96.46 30.89 9060 M.70 91.07 9680 97.88 21.24 9810 M.06 si.a 9iao 95.60 ao.H9 99M M.76 91.07 96M 97.M 21.26 9820 M.10 si.a 91M 96.66 M.M n7o M.M U.W 96M 97.98 21.26 98W M.16 tl.4t 9140 96.M 90.91 OSM M.86 Sl.M WIO W.M 21.26 9840 M.20 tLU 9160 96.M 90.09 99M M.M 91.10 WJO 96.08 21.27 9660 M.26 81.44 91M 96.71 90.99 9400 M.M 91.10 96M W.1S 21.28 OSM M.M 31.44 9170 96.7C 90.M 9410 97.01 91.11 9840 06.18 21.28 W70 M.85 S1.4ft •IM 96.81 90.94 94M 97.M 91.12 9850 96.38 21.29 96M M.40 21.46 91M 96.W 90.96 94M 97.11 91.1S 98M W.39 21.M 9eM M.45 S1.4T 9»0 96.92 90.W 9440 97.18 91.18 9870 W.84 21.80 99m M.60 21.4T 9910 96.97 90.M 9460 97.91 91.14 9880 98.89 21.81 MIO M.66 21.48 9no M.03 90.97 94M 97.96 91.15 98W 86.44 21.82 M20 M.M 21.49 9B0 90.07 90.98 9470 97.81 91.16 9700 96.48 21.88 99M M.86 21.4S 9140 W.13 90.M 94M 97.8T 91.16 9710 96.64 21.88 9940 M.70 UM 91M M.18 90.M 94M 97.49 91.17 9720 W.69 21.84 9960 M.76 tLM tMO W.23 31.M 96M 97.47 21.18 97M 96.84 31.36 90M M.M tun 9970 W.» 91.01 9610 97.69 91.19 9740 96.W 21.88 9970 M.86 S1.6S 9180 W.SS 91.01 9690 97.57 91.19 9750 98.74 21 J6 99M M.M 21.6t 99M M.38 91.09 9680 97.83 91.90 97M 98.79 21.87 99M M.M S1.64 9iW M.U 91 .OS 9640 97.87 31.31 9770 98.84 21 JK lOOM 1M.00 1144 HIO M.49 91.04 To find Square or Cube Roots of larire numbers not eoa- tained in tlie column off numliers of tlie table. Booh roots mmj MmetimM be taken at onoe from the table, b7 merelr regarding the oolnmns of powen as being oolamne of namber* ; and thoie of nambera aa being those of roota. Thna, if tte •q ft of 9BI81 ia reqd, ilrat iiiid that nnmber in the column of tquaru ; and opposite to it, In th« eolumn of oamben, ii its sq rt 160. For the evhe rt of 857876. find that namber in the eolumn of eu5M ; and opposite to it, in the eol of numbers, is its onbe rt 95. When the ezaot nnmber is not con- tained in the oolnmn of sqnares, or onbes, as the ease may be, we maj nse instead the nnmber nearest to it, if no great aoouraey is reqd. But when a oonsiderablo degree of aoonraoj is necessary, tk* following Tery oorreet methods may be need. For the squfufe root. This rale applies both to whole nnmbers. and to those which are parlor (not wholly) decimal. Flntt la the foregoing manner, take out the tabular number, which is nearest to the giren one ; and also tM tabular sq rt. Mult this tabular nnmber by 8 ; to the prod add the given number. Call the sum M» Then mult the given naml)«r by 8 ; to the prod add the tabular number. Call the sum B. Then A : B : : Tabular root : Beqd root. Sx. Let the given nnmber be 946.58. Here we find the nearest tebnlar number to bo 947 : aaA Mi Ubvlar sq rt M.7784. Henee, 947 = ub nam 8 3841 940.68 = gl 8787.68 = ▲. and 948.58 = given num. 8 2889.58 947 = tab nam. .8786.59 ^^ B. A. S787.5I B. Tab root. Beqd root. Then S787.5I : 8786.89 : : M.7784 : m!7657 +. The root as found by aetual mathematical process is also M.7667 -(-. For the cube root. This rale applies both to whole nnmbers, and to thoee which are par«v decimal. Flrat take ovt tM Ubnlar number whioh is nearest to the given one; and also its tabular onbe rt. If nit this tabular number by 3 ; and to the prod add the given number. Gall the snm A. Then mull the given anmber by 1 ; and to the prod add the tabular number. Gall the sum B. Then A : B : : Tabular root : Reqd root. Bz. Let the given nnmber be 7368. Here we fiuu cne nearest tabalar number (ia tike Mlaan •( ettftes) to be 6860; and iu tabalar cube rt 19. Hence, = tab nam. 18718 y and 7868 = given nam. 310Mr:A. B. Tab Boot. BeqdBt. 21696 — 7868 = given num. 2 14788 8859 = Ub nam. . 21696 =:B. Then, as 210M 21696 19 19.4585 Tke root as fbond by oorreet mathematioal prooess is 19.4Mi. The engineer rarely raqoiree BQCABE AND CUBE BOOTS. 67 UtilllirMof HHiTatfyi ll>r Ub pwroHi, IktHfoH, tUi pfWM ll tvMttr pnUBnbU tfp I^ DrAury To and ttte aqaBrs r»o( of n number wbleb !■ wIioIIt declaaal. hwl fln OiarH, foitntifkg from Ikejtrti ji'ummrai.Hi^ h^viudtna it, wld au or mors cIpbHra to nuJa luj rnlDlcf Ihlf UbulBt rmllo LbBHn, Jkl^ at UBDJ I>lUM la lUB riBBU7 Doa^ad [bctmaf nDDbCT ■If h( ^am J ano-IHir of wblDh la' I ; tlHnf&K, mora tha dmlmftl niat or ibH nni iij. ^qr pluu H the ton; biUbi tt .OUT. tbla la U« Tsqd vq rt or .0(a> Dornci tg iha third bamvm] TJDp]Ddad- T• Bad UlC «nb« rootof «D«ml>erwhlcliIawboll7deeliUal. Tsrj ibiipla, ud SDmn u Ua OltA mmanl loolHlia. ir iW nDBbar data not aonlUii •! Mut Bn Oiuna, aamiUDi rrom Iba Biat nuiaiil, and 1iialudlB| Fin b roo tr ,.„, Sir ,™ ! j 1 1 i 1 1 i 3 1 1 ill 11 68 ROOTS AND POWEBB. Fiftli roots and flftb powero— (Continued). Power. No. Ot Boot. Power. Rio^j f o'«r- No. or p„_^ Boot. ^«'«'* No. Of Boot. Power. No. or Boot. Power. No. Of Root. 88.2735 2.45 2824.75 4.90 86873 9.70 2609193 19.2 20511149 ^.0 459165034 54. V1.ao6-i 2.5U 2y71.84 4.95 9U392 9.80 2747949 19.4 21228258 29.2 508284376 56. 107.b20 2.55 3125.00 3.00 95099 9.90 2892547 19.6 21965275 '29.4 550731776 66. 118 bl4 2.60 3450.25 5.10 100000 10.0 3043168 19.8 22722628 29.6 601693067 57. 130.(>d« 2.65 3802.04 5.20 110408 10.2 3200000 20.0 23500728 29.8 656356768 68. lU.MIt 2.70 4181.95 5-30 121665 10.4 3363232 20.2 24300000 30.0 7149-24299 69. 167.276 2.73 4591.65 5.40 133823 10.6 3533059 20.4 26393634 30.5 777600000 60. 172.104 2.80 5032.84 5.50 146933 10.8 3709677 20.6 28629151 81.0 844696301 61. 188.(Md 2.85 5507.32 5.60 161051 11.0 3893289 20.8 31013642 31.5 916132832 62. 203.111 2.90 6016.92 5.70 176234 11.2 4084101 21.0 33554432 32.0 992436543 63. U9.4U 2.95 6563.57 5.80 192541 11.4 4282322 21.2 36259082 32.5 1073741824 64. 243.000 3.00 7149.24 5.90 210034 11.6 4488166 21.4 39135393 33.0 1160290625 66. 263.936 3.0a 7776.00 6-00 228776 11.8 4701850 21.6 42191410 33.5 1252332576 66. 286.292 3.10 8445.96 6.10 248832 12.0 49-23597 21.8 45435424 84.0 1850125107 67. 810.136 3.15 9161.33 6.20 270271 12.2 515.3632 22.0 48875980 34.5 1463933568 68. 835.54i 3.20 9924.37 6.30 298163 12.4 5392186 22.2 52521875 35.0 1564031349 69. 962.391 3.25 10737 6.40 317580 12.6 5639493 22.4 56382167 35.5 1680700000 70. 891.334 3.30 11603 650 343597 12.8 5895793 22.6 60466176 360 1804229361 7L 421.419 3.35 12523 6.60 371293 13.0 6161327 22.8 647&3487 365 19S49176B2 7*. 454.354 3.40 13501 6.70 400746 13.2 6436343 23.0 69343957 37.0 2073071593 7i 488.760 3.45 145.39 6-80 432040 13.4 6721093 23.2 74167715 37.5 2219006624 74. 525.219 3.50 15640 6.90 465259 13.6 7015834 23.4 79235168 38.0 2373046876 76- 563.822 8.55 16807 7.00 500490 13.8 7320825 23.6 84587005 36.5 7535525376 76. 604.662 3.60 18042 7.10 537824 14.0 7636332 23.8 90224199 39.0 2706784157 77. 647.835 3.65 19319 7.20 577353 14.2 7962624 24.0 96158012 39.5 -2887174368 781 693.440 3.70 20731 7.30 619174 14.4 8299976 24.2 102400000 40.0 3077056399 79. T41,577 3.75 22190 7.40 663383 14.6 8648666 24.4 108962013 40.5 3276800000 80l 792.352 3.80 23730 7.60 710082 14.8 9008978 24.6 115856201 41.0 3486784401 81. 845.870 3.85 25355 7.60 759375 15.0 9381200 24.8 1-23096020 41.5 3707398432 83. 902.242 3.90 27068 7.70 811368 15.2 9765625 25.0 130691232 42.0 3939040643 83. 961.380 3.95 28872 7.80 866171 15.4 10162550 25.2 138657910 42.5 4182119424 84. 1024.00 4.00 30771 7.90 923896 15.6 10572278 25.4 147008443 43.0 4437053125 86. 1089.62 4.05 32768 8.00 984658 15.8 10995116 25.6 155756538 48.5 4704270176 86. 1158.56 4.10 34868 8.10 1048576 16.0 11431377 25.8 164916224 44 4984209207 87. 1230.95 4.15 37074 8.20 1115771 16.2 11881376 26.0 174501858 44.5 5277319168 88. 1306.91 4.20 39.390 .8.30 1186367 16.4 12345437 26.2 1845281-25 45.0 5584059449 89. 1386.58 4.25 41821 8.40 1260493 16.6 12823886 26.4 195010045 45.5 5904900000 90. 1470.08 4.30 44371 8.50 1.338278 16.8 13317055 26.6 205962976 46.0 6240321451 91. 1557.57 4.35 47043 8.60 1419857 17.0 1.3825281 26.8 217402615 46.5 6590815232 92. 1649.16 4.40 49842 8.70 1505366 17.2 14348907 27.0 229345007 47 6956883693 93. 1745.02 4.45 52773 8.80 1594947 17.4 14888280 27.2 241806543 47.5 7.339040224 94. 1845.28 4.50 55841 8.90 1688742 17.6 15443752 27.4 254803968 48.0 7737809375 96. 1950.10 4.55 59049 9.00 1786899 17.8 16015681 27.6 J68.354383 48.5 8153726976 96. 2059.63 4.60 62403 9.10 1889568 18.0 16604430 27.8 ^>8'2475249 49.0 8587340257 97. 2174.03 4.65 65908 9.20 1996903 18.2 17210368 2M.0 •297184.391 49.5 9039207968 • 98. 2293.45 4.70 69569 9.30 2109061 1H.+ 17833868 28.2 U2500000 50.0 9509900499 99. 2418.07 4.75 73390 9.40 •2'2?«203 18 6 1 8475:^09 28.4 345025251 51. 2548.04 4.80 77378 9.aO 234«493 18.8 19135075 28.6 380-204032 62. 9683.54 4.85 81537 9.60 2476099 19..0 19813557 28.8 418195493 63. Square roots of fifth powers of numbers, j/n^, or % powers of numbers, n^^. See table, page 69. The column headed " 12 n " facilitates the use of the table in oases where, for instance, the quantity is giveti in inoheSf and where it is desired to obtain the % power of the same quantity in feet. Thus, suppose we have a % inch pipe, and we require the % power of the diameter in feet. Find ^ (the diameter, in, inches) in thecolumn headed/' 12 n," opposite which, in the column headed *'n," is 0.041666 (the diameter. In feet), and, in column headed "n%,'» 0.00035 (the % power of the diamet«r, 0.041666, in feet). Values of n, ending in or in 5, are exact values. All others end in repeat- ing decimals. Thus: n = 0.052083 signifies n«» 0.052083333 BOOTB AITD POVEBB. >qnar« roata of BfUi powers of nnmbCTM (1) Tables itT lOE^rltbioi gteatl}' facilitate multipIIcatloD anil dlTlsionuid the findlDC of powera and roots of iiumben* (2) Thelabl^pp. 78 to 81 ccinlalutlie eommOB.dMlnalor Brl«ca ■ 'fl»lin|i»ornui)ibe™. The coinmim logartitim ofatmia'-— '- •'- paDentorladeiorthalnmnberuapowerofKI. Bee (IB). ThuB:lD0O = and log lOOO (logarilbm of lOOO) = S.CWOOO. Similarly, 28.7 = 10 Lii ;bI, i lo«.28.f =1.«7S. (S) In geneiil, let A and B b« an; two uumben, and jt any Bzponi (1) log \B = log A + log B ; (a) log g = log A — log B ; (3) log A» = t (log A) ; W log y-l = ^-^ or loEB of tecton. .,„jt dividend -log of log of rractloa = log of numerator — log of deaominatoT. !) Log of quotient = logot dividend — log of divisor (1) L^ of povper =■ log of number, multiplied by ei . (4) Log of root — log of number, divided by exponent. (4) From wbat baa been aald, It followc tbat Log 100 = loglO" = 2.00 too I Log 0.1 = log »-• - l.MOOOt Log 10 ^ log 101 = i.oaooo Log 0.01 - log Iff^ - 2.00 000 Log 1 =^ log 10= =- O.OOOOOt I Log 0.001 = log lO"" = S.OOOOO 1 number, conBlstlng of an inUffral ii Index (prrarliTip tbe declmml BtmaiiBaw^i following the decimal ISO of eacU lag. the cbaracteiisLia mantMa is Klwaya positibe- The miad number, is poaiiive, and la lole number, minus l; while the r Is TKijotiue, and is Qumerically imedlalel)' followiog the decimal log !870 = 3.45 788 log 0.287 - 1.45- 7S8: " 287 = 2.45 788 " 0.0287 - 2.15 788 •' 2S.7 - 1.45 788 " 0.00287 = 3.45 788 2.87 = 0.46 788 " 0.0002B7 = 4.45 788 It win be noticed that the mantissa remains constant thr any given com- hiaatlon of signtfloaut figurea lu a number, wherever the decimal point In the number he placed ; while the cbsraeteristic depends solely upon the podtlou of the decimal pnlut in the number. (6) Let the number be resolved into two factors, one of which is m itegei power of ID, while the other is greater than 1 and less than 10. Then le indei of the power of 10 is the oharaclerlatic of the logarithm, and the logarithm of the other factor Is the mantissa. Tbns, 2370 = IDOO x 2.ST -^ l(^ X 2.87, and the Iwarlthm of 2870 (3.46 78*1 is the sum of the exponent 3 ' 3.00 000) and the log (0.45 79S) of 2.87,t * LuEBTlthms not being exact quantities, operations performed *lth them tra subject to soma ins/ionracy, especially where a logRrfthm la multiplied y a large number, the existing error being thus magnified. Logarlthmaof only five places in the mantissa usually BulDce for calculations with nuU- ben of four or five places. Greater accuracy is obtained by the \ii» of tables of logarithms carried out to seven places. t Log 1 = log 18 - log 10— log 10 = 1— 1 = ; ot 1 - 10». Log 0. 1 = log A = tog 1 — 'og 10 = — 1 = 1.0 : or ai - 10- 1. 1 0287 = 2.S7 -^ 10. Hence, log 0.287 = log 2.87 - log 10 =■ 0,45 783 - 1, which, for convenience. Is written 1^45 788. See (16). Slmilarty, log O.OIST ■ log 2.87 — log 100 - 0,45 788 — 2 = 8.45 788, LOGARITHMS. 71 (7) To find tbe lovaritbiu of a number. The short table on pages 78^ 79 gives logs of numbers up to 1000. The longer table, pages 80 to 91, giyes (1) The mantissa for each number from 1000 to 1750 (2) The mantissa for each even number fh>m 1750 to 3750 (3) The mantissa for each ^th number from 3750 to 10000 (8) Logs of numbers Intermediate of those given in the tables are found by simple proportion. The procedure necessary in these cases is explained in the examples given in connection with the tables, but it will often be found sufficiently accurate to use the log of the nearest number given in the table, neglecting interpolation. Tbe antilog^ariinm or nnm log^ {numerus logarithmt) is the num- ber correspondinfT to a given logarithm. Thus, log. 2 = 0.80 108, and antilog 0.30 l(fe = 2. (9) Mnltiplicatlon. To multiply together two or more numbers, add together their logs and find the antilog of their sum. See t'roportion (11) below. (10) AiTision. Subtract the l<^ of the divisor from that of the dividend, and find the antilog of the remainder. See Proportion (11) below. The reciprocal of any number, n, = . See page 62. Thus, recip 2 => w - = 0.5. Hence, log recip n = log - = log 1 — log n = — log n. Similarly, log recip — = log — — — = o — log . Since n«-i = ni = - , n^-i = n« = " = 1, n^-^ =n-i = - , and no-a = n-« = -j it follows that log w-i = log = log recip n ; log n-* = log zj = "^og recip 7*2, etc. • (11) Proportion. Example. 6.3023 : 290.19 = 1260.7 : ? xr w 1 xr y ^e 290.19 =2.46 269 Multiply Nos, J i* 1260.7 = 3.10 062 Add Logs. I j^^ 290 jg ^ J260.7 = 5.56 331 { Divide Nos. f Log 6.3023 = 0.79 95 Subtract Log. \ Log 58051 =4.76 381 The true value is 58049.05 + (19) Instead of subtracting the log of the divisor, we may add its coloipa- ritlim or arithmetical complement, which is log of reciprocal of divisor, = — log divisor = 10 — log divisor — 10. Thus :. 1523 _ 3.382 X 8.655 Log 1523 = 3.18 270 Colog 8.382 = 10 — log 3.332 — 10 = 10 — 0.52 270 — 10 = 9.47 730 — 10 Colog 8.655 = 10 — log 8.655 — 10 = 10 — 0.93 727 — 10 = 9.06 273 — 10 Sum of logs and cologs = 21.72 273 — 20 = Log 52.813 = 1.72 273 The true value is 52.8114 + (13) Involution, or findinf^ powers of numbers. Multiplv log of given number by the exponent of the required power, and find the anti- log of the product. Thus : 36^ = ? Log 36 = 1.55 630. 1.55 630 X 3 = 4.66 890. Antilog 4.66 890 = 46656. (14) Evolution, or finding roots of numbers. Divide log of given number by exponent of required root, and find antilog of quotient. Thus : s V46656 = ? Log 46656 = 4.66 890. 4.66 890-5-3 = 1.55 680. Antilog 1.55 630 = 36. (tJi) In finding roots of numbers, if the given number is a whole or mixed 72 LOGARITHMS. number, the division of the log is performed in the usual way, as in the preceding example, even where, as in that example, the characteristic ia not exactly divisible by the exponent of the required root. But if tl&e namber is a fraction, and the characteristic of ita log therefore nega- tive, and if the characteristic is not exactly divisible by the exponent, division in the usual wav would give erroneous results. In such cases we may add a suitable number to the mantissa and deduct the same number from the characteristic, thusj to find Vo.00048. Log 0.00048 = 4.68 124 = 0.68 124 — 4 = 2.68 124 — 6 = 6 + 2.68 124, which, divided by 8, = 2 + 0.89 375 = 2.89 375 = log 0.0783. Or, see (16) and (17). (16) To avoid inconvenience from the use of negatiTe character- istics, it is customary to modify them by adding 10 to them, afterward deducting each such 10 from the sum, etc., of the logarithms. Thus : in multiplying or dividing 7425 by 0.25, we have Multiplying. Dividing, either log 7425 = 3.87 070 = 8.87 070 log 0.25 = 1.39 794 = 1.39 794 3.26 864 4.47 276 or log 7425 = 3.87 070 = 3.87 070 modified log 0.25 = 9.39 794 — 10 = 9.89 794 — 10 13.26 864 — 10 6.47 276 + 10 = 3.26 864 = 4.47 276 In most cases the actual process of deducting the added tens may be neglected, the nature of the work usually being such that an error so great as that arising from such neglect could hardly pass unnoticed. (17) To dlTide a modified loiparithm, add to it such a multiple of 10 as will make the sum exceed the true log by 10 times the divisor. Thus : to divide log 0.00048 by 3. Log 0.00048 = 4.68 124, which, divided by 3, = 2.89 375. See (15). • Log 0.00048= 4.68 12 4 Modified log 0.00048 = 6.68 124 — 10 Add 2 X 10 20 — 20 Dividing by 3) 26.68 124 — 30 we obtain 8.89 375 — 10, which is 2.89 375 modified. (18) Except 1, any number can (like 10) be made the base of a system of logarithms. The base of the byperbolic, Napierian, or natural lograritiims, much used in steam engineering, is 1 + 1 + 1-^2 + lX-^3 + 1X2X3X4 + ' " ' " = ^'^ «^ + and is called « (epsilon) or e. M = logi oC (common log e) = 0.43 429 ; ^ =log « 10 (hyperbolic log 10) =2.30 250. For any number, n, loge n = — 1^ = 2.30259 logio n ; logjo n = M loge n = 0.43429 loge n (19) Whatever may be the base chosen for a system of logs, the man> tissas of the logs of any given numbers bear a constant ratio to each other. Thus, in any system of logs, log 4 is always = 2 X log 2, and =• K X log 8, etc., etc. (20) liOffarithmic sines, tansrents, etc. of angles are the logs of the sines, tangents, etc. of those angles. Thus, sin 80° = 0.5000000, and log sin 30° = log. 0.5 = 1.69 897, usually written 9.69 897 — 10, or simply 9.69 897. (ai) Since no power of a positive number can be negative, negative num- bers properly have no logs ; but operations with neyatl-ve nnm- bern ran nevertheless be performed by means of logs, by treating all the numbers as positive and taking care to use the proper sign ,+ or — , in the result. LOGARITHMIC CITART AND SLIDE RULE. 73 1,1- JLog». l.O- OJO- OJS 0.7- 0.0- OJi oa OJO- IJDr- J 1 r 1 1 \ 1 1 1 1 1 1 r JLog9,lJ> 0/» 0,1 OJi OJ3 0,4 OJg OM 0.7 O^ 0.9 1.0 la I I 0,9 0,4 t L_ o.e 0,8 —J L_ 1,0 L_ 1.9 I 1.4 I 2.0 — I 1.9 9.0 1__ 9.9 jro«. Mo9* E 2 3 4 S 97801 -l^ I I . f I ,1 I I I 2 r Bl C T 9 1 — I 1 M I 4 5 7891 3 4 5 07891J\ I I I I .1 I r I I + Dl i ri^-^ T 1 1 ■ I ' I I I 1 I 2 3 4 S G78»l S e 7 8 9 lA. 9 3 -T 5 e -T 7 T — 1 I ■ f 8 9 lU] J»L i3 -« ' 1 1 1 1 1 r— ij} 0.0 0.1 0.9 0.3 0.4 ojs o,e Log»» 0.7 — I r- 0:8 0.9 1.0 — I 1.1 Tb« ttOgnrfthmic Chart and th« S11d« Ral«. (1) By means of a logarithmic chart or diagram (often miscalled lo«i- rtthmic cross-section paper) logarithmic operations are performed graphi- cally, and by means of the slide rule mechanically, without reference to the logarithms themselves *. But see t. P 76. Their use greatly facili* tales many hydraulic and other engineering computations. (•) The ratio between the mantissas of the logs of any given numbers being constant for all systems of logs, the ratio between the distances laid off on the chart or slide rule is the same for all systems, and the use of the chart or rule is independent of the system of logs used. 74 LOGARITHMIC CHART AND SLIDE RULE. (2) The lofrarlttamle eliart consists primarily of a square,* on the sides oi which the distances marked 1-2, 1~3, etc., are laid off by scale according to the logs (0.30 103, 0.47 712, etc.) of 2, 3. etc. Ordinary "squared" or cross seetlon |mper may of course be used for loga- ritmnio i>lotting, by plotting on it the loo9 instead of their Not. Lines representing Nos. may be drawn in their proper places as dedired. (3) As ordinarUv constructed.^ the slide rule consists essentially of four scales. A, B, G, and D, see (17), scales A and D being placed on the ** rule," while B and C are placed upon the sliding piece, or " sUde." As in the logarithmic chart, see (2), the scales are divided loearithmically (see figure), but marked with the numberB corresponding to the logs. Scales A and B are equal, as are also scales C and D, but a given length on A or B represents a logarithm, twice as great as on C or D. See (4). Hence, each number marked on A is the aquare of the coinciding number marked on £>. (4) A single logarithmic scale is usually numbered from 1 to 10, or from 10 to 100; but it may be taken as representing any series embracing the niunbers from 10* to 10**+ ^; as from 0.1 to 1.0 (n = —1); or from 1.0 to 10.0 (n "» 0); or from 10.0 to 100.0 (n = 1); or — etc., etc. Here n and n + 1 are the cliairaeteristlcs of the corresponding logarithms. A single scale would therefore serve for all values, from to infinity ; but for convenience several contiguous scales are sometimes added, as in the log chart*. When a line reaches the limit of a square, the next square may be entered* or the same square mav be re-entered at a point directly opposite. Thus, in the case of line xH (= iTS'y. TiiTiP Trifi.i*1rAi^ between • correspondi to values of xH xttom xH from (1) (2) (3) (4) 1 and S 8} and S, S, and S. Ss and H Ito 10 10 to 31.62 81.02 to 100 100 to 1000 1 to 4.64 4.64 to 10 10 to 21.54 21.54 to 100 Note that the numbers, marked on any given scale, must be taken as 10 times the corresponding numbers marked In the next scale preceding, and the characteristics therefore as being greater by 1, and vice verm. Thus, in our figure, log 1.5 + log 2 = 1-1.5 + 1-2 = log 8 = distance 1-M. But log 15 + log 20 = (1-1.5 + 1-10) + (1-2 + 1-10), so that the characteristic ofthe resulting log is greater by 2, and the 3 representing the product of 15 and 20 is really in the second square to the right of that shown. In finding powers and roots, remember that multiplying or dividing the number by 0.1, 10, 100, etc. a. e., changing the charactensttc of its log), changes also the mantissa of the log of its power or root. Thus, 1^277 = 1.39 . . , (log = 0.14 379) ; but T>'27'== 3, aog = 0.47 712) and 1^270 = 6.46 . . , (log = 0.81 023). The chart or rule gives aU such possible roots, and care must be taken to select the proper one. Most operations exceed the limits of one scale, and fi&cility in using either instrument depends largely upon the ability to pass readily and correctly from one scale to another. This ability is best gained by prac- tice, aided by a thorough grasp of the principles involved. Where several successive operations are to be performed, a sliding runner or marker (furnished with each slide rule) is used, in order to avoid error in shifting the slide. Detailed instructions are usually famished with the slide rule. (*) A common form of chart has four or more similar squares Joined together. See (4). Our figure represents one complete square, with por- tions of adjoining squares. For actual use, both charts and slide rules are, of course, much more finely subdivided than in our figures, which are given merely to illustrate the principles. Carefully engraved charts are published by Mr. John R. Freeman, Providence. R. I. (X) Other forms embodying the same principle are : The " Reaction Scale and Gteneral Slide Rule," bv W. H. Breithaupt, M. Am. Soc. C. E. ; Sexton's Omnimeter or Circular Slide Rule, bv Thaddens Norris : The Goodchild Computing Chart ; The Thacher Calculating Machine or Cylindrical Slide Rule : The Cox Computers, designed for special formulas ; and the Pocket Calculator, issued by " The Mechanical Engineer," London. LOGABrrHMIC CHABT AND SUBB BI <5) Mvltliiltcattoii aad dlvlsiofli. For example, 1-X* in the chart, or on C or D, in the alide rule, the diatf sents by scale the logarithm (0.17 600) of 1.5, and 1-1 losaiithxn (0.30 103) of 2. If now we add these two dis by laylnflT off 1-2 ttom 1.5 on 1-X of the chart, or by placl In the figure, we obtain the distance 1-3 = .47 712 = the m or of log (2 X 1.5).* Conversely, to divide 3 by 2, we graphica cally subtract 1-2 fh>m 1-3. (•) In tbe l4»9Arftliinlc chart, the scales of both axes, 1-Y, being equal, a line 1-H, marked x, bisecting the square ai ing an angle of 45<' with each axis (tan 45° = l),t will bisect also tl sections ox all equcU co-ordinates. Thus, points In the line x, imm over 2, 3, 4, etc.. in 1-X, are also opposite 2, 3, 4, etc., respect!' 1-Y. 8ee (4). g*) If lines 2-A\ S-K, etc. (marked 2x, 8a;, etc.), parallel to m , be drawn through 2, 8, etc., on 1-Y, then points in such li mediately over any number, x, in 1-X, will be respectively oppo (*) In the slide rule, with the slide as shown, ea/:k number on 1.5 X the coinciding number on C. (t) In disenssing tangents of angles on log chart, we refer to th< measured distanoes, as shown on the equally divided scales of tog flgnres, and not tb the numbers, which, for mere convenience, are C B 10 li on lb« cljart. TJius, in )ine 1-B, tan C 1 B = ,~^ = ;;-^-, not — I C 0.38 : 2. 76 LOGARITHMIC CHART AND SLIDE RULE. numbers giving the products 2x, 2x, etc., on 1«Y; while similar lines, drawn below 1-H and through 2, S, etc., on 1-X, give. values of ^^ ?, etc., respectively. If these lines ^^ «• etc., be produced downward, they will cut 1-Y (produced) at 0.5 (= }4), 0^ . . (= V^, etc!, respectively * See (4). (8) Powers and roots. If a line z^ be drawn through 1, at an angle s — s So 1-X, whose tangent, f-^ is 2, it will give values of z*. Thus, the ver- tical through 3, on 1-X, cuts the line x* opposite 9 (= 3*) on 1-Y. Simi- larly, line x^ (tangent = 3) gives values of «' ; and line ^x (tangent = *^ gives values of a;' <*' T/'ir See (4). (9) Any equation of the form y = C.x" in which log y = log C + n log «, (such as : area of circle = ir radius*), is represented, on a logarithmic chart, by a straight line so drawn that the tangent T of its angle with 1-X is = n, and intersecting 1-Y at that point which represents the value C. Thus, the line marked v x^, (tangent = 2) is a line of squares, and, being drawn through IT (= 3.14. .) on 1-Y, it gives values of w x*. Thus, for a circle of radius 2, we find, in the line n x^ over 2, a point L opposite E, or 12.57. . . . the area of such circle.t Conversely, having area = 12.57. . . , we obtain, from the diagram, radius = 2. (10) If a chart is to be used for solving many equations of a single kind, such asy = C a:", where C is a variable coefficient, and n a constant exponent, parallel lines, forming the proper angle with 1-X, should be perma- nently ruled across the sheet at short intervals. (11) For any log, as 1-8 (= log 3), we may substitute its equal. M-N or 3-N, extending to the central diagonal line 1-H, marked x; and then, since, for instance, 1-1.2 = N-Q, 1-3 = N-K, etc., we may add any log (as 1-3) by moving upward from line x (as from N to K) or to the right, and siw^act any log (as 1-1.2) by moving downward (as from N to Q) or to the l^. This facilitates the performance of a series of operations. Thus: To multiply 1.5 by 2 (= 3). by 3 (= 9), and divide by 2 (= 4.5). F-G = 1-F = log 1.5. Add G-J = 1-2 = log 2 ; sum = F-J = log 3 = 1-3 = M-N. Add N-K = 1-3 = log 3 ; sum = M-K = log 9 = 1-9 = 9-R. Subtract R_T = 1-2 = log 2 ; remainder = 9-T = log 4.5. For an example of the application of this principle to engineering prob- lems see " Diagrams for proportioning wooden beams and posts," by Carl S. Fogh, " Engineering News^', Sept. 27, 1894. (la) If eipatiTe exponents. If a: is in the dm«or, the line will slope in the opposite direction, or downward from left to right. Thus, line 4-2 leaving 1-Y, at 4, and forming, with 1-X, the angle X, 2. 4, with tangent = ^^ ' ■ • ^ = — 2, represents the equation : j/ = - , = 4 x-*. (IS) If the lines of products, powers, and roots, C «, a?», and y^ etc., be drawn at angles whose tangents are less by 1 than those of the angles formed by the corresponding lines in our figure, the resmts may be read directly from oblique lines drawn parallel to 2-2. Lines (C x) giving multi- ples and sub-multiples of the first power of x then become horwmial lines (14)" Powers and roots by tbe slide rale. Scales C and D being twice as large as scales A and B, these scales, with their ends coinciding, form a table of squares and of square roots. See (3). By moving the slide we solve equations of the forms jy = (C x)^ and y = C x^. Thus, with the (*) In each of these lines, the product of the two numbers at its ends is = 10. Thus, in line 2-A. 2 X 5 = 10 ; in 3-K, 8 X 3.38 ... = 10, etc. The chart thus furnishes a table of reciprocals. . , (t) Even with full-size charts and slide rules for actual use, accuracy is not to be expected beyond the third or fourth significant flgure. (t) A chart of this kind, prepared by Major Wm. H. Bixby, U. S. A., atter the method of L6on Lalanne. Corps de Fonts et Chaussees, France, is published by Messrs. John Wiley & Sons, New York. Price, 25,centi. LOOARITHHIO CBABT AND SLIDE RULE. 77 slide M shown, each nmnber oa A is «= the sqaftre of (1.6 X the coinciding number on G) ; while, with 1 on B opposite 1.5 on A, each number on A is = 1.5 X the square of the coinciding number on C. (15) Since x» = *" X x, we find cubes or third powers by placing the slide with 1 on B opposite x^ on A ({. e., opposite x on D), see (3), and read- ing «■ f^om A opposite x on B. Thus, 1.5* = ?. Place 1 on B opposite 1.5 on D ; t, «., opposite 1.5* (= 2.25) on A. Then, on A, opposite 1.5 on B, find 8.875 = 1.5*. Or, turn the slide end for end. Place 1.5 on B opposite 1.5 on D, t. e., opposite 1.5* = 2.25 on A. Then, adding log 1.5 (on B) to log 2.25 on A, we find 3.375 (= 1.5') on A opposite 1 on B. (16) Conversely, to find v'iT we shift the slide (in its normal position) until we find, on B, opposite x on A, the same number as we have on I) op- posite 1 on 0, and this number will be =° f/3c7 . Or, turn the slide end for end,* place 1 on C opposite x on A, and find, on B, a number wl^ich coincides with its equal on D. This number is = i^zT See also (17), (18). (17) On the back of the slide is usually placed a scale of logs (see scale shown below the rule in figure) and two scales of angles, marked " S " and " T " respectively, for finding sines of angles greater than 0*^ 34' . . . ", and taxigents of angles between 5° 42' . . . " and 45°. (18) Placing 1 on C opposite any number a; on D (with slide in its normal pofiitiou), log X IS read from the scale of logs by means of an index on the Sack of the rule. The logs may be used in fitidlng powers and roots. ZtogB. t^ 0.0 OJf 0,4, 0,e 0.8 1,0 1,9 1^ X.e 1,8 s,o 9J» J I I I I I 1 t I I ' I « J«0«. Cfi 5 8 4 H €7891 3 3 -U » I . I I .1 I It .... 1 ■ I 1 » L ' L L 1 11 * — ^ — ^ — ' I ' ' I • 1 'I I I L r 7. ^ 4 J 078»ljA ' . ' ■ i '' i ' r' i JBl » 8 dS87891 2 8 4S87891M r^ U — ,"^ f , ?, f , ^ f J.Mfg ) ^00. tPJ IJf 9 3 4 5 7 8 llA -I > 1 1 1 1 1 1 1 r r 1 r ij> 0.0 0,1 0,2 0,3 0,4 o^ o.e 0.7 oa 0,9 ijo .1,1 (19) To find the sine or tang^ent of an angle a ; bring a, on scale S or T, as the case may be, opx>osite the index on back, and read the natural inot logarithmic) sine or tangent opposite 10 at the end of A or D : sines on S, and tangents on C. Or, invert the slide, placine S under A, and T over D. with the ends of the scales coinciding. Then the numbers on A and ]> are the sines and tangents, respectively, of the angles on S and T. Caution. Sines of angles less than 5° 45' ... " are less than 0.1. Tangents " " betw. 5° 42' . . . " and 45° are betw. 0.1 and 1.0. (90) On the back of the rule is usually printed a table of ratios of num- bers in common use, for convenience in operating with the slide rule. Thus : diameter 118 U. S. gallons 3 .. . ...... circumference = »5 = "i^nl^ ' 25 <"" * «''«° ""*""*>' of water). (31) Soaping the edges of the slide and the groove in which it runs, will often cure sticking, wnich is apt to be very annoying. If the slide is too loose, the groove may be deepened, and small springs, cut from narrow steel tape, inserted between it and the edge of the slide. (*) With the slide thus reversed, and with the ends of the scales coin- ciding, the numbers on A and Bare reciprocals (page 62), as are also those on C and D. TABLE or LOOABITHHB. TABLE OF LOOARITHMS. 79 Commoii or Brlgrs* I«oir»i4<l>iM>* 1«. No. M 81954 «7 82607 68 83250 60 83884 70 84609 71 86135 72 86783 73 86S32 74 86023 76 87606 76 88081 77 88649 78 89209 79 89762 80 90800 81 90848 82 91381 83 91907 84 92427 86 92041 86 98449 87 93961 88 94448 80 94939 00 96424 01 96904 02 96378 93 96848 94 97312 95 97772 96 08227 97 98677 98 99122 99 99668 82020 82672 83314 83947 84671 86187 86703 86891 86981 87664 88138 88705 89266 89817 90663 90902 01434 91960 92479 92993 03600 94001 94497 94987 96472 06951 96426 96806 97369 97818 98272 98721 99166 99607 82085 82736 83378 84010 84633 86248 86853 86461 67040 87621 88195 88761 89320 89872 90417 90966 91487 92012 92531 98044 93560 94051 94546 95036 96620 05999 96473 96041 97405 97863 98317 98766 99211 99651 S 82161 82801 83442 84073 84696 85309 86913 86610 87098 87679 88262 88818 89376 89927 90471 91009 91640 92064 92682 93095 93601 94101 94596 96085 05568 96047 96620 97461 97909 98362 98811 99266 82216 82866 83505 84136 84767 86369 85978 86569 87157 87737 88309 88874 89431 89982 90626 91062 91592 92116 92634 93146 93651 94161 94646 95133 95616 96094 96667 97034 97497 97964 98407 98866 99299 99738 82282 82930 83669 84198 84818 86430 86033 86628 87216 87794 88366 88930 89487 90036 90679 91115 91646 92168 92685 93196 93701 04200 94694 96182 96664 96142 96614 97081 97543 98000 98452 98900 99348 99782 6 82347 82994 83632 84260 84880 86491 86093 86687 87273 87852 88422 88986 89542 90091 90683 91169 91608 92220 92737 93247 93751 94260 94748 96230 96712 96189 96661 97127 97689 98046 98497 98946 99387 99826 82412 83068 83696 84323 84941 86661 86153 86746 87332 87909 88479 89042 80697 90146 00687 91222 91750 92272 92788 93298 93802 94300 04792 96279 96760 96236 96708 97174 97636 98091 98642 98989 99431 99869 s 82477 82542 83123 83187 83758 83821 84385 84447 86003 85064 86612 85672 86213 86272 86806 86864 87890 87448 87966 88024 88636 88692 89098 89163 89662 89707 90200 90264 90741 90794 01276 91328 01808 91866 92324 92376 92839 92890 93848 93399 93862 93902 94840 94398 94841 94890 95327 06376 95808 96866 06284 96331 96754 96801 97220 97266 97680 97726 08136 98181 98587 98632 99033 99078 09475 99619 99913 99966 Prop* 66 66 64 63 62 61 60 60 68 67 66 66 •66 64 64 63 68 62 61 61 60 49 4f 48 48* 48 47 47 46 46 46 46 44 44 For extended table of lofpaiittoms see pages 80-91. The table above, being given on two opposite pages, avoids the necessity of turning leaves. It contains no error as great as 1 in the final figure. The proportional parts, in the last column, eive merely the average difi'erence for each line. Heuce, when dealing with small numbers, and using 5-place logs, it is better to find difTer- enoes by subtraction : but where a two-page table » used, interpolation is often auneoeasary. Indeed, the first four, or even the first three, places of the man- tissas here f^ven will often be found sufficient. If rhe first number dropped is S or more, increase by 1 the last figure retained. Thus, for log 660, mantissa » 81954, or 8195, or 820. Miiltlplleatioii. Log a 6 = log a + log b. Dlvtoton. Ix>g ^ s log a — log b. Involatlon (Powers). Log of* — n. log a. BTOlntion (Roots). Log^^s^ * ^^^ Log 2870 -8.45788 u 287 = 2.46788 «l 28.7 » 1.45788 u 2.87 »= 0.45788 n sristtes. Log 0.287 = 0.45788 - 1 = 1.46788 " 0.0287 = 0.46788 - 2 = 2.45788 " 0.00287 = 0.45788 - 8 = 8.45788 " 0.000287 = 0.46788 - ■4 = 4.4578^ 80 LOQARITHMS. O^mniMi or Brim* I^OffaritliimB, Brnio » lO. 90. Log. ,1000 01 02 03 04 09 06 07 08 09 1010 11 12 13 14 15 16 17 18 19 1020 21 22 23 •• 24 25 26 27 28 29 1030 31 32 33 34 36 36 37 88 89 1040 41 42 43 44 45 46 47 48 49 00000 043 — Q87 130 173 —217 —260 —303 346 389 432 475 518 —561 —604 —647 689 732 —775 817 860 —903 945 —988 01030 072 1571^2 199:^2 — 242,t^ 42 43 44 43 43 44 43 43 43 43 43 43 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 —284 —326 —368 410 452 494 —536 —578 —620 -«62 703 745 —787 828 870 —912 953 —995 02036 —078 42 42 42 42 42 42 42 42 42 41 42 42 41 42 42 41 42 41 42 41 No. Log. 1090 02119 7: 160 J* —202 ;f 53 — 243j} 51 52 54 55 56 57 58 59 1060 61 62 63 64 65 66 67 68 , 69 1070 71 72 73 74 75 76 77 78 79 1080 81 82 83 84 85 86 87 88 89 1090 91 92 93 94 95 96 97 98 99 284 325 366 407 ■■^'1 45/ —490 -^31 -572 612 653 694 -735 —776 816 857 —898 938 —979 03019 —060 100 —141 181 —222 —262 302 342 —383 —423 —463 —503 —543 —583 —623 -663 —703 —743 782 822 862 -902 941 981 04021 060 —100 41 41 41 42 41 41 41 40 41 41 41 41 40 41 41 41 41 40 41 40 41 40 41 40 40 40 41 40 40 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 No. 1100 01 02 03 Log. ^ 04139 —179 218 —258 04 —297 05 336 06 —376 07 —415 08 —454 09 493 1110 532 11 571 12 610 13 —650 14 —689 15 727 16 766 17 805 18 844 19 883 1120 —922 21 —961 22 999 23 05038 24 -077 25 116 26 —154 27 192 28 —231 29 269 1130 —308 31 346 32 —385 33 —423 34 461 35 —500 86 —538 37 576 38 614 39 652 1140 690 41 —729 42 —767 43 —805 44 —843 45 —881 46 918 47 956 48 994 49 06032 40 39 40 39 39 40 39 39 89 39 39 39 40 39 38 39 39 39 39 39 39 38 39 39 38 39 38 39 38 39 38 39 38 38 39 38 38 38 38 38 89 38 38 38 38 37 38 38 38 38 No. Log. IISO 06070 51 —108 52 145 53 1—183 54 —221 56 56 57 58 59 1160 61 62 63 64 65 66 67 68 69 1170 . 71 72 73 74 75 76 77 78 79 1180 81 82 83 84 85 86 87 88 89 1190 91 92 93 94 95 96 97 98 99 258 —296 333 —371 408 —446 483 —521 —558 595 —633 —670 707 744 781 —819 —856 —893 —930 —967 07004 ^^41 —078 —115 151 188 —225 —262 298 335 —372 408 445 —482 518 —660 591 —628 664 700 —737 773 809 —846 —882 5 38 37 38 38 37 38 37 88 87 88 87 38 37 37 38 37 87 37 87 38 37 37 37 37 37 37 37 37 36 37 37 37 36 37 37 36 87 87 36 37 86 87 36 36 37 86 36 37 36 36 No. 1200 01 02 08 04 05 06 07 * 08 09 1210 11 It 13 14 15 16 17 18 19^ 1220 21 22 23 24 25 26 27 28 29 1230 31 32 33 34 35 36 37 .38 39 1240 41 42 43 44 45 46 liOg. s 07918 36 954 36 990 37 36 08027;;^ -099!^ — 135 on —171^ 48 49 —207 —243 —279 314 350 386 —422 —468 493 629 —565 600 —636 —672 707 —743 778 —814 849 884 —920 965 —991 09026 061 096 —132 —167 —202 —237 272 307 342 377 412 447 482 —517 .47^-687 621 656 36 36 36 85 36 36 86 86 36 86 86 86 86 86 86 36 35 86 36 36 36 35 86 35 85 85 36 85 85 85 35 35 35 35 85 35 35 86 36 35 84 35 85 Example: To find Log. 11826 : Log. 11830 = 07298 Dif. = 10 36 Log. 11820 = 07262 11826 — 11820 e= 6 Dif. for 6 under 36 = 22 Log. 11826 = 07262 + 22 = 07284 1 2 3 4 5 6 7 8 9 44 4 9 13 18 22 26 31 Z5 40 43 4 9 13 17 22 26 30 34 39 42 4 8 13 17 21 25 29 84 88 41 4 8 12 16 21 25 29 33 87 40 4 8 12 16 20 24 28 32 36 39 4 8 12 16 20 23 27 31 35 38 4 8 11 15 19 23 27 30 84 37 4 7 11 15 19 22 26 80 33 36 •4 7 11 14 18 22 25 29 32 35 4 7 11 14 18 .21 25 28 32 84 3 7 10 14 17 20 24 27 81 1 2 3 4 5 6 7 8 9 LOGABITHM8. r BrlCK* Irf»s*'"l>»»- Base = LOQAKITHU8 CMnnB*n •r Brigita LoynrlMiiii liOOABITHHS. 83 Oommoii or Brlns Ij<»s»rltliiiis. Base » 10. 9o. 1790 62 64 66 68 1760 62 64 66 68 1770 72 74 76 78 1780 82 84 86 88 1790 92 94 96 98 1800 02 04 06 08 1810 12 14 16 18 1820 22 24 26 28 18S0 32 34 86 88 1840 42 . 44 46 48 Log. 24304 853 k-403, 462' —602 551 —601 —650 699 748 797 846 895 944 993 26042 —091 139 188 —237 286 —334 382 —431 —479 627 675 —624 -672 —720 —768 —816 —864 —912 969 26007 —055 102 150 —198 245 —293 —340 387 —436 —482 —629 676 623 670 S3 49 50 49 50 49 60 49 49 49 49 49 49 49 49 49 49 48 49 49 48 49 48 49 48 48 48 49 48 48 48 48 48 48 47 48 48 47 48 48 47 48 47 47 48 47 47 47 47 47 47 Ko. 1850 52 64 56 58 1800 62 64 66 68 1870 72 74 76 78 1880 82 84 86 88 1800 92 94 96 98 1900 02 04 06 08 1910 12 14 16 18 1920 22 24 26 28 1930 32 34 36 38 1940 42 44 46 48 Log. 26717 764 —811 —868 —905 951 —998 27045 091 —138 184 —231 —277 323 —370 -416 —462 508 564 600 646 692 —738 —784 —830 875 921 —967 28012 —068 103 —149 194 —240 —285 330 375 —421 —466 —511 -656 —601 —646 —691 735 780 —825 —870 914 —959 S3 O 47 47 47 47 46 47 47 46 47 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 45 46 45 46 45 45 45 46 45 45 45 45 45 45 44 45 45 45 44 46 44 ToflDdLog. 18117: Log. 18120 ==25816 Bif 20 48 Log. 18100 = 25768 18117 — 18100 = 17 Under 48 Dif. for 10 — 24 7 = 17 u " " 17 = 41 Lttj. 18117 = ^68 + 41 =- 26809. No. 1 2 8 4 6 6 7 8 9 10 00 3 5 8 10 13 16 18 20 23 26 49 2 6 7 10 12 15 17 20 22 26 1900 52 54 66 58 1960 62 64 66 68 1970 72 74 76 78 1980 82 84 86 88 1990 92 94 96 98 2000 02 04 06 06 2010 12 14 16 18 2020 22 24 26 28 2030 32 Log. 29008 —048 092 —187 181 —226 —270 314 358 —403 —447 —491 —636 —679 —628 —667 710 754 —798 —842 886 —929 —973 30016 —060 —108 146 —190 233 276 —820 —363 —406 449 492 635 678 621 —664 —707 —750 792 34 8a5 36 —878 38 920 2040 963 42 31006 44 048 46 —091 48 —183 45 44 46 44 46 44 44 44 46 44 44 44 44 44 44 43 44 44 44 43 44 44 48 44 48 48 44 43 43 44 48 48 43 43 43 43 43 43 43 43 42 43 43 42 43 43 55 43 42 42 No. 2000 62 54 66 68 2060 62 64 66 68 2070 72 74 76 78 2080 82 84 86 88, 2090 92 94 96 98 2100 02 04 06 08 2110 12 14 16 18 2120 22 24 26 28 2130 32 34 36 38 2140 42 44 46 48 Log. 31176 —218 260 802 —846 —887 —429 -471 618 665 697 —689 —681 —723 —765 806 848 —890 981 973 32016 41 I 43 42 42 48 42 42 42 42 42 42 42 42 42 42 41 42 42 41 42 42 056 —098 189 —181 —222 263 —306 —846 887 428 469 610 —652 —593 —684 —675 715 756 797 -888 —879 919 960 33001 041 —082 122 —163 203 42 41 42 41 41 42 41 41 41 41 41 42 41 41 41 40 41 41 41 41 40 41 41 40 41 40 41 40 41 No. Log. 88244 2100 62 284 64 —825 66 —866 68 406 2160 445 62 —486 64 —626 66 —666 68 —606 2170 --646 72 —686 •74 —726 76 —766 78 -806 2180 -«46 82 886 84 926 86 966 88 84005 2190 044 92 084 94 —124 96 168 98 —203 2200 242 02 —282 04 821 06 —861 08 —400 2210 489 12 —479 14 —618 16 —667 18 696 2220 686 22 674 24 718 26 —768 28 —792 2230 880 32 869 34 908 86 947 38 986 2240 85026 42 —064 44 102 46 —141 48 —180 15 41 40 40 40 41 40 40 40 40 40 40 40 40 40 89 40 40 40 89 40 40 89 40 89 40 89 40 89 89 40 89 89 39 39 39 89 40 8f 38 39 89 89 39 39 39 38 39 39 38 48 2 5 7 10 12 14 17 19 22 24 47 2 5 7 9 12 14 16 19 21 24 46 2 6 7 9 12 14 16 18 21 23 40 2 5 7 9 11 14 16 18 20 23 44 2 4 7 9 11 13 16 18 20 22 43 2 4 6 9 11 13 15 17 19 22 42 2 4 6 8 11 18 16 17 19 21 41 2 4 6 8 10 12 14 16 18 21 40 2 4 6 8 10 12 14 16 18 20 39 2 4 6 8 10 12 14 16 18 20 88 2 4 6 8 10 11 18 16 17 19 1 2 S 4 5 6 7 8 9 10 84 LOOABITHMB. CommoB or Brlns I«oirftiltli; 10. Ho. Log. 3200 85218 02 —267 64 295 56 —834 68 372 2360 —411 62 449 64 —488 66 —526 68 564 3370 -603 72 —641 74 679 76 717 78 765 33S0 793 82 —832 84 —870 86 —908 88 —946 3390 —984 d2 36021 M 059 m 097 98 185 3300 —173 02 —211 04 248 Ort —286 08 —324 3310 361 12 —399 14 436 16 —474 18 511 3320, 22 1 24 26 ; 28 3330 32 34 36 88 2340 42 44 46 48 —549 686 —624 —661 698 —786 —773 810 847 884 —822 —959 —996 37033 —070 89 38 39 38 39 38 39 38 38 39 38 38 38 38 38 39 38 38 38 38 37 38 38 38 38 38 37 38 38 37 38 37 38 37 38 37 38 37 37 38 37 37 37 37 38 37 37 87 S7 87 No. 3850 62 64 66 68 3360 62 64 66 68 3370 72 74 76 78 3380 82 84 86 88 3390 92 94 96 98 3400 02 04 06 08 3410 12 14 16 18 3430 22 24 26 28 3480 32 34 36 88 3440 42 44 46 48 Lof. 87107 —144 —181 —218 264 291 —828 —366 401 488 —476 611 648 —585 621 —658 694 —731 767 803 —840 876 912 —949 —986 38021 057 093 —130 —166 -202 —238 —274 —810 —346 —382 417 453 489 —625 —661 596 632 —668 703 —739 —775 810 —846 881 87 37 37 36 37 87 37 36 37 37 36 37 37 36 37 36 37 36 36 37 36 37 36 36 86 36 37 36 36 36 36 36 36 36 35 36 36 36 36 a5 36 36 35 36 36 35 36 35 36 No. 3450 62 64 66 68 3460 62 64 66 68 3470 72 74 76 78 3480 82 84 86 88 3490 92 94 96 98 3500 02 04 06 08 3510 12 14 16 18 3530 22 24 26 28 3530 32 34 36 38 3540 42 44 46 48 L09. 38917 962 987 39023 068 —094 —129 164 199 —236 —270 —805 —840 875 410 446 480 615 660 685 —620 —666 —690 724 769 794 —829 863 898 —933 967 40002 —037 071 —106 140 —176 —209 243 —278 312 346 —381 —415 449 483 —518 —562 —586 —620 86 86 86 35 36 86 35 35 36 35 35 35 35 »5 36 35 35 35 35 35 35 35 34 36 35 86 34 85 35 34 35 35 34 35 34 35 34 34 35 34 34 35 34 34 34 35 34 34 34 34 No. 3550 62 64 56 68 3560 62 64 66 68 3570 72 74 76 78 3580 82 84 36 88 3590 92 94 96 98 3600 02 04 06 08 3610 12 14 16 18 3630 22 24 26 28 3630 32 34 36 38 3640 42 44 46 48 Lof. 40654 688 722 766 790 —824 —868 —892 —926 —960 993 41027 —061 —096 128 —162 —196 229 —263 296 -4J30 363 —897 430 —464 497 —631 664 697 —631 664 697 —731 —764 —797 830 863 896 929 —963 —996 42029 —062 —095 127 160 193 226 —269 —292 84 84 84 34 34 34 34 34 34 33 34 34 34 38 34 34 83 34 33 34 33 34 33 84 33 34 33 33 34 33 38 34 33 33 33 33 33 33 34 33 33 33 32 33 33 33 33 33 33 No. 3650 62 64 66 68 3660 62 64 66 68 3670 72 74 76 78 3680 82 84 86 88 3690 92 94 96 98 3700 02 04' 06 08 3710 12 14 16 18 3730 22 24 26 28 3730 32 34 36 38 3740 42 44 46 48 Log. 42826 867 890 —423 456 488 —621 663 686 -619 661 —684 716 —749 781 813 —846 878 -911 -943 976 43008 —040 —072 104 136 —169 —201 -233 —265 —297 —829 —861 —393 —426 —467 -489 —521 —653 684 616 648 —680 —712 743 776 —807 888 870 —902 32 33 33 32 33 33 32 33 33 32 83 82 33 32 32 38 32 38 32 32 ^ 32 82 32 32 33 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 32 32 31 32 32 31 82 82 81 To find Log. 23335 : LoK. 23340 = 36810 Dif. 20 37 Log. 23320 = 36773 23385 — 23820 = 15 Under 37 Dif. for 10 = 19 " " 5 =__9 " " 15 = 28 ' '^^. 23335 = -78 + 28 = 36801. 39 38 37 36 85 34 38 83 31 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 8 8 3 8 3 6 6 6 6 6 6 6 6 6 4 8 8 7 7 7 7 7 6 6 5 10 10 9 9 9 9 8 8 8 6 12 11 11 11 11 10 10 10 9 7 14 18 13 13 12 12 12 11 11 8 16 15 15 14 14 14 13 18 IS 9l 18 17 17 16 16 16 16 14 14 10 \70 19 19 18- 18 17 17 16 16 1 2 8 4 5 6 7 8 9 19 LOOABTTHMS. 85 Common or Brigrips I«ograrittams. Base « 10. No. »750 52 54 56 58 62 64 66 68 «770 72 74 76 78 »780 82 84 86 88 9790 92 94 96 98 98O0 02 04 06 08 12 14 16 16 98»0 22 24 26 28 »S30 32 84 36 38 42 44 46 48 Log. 43933 —965 996 44028 059 -091 122 —154 185 —217 —248 279 —311 —342 373 404 —436 —467 498 529 560 —692 —623 —664 —686 —716 —747 —778 —809 —840 —871 —902 982 963 994 45025 —056 086 117 —148 —179 209 —240 —271 301 —332 362 —393 423 —454 S3 32 31 d2 31 32 31 32 31 32 31 31 32 31- 31 31 32 31 31 31 31 32 31 31 31 31 31 31 31 31 81 31 30 31 31 31 31 30 31 31 31 30 31 31 30 81 30 31 30 31 30 No. $8850 52 54 56 58 2860 62 64 66 68 2870 72 74 76 78 2880 82 84 86 88 2800 92 94 96 98 2900 02 04 06 08 2910 12 14 16 18 2920 22 24 26 28 2930 32 34 36 38 2940 42 44 46 48 Log. 45484 —515 545 —576 606 —637 —667 697 —728 —758 788 818 —849 —879 909 939 969 46000 —030 —060 —090 —120 —150 —180 —210 —240 —270 —300 —330 359 389 419 —449 -479 —509 538 568 ^598 627 657 —687 716 746 —776 805 —835 864 —894 923 —953 31 30 31 80 31 30 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 30 29 30 30 29 30 80 29 30 30 29 30 29 30 29 30 29 No. 2950 52 54 56 58 2060 62 64 66 68 2970 72 74 76 78 2980 82 84 86 88 2990 92 94 96 98 3000 02 04 06 08 3010 12 14 16 18 802O 22 24 26 28 3030 32 34 86 38 3040 42 44 46 48 Log. 46982 47012 041 070 —100 129 -159 —188 217 246 —276 —305 334 363 392 —422 —451 -480 —509 538 567 596 625 654 683 712 741 —770 —799 —828 —857 885 914 943 —972 48001 029 058 -087 —116 144 —173 —202 230 —259 287 -316 344 —373 401 Cm 30 29 29 30 29 30 29 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 29 28 29 28 29 28 29 28 29 No. 3050 52 54 56 58 3060 62 64 66 68 3070 72 74 76 78 3080 82 84 86 88 3090 92 94 96 98 3100 02 04 06 08 3110 12 14 16 18 3120 22 24 26 28 3130 32 34 36 38 3140 42 44 46 48 Log. 48430 458 —487 515 ,-^44 572 —601 —629 657 —686 —714 742 770 —799 —827 855 883 911 —940 —968 —996 49024 052 080 108 136 164 192 220 248 276 —304 —332 —360 —388 415 443 471 -499 —527 554 582 —610 —638 665 —€93 —721 748 —776 803 (M 28 29 28 29 28 29 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 28 27 28 28 28 27 28 28 27 28 27 28 No. 3150 52 54 56 58 3160 62 64 66 68 3170 72 74 76 78 3180 82 84 86 88 3190 92 94 96 98 3200 02 04 06 08 3210 12 14 16 18 Log. 49831 28 —859 886 —914 941 —969 996 50024 051 —079 —106 133 —161 188 215 —243 270 297 —325 —352 379 406 433 —461 —488 —515 542 669 596 623 —651 —678 —705 —732 —759 3220 —786 22 —813 24 —840 26 866 28 893 3230 920 32 947 34 974 36 51001 38 —028 3240 —055 42 081 44 108 46 —135 48 —162 27 28 27 28 27 28 27 28 27 27 28 27 27 28 27 27 28 27 27 27 27 28 27 27 27 27 27 27 28 27 27 27 27 27 27 27 26 27 27 27 27 27 27 27 26 27 27 27 26 To find Log. 29019: Log. 29020 == 46270 Dil 20 30 Log. 29000 = 46240 »019 — 29000 = 19 Under 80 Dif. tor 10 = 16 »' " 9 = J4 " " 19 = 29 Log. 29019 = 4G240 + 29 = 46269. |3S 1 2 2 3 4 5 6 7 ^ 9 10 32 2 8 5 6 8 10 11 18 14 16 31 30 29 28 2 3 6 6 8 9 11 12 14 16 2 3 5 6 8 9 11 12 14 16 1 3 4 6 7 9 10 12 13 1 3 4 6. 7 8 10 11 13 15 14 27 1 3 4 5 7 8 9 11 12 14 26 1 3 4 5 7 8 9 10 12 13 1 2 3 4 5 6 7 8 9 10 A dasli before or after a log. de- notes that its true value is less thAu the tabular Value by less than half a unit in the last place. Thus : Log. 3128=4952667 *^ 3130=4956448 86 IX)GARITHMS. Common or Brlffss I«oir*>'itlimB. Base = 10. No. 39150 62 M 56 58 00 290 8a60--322 62 348 Log. 51188! „- 215, 27 -2421 S 268; 26 295 27 26 27 348 375 428 26 d/o —402 j 428 —455 481 -508 534 62 64 66 68 8»70 72 -^M. 74 —508 76 534 78 —661 8980 587 82 —614 84 640 86 —667 88 693 8990 —720 „. 92 —746 S 94 772 *? 96 —799 g 98 825 *^ 8800 02 04 06 24 26 28 8880 32 84 I 26 27 26 27 26 27 26 27 2C 27 *51 27 ■»'« 2fi 930 ^ 08 -957 27 8810 —983 12 52009 „- 14 035 il 16 061 ;6 18 -088; 2^ 8880 —1141 ^ 22 —140 -6 166 26 ^26 270 26 -297 -' 36 —323 26 38 —349 ^^ 38 —349 ^ 8840 —375 „- 42 —401 26 46 -453! 26 48 -H179| 5° No. 3300 52 54 56 68 3360 62 64 66 68 3370 72 74 76 78 3380 82 84 86 88 8300 92 94 96 98 8400 02 04 06 08 8410 12 14 16 18 8490 22 24 26 28 3430 32 84 86 38 3440 42 44 46 48 Log 52504 530 556 582 608 —634 —660 —686 711 737 —763 — 7«i9 —815 840 —866 —892 917 943 -969 994 58020 —046 071 —097 122 —148 173 —199 224 —260 275 —801 326 —852 877 —403 428 453 —479 504 529 —655 586 605 -631 —666 681 706 —782 -757 26 26 26 26 26 26 26 25 26 26 26 26 25 26 26 25 26 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 26 25 26 25 2o 26 25 26 26 26 25 26 25 25 25 26 25 25 3740 42 44 46 48 Log. 56229 263 —277 —301 824 348 —372 —396 419 -443 —467 490 —614 —538 661 —585 608 —632 —656 679 —703 726 —750 773 —797 820 —844 867 —891 —914 937 —961 984 67008 -031 054 —078 —101 124 —148 —171 194 217 —241 —264 287 810 —834 —857 —880 To find Log. 36114: Log. 36120 = 65775 Log. 86100 = 55751 Dif. 20 94 3B114 — 36100=»14 Under 24 Dif. for 10 = 12 (> (I 4= 6 '• " 14 = 17 Log. 36114 = 66751 + 17 = 55768. 27 1 1 2 3 3 4 4 5 5 T 6 8 7 10 8 11 9 12 10 14 A dash before or after a log. de* notes that ito true value is leu than the tabular yalue bj lees than half a unit in the last £lace. Thus : >g. 3490 = 6428264 3492 = 5480742 Comin*!! or Brines IiOK«rltkma. Base — U LOGARITHHB. LOOARlTHMa. 8S 90 10 20 n SB 40 15 flO SB 70 79 ao le ' 90 OB MO* OB » 30 ■X «o 4S «0 6B 70 811 SB w OS ;| s u - i - > - n i 35 85 31 3! I 35 S4 39 34 » 34 36 SI 34 S. M Si Nb. •BOO IS 80 40 45 SKIM «0 70 7S 80 w 6«00 10 so 40 46 «a «0 M S700 15 . 85 40 Log. 4M -sst 723 — 76T -m ~MI —086 —119 161 249 2N -«18 880 441 «H — S4( —70; -73; -8oi -ss. 891 1 38 S8 S3 S S 33 s; Ne.' •7BO 76 10 ss 45 •BSO SB 95 10 ao SB 30 M 4B «»se 80 1 Lw. -9« - K i n w 85^ -11 ■ 5 33 K 3; 32 32 32 32 31 32 3 32 32 Ho. 40 4fi ™. 60 80 90 9fi 7100 06 10 20 2fi 30 8B *^ 71S« 60 90 Taoo 16 80 35 45 Log -54 —57 —63 7S SB -94 SSOO 09 ~11 -24 33 40 -46 -82 -66 61 IS -82 -«a -91 —94. -91 S 81 30 31 30 30 No. raao «o 7300 10 so 80 40 46 7SS0 60 S6 80 86 90 95 7400 EO 3S SO 40 45 74<tO Log. 86wi —064 — OM -37! 361 4S1 ~6»! SB) 7i: —74: -801 — 95J on -216 -24! 1 30 M 90 80 30 9> 30 30 SO so 29 20 30 20 39 30 39 30 39 30 29 20 39 so 39 30 39 29 29 39 29 29 20 MOD s Its MS -as J»BSO. 3 S i 7 8 9 ■0 39 ': s: 3a 0.6 31 0.( 8 ) i 3 6.8 3 i •testbi ue Is e Ubu iHBth ta be Bnh: 1" ioa but i61 LOOABTTHUS. LOOARITHMa. Common or Brlna IiOS»'tt)»i>a> 92 eSOXBTBT. QEOMETBI. I^lnes, Fifiriire*, Solldii, defined. Strictly speaking a geometrical 11b« ii limply length, or disUnoe. The Unes we draw on paper have not only length, bat breadth and thiokneas ; still they are the most oonTeoient Bymbol we can employ for denoting a geometrioAl line. Stralirlit lines are also called rl|pb t lines. A vertical line is one that points toward the center of the earth ; and a horisontnl one is at right angles to a vert one. A. plane finrnre is merely any flat surface or area entirely enclosed by lines either straight or ourred ; which are ealled its oatline, boandary, oiroomf, or pcnphery. We often oonfoond the ootline with the tig itself a* when we speak of drawing eirolee, sqnans, «e ; for we aotaally draw only their outlines. Oeometrieally speaking, a Og has length and braadth only ; n* thickness. A solid is any body ; it has length, oreadth, and thickness. Geometrically nlmllar figs or solias, are not necessarily of the same slse; but only of precisely the same sbape. Thus, any two squares are, scien- tifically speaking, similar to each other ; so also any two circles, eobes, 4ko, no matter how diflbrenft ther may be in aiie. When they are not only of the same shape, bat of the same siie, they are said to Ibe similar, and eqaal. The qaantltles or lines are to each other simply as their leng^ttas; but the quantities, or areas, or surfaces of similar flipnreSy are as, or in proportion to, the squares of any one of the corresponding lines or aides which enclose the figures, or which may he drawn upon them : and the quantities, or solidities of similar solids, are as the enbes of any of the corresponding lines which form their edges, or the figures by which th^ are enclosed. Bem«~Simple as the following operations appear, it is only by care, and good instmrnenta, that they are made to give accurate results. Several of them can be much better performed by means of a metallic triangle haying one perfectly accurate right angle. In the field, the (ape-llne, ehain, or a ■Masuring-rod will take the place of the dividera and ruler used indoors. Te divide a si wen line, a b, into two equal pmrUu From Its ends a and h as centers, and with any rad greater than one-half of • ft, describe the area e and d, and Join e/. If the line a & is very long, first lay on eqaal dists a o and i g, each way from tba ends, so as to approach conveniently near to each other ; and then proceed as if o y were the line lo be divided. Ov ineaiare a b by a seale, and thns aaoertain its eenter. To divide a siwen line, «» a, into anj' ffiven number of equal parts. From m and n draw any .two parallel lines m o and n c, te an' indefinite dist ; and on them, tmrn m and n step off th« reqd number of eqaal parts of any convenient length : final- , ly. Join the eorresponding points thus stepped on. Or only one line, as mo, may be drawn and stepped oif, as to «; then Join «n; and draw the other short lines parallel to It. To divide a ^iren line, fa n, into two parts wbieb sball liawo a yiven proportion t^ eacb otber. This is done on the same principle as the last ; thns, let the proportion be as 1 to 8; First draw any line m o ; and with any convenient opening of the dividers, make m s equal to one step ; and •• equal to three steps. Join « n ; and parallel to it draw z c. Then m e is to c n as I is to 3. AJlGIaES. Aniples. When two straight, or right lines meet each other at any lncUn»- tion, the inclination is called an anicle; and is measured by the d^n^ees con- tained in the arc of a circle described from the point of meeting as a center. Since all circles, whether large or small, are supposed to be divided into SCO degrees, it follows that any number of degrees of a small circle will measure the same degree of inclination as will the same number of a large one. When two straight lines, as o n and a h, meet in such a manner that the inclination o n a is eqaal to the inclination o n 6, then the two lines are said to be perpendienlar to each other; and the angles on a and onh, are called rlgbt angles ; and are each measd by, or are equal to, W>, or one-fourth part of the circumf of a circle. Any angle, tMced, smaller than a right angle, is called acute or sharp ; and one c «/, laraer than a right angle, is called obtuse, or blant. When one line meets another, as in the first Fig on opposite page, the two angles on tha same side of either line are called contiguous, or a^iyacent. Thus, vus and * u w are adjacent ; also tut and tuw ; tut audit uv ; vout and wuv. The sum of two a<!yaoaat angles is always equal to two right augled ; or to 1H0°. Therefore, if we know the number of de* frees contained in one of them, and subtract it from 180°, we obtain the other. laanon o n Z QEOHETBY. 93 When two straight lines crow each other, forming four angles, either pair of those angles which point in exactly opposite directions are called opposite, or irertlcal angles ; thus, the pair a « < and vuw are .opposite an- gles ; also the pair suv and t u C9. The opposite anglet of any pair are always equal to each other. When a straight line a b crosses two parallel lines e <2, «/, the alternate angles which form a kind of Z are equal to each other. Thus, the angles don and on/ are equal : as are also con and one. Also the sum of the two internal angles on the same side of a 6, is equal to two right angles, or 180°; thus, co n + on/ =» 180°; also don -\- one = 180°. An interior angle* • In any fig, Is any angle formed intid* of that fig, by the meet- ing of two of its sides, as the angles c a b, a b c, b e a, of this triangle. All the interior angles of any straight-lined figure of any number of sides whaterer, are together eqaal to twice al many right angles minus four, as the figure has sides. Thus, a triangle has 3 sides ; twice that number is 6 ; and 6 right angles, or 6 X 9(P=b4(P; ffom which take 4 right angles, or 360° ; and there remain 18(P, which is the number of degrees in eraty plane, or straight-lined triangle. This principle furnishes ao- easy means of testing our measurements of the angles of any fig; for if the sum of all our measurements does not agree with ihc torn, given bj th« mie, It is a proof that we have committed some error. An exterior angle Of any straight-lined figure, is any angle, as a & d, formed by the meeting of any side, as a b, with the prolongation of an adjacent side, as c b; so likewise the angles c a a and b c to. All the exterior angles of any slraight-lined fig, no matter how many sides it may have, amount to 860° ; but, In (he case of a re-entering angle, as gij, the interior angle, g ij, exceeds 180°, and the "exterior" angle, g i x, being = 180° — interior angle, is negative. Thus ab d + 6cto-fca« = 360° ; and yhj+xji — gix + igie = 380°. Angles, as a, b, c, g, h, and^, which point outward, are called •alientl. From any given point, p, on a line « t, to draw a perp, p a. From p, with any oonvenient opening of the dividers, step off the •qvals po,p§. From o and g as centers, with any opening greater Ahan half o g, describe the two short arcs b and c ; and Join a p. Or still better, describe four arcs, and join a y. Or from p with any conyenient scale describe two •hori area g and e either one of them with a radius 3, and the other with a rad 4. Then from g with rad 6 describe the arc b. Join p a. tS tbe point p is at one end of the line, or very near it, ■ztfend the line, if possible, and proceed as above. But if this •aanot be done, then ftom any convenient point, w, open the divid- er* to p, and describe the semicircle, « p o ; through o to draw o «o «;JeiBf»«. Or use the last foregoing process with rada 8, 4, and 5. Front a given point, o, to let fall a perp o «» to a given line, m n. From o, measure to the line m n, any two equal dists, o e, • « ; and troxa e and « as centers, with any opening greater than half of e e, describe the two arcs a and b ; join o t. Or from any point, as d on the line, op<m the dividers to o, and the arc o g ; make i x equal to < o ; and Join o x. b>ft^c P ^^ftK V^e 94 eXOMETBT. If thm line, a b, !■ on tbe rronnd, Up«- Un«, or chaio. m»n; then Ughtea oat the striiiff, ko. u ■hown ^ m . n ; • belDg lu oeatar. Tben will • e be therMd peroT Or if SS^J.'inH'u'"**.'^^'*.** '*L"* '««'•• thenholdlnftheendof °UJif £!f . i f :5"** **■ °* ?•• '*•' "i*"^ •* »'• »"»'* *»«e four f<^t mark at «, ko»i r Inl iS'u^TJ* *?!k ***•" *' V»«»»t-*«»«l«d triangle. JwiuSd of S, 4, and », la, 16, *o : aJ«o instead of feet, we niaj use jarde, chaina, Ao. Throairb a fflTen point, a, to draw m line, a c, parallel U 6 n 10 y 8 rsTi— W «/. to anotber line. With t)>« P*rp diet, a «, from any point, n. In •/, dew^rlbe ■a arc, I ; draw a e Jut toaoblng the arc. At any point, a, In a line a b, to make an angrle «a fr^eqnal to a irlven anyle, mno. From n and a, with any oonvenlentrad, deeoribe ??/"f ««.<*«; measure s t, and make • d equal to 11; through a d draw a e. 7^^^ e n To biseet, or divide any ani^le, wxy, Into two equal parts. From X aet off any two ei^a&l dists, xr,x*. From r and « with any ra4 describe two aroe interseeting, as at o ; and Join o x. If the two sides of the angle do not meet, fis e / and g h, either first extend them until th«« do meet; or else draw lines x to, and xy, parallel to them, and at equal disu from them, so as to meet; tben proceed as before. All angles, han am,n o m, at ttaeciroamf of a semicircle, and stand' ing on its diam n m, are right angles ; or, as it is usually expressed, all angrles in a semicirele are rig^bt ang^les. An angle n « z at the center of a circle, is twice as great as an angle n n» z at the circumf, when both stand upon the same arc n x. All angles, as y dp. y e p, y ^ p, at the oiroumf of a circle, and aUndlng upon the same are. as y p, are equal to eaeh other ; or, as usually expressed. all ang^les In tbe same segment of a cfreleare equal. But ordinarily we may neglect the signs -4- and — . before eomplementa iiii supplements, and call tbe complement of an angle its dilT from W>' matt the supplement lU dvtf^ from 180°. AITGLES. 95 Aayles fln a ParaUeloffimm. A pamllelogTam is any four-aided Btraight-UBed flg< ure whose opposite sides are equal, as a b c d ; or a square, &c. Any line drawn across a parallelogram between 2 opposite angles, is called a diagoneU^ as a & orb d. A diag divides a parallelogram into two equu parts ; as does also any line m n drawn through the center of either diag ; and moreover, the line m «• itself is div into two equal parts by the diag. Two diags bisect each other ; they also divide the parallel- ogram into four triangles of equal areas. The sum if the two angles at the ends of any one side is = 180^ ; thus, dab + abc^abo-i- hed==- ISfP; and the sum of the four angles, dab,abc^bed^cdaf= 360^. The sum of the squares of the four sides, is equal to the sum of the squares of the two diags. T« reduce Minutes and Seconds to Beyrees and decimals of a Degree, etc. In any given angle — Hnmber of degrees ^ Number of minutes -!- 60. SB Kumber of seconds -^ 3600. » Hnmber of mlnntes = Number of degrees x 60. = Number of seconds -^ 60. H'nniber of seconds Number of degrees X 3600. Number of minutes X 60. Table of Hinntes and B€»conds in Decimals of a Degree, and of Seconds in Decimals of a Minute. (The columns of Mins and Degs answer equally for Sees and Mins.) Mlns. Deg. Hins. Deg. Mins'. Deg. Sees. Deg. Sees. Deg. Sees. Deg, In each equivalent, the last digit repeats indeflnitely. See * below 1 0.016 21 0.350 41 0.683 1 0.00027 21 0.00583 41 0.01138 2 0.033 22 0.866 42 0.700 2 0.00055 22 0.00611 42 0.01166 8 0.060 23 0.383' 43 0.716 3 0.00083 23 0.00638 43 0.01194 4 0.066 24 0.400 44 0.733 4 0.00111 24 0.00666 ; 44 0.01222 5 0.083 25 0.416 45 0.750 5 0.00138 25 0.00694 45 0.01250 6 0.100 26 0.433 4e 0.766 6 0.00166 26 0.00722 46 0.01277 7 0.116 27 0.450 47 0.783 7 0.00194 « 27 0.00750 47 0.01305 8 0.133 28 0.466 48 0.800 8 0.00222 28 0.00777 48 0.01333 9 0.150 29 0.483 49 0.816 9 0.00260 29 0.00805 49 0.01361 10 0.166 30 0.500 50 0.833 10 0.00277 30 0.00833 , 60 0.01388 11 0.183 31 0.516 51 0.850 11 0.00305 31 0.00861 ! 51 0.01416 12 0.200 32 0.533 52 0.866 12 0.00333 32 0.00888 I 52 0.01444 13 0.216 33 0.550 53 0.883 13 0.00361 33 0.00916 53 0.01472 14 0.233 34 0.566 54 0.900. 14 0.00388 34 0.00944 54 0.01600 15 0.250 85 0.583 55 0.916 15 0.00416 35 0.00972 55 0.01527 16 0.266 36 0.600 56 0.933 16 0.00444 36 0.01000 66 0.01555 17 0.283 87 0.616 57 0.950 17 0.00472 37 0.01027 67 0.01583 18 0.300 88 0.633 58 0.966 18 0.00500 38 0.01055 58 0.01611 19 0.816 39 0.650 59 0.983 19 0.00527 39 0.01083 59 0.01638 20 0.383 40 0.66G 60 1.000 20 0.00555 40 0.01111 60 0.01666 - Sees. Mio. Sees . Min. Sees, Min. Sees . Deg. Sees. Deg. Sees. Deg. * Each equivalent is a repeating decimal, thus : 2 minates = 0.0333333 .... degree 7 " = 0.1166666 .... " 12 " =0.2000000 .... " 12 seconds = 0.2000000 1 second = 0.0002777 50 seconds = 0.0138888 minute degree 96 ANGLES. Approzimate Measurement of Angrles. (1) The foar flnarerfl of the hand, held at right angles to the arm and at arm's length from the eye, cover about 7 degr<^ea. And an angle of 7° corre- sponds to about 12.2 feet in 100 feet ; or to 36.6 feet in 100 yards ; or to 645 feet in a mile. (S) By means of a two-foot rnle, either on a drawing or between dis- tant objects in the field. If the inner edges of a common two-foot rule be opened to the extent shown in the column of inches, they will be Inclined to each other at the angles shown in the column of augles. iSince an opening of ^ inch (up to 19 inches or about 105°) corresponds to from about U° to 1° no great accuracy is to be expected, and beyond 105° still less ; for the liability to error then in- creases very rapidly as the opening becomes greater. Thus, the last ^ inch cor- responds to about 129. Angles for openings intermediate of those given may be calculated to the nearest minute or two, by simple proportion, up to 28 inches of opening, or about 147«. Table of Angles correspondlntr to openinipi of a 2-foot rule. (Original). Correet. Ini. Deg. mio.| lD>. Deg. mln.| Ins. Deg. min.] Ids. Dsg.min.] Ins. Deg.mln.] Ins. Dag. min. H 1 12 <y* 20 24 8M 40 IS l2Ji 61 23 16K 85 14 20 Ji 115 6 1 48 21 40 61 62 5 86 S 116 » H 2 24 H 21 37 H 41 29 H 62 47 H 86 52 H 117 » 8 00 22 13 42 7 «3 28 87 41 118 30 H 8 86 H 22 60 H 42 46 H 64 11 H 88 81 H 119 40 4 11 23 27 43 24 04 58 89 21 120 52 1 4 47 5 24 3 9 44 t 13 66 35 17 90 12 21 122 • 6 33 24 39 44 42 66 18 91 8 123 20 H 6 58 H 25 16 H 45 21 y* 67 1 H 91 64 H 124 ZS « 34 25 53 45 59 67 44 92 46 125 64 H 7 10 H 26 90 H 46 88 H 68 28 H 96 88* H i 127 14 7 46 27 7 47 17 69 12 94 81 128 36 H 8 22 H 27 44 H 47 66 H 69 55 H 95 24 H 129 59 8 58 28 21 48 35 70 38 96 17 131 2ft s 9 34 6 28 58 10 . 49 15 14 71 22 18 97 11 22 132 ftS 10 10 29 35 49 54 72 6 96 6 184 M H 10 46 H 30 11 H 60 34 H 72 61 H 99 00 H 135 6S 11 22 30 49 51 13 78 86 99 65 187 36 H 11 58 Vi 31 26 H 61 63 H 74 21 H 100 61 H 189 1% 12 34 32 8 62 83 75 6 101 48 141 1 H 18 10 H 32 40 H 53 13 H 75 51 H 102 45 H 142 51 IS 46 83 17 63 63 76 86 103 48 lU 4f 1 14 22 7 33 54 11 64 34 15 77 22 19 104 41 28 146 46 14 68 34 83 55 14 78 8 106 40 148 6B 34 16 34 H 35 10 Vi 65 65 }i 78 54 H' 106 89 H 151 ir 16 10 85 47 56 35 79 40 107 40 153 41 H 16 46 H 36 25 H 57 16 H 80 27 H 106 41 H 156 Si 17 22 37 8 67 57 81 14 109 48 159 41 H 17 59 H 37 41 H 58 38 H 82 2 H 110 46 H 168 27 18 35 38 19 59 19 82 49 111 49 168 18 4 19 12 8 38 67* 12 60 00 16 83 37 20 112 53 24 180 00 19 46 39 86 tiU 41 84 26 118 58 (3) With the same table^ using: feet instead of inches. From the given point measure 12 feet toward * each object, and place marks. Measure the distauce in feet between these marks. Suppose the first column in the table to be feet instead of inches. Then opposite the distauce in feet will be the angle. ^ foot = 1.5 inches. 1 in. « .083 ft. 4 ins. = .333 ft. 7 ins. -= .583 ft. 10 ins. « .833 ft. 2 ins. — .167 ft. 5 ins. = .416 ft. 8 ins. = .667 ft. Hins. =» .917 ft. 3 ins. = .25 ft. 6 ins. >« .5 ft. 9 ins. — .76 ft. 12 ins. = l.O ft. (4) Or, measure toward * each object 100 or any other number of feet, and place marks. Measure the distance in feet between the marks. Then Sine of half _ half the distance between the marks the angle ~* the distance measured toward one of the objecta* Find this sine in the table pp. 98, etc. ; take out the corresponding angle and multiply it by 2 (0) See last paragraph of foot-note, pp 152 and 153. _ * If it Is inconvenient to measure toward tbe objects, measare directly /Vom them. SnfTBS, TAKQENTS, B70. 97 Sines, Tans^nta, Ac. Sine* a », of any angle, a e 5, or vUeh is th* same thing, the sine of any oiroolar aro, • », vhieh subtends or measures the angle, ix.a straight line drawn from one end, as a, of the aro, at right •ftgles to, and terminating at, the rad c 6, drawn to the other end b of the are. It is, therefore, eqoal lo half the chord a n, of the aro a 5 n, which is equal to twice the aro a b ; or, the sine of an angle ia •lw»n equal to half the obord of twice that angle; and Tioe vena, the ohord of an angle is alwajt a Ml to twioe the sine of half the angle, e sine < c of an angle ( c b, or of an are fa ft, of iW, is equal to the rad of the aro or of the oirele ; and this sine of 90° is y ter than that of any other angle. Cosine e < of an angle acb^ Is that part of the rad which lies between the sine and the oenter of the oirole. It is always equal to the sine y a of the complement tcaotaeb; or of what a e b wants of being 90°. The prefix co be- fore sines, Ao, means oompiemeni ; thus, cosine means sine of the complement. Tersed sine «b of any angle • e 6, is that part of the diam whieh lies between the sine, and the outer end 6. It is T«ry common, but erroneous, when ■peaking of bridges, Ao, to call the rise or height « fr of a caronlar areb a 6 n, its Tersed sine; while it is actually the versed ■ineofonly half the arch. This absurdity •hoald.oease ; for the word rise or height is not only more ezpressiTe,but is correct. Tanicen tbworad, of any angle « « fr. is a line drawn from, and at right angles to, the end 6 or a of either rad c 6, or c a, which forms one of the legs of the sn^ ; and terminating as at to, or d, in the prolongation of the rad which forms die other leg. This last rad thns pro- lonfBd, that is, c w, or e d, as the case may W, is the secant of the angle • e i. The angle (eft being loppeaed to-be equal to 90°, the angle tea becomes the complement of the angle a o ft, or what a e ft wanta of being 90° ; and the sine y a of this complement ; its versed sine t y ; its tangent < o; and its seoaat e o, are respeotirely the eo-sine, co-rersed sine ; co-tangent; and oo-«ecant, of the angle a e ft. Or, viee versa, the sine, 4o, of aeb, are the cosine, Ac, of tea; because the an^le a e ft is the oomple* ment of the angle tea. When the rad e ft, e a, or c t, is assumed to be equal to unity, or 1, the cor> responding sines, tangents, Ac. are called natural ones ; and their several lengths for diff angles, for said rad of unity, have been calculated ; constituting the well-known tables of nat sines, fto. In any eirele whose rad is either larger or smaller than 1, the sines, Ac, of the angles will be in the amme proportion larger or smaller than those in the tables, and are consequently found . by mult tlM ■iae. M, of the table, by said larger or smaller rad. The followinir table of natural sines, Ac. does not contain nat Tened sines, co-versed sines, secants, nor cosecants, but these may be found thus ; Cnr any angle not exceeding 90 degrees. Vened 9bu. From I take the nat cosine. Oo-verted Sine. From 1 take the nat sine. Seeant. Divide 1 by the nat cosine. OoaeeaiAt. Divide I by the nat sine. Wmr «Bftfe« ezeee4bur M^ t to find the sine, eosine, tangent, ootang, secant, or coseo, (but not the versed sine or co-versedsine), take the angle trota 180° : if between 180° and 370° take 180° fk-om the angle : if bet 270° and 360°, Uke the angle from 860°. Then in each ease take trom the tebie the sine, ooeine, tang, or ootang of the remainder. Find Its leoant or coseo as directed above. Far the ^ ttnm ; if between 90(^and 270°, add cosine to 1 ; if bet 270° and 360°, take eosine from 1. (The ddem needs sines, Ae, ezoeoding 180°. To find tbo nat sine* cosine, tans, secant* Tersed sine, ^fcc, of an anvle containing seconds. First find that due to the given deg sad min ; tbea the next greater one. Take their diff. Then as 60 see are to this diff, so are the see only of the given angle to a dec quantity to be added to the one first taken out if it ia a sine, tang, secant, dec ; or to be subtracted from it if it is a cosine, cotang, cosecant, &c. The tjanfpents in the table are strict triiponometrical ones ; that is, tsBcents to given anglts ; and which must extend to meet the secants of the angles towbich they belong. Ordinary, or ipeometrical tangents, as those on p 162, may extend as far as we please. In the field practice of railroad earvea* two trigonometrical tangents terminate where they meet each other. Iseb oftnese tangs is the tang of half the curve. It is usually, but improperly, called '' the tang of the eurM. ' ** Apex dist of the curve," as suggested by Mr Shank, woald be better. 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PH ^ ^ 04 I- s 3 TABUS OF CHOBDS. 143 below, fkinkisbes the meaoBoflaying down angles on paper more accurately than by an ordinary protractor. To do this, after having drawn and measured the first side (say ac) of the figure that is to be plotted ; from its end c as a center, describe an arc ny of a circle of sufficient extent to subtend the angle at that point. The rad en with which the arc is described should be as gjeat as conyenience will permit ; and it is to be assumed as unity or 1 ; and must be decimally divided, and subdivided, to be used as a scale for laying down the chords taken fh>m the table, in which their lengths are given in parts of said rad 1. Having described the arc, find in the table the length of the chord n t corresponding to the angle act. Let us suppose this angle to be 46^; then we find that the tabular chord is .7654 of our rad 1. There- fore fiom n we lay oif the chord nt, equal to .7654 of our radius-scale ; and the lint et drawn through the point t will form the reqd angle act of 46^. And so at each angle. The degree of accuracy attained will evidently depend on the length of the rad, and the neatness of the drafting. The method becomes preferable to the com- mon protractor in proportion as the lengths of the sides of the angles exceed the rad of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not much exceeding the same limits, the protractor will usually be preferable. The di- viders in boxes of instruments are rarely fit for accurate arcs of more than about 6 ins diam. In practice it is not necessary to actually describe the whole arc, but merely the portion near t, as well as can be Judged by eye. We thus avoid much use of the India-rubber, and dulling of the pencil-point. For larger radii we may dis- pense with the dividers, and use a straight strip of paper with the length of the rad marked on one edge ; and by laying it from c toward «, and at the same time placing another Jtrip (witii one edge divided to a radius-scale) from n toward t, we can by trial find their exact point of intersection at the required point t. In such mat* ters, practice and some Ingenuity are very essentlial to satisfactory results. We can' not devote more space to the subject. m' » CHORDS TO A RADIUS 1. M. OO 1° HP SO 4° 5° e° 70 80 90 10° M. 0' .0000 .0175 .0849 .0584 .0098 .0872" .1047 .1221 .1395 .1669 .1748 0' 2 .0000 .0180 .0855 .0589 .0704 .0878 .1063 .1227 .1401 .1675 .1749 8 4 .0012 .0186 .0061 .0585 .0710 .0884 .1068 .1288 .1407 .1581 .1756 4 6 .0017 .0192 .0M6 .0541 .0715 .0890 .1064 .1288 .1418 .1587 .1761 6 8 .0023 .0198 .0872 .0547 .0721 .0896 .1070 .1244 .1418 .1692 .1766 6 10 .0029 .0204 .0878 .0668 .0727 .0901 .1076 .1860 .1484 .1596 .1772 10 IS .0035 .0200 .0884 .0558 .0738 .0907 .1082 .1256 .1430 .1604 .1778 18 14 .0041 .0215 .0890 .0564 .0739 .0913 .1087 .1262 .1436 .1610 .1784 14 16 .0047 .0221 .0896 .0570 .0745 .0919 .1093 .1267 .1442 .1616 .1789 16 18 .0052 .0227 .0401 .0576 .0750 .0025 .1009 .1278 .1447 .1621 .1706 18 SO 2S ] .0058 .0238 .0407 .0682 .0766 .0981 .1105 .1279 .1468 .1627 .1801 20 -.0004 .0239 .0413 .0588 .0762 .0936 .1111 .1285 .1459 .1683 .1807 22 24 .0070 .0244 .0419 .0598 .0768 .0942 .1116 .1291 .1465 .1639 .1813 24 ss .0076 .0250 .0426 .0599 .0774 .0948 .1182 .1296 .1471 .1645 .1818 26 28 .0081 .0256 .0430 .0605 .0779 .0954 .1128 .1302 .1476 .1660 .1824 28 80 .0087 .0262 .0486 .0611 .0785 .0960 .1134 .1808 .1482 .1666 .1830 30 S2 .0008 .0268 .0442 .0617 .0791 .0965 .1140 .1314 .1488 .1662 .1836 32 94 .0000 .0273 .0448 .0622 .0797 .0671 .1145 .1320 .1494 .1668 .1842 34 SS .0105 .0279 .0464 .0628 .0808 .0977 .1151 .1325 .1500 .1874 .1847 86 88 .0111 .0285 .0460 .0684 .0808 .0983 .1157 .1831 .1505 .1679 .1858 36 40 .0116 .0291 .0465 .0640 .0614 .0989 .1168 .1887 .1511 .1685 .1859 40 42 .0122 .0297 .0471 .0646 .0620 .0994 .1169 .1343 .1517 .1691 .1865 42 44 .0128 .0303 .04n .0651 .0826 .1000 .1175 .1349 .1523 .1697 .1871 44 48 .0134 .0308 .0483 .0857 .0832 .1006 .1180 .1366 .1529 .1703 .1876 46 48 .0140 .0314 •fPMHp .0463 .0838 .1012 .1186 .1360 .1534 .1708 .1882 48 M) .0146 .0320 .0404 .0660 .0843 .1018 .1192 .1366 .1540 .1714 .1888 50 (2 .0151 .0386 .0500 ,0675 .0849 .1023 .1198 .1372 .1546 .1720 .1894 5S 64 .0157 .0382 .0606 .0681 .0856 .1029 .1204 .1.378 .1552 .1726 .1900 54 M .0163 .0387 .0512 .0686 .0861 .1035 -.1209 .1384 .1658 .nn .1905 56 68 .0160 .0848 .6618 .0092 .0867 .1041 .1215 .1389 .1561 .1737 .1911 58 •D join U»49 .0524 .0096 .0872 .1047 .1221 .1396 .1569 .1743 .1917 60 144 TABLE OF CHORDS. Table of Cbords, In parte of aradl; for protractlng^-Gontinued. M. 11° 12° 13° 14° 15° 1«° 17° 18° 1©° 20° M. 0* .1917 .2091 .2264 .2437 .2611 .278S .2966 .3129 .3301 .3478 0' 2 .1»2S .2096 .2270 .2443 .2616 .2789 .2961 .8134 .8807 .8479 2 4 .1928 .2102 .2276 .2449 .2622 .2796 .2968 .8140 .8812 •••Cm 4 6 .19S4 .2108 .2281 .2455 .2628 .2801 .2973 .3146 .3318 .8480 6 8 .1940 .2114 .2287 .2460 .2634 .2807 .2979 .8152 .3824 Jt496 8 10 .1946 .2119 .2293 .2466 .2639 .2812 .2986 .3167 .8330 .8502 20 n .1962 .2125 .2299 .2472 .2645 .2818 .2901 .S16S .8386 .8607 12 u .1957 .2131 .2305 .2478 .2651 .2834 mMK^9 .S169 .8341 .8618 14 16 .1963 .2137 .2310 .2484 .2657 .2830 .8002 .8176 JU47 .S6I» 16 18 .1960 .2143 .2316 .2489 .2662 .2836 .3008 .8180 .3353 .8526 18 20 .1975 .2148 .2322 .2495 .2668 .2841 .SOU .3186 .8366 .8630 20 22 .1981 .2154 .2328 .2501 .2674 .284T .9019 .8192 .8364 .3536 22 M .1986 .2160 .2333 .2507 .2680 .2853 .3026 .8198 .8370 .3542 34 26 .1992 .2166 .2339 .2512 .2685 .2858 .3081 .8208 .3376 .3547 36 28 .1998 .2172 .2345 .2518 .2691 .2864 .3087 .8200 .8381 .3553 38 SO .2004 .2177 .2351 .2524 .2530 .2697 .2870 .8042 .8215 .3387 .3659 80 32 .2010 .2183 .2357 .2703 .2876 .3048 .3221 .3398 .3565 S3' Si .2015 .2189 .2362 .2536 .2709 .2881 .3054 .3226 .3398 .3570 84 36 .2021 .2195 .2368 .2541 .2714 .2887 .3060 .3233 .3404 .3576 86 38 .2027 .2200 .2374 .2547 .2720 .289S .3065 .8288 .3410 .3688 88 40 .2033 .2206 .2380 .2553 .2726 .2890 .3071 .8244 .3416 .3587 40 42 .2038 .2212 .2385 .2559 .2732 .2904 .3077 .8249 .3421 .3693 43 44 .20U .2218 .2391 .2564 .27.'57 .2910 .3088 .8255 .8427 •oOW 44 46 .2050 .2224 .2397 .2570 .2743 .2916 .3088 .8261 .3433 .8606 4C 48 .2056 .2229 .2403 .2576 .2749 .2922 .3094 .3267 .8439 .8610 48 60 .2062 .2235 .2409 .2582 .2755 .2927 .3100 .3272 .3444 .3450 .3616 60 52 .2067 .2241 .2414 .2587 .2760 .2933 .3106 .3278 .3622 63 54 .2073 .2247 .2420 .2593 .2766 .2989 .3111 .8284 .3456 .8626 66 56 .2079 .2253 .2426 .2599 .2772 .2945 .3117 .3289 .8462 .3633 5ft 58 .2085 .2258 .2432 .2605 .2778 .2950 .3123 .3295 .3467 ..H639 58 60 .2091 .2264 .2487 .2611 .2783 .2956 .3129 .8801 .3473 .3645 60 M. 21° 22° 28° 24° 25° 26° 27° 28° 2»° so° "-. 0' .3645 .3816 .3967 .4158 .4329 .4489 .4609 .4838 .5008 .5176 0' 3 .3650 .3822 .3898 .4164 .43^4 .4606 .4675 .4844 .5013 .5182 2 4 .3656 .3828 .3999 .4170 .4340 .4510 .4680 .4850 .5019 .5188 i 6 .3662 .3833 .4004 .4175 .4346 .4616 .4686 .4855 .6034 .5193 • 8 .3668 .3839 .4010 .4181 .4352 .4523 .4608 .4861 .5030 .5199 8 10 .8673 .3845 .4016 .4187 .4357 .4527 .4697 .4867 .6036 .5204 10 12 .3679 .3850 .4022 .4192 .4363 .4538 .4703 .4872 .5041 .5210 12 14 .3686 .3856 .4027 .4198 .4369 .4539 .4708 .4878 .6047 .5816 14 16 .3690 .3862 .4033 .4204 .4374 .4544 .4714 .4884 .5063 .5221 16 18 .3696 .3868 .4039 .4209 .4.180 .4550 .4720 .4888 .5058 .6227 18 ao .3702 .3873 .4044 .4215 .4386 .4556 .4725 .4885 .6064 .5233 30 22 .3708 .8879 .4050 1 .4221 .4391 .4661 .4731 .4901 .5070 .5238 22 24 .3713 .3885 .4056 .4226 .4397 .4567 .4787 .4906 .5075 .5244 34 26 .8719 .3890 .4061 .4232 .4403 .4573 .4742 .4812 .5081 .5249 36 28 .3725 .3886 .4067 .4238 .4408 .4578 .4748 .4917 .5086 .5255 38 SO .3730 .8902 .4073 .4244 .4414 .4584 .4754 .4923 .6092 .5261 SO 32 .3736 .3908 .4070 .4249 .U20 .4590 .4759 .4929 .5098 .5266 S3 34 .3742 .3913 .4084 .4255 .4425 .4595 .4765 .4934 .5108 .6272 34 36 .3748 .3919 .4090 .4261 .4431 .4601 .4771 .4940 .5109 .5277 36 88 .3753 .3936 .4096 .4266 .4487 .4607 .4776 .4946 .5115 .52b3 80 40 .3759 .3980 .4101 .4272 .4442 .4612 .4782 . .4061 .5120 .5269 40 42 .3765 .3936 .4107 .4278 .4448 .4618 .4788 .4957 .5126 .5294 48 44 .8770 .3942 .4113 .4283 .4454 .4624 .4793 .4963 .6131 .5300 44 46 .3776 .3947 .4118 .4289 .4459 .4629 .4799 .4<M8 .5137 .5306 40 48 .3782 .8953 .4124 .4295 .4465 .4635 .4805 .4974 .5143 .5311 40 .1. .3788 .3959 .4130 .4800 .4471 .4641 .4810 .4979 .5148 .5317 60 52 .3798 .3065 .4135 .4.HG6 .4476 .4646 .4816 .4985 ..M54 ..^322 fit 54 .3799 .3970 .4141 .4312 .4482 .4652 .4822 .4991 .5100 .58?8 M 56 .9806 .8976 .4147 .4317 .4488 .46.')8 .4827 .4996 .6166 .5834 60 58 .3810 .3982 .4153 .4323 .4493 .4663 .4888 .6003 .6171 .5839 60 00 .3816 .3987 .4158 .4329 .4499 .4669 .4888 .5008 .6176 .5846 00 TABLE OF CHOBDB. 145 Tftble of ehovdOflii parte off a rad 1^ for protractlnv— ContliraeC M. 81° as*" Sso Z4P 99° 86° 87° 88° 89° 40° M. •• .5846 .5613 .5680 .5847 .6014 .6180 .6346 .6511 .6676 .6840 0' 3 .5850 .5618 .5686 .5868 .6030 .6186 .6363 .6517 .MH'X •OBVQ 2 A .5856 .5534 .5601 .6868 .6035 .6191 .6357 .6633 .6687 .6851 4 « .5868 .5630 .5697 .5864 .6081 .6197 .6363 .6538 .6693 .6867 6 8 .5867 .6685 .5708 .5870 .6036 .6303 .6368 .6633 .6606 .6863 8 M .5878 .5541 .5706 .5676 .6042 .6306 .6874 .6630 .6704 •0888 10 13 .5878 .5646 '.5714 .5881 .6047 .6314 .6379 .6544 .6709 .6873 12 14 .5884 .5562 .5719 .6886 .6063 .6310 .6385 .6560 .6715 .6879 14 U .5880 .5667 .5786 .5893 .6058 .6335 .6390 .6730 16 18 .5395 .5568 .5780 .5897 .6064 .6280 .6396 .6661 .6725 .6890 18 90 .5401 .5569 .6796 .5803 .0070 6236 .6401 .6666 .6731 .6895 20 S .5406 .S6T4 .5743 .5600 .6075 .6241 .6407 .6673 .6736 .6901 22 M .5413 .5580 .5747 .5814 .6081 .6247 .6412 .6677 .6743 .6906 24 » .5418 .5686 .6758 .5830 .0086 .6353 .6418 .6683 .6747 .6911 26 » .M2S .5501 .6768 .5936 .6002 .6258 .6438 .6588 .6763 .6917 28 JO .54*29 .5507 .6764 .6981 .6097 .6263 .6439 .6694 .6758 .6923 80 n .5484 .5608 .6769 5986 .6103 .6260 .6484 .6589 .6764 .6838 82 a .5440 .6606 .6775 .5843 .6108 .6374 .6440 .6605 .6769 .6933 81 » .5446 .5613 .6781 .5047 .6114 .6280 .6445 6610 .6775 .6039 M 18 .5451 .5619 .6786 .6963 .6119 .6386 .6451 .6616 .6780 .J944 38 40 .5457 .5625 .6793 mngg\ •OWOV .6135 .6391 .6456 .6631 .6786 .6950 40 43 .5463 .5630 .6797 .5964 .6130 .6396 .6463 .6637 .6791 .6955 42 44 •9voO .6686 .6806 .6870 .6136 .6303 .6467 .6632 .6797 .6061 44 46 .5474 .5641 .6808 .5075 .6143 .6307 .6473 .6638 .6803 .q8od 46 48 .5479 .5647 .5814 .5061 .6147 .6313 .8476 .6643 .6806 .6971 48 fiO .5485 .5653 .5820 .5866 .6153 .6318 .6484 .6649 .6654 .6613 .6977 50 51 .5490 .5668 .6826 .5983 .6158 .6334 .6489 .6619 .6983 52 64 .5486 .5664 .6861 .5087 .6164 .6330 .6495 .6660 .6824 .6988 54 M .5502 .5660 .5886 .6006 .6169 .6336 .6600 .6665 .6829 .6993 56 W .5507 .5675 .6648 .6000 .6175 .6841 .6606 .6671 .6835 .6999 66 40 .5513 .5680 .6847 .6014 .6160 .6846 .6611 .6676 .6840 .7064 60 0' 3 4 6 8 10 i7 14 16 U 21 24 28 28 10 HT J4 16 18 40 46 46 50 IS' 54 M 41° 48° .7004 .7010 .7015 .7020 .7026 .7081 , .7167 I .7171 I .7176 .7184 I .7188 .7186 .7200 .7206 .7211 .7216 .7222 .7227 .7232 .7238 .7343 .7249 .7081 .7254 .7097 .7280 .7102 .7265 .7106 .7270 .7113 .7276 .7118 .7124 .7129 .7135 .7140 .7281 .7387 .7282 .7388 .7803 .7146 .7151 .7156 .7162 .7187 .7806 .7314 .7819 .7126 .TIM .7380 .7335 .7341 .7346 .7362 .7357 44' .7482 .7486 .7606 .7608 .7614 .7518 .7362 .7368 .7878 .7379 .7384 .7390 .7385 .7400 .7406 .7411 .7417 .7432 .7427 .7433 .7488 .7524 .7580 .7536 .7541 .7546 .7551 .7557 .7562 .7568 .7573 .7578 .7584 .7588 .7596 .7600 .7443 .7448 .7464 .7460 .7466 .7471 .7476 .7481 .7487 .7493 .7605 .7611 .7616 .7631 .7637 .7683 .7638 .7648 .7648 .7664 45° 48° .7664 .7816 .7659 .7820 .7664 .7826 .7670 .7831 .7675 .7836 .7681 .7841 .7686 .7847 .7691 .7852 .7687 .7857 .7703 .7868 .7707 .7868 .7713 .7873 .7718 .7879 .7733 .7884 .7739 .7890 .7784 .7895 .7740 .7900 .7746 .7906 .7750 .7911 .7756 .7916 .7761 .7933 .n66 .7987 .7773 .7933 .7777 .7938 .7783 .7948 .7788 .7948 .7793 .7954 .7799 .7959 .7804 .7964 .7809 .7970 .7815 .7975 47° 48'= .7975 .7960 .7966 .7991 .7996 .8003 .8007 .8013 .8018 .8033 .8028 .8084 .8030 .8044 .8050 .8065 .8060 .8071 .8076 .8083 .8067 .8093 .8098 .8103 .8108 .8118 .8119 .8134 .8139 .8136 .8135 .8140 .8145 .8151 .8156 .8161 .8167 .8173 .8177 .8183 .8188 .8193 8198 .8204 .8209 .8314 .8320 .8235 .8230 .6236 .8341 .8246 .8351 .8257 .8263 .8367 .8273 .8278 .8383 .8389 .8394 49° 59° .8394 .8453 .8299 .8458 .8304 .8463 .8310 •o40d .8315 .8473 .8320 .8479 .8336 fUAL .8331 .8489 .8336 .8495 .8341 .8500 .8347 .8505 .8353 .8510 .8357 .8516 .8363 .8521 .8368 .8526 .8373 .8531 .8378 .8537 .8.^84 .8543 .8389 .8547 .8394 .8552 .8400 .8558 .8405 .8563 .8410 ■8668 .8415 .8573 .6431 .8579 .8436 .8584 .8431 .8589 .8437 .8694 .8443 .8600 .8447 .8605 .8453 .8610 V 3 4 6 8 10 13 14 16 IB 20 33 34 36 38 30 82 34 36 98 40 42 44 48 48 50 ~M 54 68 58 10 146 TABLE OF CHORDS. VsMe of ebordSy in parts of a rad 1 ; for ^rotrmmUmg >- Contiiiiisd M. n° 6SO MP 54'' Ofto 56° 57« Sfio 59° •o° 0' MIO .8767 .8934 .9060 .9286 .9889 .9648 .9696 1.0000 3 .8615 .8778 .8939 .9066 .9340 .9396 .9648 .9701 .9864 1.0006 4 .8621 .8778 .8984 .9090 .9345 .9400 .9568 .9706 .9860 1.0010 « .8636 .8783 .8940 .9096 .9260 .9405 .9569 .9711 UCMBJ 1JW16 8 .8681 .8788 .8946 .9101 .9256 .9410 .9564 .9717 •vonP 1.0030 10 .8686 .8794 .8960 .9106 .9281 .9416 .9669 .9733 .9674 1.0036 13 .8642 8790 .8966 .9111 .9266 .9430 .9674 .9737 .9879 1.0060 14 .8647 .8804 .8960 .9116 .9271 .9436 .9679 .9782 .9884 1.0066 16 .8662 .8809 •8D0D .9131 .9276 .9480 .9684 .9737 •VSoV 1.0040 18 .8667 .8814 .8971 .9136 .9281 .9486 .9689 .9742 ■INNM 1.0046 30 .8668 .8830 .6976 .9183 .9287 .9441 .9694 .9747 .9899 1.0060 38 ■8Od0 .8836 .8961 .9187 .9292 OMf .9763 .9904 1.0065 34 .867S .8880 •cWBo .9143 .9297 .9461 •9604 .9767 .9909 1.0060 as .8678 .8885 .8993 .9147 .9302 .9466 .9610 .9763 .9914 1.0065 38 .8684 .8841 .8897 .9163 .9807 .9461 .9616 .9767 .9919 1.0070 M .8688 .8846 .9003 .9167 .9312 .9466 .9630 .9773 .9934 1.0076 83 •OWv .8851 .9007 .9168 .9817 .9473 .9626 .9778 .9939 1.0060 M .8690 .8866 .9013 .9168 .9823 .9477 .9680 .9788 ■VvV* 1.0066 86 .8706 .8861 .9018 .9178 .9828 .9483 .9685 .9788 .9989 1.0061 88 .8710 .8867 .9038 .9178 .9833 .9487 .9640 .9798 .9946 1.0096 40 .8716 .8872 .9038 .9183 .9888 .9493 .9646 .9798 .9960 1.0101 43 .8720 .8877 .9088 .9188 .9843 .9497 .9660 .9808 .9955 1.0106 44 .8736 .8882 .9088 .9194 .9348 .9503 .9666 .9608 .9060 1.0111 46 .8781 .8887 .9044 .9199 .9853 .9607 .9661 .9618 .9965 1.0116 48 .8786 .8888 .9049 .9304 .9869 .9512 •VOBo .9818 .9970 1.0131 60 .8741 ftflOfi .9064 .9309 .9364 .9518 .9671 .96X8 .9976 1.0136 63 .8747 .8908 .9069 .9314 .9869 .9623 .9676 J638 .9980 1.0181 64 .8762 .8908 .9064 .9319 .9874 .9638 .9681 .9668 .9986 1.0186 66 .8757 .8914 .9069 .9335 .9379 .9638 .9686 .9888 .9990 1.0141 68 .8762 .8019 .9076 .9330 .9884 .96a6 .9691 .9648 .9996 1.0146 60 .8767 .8924 .9080 .9336 .9880 .9548 •VQVD .9648 1.0000 1.0161 9 3 4 6 8 10 13 14 1« 18 IS 94 16 SB 10 ss 84 44 6S 64 M. en.o 62° •8° 64° 65° e^° •7° •SO er> 700 M. 0' 1.0151 1.0801 1.0450 1.0698 1.0746 1.0693 1.1089 1.1184 1.1838 1.1473 0- 3 1.0156 1.0306 1.0455 1.0608 1.0761 1.0898 1.1044 1.1189 1.1888 1.1476 s 4 1.0161 1.0811 1.0460 1.0608 1.0756 1.0903 1.1048 1.1194 1.1888 1.1481 4 6 1.0166 1.0316 1.0466 1.0613 1.0761 1.0907 1.1063 1.1198 1.1S43 1.1486 e 8 1.0171 1.0321 1.0470 1.0618 1.0766 1.0912 1.1068 1.1203 1.1S47 1.1491 s 10 1.0176 1.0826 1.0475 1.0623 1.0771 1.0917 1.1063 1.1208 1.U63 1.1496 M 13 . 0181 1.0331 1.0480 1.0628 1.0775 1.0923 1.1068 1.1213 1.IS67 1.1500 IS 14 1.0186 1.0336 1.0485 1.0683 1.0780 1.0927 1.1073 1.1218 1.1963 1.1606 U 16 1.0191 1.0841 1.0490 1.0688 1.0785 1.0982 1.1078 1.1222 1.1866 1.1610 16 18 1.0196 1.0346 1.0495 1.0643 1.0790 1.0937 1.1082 1.1227 1.1371 1.1614 U 20 1.0301 1.0361 1.0500 1.0648 1.0795 1.0942 1.1067 1.1232 1.1876 1.1619 33 1.0206 1.0356 1.0504 1.0653 1.0800 1.0946 1.1093 1.1237 1.1381 1.1634 S8 34 1.0211 1.0361 1.0609 1.0658 1.0605 1.0951 1.1097 1.1242 1.1386 1.1529 S4 26 1.0216 1.0866 1.0614 1.0662 1.0810 1.0956 1.1102 1.1246 1.1390 1.1683 38 1.0221 1.0870 1.0619 1.0667 1.0615 1.0961 1.1107 1.1351 1.1395 1.1538 36 80 1.0236 1.0876 1.0534 1.0672 1.0620 1.0966 1.1111 1.1366 1.1400 1.1643 80 83 1.0231 1.0380 1.0529 1.0677 1.0824 1.0971 1.1116 1.1261 1.1406 1.1548 83 84 1.02S6 1.0385 1.0534 1.0682 1.0829 1.0976 1.1121 1.1266 1.1409 1.1562 84 86 1.0241 1.0390 1.0539 1.0687 1.0834 1.0980 1.1126 1.1271 1.1414 1.1667 86 88 1.0246 1.0896 1.0644 1.0692 1.0839 1.0985 1.1131 1.1275 1.1419 1.1662 Si 40 1.0251 1.0400 1.0648 1.0697 1.0644 1.0990 1.1136 1.1280 1.1434 1.1567 46 43 1.0256 1.0406 1.0554 1.0702 1.0649 1.0995 1.1140 1.1285 1.1439 1.1571 4S 44 1.0361 1.0410 1.0659 1.0707 1.0654 1.1000 1.1145 1.1290 1.1433 1.1576 44 46 1.0266 1.0416 10664 1.0712 1.0859 1.1006 1.1150 1.1295 1.1438 1.1681 4ft 48 1.0271 1.0420 1.0568 1.0717 1.0863 1.1010 1.1165 1.1299 1.1443 1.1586 4ft 60 1.0376 1.0425 1.0574 1.0721 1.0868 1.1014 1.1160 1.1304 1.1448 1.1690 60 63 1.0281 1.0430 1.0579 1.0726 1.0673 1.1019 1.1165 1.1309 1.1453 1.1506 63 64 1.0286 1.0435 1.0584 1.0781 1.0678 1.1024 1.1169 1.1314 1.1467 1.1600 64 66 1.0391 1.0440 1.0589 1.0736 1.0683 1.1029 1.1174 1.1319 1.1462 1.1606 M 16 1.0396 1.0445 1.0598 L0741 1.0888 1.1034 1.1179 1.1833 1.1467 1.1600 6B •0 1.0801 1.0460 1.0666 1.0746 •- 1.0693 1.1039 1.1184 1.1828 1.1473 l.ljSU •ft TABLE OF CHORDS. 147 Table of Cbovda, in parte of a rad 1 } i for protractlnfT— -Continued M. 71° TSB® 7SO 740 750 7«o 770 78° 7V> 80° ML 0' 1.1614 1.1756 1.1896 1.2036 1.2175 1.2313 1.2450 1.9586 1.2722 1.2856 » •i 1.1619 1.1700 1.1901 1.2041 1.2180 1.2318 1.2455 1.2691 1.27a 1.2860 3 i 1.1624 1.17« 1.1906 1.2046 1.2184 1.2322 1.2459 1.2505 1.2731 1.2865 4 • 1.1628 1.1770 1.1910 1.2050 1.2188 1.2327 1.2464 1.2600 1.2735 1.2869 « • 1.163S 1.1775 1.1916 1.2056 1.2194 1.2882 1.2468 1.2604 1.2740 1.2874 8 10 1.1638 1.1642 L1770 1.1704 1.1920 1.9060 1.2198 1.2886 1.2473 1.2609 1.2744 1.2878 10 u 1.1934 1.9004 1.3303 1.2841 1.3478 1.M14 1.2748 1.2882 IS 14 1.1647 1.1T80 1.1939 1.2060 1.2208 1.2346 1.24«i 1.2618 1.2763 1.2887 14 U 1.1663 1.170S 1.19S4 1.3073 1.2212 1.2360 1.24H7 1.2623 1.2757 1.2891 10 18 i.nsT 1.1706 1.1IS8 1.3078 1.2217 1.3364 1.2491 1.9627 1.2763 1.2896 18 30 1.1661 i.isa 1.1948 1.3086 1.9991 1.3869 1.9496 1.9I83 1.27M 1.2900 n n 1.1666 1.1807 1.1949 1.3067 1.3236 1.2364 1.2500 1.2636 1.2771 1.2905 39 34 1.1671 1.1813 l.MM 1.2003 1.9381 1.3368 1.2506 1.9641 1.2776 1.2909 84 3S 1.1676 1.1817 1.1W7 1.9007 1.3236 1.2873 1.2600 1.3646 1.3780 l.»I4 a a 1.1680 lun 11063 1.2101 1.8340 1.2377 1.2514 1.2660 1.2784 l.»18 a » 1.U86 1.18M 1.1866 1.3106 1.32a 1.2389 1.K18 1.9664 1.1789 1.2933 so n l.ltM 1.1BS1 1.1971 1.3111 1.93a 1.2886 1.2523 1.2659 1.3798 1.2937 88 M. 1.1604 LUM 1.1976 1.3116 1.2254 1.2891 1.2528 1.2663 1.2798 1.2931 84 M LU99 1.1840 1.1980 1.2120 1.2268 1.2896 1.2539 1.2668 1.2802 1.2936 M M L1T04 1.1846 1.1986 1.2124 1.22tt 1.2400 1.2687 l.a72 1.2807 1.29a M 40 i.no0 1.1860 1.1990 1.2129 1.3967 1.2406 1.2641 i.an 1.2811 1.2945 a 43 LHU 1.1864 1.1994 1.2134 1.2272 1.2409 1.2546 1.9B8I i.aie 1.2949 48 44 1.1718 1.1659 1.1900 1.2138 1.2277 1.2414 1.2550 1.2686 1.2820 1.2954 44 a Ln23 1.1864 1.9004 1.2143 1.2281 1.2418 1.2555 1.2690 1.2825 1.2958 a a 1.1727 1.1868 1.3006 1.2148 1.2286 1.2428 1.2559 1.2695 I.28» 1.2962 a w 1.17S2 1.187S 1.201S 1.2152 1.2290 1.2428 1.2564 1.2690 1.3838 1.2967 60 it 1.1TS7 1.1878 I.90I8 1.2157 1.2296 1.2432 1.3668 1.2704 1.2838 1.2971 68 u 1.174S 1.188t 1.9022 1.2161 1.2299 1.2437 1.2573 1.2706 1.3842 1.2976 64 M 1.1746 1.1887 1.9037 1.2166 1.3304 1.2441 1.2577 1.2713 1.2847 1.2980 66 M 1.1761 1.1803 l.aOS2 1.2171 1.2309 1.2446 1.2582 1.2717 1.2861 1.2985 68 m L1756 1.1896 1.3066 1.2176 1.3311 1.2450 1.2586 1.2722 1.2866 1.2989 M 0' 9 4 • 8 10 18 14 16 18 80 "m" 84 a a M 38 84 48 44 a a w IT H 16 «1« .3903 1. 1. 1. 1.9008 1.M07 1.9011 1.8015 1.3030 1.S024 i.soa i.soa i.9oa 1.3048 i.soa 1.8061 1.3056 1.3060 1.3064 1.3068 1.8073 1.S0T7 1.3068 1.8086 1.3000 1.! 1. 1.3104 1.3106 1.8118 1.1117 1.S181 1.8181 1.3ta 1.3ia 1.3134 1.3ia i.8ia 1.3147 1.3158 1.3156 1.3161 i.3ia i.3ia 1.9174 1.3178 IJia 1J187 1.3191 1.S1M 1.3800 1.3904 1. 1.8213 1.3318 1.8828 1.33a 1.8881 1.3336 1.3239 1.9844 1.83a i.3a8 i.8a9 i.3a7 i.3ai 1.32tt 1.8270 1.3274 1.3379 1.3388 1.3287 1.3293 Lsao 1.3800 1.8306 1.38a l.ai8 l.ttl8 l.a28 i.aa 1.3W1 1J886 1. 1.044 1.83a 1.3868 1.3367 1.3M1 1.3865 1.3370 l.a74 1.3878 1.8383 840 1.1 1.3387 i.8ai 18896 1.3400 1.3404 1.3409 1.3413 1.3417 1.3481 1.84a 1.3430 1.8484 l.S4a i.Ma 1.3U7 1.3468 1.84S6 1.8460 1.8466 1. 1.8473 1.3477 1.3a8 1.34a 1.3490 1.3486 1.3499 1.8608 1.3508 l.ai2 85^ 1.3612 1.3516 1.85W 1.3525 l.a29 1.3533 1.3538 1.3542 1.3546 1.3560 1.8665 1.85a 1.3663 1.3667 l.a72 l.tt76 1.8580 1.8586 1.85a 1.86a 1.3697 1.8a2 1.3606 1.K10 1.3614 1.3619 1.8623 1.3627 1.3631 1.3636 1.3640 8«° 1.86W 1.3644 l.S6a 1.3668 1.3657 1.3M1 1.3665 1.M70 l.a74 1.M78 1.3682 1.3687 Lsai 1.36M IJMW 1.3704 1.37a 1.3712 1.3716 1.8721 1.87a 1.37a 1..H73S 1..17a 1.3742 1.37a 1.8750 1.3754 1.37a 1.3783 1.3767 870 880 880 1.8767 1.88a i.ai8 1.3771 1.3897 1.4028 1.3776 1.3902 i.4oa 1.37a 1.39M 1.4031 1.3784 i.aio 1.4035 1.87a i.ai4 i.4oa 1.8792 i.ai8 1.4043 1.3797 1.3922 1.4047 1.3801 i.a27 1.4051 1.3806 i.3ai 1.4055 1.8800 1.3966 i.4oa i.ais i.aso 1.4064 i.ai8 1.3943 1.4068 i.Mn 1.3947 1.4072 i.a26 1.3952 1.4076 1.8830 1.3966 1.4080 1.3»4 l.S9a 1.4084 1.38a 1.3964 1.4089 1.38a 1.39a 1.4O03 1.8847 l.a72 1.4097 1.3861 l.a77 1.4101 1.3855 i.3ai 1.4105 1.3860 1.3985 1.4109 1.8864 1.38a 1.4113 1.3868 1.3993 1.4U7 1.3872 i.sa7 1.4122 l.a76 1.4002 i.4ia 1.3881 1.4006 1.41M 1.3885 1.4010 1.4134 1.3889 1.4014 i.4ia 1.8808 1.4018 1.4148 0' 8 4 6 8 10 18 14 16 16 M 22 24 a a a 88 34 M a 40 48 44 a a a la 54 a a a F0LYG0N8. m. HfiuaH. BipUoam. Dctiun. nsBlar. Of coarvf tfin aambn af poljfOQK U IbBoLH. ' T»I>I« orBeroluP Polygons, X ■.itTk- *^ar M tiiugla. Deongon, UndKBgon. J .«.„ Ji77»M eo° ISO" Ii;i96152 :»so6si 108° isn° H7° is.sese' 180° 90" 60° .,.».„ 40° 32°43.«3M' ^^rr^ij'/x'K.rf^ES'JKS^ ''°*'' "' "■ ■«•■ • 'X p^ ' *"" nx.«,^«,l. S« Bf lawriar astf «, ■ b <!. m, ar mar poljB«. respUr « In. ■■Ur = iaa°x TBIASTOIiES. *A »/K /K* h\i> IV^ E 7 f\ /^ \/r \i^ ^ ^ r\ \ i<B^; <c IbDH biTlii) itoml^t TBIANaLES. 149 ^•o find area, baTlnflr one aide and tbe A angles at its ends. Add the t anglM together; take the sam from lW>f the rem will be the angle opp the given ilde. Find the nat BUte of tfals angle ; also find the nat ainea of the other angles, and mult them together. Then ai the nat alne of the alngle angle, ia to the prod of the nat sinei of the other 2 anglea, ao ia the tfumre of the given side to tUnM* tbe reqd area. To find area, bavlngr two sldes^ and tbe Inelnded ang^le* Ifnlt together tbe two eidee, and the nat sine of the tnoloded angle ; dlr by 2. Ez.~8ides 650 ft and 980 ft; included angle W* 20'. By the table we find the nat tine .9856 1 therefore* ^j s= 397988.6 aqnare ft area. To find area^ baTlnc tbe tbree ang^les and tbe o perp belybt, a b. Find tbe nat sines of the three angles ; mult together the sines of the anglae d and : dlT the sine of the angle h by tbe prod ; mult the qnot by the squari of the perp height a & ; dlr by 2. To find any side, as tf o> baTing^ tbe tbree angles, d, h and Of and tbe area. (Sine of d X rine of o) | sine of b 1 1 twlee the area t aware of d o. The perp height «fmm eqvilatenU irlansle is eqaal to one aide X .860025. Hence one of its Bidea is equal to the perp height div by .8660-25 or to perp height X 1.1M7. Or, to find £ at4«i BHdt the sq rt of its area by 1.61967. The side of an equilateral triangle, mult by .658037 = side of* I of ue same area } or mult by .742517 it gives the diam of a eircle of the same area. n C a B The following apply to any plane triangle, whether oblique or right-angled S. The three angles amount to 180°, or two right angles. 9l Any Mcterior angle, as A C n, is equal to the two interior and opposite aoes, A and B. C The greater side is opposite the greater angle. 4i Tha sides are as the sines of tbe,opposite angles. Thus, the side a is to the Mm 6 as the sine of A is to the sine of B. ik If any angle as s be biseeted by a line • o, tbe two parts me, o n of thfi eppaeite side m n will be to eaeh other as the other two aides »m, an; •r, »•:« n::s m:s n. 4L If ttnes Iw drawn tnm eaoh angle r • < to the ~ eenter of tbe onposite side, they will eross eaoh other at one punt, a, and the abort part of each of the lines will be tbe third part of the whole line. Alao, « is the eea of sntT of the triangle. T. If lihoa be drawn bisecting the three angles, they will meet at a point perpendionlarly equidistant from eaeh aide, and consequentlj the centev ai^ V — a^ f of tke sreateet etr<de that ean be drawn in the triangle. •^ ^^* 8. If a line « n be drawn parallel to any side e a, «iie two trianglM ran^re€i, will be similar. •. To divide any triangle aer into two equal parts by a line s n parallel to any en* of its sides c a. On either one of the other aides, as a r, as « diam, dsMrIb* a samiairele a o r/ and find its middle e. From r (opposite e a), with radiusre, deaerilM theareon. From n draw n s. par- Q allel to e a. y\ 10. To And the grcatast parallelogram that ean be y^ \ drawn in any jriven triangle onh. Bisect the tbree sidea at a e s, and join <V^ jf o e> « «i a 0* Then either aehe, aeeo, or a ean, eaoh equal to half the ^\ y^\ triangle, will be tbe reqd parallelogram. Any of these parallelograms can ^ \^ \ plainly t>e converted into a rectangle of equal area, and the greatest that ean be % t 1% drawn in the triangle. * lOX. If a line a e bisects any two sides o i, o n, of a triangle, it will be par* allel to the third aide n b, and half as long as it. 11. To find the greatest square that ean \m drawn in any triangle a ae r. From an angle as a draw a perp a n to the opposite side «r, and find its length. Then 9 n, or a side v I of the square will = . BeBU~*If the triangle la such that two or three suoh perps ean be drawn, thM two or three equal squares may be found. an r ;\5(\«5-«;''t.^- 150 FLANE TBIGOKOMETBT. Bifflit-aiiirle^ Tri»iiirlefl« 4.U the foregoing appw also to right-angled triangles : hat what foUew the right angle A, and the othen B and C ; and eali oppoelte to them a, i, and e. Then Is ft = a X Sine B = aXOoeC = eXCotOs«X Tana S, cs«XSineO = aXGoaB-=»XTangO. them only. >e sidM nwMtlfelf e h Also Sine of = -; OoeO = ~/ a Tang I TangOi h § 5 And Sine ttrBs-zOoeBs-/ Tang B = j. - ,. -w ^ _.\»* **■;••' 4. <>''**° = ': CoiA=0. Tang A rrlndnHy. SeeAstalBl^. 1* If from the right angle o a line o w be drawn perp to the hypothenuie or long side * «, then the two small triangles owh.owg, and the large one oka. will be similar. Or Mr : 10 : : IP o : w A; and gwXwhszwoi. t. A line drawn from the right angle to the oeater of the long side will be hair as long as sa>d side. 8. If on the three sides oh, og, gh me draw three sqnarae (, u, m, or three oireles, or triangles, or any other three figs that mm siadlar, thtp the area of the largest one is eq^oal to the sum of the areas of the (wo othfsn. 4* In a triangle whose sides are as S, 4, and 6 Cas are thoee of the Irt* angle ABC), the angles are rery approximately MP; 5tor4S.nw; nad 36° 52' 11.62'/. Their Sines, 1. ; .8} and .6. Their Tangs, inOnitj ; l.SaM : and .73. ft. One whose sides are as 7, 7, and 9.9, has rery appror one angle of 90» and two «r W* eaoh, near enoogh for all prsctical purposes. ' «\ h ^^ •■ ;\ ^ u /. 9 ►-•- PLANE TEIGONOMETEY. P&Aira trigonometry teaohee how to find certain unknown parts of plane, or straight • aldnd M> •ni^, by means of other parts which are known ; and thus enables us to measure inaooessiUe dla> tanoes, Ao. A triangle oondsu of six parts, namely, three sides, and three ancles ; and If we know any three of theee. (except the three angles, and in the ambiguous case under "Case S,") we can flad the other three. The following four oases include the whole sulyeot ; the student shon^i oommlt then le memory. ^ ■ ' C pH<» va Case 1. HaTlna: any two angles, and one side^ ^ **' to find the oilier sides and an^le. Add the two angles together ; and subtract their sum from 180^; the rem •vill be the third angle. And for tbe sides, as Sine of the angle . Sine of the angle . , ^„ .^ . .^^ ,,j- opp the given side • opp the reqd side • • «»»•«» "<» • '^l*^ •»<»* Use the tide thus found, as the given one ; and in the same manner And Ihe third side. Case 2. HaTlngr two sides, ba,ae, Vi^ X, and the ani^le a be, opposite tooneof tiiem, to find the other side and angles. Side a c opp The other Sine of the Sine of angle hdaor the given an* I given side I * given angle I icaopposite the other ^tr gle a b c ba ab e given side b a. Having fonnd the sine, take out the oorreeponding angle from the labia af nat sines, but, in doing so, if the side • e opp the given aagto Is shorter than the other given side b a, bear in mind that an angle and Its snp« plement have the same sine. Thus, in Fig X, the sine, ai found above, is opp the angle & e a in the table. But a e, if sJtortsr than b a, can evidently be laid off in the opp direction, a d, in which case I «I • is the sappltment of ( c s. If a c is as long as, or longer than, b a, there can be no doubt ; for In that i It oannot be drawn toward b, but only toward n, and the angle A « « will ftMind ec onoe in th« table, opp the sine as fonnd abovib PLJLKE TRIOONOMETBT. 161 When th« two angtei, ahe,heo, have been (band, find th* remalalnK side hj Cue 1* IW the remaining angle, hae, add together the angle abc flrtt given, and the one, i e s. M abOTO. Oedoet their aam from 180<*. Case 3. KaTlniT ^wo sides, and the an^le included between tbem. Take the angle trem 180''; the rem will be the sum of (he two uDknown angles. Dlr thU sum bf t; and find the nat tang of the qaou Then as The »m of the . mw«|_ ^nr . • Tang of half the earn of . Tang of half two giTon sides • ^""■i^«"' . . the two unknown angles • their dlff. Take flrem the table of nat tang, the angle opposite this last tang. Add this angle to the half sum •f the two unknown angles, and it will give the angle opp the longest given side ; and subtraot it firem the same half sum, for the angle opp the shortest given side. Having thus found the angles, lad the third side by Case 1. As a praetieal example of the use of Case S, we oan asoertain the dist n m across a deep pond, by measuring two lines n o and mo; and the angle n e m. From these data we may calculate nm ; or by drawing the two sides, and the angle on paper, by a soale, we can afterward measure » m ea •he drawing. €ase 4. Kaviuir ^b® tbree sides* lb And tte three aaglM; upon one side • ( as a base, draw (or suppose to be drawn) a perp eg tnm the oppoaita angle c Find the diff between the other two sides, a c and c b ; also theLr sum. Then, as Sum of the , . Diff of other . Diff of the two other two sides • • two sides • parts ag and bg, of the base. The base Add half this diff of the parU, to JuU/ the base a &; the sum will be the longest part ag; which taken tnm the whole base, gives the shortest part g 6. By this means w« get in each of the small tri- angles a eg and egb, two sides, (namely, a c and a gi and c b and gb;) and an angle (namely, the right angle cga,megb) opposite to one of the given sides. Therefore, use Case 2 for flnding the a and e. When that is done, take their sum fMm WV>, tor the angle • c *. Or* Sd ■§•<« t call kalf the sum of the three sides, si and call the two sides which form either angle, mt and m. Then the nat sine of hiOf that angle wUl be equal to \ /C — *»)XJs -«> Fiir.i. Tig.fi. Ex. 1. To find tbe dlst from a to an Inae* eesslble objeet e. Measure a line ab; and from its ends measure the angles eab and eba. Thus having found one side and two angles of the triangle a > c, ealenlate a c by means of Case 1. Or if extreme aqonracy is not read, draw the line a I on paper to any convenient scale ; then by means of a protraeter lay off the angles c ab,eba; and draw a e and eb; thaa measure • e bj the same scale. Ex. 3. To find the helgrli^ of a veffioal objeet, n a. Place the instmmeni for measnrlng eagles, at any oenve. nlent spot o ; also meas the distea ; orif oa cannot be actually measd in consequence of some obstacle, calculate it by the same process as a e in Fig 1. Thm, first directing the instra< ment horizon tally,* as o s, measure the angle of depreesioa, to a, say liP ; also the angles o n, say 80°. These two anises added together, give the angle a on, 42°. Kow. in the small triangle o « a we have the angle o « a equal to 90O, because a n is vert, and o a hor ; and ninoe the three angles of any triangle are equal to 180p, if we subtract the angles ota <90O), and s e « (12°) from 180°. the rem (78°) will be the angle o a « or o a «. Therefore, in the triangle one, we have one side o a; and twe angles a on, and o a «i, to calculate tbe side a n by Case 1. i dlsts on sloping ^ronnd must be measured hor- Ison tally. The graduated hpr clrole of the instrument evideafly meaa> fr-rj *-'-*-'*TtP \ ures the angle between two ob}eets horl 1 :^- /\r \ tonully, no matter bow much hlirher one — ^i^/. \ of them may be than the othf>r ; one pes* haps requiring the telescope of the iastra* ment to be directed upward toward it; and the other downward. If. thereforek the sides of trianglen lying upon sloping C \ ground, are not also meiuid hor, there can be no accordance between the two. Tba« PLANE TBIOONOMETKY. PLANE TRIGONOMBTBY. 153 its sngle iftt of incUuftUoa with the horison foand u before i in whioh cue the dut a n is caloolated. Or if the vert height c n is sought, the point o may first be found bj sighting upward along a plumb-line held abore the head. Ex. 3. To iind tlie approximate belifht^ 9 00; of a moantain. Of whioh, perhaps, only the very summit, x, is visible abova interposing forests, or other obstacles ; but the dist. mi, of whioh is known. In this case, first direct the instrument hor, as m k; and then meainre the anglb i m x. Then in the triangle i m z we have one ^de mi: the measd angle <ms, and the angle mix (90°), to find ir by Case 1. But to this » z we must add 1 0, equal to the height y m of the -instrument above the ground; and also o «. Now, o s is apparently due entirelv to the curvature of the earth, whioh is equal to very nearly 8 ins, or .667 ft in one mile : and iaoreases aa the squares of the dists; being 4 times 8 ins in 2 miles ; 9 times 8 ins is S mflM, ito. Bat thts It MBMVhat dinlnlshed bv the refraotion of the atmosphere ; whioh variee with temperature, moisture, &o ; but alwaya teaos to make the obieet x appear higher than it ■otoallj is. At an average, this deoeptive elevation amovmts to aboat-=-th part of the enrvatuie of the earth ; and like the latter, it varies with the ■qnarea of the dists. Consequently if we subtract -=- part from 8 ins, or .667 ft, we have at onoe the combined effect of curvature and reft-action for one mile, eqaal to 6.867 Ins, or .5714 ft; and for other dists, as shown in the following table, by the UM of which we avoid the neoessity of making »q}arate allowances for curvature and refraction. Table of allowances to be added for carvature of tbe eartb ; and for refraction ; combined. Fig.7. Dist. Allow. Dist. Allow. Dist. AUow. Dist. Allow. inyarda. feet. in miles. feet. in miles. feet. in milee. feet. 100 .002 .036 6 20.6 20 229 150 .004 xt .143 7 28.0 22 277 200 .007 y^ .321 8 86.6 25 357 800 .017 1 .572 9 46.3 30 614 400 .080 11^ .803 10 57.2 35 700 500 .046 \Xc 1.29 11 69.2 40 916 600 .066 ' 1% 1.75 12 82.3 45 1168 700 .090 2 2.29 13 96.6 60 1429 800 .118 2H 3.67 14 112 55 1729 goo .140 3 5.14 15 129 60 2058 1000 .185 3K 7.00 16 140 70 2801 1200 .266 4 9.15 17 165 80 3659 1500 .415 4^ 11.6 18 185 90 4631 2000 .738 6 14.3 19 206 100 5717 , If a person whose eye is 5.1i ft, or 112 ft above the sea. sees an object just at the sea'b korixoB, that object will be about 3 miles, or 14 mites distant from him. A borlBOntal line is not a leirel one, for a straight line cannot be a level one. The carve of the earth, as exemplified in an expanse of quiet water. Is level. In Fig T, If we suppoee tiie enrved line tp»gio represent the sarfaoe of the sea, then tbe points ty » and g aae on a level with each other. They need not be equidistant ft-om the center of the earth, for the sea at the poles is about IS miles nearer it than at the equator ; yet its surface is everywhere on a level. Up. and down, refer to sea level. IjCTcI means parallel to the curvature of the sea ; and boriaontal means tangential to a level. Ex. 4. If tbe inaccessible irert beiffbt e d, Flip 8, A $o lUuated thai v>* cannot reach it at aU, then place the instrument for measuring angles, at any oonveoient spot n ; and in range between n and d, plant two staffs, whose tops o and i shall range praeiaely with n, though they need not be on the same level or hor line with it. Measure n o : also from n meaaore the angles on d and one. Then move the instrument to the precise spot previously • — I — ' ■ — ' ~i which he had no idea. For allowance for curvature and refraction see above Table. A triangri® wbose sides are as 3, 4, and 5, is right angled ; and one 'hose sides are as 7 : 7 ; and 9. 9 ; eontains 1 right angle ; and 2 angles of iffi each. At it is fre* <|eently' necessary to lay down angles of 45° and 9QP on the ground, these proportions may be used for the purpose, by shaping a portion of a tape-line or chain into suoo a triangle, and driving a stake at eaehani^ 154 PLANE TBIQOKOMETBY. ipted by tbe top o of the lUff; and trvm o mearan th« aaftat <• 4 kdA40c tract tbe angle < o e ftom 180° ; tbe rem will be tbe angle e • n. Cenaeqaent- ly in tbe triangle nee, we bare one side n o, and two angles, «no and e o n, to find by Case 1 tbe aide o e. Again, take tbe angle iod from 180° ; tbe remainder will be tbe angle n o d, ao that in tbe triangle dno we bare one side n o, and tbe two angle* dno and » d, to find br Case 1 tbe tide od. Finally, in tbe triangle cod, we hare two aides CO and od, and tbcir included angle cod, to find d, tbe reqd rerfe bfligbt. Figr.a. Figr.9. Jttd were in a valley, or on a bill, and tbe obserrationi reqd to be made tnm either hlgta«r •r lower groond, tbe operation would be precisely the same. £x. 5. See Sx 10. To find (be dlst ao. Tig 9, betwe«M two oiitirely inaceemiMe oliJecUi, Meaiwre asldenm; at n measure the angles a nm and onm: also at mnMasore the angles o mm, and • M fk This being done, we have in tbe triangle anm, one side n m, Fig 9, and tbe anglee •«»••, and nma; benoe, br Case 1, we can calculate the side an. _ ▲gain, in tbe triangle o m n we have one side n m, and P the two angles omn, and mno; hence, by Case 1, we can •alenlate the side n e. This being done, we have in the triangle ano, two sides an, and n o ; and their included angle a n o ; hence, br Case 8, we can oalcnlate tbe side ao, which is the reqd dist. It Is plain that in this manner we may obtain also the position or direction of tbe inacces- sible line a o ; for we ean calculate tbe angle nao; and can therefrom deduce that of ao; and thus be enabled to ran a line parallel to it, if required. By drawing n m on pa- T!itr If) per bT a scale, and laying down the four measd angles, 'iK- -lu* Che dist a • may be measd upon tbe drawing bj tbe same scale. If the position of the inaccessible dist c n. Fig 10, be such that we can place a stake p in line with it, we may proceed thus : Place the instrument at any suitable point «, and take tbe angles ptc and cnn. Also find the angle eps, and measure tbe distps. Then In the triangle p t c find « e by Case 1 ; again, the exterior angle n e «, being equal to tbe two interior and opposite angles cp «, and j> « c, we have in the triangle eon^ one side and two angle* to find e n by Case 1. Ex. 6. To flnd a dlst ah, Flgr II9 of whieh the ends only Mre accessible. From a and 6, measure any two lines a e, & c meeting at e ; also measure the angle a eh. Then in the triangle aft c we have two sides, and tbe included angle, to find the third side a 6 by Case S. Ex. 7. To And tbe vert beigbt o nt^ of a FfflT- U. bill, above a i^iven point i. Flaoe the instrument at i ; measure a m. Directing the instrument hor, as an, take tbe angle nam. Then, since a n m is 9P Fig 12, we bare one side a m, and two angles, nam and a n m, to find n m by Case 1. Add n o, equal to a <, the height of the instrument. Also, if tbe bill Is a long one, add for cnrrature of the earth, and for reh-action, as explained in Example 3, Fig 7. Or tbe instrument may be plaoed at the top of the bill ; and an angle of depression measured ; instead of tbe angle of elevation nam. Bxu. 1. It is plain, that if tbe height o m be previously known, and we wish to ascertain tbe dist from its Bum- TiMir 72 mit m to any point i, the same measurement as before, * ' * of the ancle nam, will enable us to calculate a m by Case 1. So in Ex. 2, if the height na be known, the angles measd in that example, wfU enable «k to compute the dist a ; so also In Figs S, 4, 6, and 7 ; La all of which tbe process is so plain as to raqnire no further explanation. Bbm. 2. Tbe height of a vert object by UieanS Of its SbadOW. Plant one end of a straight stick vert in the ground ; and measure ts shadow ; also measure tbe length of tbe shadow of the object. Then, as the length of the shadow of the stick is to tbe length of the stick abovt PLANE TRIGONOMETRY. 156 gnvaA, lo to tlM toagtli of IIm ahadov of tht ol^oot, to its helgbt moBk bo eqvftUy iaolinod. If the ob|}«et It inoHiMd, the itiek xu 1 my Rem. 8. Or tb« beiffbtof a irert object mn^ '^£r* Ji^H , Fig l^^whoee distance r m is known, may be found by ZJ^ Iti rellection in a vessel of water, or in a piece of .'"y^ looking f iUB plaoed perteotW borixontal at r ; fttr •■ r als to tlM balglUI [^ a < of the eye above the refliMtor r, w to r m to^^ ^ 13*1 <» "i^Xd. the height m n of the ol^eot above r. Rem. 4. Or n pl»nied pole, or a rod held yert staod at a proper dlit baok tnm It, and keeping the ^ee eteadj, let marks made at o and e, where the lines of sight i n aad iae strifea tht rod. Then ieistoeo, soisimtomn. »r let c. Fig 12K by an assistant. T "•"fir.. Pifir.l2> -ksbe I 6L-->* sn •m^.Mex::^ — ' 1, flff.lS. The following examples may be regarded as tabetitntei for strict trigonome- try : and will at times be nsefhl. in ease a table of sines, fto, to not at hand for making trigenometrieal ealoulations. Ex. 8. To And tbe dlst a h^ of wbicb one end only Is accesftlble. Drive a stake at any eonvenient point a ; ft!>om a lay off any angle i a e. In the line « e, at any coDvenient poini c, drive a stake ; and fh>m c lay off an angle acd, eqaal to the angle b ac. In the line e d. at any oonrenient point, as dt drive' a stake. Then, standing at d, and looking at h, plaoe a stake o in raoft with d h ; and at tbe name time in the line a c. Measure ao,oc, and cd\ from the principle of similar triangles, as o e \ e d I X a o X Ah. Fiff.lfi. Or tbnss VIg 14, » A being tbe dtot, plaoe a stake at n ; and lay off tbe angle b n m VP. At any convenient dlst n tn, place a stake m. Make the angle it m y =90° ; and plaoe a stake at y, in range with h n. Measure n y and n m ; then, fh>m tht principle of similar trianglea, as n]f:tt»»t:nn»:nA. Or tbns. Fig 14. Lay off the angle hnm=^ 90°, placing a stake m, at any ooaventent dtot n m. Measure n m. Also measure the angle n m A. Find nat tang of » m A by Table Mult thto nat tang by n «. The prod will ben A. Or tbns. Lay oflT angle A n m » 90^. From m measure the angle n m A, and lay off angle n m y equal to tt, plaolag a ttnkt at y la raagt with A n. Then to n y = n A. Or tbns, without measurlnir any ang^le ; t « being the dlst. Make it v of any convenient length, in range with ( u. Measure any v o ; and o % equal to It, in range. Measure u o ; and «« equal to it in range. Plaoe a stake s in range with both X y, and ( o. Then will y jt be both equal to t u, and parallel to it. Or tbna, witbont meiisarlnir ^ny anffle. Drive two stakes I and «, in range with the object s. From ( lay off any eonvenient diet t x, in any direction. From « lay off w w parallel to < s, placing 10 in range with z <. Make « v equal to ( «. Measure w •, v s, and X t. Then, as vpifxvaBx xett xt0. Or tbiifl. At a lay off angle oac » S^ 48^ Lay v « off 00 at right angles to ao. Measure oe. Then _, • _ 00 » lOoe, too long only 1 part in 935.6, or 5.643 feet Ylg. 16, in a mile, or .1069 foot (full U Inches) in 100 feet. PLANE TRIGONOMETRY. Ex. lO. See Bx. 4. To And U* •ntlreir iDMcewlble dlBt — ------ lu dlr«ei FABALLELOGBAHB. 167 Square. PARAI«IiEI.OOBAMB. Rectangle. Bhombus. Rhomboid. ]^""*--, 8 A PAKALLELOORAX is any figure of four straight sides, the opposite ones of wbtch are parallel. There are bat four, as in the above figs. l%e rhombus, lilce the rhom- bf^odron. Fig 3, p 106, is sometimes called ** rhomb." In the square and rhombus all the foar sides are equal ; In the rectangle and rhomboid only the opposite ones are equal. In any parallelogram the four angles amount to four right angles, or 360^ ; and any two diagonaUy opposite angles are equal to each other ; hence, having one angle given, the other three can readily be found. In a square, or a rhombus, a diag divides each of two angles into two equal parts ; bat in the two other parallel- ograms it does not. To flnd tbe area of any parallelosram. Mnltlply any ilde, m 8, bv the perp height, or dUt p to ihs opposite aide. Ovk multiply tocathar two sMm and nmt alne of their inoladad aagla. The 4Smm a b of any s^aare is equal to one side molt by 1.41421 ; and a side is eqaal to diacooal ^^^31 ; er, to diag mult by .707107. '31ie side ef a B««are eqval tn area to a aUrem elrele» is equal to dSam X .89Stn. Tke dide of file sreateet aoaare, tMat can h*in»erib«d in •^MM HreU, is eqnal todlaoi X .707107. Tha side of a sanara molt by 1.51967 gives the aide of an equi- lateral trtanue of the same area. All paraUelosraau as a. aad C, whiek littve eq^al baaea» a c, and eqnal psrp heights n e, haTe also equal areas ; and the area of ea«h Is twice tbat of a tri> angle baring the same base, and perp height. The area of a ■raare laserlbed In a elrele i« equal to twioe tbe square of the In every parallelosranM the 4 squares drawn oh its sides have a united area «qu^ to that of tha tvo squares drawn on iu 2 diags. If a Inrcer aqnare be drawn on tha diag a 6 of a a mailer square, ite area will be twioe tbat of said smaller square. Either dlas of any parallelosram tfridea IMato two eqnal triangles, and the S diags div it inte 4 triangles of eonal areas. The two ly MiraUelo|trani divide each other Into two equal parte. Any Une drawn throach iter of a 41aC divides the parallelogram into two equal parte. 1.— The urea of any fiff whatever as B that la eneloeed bylbnr atralcht . __j may be found thus : Mult together the two diags mm,nb: and the nat sine of tbe least angle «oi;ori»e«H fbnnad by their interseotion. Div the prodnet by 3. This Is useful Id land surveying, whan ohataelaa, as is often tha aaaa^ make it dilBauU to measara tha sides of the flg or flald ; while Ik may be easy to measure the diags ; and after finding their point of interseotion o, to measure the re* qnbed angle. Bnt If the flgr 1* to be drawn, the porta o «, o 6, o n, o m of the diags must also be measd. Boh. 9.— The sidee of a parallelogram, trlani^e, and many other !«■ may he Ibnnd, when only the area an4 aanlea are ftven, thus : Assume some partloular one of ite •ides to be of tbe length 1 ; and oaleulaw what ite area would be if that were the ease. Then as the sq rt of the area thus found is to this side 1, so Is the sq rt of the aotual given area, to the oorre* •pondtaig aotoal aide of the fig. On a iriTen line tcr0e,to ^vww a M|aare^ From w and x, with red ts x, describe the aros xrp and to r e. From their intersection r, and with rad equal to H of w«. deaeribe M»». From ts and s draw tvn and 0m tangential to «s«, ending at the other aros j Join n «i. the 158 TRAPEZOIDS AND TBAPBZIUM8. TBAPEZOmS. fi t m n a « e at A trupmM menm,l» Miy flfwe with tour ttrmighl ildM, only two of ▼bioh, m me mad » *, art paraUd. To And tbe area of any trapoaold. Add toffBthar the two panlM tidoaf a « aad m n; malt ika aaai by tha parp diat • i tliam ; div Um prod bj S. Saa tha faUowiog mloa far trapaaiaKB, whlah ara all aqnally totoapasolda; alM laa BaoMrlu aftar Parallalofraau. TRAPCZIUMS. A trapaaiam a & e o, ia any flg with foor atralght ildaa, of which no two ara parallal. To find the area of any trapoBlnm, taaTlnir griven tbe diac 5o, or a e, between eliber pair of opposite an^lee; and alia the two perpe, n, ft, fW>ni the other two anirlee. Add togathar thaae two parpo ; molt the som by the diag; dlT the prod by i. SlaTiniT the fonr sides i and either pair of opposite anirlcs* mm a be, a o eg or bao, and beo, Conaider the trapeiiam aa diridad into two trlanglaa, in aaeh of whieh ara givaa two lidae and tte Inoladed ancle. Find tbe area of eaoh of theae triangfea as direoted under the preoading head " Trt* aaglea," and add them together. HaTlnfp the fonr angples, and either pair of opposite sides. Begin with one of the aidea, and the two anglea at its enda. If the aam of these two aaglea exeeeds 180O, aabtraet aaeh of tbem from 180°. and make use of the rema Inataad of the angles tbemaalTaa. Than oonslder this side and its two adjaoant anglea (or the two reau, as tha oaae aMT be) aa tbn— af atriangia; and And ila area aa diraeted far thai aaaa under tha praead lag head "friangla." D* a* aama with the alhar glvao aida, and ita twa adjaoent angles, (ar their reau, aa tha oaae may ha.) Subtraot the least of the areas thus, found, from the greatest; the rem will be the raqd area. Havinff three sides ; and the two included anfrles. Mult together the middle side, and one of the adjaoent sides ; mult tbe prad by the uat sine of their ineloded angle ; call the result a. Do the same with the middle aide and its other a^aaaut aida, and the nat sine of the other included angle; call the result b. Add the two anglea together ; fln4 the diir between their sum and 180(>, whether greater or less ; find the nat sine of this diff; malt together the two given sides whieh ara appostta one another ; molt the prod by the nat aine just found ; eall the result e. Add together the results a and ft ; then, if the sum of the two given angles is lass than 180°, subtract e from the anm of a and 6 ; Aof/the rem will be the area of tha trapeiTum. Bat if the aum of the two given anglea be greater than 180°, add together the three reanlta a, ft, and a; half their aum will be the area. Havlnff the two diayonalSy and either ann^le formed by their intersection. Sea Bamarka affear Parmllalegrams. In railroad measurements Of ezearation and embankment, the trapeslum imno frequently ooours ; as well as the two 6-sided figures { a» « o < and { m n o a ; in all of which m n represents the roadway ; rt.rc, and r ( the center- depths or heights ; I u and o v the lide-deptha er heigbta, aa given by the level ; Im and no the aide- alopea. The aame general rule for area appliea to all three of theae flga ; namely, mult the extreme hor width « « by ko^ the center depth r «, r e, or r t. an the oaae may be. Also molt one fawih of the width of roadway m n, by the mm of tbe two aide-depths I u and «. Add the two proda together ; the sum is the reqd area. Thia rule appliea whether tbe two side- slapas at I and n o have the same angle of inelination or DOC IB ndlvMtd work* 0t«H tka nIC* way hor width, eeatar depth, and aida depths of a prismOld ara respectively tm tIm half nm» «| ttia aorreaponding end ones, and thus ean be found without actual meaaurament. 1 POLYGON& 169 To draw a hezason, eacb nide of whteh shall be eqaal to a ffiven line, a b. From a and h, with rad a h, dosoribe the two arcs; from their Jntersectien, i, with Um oaBe rad, deaoribe aolreloi aroand the oireumf of which, step off the same rad. Side or a bexagon ts^nnX ^7795. T» draw side an oetaflpon, with each equal to a grlven line, e e. Prom c and e draw two perps, cp, ep, Aiso prolong c« toward / and g; and ftrom c and e, with rad equal e «, draw the two onadraats : and find their centers h h : join e A, and e h ; draw « • and h t parallel to e j> ; and make each of them equal to c 0; aaka c Qt and « o, each equal to h h ; Join oo^o*, and o <. tSlde of an oetaffon ^nnX .41421354. To draw an oetaffon in a irlTen oqnare. Vrom each comer of the square, and with a rad equal to half its diag, deicribe the few arcs; and Join the points at which they out the sides of the •qaare. To draw anjr reirnlar |M>1yson, with each side e^inal to «n n« IHr MQ degrees by the anmber of sides ; take the qoot fh>ro IBffi ; div the Km br t. Thil will give the angle c m n, or e n m. Mm and n la; down these ancles hr » protractor: the side* of these angles will meet ata point, c. f^m which desoribe the circle m m y ; and aronad it* drcumf step off disu equal to mn. In any circle* m m y, to draw any reffular polycfon. JHfWlP'tj the number of sides ; the qoot will be the aa^^le m c m, aithe cen ler. Ltf eff this angle bj a protraeiw ; and its chord m n will be one side ; which atep dff arooad tbrcironmf. To reduce any polyiron, asa50^e/a^toa triani^Ie of the same area. W Fig. 2. If- *• ai^oco the side /a toward w; and draw b g parallel to a c, and join g c. we get equal trl* inclas a e'fr and a eg, both on the same base a c ; and both of the same perp height, inssmuch aa Iherare between the two parallels a c and g 6. But the part a e i forms a portion of both these irt* aa^ or in other iravde. Is eommpn to botk. Tber«rore, if it be tak«i away from both triangles, IheremalnlBC parts, < e 6 of one of them, and < y a of the other, are also equal. Therefore, if the •srt7e b be left off from the p^ygon, and the part igabe Mken into it, the polygon g/edcigviM ■Me the « »i T«*> area as a/« d e 6 a; but it will have but five sides, while the other has six. Again, tt«s Indrawn parallel to 4/, and d* joined, we have upon the same base es, aud between the same mut^MM e a aadd/. the two equal triangles e • d. and e •/. with the part eot common to both ; and iMmMBay the rewaintaig part e o d or one. and o «/ of the other, are equal. Therefore, if o «/ be AaffftMn the polygon, and so d be taken into it, the new polygon gad eg, Fig 2. will have the same Mas a/ e d eo ; but It has but fbor sides, while the other has five. Finally, if g t, Fig 2, be ttmZJl u>wa«d)t: aad d » drawn parallel to c s : and c n joined, we have on the same base c «, and tSMsa lAe aaMt paraMtlt e s and d n, the two equal triangles etn, and ttd, with the part c s I MHaM le hoth. Tberefore, If we leave out c d (, and take ltt.s f n, we have tbe triangle gne equal •theaolfBOBjadcy.Pigi; orto o/«dc6a, FIgl. , , „ TM/ffT^P'* method it applicable to polygons of any number of aides. Wtel 160 POLYGOKS. IU^hede fg, to a ■mailer To reduee a larire nlmllar one. From Any interior point o, which had better be near the center, draw line* to all the angles a, h, c, ko. Join these lines by others parallel to the sides •f the fig. If it should be reqd to enlarge a small fig, draw, from any point • within it, lines extending beyond its angles ; and Join these lines by others fsnllsl to the sides of the small fig. To redaee a map to one on a smaller seale. The best meth9d is by dividing the large map into squares by faint lines, with a rery soft leadi penoil; and then drawing the rednoed map upon a sheet of smaller squares. A pair of proportional dividers will assist mueh in nzing points intermediate uf the sides of the squares. If the large map would be injured by drawing and rubbing •n# the squares, threads may be stretched across it to form the aqnares. In a reetanfpnlar tk§;^ ghsd, Bepresenting an open panel, to find the points • o o o In Ua •ides ; and at equal dists firom the angles g. and « ; Cor inserting a diag piece o o o o, of a given width 1 1, measured at right angles to its length. From g and « as centers, describe several ooncentrio arcs, as in the Fig. Draw upon transparent paper, two parallel lines a a, c e, at a distance apart equal to II; and placing these lines on top of the panel, move them about until it 18 shown by the ares that the four dists g o, go, t o, s o, are equal. Instead of the transparent paper, a strip of common paper, of the width { I may be used. Rbm. Many problems which would otherwise be very diflBcult, ■Bay be thus solved with an aoouraoy suffloient for praotieal purposes, by means of transparent paper. To find tbe area of any irreffnlar poly* §fon, anb e m. Div it into triangles, as anhfame, and a b e; in oaoh of wliloh find the perp dlst o, between its base a &, a e, or 6 e; and tbe opposite angle n, m, or a ; mult eaoh base by its perp dist; add all tbe prods together ; div by 2» *" To find approx tbe area of a lon^r tr^ reg^nlar fiK, as a 6 e d. Between it* ends «&,« 4, mc:r apace off equal dists, (the shorter they are the more accurate will be the result,) through whioh draw the intermediate parallel lines 1. 2, S, &o, across the breadth of the fig. Measure the lengths of these intermediate lines : add them together : to the sum add ht^/ the sum of the two end breadths • 6 and c d. Mult. the entire sum by one of the equal. spaces between the parallel lines. The prod will be the area This rule answers as well if either one or both the ends terminate in points, as at m and n. In the )ast of these cases, both a b and c d will be included In tne kntormodiate linos ; «nd kalf the two end breadths will be 0, or nothing. To find tbe area of any irre^nlar fiynre. Draw around it lines whioh shall enclose within them (as nearly as ean be judged by the eye) as much spaoe not belonging to the flgnro as they exclude space belonging to it. The area of the simpUflod flgnro thus formed, being in this manner rendered equal to that of the eom- plicated one, may be calculated by dividing it into triangles, Ao. By using a piece of fine thread, the proper position for the now bovndary lines may be found, before drawing them in. Areas of irregular figures may be found from a drawing, by Inyinc noon it a piece of transparent paper garefnUy ruled into small squares, eaoh of agivon area, say u M, or 100 sq. ft. eaoh ; apd by first oounting the whole squares, and then adding the fHkoUona of squares. cn dBCLESb 161 CIBCIiES. A •iNto Is Um area Ineladed within s onrred Him or aueh a eharMtw fhst evwy pofnt In it ts «|a«Uy ditunt from » c«rt«iD {lOiDt within It, cilUbA ita oontor. Tb« oorred line ItMlf la eaUed tlio airouBferoaoe, or peripherj of the circle ; or verj common! j It la called tbe oirole. T* And tbe circnmrerenee. Malt dlam bj S.1416, which givea too maoh by only .148 of an Inoh In a mlla. Ov, aa 113 la to SM - to is diam to elreaaif ; too graat 1 Inch in 186 niUea. Or* molt dlam h7 9^i too grpat bj about 1 part in UBS. Or* mnlt area by IS.MW, and take aq root of prod. To find tbe diam. DiT the •Irounf by S.14I6 ; or. aa SS5 la to US, ao la cireumr to diam ; or, molt the elrenmf. by 7: aaddlT »k» prod by tt, whish (Ivao thediaih toe anali by only abont om part ia S48&; or, mnlt the area by l.STSl; aad take th* aq rt of tiie prod. The dlam la to the olroamf more exactly aa 1 to S. 14159366. To find tbe area of a cflrele. Square the dlam; malt tbia aqoare by .7864; or more accarately by .786S9816; ^r aqnare the dr- eanf; mnlt thla aquare by .071)68 : or more accurately by .07957747 ; or mult half the diam by half the eirenmf ; or refer to the following table of areaa of olrdea. Alao area = an of rad X S.I416. The area of a drele la to the area of anr etreumaorlbed atraight-alded flg, aa the circumf of the drsle la to the elrenmf or periphery of the ig. Tbe area of a aquare Inaeribed in a circle, ia equal to twice the aqnare of the rad. Of a circle in a square, =r square X .7864. It Is eonvenient to remembatv In rmmdlnt off a aquara ooroer a h «, by a quarter of j a drele, that the shaded area • b c la equal to about 1 pan (correctly .3146) of the " wholA aqnare ahed. o To find tbe dlam of a circle eqoal In area to a ylTon sqaare. Mnit one aide of the aqnare by 1. 128S8. To find tbe rad of a circle to drcamscrlbe a i^lTcn eqaare. Mult one aide by .7071 ; or take H tbe diag. To find tbe side of a square equal In area to a fflYcn circle. Malt the diam by .8863S. To find tbe side of tbe (rre^^^st square in a siven circle. Malt dlam by .7071. The area of the greatest aquare that can be inscribed in a drele la equal to toiae the equare of tbe rad. The diam X by 1.3468 glvea tbe aide of an eqallatoral trianglf of equal area. To find tbe center e, of a nrf Ten dr^sle. Draw any chord a b ; and from the middle of it o, draw at r^ght angles t* it, a dlam d g ; find tbe center e of thla diam. 11 To describe a circle tbrongb any tbree points, abe, not in a straiipbt line. Join the pointo by the linea a6, ie; from the centers of these linea draw the dotted perpa meeting, as at o, which will be the center of the circle. Or from b, with any convenient rad. draw the arc m n; and from, a and c, with the aame rad. draw arcs y and jr; then two linea drawn through the iatoraeotiona of these area, will meet at the center o. To describe a circle to toucb tbe tbree ancles of a triangle is plainly the same as this. To inscribe a circle In a trianirle draw two lines blaeeting any two of tbe anglea. Where theae linea meet ia the eentor of the drele. 162 OEMXJLBBm T9 4i»W a tonyent* i€i,fm circle, firom any i^lven point, e, in its circnnMi. Through the center n, and the glren point «. dr»w n e ; "»*^ » •9"*J J* e n ; from n and o, with any rad creatar than half of o n, dewnrihe tha twa oairs of arc <<: Join their IntarMoUona iU Here, and in the following three flgt. the («n««nt« are ordinary vrjuo- mtrical one*; and may end where we pleaae. But the mgonometrum tangent of a given angU, must end in a Meant. Or ftom c lay off two equal distances c c, e < ; ana draw i i parallel to c t. To draw a tangr, « « ft, to a circle, ftnom a point. a, wblcii la onUiide of tlie circle. Draw a e, and on it deacrihe a •emiolrcle ; through the intaneetieB, «, drma a • 6. Here e is the oenter of the oirole. To draw a tangr* gh,ttonk a circnlar arc,sr«0» Of which n a is the rise. With rad g a, describe an are, • • o. lUH f « •qual ta • a. Through t draw g h. To draw a tani; t6 two circles. First draw the line m «, just touching the two •irales; this gives the direction of the Ung. Then from the centers of the circles draw the rsdil. o •^V^rP to n» n. The potato ( t are the Ung points. If the tang is in the position of the dotted line, • y, the ope- ration is the same. If any two chords, as a b, o c, cross eacli otkier, then as on : n 6 :: o n : n c. Hence, n ft X a n = onX ne. That f is the product of the two parts of one of the lines, is «- tlkS pro- h 4uct ofthe two parts of the other line. ..S'!: Jit til ! «i.„ 1 ...li ; 1 "i... I: - •~ .^loi,...,. *~ i s 1 Is II si 1 i ES 1 j 1 J J 1 1 i afss i TABLB 1 OP CIBCI.E»-(aaDtlDiwd). BDlbtU 1 fliaftUiB. Ac ,„ Cfrwmf, *,^ m... C.r.u«r. w Dlua. Inoul. .™K ,„™r .* raiia " iiiJ T 1 in a» IS B-iiil rai.«4 M.D11 W M» a> SS.IIWI WS.63 S^ » MJliJS Ml. 18 Itt.* m:S« ji »O.IHi wJ:" H SI. 180 M in »1.1UM ^ ISIJM H i>i no N eee.13 !! Sl.tlS 671 .M M 1? lU JO, H SBa:*) M « 8U.1] )i TO0.M M u I» M M H l» Ml KifflM t1b1« 8 m^m Us' 8 "'so* '^ Sri »? 1 la >ii ■ l-ssj? !«:« 1 iitMa i 1 ^i li g 1? H Ml >K 1 « M.SM St.OM !»: ».IH ^ 3M. "'sa* s Iffl.BM 3(1. w™ ISO.JIO go.fu OoiBl K 130.™ 1 m.im N «;™ ^ Bie.a tl.TK oi:?™ K.1BS 1 1 02.102 ulii S taiTM toa; i KJIVV iS'^ SS mieli 193.™ »~ 0J.5M 118. ^ W 1 «!.■ H imIiw ■a US. ^ ItMl 1H.J70 •0 «!. afA 19S.17I g 1 1«.S«1 11 Sl.Tl* H6.0S7 g «■§! 9«:Bt S ™- M." IM.W) B M ISfilJS K 1 n.ao i»7.ta ft 1»7.™ S IM.SI9 1 1 U W.3S J sra'.w ^ 1 !«'» sm:. 11 TW LOO. 1 1 ilil imi' ^ Hi IS: 11* i!™ ■ s !S:™ 1 836. ■ « 3 J.Hi H 1 :m 1^1 \yxf, Si! fS! IHltH iint.1 11 SIS I.7SI g lomi 11 MM S?* Mil g u m S.IM 1 !.*» 1 idKa won g 11 B«0 1 ! tiBOO i ili IfSli "i< iiS !«■ J "iS 1 nun s si iil 1 ■is 7ds: S KJS StSi I ^ ' H mi:!: M r: 166 CtBCLEB. TABIDS 3 OF cmCIiES. IMameters in anita and tenths* DUu Ctreamf. Area. mm. Cireanf. Area. Dia. Ciroinf. Area. •.1 .814159 .007854 6.3 19.79208 81.17245 12.5 89.26991 122.7185 .2 .628319 .031416 .4 20.10619 82.16991 .6 39.58407 124.6898 ^ .942478 .070686 .6 20.42085 83.18307 .7 39.89823 126.6769 .4 1.256637 .125664 .6 20.73451 34.21194 .8 40.21239 128.6796 Jb 1.570796 .196360 .7 21.04867 35.25652 .9 40.52655 130.6981 A 1.884956 .282743 .8 21.36288 36.31681 18.0 40.84070 132.7323 .7 2.199115 .384845 .9 21.67699 37.89281 .1 41.15486 134.7822 .8 2.513274 .502655 7.0 21.99115 38.48451 o 41.46902 136.8478 .9 2.827433 .636173 .1 22.30531 30.59192 .8 41.78318 138.9291 1.0 3.141593 .785398 .2 22.61947 40.71504 .4 42.09734 141.0261 .1 3.455752 .950332 .8 22.93363 41.85387 .5 42.41150 143.1388 ^ 3.769911 1.13097 .4 23.24779 43.00840 .6 42.72566 145.2672 ^ 4.084070 1.32732 .5 28.56194 44.17865 .7 43.03982 147.4114 .4 4.398230 1.53938 .6 23.87610 45.36460 .8 43.35398 149.5712 .5 4.712389 1.76715 .7 24.19026 46.56626 .9 43.66814 151.7468 ,6 5.026548 2.01062 .8 24.50442 47.78362 14.0 43.98230 163.9880 .7 5.34070» 2.26980 .9 24.81858 49.01670 .1 44.29646 156.1460 .8 5.654867 2.54469 8.0 25.13274 50.26548 .2 44.61062 158.3677 .9 5.969026 2.83529 .1 25.44690 51.52997 ^ 44.92477 160.6061 2.0 6.283185 8.14159 .2 25.76106 62.81017 .4 45.23893 162.8602 .1 6.597345 3.46361 .8 26.07522 54'.10608 .5 45.55309 165.1300 ;2 6.911504 3.80133 .4 26.38938 55.41769 .6 46.86725 167.4165 .8 7.225663 4.15476 .5 26.70354 66.74502 .7 46.18141 169.7167 A 7.539822 4.52389 .6 27.01770 58.08805 .8 46.49657 172.0336 Jb 7.858982 4.90874 .7 27.33186 59.44679 .9 46.80973 174.3662 A 8.168141 5.30929 .8 27.64602 60.82123 15.0 47.12389 176.7146 .7 8.482300 5.72555 .9 27.96017 62.21139 .1 47.4.3805 179.0786 ^ 8.796459 6.15752 9.0 28.27433 63.61725 .2 47.76221 181.4584 .9 9.110619 6.60520 .1 28.58849 &'>.03882 .8 48.06637 183.8539 3.0 9.424778 7.06858 .2 28.90265 66.47610 .4 48.38053 186.2660 J 9.738937 7.54768 .8 29.21681 67.92909 .5 48.69469 188.6919 ^ 10.05310 8.04248 .4 29.53097 69.39778 .6 49.00885 191.1345 ^ 10.36726 8.55299 .5 29.84513 70.88218 .7 49.32300 193.5928 .4 10.68142 9.07920 .6 30.15929 72.38229 .8 49.63716 196.0668 .5 10.99557 9.62113 .7 30.47345 73.89811. .9 49.95132 198.5565 A 11.30973 10.17876 .8 30.78761 75.42964 16.0 60.26648 201.0619 J 11.62389 10.75210 .9 31.10177 76.97687 .1 60.57964 203.5831 .8 11.93805 11.84115 10.0 81.41593 78.53982 .2 60.89380 206.1199 .9 12.25221 11.94591 .1 31.73009 80.11847 .8 61.20796 208.6724 4.0 12.56637 12.56637 .2 32.04425 81.71282 .4 61.52212 211.2407 .1 12.88053 13.20254 .8 32.35840 83.32289 .5 61.83628 213.8246 .2 13.19469 13.85442 .4 32.67256 84.94867 JR 62.15044 216.4248 .3 13.50885 14.52201 .5 32.98672 86.59015 .7 62.46460 219.0307 .4 13.82301 15.20531 .6 33.30088 88.24734 S 52.77876 221.6708 .5 14.13717 15.90481 .7 33.61504 89.92024 .9 63.09292 224.3176 .6 14.45133 16.61903 .8 33.92920 91.60884 17.0 63.40708 226.9801 .7 14.76549 17.34945 .9 34.24336 93.31316 .1 63.72123 229.6583 ^ 15.07964 18.09557 11.0 34.55752 95.08318 .2 64.08539 232.3522 .9 15.39380 18.85741 .1 34.87168 96.76891 S 64.34955 235.0618 6.0 15.70796 19.63495 .2 35.18584 98.52035 A 64.66371 237.7871 .1 16.02212 20.42821 .8 35.50000 100.2875 £ 64.97787 240.5282 ^ 16.33628 21.23717 .4 35.81416 102.0703 .6 65.292as 243.2849 .8 16.65044 22.06183 .5 36.12832 103.8689 .7 55.60619 246.0574 -.4 16.96460 22.90221 .6 36.44247 105.6832 .8 65.92035 248.8456 ^ 17.27876 23.75829 .7 36.75663 107.5182 .9 56.23451 251.6494 j6 17.59292 24.63009 .8 87.07079 109.3588 18.0 56.54867 264.4690 .7 17.90708 25.51759 .9 37.38495 111.2202 .1 56.86283 267.3048 .8 18.22124 26.42079 ISO 37.69911 113.0978 J2 57.17699 260.1558 .9 18.53540 27.33971 .1 38.01327 114.9901 A 67.49116 268.0220 «.o 18.84956 28.27433 .2 38.32743 116.8967 A 67.80580 265.9044 .1 19.16372 29.22467 .8 38.64159 118.8229 Jb 68.11946 268JN)25 .2 19.47787 80.19071 .4 88.96575 120.7628 ^ 6&48862 271.71168 CIBGI«EB. 167 TABIiS 8 OF €IB€I«BiM00BtiBiw4). Dittinetem in unite and tenths. Ma. droinf. Atmu DIft. Ctreamf. Area. Mft. Ctreanf. kntu 18.7 68.74778 274.6459 24.9 78.22566 486.9647 81.1 97.70B53 759.6460 .8 59.06194 277.59U 86.0 78.53982 490.8789 .2 98.01769 764.6880 .9 59.37610 280.5621 .1 78.85388 494.8087 .8 98.38185 769.4467 19.0 59.69026 283.5287 .2 79.16818 498.7592 .4 98.64601 774.8712 .1 60.00442 286.5211 .8 79.48229 502.7255 .5 98.96017 779.3118 .2 60.31858 289.5292 .4 79.79645 506.7075 .6 99.27438 784.2672 ^ 60.63274 292.5530 .5 80.11061 510.7052 .7 99.58849 789.2388 . A 60.94690 205.5925 .6 80.42477 514.7185 A 99.90266 794.2260 J5 61.26106 298.6477 .7 80.73803 518.7476 .9 100.2168 799.2290 .6 61.57582 301.7186 .8 81.05309 522.7924 88.0 100.5310 804.2477 .7 61.88986 304.8052 .9 81.36725 526.8529 .1 100.8451 809.2821 JR 62.20363 307.9075 86.0 81.68141 580.9292 .2 101.1503 814.3322 S 62.51769 311.0255 .1 81.99557 535.0211 .8 101.4734 819.3980 80.0 62.83185 314.1598 .2 82.30973 539.1287 .4 101.7876 824.4796 .1 68.14601 317.3067 .3 82.62389 5482521 .5 102.1018 829.6768 J2 68.46017 320.4730 .4 82.93805 547.8911 .6 102.4159 834.6898 Jl 68.77438 323.6547 .5 83.25221 55L5459 .7 102.7301 839.8184 .4 6108848 326.8513 .6 83.56686 565.7163 .8 106.0442 844.9628 A 64.40266 380.0636 .7 83.88052 569.9025 .9 103.8584 850.1228 JS 64.71681 383.2916 .8 84.19468 564.1044 88.0 103.6726 855.2986 .7 66.03097 336.5353 .9 84.50884 568.3220 .1 103.9867 860.4901 .8 65.34518 339.7947 87.0 84.82300 572.6653 .2 104.3009 865.6973 .9 65.65929 343.0698 .1 85.13716 576.8043 .8 104.6150 870.9202 tl.O 65.97S45 346.3606 .2 85.45132 581.0690 .4 104.9292 876.1588 .1 66.28760 849.6671 .3 85.76548 585.3494 .5 105.2434 881.4131 .2 66.60176 852.9894 .4 86.07964 589.6455 .6 105.6575 886.6831 ^ 66.91592 356.3278 .6 86.39880 593.9574 .7 105.8717 891.9688 .4 67.23008 359.6809 .6 86.70796 598.2849 .8 106.1858 897.2708 ^ 67.54tt4 363.0608 .7 87.02212 602.6282 .9 106.5000 902.5874 ^ 67.85840 366.4354 .8 87.33628 606.9871 84.0 106.8142 907.9208 .7 68.17256 369.8861 .9 87.65044 611.8618 .1 107.1288 918.2688 .8 68.48672 873.2526 88.0 87.06459 615.7522 .2 107.4426 918.6331 .9 68.80088 376.6848 .1 88.27875 620.1582 .3 107.7666 924.0181 M.0 69.U504 380.1327 .2 88.59291 624.5800 .4 108.0708 929.4088 .1 69.42920 388.5963 .8 88.90707 629.0175 .5 108.8849 934.8202 .2 69.748SS 887.0756 .4 89.22123 638.4707 .6 108.6991 940.2478 ^ 70.06788 300.5707 A 89.58539 637.9397 .7 109.0138 945.6901 A 70.37168 394.0814 .6 89.84955 642.4243 .8 109.3274 951.1486 Jb 70.68688 397.6078 .7 90.16371 646.9246 .9 109.6416 956.6228 j6 70.99999 401.1600 .8 90.47787 651.4407 86.0 109.9557 962.1128 .7 71.81415 404.7078 .9 90.79203 655.9724 .1 110.2699 967.6184 ^ 71.62881 408.2814 88.0 91.10619 660.5199 J2 110.5841 973.1397 .9 71.94247 411.8707 .1 91.42035 665.0830 .8 110.8982 978.6768 tt.O 72.26668 415.4756 .2 91.73451 669.6619 .4 111.2124 984.2296 .1 72.57079 419.0068 A 92.04866 674.2565 .6 111.5265 989.7980 .2 72.88496 422.7827 A 92.86282 678.8668 .6 111.8407 995.3822 ^ 78.19911 426.3848 A 92.67698 683.4928 .7 112.1649 1000.9821 A 78.51827 480.0526 A 92.99114 688.1345 .8 112.4690 1006.5977 & 78.82M8 488.7861 .7 93.30530 692.7919 .9 112.7832 1012.2290 A 74.14169 487.4854 .8 98.61946 697.4650 86.0 113.0973 1017.8760 .7 74.45695 441.1608 .9 98.93362 702.1538 .1 113.4115 1023.5387 ^ 74.76001 444.8809 80.0 94.24778 706.8583 .2 113.7257 1029.2172 .9 75.06406 448.6278 .1 94.56194 711.5786 .3 114.0898 1034.9118 M.0 75.30822 452.8808 .2 94.87610 716.3145 .4 114.3540 1040.6212 .1 75.71238 466.1671 .8 05.19026 721.0662 .5 114.6681 1046.3467 a. 76X>2I64 459.9606 .4 95.50442 725.8886 .6 114.9828 1052.0880 z 76.84090 468.7698 .5 95.81858 730.6166 .7 115.2965 1057.8449 A 76.66418 467.5947 .6 96.13274 735.4154 .8 115.6106 1063.6176 J» 76.90182 471.4862 .7 96.44689 740.2299 .9 115.9248 1069.4060 A 77.S8n8 475.2916 .8 96.76105 745.0601 87.0 116.2889 1075.2101 a 77J0li4 479J686 .9 97.07521 749.9060 .1 116.5531 1081.0299 M 97.01160 4K.DG18 81.0 97.38937 754.7676 .2 116.8672 1086.8664 168 CIBCLES. TABIiE 3 OF cmCIiKIMOontiaiMd). Diameters in iiniUi and tenths. Dis. Ciroumf. Are*. DU. Cirenaf* Area. DU. 49.7 arcamf. ▲res. 87.3 117.1814 1092.7168 48.5 136.6593 1486.1697 186.1372 1940.0041 .4 117.4956 1098.5835 .6 136.9734 1493.0105 .8 166.4513 1947.8189 A 117.8097 1104.4662 .7 137.2876 1499.8670 .9 166.7655 1965.6493 .6 118.1239 1110.3645 .8 137.6018 1606.7393 60.0 167.0796 1963.4964 .7 118.4380 1116.2786 .9 187.9159 1513.6272 .1 167.3938 1971.3572 .8 118.7622 1122.2083 44.0 138.2301 1520.5308 J2 157.7080 1979.2348 .9 119.0664 1128.1538 .1 138.5442 1527.4502 .5 158.0221 1987.1280 88.0 119.3805 1134.1149 .2 138.8584 1534.3853 .4 158.3363 1995.0370 .1 119.6947 1140.0918 .8 139.1726 1541.3360 .5 168.6504 2002.9617 .2 120.0088 1146.0844 .4 139.4867 1548.3025 .6 168.9646 2010.9020 .8 120.3230 1152.0927 .5 139.8009 1555.2847 .7 169.2787 2018.8581 .4 120.6372 1158.1167 .6 140.1150 1562.2826 .8 169.6929 2026.8299 .5 120.9513 1164.1564 .7 140.4292 1569.2962 .9 159.9071 2034.8174 .6 121.2655 1170.2118 .8 140.7434 1576.3255 61.0 160.2212 2042.8206 .7 121.5796 1176.2a30 .9 141.0575 1583.3706 .1 160.5364 2050.8395 .8 121.8938 1182.3698 46.0 141.3717 1590.4313 .2 160.8495 2058.8742 .9 122.2080 1188.4724 .1 141.6858 1597.5077 .3 161.1637 2066.9245 89.0 122.5221 1194.5906 .2 142.0000 1604.5999 .4 161.4779 2074.9906 .1 122.8363 1200.7246 .8 142.3141 1611.7077 .5 161.7920 2083.0728 J2 123.1504 1206.8742 .4 142.6283 1618.8313 .6 162.1062 2091.1697 Ji 123.4646 1213.0396 .6 142.9425 1625.9705 .7 162.4203 2099.2829 A 123.7788 1219.2207 .6 143.2566 1633.1255 .8 162.7345 2107.4118 .5 124.0929 1225.4175 .7 143.5708 1640.2962 .9 163.0487 2115.5663 .« 124.4071 1231.6300 .8 143.8849 1647.4826 62.0 163.3628 2123.7166 .7 124.7212 1237.8582 .9 144.1991 1654.6847 .1 163.6770 2131.8926 J& 125.0354 1244.1021 46.0 144.5133 1661.9025 .2 163.9911 2140.0848 .9 125.3495 1250.3617 .1 144.8274 1669.1360 .3 164.3063 2148.2917 40.0 125.6637 1256.6371 .2 145.1416 1676.3853 .4 164.6196 2166.5149 .1 125.9779 1262.9281 JS 145.4557 1683.6502 .6 164.9386 2164.7587 .2 126.2920 1269.2348 .4 145.7699 1690.9308 .6 165.2478 2173.0082 .8 126.6062 1275.5573 .6 146.0841 1698.2272 .7 166.6619 2181.2785 .4 126.9203 1281.8955 .6 146.3982 1705.5392 .8 166.8761 2189.5644 .6 127.2345 1288.2493 .7 146.7124 1712.8670 .9 166.1908 2197.8661 .6 127.5487 1294.6189 .8 147.0265 1720.2105 68.0 166.5044 2206.1884 .7 127.8628 1301.0042 .9 147.3407 1727.5697 .1 166.8186 2214.5165 .8 128.1770 1307.4052 47.0 147.6549 1734.9445 .2 167.1327 2222.8658 .9 128.4911 1313.8219 .1 147.9690 1742.3351 .8 167.4469 2231.2296 41.0 128.8053 1320.2543 .2 148.2832 1749.7414 .4 167.7610 2239.6100 .1 129.1195 1326.7024 .8 148.5973 1757.1635 .5 168.0752 2248.0059 Jl 129.4336 1333.1663 .4 148.9115 1764.6012 .6 168.3894 2256.4175 .8 129.7478 1339.6458 .5 149.2257 1772.0546 .7 168.7035 2264.8448 .4 130.0619 1346.1410 .6 149.5398 1779.5287 .8 169.0177 2273.2879 .5 130.3761 1352.6520 .7 149.8640 1787.0086 .9 169.3318 2281.7466 .6 130.6903 1359.1786 .8 150.1681 1794.5091 64.0 169.6460 2290.2210 .7 131.0044 1365.7210 .9 160.4823 1802.0254 .1 169.9602 2298.7112 .8 131.3186 1372.2791 48.0 160.7964 1809.5574 .2 170.2743 2307.2171 .9 131.6327 1378.8529 .1 151.1106 1817.1050 .8 170.5885 2315.7386 4S.0 131.9469 13)85.4424 J2 161.4248 1824.6684 .4 170.9026 2824.2769 .1 182.2611 1392.0476 .8 151.7389 1832.2476 .5 171.2168 2882.8289 .2 132.5752 1398.6685 .4 152.0531 1839.8423 .6 171.5810 2341.8976 Ji 132.8894 1405.3051 .6 162.3672 1847.4528 .7 171.8451 2849.9820 .4 183.2035 1411.9574 .6 162.6814 1855.0790 .8 172.1593 2358.5821 Jb 133.5177 1418.6254 .7 152.9956 1862.7210 .9 172.4784 2967.1979 A 183.8318 1425.8092 .8 153.3097 1870.8786 66.0 172.7876 2375.8294 .7 184.1460 1432.0086 .9 153.6239 1878.0519 .1 173.1018 2884.4767 .8 184.4602 1438.7288 48.0 153.9380 1885.7410 .2 173.4159 2893.1396 .9 134.7743 1445.4546 .1 154.2622 1893.4457 .8 173.7801 2401.8188 48.0 185.0886 1452.2012 .2 154.5664 1901.1662 .4 174.0442 2410.6136 .1 1S5.4026 1458.9685 .8 154.8805 1908.9024 .6 174.8584 24192227 JZ 186.7168 1465.7416 .4 155.1947 1916.6543 .6 174.6726 2427.9485 J 186.0810 1472.6352 .6 155.5088 1924.4218 .7 174.9867 2486.6899 4 186.3451 1479.8448 .6 155.8230 1932.2061 .8 175.3009 2445.4471 GIBCLES. TABIA 2 OF €lB€I<iaiMOoi»tliiii«dX I^lamet^vs in nnlts waA tentha. 169 ma. 56.9 175.6160 56.0 175.9292 .1 176.2433 .2 176.6576 .3 176.8717 .4 177.1858 .5 177.5000 .6 177.8141 .7 178.1283 r ^ 178.4425 .9 178.7566 67.0 179.0708 .1 179.3849 .2 179.6991 .8 T80.0133 .4 180.3274 .5 180.6416 .6 180.9557 .7 181.2699 .8 181.5841 .9 181.8982 68.0 182.2124 .1 182.5265 .2 182.8407 .3 183.1549 .4 188.4690 .5 188.7832 .6 184.0973 .7 184.4115 .8 184.7256 .9 185.0398 69.0 185.3540 .1 185.6681 .2 185.9823 ^ 186.2964 .4 186.6106 .6 166.9248 .6 187.2389 .7 187.5531 .8 187.8672 .9 188.1814 io.o 188.4956 .1 188.8097 .2 189.1289 .3 189.4880 .4 189.7522 .5 190.0664 .6 190.3805 .7 190.6947 .8 191.0068 .9 191.8280 §1.0 191.6672 .1 191.9518 .2 192.2666 ^ 192.6796 .4 192.8868 ^ - l'(PS.20'99 .6 VfSi.S^ki .7 .8 Cireaiif. Areft# 2454.2200 2463.0086 2471.8130 2480.6330 2489.4687 2498.3201 2507.1873 2516.0701 2524.9687 2533.8830 2542.8129 2551.7586 2560.7200 2569.6971 2578.6899 2587.6985 2596.7227 2605.7626 2614.8183 2623.8896 2632.9767 2642.0794 2651.1979 2660.8321 2669.4820 2678.6476 2687.8289 2697.0259 2706.2386 2715.4670 2724.7112 2733.9710 2743.2466 2752.5378 2761.8448 2771.1675 2780.5058 2789.8599 2799.2297 2808.6152 2818.0165 2827.4384 2836.8660 2846.8144 2855.7784 2865.2582 2874.7536 2884.2648 2898.7917 2903.8343 2912.8926 2922.4666 2982.0568 2941.6617 2951.2828 2960.9197 2970.6722 2980.2406 2989.9244 21^.6241 6009.8896 6019.0706 Dift. 62.1 .2 .3 .4 .5 .6 .7 .8 .9 68X) .1 .2 .8 .4 .5 .6 .7 .8 .9 64.0 .1 .2 .8 .4 .6 .6 .7 .8 .9 66.0 .1 .2 .8 .4 .5 .6 .7 .8 .9 66.0 .1 .2 .3 .4 .6 .6 .7 .8 .9 67.0 .1 .2 .8 .4 .6 .6 .7 .8 .9 68.0 .1 ,2 Cireumf. 195.0929 195.4071 196.7212 196.0364 196.3495 196.6637 196.9779 197.2920 197.6062 197.9203 198.2346 198.5487 198.8628 199.1770 199.4911 199.8053 200.1195 200.4336 200.7478 201.0619 201.3761 201.6902 202.0044 202.3186 202.6327 202.9469 203.2610 203.5752 203.8894 204.20a'> 204.5177 204.8318 205.1460 205.4602 205.7743 206.0885 206.4026 206.7168 207.0310 207.3451 207.6593 207.9734 208.2876 208.6018 208.9159 209.2301 209.5442 209.8584 210.1725 210.4867 210.8009 211.1160 211.4292 211.7483 212.0575 212.3717 212.6858 213.0000 213.3141 213.6283 213.9425 214.2566 Area. DU. 8028.8178 68.8 3038.5798 .4 3048.3580 .6 3058.1520 .6 8067.9616 .7 3077.7869 .8 3087.6279 .9 S097.4847 69.0 3107.3571 .1 3117.2453 J2 8127.1492 .8 3137.0688 .4 3147.0040 .5 3156.9560 .6 3166.9217 .7 3176.9042 .8 3186.9023 .9 3196.9161 70.0 3206.9456 .1 3216.9909 .2 3227.0518 .3 3237.1285 .4 3247.2209 .6 3257.3289 .6 3267.4527 .7 3277.5922 .8 3287.7474 .9 3297.9183 llJO 3308.1049 .1 3318.3072 .2 3328.5253 .8 3338.7590 .4 3349.0086 .5 3859.2786 .6 3869.5545 .7 3379.8510 .8 3390.1683 .9 3400.4913 72.0 3410.8350 .1 3421.1944 .2 8431.5695 .3 3441.9603 .4 3452.8669 .6 3462.7891 .6 8473.2270 .7 3483.6807 .8 3494.1500 .9 3504.6351 78.0 8515.1359 .1 3625.6524 .2 3536.1845 .8 8546.7324 .4 3557.2960 .6 3567.8764 .6 3578.4704 .7 8589.0811 .8 3899.7075 .9 3610.8497 74.0 3621.0075 .1 3631.6811 .2 3642.3704 .3 3658.0754 .4 Circomf. 214.5708 214.8849 215.1991 216.5133 215.8274 216.1416 216.4557 216.7699 217.0841 217.3982 217.7124 218.0265 218.3407 218.6548 218.9690 219.2882 219.5973 219.9115 220.2266 220.5398 220.8540 221.1681 221.4823 221.7964 222.1106 222.4248 222.7389 223.0531 223.3672 223.6814 223.9956 224.3097 224.6239 224.9880 225.2522 225.5664 225.8805 226.1947 226.5088 226.8230 227.1871 227.4518 227.7655 228.0796 228.3938 228.7079 229.0221 229.3363 229.6504 229.9646 280.2787 230.5929 230.9071 231.2212 231.5354 231.8495 232.1687 232.4779 232.7920 233.1062 233.4203 233.7345 Area. 3663.7960 3674.5324 3685.2845 3696.0623 3706.8369 3717.6361 3728.4500 3739.2807 3750.1270 3760.9891 3771.8668 3782.7603 3793.6695 3804.5944 3815.5360 3826.4913 3837.4633 3848.4510 3859.4544 3870.4736 3881.5084 3892.5690 3903.6252 3914.7072 3925.8049 3986.9182 3948.0473 3959,1921 3970.3526 3981.5289 3992.7208 4003.9284 4015.1518 4026.3908 4037.6456 404S.9160 4060.2022 4071.6041 4082.8217 4094.1550 4105.5040 4116.8687 4128.2491 4139.6452 4151.0571 4162.4846 4173.9279 4185.3868 4196.8615 4208.3519 4219.8579 4231.8797 4242.9172 4254.4704 4266.0394 4277.6240 4289.2243 4300.8403 4312.4721 4324.1195 4335.7827 4347.4616 170 TABUB S OF €lII€IdB»-(OcmtlBiMdX Dtentetem In unite and tenths. M«. Clrennf. Area. DU. 80.7 CirewBi; Area. DU. Cirenni: Arab 74.5 284.0487 4359.1562 288.5265 6114.8977 86.9 278.0044 5931.0206 .6 284.3628 4370.8664 A 258.8407 5127.5819 87.0 278.8186 5944.6787 .7 234.6770 4382.5924 .9 254.1548 5140.2818 .1 273.6327 6968.8525 .8 234.9911 4384.8841 81.0 254.4690 5152.9974 .2 273.9469 6972.0420 .9 235.3053 4406.0016 .1 254.7832 6165.7287 .8 274.2610 6985.7472 75.0 235.6194 4417.8647 .2 255.0973 5178.4767 .4 274.5762 5999.4681 .1 235.9336 4429.6535 .8 255.4115 5191.2884 .5 274.8894 6013.2047 a, 286.2478 4441.4580 .4 255.7256 6204.0168 .6 276.2035 6026.9570 ^ 236.5619 4453.2788 .5 256.0398 5216.8110 .7 275.6177 6040.7250 A 236.8761 4465.1142 .6 256.3540 5229.6208 .8 276.8818 6054.5088 Jb 287.1902 4476.9659 .7 256.6681 5242.4463 .9 276.1460 6068.3082 .6 237.5044 4488.8832 .8 256.9823 5255.2876 88.0 276.4602 6082.1284 .7 237.8186 4500.7168 .9 257.2964 6268.1446 .1 276.7743 6096.9542 .8 238.1327 4512.6151 89.0 257.6106 6281.0178 .2 277.0886 6109.8008 .9 238.4469 4524.5296 .1 257.9248 6293.9066 .8 277.4026 6123.6631 fl.0 238.7610 4536.4598 .2 258.2389 6306.8097 .4 277.7168 6137.5411 J 239.0752 4548.4067 .8 258.5531 6819.7295 .6 278.0309 6151.4348 2. 239.3894 4660.3678 .4 258.8672 6332.6650 ,6 278.3451 6165.3442 ^ 239.7035 4572.3446 .6 259.1814 6345.6162 .7 278.6593 6179.2698 A 240.0177 4584.3377 .6 259.4956 5358.5882 .8 278.9734 6193.2101 Ja 240.3318 4596.3464 .7 259.8097 6371.5658 .9 279.2876 6207.1666 .6 240.6460 4608.3708 .8 260.1239 6384.6641 89.0 279.6017 6221.1380 .7 240.9602 4620.4110 .9 260.4380 6897.6782 .1 279.9159 6235.1268 A 241.2748 4632.4669 88.0 260.7522 6410.6079 .2 280.2301 6249.1804 .9 241.5885 4644.5384 .1 261.0663 6423.6534 .8 280.6442 6263.1498 77.0 241.9026 4656.6257 .2 261.3805 6436.7146 .4 280.8584 6277.1849 .1 242.2168 4668.7287 .3 261.6947 6449.7915 .6 281.1725 6291.2356 .2 242.531C 4680.8474 .4 262.0088 6462.8840 .6 281.4867 6305.3021 .8 242.8461 4692.9818 .5 262.3230 6475.9923 .7 281.8009 6319.3843 .4 243.1593 4705.1319 .6 262.6371 6489.1163 .8 282.1160 6333.4822 Jb 243.4734 4717.2977 .7 262.9513 6502.2561 .9 282.4292 6347.6958 .6 243.7876 4729.4792 .8 263.2655 5516.4115 90.0 282.7483 6361.7251 .7 244.1017 4741.6765 .9 263.5796 6528.6826 .1 283.0575 6375.8701 .8 244.4159 475S.8894 84.0 263.8938 6641.7694 .2 283.3717 6390.0909 .9 244.7301 4766.1181 .1 264.2079 6554.9720 .8 283.6868 6404.2073 18.0 245.0442 4778.3624 .2 264.5221 5568.1902 .4 284.0000 6418.8995 .1 245.3584 4790.6225 .8 264.8363 5581.4242 .5 284.3141 6432.6073 a 245.6725 4802.8988 .4 265.1504 6594.6789 .6 284.6283 6446.8309 A 245.9867 4815.1897 .5 265.4646 5607.9392 .7 284.9425 6461.0701 A 246.3009 4827.4969 A 265.7787 6621.2208 .8 285.2566 6475.3251 6489.6968 .6 246.6150 4839.8198 .7 266.0929 5634.6171 .9 285.6708 .6 246.9292 4852.1584 .8 266.4071 5647.8296 91.0 286.8849 6503.8822 .7 247.2488 4864.5128 .9 266.7212 6661.1578 .1 286.1991 6518.1848 .8 247.5575 4876.8828 85.0 267 0354 6674.5017 .2 286.5188 6532.6021 S 247.8717 4889.2685 .1 267.8495 6687.8614 .3 286.8274 6546.8856 99.0 248.1858 4901.6699 .2 267 6637 6701.2367 .4 287.1416 6561.1848 a 248.5000 4914.0871 .8 267.9779 6714.6277 .6 287.4657 6575.6498 .2 248.8141 4926.5199 .4 268.2920 6728.0346 .6 287.7699 6589.9804 .3 249.1283 4938.9685 .5 .268.6062 6741.4569 .7 288.0840 6604.8268 .4 249.4425 4951.4328 .6 268.9203 6754.8951 .8 288.3982 6618.7388 .6 249.7566 4963.9127 .7 269J2345 6768.8490 .9 28a7124 6633.1668 .6 250.0708 4976.4064 .8 269.5486 5781.8185 92.0 289.0265 6647.6101 .7 250.3849 4988.9198 .9 269.8628 5795.8038 .1 289.8407 6662.0602 .8 250.6991 5001.4469 8A.0 270.1770 6808.8048 .2 289.6548 6676.6441 .9 251.0133 5013.9897 .1 270.4911 5822.8215 .8 289.9690 6691.0347 io.o 251.3274 5026.5482 .2 270.8053 6835.8539 .4 290.2882 6705.5410 .1 251.6416 5039.1225 .8 271.1194 6849.4020 .5 290.5978 6720.0680 .2 261.9557 5051.7124 .4 271.4336 6862.9659 .6 290.9116 6734.6008 .8 252.2699 5064.8180 .5 271.7478 5876.6454 .7 291.2256 6749.1542 .4 252.5840 5076.9394 .6 272.0619 6890.1407 .8 291.5898 6768.7288 ^ 252.8982 5089.5764 .7 272.3761 5908.7516 .9 291.8540 6778iKW2 A 253.2124 5102.2292 .8 272.6902 6917.8788 98.0 292.1681 6792.9087 CIBGLE8. 171 TABIDS 9 OF ClBCIiES-<Ooiittniiad). Blameters in nnlts and tenths. Ma. Clrcnnf. Area. ms. Gtrennf. ArMU Dia. Cirenmf. Area. iM.1 292.4823 6807.5250 05.5 800.0221 7163.0276 97.8 307.2478 7512.2078 a, 292.7964 6822.1569 .6 300.8363 7178.0366 .9 307.5619 7527.5780 .3 293.1106 6836.8046 .7 300.6504 7193.0612 98.0 307.8761 7542.9640 .4 298.4248 6851.4680 .8 900.9646 7208.1016 .1 308.1902 7558.3656 .6 293.7389 6866.1471 .9 301.2787 7223.1577 .2 308.5044 7573.7830 .6 294.0531 6880.809 96.0 801.5929 7238.2295 .3 308.8186 7689.2161 .7 294.3672 6895.5524 .1 801.9071 7253.3170 .4 309.132'3: 7604.6648 .8 294.6814 6910.2786 .2 302.2212 7268.4202 .0 309.4469 7620.1293 .9 294.9956 6925.0205 .3 302.5354 7283.5391 .6 309.7610 7635.6095 M.0 295.3097 6939.7782 .4 802.8495 7298.6737 .7 310.0752 7651.1054 .1 295.6239 6954.5515 .5 803.1637 7313.8240 .8 310.8894 7666.6170 .2 295.9880 1 6969.3406 1 .6 803.4779 7328.9901 .9 310.7035 7682.1444 .3 296.2522 6984.1453 .7 803.7920 7844.1718 99.0 311.0177 7697.6874 .4 296.5663 6998.9658 .8 304.1062 7859.3693 .1 311.3318 7713.2461 .5 296.8805 7013.8019 .9 304.4203 7374.5824 .2 311.6460 7728.8206 .6 297.1947 7028.6538 97.0 304.7345 7889.8113 .8 811.9602 7744.4107 .7 297.5088 7043.5214 .1 305.0486 7405.0559 .4 312.2743 7760.0166 .8 297.8230 7058.4047 .2 805.8628 7420.3162 .5 812.5885 7775.6382 .9 298.1371 7073.3037 .3 305.6770 7435.5922 .6 812.9026 7791.2754 •5.0 298.4513 7088.2184 .4 305.9911 7450.8839 .7 813.2168 7806.9284 .1 298.7655 7103.1488 .5 306.3053 7466.1913 ,8 813.5309 7822.5971 .2 299.0796 7118.0950 .6 306.6194 7481.5144 .9 313.8451 7838.2815 .3 299.3938 7133.0568 .7 306.9336 7496.8532 100.0 314.1593 7853.9816 .4 299.7079 7148.0343 Cirenmferenees when the diameter has more than one place of decimals. Dian. 1 Giro. Dlun. Circ. Diam. Clro. 1 Diam. Giro. Diam. Giro. .1 .314169 .01 .031416 .001 .003142 .0001 .000314 .00001 .000031 .2 .628319 .02 .062832 .002 .006283 .0002 .000628 .00002 .000063 .8 .942478 .03 .094248 .003 .009425 .0003 .000942 .00003 .000094 .4 1.256637 .04 .126664 .004 .012566 .0004 .001257 .00004 .00012$ Ji 1.570796 .05 .157080 .005 .015708 .0005 .001571 .00005 .000157 .6 1.884956 .06 .188496 .006 .018850 .0006 .001886 .00006 .000188 .7 2.199115 .07 .219911 .007 .021991 .0007 .002199 .00007 .000220 ^ 2.513274 .08 .251827 .008 .025133 .0008 .002513 .00008 .000251 3 2.827433 .09 .282743 .009 .028274 .0009 .002827 .00009 .000283 Examples. Diameter = 3.12699 Circumference «■ Cire for dia of 3.1 .02 .006 iK)09 .00009 M Snm of 9.788937 .062832 .018850 .002827 .000283 9.823729 Clrcnmfte — Diameter — Dia for circ of 9.823729 9.738937 .084792 .062832 .021060 .018860 .003110 .002827 .000283 .000883 Sum of 3.1 .02 .006 .0009 .09009 3.19699 172 CIRCLES. TABUB a OF CIBCIiKS. Diams in unite and twelfths) as in feet and inehea. Dia. Circumf. Area. Dia. Cirenmf. Area. Dia. Clrcamf. Area* irt.in. Feet. Sq. ft. Ft.In.l Feet Sq.ft. Ft.In. Feet. Sq. ft. 5 ' 15.70796 19.63495 10 31.41593 78.53982 1 .261799 .005454 1 15.96976 20.29491 1 31.67773 79.85427 2 .523599 - .021817 2 16.23156 20.96577 2 81.93953 81.17968 8 .785398 .049087 3 ' 16.49336 21.64754 3 32.20132 82.51589 4 1.047198 .087266 4 16.75516 22.34021 4 32.46312 88.86307 5 1.308997 .136354 5 17.01696 28.04380 5 32.72492 85.22115 6 1.570796 .196350 6 17.27876 23.75829 6 32.98672 86.59015 7 1.832596 .267254 7 17.54056 24.48370 7 33.24852 87.97005 8 2.094395 .349066 8 17.80236 25.22001 8 33.51032 89.8606S 9 2.356195 •441786 9 18.06416 25.96723 9 33.77212 90.76258 10 2.617994 .545415 10 18.32596 26.72535 10 84.03392 92.17520 11 2.879793 .659953 11 18.58776 27.49439 11 84.29572 98.59874 1 3.14159 .785398 6 18.84956 28.27433 11 34.55752 95.08818 1 3.40339 .921752 1 19.11136 29.06519 1 34.81982 96.47858 2 3.66519 1.06901 2 19.37315 29.86695 2 36.08112 97.98479 8 3.92699 1.22718 3 19.63495 30.67962 8 85.34292 99.40196 4 4.18879 1.39626 4 19.89675 31.50319 4 35.60472 100.8800 5 4.45059 1.57625 5 20.15855 32.33768 5 35.86652 102.8690 6 4.71239 1.76715 6 20.42035 ^33.18307 6 36.12832 103.8689 7 4.97419 1.96895 7 20.68215 34.03937 7 36.39011 105.3797 8 5.23599 2.18166 8 20.94395 34.90659 8 36.65191 106.9014 9 6.49779 2.40528 9 21.20575 35.78470 9 36.91371 108.4840 10 6.76959 2.63981 10 21.46755 36.67373 10 87.17551 109.9776 11 6.02139 2.88525 11 21.72935 37.57367 11 37.43731 111.5320 S 6.28319 3.14159 7 21.99115 38.48451 12 37.69911 113.0973 1 6.54498 3.40885 1 22.25295 39.40626 1 37.96091 114.6736 2 6.80678 3.68701 2 22.51475 40.33892 2 38.22?71 116.2607 3 7.06858 3.97608 8 22.77655 41.28249 8 38.48451 117.8588 4 7.33038 4.27606 4 23.03835 42.23697 4 38.74631 119.4678 5 7.59218 4.58694 5 23.30015 43.20235 5 39.00811 121.0877 6 7.85398 4.90874 6 23.56194 44.17865 6 39.26991 122.7185 7 8.11578 5.24144 7 23.82374 45.16585 7 39.53171 124.3602 8 8.37758 5.58505 8 24.08554 46.16396 8 39.79351 126.0128 9 8.63938 5.93957 9 24.34734 47.17298 9 40.05631 127.6763 10 8.90118 6.30500 10 24.60914 48.19290 10 40.31711 129.3507 11 9.16298 6.68134 11 24.87094 49.22374 11 40.57891 131.0360 S 9.42478 7.06858 8 25.13274 50.26548 18 40.84070 132.7328 1 9.68658 7.46674 1 25.39454 51.31813 1 41.10250 134.4894 2 9.94838 7.87580 2 25.65634 52.38169 2 41.36430 136.1575 8 10.21018 8.29577 3 25.91814 53.45616 8 41.62610 137.8865 4 10.47198 8.72665 4 26.17994 54.54154 4 41.88790 189.6263 5 10.73377 9.16843 5 26.44174 55.63782 5 42.14970 141.8771 6 10.99557 9.62113 6 26.70354 56.74502 6 42.41160 143.1888 7 11.25737 10.08473 7 26.96534 57.86312 7 42.67:^30 144.9114 8 11.51917 10.55924 8 27.22714 58.99213 8 42.93510 146.6949 9 11.78097 11.04466 9 27.48894 60.13205 9 43.1^90 148.4893 10 12.04277 11.54099 10 27.75074 61.28287 10 43.45870 150.2947 11 12.30457 12.04823 11 28.01253 62.44461 11 43.72050 152.1109 4 12.56637 12.56637 • 28.27433 68.61725 14 48.98230 158.9388 1 12.82817 13.09542 1 28.53613 64.80080 1 44.24410 155.7761 2 13.08997 13.63538 2 28.79793 65.99526 2 44.50590 157.6250 8 13.35177 14.18625 3 29.05978 67.20063 8 44.76770 159.4849 4 13.61357 14.74803 4 29.32153 68.41691 4 45.02949 1 61.8557 5 13.87537 15.32072 5 29.58333 69.64409 6 45.29129 168.2374 6 14.13717 15.90431 6 29.84513 70.88218 6 45.55809 165.1801 7 14.39897 16.49882 7 30.10693 72.13119 7 45.81489 167.0831 8 14.66077 17.10423 8 30.36873 73.39110 8 46.07669 168.9479 9 14.92267 17.72055 9 30.63053 74.66191 9 46.88849 170.8738 10 15.18486 18.84777 10 30.89233 75.94364 10 46.60029 172.8094 U 15.44616 18.98591 11 31.15413 77.23627 11 46.86209 174.7665 OIBOLEB. 173 Mmmam In nnlt* and tw«lftiift| as tn ft«i and ineliea. Miu Cirenaf. Arcs. ma. Cireoinf. Ar«ft. Dte. Olreimf. IrMU FUn, Feet. Sq.ft. Ftln. Feet. Sq. ft. Ft.In. Feet. Sq.ft. 16 47.12389 170.7146 20 62.88185 314.1598 25 78.53982 490.8739 1 47.38589 17&6835 1 63.09865 816.7827 1 78.80162 494.1518 2 47.64749 180.6634 2 63.35545 819.4171 2 79.06342 497.4407 3 47.90929 182.6542 3 63.61725 322.0623 8 79.32521 500.7404 4 48.17109 184.6558 4 63.87905 324.7185 4 79.58701 504.0511 5 48.43289 186.6)S84 5 64.14085 827.8856 6 79.84881 607.8727 6 48.60469 188.6919 6 64.40265 830.0636 6 80.11061 510.7052 7 48.95649 190.7263 7 64.66445 832.7525 7 60.37241 514.0486 8 49.21828 192.7716 8 64.92625 335.4523 8 80.68421 517.4029 9 49.48008 194.8278 9 65.18805 838.1630 9 80.89601 520.7681 10 49.74188 196.8950 10 65.44985 340.8816 10 81.15781 524.1442 11 50.00868 198.9730 11 65.71165 843.6172 11 81.41961 527.5312 le 50.26548 201.0619 21 66.97345 346.3606 28 81.68141 530.9292 1 50.52728 203.1618 1 66,23525 349.1149 1 81.94321 534.3380 2 60.'^3908 206.2725 2 66.49704 351.8802 2 82.20501 537.7578 8 51.06068 207.3942 3 66.75884 354.6564 3 82.46681 541.1884 4 51.31268 209.5268 4 67.02064 357.4434 4 82.72861 544.6300 5 51.67448 211.6703 5 67.28244 360.2414 5 82.99041 548.0825 ,6 51.83628 213.8246 6 67.54424 363.0503 6 83.25221 551.5459 7 52.09808 215.9899 7 67.80604 365.8701 7 83.51400 555.0202 8 52.85988 218.1662 8 68.06784 368.7008 8 83.77580 558.5054 9 52.62168 2W.3533 9 68.32964 371.5424 9 84.03760 562.0015 10 52.88348 2X>..5513 10 68.59144 374.3949 10 84.29940 565.5085 11 58.14528 224.7602 11 68.85324 377.2584 11 84.56120 569.0264 17 58.40708 226.9801 22 69.11504 ' 380.1327 1 27 84.82300 572.5558 1 53.66887 229.2108 1 69.37684 383.0180 1 85.08480 576.0960 2 58.93067 231.4525 2 69.68864 385.9141 2 85.34660 579.6457 8 54.19247 233.7050 3 69.90044 388.8212 8 85.60840 583.2072 4 54.45427 235.9685 4 70.16224 391.7392 4 85.87020 586.7797 5 54.71607 238.2429 5 70.42404 394.6680 5 86.13200 590.3631 e 54.97787 240.5282 6 70.68583 397.6078 6 86.^9380 593.9574 7 55.23967 242.8244 7 70.94763 400.5585 7 86.65560 597.5626 8 55.50147 246.1315 8 71.20943 403.5201 8 86.91740 601.1787 9 55.76327 247.4495 9 71.47123 406.4926 9 87.17920 604.8057 10 56.02507 249.7784 10 71.73308 409.4761 10 87.44100 608.4436 , 11 56.28687 252.1183 11 71.99483 412.4704 11 87.70279 612.0924 18 56.54867 254.4690 28 72.25663 415.4756 28 87.96459 615.752? 1 56.81047 256.8307 1 72.51843 418.4918 1 88.22639 619.4228 2 57.07227 259.2032 2 72.78023 i 421.6188 2 88.48819 623.1044 8 57.38407 261.6867 8 73.04203 424.5568 3 88.74999 626.7968 4 57.59587 263.9810 4 73.30383 427.6057 4 89.01179 630.5002 6 57.85766 266.8863 5 73.56563 1 430.6654 5 89.27359 634.2145 6 58.11946 268.8025 6 73.82743 433.7361 6 89.53639 637.9397 7 58.88126 271.2296 7 74.08923 436.8177 7 89.79719 641.6758 8 58.64806 273.6676 8 74.35103 439.9102 8 90.05899 645.4228 9 68.90486 276.1165 9 74.61283 443.0137 9 90.32079 649.1807 10 59.16666 278.5764 10 74.87462 446.1280 10 90.58259 652.9495 11 59.42846 281.0471 11 75.13642 449.2532 11 90.84439 656.7292 t» 50.69026 288.5287 24 75.39822 452.3893 29 91.10619 660.5199 1 59.96206 286.0213 1 76.66002 455.5364 1 91.36799 664.3214 2 60.21886 2885247 2 76.92182 458.6943 2 91.62979 668.1339 8 60.47566 291.0891 3 76.18362 461.8632 8 91.89159 671.9572 4 6a7S?46 293.5644 4 76.44542 465.0430 4 92.15338 676.7915 8 60.99926 296.1006 5 76.70722 468.2337 5 92.41518 679.6867 8 61.2fa06 298.6477 6 76.96902 471.4352 8 92.67698 683.4928 7 81.52286 801.2056 7 77.23082 •474.6477 7 •92.93878 687.8597 8 61.78486 808.7746 8 77.49262 477.8711 8 98.20058 691.2377 9 8Z0IM6 806.3544 9 77.75442 481.1055 9 93.46238 695.1266 10 62.80895 808.9451 10 78.01622 484.3607 10 93.72418 699.0262 11 82.G99D5 811.54ff7 11 78.27802 487.6068 11 98.98598 702,9868 174 CDtBOUBIL TABUB S 0F CMMCIMM (OontlmMdr). DlaoM in mats wad tweUftb*; m in Wft and immU Ma. Clreuif. Am. Ua. Cireunf. ArtA. Utu Ctreamf. Aim. Vt.In. Fe«t. Sq.ft. FUn. FMt. Sq.ft. Vt.Tn. Feet. 8q.ft •0 94.24778 706.a'j88 t6 109.9657 962.1128 400 125.6687 1266.6871 1 94.50958 710.7908 1 110.2175 966.6997 1 126.U266 1261.8785 2 94.77188 714.7841 2 110.4793 971.2975 2 126.1878 1267.1809 8 95.08318 718.6881 8 110.7411 975.9063 3 126.4491 1272.3941 4 95.29498 722.6536 4 111.0029 980.6260 4 126.7109 1277.6688 5 95.55678 726.6297 6 111.2647 985.1566 • 5 126.9727 1282.9684 6 95.81858 780.6166 6 111.6265 989.7980 6 127.2345 1288.2498 7 96.08038 734.6145 7 111.7883 994.4504 7 327.4963 129&6662 8 96.34217 788.6233 8 112.0601 999.1187 8 127.7681 1298.8740 9 96.60397 742.6431 9 112.8119 1003.7879 9 128.0199 *1804.2027 10 96.86577 746.6787 10 112.6737 1008.4731 10 128.2817 1809.5424 11 97.12757 750.7152 11 112.fi3r>5 1013.1691 11 128.6435 1314.8929 SI 97.38937 764.7676 M 118.0973 1017.8760 41 128.8063 1820.25tt 1 97.65117 758.8810 1 113.3591 1022.6939 1 129.0671 1825.6267 2 97.91297 762.9052 2 113.6209 1027.3226 2 129.8289 1831.0099 8 98.17477 766.9904 8 113.8827 1032.0623 3 129.5907 1886.4041 4 98.43657 771.0865 4 114.1445 1036.8128 4 129.8626 1841.8091 5 98.69887 775.1984 5 114.4063 104L6748 5 130.1143 1847.2251 6 98.96017 779.8118 6 114.6681 1046.8467 6 130.8761 1862.6625 7 99.22197 783.4401 7 114.9299 1051.1800 7 130.6379 1868.0808 8 99.48877 787.6798 8 115.1917 1055.9242 8 130.8997 1363.6885 9 99.74557 791.7304 9 115.4635 1060.7293 9 131.1616 1868.9981 10 100.0074 795.8920 10 115.7153 1065.5458 10 131.4238 1874.4686 11 100.2692 800.0644 11 115.9771 1070.3728 11 131.6851 1879.9600 tt 100.5310 804.2477 87 116.2389 1075.2101 42 131.9469 1886.4424 1 100.7928 808.4420 1 116.5007 1080.0588 1 182,2087 1890.9458 2 101.0546 812.6471 2 116.7625 1084.9185 2 132.4705 1896.4698 8 101.8164 816.8632 8 117.0243 1089.7890 3 182.732S 1401.9848 4 101.6782 821.0901 4 117.2861 1094.6705 4 132.9941 1407.5208 5 101.8400 825.8280 6 117.5479 1099.5629 5 133.2569 1418.0676 6 102.1018 829.6768 6 117.8097 1104.4662 6 133.5177 1418.6254 7 102.3636 833.8365 7 118.0715 1109.3804 7 133.7796 1424.19a 8 102.6254 838.1071 8 118.3338 1114.8055 8 134.0413 1429.7787 9 102.8872 842.8886 9 118.6951 ni9StAib 9 134.8031 1436.8642 10 103.1490 846.6810 10 318.8569 1124.1884 10 184.6649 1440.9656 11 103.4108 850.9844 11 119.1187 1129.1462 11 134.8267 1446.5780 S8 103.6726 865.2986 88 119.3805 1134.1149 48 185.0885 1452.2012 1 103.9344 859.6237 1 119.6423 1139X)946 1 135.3603 1457.8858 2 104.1962 863.9598 2 119.9041 1144.0851 2 185.6121 1463.4804 8 104.4580 868.3068 3 120.1659 1149.0866 3 185.8739 1469.1364 4 104.7198 872.6646 4 120.4277 1154.0990 4 136.1357 1474.8082 5 104.9816 877.0334 5 120.6895 1159.1 2*?? 5 1S6.3975 1480.4810 6 105.2434 881.4131 6 120.9513 1164.1564 6 186.6593 1486.1697 7 105.5052 885.8037 7 121.2131 1169.2015 7 136.9211 1491.8698 8 105.7670 890.2052 8 121.4749 1174.2575 8 137.1829 1497.5798 9 106.0288 894.6176 9 121.7367 1179.3244 9 137.4447 1508.8012 10 106.2906 899.0409 10 121.9985 1184.4022 10 137.7065 1509.0835 11 106.5524 903.4751 11 122.2603 1189.4910 11 137.9688 1614.7767 S4 106.8142 907.9203 89 122;5221 1194.5906 44 138.2301 1520.6308 1 107.0759 912.3763 1 122.7839 1199.7011 1 138.4919 1526.2969 2 107.3377 916.8433 2 123.0457 1204.8926 2 138.7687 1532.0718 8 107.5995 921.8211 8 123.3075 1209.9550 8 189.0166 1687.8587 4 107.8613 925.8099 4 123.5693 1215.0982 4 189.2778 1548.6666 5 108.1231 930.8096 6 123.8811 1220.2524 6 189.6891 1549.4651 6 108.3849 934.8202 6 124.0929 1226.4175 6 189.8009 1556.2847 7 108.6467 989.3417 7 124.3547 1230.5935 7 140.0627 1561.1152 8 108.9085 943.8741 8 124.6165 1285.7804 8 140.3245 1666.9566 9 109.1703 948.4174 9 124.8783 1240.9782 9 140.6863 1572.8069 10 109.4321 952.9716 10 125.1401 1246.1869 10 140.8481 1578.6721 U 1 109.6989 957.6867 U 125.4019 1251.4065 11 141.1099 1584.5462 GIBCIiBS. 175 TAIUUB 8 OF €IB€I<BI^(Coiitila«0d). Wiaumm in unite mnd twelftlm; m in feet mnd incli DIa. Cireunf. Area. Dia. drenmf. Area. Dia. Circnnf. Area. Ftln. Feet. Sq.ft. Ft.Iii. Feet Sq.ft. Ft.Tn. Feet. Sq.ft. 46 141.8717 1590.4313 50 157.0796 1968.4964 56 172.7876 2375.8294 1 141.6885 1596.3272 1 157.3414 1970.0458 1 173.0494 2383.0344 2 141.8953 1602.2841 2 157.6032 1976.6072 2 173.3112 2390.2502 8 142.1571 1606.1518 8 157.8650 1983.1794 8 173.5730 2397.4770 4 142.4189 16140805 4 158.1268 1989.7626 4 173.8348 2404.7146 5 142.6807 1620.0201 5 168.3886 1996.8567 5 1740966 2411.9632 6 142.9426 1625.9705 6 158.6504 2002.9617 6 1743584 2419.2227 7 143.2048 1681.9319 7 158.9122 2009.5776 7 1746202 2426.4931 8 143.4661 1687.9042 8 159.1740 2016.2044 8 174.8820 2433.7744 9 148J279 1648.8874 9 159.4358 2022.8421 9 175.1438 2441.0666 10 143.9897 1649.8816 10 159.6976 2029.4907 10 175.4066 2448.8607 11 144.2515 1655.8866 11 159.9594 2036.1602 11 175.6674 2456.6887 46 144.5133 1661.9025 51 160.2212 2042.8206 66 175.9292 2463.0086 1 144.7751 1667.9294 1 160.4830 2049.5020 1 176.1910 2470.3446 2 145.0369 1678.9671 2 160.7448 2056.1942 2 176.4528 2477.6912 8 145.2987 1680.0158 8 161.0066 2062.8974 8 176.7146 2485.0489 4 145.5605 1686.0753 4 161.2684 2069.6114 4 176.9764 2492.4174 5 145.8223 1692.1458 5 161.5302 2076.8364 5 177.2382 2499.7969 6 146.0841 1698.2272 6 161.7920 2083.0723 6 177.5000 2507.1878 7 146.8459 1704.8195 7 162.0538 2089.8191 7 177.7618 25145886 8 146.6077 1710.4227 8 162.3156 2096.5768 8 178.0236 2522.0008 9 146.8696 1716.5368 9 162.5774 2103.8454 9 178.2854 2529.4239 10 147.1818 1722.6618 10 162.8392 2110.1249 10 1785472 2536.8579 11 147.8931 1728.7977 11 163.1010 2116.9153 11 178.8090 25443028 47 147.6649 17849445 63 163.3628 2123.7166 67 179.0708 2551.7586 1 147.9167 1741.1023 1 163.6246 2130.5289 1 179.3326 2569.2254 2 148.1785 1747.2709 2 163.8864 2137.8520 2 179.5944 2566.7030 8 148.4403 1753.4505 3 164.1482 2144.1861 8 179.8562 25741916 4 148.7021 1759.6410 4 1644100 2161.0310 4 180.1180 2581.6910 5 148.9689 1765.8423 5 1646718 2157.8869 5 180.3798 2589.2014 6 149.2257 1772.0546 6 164.9336 2164.7537 6 180.6416 2596.7227 7 149.4875 1778.2778 7 166.1954 2171.6314 7 180.9034 2604.2549 8 149.7492 17845119 8 165.4572 2178.5200 8 181.1662 2611.7980 9 150.0110 1790.7569 9 165.7190 2185.4195 9 181.4270 2619.3520 10 150.2728 1797.0128 10 165.9808 2192.3299 10 181.6888 2626.9169 11 150.5346 1803.2796 11 166.2426 2199.2512 11 181.9506 2634.4927 48 150.7964 1809.5574 68 166.5044 2206.1834 58 182.2124 2642.0794 1 151.0582 1816.8460 1 166.7662 2213.1266 1 182.4742 2649.6771 2 151.3200 1822.1456 2 167.0280 2220.0806 2 182.7360 2657.2856 8 151.6818 1828.4560 3 167.2898 2227.0456 3 182.9978 26649051 4 151.8436 18347774 4 167.5516 2234.0214 4 183.2596 2672.5354 152.1064 1841.1096 5 167.8134 2241.0082 5 183.5214 2680.1767 6 152.3672 1847.4528 6 168.0752 2248.0059 6 183.7832 2687.8289 7 152,6290 1853.8069 7 168.3370 2255.0145 7 184.0450 2695.4920 8 152.8908 1860.1719 8 168.5988 2262.0340 8 184.3068 2703.1669 9 153.1626 1866.5478 9 168.8606 2269.0644 9 1845686 2710.8508 10 163.4144 1872.9346 10 169.1224 2276.1057 10 184.-8304 2718.5467 11 153.6762 1879.3324 11 169.3842 2283.1679 11 185.0922 2726.2534 48 153.9380 1885.7410 64 169.6460 2290.2210 69 185.3540 2733.9710 1 1541998 1892.1605 1 169.9078 2297.2951 1 185.6158 2741.6996 2 164.4616 1898.5910 2 170.1696 2304.3800 2 185.8776 2749.4390 8 1647234 1905.0323 3 170.4314 2311.4759 3 186.1394 2757.1893 4 1549852 1911.4846 4 170.6932 2318.5826 4 186.4012 2764.9506 5 165.2470 1917.9478 5 170.9550 2325.7003 5 186.6630 2772.7228 6 156.6068 19244218 6 171.2168 2332.8289 6 186.9248 2780.5058 7 156.7706 1980.9068 7 171.4786 2339.9684 7 187.1866 2788.2998 8 166.0824 1987.4027 8 171.7404 2347.1188 8 187.4484 2796.1047 9 166.2942 1948.9095 9 172.0022 2354.2801 9 187.7102 2803.9205 10 156.6660 1960.4273 10 172.2640 2361.4523 10 187.9720 2811,7472 11 156.8178 1966.9569 11 172.5258 2368.6854 11 188.2338 2819.5849 176 CIBCLB8. TABI<E S OF €IRCIiE8->(Gonttiiii«d). Dlamsi In units and twelfths; a4s In feet and inches. DIa. Circumf. Area. Dia. Circumf. Area. Dia. Circomf. Ares. Ft.Iu. Feet. Sq. ft. Ft. 111. Feet. Sq.ft. Ft.Iij. Feet. Sq.ft. 60 188.4956 2827.4334 65 204.2085 8318.3072 70 219.9116 8848.4510 1 188.7574 2885.2928 1 204.4658 3826.8212 1 220.1733 8857.6194 2 189.0192 1 2848.1632 2 204.7271 3335.3460 2 220.4861 3866.7988 3 189.2810 2851.0444 8 204.9889 8848.8818 8 220.6969 8876.9890 4 189.5428 2858.9:^ 4 205.2507 8362.4284 4 220.9587 8886.1902 5 189.«04() ' '.^866.8397 5 205.5126 3360.9860 5 221.2206 3894.4022 6 190.0664 ' 2874.7536 6 205.7748 3369.6546 6 221.4823 8903.6262 7 190.3282 2882.6786 7 206.0861 8378.1889 7 221.7441 8912.8591 8 190.5900 ; 2890.6143 8 206.2979 3386.7241 8 222.0069 3922.1089 9 190.8518 ! 2898.5610 9 206.5597 3895.8263 9 222.2677 3981.8506 10 191.1136 2906.5186 10 206.8215 8403.9876 10 222.6296 8940.6262 11 191.3754 ; 2914.4871 11 207.0833 3412.5605 11 222.7918 8949.9087 •1 191.6372 2922.4666 66 207.3451 3421.1944 71 223.0681 3969.1921 1 191.8990 2930.4569 1 207.6069 8429.8392 1 228.8149 8968.4915 2 192.1608 2938.4r)81 2 207.8687 3438.4950 2 223.6767 8977.8017 3 192.4226 , 2946.4703 3 208.1806 8447.1616 3 228.8885 3987.1229 4 192.6843 2954.4934 4 208.3928 8455.8392 4 224.1008 8996.4549 n 192.9461 , 2962.5273 6 208.6641 8464.5277 5 224.8621 4006.7970 C 193.2079 i 2970.5722 6 208.9159 8473.2270 6 224.6239 4016.1618 7 193.4697 2978.6280 7 209.1777 3481.9873 7 224.8867 4024 5165 8 19:^.7815 2986.6947 8 209.4895 3490.6686 8 225.1475 4088.8022 9 193.9933 2994.7723 9 209.7018 8499.8906 9 1 225.4093 4048.2788 10 194.2551 3002.8608 10 209.9631 8508.1386 10 1 225.6711 4052.6768 11 194.5169 3010.9602 11 210.2249 *351 6.8876 11 225.9329 4062.084S 62 1 94.7787 i 3019.0705 67 210.4867 8525.6524 72 226.1947 4071.5041 ] 195.0405 3027.1918 1 210.7485 8534.4281 1 226.4566 4080.9848 2 195.3023 3035.3289 2 211.0108 3643.2147 2 226.7188 4090.3766 8 195.5641 3048.4670 8 211.2721 ; 8552.0128 1 8 226.9801 4099.8275 4 195.8259 3051.6209 4 211.5339 3560.8207 4 227.2419 4109.2906 5 196.0877 3059.7858 5 211.7957 a569.6401 5 1 227.5037 4118.7648 6 196.3495 3067.9616 6 212.0575 8578.4704 6 227.7656 4128.2491 7 196.6113 8076.1483 7 212.3198 3587.8116 7 228.0273 4187.7448 8 196.8731 :S084.8459 8 212.5811 8596.1687 8 228.2891 4147.2514 9 197.1349 8092.55441 9 212.8429 8606.0267 9 228.6509 4156.7689 10 1 197.3967 3100.7738 10 213.1047 3618.9006 10 228.8127 4166.2978 n 1 197.6585 3109.0041 11 213.8665 8622.7864 11 229.0746 4175.8866 68 197.9203 3117.2453 68 213.6283 3631.6811 78 229.8868 4185.8868 1 198.1821 3125.4974 1 213.8901 3640.6877 1 229.5981 4194.9479 2 198.4439 3183.7605 2 214.1519 8649.6068 2 229.8699 4204.5200 3 198.7057 3142.0344 3 214.4187 8(h')8.4887 3 280.1217 4214.1029 4 198 9675 3150.3193 4 214.6755 3667.3781 4 230.8886 4228.6968 6 199.2293 3158.6151 5 214.9373 8676.8284 5 280.6458 4283.8016 6 199.4911 3166.9217 6 215.1991 8685.2845 6 230.9071 4242.9172 7 199.7529 3175.2393 7 215.4609 3694.2566 7 231.1689 4252.5488 8 200.0147 8183.5678 8 216.7227 8708.2396 8 231.4307 4262.1818 9 200.2765 3191.9072 9 215.9845 ^ 3712.2385 9 281.6925 4271.8297 10 200.5383 3200.2575 10 216.2463 ; 3721.2388 10 231.9643 4281.4890 11 200.8001 3208.6188 11 216..'>081 1 3730.2540 11 282.2161 4291.1592 64 201.0619 3216.9909 60 216.7699 3739.2807 74 282.4779 4800.8408 1 201.3237 8225.3739 1 217.0317 3748.8182 1 282.7397 4310.6824 2 201.5855 3283.7679 2 217.2935 3757.86(>6 2 238.0015 4320.2858 S 201.8473 3242.1727 8 217.5558 3766.4260 3 288.2683 4829.9492 4 202.1091 3250.5886 4 217.8171 3776.4962 4 283.6261 4839 6789 5 202.3709 3259.0151 5 218.0789 3784.5774 5 288.7869 4849.4096 6 202.6327 3267.4527 6 218.3407 3798.6696 6 234.0487 4859.1562 7 202.8945 3275.9012 7 218.6025 3802.7726 7 234.8105 4368.9186 8 203.1563 3284.3606 8 218.8643 3811.8864 8 234.6728 4878.6820 9 203 4181 3292.8809 9 219.1261 8821.0112 9 2^.8341 4388.4618 10 203.6799 8801.8121 10 219.1^79 38.30.1469 10 235.0959 4896.2S15 11 203 9417 8309.8042 11 219.6497 3839.2936 11 235.3576 4408.0626 CIRCLES. 177 TABI.1: S OF €IB€I.Efll-(ContIniied). Dlams In imtta and twelftbat w In feet and Inelies. PU. Cirvnnif. JLrfMU Dia. Cireunf. Area. Dia. Clrenmf. ArMU run. teeU Sq.ft. FUn. Veet. 8q.ft. Ft.In. Feet. Sq.ft. 96 235.6194 4417.8647 80 251.8274 5026J>482 86 267.0354 6674.5017 1 285.8812 4427.6876 1 251.5892 5037.0257 1 267.2972 5685.6337 2 236.1430 4437.5214 2 251.8510 6047.5140 2 267.5590 5696.7765 8 236.4048 4447.8662 8 252.1128 5068.0188 8 267.8208 5707.9302 4 286.6666 4457.2218 4 252.3746 5068.5284 4 268.0826 5719.0949 5 236.9284 4467.0884 6 2524»64 5079.0445 6 268.3444 5780.2706 6 2S7.1902 4476.9659 6 252.8982 5089.5764 6 268.6062 5741.4569 7 287.4520 4486.8548 7 253.1600 5100.1193 7 268.8680 5752.6543 8 2S7.7138 4496.7536 8 253.4218 5110.6731 8 269.1298 5763.8626 9 287.9756 4506.6637 9 253.6886 5121.2378 9 269.8916 5775.0818 10 238.2374 4516.5849 10 253.9454 5131.8184 10 269.6534 5786.3119 11 288.4992 4526.5169 11 254.2072 5142.3999 11 269.9152 5797.5529 n 238.7610 4586.4598 81 254.4690 5152.9974 86 270.1770 5808.8048 1 289.0228 4546.4136 1 254.7808 5163.6057 1 270.4388 5820.0676 2 289.2846 4556.3784 2 254.9926 5174.2249 2 270.7006 5831.3414 8 289.5464 4566.3540 8 255.2544 5184.8551 8 270.9624 5842.6260 4 289.8082 4576.3406 4 255.5162 5195.4961 4 271.2242 5853.9216 6 240.0700 4586.3380 5 255.7780 5206.1481 5 271.4860 5865.2280 6 240.8318 4596.3464 6 256.0398 5216.8110 6 271.7478 5876.5454 7 240.5936 4606.3657 7 256.8016 5227.4847 7 272.0096 5887.8787 8 240.8554 4616.3959 8 256.5634 5238.1694 8 272.2714 5899.2129 9 241.U72 4626.4370 9 256.8252 5248.8650 9 272.5332 5910.5680 10 2a^790 4636.4890 10 257.0870 5259.5715 10 272.7950 5921.9240 11 241.6408 4646.5519 11 257.8488 5270.2889 11 273.0568 5983.2959 17 241.9026 4656.6257 81 257.6106 5281.0178 87 278.8186 5944.6787 1 242a644 4666.7104 1 257.8724 5291.7565 1 278.5804 5956.0724 2 242.4262 4676.8061 2 258.1342 5302.5066 2 278.8422 5967,4771 8 242.6880 4686.9126 8 258.8960 5313.2677 8 274.1040 5978.8921 4 242.9498 4697.0801 4 258.6578 5324.0396 4 274.8658 5990.3191 6 248.2116 4707.1584 6 258.9196 5334.8225 5 274.6276 6001.7564 • 248.4784 4717.2977 6 259.1814 5345.6162 6 274.8894 6018.2047 7 248.7862 4727.4479 7 259.4432 5356.4209 7 275.1512 6024.6689 8 248.9970 4787.6090 8 259.7050 5367.2365 8 275.4130 6086.1340 9 244.2588 4747.7810 9 259.9668 5378.0630 9 275.6748 6047.6149 10 244.5206 4757.9639 10 260.2286 5388.9004 10 275.9366 6059.1068 11 244.7824 47681577 11 260.4904 5399.7487 11 276.1984 6070.6087 38 246.0442 477&3624 88 260.7522 5410.6079 88 O: 276.4602 6082.1284 1 245.8060 47885781 1 261.0140 5421.4781 1 276.7220 6093.6480 2 246.6678 4798.8046 2 261.2758 5432.8691 2 276.9838 6105.1885 8 245.8296 4809.0420 8 261.5376 5443.25U 8 277.2456 6116.7800 4 246.0914 4819.2904 4 261.7994 5454.1589 4 277.5074 6128.2878 5 2463582 4829.5497 6 262.0612 6465.0677 5 277.7692 6189.8556 6 246.6150 4839.819B 6 262.3230 5475.9923 6 278.0309 6151.4348 7 246.8768 4850.1009 7 262.5848 5486.9279 7 278.2927 6163.0248 8 247.1386 4860.3929 8 262.8466 5497.8744 8 278.5545 6174.6258 9 247.40Q4 4870.6058 9 2631084 55088318 9 278.8163 6186.2877 10 247.6623 4881.0096 10 263.3702 5519.8001 10 279.0781 6197.8605 U 247.9240 4881.8348 11 263.6320 5580.7793 11 279.8899 6209.4942 n 24&1868 4901.6699 84 263.8938 5541.7694 89 279.6017 6221.1889 1 24&4476 4912.0165 1 264.1556 5552.7706 1 279.8635 6232.7944 2 248.7094 4922.8739 2 264.4174 5563.7824 2 280.1253 6244.4608 8 248.9712 4982.7423 8 264.6792 5574.8058 8 280.3871 6256.1882 4 249.2K0 4943J215 4 264.9410 5585.8390 4 280.6489 6267.8264 5 249.4948 4958.5117 5 266.2028 5596.8887 5 280.9107 6279.5266 6 249.7566 4968.9127 6 265.4646 5607.9892 6 281.1725 6291.2856 7 250.0184 4974.8247 • 7 265.7264 5619.0057 7 281.4343 6302.9566 8 250.2802 4984.7476 8 265.9882 5630.0881 8 281.6961 6314.6885 9 250.^420 4996.1814 9 266.2500 5641.1714 9 281.9579 6326.4813 10 250^088 5005.6261 10 266.5118- 6652.2706 10 282.2197 6888.1860 11 25L0668 6016.0817 11 266.7736 5663.3807 11 282.4815 6849.9496 1? I/O CIBCLES. TABUS 8 OF €IRCI<B»-<CoBtinQed% Dlams in anlts and twelfUisi m in Wtet nnd lnck( DIa. Cirenmf. Area. Dia. Cireunf. Area. Dla. dreumf. Area. Ft.In. Feet. Sq. ft. Pt.In. Feet. Sq. ft. Ft.In. Feet. Sq. ft. •0 282.7433 6361.7251 98 5 29a4771 6858.9134 96 9 908.9491 7851.7686 1 283.0051 6378.5116 6 293.7889 6866.1471 10 804.2109 7864.4881 2 283.2669 6885.8089 7 294.0007 6878.8917 11 804.4727 73770196 3 283.5287 6397.1171 8 294.2625 6890.6472 97 804.7345 7889.811S 4 283.7905 6408.9863 9 294.5243 6902.9135 1 804.9963 7402.5140 5 284.0623 6420.7663 10 294.7861 6915.1908 2 8a').2581 7416.2277 6 284.3141 6432.6078 11 295.0479 6927.4791 8 905.5199 7427.9522 7 284.5759 6444.4592 04 295.8097 6989.7782 4 805.7817 7440.6877 8 284.8377 6456.3220 1 295.5715 6952.0682 5 806.0485 7458.4840 9 285.0995 6468.1957 2 295.8333 6964.4091 6 .306.8053 7466.1913 10 285.3613 6480.0803 8 296.0951 6976.7410 7 306.5671 7478.9595 11 285.6231 6491.9758 4 296.3569 6989.0887 8 806.8289 7491.7386 tl 285.8849 6503.8822 5 296.6187 7001.4874 9 807.0907 7504.6286 1 286.1467 6515.7995 6 296.8805 7013.8019 7026.1774 10 807.3525 7517.8294 2 286.4085 6527.7278 7 297.1423 11 807.6143 7530.1412 8 286.6703 6539.6669 8 297.4041 7038.5638 98 807.8761 7542.9640 4 286.9321 6551.6169 9 297.6659 7050.9611 1 908.1879 7555.7976 5 287.1989 6563.5779 10 297.9277 7063.8693 2 808.3997 7568.6421 6 287.4657 6575.5498 11 296.1895 7075.7884 8 808.6615 7581.4976 7 287.7175 6587.5325 Wi 298.4513 7088.2184 4 808.9238 7594.8689 8 287.9793 6599.5262 1 298.7131 7100.6593 5 809.1851 7607.2412 9 288.2411 6611.5808 2 298.9749 7118.1112 6 809.4469 7620.129S 10 288.5029 6623.5468 8 299.2367 7125.5739 7 309.7087 7688.0284 11 288.7647 6685.5727 4 299.4985 7138.0476 8 809.9705 7645.9884 fS 289.0265 6647.6101 5 299.7603 7150.6321 9 810.2323 7658.8598 1 289.2883 6659.6588 6 800.0221 7163.0276 10 810.4941 7671.79n 2 289.5501 6671.7174 7 .300.2839 7175.5340 11 810.7559 7684.7888 •« 289.8119 6683.7875 8 300.5457 7188.0518 99 311.0177 7697.6874 4 290.0737 6695.8684 9 300.8075 7200.6794 1 311.2795 7710.6519 5 290.3355 6707.9603 10 301.0693 7213.1185 2 311.5418 7723.6274 6 290.5973 6720.0630 11 801.8811 7225.6686 8 811.8031 7736.6187 7 290.8591 6732.1767 96 301.5929 7238.2295 4 812.0649 7749.6109 8 291.1209 6744.8013 1 301.8547 7250.8018 5 312.3267 7762.6191 9 291.3827 6756.4368 2 302.1165 7263.8840 6 312.5885 7775.68R2 10 291.6445 6768.5882 8 302.3783 7275.9777 7 812.8503 7788.6681 11 291.9063 6780.7405 4 302.6401 7288.5822 8 813.1121 7801.7090 M 292.1681 6792.9087 5 302.9019 7301.1977 9 313.3739 7814.7606 1 292.4299 6805.0878 6 303.1637 7313.8240 16 313.6857 7827.8286 2 292.6917 6817.2779 7 303.4255 7326.4613 11 318.8975 7840.8971 S 292.9535 6829.4788 8 303.6873 7339.1095 100 314.1593 7858.9816 4 293.2153 6841.6907 Diam. Ciroamf, Diam, Ciroamf, Diam, iMk. Ibot. 1 Ineh. ■ Ibot. Ineh. 1-64 .004091 7-32 .057269 27-64 1-32 .008181 15-64 .061359 7-16 8-64 .012272 ili .065450 29-64 1-16 .016362 .069640 16-32 «-64 .020463 0-82 .073631 81-64 8^ .024644 10-64 .077722 8^ 7^ .028634 6-16 .081812 Hu .032726 21-64 .086908 17-32 U)36816 11-32 .089994 86-64 6-32 .040908 23-64 .094084 9-16 11-64 .044997 1^ .098176 87-64 8-16 .049087 .102266 19-32 IM4 .068178 13-32 .106366 39-64 Giroamr, _lbat._ .110447 .114637 .118628 .122718 026809 030900 034990 039081 048172 047262 061868 056448 059534 Diam. 6-8 41-64 21-32 43-64 11-16 46-64 28-82 47-64 Jii 26-32 61-64 13-16 Ciroamf, .163626 067715 .171806 .176896 079987 084078 .188168 092269 .196360 .200440 .204531 .208621 .212712 Diam, Inelu 63-64 27-32 65-64 7-8 67-64 29-32 69-64 15-16 61-64 81-32 68-64 1 Cireomr* .216808 .220808 .224064 .229074 .238161 .237266 .241346 .246487 .249688 .263618 .267700 .261799 dBCULAB ARCS. CIBCVI.AB ARCS. 179 S^itf.l BnlM for Fig. 1 apply to all arei •qnal to, or l€w than, a Bemi-circle. ** " Fig. i «« *• «• or greater than, a ■emi-cirelt^ Cltordy a b, ot -vrlfcole aircy mdb, 2 X \/raditi«s — (radiua — rise)^. Fig. 1. 2 X \/iadia«> — (rise — radiiis)^. Fig. 2. 2 X \/rise X (2 X radius — rise). Figs. 1 and 2. 2 X radius X >ine cf}4acb. Figs. 1 and 2. rise — 2 X Figs. 1 and 2. tangent of a b d.* 2 X dbl X cosine of a&d.* Figs. 1 and 2. 2 X >/db9 — rise*. Figs. 1 and 2.§ approximately 8 X db^ — 3 X Length of arc adb^. Fig. 1. — 2 « radius X JjmiMjgOkf adb, arc a d 5 in degrees 360 . Figs. 1 and 2; •^ .01746 X radius X arc a d b in degrees. Figs. 1 and 2. drenmference of circle — length of mnaU arc subtending angle aeb. Fig. 2. . 8 X d&§ — ohordaft.** ^ , approximately 5 Fig. 1. •abdis — ^ofttie angle a b, subtended by the arc. In Fig. 2 the latter angle exceeds 180°. 2<I6 — chord of dib^ or of half ad&— \/rlBe« + (i^ab)*. Figs. 1 and 2, flf rise — ^ chord, .4 « ..833 « .8 « ••If rise — .6 chord .4 « .833 ** .8 •* multiply the rsaolt l^ 1.036 1.0196 1.0114 l.t083 multiply the rasnli by 1.012 1.0066 1.00B8 1.0t28 If rise — .26 chord, .2 « .126 « .1 « If lisa — .26 chord .2 « .126 « .1 « multiply the result by 1.0044 1.0021 1.00036 1.00016 multiply the result Hr 1.0015 1.0007 1.00012 1.00006 180 OIBGULAB ABGB. Ooattnwd from p. 179. Bolts for Fig. 1 appij to all arcs equal to or less than a semi-circle. M u pig^ 2 ** ^ ** or greater than a 8emi<clrclo. R adimiy eOfC^pi or cbp . (H «<>)« + ri»e« ^ ij-jga. 1 and 2. 2 X rise . ^^§_ , Pigs. 1 and 2. 2 X rise %ab , Figs. 1 and 2. sine of ^ a e 6 1 — cosine of ^ a e 6 - ^<^^? , ngs. 1 audi, sineof >^6e<i| risedc 1 4- cosine of ^ a o d f , FIg.x Rifle* or middle ordliisite» d9p radius — \/radius« — Q^ab]^, Fig. 1. radius + \/ndiwfl — Q^aS^, Fig. 2. radius X (1 — cosine of 6 e d ||), Fig. 1. radius X (1 + cosine of b e d ||),t Fi^. 2, ^^^ , Figs. 1 and 2. 2 X radius liab X tangent cf abd,* Figs. 1 and S. approximately ^^^^ ' '*«• 1- 2 X radius When radius — chord a b, the resftit is 6.7 parts In lUO too shwrt. ** *^ — 3X chord a b, the result is 0.7 parts in 100 too ahoft; Side ordimatey as n <» = >/radiu8> —en* + rise — radlni, Figfc 1 and S. = proximately /^ ^^ . Fig. l.t * a b d is s 3>^ of the angle acb^ subtended by the arc. t Strictly, this should read 1 mimu cosine; but the ooslBes of angles between 90* and 270^ must then be regarded as mimu or negative. Our rule, therefore, amonnta to the same thing. ^db '^ chord of dib, or of half adb, — \/rUe» + (^a^)'- Xig>- 1 and 2. I be d — half the angle eob subtended by the are, la Fig. 2, the latter angle exceeds 180°. \ When radius = chord a b, this makes de 6.7 parts in 100 too short '< «< = 3 X chord a b, this makes d e 0.7 parts in 100 too short The proportionate error is greater with the side ordinates. CflBCDLAB ABGB. 181 Angley acb, sabtended lay Arc* adb. An angle and its supplement (as 5 e « and bed, Fig. 2) have the same «ine, the same cosine and the same tangmU. CAUtlon. The following sines, etc., are those of only half aob. fflneof J^oc6 — H?^ . Figs.land2. radius radius — rise rise — radius radius , ng.2. Cosineof Jiac6 J^aST" *^«-^* Tangent of >^ a c6 ^,^"^^ , Tig.l; - ^ ^**^^. , Fig. i ^* radius ~ rise ® * rise — radius ' Versed sine of ^ a« 6 ■— rise radius , Figs. 1 and 2. Vo dMwrilM ttie mve sf m elrde too Isury* ftnr Um dl-rtders. Let a c 1m the choordy and o b the height, of the required arc, as laid down om the drawing. On a separate sMp af paper, «• m n, drawa c. o h. and aft. •Ibo b e, parallel to the chord a c. It Is well to make b«,and b e, each a little longer than a b. Then cut off the paper earefhUy along the lines 8 h and 6 «, so as to leare renaaining only the strip tabemn. Now, if the straight sides s b and 6 e be applied to tlie drawing, so that any narts of them shall touch at the same time the points a and 6, or b and e, the point h on the strip will be in ttie circumference of the arc, and may be prldced off. Thus, any number of points in the arc may be found, and afterward united to form the corre. 31d Hi ottiodt Draw tteOMn a b; the rise re; and a 0^6 a From c with radios e r describe a drele. Make each of the arcs o I and i I equal to ro or r i; and draw c C cL DiTide eC, eZ, er, each into half as many equal parts as the curre is to be divided into. Draw the lines 61, 52, 2>3; and a4, a5, a6, extended to meet the first ones at e, «, A. Then e, «, A, are points in one half the curve. Then for the other half, draw simUar lines flrom a to 7» 8, 9; and others from b to meet them, as before. Trace tte ennro by hand. 182 CIRCULAR ARCS. ^It DMj firaquentlj b* of um to 'afhattaiABjMedoi^nol ' azMeding 29<*, or in o<:her wordi, whou cluyrd be it of Uad tiadUm Umm iUriM, th* nUddle oratnate a o, will be one-half of a c, quite near enovgh fbr manj pap* poses; b c and < e boinir tangenta to the arc.f And Tica Tena, if in tnch an arc we make o c equal a o, then will o be, rwj nearly, the point at which tangents fh>m th« ends of the arc will meet. Also the muUlle oxdlnate n, ot thm ikmlt uno ob,or ott will be approximately 3^ of a ft, the middle ordinate of the whole arc. Indeed, this last obserTadon will apply near enough for many approximate uses even if the arc be as great as 46°; for if in that case we take ^ of o a fbr the ordinate n, n wlU then be but 1 part in 1U3 too small; and therefore the principle may often be used in drawings, for finding points in a curve of too great radius to be drawn by the diTiders ; for in the same manner, V^ of n will be the middle ordinate for the arc n h or n o; and so on to any extent. Below will be f>uud a table bjr nrldelk tbe rlae or middle ordliuite ot a ludf mrc can be obtained with greater accuracy when required for more exact drawings. CIRCUIjAR arcs in FBSMiUKlIT ITSIB. The fifth column is of use for finding points for drawing arcs too \argB fbr tiM beam-compass, on the principle giren above. In even the largest cfllce drawings it will not be necessary to use more than the first three decimals of the fifth column ; and after the arc is subdirided into parts smaller than about 86° each, the first two decimals .25 will generally su£Bce. OriginaL BlM For ForriM BiM For Fer in De(r«ei For nA length of of half In Dogreei For rad length of rlMoff paru in whole mult rise aro malt aro paru in whole multrlM aro nalt halfara of •ro. by oborA mnltriM of are. iv ehord bibIS dioid. • by by sherd. bj ti—hf 1-60 o / 9 9.76 313. 1.00107 .2601 u o / 66 8.70 6^ 1.04116 • .2688 1-46 10 10.76 263.626 1.00132 .2501 63 46.90 6.626 1.06366 .2649 1-40 11 26.98 200.6 1.00167 .2602 .165 68 63.63 6.70291 1.06288 .2667 1-36 13 4.92 163.625 1.00219 .2502 1-6 73 44.89 6. 1.07260 .26t6 1-30 15 16.38 113. 1.00296 .2503 .18 79 11.73 4.36803 1.08428 .2676 1-26 18 17.74 78.626 1.00426 .2504 1-6 87 12.34 3.626 1.10847 .2693 1-20 22 60.54 60.6 1.00666 .2506 .207107 90 3.41422 1.11072 .2699 1-19 24 2.16 46.026 1.00737 .2607 .226 96 64.67 2.96913 1.12997 .2616 1-18 26 21.65 41. 1.00821 .2508 .2^6 106 16.61 2.6 1.16912 .2639 1-17 26 60.36 36.626 1.00920 .2609 116 14.69 2.15289 1. J 9083 .2666 1-16 28 30.00 82.6 1.01088 .2510 .3 123 6130 1.88889 1.22496 .2692 1-16 30 22.71 28.626 1.01181 .2611 ^ 134 46.62 1.626 1.27401 .2729 1-14 32 31.22 26. 1.01366 .2613 144 30.08 1.43827 1.32413 .2766 1-13 34 69.08 21.626 1.01671 .2516 .4 154 38.35 1.28125 1.^322 .2808 1-12 37 60.»6 18.6 1.01842 .2517 .426 161 27.52 1.10204 1.42764 .2838 1-11 41 13.16 16.626 1.02189 .2620 .45 167 66.93 1.11728 1.47377 .2868 1-10 46 14.38 •18. 1.02646 .2625 .476 174 7.49 1.06402 1.62162 .2899 1-0 60 6.9II 10.625 1.03260 .2630 .6 180 1. 1.67080 .2929 V At 29° o • thus fbond will be bat about 8 parti too tiiort in 100. MENaUKATION, 183 bniStbB af elpenlH »f«s. If itrc«zce«da aaeialelrel«,H*p IS4 riMolii lu obon) ud bdibb dlrtd> Iha fal«tt bj lb« Uud. Ttaa In Uu MoBn dT balibli Iki MiUpIj llu Uit EiBbir bj ili> Itatlh of U> Jru lEonL * « omn <> Dt>U TABLE OF CIKOVLAB ABCS. H«nn». Uvi^i. P'lbu. I'Oiiftb*. H'ibli. l^nctbB. B'lliUr L«DBI^ 184 MESSUBATIOH. TABI.B «F CIKCIJI.AB ARCH— n arc of 1° if tbe eartb's Krent circle Is but 4.3354 feet loBcrr tbni lt> 1. lu lsiijiUiiiO.lt lindi>riiuniumnn. ■ulli'i«|»virli>lnil:^>HI.b10Siiill«. Polir 3*«><fT. MENBUKATIOI'. 185 T« Bad tbe Ie>|rUi of > circular src br tbe followliic teUe- I'EireTBS or circdi.ak abcs to bad i mi 186 MENSURATION, CmCVttAR BBCTORSy BINGMS, SBOmSRVS, SSTCX ^ * Area of a eironiar ■eetor, adbe^ Fig. A, arc adh X radlua o a. — area of entire drole X Fig. B. aro g d 6 In degrees; S60 Area of a clrc«lar ving. Fig. B, .—1 area of larger circle, d, — area of smaller one, a b. 1^ — .7854 X (sam of diams. cd + ah)X (cUfil of diams. e d^a 6.) — 1.5708 X thickness e « X *<i°^ <^ diameters « d and a h. To And. the rmdi«a of a clrele -vrhleli aliall have the aanie as a giyrevk elrciilar rln|^ c» dab. Fig. B, Draw any radius n r of the outer circle ; and from where said radins cuts tht bner circle at t, drew < « at right angles to it. Then will t « be the required ladins. Bresultl&y ea^mbd, of a circular rl»|ft Flf. ^ iM. V^ difference of diameters e d and a &. « ^ (diameter ed—w 1.2732 area of circle a 6.) Area of a eirenlar xone abed^ 0m area of circle m n — areas of segments am 5 and end, (for areas of segments, see below.) A circular Inne is a crescent-shaped figure, comprised between two arcs abe . and a o e of circles of different radii, a d and AM. of a drcvlar lume uheo ^ area of segment ahe — area of segment a oc^ (fix arcM of segments sea bcloir.) Pig.D. V»flndflio «f »olreiilMP it^mbodf Figi.O^Di. Area of Segment adbn, Fig. A (at top of page) ■■ Area of Sector a d 5 e — Am of Triangle a 5 0. •^^iiAroadb X tadinaa* — en X cbordafty. Vmwinff the area of a aeKment required to bo ent mtt gkvewk clrelcy €0 flnd tta chord suad rise. ^ IHTide the area hj the square of the diameter of the clrele : look for the qnotleot In th9 column of areas in the table of areas, opposite; taice out from the table Che corresponding number In the column of risei. Mnltipljr this nninbar bgr the diameter. The product will be the required rise, Thea ahord — 2 X V^ (dUmeter — rte) X MENSURATION. 187 TABUB OP AREAS OF CIB€UI«AR SEOlIEjnni, Fiffi C, Dl ' If the seyment exeeeda a semieirelef it* are« i* = %nm <a eireie— i of • aegmant whose riie Is = (dUm of eirelt — rise of giren segment). Dlaai of eird* * (eqiian ef hair ohord t> rise) 4* rise, whether the segment exeeeds a eemieirole or not. Rise Area= Rise .Areas Rise Areas Rise Area» Rise Areap* dlrhf (sqnare diYby (sqaare of diam) dlTby (Bonare of diam) diT by^ (square of diam) dirby (sqnare diamef ef diam) diam of diam of liaaof diam of of diam •irele. malt by oirole. moltbj eiioto. moltby einia. mult by oirole. .25^ BMritby .001 ..000042 .064 .021168 .127 .057991 .190 .103900 .166149 .002 .000119 .065 .021660 .128 .058658 .191 .104686 .254 .157019 .003 .000219 .066 .02'2;55 .129 .059328 .192 • .106472 .255 467891 .004 .000337 .067 .022663 .130 .059999 .193 .106261 .256 .168768 .005 .000471 .068 .023156 .131 .060673 .194 .107051 .257 469686 .006 .000619 .060 .023660 .132 .061349 .196 .107843 .258 460511 .007 .000779 .070 .024168 .133 .062027 .196 .108636 .269 461386 .008 .000952 .071 .024680 .134 .062707 .197 .109431 .260 462268 .009 .OOllSft .072 .025196 .136 .063389 .198 .110227 iS61 .168141 .010 .001329 .073 .025714 .136 .064074 .199 .111025 .262 464020 .011 .001633 .074 .026236 .137 .064761 .200 .111824 .263 464900 .012 .001746 .076 .026761 .138 .065449 .201 .112626 i264 .166781 JQIS .001969 .076 .027290 .139 .066140 .202 .113427 .266 .166688 mt .002199 .077 .027821 J40 .066833 .203 .114231 .266 487646 XH6 .002438 .078 .028356 .141 .067528 .204 .115036 .267 .188481 Me .002685 .079 .028894 .142 .068225 .205 .115842 .268 .109816 .017 .002940 .080 .029435 .143 .068924 .206 .116651 .260 .170202 .018 .008202 .081 .029979 .144 .069626 .207 .117460 .270 471090 .019 .003472 .082 .030526 .146 .070329 .208 .118271 .271 .171978 .020 .003749 .083 .031077 .146 .071034 JHOd 419084 .272 .172868 joai .004032 .084 .031630 .147 .071741 .210 419898 .273 .173768 JOZ .004322 .086 .032186 .148 .072450 .211 420718 .274 474660 JOSS .004619 .086 .032746 .149 J073162 .212 .121530 .276 .176542 J024 .004922 .087 .033308 .160 .073876 .213 422348 .276 476486 J0fi6 .005231 .088 .033873 .181 .074590 .214 423167 .277 477830 JM .005546 .089 .034441 .152 .076307 .216 .123988 .278 478226 Ml .005807 .090 .035012 .163 .076026 .216 424811 .279 479122 xas .006194 .091 .035586 .164 Wfl747 .217 .126634 .280 480020 M9 .006627 .092 .036162 .165 .077470 .218 426469 .281 .180918 J06O .006866 .003 .036742 .166 .078194 .210 .127286 .282 481818 JOSL .007209 .094 .037824 .157 .078921 .220 428114 .283 482718 M2 .007660 .096 .037909 .168 .079660 .221 428948 .284 488619 JOBS .007913 .006 .038497 .169 .080380 .222". .129778 .286 484622 J084 .008273 .097 .039087 .160 .081112 .223 430606 .286 .186426 j066 .008638 .098 .039681 .161 .081847 .224 431488 .287 486329 JOM .009006 .099 .040277 .162 .062682 .225 .132278 .288 487236 .037 .009388 .100 .040875 .163 .088320 .226 483109 .289 488141 JOSS .009764 .101 .041477 .164 .084060 .227 .133946 .290 489048 .080 .010148 .102 .042081 .165 .084801 .228 434784 .291 .189956 J040 .010638 .103 .042687 .166 .085545 .229 .136624 .292 .190866 041 .010932 .104 .043296 .167 .086290 .230 .136466 .293 .191774 J042 .011831 .106 .043908 .168 .087037 .231 .137307 .294 492685 .048 ^11734 .106 .044623 .169 .087785 .232 .138151 .296 493597 J044 .012142 .107 .045140 .170 .088536 .233 438996 .296 494509 .046 .012555 .108 .045759 .171 .089288 .234 439842 .287 .196428 .046 .012971 .109 .046381 .172 .090042 .235 140689 .298 496337 j047 .013303 .110 .047006 .173 .090797 .236 .141538 .299 497262 .048 .013818 .111 .047633 .174 .091556 .237 .142388 .300 .198168 J04» .014248 .112 .048262 .175 .092314 .238 443239 .301 .199086 .060 .014681 .118 .048894 .176 4»8074 .239 .144091 «302 .200008 JO&I .016110 .114 .049529 .177 .093837 .240 .144945 .308 .200922 M>2 .016661 .115 .060165 .178 .094601 .241 .145800 .804 .201841 J06» .016008 ai6 .060805 .179 .095367 .242 446656 .366 .202762 J064 .016468 J17 .061446 .180 .096135 .243 .147513 .306 .203688 j06§ .016013 .118 .062090 .181 •090804 .244 448371 .307 .204606 iNM M79n 419 .062737 .182 .097675 .246 .149231 .308 .206628 jm .017881 .120 .063886 .183 .098447 .246 .160091 .309 .206462 MB .018907 .121 .0640:7 .184 .099221 .247 460953 .310 .207376 JOM .018766 .122 .064690 .186 .099997 .248 461816 .311 .208302 JIMO .019188 428 .066846 .186 .100774 .249 452681 .312 .209228 jOd .oime J24 .066004 .187 .101553 .250 463546 .313 .210166 jm /mm 096 .066664 .188 .102.334 .261 .154413 .314 .211083 Ml iMMBI Jfl6 .087827 .189 J03116 1 .262 .166281 .816 .212011 188 MENSURATION. TABLK OF AKEAS OF CIRCDE.AK SBONEVTS-tCoHTHiDH: Urn 1« A««_ dl.BT irdi™ «lui< orai>~i ^nli. "^imi .363 i!73« .380 .383603 '427 mint. Biujt 56e730 .284H9 .320940 406 5677 2S £li .314S0'J .302 JiU .369723 isieeea .3607a il7«» .!BaB3.; .SOS 4«9 .3fll7M xa ^86W XSi ^wee isao .ae; !363IU Mt 33SMM .Ml su SMHli -390 592390 M» iaZMTS iVi !36fl711 .tzg^S .8M .168385 .330S6fe .367710 MS .2M1M .366 JJB9SM •402 .331861 jm ,236094 too .ismi !38383« .?I070S ^1 .2«S249 ■406 .834829 JS2 iersi 406 .33S .300238 !4S1 571 .301221 .837810 sitm .S3B '409 xe JSl !»l^04e 504171 ^143 5T7T01 jaa M17SB xss MO Ml liwai is78 .i710Bl A\t 1307126 JOSllD 462 I4477S 4m1 asijoo MS .affi3«9 .309096 M673S .382700 Ma .238319 .380 464 M* jawss .381 .ii*sa -418 aiiow 466 !492 :384eM .M0Z19 .848766 ,3S68» !3S3 .38Se»0 Ml .3S4 ^77748 J21 468 .381390 1213074 .38* .316017 ■.3fil7«. 5<W«90 MS .383 562142 .389300 Xba .24U80 jHoero J»i 3MaS6 .28HM3 426 .317981 !4«2 .364736 !499 .S913W Mi .MflSM .380 .118970 4«3 566733 JiOO xMm ELLIPSE (page 139). Focal dlBMiice^/0 = HENSURATIOir. 189 THK BIiI.IPSfi» An «B!tM« Is m enrra, • «««, Fig I. formed by an obllqae Mctioa of eltlMr • oone or s eylinder, paaa* Ins throngh Ita ourred Mrfaee, withoat cattiog the base, lu nature la luoh that if t«o linei, aa n/ and n g. Fig. 8, be drawn from any point n in Ita periphery or etraamf, to two oertain points/ nnd g, in iu long diam o w, (and called the foei of the eiUpie,) their ram will be eqnal to that of any other two lines, as i/, and b g, drawn from any other point. a« 6, in the clreumf, to the fooi/aad ^j slao the snm of any two snch lines will be equal to the long diam « w. The line e w diriding the ellioso Into two eqnal parte lengthwise, is oalled its transverse, or major axis, or long diam ; and • i, whieh dirtdee it equally at right-angles to e io, is called the oonjogate, or minor axis, or diort 41ain. To find the position of the tool of an ellipse, from either end, as 6, of the short diam, memsnre olf the diets ft /and 6 g. Fig S, each equal to o c, or one-haif the long diam. The parameter of an ellipse is a oertain length obtained thus ; as the long diam i short diam : : short diam : parameter. Any line r v, or • d, Fig S, drawn from the eireamf* to, and at right angloa to, eliher diam, is ealled an ortUnau; and the parts e v and 9W,b» and • «, of that diam* between the ord and the eiroumf, are oalled al^teUam, or a&seiseei^ To flnil tlie leufftli of any ordinate, rvovsd, drawn to eitbetf dianif e W or h a* Knowing (h« ahecisa, « • or « a, and tiM two diams, e w, ft •{ ew*:fta<::cvXvwiFA ftd^i««!*::fr« X « a:g<i>. To lind the elreumf of an elHpse. Mathe— HelnM have fhmisked praodeal men with no simple working rale Ibr this pvrpoae. The •e-ealled appvMdmate mlea do not deserre the name. They are as foUowa, D being the long diam ; 4 the aiiorteino. RvLB 1. Circamf =8.141« R±A. • Rvlb S. S.M16 / f^^^-\ • Buu t. «.2ai6y' DS^hP: thte if tiie nme aa Bnle 2, bnt In a dllT shape. Sou4.2X|/ DS+ 1.1874 A Now, in an elUpse vhoae long and short dlams are 10 and S, the oirenmf Is MtnaUy 11, very approximately; bnt rule 1 (ires it = 18.85 ; rale 2, or 3, == 22.65 ; and rule 4. =: 30.68. Again, if the diams-be 10 and 6, the dr. •omf aotnallT = 25.50; but rule 4 gives 24.72. These examples show that none of the rales nsnaUy SiT0n are reliable. The following one by the writer, is snfflclently exact for ordinary pnrpoaes; Ml Mag iasrrer probably more than 1 part in 1000. When D la not more than 6 ttaass as long as 4, If D ezeeeda 5 times if, then in- fr stead of dividing (D — d^ by 8.8, div i^ by Si m the number in ibis table. o The following rule originated with Mr. M. Arnold Pears, of New South Wales, Australia, s;«S«««SSm68SSS!:fl« stetSkeisteCaieiSeisieiee^ee and was by him kindly communicated to the author. Although not more accu* rate than our own, it is much neater. 3.1416 d + 2(D — d) — d(D — d) Circumf V<(D -f d) X (D + 2d) The following table of senii»elllptle arcs was prepwvd by oar niik To nse this table, div the height or rise of the are, by its span or ehord. The qnet will be the height of an are whose span is 1. Find this quot in the oolnmn of heights ; and Uke out the oorresponding number ft*om the ool. of lengths. Halt this number by the actoal span. The prod will be thereqd lenRth. When the height becomes .500 of the chord fas at the end of the table) the ellipse beeomee a eirole. When the height exceeds .500 of the chord, as in a b e, then take a o, or half the ehord, as the rise ; and dir this rise by the long diam 6 d, for the qnot to be looked ror in the ool of heights ; and to be mult by long diam. We tfens get the aro had, which is evidently equal to a 6 c 190 MENSUIUTIO>. TABI.E OF I^ENOTHB OF 8EMI.EI«I«IPTI€ ABCB. ftnrlglnal4 Height Lengtl^a Hdght Lengths Height Length v Height Lengths •I'SlAn. spanxby . •A'lpftn. ■pan X by •fr span. ■pan X by 4- ■pan. ■pan X by JOOb 1.000 .130 1.079 .266 1.219 .880 1.390 M 1.001 .136 1.084 .260 1.226 .385 1.897 .015 1.002 .140 1.089 . .266 1.233 .890 1.404 .02 1.003 .145 1.094 ^0 1.239 .396 1.412 026 1.004 .160 1.099 .276 1.245 .400 1.419 .03 1.006 .166 1.104 .280 1.262 .406 1.425 .036 1.008 .160 1.109 .286 1.259 AIO 1.434 X)4 1.011 .166 1.116 .290 1.265 .416 1.441 X)46 1.014 .170 1.120 .295 1.272 .420 1.44P .06 1.017 .176 1.126 .300 1.279 .425 1.456 .066 1.020 .180 1.131 .306 1.286 .430 1.464 .06 1.023 .186 1.137 .310 1.292 .436 1.471 .066 1.026 .190 1.142 .316 1.298 .440 1.47» ..07 1.029 .196 1147 .320 1.306 .446 1.486 .076 1.032 .200 1.153 .326 1.312 .460 1.494 .08 1.036 .206 1.169 .330 1.319 .455 1.50i .086 1.039 .210 1.166 JXif> 1.325 .460 1.509 .09 1.043 .216 1.171 .340 1.332 .465 1.517 .096 1.046 .220 1.177 •346 1.339 .470 1.624 .100 1.061 .226 1.183 .350 1.346 476 1.582 .105 1.066 .230 1.189 .365 1.368 .480 1.540 aio 1.069. .236 1.196 .360 1.361 .486 1.547 J16 1.064 .240 1.202 .365 1.368 .490 1.556 .120 1.069 .1?45 1.207 .370 1.376 .495 1.568 .126 1.074 .260 1.213 .375 1.382 .500 1.571 Area of an ellipse = prod of dlam^ X .78M. Bz. D = lO ; d = «. Then 10 X 6 X .T§6« c 47.124 area. The area of an elUpiie la a mean proportional between the areae of two cirelae, d«* ■eribed on its two dlama ; therefore it may be found by mult together the areaii of.thote two -eirolaa ^ and taking the aq rt of the prod. The area of ah ellipse ii therefore always greater than that of th« eircolar seotion of the cylinder f^om which it may be supposed to be derived. Dlam of circ of same area as a given ellipse = i^Long diam x ahort diaml To find tbe area of an elliptic segment wbose iNwe is paral. lei to eitlier dlam. DIt the height of the segment, bT that diam of which wid height !■ a part. From the table of circular segments take out the tabular area opposite the qnot. If nil together this area, the long diam, and the short diam. To drair an ellipse. Having its long and short dtaas a b and e d, Pig. 4. BoLB 1. From either end of the short diam., as c, lay off the dists. ef, ef, each equal to « a, or to one-half of the loug diam. The points/, /' are the foci of the ellipse. • Prepare a string, fn/.orfgf. with a loop at each end ; the total length of string from end to end of loop, being equal to the long diam. Place pins at /and/'; and placing the lloops over them, trace the curve by a pencil, which in every position, as at n, org, keeps the string/' n /, or /' gf stretched all the time. Note. Owing to the diflDoulty of keeping the string equally stretched, this method is not as satisfactory as the following. Bulb 2. On the edge of a strip of paper «0 «, mark w I equal to half the short diam. ; and IS a equal half the long diam. Then in whatever position this strip be placed, keep- ing I on the long diam., and s on the short diam., te will mark a point in the eircumf. of the ellipse. We may thna obtain at many each polnu as we please ; and then draw the curve through them by hand. Bdlb 8. From the two foci / and /', Fig. 4, with a rad. equal to any part whatever of the long diam. describe 4 short arcs, o o o o; also with a rad. equal to the remaining part of the lon^ diam., describe 4 other arcs, iiii. The intersections of these four pairs of ares, will give four points in tha eircumf. In this manner any number of such pointt may be found, and the curve be drawn by hand. To draw a tanarent 1 1, at any point n of an ellipse. Draw n / and n /', to the foci ; bisect the angle / n /' by the line xp ; draw < n ( at right angles to xp. To draw a Joint n p^ of an elliptic arcli, f^om any point a, im tbe arcb. Proceed as ic the foregoing rule for a tangent, only omitting (I; np will be required joiac I?ig-4. IfBNSUBATHnr. 191 To draw an OTal, or felse ellipse. When only tbo long diam a b It given, tbe fbllowing will give agreeable caires, of wbicb tbe span a h wiU not exceed abont tbree times tbe riie e o. On a & d»> ■eribe two Intersecting circle* of any rad; through their Interseetiona t, 9, draw ay; make • g and r • each eqnal to tbe dtam of one of the eirelea. Tbrongb the center* of tbe circles, draw «f,*h,gd,gU FroB edeioribeA<y; and from y dMoribe d o I. "Wiieii the span, «nn^ and tlio rise* s t, are boUi yliren. Make any f w and mr, eqnal to each otbei;^ but each less than t ». Draw r w; and throngn its center o draw tbe perp toy. Draw y r «• Make n « equal mr, and draw tfxb. From sand r describe n e and m m; and fh>m y describ* ate. By making « d eaaal to « y, we obtain the center Ibr tbe other side of the oral. Tbe beaaty of tbe canre will depend npon what portion of I « is taken for m r and t m. When OB oval le verf flat, more than three cen- ters are reqnired for drawing a gracefbl enrre ; bat the flnflng of these centers Is qaite aa tron* bleseme as to draw tbe oorrect ellipse. €tai the §:!▼«>■ line, a 9, to draw a cyma reeta^ aes. Find the eenter e, of a ». From «, e, and $, with one-half ef • • aa rad, draw the fonr small arcs ato. o. The inter* o, «, are the oenters Ibr drawing the oyma, with I ra4. By rerersing the position of the ares, w« oreyee, 4 </. 192 MENSURATION. THB PAIIABOI.A* The eommoii or eonle iiarabola, o b e. Fif 1, is a onrre formed by oatting • oone in a dlreetlon b a, parallel to ita lida. •arred line obe itself is called theptrimt«r of the parabola ; the line o e is called ita bcwe ; ft • iti height or axta ; b its apex or vertex i any line e s, or o a. Fig S, drawn from theonrve, to, and at right angles to, the axis, is an ordinate ; and the part s 6, or a i, of the axis, between the ordinate and the apax b, is an abscissa. The /ooms of a parabola is that point in the axis, where the abaoisaa 6 «, is oqual to one-half of the ord e ». The dist from apex to focus, called the focal diet, is found thus: square auy oid, as o a; div this sauare br the abscissa i a of that ord; diy the quot by 4. The Cature of the parabola is such that its absoiBsas, as 6 s, 6 a, fto, are to each other as, or in proportion », the sanares of their respective ords s s, o a, Ac; that is, as i s : ba : : ss* :o<i>;orbs:ss>::b«: • a* . If the square of any ord be divided by iu abscissa, the qnot will ho a constant qnantltj ; that Is, it wHl bo equal to the sqoaro of any other ord dlTlded by Its abscissa. This qnot or oonstantqaan* tfty Is also equal to a eertsln quantity oallod the pmrameter of the parabola. Thersfbra tho p^'^nwtsr may be found by squaring s s, or e a, (one>ha^ of the base,) and dividing said square bv tho height i s. or b a, as the case may be. If the square of any ord be divided by tho panoMtar, tbt qnot wff he the abscissa of that ord. To And (lio lenyth of a parabolle enrre. The approximate rule given by various pocket-books, is as IbUows t Length — 2 X V(H '^>^e)a + \% Umes the (Height^ (g Where the height does not exceed 1-lOth of the base, thls'mle may, for praetlMi purposes, be called exact. With ht = )^ base, it gives about H par oeat tos Bueh; ht s M base, about 3^ percent; htsbase, about 8K per coot; ht = %«tee the base, about 11% percent; ht= 10 X base, or more, about 15)t( per oeat The flillewlas \ij the writer U eo r reel within perhaps 1 part in aOO, in all eases ; and will therefore answer for many purposes. Let a d b. Fig S, orik a d. Fig 4, be the parabola. In whioh are given the base abvtndt and tte height c li or c a. Imagine the eonpleteflg ad bs, or » a 4< b, to be drawn ; and in sttAsr ease, aaanms Us loMi^ dlam a b to be the chord or base; and one- half the short diam, or e <i, to be the heightt of a circular arc. Find the length of this circular are, by means of the rule and table given for that pur* pose. Then div the chord or Immo a b, or n d of the parabola, by its height c d or e a. Look for the qnot in the column of bases in the following table, and take from the table the correspondiag multiplier. Mult the length of the eireolar aro by this ; the prod will be the length of are a d b, or n a cl, as the case may be. For bases of parabolas less than .05 of the hdght, or greater than lOtimea the height, the multiplier is 1, and is very approx> imate; or in other words, the parabola will be of almost exactly the same length as the eiroular are. To find the area of a |»arabola ta a n l^. Mult iU base m n, Fig 5, by its height a h ; and Uke %^^M of the prod. The area of any segment, as « b v, whose base tt v is parallel to as n, is found in the same way, using u « and s b, instead of iw i» and a b. To find the area of a parabolic aone, or fl^as- tam, as t>» n t« V. RuLx 1. First find by the preceding rule the area of the whole pambola m b n ; then that of the segment « b « ; and subtract the last mm the flmt. RuLK 1. From the cube of m n, take the eubo of « v; eall the difP %, From the square of m n, take the square of m « ; eall the dlff «. Div e bf «. Mult the quot by ^ds of the height • s. MENSURATION, 193 1 Table lor I^enytlis off Parabolic Curves. See opp page. (Original.) Baa«. Mole BM6. Molt. Bue. Molt. , Base. Molt. .05 1.000 1.10 .999 2.15 .949 8.20 .983 .10 1.001 1.16 .997 2.20 .951 3.30 .984 J6 1.002 1.20 .995 2.25 .954 3.40 .986 .20 1.004 1.25 .993 2.30 .956 3.50 .986 .25 1.006 1.80 .990 2.S5 .958 3.60 .987 JSO 1.007 1.35 .987 2.40 JMM) 8.70 .988 JB6 1.007 1.40 .984 2.45 .002 3.80 .989 AO 1.008 1.45 .980 2.50 .963 3.90 .990 .45 1.009 1.50 .977 2.55 .965 4.00 .991 .60 1.010 1.55 .974 2.60 .967 4.25 .992 .65 IMO 1.60 J>70 2.65 .969 4.50 .993 jOO 1.010 1.65 .966 2.70 .970 4.75 .994 .66 1.011 1.70 .963 2.75 .972 6.00 .996 .70 1.011 1.75 .960 2.80 .973 5.25 .996 .76 1.010 1.80 .957 2.85 .975 6.50 .997 .80 1.009 1.85 .953 2.90 .976 6.76 .908 .85 1.008 1.90 .950 2.95 .978 6.00 .998 .00 1.006 1.95 .946 3.00 .979 7.00 .999 .96 1.004 2.00 .942 306 .980 8.00 1.000 1.00 1.002 2.05 .944 3.10 .981 10.00 1.000 1.05 1.001 2.10 .946 3.15 .982 To draw a parabola) having base o t and height « o. ••«, Flc6. Make e I eqoal to the height «e. DraweCand • I; and dlride each ofthem into aoT number of equal parte; BmnberlDg them as in the Fig. Join 1,1; 2, 2 ; 3, 3, Ao ; then draw the oorve by hand. It will be obeenred that Um itttereeetions of the lines 1,1; 1, 3, &o, do not give pointi in the eurre ; but a portion of each of those lines forms a tan. gent to the eurre. By increasing the number of diri^iona on e < and « t, an almost perfect oorre is formed, scaroelj teqnlring to be tooohed up by hand. In practice it is best first to draw onlr the center portions of the two lines whioh •rasa eaeh other Just aboTO o ; and trom them to work down* ward; aetnally drawing oalj that small portion of eaeh low« Une, whioh is neoessary to indioate th« bo drawn Fifir.tt. Or the i»araboIa ma tbasx Let ft «, Fig T, be the base ; and a d the height. Draw th» leetangie hnine; dir each half of the base into an j nom. ber of equal parts, and number them ftom the center each vmT. DIt n h, and m e into the same number of equal parts ; ■ad number them from the top, downward. From the points on b e draw rert lines ; and trom those at the sides draw lines to d. Then the interseetions of lines 1,1; 2. 3, ke, will form points in the parabola. As in the pre- esding ease. It is not necessary to draw the entire lines ; but merely portions of them, as shown be. teeeu d and c. Or a parabola may be drawn by first dlT the height a h. Fig 5, into any number of parts, either equal or unequal; and then ealoulating the ordi. aatea u»,Ao; thus, as the height a h : square of half base am : : any absciss b s : square of iu erd « «. Take the sq rt for ««. I. —When the height of a parabola is not ir than 1.10th part iu base, the eurre eoin- ■o very eloeely with jlhat of a drcntar are, that in the preparation of drawings for suspen> rieo bridges. Ac., the eironlar are may be em. ployed ; or if no groat aoenraoy is veqd, the olrole ■ay be need eren when the hMghfe la aa great •• «e^«igfath of the base. To dra^w a tangr^nt w v, TIk- 5, to a parabola, from any point v. Draw V » perp to axis a h ; prolong a h until b w equals s b. Join v> v. 13 194 MENSURATION. a Tlie Cycloid, ^^h i-the curve deacribed by a point a in the circumference of a circle, .^'d'ix^fonr^^^S^^o.uLn'S the clro.e.^roU^^^ d h cycloid. Tlie vertex of the cycloid is at e. Base, a 6, =s circumference of generat- ing circle a u =s diameter, cd, of generat- ing circleXir = 3.1416«i. Axis, or taeli^lit, cd=^an. lieuiTtli, oc6, = 4cd. I, a c 6 d = 3 X area of generating circle, o n = 3?^ = ca8 X 3ir = cci« X 2.3562. Center of sravity of surface at g. cg = t\ c d. Center of gravity oi cydoid (curved line a c 6) in axis c d at a point (as ») distant J c d ttom c. To draw a tangent, «o, from any point e in a cjrcloid; draw « » at right anTlM to the axScd; one d describe the generatingcircle dc<; join /c; from J draw CO parallS to / c. The cycloid is the curve of a uickest descent ; So thit a ESdy would fall from"^ h to c along the curvelm c, in less time than along the inclined plane 6 ic, or any other line. TKE REGVIiAB BOBIES. A revnlar body, or reffular polyhedron, is one which has all its dies, and its solid angles, resnectively similar and equal to each other. There 'e but five such bodies, as follows : ■ides are Name. Tetrahedron ......... Hexahedron or cube Octahedron Dodecahedron Icosahedron • Bounded by 4 equilateral triangles. 6 squares. 8 equilateral triangles, 12 " pentagons, |20 " triangles. Surface (—sum of surfaces of all the faces). Multiply the square of the length of one edge by 1.7320 6. 3.4641 20.6458 8.6602 Tolnme. Multiply the cube of the length of one edge by .1178 1. .4714 7.6681 2.1817 Ouldinus' Tbeorem. Fig. A. Fig. B. I To find the volume of any body <as the irregular mass a 6 c w. Fig A, or the rinft abom^ Fig B), generated by a complete or partial reyofution of any figure (as _ ahca) around one of its sides (as/ie, Fig A), or around any other axis (as a;v,FigB). volume =3 surface ahcaY. length of arc described by its center of grar^ ity G. If the revolution is complete, the arc described is = circumference = radius G* X 2ir = radius o G* X 6.283186 ; and Tolume =surface a6ea X radius oG*X 6.283186. If the revolution is incomplete, complete . incomplete . . circumference . mo revolution ' revolution * ' found as above * described * Measured perpendicularly to the axis of revolution. HEMBirRA.TION. PABA1.1.EI.OPIPEDS &r^^f^ nlt^Fig 1,Dhl£h)u iglM right iDgleB, each pair of ;1*>> right 1 nil 1(1 ildco eqoil rhombn , , loalled-'itaomb"; iba EJumbia prism. Fig 4; Ita lluiei, rbomJ loibolds. well pilr ot oppoilte bon aqosl, but not *11 ila Kwes eqi (rrm. Fig 3, UB, p 15?. Is ^ ^WTVrxJvuJar dl '^ Cs tlig oppodi A piiBm ig aoy solid irhaM >Dd equal ; and whose iida art pwaUeiogTami, »a Flga G to 10. Cansequ«ntlT the for^ n faint pBrBllelopipeds are prlnns. A HgU prism is i>d« wh«e Bldu are perpeodic- bnn the cuds are equal, aod the anglea included bati eqnjd, the prJam la aaid to ' -"■ "- T«Inni« cf mMT prii ngnlu or Irr^ular. right or oblique) ,^., lataDOe,p.totb<otheTend. — area of cfOM NCtioii perpeDdlculat ta tbe ddea x utnal length, aft, Figi H 8 X TfduiFi* of prnmld vhoae biae aod height are ^ those ol the prism. idlcnlar to Iti nlH*. ly pirallelog J Dumber nf sldi " lanale ; any piraUelognii 1u^> 01 a reffiUar paljgo reffiUar paljgan of goflenKthaofporolWedgea, "S*""*- i~f + Ti + S~i + T^ "fe* of <!«™ section nDmberotauchedgea ^ ^SH^rf^ 196 MEKSURATION. fl ■ # I dL g Fig. 10J4 This rule may be used for aacertainins beforehand, the Quantity of earth to be removed from a "borrow pit." The irregular surface of the ground is first staked out in squares; (the tape-line being stretched horizontally ^ when meas* uring o£f their sides). These squares should be of such a suse that without material error each of them may be considered to be a plane surface, either horizontal or in- clined. The depth of the horizontal bottom of the pit being determined on, and the levels being taken at every ^b corner of the squares, we Hre thereby furnished with the lengths of the four parallel vertical edges of each of the resulting Arnstums of earth. In Figs 10^ y may be sup- posed to represent one of these Arustums. If the frustdm is that of an irregular 4-sided, or polyg- onal prism, first divide its cross section perpendicular to \ts sides, into tri- angles, by lines drawn frpm any one of its angles, as a, Fisr 10^. Calculate the area of each of these triangles separately ; then consider the entire frustum to be made up of so many triangular ones; calculate the volume (•;\ of each of these by the preceding rule for triangular frustnms; and add them together, for the volume of the entire frustum. Tolnme of any frnstam of any prism. Or of a cylinder. Consider either end to be the base ; and find its area. Also fipd the center of gravity c of the other end, and the perpendictUar distance n c, from the base to said center of g^ravity. Then Volame of frnstam = area of base X»«, Fig 10^. The slant end, c, is an ellipse. Its area is greater than that of the circular end. Snrfaee of any prism. Figs 5 to 10, whether right or oblique, regular or irregular / circumference measured s^ «-*„-i iA«»ti, >. A i »tt™ of the areas " Vperpendicular to the sides ^ *®^"" lengin, a <> j + of the two enda. CTIilHTBERS. . If A cylinder is any solid whose ends are ^h^-^_^ jC ^ parallel, similar, and equal curved fignires ; and whose sections parallel to the ends are everywhere the same as the ends. Hence there are circular cylinders, ellip- P tic cylinders (or cylindroids) and many others ; but when not otherwise expressea, the circular one is understood. A right cylinder is one whose ends are perpen- dicular to its sides, as Fig. 11 ; when otner- Fig. 11. Fig. 12. wise, it is oblique, as Fig 12. If the ends of a right circular cylinder be cut so as to make it oblique, it becomes an elliptic one ; oecause then both its ends, and aJl sections parallel to them, are ellipses. An oblique circular cylinder seldom occurs ; it may be conceived of by imagining the two ends of Fig 12 to be circlet^ united by straight lines forming its curved sides. A cylinder is a prism having an infinite number of sides. Volume of any cylinder (whether circular or elliptic, Ac, right or obliqa^ = area of one end X perpendicular distance, j9, to the other end, -{rJZ^^^Zi^ X actual length, « 6. Figs U and 12. ^ 3 X volume of a cone whose base and height are » those of the cylinder. Snrface of any cylinder (whether circular or elliptic, &c, right or oblioue) (circumference ^ g^m ^f ^^jje areas measured perpendicularly X actual length, o 6 1 + ^f the two ends to the sides, as at c o. Fig 12, f RIfirlit circular cylinder whose lieiirb^ " diameter. Volume = H X volume of inscribed sphere. Curved surface = surface of inscribed spltere. Area of one end == \ surface of inscribed sphere =«= \ curved surface. Entire surface = U X surface of inscribed sphere =« IJ X curved surfkee. CJONTENTB OP CTUNDBRS, OB PIPEa. 197 ContentB for one fi»ot tn lenstti, in Cub Ft, and in U. 8. Gallons of Ml oab ins, or 7.4806 Galls to a Cub Ft. A e«1» Rof water wei«lu aboat 62M lbs ; and a gallon altoat 6H IlM. IHaaw »• 8» or 10 Hmm m svMt* «iTe i, 9. or 100 times tbe (Mutant. For in. in For I ft in For 1 ft. im length. lengtH. length. Dlam. Dlam. in deoi- Diam. in Dlam. in deci- Diam. Dlam. in deci- in -•3 • ^ ■ -5 *s 2 ^ v^^ ^ 9 Ins. malsof • H * o a Ins. mals of ^a^ o a in mal* of 8?*i a afoot. h afoot. ii Ins. afoot. ^s^ 5 . •§sS' 5" ■pg- =50 •Ss^ ^0 t "3 ^n i «3 5s «3 *a ^Yt .0206 .0003* .0025 % .5625 .2485 1.869 19. 1.683 1.969 14.73 .0260 .0005 .0040 7. .6833 .2673 1.999 H 1.626 2.074 16.61 ,-!i .0313 .0008 .0057 ' ^ .6042 .2867 2.146 20. 1.667 2.182 16.32 .0366 .0010 .0078 .6260 .3068 2.296 34 1.708 2.292 17.15 ».^ 0417 .0014 .0102 yi .6466 .3276 2.460 21. 1.750 2.406 17.99 .0409 .0017 .0129 8. .6667 .3491 2.611 H 1.792 2.621 16.86 nM .0521 .0021 .0180 § .6876 .3712 2.777 22. 1.833 2.640 19.76 .0673 .0026 .0193 .7083 .3941 2.948 H 1.875 2.761 20.66 4 .0625 .0031 .0230 % .7292 .4176 3.125 23. 1.917 2.885 21.68 .0677 .0036 .0209 9. .7500 .4418 3.306 }4 1.968 3.012 22.63 is-fi .0729 .0042 .0312 H .7708 .4667 8.491 24. 2.000 3.142 23.60 .0781 .0048 .0359 .7917 .4922 3.682 25. 2.083 3.400 25.60 1. .0633 .0065 .0408 74 .8126 .5185 3.879 26. 2.167 3.687 27.66 8 .1042 .0085 .0638 10. .8333 .5464 4.060 27. 2.260 3.976 29.74 .1260 .0123 .0918 i .8542 .5730 4.266 26. 2.333 4.276 31.90 H .1458 .0167 .1240 .8760 .6018 4.498 29. 2.417 4.687 34.31 2. ^* .1667 .0218 .1632 Z4 .8968 .6303 4.716 30. 2.600 4.009 36.72 /4 .1876 .0276 .2066 11. .9167 .6600 4.937 31. 2.683 6.241 39.21 .2063 .0841 .2650 H .9375 .0903 5.164 32. 2.667 6.585 41.78 5i .2-292 .0412 .3085 .9683 .7213 6.S96 33. 2.760 6.940 44.43 a. ^* .2500 .0491 .3612 7* .9792 .7680 5.638 34. 2.833 6.306 47.15 .2708 .0670 .4300 12. 1 Foot. .7854 5.876 36. 2.917 6.681 49.98 .2917 .0668 .4906 H 1.042 .6522 6.376 36. 3.000 7.060 62.68 5k .3125 .0767 .5738 18.^ 1.083 .9216 6.896 37. 3.068 7.46T 66.86 i. * .3333 .0873 .6628 u^ 1.126 .9940 7.436 36. 3.167 7.876 68.92 .8542 .0986 .7360 1.167 1.069 7.997 39. 3.260 8.206 62.06 .3750 .1104 .8263 H 1.208 1.147 8.678 40. 3AS3 8.727 65.28 5i .9958 .1231 .9206 15. 1.250 1.227 9.180 41. 3.417 9.168 68.68 5. ^* ^167 .1864 1.020 H 1.292 1.310 9.801 42. 3.600 9.621 71.97 .4375 .1508 U26 16.^ 1.383 1.396 10.44 43. 3.683 10.085 76.44 .4583 .1650 1.234 H 1.375 1.485 11.11 44. 3.667 10.659 76.99 X* .4792 .1808 1.340 17. 1.417 1.576 11.79 46. 8.760 11.046 82.62 «. .5000 .1903 1.469 u t.458 1.670 12.49 46. 3.833 11.641 86.33 .5208 .2131 1.594 18.' " 1.600 1.767 13.22 47. 3.917 12.046 90.13 .5417 .2804 1.724 }4 1.642 1.867 13.96 48. 4.000 12.666 94.00 TaMo oontlniied, bat wtth tbe dlanui In feet. Gab. U.S. Dlam. Onb. U.S. DU. Gab. U.S. Dia. «ab. U.S. Feet. Feet. Gallfl. Feet. Feet. Oallfc Feet. Feet. Galls. Feet. Feet. Galla. 4 12^ 04.0 7 S8.48 287.9 12 113.1 846.0 24 452.4 8884 1^ 14.19 106.1 41.28 808.8 18 132.7 992.9 25 490.9 8672 xt 10.90 119U) 23 44.18 330.5 14 153.9 1152. 26 530.9 3972 /i 17.72 182.6 & 47.17 852.9 16 176.7 1822. 27 672.6 4288 % 19.0S 146.0 8 60.27 876.0 16 201.1 1604. 28 616.8 4606 W 21.66 161.9 M 66.75 424.5 17 227.0 1698. 29 660i» 4941 I4 28.76 177.7 0^ 63.62 475.9 18 254.6 1904. 30 706.9 5288 /• 25.97 1912 K 70.88 580.2 19 283U$ 2121. 31 764.8 6646 f 2&27 211A. 10 78.54 687.5 20 814.2 2850. 32 8012 6018 l^ 80.68 22BA K 86.59 647.7 21 346.4 2591. 33 865.3 6398 xc 88.18 248^ 11 06.08 710.9 22 380.1 2844. 84 907.9 6792 % 8&78 287.7 % 108.87 777.0 28 415.5 8108. 85 962.1 7197 198 CONTENTS AND LININ08 OF WELI*. COSTENT8 AKD LIJriHeB OF VELIA. For lIuH WlBe u irul u IkaH In Ih. U-Ut. Ibr »>• n» JiU iC Unliil. Uli ml tbm onuM OM ka|f dT lh> inuH dim ; u< khU IMM In 4, Tkm, iH- Un gDl ill [u (loli tvA of d>pi£ id • vallfll r««tlB«w, llmUbautrniiaillauW*lbo«a«»a<U«lli*dtioar]A4fcfaK; nu»J/, A.Hi. Tbn t.Mi X t ~ n.aU anb Jill ngd tor UnHlft^Um. BHItattlK uoni llnlDf •rnUlu HliU ar pUiHiiH, BiU tbg laJmUr gguilu •hkhIH half U» ininr OiH. br 1. TllL thi HnMi tf IKH nlUMf Ht aKik HM gf «iplk ors will of II 6 diuD. wlU la LOn X I = I.IM. Ir tht nil li r™ eMh"(D« pm- »lu»«. "inf Diull. Hllll(. s; #i -.. f l.Wl s '.(Mi Inn K i.Ba JIUl W Pt I.BU s iuM 3IU « :<! "■b ;lm In M K ■^! J ii: ■i JM :!>9H ,*W :»!» 5 .CHU iT e-ioi K .«B 1 'l i i 1 ti ioK & ™ .MT ') a ■» "lis [JSJ s "'a '1 410 tu H 'a jg m J» ^ '1 IS rm M6 !S !:3S K .._. S M !« nS 31 N .VH «J M ^1 ■» ^ i "'s '^ ■^ Si «^ |^« >m j™ icSi K J i 1 i f "si a." i:i Ip J *■"• .TO TH ew A at>7d ^«oai .b"U !«■ ItV«r«bea u« named In ■ ' CYLINDRIC CNOULAB, ETC. 199 CIBCUI^B CVLIITDBIC UNQITI^B. ■ the enttlng plaoe dvea mot eat tbe baae. Flp l^ 14 1 -m ]ft J I perp u ildw, u z, '^ jr'n,al,nieai<Jili>iieChB>id«. Add arena *t ends If required. r«r area* of SAetlaiis perpendioulir to tbe ildM, see GIrelH. r«r areaa of aecUoaa oblfqu* to tbs ildea, •»• Tbu ElltpN. II. Wb«n the enttlns plane to>«taea tbe baae. Flgi A l« IX .^--* Talome FlgA-(^at X*ra*a<l«t otbau)- («ta.ih« -«»WX«ii. FigD-H>">(>f (sIrcIayM X"" ~ ^ TolDins of cfUiider c y m n. Kg A - foi X "in - o< X length of are imh ) |^. (^,, FisB-»,X-. nBgoU FlgC — (lift Xn>n + 00 X length of «jc dm* )— —. oflj) "- 200 PYRAMIDS AND COVES, PTH^ttlDft AND COITEB. 4 5 A pjrainid, Fin. 1, 2, 8. Is any solid which has, for Its base, a plane figure of any number of sides, ana, for its sides, plane triangles all terminating at one point d, called its apex, or top. When the base is a regular figure, the pyramid Is regular ; otherwise irr^uCar. A cone, Figs. 4 and 5, is a solid, of which th6 base is a curved figure; and which may be considered as made or generated by a line, of which one end is stationary at a certain point d, called the apex or top, while the line is being carried around the circumference of the base, which may be a circle, ellipse, or other curve. A cone may also be regarded as a pyramid with an infinite numoer of sides. The axis of a pyramid or cone. Is a straight line eZ o In Figs. 1, 2, 4 ; and diiA Fi^s. 8 and 5, from the apex e2, to the center of gravity of the base. When the aj^s is perpendicular to the base, as In Figs. 1, 2, 4, the solid is said to be a right one ; when otherwise, as Figs. 3. 5, an oblique one. When the word cone is used alone, the right circular cone. Fig. 4, is understood. If such a cone be cut, as at 1 1, obliquely to its base, the new base 1 1 will be an ellipse; and the cone dtt becomes an oblique elliptic one. Fig. 6 will represent either an obUtiue <^ular eoiie, or an oblique elliptic one, according as its base Is a circle or an ellipse. V oliune or pyramtd or co£e, regular or Irregnlai^ right or obliqu«. Volume mm ^ «rea of base X perpendicular height d o. Figs. 1 to i. -» ^ volume of prism or cylinder having same area of base and same perpendicular height. — K volume of hemlsphJBre of same base and same height Or, a oone. hemisphere and cylinder, of the same base and same height, havt volumes as 1, 2 and 3. Area of anrlkec of sides of right regular pyramid or right dicular ooiM. Area — J^ circumference of base X slant height.*^ In the cone, this becomes I Add area of bass Area of sarfoce of oblique elliptic eone, dtt, Fig. 6i, cut from a rieht circular cone, dss. From the point c where the axis d o of the right circular cone cuts the elliptio base t L measure a perpendicular, r, in any direction, to the curved surface of the cone. Let v = the volume of oblique elliptio cone, dti; let a — the area of its elliptic base t (.and let A = the height d u measured perpendicularly to said nase. Then Carved snrlkiee = = . r r Add area of base if required No measurement, has been devised for the surface of an oblique circular cone. *In the pyramid, this slant height must be measured along the middle of one of the sides, and not along one of the edges. PYRAMIDS AND CONES. 201 To And thm surfiwe of mat IrvcffiKlar p jramld. Whether right or oblique, each side must be calculated as a separate triangle (i p. 148); and we several areas added together. Add the area of base if required. FRUSTUMS OF PYRAMIDS AND CONES. Flff.0. Fig. 7. Frastam at pjnunld (Fig. 6) or of oono (Fig. 7) with haw aad Uff pnaUeL Tolmne (regular or irregular, right or oblique) my ^, perpendicular v- / area i area i / area v/ area \ — >* P^ height oo ^ ^of top •" ©f base t" V of top -^ of base/ ^ w vr perpendicular w / area' i "«» i * ^ areaof aBection \ — X X *helght oo X V of top + of base + l^^^ to, and midway I >» ' between, base and top / »^ (for ffmstam of right or oblique circular cone only; Hee Fig. 7) « X "^SSS^ X M4M X (•<• 4 •»* + •« . o.) of frustum of righi fgiAjur pjmunid or ooue, with top And base paiallelt 9|0k 6 and 7. J. /diemnferenoe _i oirouinlbrenoeX v^ dant • >^\ oftop T ofbaM y X iMigiitfC Aid MiM «f top and Inuo If nq«li«4. Im tlM finuitoaA of a vl|^t etreolar oono^ tUibMoaat "^ Vof top T^ of basej X hdght f f (ir * 8.1416) . Add areas of top and base !f reqafawd. of IwegiUsur or o1»liq«« pjnroiBld or ooim. Sorlhee •• ■an of smrfiwes of sldsi, each of which must be treated as a trapeasoid. •In the frustum of the jpframld (fig 8), this slant height must be measured along of Ite MM (M at <s), Mid net along one of tha edgsib 202 PBI6HOID& PBIBHOIDB. Flff.L VtK.2. A prUnnoUl is sometimM d<iHwtl M AfBlid bttdng Ibr Hi ends two paralWI plane figures, connected by other plane flfiuns on which* and through every point of which, a straight line may be drawn nom one of tho two parallel ends to ^s other. These connecting planes msj bo parallelograms or not. and parallel to each other or not. Tbla doflnltlon iroiild Imolndo the cube and all other parallelopipeds; the prism : the cylinder (considered as a prism baring an infinite namber of sides); the pyramid and cone (in whieb one of the two parallel endl^ i« theonelbiminiftiio apex, is considered to be infinitely small), and their frnstams with top and boso parallel ; and the wedge. But the use of the term prlanaold is frequently restxietod to siz-eided aolidd, in which the two parallel ends are unequal quadrangles; and the connecting plane^ trapezoids; as in Figs. 1 and 2; and, by soma writers, to cases where the patalkl quadrangular ends are rtetatiffies. The following •'prlsmoldal fbrmnla** i^Uas to all tbo ftregolng •olidi^ and to others, as noted below. Let A — the area of one of the two parallal ends. a — <* ** the other of the two panUlel ends. M — « *< a cross section midway between, and panllil to^ Hm tm parallel ends. L — the peipendicnlar distance between tfao two psnlU < Then Tolmiae — L X ^ L X mean area of enm section. The following six flgnrss repvstent a few of the irregular solids which ftlltBderlht aboye broad definition of '< prismoid,*' and to which the prismoidal formnla appUiC They may be regarded as one-chain lengths of raihroad cutttnga; a o being^the loogUv sr perpendicular (horiaontal) distance between the two parallel (Tertloal) ' WEDGES. 203 The prismoldal ft»rmii]» applies also to the qihere) hemiiphere, and ether qpE«rlcel segmeiite; also to any aeotlf joe each aeafroi^aiid onidbct ai the In which the ddee ad^ ae, or od, <<i^ are itraiffhii tM ttuj are onty when the •atttng plane ade paaaes Umugh ike apes or top a. Also to ih» cylliiidev when a plane paraUd to the tides passes through both ends; but not if the plane «s is obHquet as in the fig., though never erring more than 1 in 142. In tl&la last case we must imagine the plane to be extended until it cuts the side of the cylinder likewise extended ; and then by page 199 find the solidity of the uegnlathus formed. Then find the solidity of the snuUl nngnla above to, also thus formed, and subtract it fh>m the large one. This very extended applicability of the prismoidal formula was first discorered, and made known* hy KUwood Morris, a B., of Philadelphia, in 1840. WEDGES* m n m SI m m Fiff.]a m Fiff.n. b neaally defined to be a solid. Figs. 8 and OjjKenerated by a plane triangle, anei, moving; parallel to itself; In a straight line. This definition requires that the twe triamgnlar ends of the wedge should be parallel; but a wedge may be shaped as in ng. 10 or 11. We wouid therefore propose the following definition, which embraces sll the figs.; besides vuious modifications of them. A solid of five plane faces ; one sf which is a parallelogram abed, two opposite sides of which, as a e and h d, are onlted by means of two triangular foces aen, and frdm, to an edge or line « m, parallel to the other opposite sides ab and ed. The parallelogram abed maj be eitlier rectangular, or not ; the two triangular Ikces may be similar, or npt ; and the with r^ard to the other two fhces. The following rale appUss equally to all : SunoTleDgths — K X oftheSedges peiphtj^from edgetobaok width of back {abed^ massed neip to « it 204 lOENBURATIOV. SPHERES OR GLOBES. A Sphere Is a solid generated by the revolation of a semicircle around its diameter. E^ery point in the surface of a sphere is equidistant (h)m a certain point called the center. Any line passing entirely throns;h a sphere, and through its center, is called its axis, or diameter. Any circle described on tlie surface of a sphere, fh>m the center ol the sphere as the center of the circle, is called a great eirde of that sphere i in other words any entire circumference of a sphere is a great circl«f. A «phere has a greatei content or solidity 'than any other solid with the same amount of surface ^so that i| the riiape of a sphere be any way changed, its content will be reduced. The inter- section of a sphere with any plane is a circle. Tohune of sphere — J TT radios* — )^ TT diameter* , ^ circumference * ■" •« zr5 — 4.1888 — 0.5236 radius' diameter* -» 0.01689 circomferenoe* — 3^ diameter X area of surface "" ^ diameter X area of great circle «- % Tdlume of circumscribing cylinder ^ 0.6236 Tolnme of circumscribing cube. ot avtrfiace of sphere — 4 TT radius* — w diameter* circumference* — 12.6664 radius* — 8.1416 diameter* •^ 0.8183 circumference* — diameter X circumference ■- 4 X area of great circle ^ area of circle whose diameter is equal to twloe diameter of — curved surfkce of circumscribing cylinder 6 X volume diameter. Badlw of sphere s t = * f volume = O.e2036 'v^volna* = / Area of surface 47r = ^.07968 X anaof Boxflwe Gireiinalbrenee of sphere =s \/6 TT* volume a« ^TT Area of surfisoe _^ area of snrikoe ~^ diameter. =r '^/59.2176 VolWM =s ^8.1416 are* of ioifiMe MEKBUBATION. J_ j_ 1 1 ill IMl ,1» r.Bie S "!:?; T-« 7.Ma IMI H iUJS n.f :«n JllS^ ,u» a.<>i< K iw-as s-ia n.«ai SBIPJ i*-ti -us «.im ,i.S .MS t.l7» 1.1U ».8HI U-ll W.IM ttJi '.jta io!k> ■m'.ta in. 41, ( JS4« HI.U n.ii .m HO ■i» ■:£> iiiin KtM ■-% _5g li "Mn "iS auH SSsi »ii Si SiS! iSS 11 B M.MI i: MO 11 M.I54 198 S» ] n,4ii n.oia Is lIliiB 'E «I.,. "is II tSoi !:i ..! ►! „.ll .HI .OW ,..a MIW 1 ,>» S71il 1 ra.Dw SS1.HI 1 s. M!M 1 in «i:ws •W.SO 1 m.«i 1 «IS,7S 1 IS.OM OHM 1 im i-S W ss SSI 7-11 « ei.jos •W&l 1 ill. 16 1 ms.ia 1 11-11 1 S' siis "'w '":» 1 11 U M.Ml »:» H 7m!i7 S 1<7.M .1 «, K 17>.SI \x H T>l.71 |] £ 1 s iMa j* Jg-JJ ; 1 l^s ISO 8 iii 1 s u! ' Mi-n ■,.»is?:S : UENBUBATION. 8PB EBEI 1 i 1 •1 1 f J ii ii 1 1 i IS § J \ \ » J IS.' 1 I s 11 1" -(OoRninmi,) f 1 1 I j I I :i j: 208 ■BOXENTS, STG., OF SFHSRIB. To find the solidity of a splierieal seviiieiit. RiTM 1. Bqaar* the radon, of its baie; multthla tqnarebjS; to the prod add the iquare of ita hole ht o « ; mult tke bud by the helghfe o « : and mult this last prod by .5286. Bulb S. Malt the diam ah ofth* 4)ker«byS; flrom the prod take twiee the height o « of the Mcmeat; mult the rem by the Mioare ef the height o « ; and malt thle prod br .&SS6. The ■oUdtty of a sphere being HAa that of Its draamwnibiiic ^Un- der, If we add to any solidity In the Ubie. Ita half, we obtain that of a cylinder of the same dlam as the sphere, and whose height equals ita dlam. To And the enrved sarftM^e off a ■ptaerleal seirneiit. RvLi 1. Mult the diam a b of the sphere fk«m whleh the segment is out, by S.141C; MBit the prod bT the height e « of the seg. Add area of base If reqd. Ban. Having the diam n f •f the seg, and ita height o «, the diam a 6 of the sphere may be found thus: Div the square of half the dlam n r, by Ita height o • ; to the qoot add the height o : Bulb 3. The eurvmi surf of either B segment, last Fig, or of a lone, (nest Fig,) bears the same proportion to the surf of the whole ■phere, that the height of the aeg or tone bean to the diam of the sphere. Therefore, first find the snrf of the whole sphere, either by rule or from the preoeding table ; mult it by the height of the aeff or Bone ; dir the prod fatr diam of s|riiere. Bin^ S. Molt the oiroumf of the splierB by the height e • of the sag. To find tbe solidity of * spberieal cone. Add together the square of the rad • d, the square of rad o &, and H<1 of ^^^ square of the perp height «o; mult the earn by 1.&706; and mult this prod by the height «•. • To find the carved snrflsee off a spiier- ical sone. BvLB 1. Mult together the diam m n of the sphere ; the height e of tbe sone, and the number S.U16. Or nee preoeding Rule t tor surf of segmenta. Bale S. Mult the etroamf of the sphere, by the bf^lghtof the zone. • To find the solidity off a hollow spher- ical shell. Take f^m the fbregolng table the loltditlee of two aphorae haTlBf the diams a &, and e <L Snbtraot the least fhmi the grMtMi. B«i« a c or » 4 U the Ihiokneie «r tha ahMtt. THE ElililPSOID, OR SPHEROID, Is a solid generated by the rerolution of an ellipse around either Ita long or ita short dlam. When around the long (or transverse) diam, as at a. Fig 1, it is an oblons" or pr< late spheroid; when around the short (or co^Jaffate) one, as at m, in Fig % it is oblate. Fiir.i. Flg.2. For the solidity in either case, mult the fixed diam or ^tU br the •quare of the revolving one ; and mult the prod by .5336. ^ —— ■ — — . . » ___^_^ *This rule applies, whether the zone includes the equator (as in our figure) or not, as in the earth's temperate zoues. PAAABOliOWa, THE PAHABOLOID, OH PARABOI.IC COKOID, r lU ■olldltj' mult the ires of Ita bue, bj batr lu belaht, re. Oi Pop tbe •olldlt^ af a (TiutaB, I. (bl mil ot wlloll ««r.r]J Hi IMUjiril IiMHIIMIwlM To and the anraMm ofK pBraboloM, To And IM sarlkee. 1b« dirt a« rroBi tbe«at4r ortha oIhU w tb« cvrtUr ar Eb« iplndle. CkQ To Nnd tbe ■oIMIly of » mldfUe cone ofn elrnilsp Bplndlo, ((—'?)«.•)-(••«—'■■■))"- Tvlnme- of ,),iei, rfng ia mads '< dlamBlflm, ooand 6t XI-MIWB. -__-.__ _ drcumfflivrMfl of bar ^ 1 Bum of Inner HOd out^r w a tjifau 210 SPECIFIC GRATITT. SPEOmO GEAVm. 1. The specific gravity, or relative density, D*. of a sabstancei is the ratio between the weight, W, of any given volume of that substance and the weight, A, of an equal volume of some substance adopted as a standard of w comparison. Or: D = -^. 2» For ffaseous substances, the standard substance is air, at a temper- ature of 0° Cent. =s ^29 Fahr., with barometer at 760 millimeters = 29.922 incnes. 3. For solids and liquids, the standard substance is distilled water, at its temperature (4^ Cent = 39.2° Fahr.) of maximum density. 4. For all ordinary purposes of civil engineering, any clear fresh water, at any ordinary temperature, may be used. Even with water at SOP Cent., = 86^ Fahr., the result is only 4 parts in 1000 too great. 5. When a body is immersed in water, the upward forc& or '* buoyancy*** exerted upon it by the water, or the **loss of weight " of the body, due to its immersion, is equal to the weight of the water displaced by the immersion of the body f ; or, if W = the weiffht of the body in air, u; = its weight in water, D ■= its relative density or specific gravity, A = the weight of water displaced ; then A =- W — ic ; and D = -r- — tT? • ' A W — w 6. Since the volume, V, of a body, of given weight, W, is inverselv aa iti density, or specific gravity, D ; the specific gravity is equal also to the ratio between the volume V, of an equal weight of the standard substance, to the volume, V, of the body in question ; or D = ^'. 7. The specific gravities of substances heavier than water are ordi- narily determined by weighing a mass of the substance, first in air (obtain- ing its weight, W), and then when the mass is completely submerged in water W (obtaining its diminished weight, w). Then D = -^ , as in If 5. 8. If the body Is lighter than water » it must be entirely immersed, and held down against its tendency to rise. Its weight, «>, in water, or ita upward tendency, is then a negative quantity, and means must beprovided for measuring it, as by making it act upward against the scale pan. We then have^ A = W — (— w) = W + M? ; or Loss due to immersion = weight of body in air, pltis its buoyancy. 9. Or, first allow the body to float upon the water, and note the resulting di»- placeraent, t>, of water, as by the rise of its surface level in a prismatic vessel. Then immerse the body completely, and again note the displacement, V. Now V, the volume displaced by the body when floating, and V, tne volume displaced by the body when completely immersed, are proportional respectively to the weight, W, of the body, and to the weight, W — tr, of a mass of water of equal volume with the body. Hence D == = ^^. W — w V 10. Or, attach to the light body, b, a heavier body, or sinker, S, of such den* sity and mass that both bodies together will sink in water. Let W be the weight of the light body, 6, in air ; Q the weight of both bodies in air, and q their combined weight in water. Then Q — ^ = the weight of a mass of water of equal volume with the two bodies, and Q — W =». the weight, S, of the sinker in air. By immersing the sinker alone, find the weight, fc, of water equal in volume to the sinker alone, — loss of weight in sinker, due to immersion. Then, for the weight, A, of water of equal volume with the light body, fr, or tor * Strictly speaking, " specific gravity " refers to weight, and " relative density »» to mcus (see Mechanics, Art. 14 a); but, as specific gravity and density numerically equal, they are often treated as identicaL t See Hydrostatics, Art. 18. 1 SFifiOIPIC GRAVITY. 211 the low of weight of b, due to immersion, we have A = Q --0 — k ; and, for the specific gravity, D, of the light body, 6, we have D =- a_ ^ . =" ^_^ where to — the (unknown) buoyancy of b. 11. A granular body, as a mass of saw-dust, gravel, sand, cement, etc., or a porous body, as a maas of wood, cinder, concrete, sandstone, etc., is a com- posite body, consisting partly of solid matter and partly of air. Thus, a cubic foot of quartz sand weighs about 100 fi>s.; while a cubic foot of quartz weighs about 165 lbs. 12. The specific sraTltj of porous substances is usually taken as that of the composite mass of solid and air. Thus, a wood, weighing (with its contained air) 62.5 ttw. per cubic foot, or the same as water, is said to have a specific gravity of 1. The absorption or water, when such bodies are immersed forthe purpose of determining their specific gravities, may be prevented by a thin coat of varnish. 13. The specific grravity of granular substances is sometimes taken as that of the solid part alone. Thus, Portland cements ordinarily weigh (in air) from 75 to 90 fts. per cubic foot, which would correspond to specific gravities of from 1.20 to 1.44 ; out the specific gravity of the solid portion ranges from 8.00 to 3.25 ; and the latter figures are usually taken as representing the speciflo gravities. 14. In determining the specific gravities of substances (such as cement) which are soluble in water or otherwise affected by it, the substances are weighed in some liquid (such as benzine, turpentine or alcohol) which will not affect them, instead of in water. The result, so obtained, must then be multi' plied by the ratio between the density of the liquid and that of water. 15. The specific ifpavity of a liquid is most directly determined by weighing equal volumes of the liquid and of water. 16. Or weigh, in the liquid, some body, whose weight, W, in air, and whose specific gravity, d, are known. Let u/ = its weight m the liquid. Then, for the specific gravity, D, of the liquid, we have d(W— «/) W:W — «/ = d:D; or D = -^— ^^^ — •'. 17. Or, let the body, in f 16 (weighing W in air), weigh %o in water, and (as before) to' in the liquid in question. Then, since specific gravity of water =s 1, we have W — m;:W — u/^lrD; orD = 3~"^ » w — to 18. The specific gravities of liquids are commonly obtained by observing the depth to which some standard instrument (called a hydrometer) sinks when allowed to float upon the surface of the liquid. The greater the depth, the less the specific gravity of the liquid. In Beaum4('s hydrometer tne depth of immersion is shown by a scale upon the instrument. The graduations of the scale are arbitrary. For liquids heavier than water, 0^ corresponds to a specific gravity of 1, and 76^ to a specific gravity of 2. For liquids lighter than water, 10° correspond to a specific gravity of 1, and 60° to a specific gravity of 0.745. 19. In Twaddell's hydrometer, used for liquids heavier than water, .- ,^ 6 X No. of degrees + 1,000 specific gravity = -^ J — Thus, if the reading be 90°, ,- 4* 5 X 90 -f 1,000 1,450 , ^„ specific gravity j^^^^-i- ^ ^^ 1.46. 20. In Nicholson's hydrometer, largely used also for solids, the specific gravity is deduced from the weights required to produce a standard depth of immersion. It consists of a hollow metal float, trom which rises a thin but stiff ▼ire carrying a shallow dish, which always remains above water. From the float is suspended a loaded dish, which, like the float, is always submerged. On tile wire supporting the upper dish is a standard mark, which, in observations, is alwavB brought to the surface of the water. The specific gravity is then deter- miiied by means of the weights carried in the two dishes respectively. 21. The determination of the specific graYltles of ffaseous sub- ■tanees requires the skill of expert chemists. 212 8PE0IFIC GRAVITY. Table of speelfle ^mvitiefl, and w«lirlita* In this table, the sp gr of air, and gases also, are oompared with that of watec instead of that of air ; which last is usual. Th« specific gravity of any substance Is « its weicht in fframs per enbie «eiitlmetre. »••••••• <« « Air, atmoapbario ; MfiO° Fkh, and ander tbe pnMve ef oat atmMph«r> or 14.7 Afl per aq iaoh, weigh* j\j part as mooh aa water at 00° Aleobol, pure " of oommeroe " proof spirit ^ * ', ▲ab, perfcotly dry. V.V.V.V.V.V.'.'aTe'iife. * 1000 ft board meaaore weighs 1.748 tons. Aab, American white, dry " 1000 ft board meaaare weigha 1.414 tona. Alabaater, fklaely ao ealled; bat reaUy MarUea " real; a eonpaot white plaster of Paria aTerage.. Alamlnlom Antimony, caat,'6.86 to 6.74 averace .. " natlTe •• .. Anibraoite. See Coal, below. Aaphaltom, 1 to 1.8 Baaalt. See Limeatonea, qnarrled Bath Btone, Oolite. ..................................... lUamoth, oast. Alaonatlre gltamen, aolid. See Aaphaltom. rasa, (Copper and Zinc,) oast, 7.8 to 8.4 " " rolled «« Brooie. Copper 8 parte; Tin 1. (Gun metal.) 8.4 to 8.6 '• Brick, beat pressed " common hard ** ■eft, inferior Brickwork. See Masonry. Boxwood, dry Oaloite, transparent.. , Carbonic Acid Oas. is IM times as beary aa air " .. Cement. (See T IS.) •• Portland, 8.00 to J.tft.-. » •• Natural, 2.75 to 8.00 Chalk, S.'i te 3.8. Bee Limestones, quarried m Charcoal, of pines andoak«.~ Cherty, perfectly dry Chestnut, perfectly dry ......^ Goal. See also page S15. Anthracite, 1.8 to 1.7 " piled loose Biinmlnons, 1.8 to 1.4 m««*.««m..... " piled loose ....M...M €oke ** piled loose In ooUag, coals swell from 86 to 60 per sent. Copper, oast, 8.6 to 8.6 , " roUed, S.8to».0 Crystal, pure Qnarti. See Quartz. .1 4» .*••••••*...«•.•..• Cork. Diamond, 8.44 to 8.66 ; asaaUy8.61 to 3.66 ■arth ; common loam, perfectly dry, loose " •' " shaken " ** " moderately rammed.... *' slightly moist, looae. " more moiat, " • " •« ahaken " *' moderately packed • • " aa a aofl flewinr. mad " aaaaoftmud, well preaaed into a box........ il 4< 4< M «4 M M M U Ether Blm, peribctly 4rr. 1000 ft board measnre weicbs 1.803 teas. Bbeny, dry Emerald, 3.68 to 2.76 Fat. .average. flint •• Feldspar, i.5tot.8 •* Qarnet,8.5to4.8; Preoions, 4.1 to 4.8 ** Qtaas, 8.6 to 8.46 « " oommon window .' *' " Mill viUe, Kew Jersey. Thiek flooring glass " Oranite, 8.66 to 8.88. See Limestooe. 160 to 180 " ATerage BpOr. .00188 .798 .884 .916 .768 .61 8.7 8.81 2.6 6.70 6.67 1.4 8.9 8.1 9.74 8.1 8.4 8.6 J6 8.788 .00187 8.19 2.87 840 0.67 0.66 1.60 1.80 1.00 S.T &9 .96 8.68 •••• •»••! .716 .66 1.28 8.7 .08 9.6 9.66 9.W 9.69 946 9.79 ATerage Wtof a Cab Ft. Lbs. .6766 4i>.48 63.1 67.8 4T. 88. US. 14i. 163. 418. 41«. 87.8 181. 181. 607. 804. 694. 629. 160. 196. 100. 60 169.9 TSteSO 60 to 66 1B6» 16 to 90 48. 4L nteVM 4TtoM Tswaa «4toU CiJ6 79 to 80 89to 99 90 to 100 70 to 76 66 to 68 76 to 90 00 to 100 104 to 119 UOtoUO 44.6 86. W.1 0B. tot. 10k 106. 167. 160. t ITOw 8PEOIFIO GBAVITY. 213 T»¥le of speelflc frnkvttlea, mnA welffbtfi— (Ooutiiiiifld.) The specific gravity of any anbatance is » ttm welfllt in grains per cubic eentiHietre. (I «< ftneiMt oommoa* t.68 to 2.76 ** In looM piles " Hornbtondlo '* " quarried, in loose piles. Oyponia, Plaster of Paris, 2.24 to 2.80 ** in irregular lamps " '* gronnd, loose, per straok Iraahel, 70 " M «• well shaken '* *' 80.... •• " '• " Oaloined, loose, per stniokbaVhVtt to ral *.'.*. II " 1! GtMnstone. trarr *>8 to 3.2 «' ,. '* " fnarried, in loose piles ** Oravel, abont the same as sand, which see. Gold, oast, ptixv, or 34 earau '* " native, pure, 19.3 to 10.34 '* .. " *' freqaentiy oontaining silrer, 15.6 to 19.8 *' " pure, hammered, 19.4 to 19.6. > " OnttaPeroha ** .. HomUende, blaok, 8.1 to 8.4 '* Hydrofm G«s, is 14)^ times lighter than air ; and 16 times lighter than o^gea average.. Hendoek, perfeotljdrr. " 1000 reet board measure weighs .930 ton. Hlekorj, perfeotly dry. " 1000 feet board measure weighs 1.971 tons. Inn, and steel. •• Pig and oast iron and cast steel •* Wvoaght iron and steel, and wire, 7.6 to 7.9 •..••.. Ivory ' lee, .911 to .922 fiidiarobber '* Lignum vita, dry *< Lard " .. Lead, of eoBaMree,U.80ta 11. 4T; either rolled or east '• UmMtanee and Marbles, 3.4 to 2.M,U0 to 17&8 " " •* ordinarily about ** ** ** quarried in irregular fragments. 1 oub yard solid, makes abont 1.9 cab yds perfeotly loose : or about 1^ yds piled. In this last oase. 571 of the pile is solid; aod the Nmaining .429 part of it is voids piled.. UmBt qafBk, ground, loose, per straok bushel 62 to 70 lbs •• •• " well shaken. •• »• ....80 " ♦• " " thoroughly shaken, '* ...MH " ICahogaay, Spanish, dry*..... ....« «•.•... ...average. • ** Honduras, dry " Ibpte, di7« ♦' .. MarMei, sea Limestones. Maaoiuy, of granite or limestones, well dressed throngheal. ** *' " weU>scabbled mortar rubble. About 4 of the mass will be mortar - f ** wen-seabbled drr nibble M •< M roughly soabbled morur rabble. About H to Mi P^^ri will be mortar M M M rsntfily soabbled drv rabble ▲t 156 lbs per eub n, a cub y trd weighs 1.868 tons ; and 14.46 oub ft, 1 ton. Masouy of sandstone ; about H part less than the fbregolnf . *' briokwork, press e d briok, fine Joints average. . medlam quality •« (1 M <t •• " •* eoarse; infbrlor soft bricks " At 135 fl>s per eub ft, a oub yard weighs 1.607 tons; and 17.98 eub fl. 1 ton. IbraaiT.atSSOFah » 60° " •< tijo •« llka.2.75toS.l.... Mortar, hardened, 1.4 to 1.9k.. Mad, dry, close moderately pressed. fluid Average BpGr. 3.69 '2.8*" *2.'27' 8. 19.268 19.32 19.6 .96 3.35 A .86 7.2 7.76 1.82 .92 .98 1.38 .96 11.88 3.6 2.7 .86 .66 .79 19.62 13.58 13.88 2.93 1.66 Average Wt of a Cub Ft. Lbs. 168. 96. 176. 100. 141.6 8Z 66. 64. 52 to 66 187. 107. 1204. 1206. 1217. 61.1 203. .00531 25. 53. 450. 4T6ta4 114. 57.4 58. 83. 59.3 709.6 164.4 168. 96. 61. 64. 76. 63. 86. 40. 166. 154. 138. 150. 135. 140. 135. 100. 849. .846. 8S6. 183. 103. 80 to 110 110 to 130 104 to 120 • Green timbers asually weigh from one-fifth to nearly one-half more than 4fT;and ordinary building timbers when tolerably seasoned about one-sixth morethao perfectly dry 214 SPEOIFIO GRAVITY. Table of speelflc ffravitleB, and wetybUi— (Oontinaed.) The specific gravity of any sQbfltance is » its weiifllt in i^rams per cnbie centimetre. ATenge . Sp Or. Naphtlia Viirog«D Gas is about -^ part lighter than air Oak. live, perfeotly dry, .88 to 1.02* averafQ.. " r«d. blacli, 4o« " .. Oils, irhale; olive •• " oftarpentine " Oolites, or Boestones, 1.9 to 2.6 " Ozygeu Oas, a little more than JL part heavier thau air Petroleum Peat, dry, unpressed Pine, white, perfectly dry, .86 to .46* 1000 ft board measure weighs .080 ton.* " yellow, Northern, .48 to .62 1000 ft board measure weighs 1.276 tons.* • " Southern, .64 to .80 1000 ft board mean u re weighs 1.674 tons.* Pine, heart of long-leafed Southern yellow, luueai. ... 1000 ft board measure weighs 2.418 tens. Pitch Plaster of Paris ; see Gypsum. Powder, slightly shaken Porphyry, 2.66 to 2.8 Platinum 21 to 22 " native, in grains 16 to 19 Qnarti, common, pure 2.64 to 2.67 *' " finely pulveriied, loose ** *' " " well shaken " " " " well packed " quarried, loose. One measure solid, makes full IK broken and piled Baby and Sapphire, 8.8 to 4.0b^ Bosin 8alt...... Sand, pure quarts, perfectly dry, loose •* <• ** •* •* slightly shaken •« «« rammed, dry.... Natural sand consists of grains of differeat sixes, and weighs more, per unit of volume, than a sand sifted from it and having grains of uniform site. Sharp sand with very large and rery ■mall grains may weigh as much as < Sand is very retentive of moisture, and, when in large bulk, its natural moisture may diminish its weight from 6 to 10 per eent. ** perfectly wet, voids full of water ->»- Sandstones, fit for building, drv, 2.1 to l.YS 131 to 171. '* quarried, and piled, 1 measure solid, makes about IH piled... Serpentines, good 2.5 to 2.66 Bnow, fresh fallen ** moistened, and compacted by rain... Sycamore, perfectly dry. 1000 ft board measure weighs 1.S76 tons. Shales, red or black 2.4 to 2.8 average.. ** quarried, in piles " .. Slate t.Tto2.9 • ** .. Silver " .. Soapstone, or Stea|ite 2.66 to 2.8 *' .. Steel, 7. T to 7.9. The heaviest oon tains least earbon " .. Steel is not heavier than the iron from which it is made; onless the iron had impurities which were expelled daring its oonversion into steel. Svlphur ...., •.••.....•...•«••■■.•.....•..••..••..... average.. Spruce, perftiotly drr. • " .« 1000 ft board raeasore weighs .990 ton. Spelter, or Zinc 6.8 to 7.2. Sapphire; and Ruby, 3.8 tQ i...« Tallow " Tar " Trap, compact, 2.8 to 3.2 ** " quarried; in piles " Topaz. 8.46 to 8.66 " .95 .77 .92 .87 2.2 .00186 .678 •"•'-'•■ .65 .72 1.04 1.16 1. 2.78 21.6 17.5 8.66 S.9 1.1 M U t.41 • • • • S.6 •••••• .60 S.6 2.8 10.6 S.TS 7.66 1. .4 7.00 8.9 .94 1. 8. *S!65* Average Wt of a Cob Ft. Lbs. 6X.9 .0741 50.3 48. 82 to4B 57.3 54.8 137. .0648 54.8 20 to SO 25. 34.8 45. 66. Tl.T 82.8 170L 1343. 185. 90. 105. 112. 94. 88.t 60 to 70 90 to 108 92 to 110 100 to 180 117. U8 to ISO 111. 88. 182. 6 to IS 16 to 50 87. 161. 92. 17i. 656. 17QL 480. 1S5. 487.6 68.8 02.4 187. 107. * Green timbers usually weigh from one-fifth to nearly ODe-half more than dry ; and ordinary building timbers when tulerably seasoned about cue-sixth more than perfectly dry. WEIGHT OF COAL. 215 Table of apeclfle gravities, and weiffbta— (Continued.) The specific gravity of any substance is » its welgrli^ In yrams per eubie eentimetre. Tin, oast, 7.2 to 7.5 arerage. Turf, or Peal, dry, unpreaaed Water. Sm pagA 3*i6. Wax. bees average. Wine*. .993 to 1.04 •» , WalDOt, blaok, perfectly dry. " . 1000 ft board measnre weighs 1.414 tons. Zlno, or Spelter, 6.8 to 7.2.... < «« . Zirooo, 4.0 to 4.9 ** . Average 8pOr. 7.35 OQR •wo .61 7.00 4.45 Averace Wt of a Cub Ft. Lbs. 459. 90 to 80 68.417 eo.5 63.8 38. 4S7.6 S|Miee oeenpied by eoal. In cubic feet per ton of 2240 pounds. PennsylTanla Anthracite. Hard white ash* Free-burning white ash *. Shamokin * , Schuylkill white ash *. " red " *. Lykens Valley * Wyoming free-bumingf * Lehigh t Lehigh ; Reading C. & I. Co. *... Lehigh : f Lump, 40.5 ; cupola, 40 Bro- ken. Egg. Stove. Nut. Pea. Buck- wheat. f 38.6 39.2 39.8 40.5 41.1 ' 39.4 39.6 39.6 39.6 89.8 39.8 39.0 39.6 40.2 40.8 41.6 ' 39.6 39.6 39.6 41.2 41.9 42.4 39.3 39.9 40.5 41.2 41.9 39.0 39.9 42.6 45.7 46.5 47.7 39.6 40.3 40.9 41.6 42.3 40.0 40.5 41.1 41.7 42.3 {44.2 44.8 45.2 45.7 46.2 46.7 443 44.3 45.0 46.1 46.5 40.0 39.8 39.4 39.4 38.8 38.5 38.4 42.1 41.4 38.5 38.8 40.1 40.3 40.3 40.5 0.3; du Lst, 39.] .• Aver- age. 39.8 39.6 40.2 40.7 40.6 43.6 40.9 41.1 45.7 45.1 39.7 40.0 39.7 3itaininoas« From Coxe Bros. & Co. f Pittsburg 48.2 Erie 46.6 Hocking Valley 45.4 Ohio Cannel 45.5 Indiana Block 51.1 Dlinols 47.4 From Jour. XJ. S. Ass'n Charcoal Iron Workers. Vol. Ill, 1882.g Pittsburg 47.1 Cumberland, max 42.3 min 41.2 Blossburg, Pa 42.2 Clover Hill, Va 49.0 Richmond, Va. (Midlothian) 41.0 Caunelton, Ind ,....47.0 Pictou,N. S 45.0 Sydney, Cape Bretou.47.0 Logarithm. 1 cubic foot per ton of 2240 pounds = 0.89286 cubic foot per ton of 2000 pounds 1.950 7820 2240 (exact) pounds per cubic foot 3.350 2480 1 cubic foot per ton of 2000 pounds = 1.12 (exact) cubic reet per ton of 2240 pounds 0.049 2180 aOOO (exact) pounds per cubic foot .3.801 0800 1 pound per cubic foot = 2240 (exact) cubic feet per ton of 2240 pounds 8.850 2480 2000 " " " 2000 " 3.301 0300 •From Edwin F. Smith, Sup't A Eng'r, Canal Div., Phila. and Reading R. R. fFrom very oarefiil weighings in the Chicago yards of Coxe Bros. & Co. Kote the irregular variation with size of anthracite In Coxe Bros.' figures. ^Quoted from ITie Mining Record. On the authority of *• many years' experi- ence" of "a prominent retail dealer in Philadelphia," the Journal gives also figures requiring from 4 to 13 per cent, less volume per ton than those here quoted from the Journal and from other authorities. 216 WEIGHTS AND KEASITKE8. WEIGHTS AND MEASURES. United States and Brttisb measures of lengrtli and weiirbt» of the same denomination, may, /or all ordinarp pttrposeSf be ooncidered as equal ; but the liquid and dry measures of the same denomination differ widely in the two countries. Ttaie standard measure of leng^tb of both coun- tries is theoretically that of a pendulum vibratiiig seconds at the level of the sea, in the latitude of Loudon, in a vacuum, with Fahrenheit's thermometer at 629. The length of such a pendulum is supposed to be divided into S9.1393 equal parts, called inches ; and 36 of these inches were adopted as the standard yard of both countries. But the Parliamentary standard having been destroyed by fire, in 1834, it was found to be impossible to restore it by measarement of a pendulum. The present British Imperial yard, as determined, at a temperature of 629 Fahrenheit, by the standard preserved in the Houses of Parliament, is the standard of the United States Coast and Geodetic Survey, and Is recognized as standard throughout the country and by the Departments of the Govern- ment, although not so declared by Act of Congress. The yard between the 27th and 63d inches of a scale made for the U. S. Coast Survey by Troughton, of Lon- don, in 1814, is found to be of this standard length when at a temperature of 59^.62 Fahrenheit : but at 629 is too long by 0.00083 inch, or about 1 part in 43373, or 1.46 inch per mile, or 0.0277 inch in 100 feet The Coast Survev now uses, for purposes of comparison, two measures pre- sented by the British Government in 1855, as copies of the Imperial fltandsrd, namely : ** Bronze standard, Ko. 11 ;" of standard length at 62^.25 Fahr. " Malleable iron standard, No. 57 ;" " " " 62«>.io " See Appendix No. 12, Beport of U. S. Coast and Geodetic Survey for 1877. Tbe legral standard of ireielit of the United States is the Troy pound of tbe Mint at Philadelphia. This standard, containing 5760 Sains, is an exact copy of the Imperial Troy pound of Grea* ritain. The avoirdupois or commercial pound of the United States, con- taining 7000 grains, and derived from the standard Troy pound of the Mint, is found to agree within one thousandth of a erain with the British avoirdu|M>fs pound. The U. S. Coast Survey therefore declares the weights of the two ooun- tries identical. Tlie Ton. In Revised Statutes of the United States, 2d Edition, 1878, Title XXXiy, Collection of Duties upon Imports, Chapter Six. Appraisal, says : "Sec. 2951. Wherever the word 'ton' is used in this cnapter, in reference to weight, it shall be construed as meaning twenty-hundredweight, each hundred- weight being one hundred and twelve pounds avoirdupois." This appears to be the only U. S. Government regulation on the subject. The ton of 2240 ft>s (often called a sross ton or Ions ton) is commonlj used in buying and selling iron ore, pig iron, steel rails and other manufactured iron and steel. . Coke and many other articles are bought and sold by the net ton or sliort ton of 2000 lbs. The bloom ton had 2464 ftis, = 2240 fira -^ 2 hundredweight of 112 S>s each ; and the pig iron ton had 2268 fi>s, == 2240 lbs + a "sandage" of '28 fcs, or one "quarter," to allow for sand adhering to the pigs, but some furnace men allowed only 14 lbs. In electric traction work the ton means 2000 lbs. As a measure, the ton, or tun, is defined as 252 gallons, as 40 cubic feet of round or rough timber or in ship measurement, or as 60 feet of hewn timber. 252 U. S. gallons of water weigh about 2100 Ha ; 252 Imperial gallons about 2500 lbs ; SO cub ft yellow pine about 2500 Sts. Tbe metric system * was legalised in the United States in * The metric system, as compared with the English, baH much the same advantagea and disadvantages that our American decimal coinage has in comparison witiii the English monetary system of pounds, shillings and pence. It will enormously facili- tate all calculations, but, like all other improvemeute, it will necessarily eause some inconvenience while the cliange is being made. The metric system has also tMa ftir- ther and very great advantage, that it bids fair to become univei-sal among of viliaeo rations. WEIGHTS AND MEASURES. 217 1866, but hM not been made ot)llgfttorT. The gorernment has since ftirnished very exact metnc standards to the several States. The use of the metric system has been permitted In Great Britain, beginning with August 6, 1897. and in Ruflsia, beginning with 1900. I to use is now at least permissive in most civil- ised nations. Tlie laetrle nnlt of lenytb is tlie metre, er nueter, which waa fntended to be one ten-millionth I j of the earth's quadrant, f. c, of Ihat portion of a meridian embraced between either pole and the equator. This lengtn was measured, and a set of metrical standards of weight and measure were prepared in accordance with the result, and deposited among the archives «f France at Paris (MHre des Archives.. Kilogramme des Archives, etc.). It has since been discovered that errors occurred in the calculations for ascertaining the length of the quadrant ; but the standards nevertheless remain as originally preparM. Tlie metric measures ef surface and of capacitv^ are the squares and cubes of the meter and of ito (decimal) fractions and multiples. Tlie metric unit of welarlit is tlie grramme or grram, which is the weight of a milliliter or cubic centimeter * of pure water at its tempera* tore of maximum density, about A.5^ Gentisrade or 40^ Fahrenheit. By the concurrent action of the principal governments of the world, an In- temational Bureau of Weiyiits and MeasuriMi has been estab- Ushed, with its seat near Paris. It has prepared two ingots of pure platinum- ixidium, from one of which a number of standard kilograms (1000 grams) havf been made, and from the other a number of standard meter bars, both derived from the standards of the Archives of France. Of these copies, certain ones were selected as international standards, and the others were distributed to the different governments. Those sent to tne United States are in the keeping of the U. 8. Coast Survey. The detennination of the ei|niTalent of tbe meter in Eng^Iisii measure is a very difficult matter. The standard meter is measured from end l» sfuf of %pkUiiuan bar and at the freexbtng point ; whereas the standard yard is measured hehown two lines drawn on a silver seale inlaid in a brmize bar. and ai ^aP FiihrenheU. Tbe United States Ooast Surweyf adopts, as the length of the meter at 62° Fahrenheit, the value determined by Capt. A. R. Clarke and Col. Sir Henry James, at the office of the British Ordnance Survey, in 1866, vis. : S9.37(M82 inches (= 8.2808666 + feet « 1.0986222 + yards) ; but the lawftil equiwaient, established by Congress, is 39.87 inches (=t 3.28083 feet = 1.098611 yards). This value is as accurate as any that can be deduced from existing data. Tbe ffram Weislis, by Prof. W. H. Miller's determination,! 15.43234874 Sains. An examination made at the International Bureau of Weights and easures in 1884 makes it 15.43236639 grains. The leeal value in the United States is 15.432 grains. • 1 centimeter =» r^ meter = 0.3937 inch. 1 milliliter {^^ liter) or cubis centi- meter =3 0.061 + cubic inches, t Anpendix No. 22 to report of 1876, page 6. X Philosophical Transactions, 1866, pp. &3y ets. 218 rOEBIGN COINS. Approximate Talses of Foreign Coins* in U. S. Honey. The references 0, ^, ^ and *) are to foot-notes on next page. From Circular of U. S. Treasury Department, Bureau of the Mint, Jan. 1, 1887; from " Question Mon6taire," by H. Costes, Paris, 1884; and from our 10th edition. Argentine Repub.— Peso = 100 Centavos, 96.5 ots.** Argentino = 5 Pesos, $4.82. Austria.— Florin = 100 Kieutzer,47.7 cts.,2 3o.9 cts.s Ducat, $2.29. Maria Theresa Thaler, or Levantin, 1780, $1.00.2 Rix Thaler, 97 cts.* Souverain, $3.57.* Belgium.i— Franc = 100 centimes, 17.9 ct8.,« 19.3 ots.* Bolivia— Boliviano = 100 Centavos, 96.5 cts.,* 72.7 cts.« Once, $14.95. Dollar, 96 cts * Brazil.— Mil reis = 1000 Reis, 50.2 cts.,* 54.6 cts.3 Canada. — English and U. S. coins. Also Pound, $4.* Central America.*— Doubloon, $14.50 tu $15.65. Reale, average S^ cts. See Honduras. Ceylon.— Rupee, same as India. Chili.— Peso = 10 Dineros or Decimos = 100 Centavos, 96.5 cts.,* 91.2 ct«.» Con- dor = 2 Doubloons = 5 Escudos = 10 Pesos. Dollar, 93 cts.* Cuba.— Peso, 93.2 cts.* Doubloon, $5.02. Denmark.— Crown = 100 Ore, 26.7 ct8.,« 26.8 cts.a Ducat, $1.81.* Skilling, % ct* Ecuador.— Sucre, 72.7 cts.» Doubloon, $3.86. Condor, $9.66. Dollar, 93 cts.* Eleale 9 cts * Egypt.— Pound = 100 Piastres :« 4000 Paras, $494,3.* Finland.— Markka = 100 Penni, 19.1 cts.* 10 Markkaa, $1.93. France.1— Franc =100 Ceniimes, 17.9 ct8.,« 19.3 cts.8 Napoleon, $3.84.* Livre, 18.5 cts.* Sous, 1 ct.* Germany.— Mark = 100 Pfennigs, 21.4 cts.,2 23.8 cts.* Augustus (Saxony), $3.98.* Carolin (Bavaria), $4.93.* Crown (Baden, bf,varia, N. GermanyX $1.06.* Ducat (Hamburg, Hanover), $2.28.* Florin (Prussia, Hanover), 66 eta.* Groschen, 2.4 cts.* Kreutzer (Prussia), .7 ct. Maximilian (Bavaria). $3.30.* Rix Thaler (Hamburg, Hanover), $1.10* (Baden, Brunswick), $1.00* (Prussia, N. Germany, Bremen, Saxouy, Hanover), 69 cts.* Great Britain. — Pound Sterling or Sovereign (£) = 20 Shillings = 240 Pence, $4.86.65.* Guinea = 21 Shillings Crown = 6 Shillings. ShilUng (*), 22.4 cts.,s 24.3 cts. (^ pound sterling). Penny (d), 2 cts. Greece.!— Drachma = 100 Lepta, 17 cts.,« 19.3 cts.* Hayti.— Gourde of 100 cents, 96.5 cts.s* Honduras.— Dollar or Piastre of 100 cents, $1.01. See Central America. India.— Rupee = 16 Annas, 45.9 cts.,^ 34.6 cts.* Mohur = 16 Rupees, $7.10. Star Pagoda (Madras), $1.81.* Italy, etc.i— Lira = 100 Centesimi, 17.9 cts.,2 i9.3cts.* Carlin (Sardinia), $8.21.* Crown (Sicily), 96 ctfi.* Livre (Sardinia), 18,6 cts.* (Tuscany, Venice), 16 sts.* Ounce (Sicily), $2.50.* Paolo (Rome), 10 cts.* Pistola (Borne), $3.37.* Scudo* (Piedmont), $1.36 (Genoa), $1.28 (Rome), $1.00 (Naples, Sicily), 95 cts. (Sardinia), 92 cts. Teston (Rome). 30 cts.* Zecchino (Rome), ^.27.* Japan.— Yen = 100 Sen rgold), 99.7 cts.* (silver), $1.04^, 78.4 cts.* Liberia.— Dollar, $1.00.* * Mexico.— Dollar. Peso, or Piastre = 100 Centavos (gold), 98.3 cts. (silver), $1.05,« 79 cts.* Once or Doubloon = 16 Pesos, $15.74. Netherlands.— Florin of TOO cents, 40.5 cts.," 40.2 cts.« Ducatoon, $1.32.* Guilder, 40 cts.* Rix Dollar, $1.05.* Stiver, 2 ctfl.* New Granada.— Doubloon, $15.34.* Norway.— Crown = 100 Ore = 30 Skillings, 26.7 ct8.,« 26.8 cts.« Parascuay .—Piastre = 8 Reals, 90 cts. Persia.— Thoman = 6 Sachib-Kerans = 10 Banabats = 25 Abassis — 100 Scahia, $2.29. Peru.— Sor= 10 Dineros = 100 Centavos, 96.5 cts.,a 72.7 cts.* Dollar, 93 eta.* Portugal.— Milreis = 10 Testoons = 1000 Reis, $1.08.* Crown = 10 Milreis. Moidore, $6.50.* Russia.- Rouble = 2 Poltinniks = 4 Tchetvertaks = 6 Abassis = 10 Griviniks = 20 Pietaks = 100 Kopecks, 77 cts.,« 58.2 cts.* Imperial =-« 10 Roubles, $7.72. Ducat = 3 Roubles, $2.39. Sandwich Islands.- Dollar, $1.00.* Sicily.— See Italy. Spain.— Peseta or Pistareen = 100 Centimes, 17.9 cts.,* 19.3 cts.* Doubloon (new) = 10 Escudos = 100 Reals, $5.02. Duro = 2 Escudos,* $1.00.2 Doubloon (old), $15.65.* Pistole = 2 Crowns, $3.90.* Piastre, $1.04.* Reale Plate, 10 cta.^ Beale vellon, 6 cts.* 1, 2, 3, 4. See foot-notes, next page. FOBEIGN COINS. 219 (Foreign Coins QnUinMd. Small flsnreft Oi *» 'i *) ^^^ ^ M^ noUs.) Sweden.— Crown = 100 Ore, 25.7 ct8.,« 26.8 cta.» Ducat, $2.20.* Rix Dollar, $1.05.« Switzerland.!— Franc = 100 Centimes, 17.9 et8.,2 19.3 ct8.« Tripoli.— Mahbub = 20 Piastres, 65.6 ct8.» Tunis.— Piastre = 16 Karobs, 12 cts.2 10 Piastres, f 1 .16.6. Turkey.— Piastre = 40 Paras, 4.4 cts.' Zecchin, J1.40.* United States of Colombia.— Peso = 10 Dineros or Decimos = 100 Centaros, 96.5 cts.,« 72.7 ct8.3 Condor = 10 Pesos, $9.65. Dollar, 93 5 cts.* Uruguay.— Peso = 100 Centavos or Centesimos (goldl, $1.03 (silver^ 96.5 cts.s Venezuela.— Bolivar — 2 Decimos, 17.9 cta.,2 19.3 cts.* Venezolano = 5 Bolivars. Standard Blameiers and Welgrbte of United States Coins. Valae. Diam«ier. Wetgbt. €k>ld, 10 per cent, alloy : Double Eagle Eagle TTfLlfFagle . . 1 20 10 '5 2.50 1.00 0.50 0.25 0.10 0.05 0.01 Inches. 1.350 1.060 0.848 0.700 1.500 1.205 0.955 0.705 0.835 0.750 Millimeters. 34.29 26.92 21.54 17.78 38.10 30.61 24.26 17.91 21.20 19.09 Grains. 516.00 258.00 129.00 64.50 412.60 192.90 96.45 38.58 77.16 48.00 Grams. 33.436 16.718 8.359 Quarter "kagle Silver, 10 per cent alloy : Standard Dollar TTalf Dnllfif . . 4.180 26.729 12,50 Quarter Dollar Dime JHlnor Five Cents, 75^^ copper, 25^« nickel . . .• 6.25 2.50 5.00 One Cent, 95^^ copper, 5^ tin and zinc 3.11 Perfectly pure sold is worth $1 per 28.22 grs = $20.67183 per troy oe =* $18.84151 per avoir oz. Bttandard (U. 8. coin) is worth $18.60465 per troy oz = $16.95736 per avoir oz. It consists of 9 parts by weight of pure gold, to 1 part alloy. Its value is that of the pure gold only ; the cost of the alloy and of the ooini^ being borne by Government. A cable f€»ot of pure cold irelgphs about 1204 avoir lbs ; and is worth $362963. A cubic ineh weighs about 11.148 avoir oz ; and is worth $210.04. Pure gold is called fine, or 24 earat gold ; and when alloyed, the alloy is sup- posed to be divided into 24 parts by weight, and according as 10, 15, or 20, 4&c, of these parts are pun gold, the alloy is said to be 10, 16, or 20, Ac, carat. The averaipe fineness of California natlTe void, by some thou- sands of assays at the U. S. Mint in Philada., is 88.5 parts gold, 11.5 silver. Some from Georgia, 99 per cent. gold. •Pure sllTer fluctuates in value : thus, during 1878-1879 it ranged between $1.05 and $1.18 per troy oz., or $.957 and $1,076 per avoir, oz. A cubic inch weiglfs about 5.528 troy, or 6.065 avoir, ounces. 1 France, Belgium, Italy, Switzerland, and Greece form the Latin Union. Their coins are alike in diameter, weight, and fi^ieness. t __ 19.3 times the value of a single coin in francs as given by Costes. » Par of exchange, or equivalent value in terms of U. S. gold dollar.— Treasury Giicalar. « Erom our 10th edition. 220 WEIGHTS AND MEASURES. Troy Weifrbt. U. S. and British. 24 grains 1 pennyweight, dwt. 20 pennyweights 1 ounce = 480 grains. 12 ounces 1 pound = 240awtB. = 5760 grains. Troy welcht is nsed for grold and silver. A carat of the jewellers, for precious stones is, in the U. S. = 3.2 grs. ; in London, 3.17 grs. ; in Paris, 3.18 grains., divided into 4 jewellers' grs. In troy, apothecaries' and avoirdupois, tbe grain is tbe same. Apotbecaries' Weiffbt. U. 8. and British. 20 grains 1 scruple. 3 scruples 1 dram = 60 grains. 8 drams 1 ounce = 24 scruples = 480 grains. 12 ounces 1 pound = 96 drams = 288 scruples = 5760 grains. In troy and apothecaries' weights, the grain, ounce and pound are the same. Avoirdupois or €oniniereial Weiffbt. U. 8. and British. . 27.34875 grains - 1 dram. 16 drams 1 ounce = 437V grains. 16 ounces 1 pound = 256 drams = 7000 grains. 28 pounds 1 quarter = 448 ounces. 4 quarters ~ 1 hundredweight = 112 fl)8. 20 hundredweights 1 ton = 80 quarters = 2240 fts. A stone «> 14 pounds. A quintal = 100 pounds avoir. Tbe standard of tbe avoirdupois pound, which is the one in common commercial use, is the weight of 27.7015 cub ins of pure distilled water. at its maximum density at about 39°.2 Fahr, in latitude of London, at the level of the sea ; barometer at 30 ins. But this involves an error of about 1 part in 1362, for the IS) of water = 27.68122 cub ins. A troy lb = .82286 avoir ft. An avoir ft = 1.21528 troy ft, or apoth. A troy OS. = 1.09714 avoir, oz. An avoir, oz. = .911458 troy oz., or apotb. IiOn§: Measure. U. 8. and British. 12 inches 1 foot = .3047978 metre. 3 feet 1 yard = 36 ins = .9143919 metre. 5^ vards 1 rod, pole, or perch =» 16U feet = 198 ins. 40 ro^s 1 furlong = 220 yards -= 660 feet. Sfurlongs 1 statute, or land mile = 320 rods = 1760 y^ =.6280 ft « 63360 iiM. 3 miles 1 league = 24 fUrlongs = 960 rods = 5280 yds = 15840 it. A point =y, inch. A line = 6 points =*t^ inch. ^ palm = 3 ins. A banS = 4ins. Aspan = 9ins. A fatbom = 6 feet. A cable's lenKtb = 120 fathoms = 720 feet. A Gnnter's surveying cbain is 66 feet, or 4 rods long. It has 100 links, 7.92 inches long. 80 Gunter's chains = 1 mile. A nautical mile, geoffrapbical mile, sea mile, or knot, is variously defined as being = the length of metres feet statute miles 1 min of loniritude at the equator = 1856.345 6087.16 1.15287 1 « latitude « " = 1842.787 6045.95 1.14507 1 ^^ lauiuu ^^ ^ 1861.655 6107.85 1.15670 1 '« «* atlat46° = 1862.181 6076.76 1.15090 1 "a great circle Qf a true') (value adopted .by U. S. Coa»t mhere whose surface area is V -=< and Geodetic Survey fqutl To that of the earth j ll853.248 6080.27 1.15157 British Admiralty bnot = 1853.169 6080.00 1.15152 The above lengths of minutes, in metres and feet, are those published by the U. S. CoMt and Geodetic Survey in Appendix No 12, Report for 1881, and are calculated from Clarke's spheroid, which is now the standard of that Survey. At the equator, 1° of lat =-- 68.70 land miles; at lat 20° = 68.78 ; at 40° = 69.00 ; at 60° - 69.23 ; at 80° = 69.39 ; at 90° = 69.41. WBiaHT8 AKD MEASURES. 221 I^en^tlis of a D flg r— of Ii«B9itiide In Afferent liatltndefl, and at tllC level or tMke iteat The** Itngthi are In oommon land or statate mlleii, •r 5S80 n. SioM the flgure of the earth has nerer been prteUtli/ aaeertained, these are but oloee ap proximatlene. Intermediate onee may be fouid eorreettj bj simple proportion. !<> of tongituM * te 4 mine ef oItU or eloek tUM| 1 mln of InngltiiilB to 4 eeoi of tine. Degofi , Lat. ^ iilSB. Dec of Lat. Mike. Dec of Lit. MUea. Dec of Lat. miM. Dec of Lat. MUes. Dec of Lat. MUes. 1 W.16 14 67.12 28 61.11 42 61.47 66 88.76 70 28.72 a 1 ie.i2 16 66.50 80 69.94 44 49.88 68 86.74 72 21.43 4 1 M.N 18 65.80 S3 58.70 46 48.13 60 84.67 74 19.12 6 ( B6.76 20 66.02 34 67.39 48 46.88 62 82.56 76 16.78 8 B&tt 22 64.16 36 56.01 50 44.54 64 30.40 78 14.42 10 118.12 24 63.21 88 64.56 63 43.67 66 28.21 80. 12.05 13 17.66 96 62.90 40 53.06 54 40.74 68 26.98 82 9.66 InelieB redaeed to Deeimals of a Foot. Ao errors. Ina. ] root. las. Foot. IDI. Foot. Ins. Foot. Ins. Foet: Itti. Foot. • .0000 % .1867 4 .8383 6 .5000 S .6667 10 .8833 1-SS .0026 .1693 .3359 .5026 .6693 .8859 1.16 .0062 .1719 .8886 .6052 .6719 .8886 8-n .0078 .1746 .8411 .5078 .6746 .8411 Ji . .0104 H .•771 H .9488 H .5104 H 41771 H .8438 OUO .1797 a Jig 4 mOVfn .6130 .6797 .8464 S-16 . 0166 .1828 .3480 .5156 41823 .8490 f-tt 0182 .1849 .8516 .6182 41848 .8616 Ji : 0208 H .1876 H .3542 H .6208 H .6875 H .8643 0284 .1901 .3568 .5284 .8801 .8568 fr-16 0280 .ion .3594 .6200 .6927 .8694 11-S9 0286 .1953 .3620 .6286 .6953 .8620 H 0313 H .1979 H .3646 H .5313 H .6879 H •oDvO ust 0339 .2006 .8672 .5339 .7006 .8672 7«1« 086& .2031 .3698 .6866 .7031 .8688 U^ 0381 .2067 .3724 .5391 .7057 .8724 .^ 0417 H .2083 H .3750 H .6417 H .7083 H .8750 17-SS 0443 .2109 .8776 .5443 .7109 .8776 9-M 0469 .9186 .8802 .5469 .7135 .8802 IMS 0485 .2161 .8828 .5495 .7161 .8828 nji : 0621 H .2188 H .3854 H .5521 H .7188 H .8854 0647 .2214 .8880 .5647 .7214 .8880 ii.i« 0573 .2340 .8906 .5573 .7240 .8806 ss-ss 0680 .2966 .3932 .6599 .7266 .8692 H 0626 H .2392 H .8958 H .5625 h .7292 H .8958 Sft^ 6661 .2318 .8964 .5651 .7818 .8964 lft.lC oon ^2844 .4010 .5677 .7344 .9010 S7-» 0703 . .2370 .4036 .5703 .7370 J8006 y • 0729 % .2396 X .4063 X .6729 H .7396 }i .9063 f^ 0765 .2432 .4069 .6755 .7422 .9089 mi . 0781 •9vfto .4115 .6781 .7448 .9115 • Sl-SS 0807 .2474 .4141 .6807 .7474 .9141 1 06SS S .2509 .4167 y 4i688 9 .7500 11 .9167 1« 0869 .2626 .4193 .6859 .7526 .9193 1-lC 0885 .3563 .4219 .7562 .9219 8-S2 0911 .2678 .4245 .6911 .7578 .9246 H 0888 H .2004 H .4271 H .5038 H .7604 H .9271 5-St 096A .3660 .4297 .5964 .7680 .9297 S-I6 0800 .3866 .4323 .6990 .7656 .9823 7-8i 1016 .3683 .4.')49 .6016 .7682 .9349 3< • 1042 H ■S& H .4876 3i .6043 H .7708 H .9375 9-Si 1068 .4401 .8068 .7784 .9401 6-16 1684 .2768 .4427 .6094 .7760 .9427 11-32 1198 .2786 .4453 .6120 .7786 .9468 K 1148 H .2811 H .4479 H .6146 H .7813 H .9479 lS-3t2 1172 .2889 .4505 .6172 .7889 .9506 7-16 1198 .2666 .4531 .6198 .7865 .9531 « 16-32 1224 .«9l .4567 .6234 .7881 .9557 ^ 1260 H S& H .4583 H .6250 H .7917 H .9583 17-.% 1276 .4809 .6276 .7948 .9609 9-16 UOS .2M9 .4635 .6302 .7969 .9636 19-32 1828 :SSi .4661 .6828 .7995 .9661 2i.l^i : 1864 H H .4688 H .6354 H .8021 H .9688 1380 .lOiV .4714 .6380 .8047 .9714 11-16 1406 .8978 .4740 .6406 .8073 » .9740 SS.S2 108 J089 .4766 .6432 .8099 .9766 9i 1468 H .8136 h .4792 H .6456 H .8125 h .9792 25-S3 1484 .8161 .4818 .6484 .8151 .9618 13-16 1610 .8177 .4844 .6510 .8177 OtlAA •von 27-S2 1686 .8908 .4870 .65.<{6 .8203 .8229 .9870 H 1668 H .8228 H .4896 K .6bea X H .9896 n.n . 1689 41256 .4922 .6589 .8255 .9922 16-16 1816 . .8281 .4948 .6615 .8281 .9948 n« • 1641 .8807 .4974 .6641 .8307 .9974 WEIGHTS AND MEABUBBS. — —"■H-Ij » iq ill = 10a» aq tOl. rodi = W40 iq Ida = UMt K Ml- Cnblp. or Solid M^amare. A CBbt* a M Dik THd, or i.Ma» ■•knlg^ (I. HI iu^>llln. •> ««HHn, n. A tim i.iw> ai^ ci iennijtiu A cnbl« luch Is midaI to l.nuta snlllllni; e.r.ie3S«e3 arellLLnir a A cubic yard la emnMl l4 1 aphere I toot In diameter, tiontnlna A sphere 1 Inek In diameter, eonlnlna WEZGHTS AUTD HBASimiiB. 22a cylinder 1 foot In diameter, .02909 oub yard. .7854 cub foot. I35T. 1712 cub inches. .63112 U. S. di7 bushels. 2.5245 U. S. dry pecks. a0.1958 U. S. dry quarts. . 40.3916 U. S. dry pints. 5.8752 U. S. liaaid gallons. 28.5008 U. S. liquid quarts. A eylinder 1 ineli in diameter, and .005454 cub foot. 9.4248 cub inches. .2805 U. 8. dry pint. .3264 liquid pint. 1.3056 U. S. gill. and 1 f<N>t bisrta, coui^Jiins 47.0016 U. S. liquid piuta. 188.0064 U. b. liquid gills. 4.8947 Brit imp gallons. 19.5788 Brit imp quarts. 39.1575 Brit imp pint*. 156.6302 Brit imp gills. 222.S95 decilitres. 22.2395 litres. 2.22395 decalitres. .222895 hectolitre. 1 foot liiji^li, contains .2719 Brit imp pint. 1.0677 Brit imp gill. 15.4441 centilitres. 1.54441 decilitres. .164441 litres. I«iqald JHeasnre. u, g. only. The iMMda of this measure in the U. S. is the old Brit wine gallon of 231 oub ins; or 8.3S888 Ibr aToir of pure water, at its max dennity of about 39^.2 Fabr ; the barom at 30 ins. A cylinder 7 in» iiam, and 6 ins high, contains 230.904 cob ins, or almost precisely a gallon ; as does also a oube of t.lS68 ina on an edge. Also a gallon = .13368 of a cub ft ; and a cub ft contains 7.48052 galls ; nearly 1H gall*-. TUs bastfl howerer InTolres ab err«r of about 1 part in 1363, for the water adtn- 63 gallons 1 hogshead. 2 hogsheads 1 pipe, or butt. 2 pipes. 1 tun. In the U. S. and Great Brit. 1 barrel of wine or brandy = 31i^ galls ; in Pennsylvania, a half barrel, 16 galls; a double barrel, 64 galls; m puncheon, 84 galls; a tierce, 42 galls. A liquid Beasore barrel of 81^ galls contains 4.211 cub ft = a oube of 1.615 ft on an edge ; or 3.38v U. S. struck bosbals. A sill = 7.21875 oub ins. The followlns cyliinders contain some o.' these measure* very approximately. ally weighs 8.3450(tti tbi cub ins. 4glUa Ipint =28.875. 2 pints 1 qnart = 57.750 = 8 gills. 4 qxaaU 1 gallon = 231 . =8 pints— 32 gills DIam. Height, enb ins. Ins. Ins. Omj.21875) IH 3 ><pint 2« 3« Pint 3« 3 quart S^ 6 Diam. Ins. Gallon 7 . 2 gallons 7 . 8 gallons 14 . 10 gallons 11 . Height. Ins. 6 . 12 . 12 . 15 Apotbecaries* or Wine Measure. 1 Gallon mnt... 1 Fluid ounce . . 1 Fluid drachm. IMmim Symbol. Pints. Floid .ounces. FJoid draohms. Minims. Coble inches. Cong* m 8 1 • ■ • • • • • • • • • • 128 16 1 • • • • • • • • 1024 128 8 1 • • • 61440 7680 480 60 1 231 28.875 1.8047 0.2256 0.0088 Weight of water4 Pounds, av. Grains. 8.345 1.043 Ounces, av. 1.043 68415 7301.9 456.4 57.05 0.96 To redoce U. H» liquid measnres to Brit ones of the same denomina* tlon, divide by 1.30032; or near enough for common use, by 1.2; or to reduce Brit to U. S. multiply by 1.2. Dry Measure. U. S. only. Tlie basis of tliis is the old British Winchester struck bushel of 2150.42 cub las; or 77.627418 pounds avoir of pure water at its max density. Its dimensions by law are 18^ ins iaaer diam ; 19>t id> outer diam; and 8 ins deep ; and when heaped, the cone is not to be less than 6 ins Ugh ; which makes a heaped bushel equal to 134 struck ones ; or to 1.55556 cub ft. Bdge of a cube of equal capacity. 2 pints 1 qoart, =67.2006 cub ins = 1.16365 liquidiit 4.066 ins. 4 quarts 1 gallon. = 8 pints, = 268.8026 cub ins, :^ 1.16:i65 liq gal 6.454 " 2 gallons 1 peek, = 16 pints, = 8 quarts, = 537.6050 cub ins 8.131 " 4 pe<d(s 1 stmok bushel, = 64 pinls, = 32 quarts, = 8 gals, = 2150.4200 cub ins. 12,908 " * Abbreviation of Latin, Congius. t Abbreviation of Latin, Ootarios. } At its maximum density, 62.426 pounds per eubio foot, correspouding to a temperature of 4° Ceotigrade = S9.2P Fahrenheit. 224 WMGH1B AKD MBA8X7BBS. A 9trnck bnshel =» 1.24445 cub a. A cub ft * .80356 of a struck bushel. Xhe dry flour barrel = 8.75 cub ft; =8 struck bushels. The dry barrel la not, howe%'er, n legMliied measure; and no great attention is given to its capacity; consequently, barrels rar^ cunsiderablT. A barrel of Qour conuins by law, liW Its. In ordering by tbe barrel, the amount of its contents sboald be specifled in pouods or galls. To reduce IJ. S. dry measures to Brit imp ones of the same name, di? by 1.031516 ; and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposes use 1.033. Brltlsb Imperial Measure, botb liquid and dry. This system is established throughout Great Britain, to the exclusion of tbe old ones. Its basis is the imperial gallon of '277.274 cub ins, or 10 lbs avoir of pure water at the temp of 62^ Fahr, when the barom is at 30 Ids. This basis Involves an error of about 1 part im 18S6, for 10 lbs of the watar =:only 277.128 cab ina. Aroir Ihe. of water. Oob. las. Cab. ft. Edge of a cube «f equal capaeity. Inches. Acllla 1 pint 1.25 8.50 6. 10. 80. -1 80. I Dry 820. { meaa. 84.6688 e».8l85 188.687 877.874 554.648 9818.188 8878.768 in45.686 8.8605 Ipinta 1 quart S quarts 1 pottle 8 Dottles I Kallon 4.1079 6.1756 6.6908 S fftllODB 1 p6C!C ••••••••••• ■•• 8. 2157 4 Dooki 1 buhel.a.a •••••••••• 1.8R87 6.1847 10.2694 1*041? 4 basbelsl coomb 8 coombs 1 quarter 1 6i0. TiM) imp gall = .16046 cub ft; *Dd 1 Ottb ft =<.9B918 galls. Measure. Symbol. Pints. Fluid ounces. Fluid drachms. Minims. Oubic iochM. Weight of watar4 Pounds, AT. Graimt. 1 Gallon 1 piDt Of fl. OS. fl.dr. mill* 8 1 • • • • • • • • 160 90 1 • ••• • • ■ • 1280 160 8 1 • • • • 78800 9600 480 60 1 877.274 86.669 1.783 0.217 0.0086 10 1.85 Ounces, ar. • ••• 70068 •750 487.5 54J875 0.9114 1 Fluid ounce . . . 1 Fluid drachm.. 1 Minim The weight of water aflbrds an easy way to find the cubic contents of a tressel. To' obtain the slae of commerelal measai^ea by means Qf tlio * welg^bt of water. At the common temperature of fh>m 70*^ to 75° Fah, a cub foot of ftesh water weighs wrr appnud> mately 6214 \bi avoir. A cubic half foot, (6 ius on each edge,) 7.78125 0>a. A cub quarter foo^ (8 ins on each edge.) .97266 n>. A cab yard, 1680.75 lbs; or .75034ton. ▲ cub half yd, (18 ins on each «das,) 210.094 lbs ; or .0938 ton. A cub inch, .036024 0) ; or .576384 ounce ; or 9.2222 drams ; or 252.170 grama. An Inch square, and one foot long, .432292 Bk. Also lib = 27.76908 cab ins, or a cube of 8.096 ins on IB edge. An onnce, 1.785 «ub ins ; a ton, 85.964 cab ft, all near enoof h for common me. Original. Uquld Measures. i^^\^^«^- of Water. V. S.Gill 26005» U. 8. Pint 1.0409 U. S. Quart 2.0804 U.S. Gallon 8 lbs 5l 01 8.8916 U. S. Wme Barrel, 31 H Gail 969.1810 Dry Measures. U. S. Pint 1.2104 U. 8. Quart 2.4208 17. S. Gallon 9.6834 V. 8. Peck 19.3668 U. a. Bushel, struck 77.4670 ' * Or 4 ounces ; 2 drams ; 15.6625 grs. I«lqal€l and I>ry. Um AT*ir. ^ ot Water. British Imp Gill S1914* *' Pint 1.94858 " «• Quart 9.49715 •• •' Gallon 9.9886 " •• Peek..M. 19.9779 " Bushel 79.9088 * 4.9949 ; or rery nearly 5 onnoas. Metric Measnires. Centilitre .03196t pMilltre siMt Litre J.1981 Decalitre, or Centlatere 91.9606 Btere (eubio meter) 9198.0786 t Or 5.6271 drams; or 153.866 gra. { 3.5169 onnoes. * Abbreviation of Latin, Congius. t Abbreviation of Latin, Ootarius. t At the standard lemperatore, 929 Fahrenheit a about 16.r> Oentlf rada. WEIGHTS, AND UEABUBB0. 225 Metrle Measures of I^eni^^b. By U. 8. and Brltfsli StaaiUrd. Ins. Ft. Yds. Miles. Millimetre* .089370 .89370428 8.9370428 89.370428 393.70428 Road measures. .008281 .082809 .8280869 3.280869 32.80869 328.0869 3280.869 82808.69 CeTltim«tre+t--,T,---r r ^,,r,r-,„r „f • ]|[)ACini6tTA .1093628 1.093628 10.93623 109.3623 1093.628 10936.23 Metret Dnftiunetrft ") Hectometre .0621875 Eflometre .6218760 Kyriametre j 6.213750 • N«arl7 the ^ part of ao inoh. t Full K inob. } Yerj nearly 8 ft, 3H ioB. wbioh is too long hj onlj 1 part in 8616. Hetrlc Square Measure- By U. S. m4 British Slradard. 8q Millimetre 8q Centimetre Sq Decimetre Sq Metre, or Centlare., Sq Decametre, or Are. D«care (not nsed) Hectare 8q Kilometre 8q Myriametre Sq. Ins. .001550 .155003 1S.500B 1550.03 155008 .3861090 so miles. 38.61090 " Sq. Feet. .00001076 .00107641 .10764101 10.764101 1076.4101 10764.101 107641.01 10764101 Sq.Yd8, .0000012 .0001196 .0119601 1.19601 119.6011 1196.011 11960.11 1196011. Acres. .000247 .024711 .247110 2.47110 247.110 24711.0 Metric Cubic or Solid Measure. Aaevrdlns to V. 8. Standard. Only thoM marked '« Biit" are Britiah. Mill1]itr«,oroab Centimetre.... Centmtre Decilitre Litis, or cubic Dscimetre.... Decidltre, or Coitiatere.... Hectolitre, or Decistere Kflolitre, or Cubic Metre, or Stere [friolitie, or Decastere Cub Ins. .0610254 .610254 6.10264 61.0254 610.254 Cub Ft. .858156 8.53156 86.3156 863.156 riiiaoid. (Dry. J Liquid. (Dry. J Liquid. (Dry. .0084537 gill. .0070428 Brit gill. .0018162 dry pint .084537 ffUl. .070428 Brit gill. .018162 dry pint. .84537 gill = .21184 pint. .70428 Brit gill = .17607 Brit pint. .18162 dry pint. { Liquid, Dry. 2.1134 pints. fUpi .11351 peck = .9081 dry qt « 1.8162 dry pt 1.05671 quart » 2.1134 pii .88036 Brit quart = 1.7607 Brit )ints. (Liquid. (Dry. (Dry. I Liquid. (Dry. 2.64179 U. S. Uquid gal. 2.20000 Brit gaL .283783 bush ^ 1.1851 peck « 9.061 dry qts. 26.4179 U. S. Uquld gal. 22.0090 Brit gal. 2.83783 bush. 264.179 U. S. liquid gal. 220.090 Brit gal. 28.3783 bush. Onb yds, 1.8080. Liquid. 2641.79 U. S. Uquid gal. 283.783 busb. r Liquj iDry. } I Cub yds, 18.060. 15 226 WXI6H1S Ain> KBAMUMBB* Metric Welflrhta* redoeed to eonnnon Commercial or AtoIc Welfffitt of 1 poand = 16 ounces, or 7000 yralns. MiUigramme.. GentigrEunme. Decigramme .. Gramme Decagramme Hectogramme Kilogramme Mynogramme Quintal* Tonneau; Millier; or Tonne. Grains. .015432 .15482 1.6482 15.432 Pounds aT. .022046 .22046 2.2046 22.046 220.46 2204.6 The graniiM is the YtaaHa of Tr«neh wdgtatt r u>d !■ the welf ht of a cab eendmetre of ^*«^^ Vater at its max deniity, at lea level, la lat of Parle ; barom 29.922 ins. k Frencb Measures of tlie *' Systeme Usuel.** This iTstem wae In nse from about 1812 to 1840, when It was forbidden by law to nse eren its naoMB. This was done in order to expedite the general nse of the tables which we have before glTen. But ss the Systema Usnel appears In books pnbUshed daring the above interral, we add a taUa of sobw oC its valnes. Measures of liOiiflrtli* Ugnensml, orliae Pouee vsael, or inch, = 12 Ugnes. Pled nsnel, or foot, =12 peaces .. JLnne nsael, or elL Toise asnel,=6pieds Yards. .8M&4 i.si2se 2.18727 PecC. .09118 1.09862 8.9S706 6.M181 .09118 i.oasa U.lStM 47.346 78.T4in Weights, VsueL Qrala nsnel... GrosnsoeL... Once nsnel... Marensnd... Lirre nsnel, I 1,5 or pound, ^75 grains. 60.297 '• 1.10268 arotr os. .66129 avoir lb. 1.10268 avoir n>. Onbio, or Solid. TTsueL = 1.7606 British pis*. S.TSU British sate. 1811, or before the '*8jsteme nend," the Old System, " Systeme Anolen," was in Frencli Measures of tbe '* Systeme Anden.** LlneaL Point anclen, .0148 Ins. •.....•••....., ........... Ligne anoien, .06881ns Pouoeanden. 1.06677 ins =.0888 ft Pled anoien, 12.76^2 ins = 1.06677 ft Anne anoien, 46.8989 lns=8.90782ft=l.S0261 yds Toise anoien. = 6.3946 ft= 2.1816 yds Leagne= 2282 toises= 2.7687 miles Sqna Sq. ins. .00789 1.1359 Sq.ft. 1.1859 40.8908 Sq. yds. 4.6484 Onbio. 0. ins. .0007 1.2106 C.ft. 1.2106 261.482 G.yda. •.68a There is, however, much oonfosion about these old measures. Dliferent measnfas had the same same in diibreBt provinces. ^^Ml 1 I I I .. - ^ I l ' • The m99tr4¥foU qniatal is 100 avelrdapois p«aui4s. WEIGHTS AND MEASURES. 227 Biuwlan. Foot; same as U. 8. or British foot. Sacblne = 7 feet. Temi * 50C sachine » 3600 feet ai 116^ yards » .6629 mile. Pood » 86.114 lbs avoirdapoisi Spanlsb. Tlie eastellano of Spain and New Granada, for weighing gold, is varlouslf estimated, from 71.07 to 71.04 grains. At 71.0S5 grains, (the mean between th« two,) an avoirdapois, or common commercial oaoce contains 6.1572 castellano; and a lb aToirdupois contains 98.51ff. Also a troy ounce =s 6.7553 casteliano ; and a troy lb » 81.064 castellano. Three U.S. gold dollars weigh about 1.1 castellano. Tlio Spanisli nuirlL, or mareo^ for precious metals, itf South America, may be taken in practice, as .5065 of a lb aroirdupois. In Spain, .5076 lb. In other parts of Europe, it has a great number of values : most of them, however, being oetween JH and .54 of a pound avoirdupois. The .6065 of a lb =3 8545^ Sains ; and J5076 9) «■ 8553.2 grains. 1 marco = 60 castellanos = 400 tomine =» 90 S^nish jjroM-grains. The arroba has various vslues in difl^rent parts of Spain. That of Cas- tile, or Madrid, is 25.4025 lbs avoirdupois; tlie tonolada of Castile =- 2082.2 fts avoirdupois ; tlie quintal = 101.61 lbs avoirdupois ; the libra » 1.0161 fta avoirdupois; tbe eantara of wine, Ac, of Castile a 4^268 U. S. gallons; that of Havana a 4.1 gallons. "nie wara of Castile =3 82.8748 inches, or almost precisely 82j^ inches; or 2 feet 8Ji inches. Tbe iianeyada of land since 1801 » 1.5871 acres = 69134.08 sqaare feet. Tbe ftmeffa of corn, Ac « 1.69914 U. & struck bushels. In California, tbe vara by law »» 88.872 U. S. inehee ; and tbe leipui - 6001 varaa; or 2.6888 U. SL miles. fit iill^lfii I III -1 Slj, Ppini.1T! mm ^i' 11 11 I i i i I ! S S a i ,- I i| J m \ 1 1 CONTER8IOH TABLES. l|5 till? 82 I I SiiSl it iliiljlliiilhiiiiiii sjSI=»llsl J,=«^a Bill a Pa-., il rs >?:. I I iiiiiiijiiiifiiiii «io«SifS-3i3S . t lliiiliisi iiiHIIiiiiiii mil III '" i iiliiii I \ ! II li 232 CONVERSION TABLES. o I Xioco I :j55 s K00O> qXOO •^ -a I •I I T o §8 3 C9 Q 0» b> Q Q O) Q t^ lOQeoe^QpCQQOi • 00 ooOcoudT-iScboo y • • •^'^' a| mO • ^ • to 2 fl S »0 CO 4— !—♦ — «- ■!-■■»■ 00 ■♦-» ■ QQQQOOC^OOiH COOC^t^CO •ScoodkOtH a* CONCO"^ ■ ■ • • a d O »oeoco^ QOIWCO C o II d,^ I o e iHv-ti-HOO toco CO lo h0>0^coo ^S^SS; S «25^S 3 oodooOco • • ■ • • dd CO to 00 8 03-3 ®2 •S S o 3 s o i-iO> ^00( » O' 8'°'^°^ It (^ a B B lOOO Cv^KjiH^r^OO •HOOCOCOCOUd oob-i-iusco^ »HOO>0"^eoo |v.C^00O0N>0O ■ •#••• d d d d d OOIQCO S^ Ok oxooo d O OOOOtII •-i eociioid d !dd : "g •9 ©5 cO'^ddd llllll • CO • • 01 * COfHOOO COQiOi-HOO co©i-i( • POO* < •I-* « 0( II iH o e fa CONVERSION TABLES. 233 >0(X)»«'HOOQC'-<'-< Soo>^o>Tt<QOcor;t •<OC0aDTHOOQC»QQ O «0 "(tJ CO Csi Ci rH O CC CO iH ^ o :a 8 P OQ s o •« o s 08 OQ s 03 oa n QQtM 09 OQ O n aQ>4 •i—i— i-b-- •M QO I QQOXQOOOOCOO • 35" 685"* CO to CO COl^O oo>ooo »HOW a a C0«Or-» OCOO) coded a aa 5 * CO „ «Oi-i llQiO '^ CO aa a go* OQ OQ oocoooco^ t^«OC40e^O ■s OQ OQ 08 OiM S « OQ d 9 crSoS coo V 09 iz; CQ o 03 m a u bl fl •^^ 03 V a I I. SioiO< H«><0< 2iOiO< 0»0»0< OtHCsicOCQ (OOO Orl d a OQ a O »4 ». ^ ** « :2 .s^aa n CO 00 fecoN "geoo-i-O gddod Is .2 a ■D •it* fljCO-4— Q oootp »oio« OitHCO lOUSO i-I d d CO ci r-i CO o CO « • • « S S b .s^g.as-§:3 .eo»-i .oo<o ■ Oco cocsjq. OSCOrHOOOd CO oo»-i a E a MO>00< 00«Di O)00( UdOOCOO tH coo^o • • • , CO CO »^ CO ^Saa n CO gcOi-rM ^XCO«0- OO flOOO O H 9 234 CONYBBSIOM TABLES. gOft 6 10 to lO 00 o s o |2t«tOC4>OiO 'T CO 00 CO tJ* ''J* a09«D«00 r^co^Qoe^C4 '^t^O^V'O'O loor^io^,^ coddo'^'ci 00 CO IO0Q9C0 t»giO^ 8eo^^<«4< COQOOOQO C40)iOt« (ocoqoooqo CO cioiHiHOeo 00 e 3 OS I : I t.^ gs t^ P<?«. W<N ^00 3 S 1^ 9 SI eo _, »o tHOOCOOO COCOIQCO Ctor^dd '3r:4xddc6d '3t>'-< ^2 OD ^ 9 o-^ 1 i »^ »o *o o d I *o (9 S « .S C3 o U I H X • • • Oeoo^ i 1 9 "la o'cro' OB n B CO ^+-♦-00 CcDoad 5! • '^-l- «o COiHO'* ca .sa s a o^ o* c o* QD BB OQ in S*oo lO© flOo » OQ II U •COiO • S OQ OQ ■«*'iO SS'S'.S »4 ^« - «>0( <? i-*N<0 eooioo OCQiO k6deo'^'c4C« J ri ilia O g 09 J I 9 00 flieo 50 O M^ is s OONVEBSIOK TABI.E8. 235 SS25S::22§ 73 • -00 • -I h -b-oiOi-ioo rHWOOWeO v*^ COCOOOtt lOiO'^eoeo •-§1 3 GQ O. ill Si 1 u o o CO 1 •lOCD a o 1l s ;; I tr " S • 1° SS •"5 • • • ooo< 236 OONVERfilON TABLES. ,3 OWN si th<6.-)QO'-' CO IQ 00 to CO CO OS »^co 1^ ^^ a a a • • * * ■ a c9 •co^ »ococo oot^t- iOCOOOO C^'-H00»O OiO«0> CO»-it«<* COOOOto U3h->HiO tqt>;<Dp coh-iO 1-tOOb- lO-^CO <-HOOCD 00 •-•»-< c fl h>ieeo 00t<-«0 *^coco CSC OOD ^8 CO lOlO 1 Ud to o o I to to « o •J I. So«D<o qCOOO s II §9 o I oco «^ el eoOMO C0O9O • • • • coo«^ce 10 B I *i4 •I ^ o . s» - •^ s 1 1 . (Metric) or Metrl .62 .10231... 984206.. t §2 Quintal 100 Tonne 2.204 i.ooe !i i i m s a lis I II as = s! la > B in i ip u ^§PwfW^ I I il iiii Niiiiili i* 238 €X>NyEBSION TABLES. OONVEBSIOH TABI.E8. 239 oooa» • • • f i 9 OD ■ CO ^B • • • • • • ■i I S i ess h-Oft<0 US I Is & l(0 »COK a" ^ • • • si • * fl ^ gH-eoo "^3 * « • s eoioeo co«^ eidcsi .8 O .QCO Cjeo ^ OD |45«ooo Si ^ is slii 53 ip ^ Si m " ! i "^ I J ii i i 11 ill ii 11 ill I li S M ii 11 H ii Hi • a 5 S r I s s s I COKYEBSION TABLES. 241 d o a H O .2 ^ § a. > -s >« ^ .S3 •« o 4 a s a OS 2 • • • a ^ e3 3 O C O 0.W CO • • ■S • • Ml '>0 -•ooco fN • • • qOOCO fl '^ i-<00 IQIO ■ • • • »H0 iHiH COCO '^1-4 COCO CON to Sj3 OB .4 coo .^co OcOO CQ l| OB .4 Ml nooo 1— I o • ■ o B to « o OQ O «CSIO HpC4 00 1« <0 CO QO ? ooooo oe^Mcsi C4ioaaco iOOcOaO s o c aa kl K OQ a S'g' II s $ « s s-g 3 a e3< f4 08 w * 00 oo csi S COcO»^M CO COO* CO [O'-Io ^ e o a 16 242 CONVEK8ION TABLES. 00 WftCO lHi-tt-t ^iHO»-« 2 •s SPSS u u V o s CO coot*. 62* ^M JgO O coco ^00 GC "^W 00 0)0 o «o I s :9 o o I ■3 m 1 I I SCO MC4ao OrH,-;C<iWiH 08 O C3 fl ■ • • • • Q^ • • • • • c» •»COO»C^ esxooxQto a)0)a>9^C0 ab. i-( 0» r-t lO O'-^OO'* • • ■ • • o oooooioo '^OOUdOkO OOO'^'^Q t^i4«COiOO • • • • • »Hi-li-lNO Ada « H «co S3 CO no ^O-f- s ;9 0Q& « S A S sgaa§^ S3 • • • • 3 • • • • a • • • • « -ow^ M 'b-OOr-" Sco^oio C«SCDC4C0 ONNO- gjHood" o a 28 ^ooomim«> Z<S6c\ pi ill i 11 i 1 , 11 11 ii iiiii 1 1 ii ii i iill inny MMinM-iUN;: iiiiipiiiii ii i i ill Ii I Hi ill I is i i ii i! i ill m I flifiiiiiilJiia! iiiiiliiiiii' Mil .fi I §rf CONVERSIOIT TABLES. dli s Sp ill B| ^i. -Ill, j«l. gigs 1^ gsSSIU i : "=" s « .a H? !|i Si 8 .sp* liil ■ -- iii i I ill il ill il-5 '■ .i i |l ifi 6 |?l .sl'sSfafi 111 i s ^lA's.Klp'sllplfllli =11 I.I I i '°° si i' S- *" CX>NTBBaiOS TABLES. ii g§i ig M s III i III H 11 I sfi % s °° Lr-= " ■1=1 hi si '^ 1 ij lis Me I ill a s ill i iiii liii 111 ii ii ' ,2S ii I ill *5 ill I It ■ . S'Ss .1 ". Sttl|S ii ,. jils BS3 ri Is 246 OOXYEBSION TABLES. teoe^ d fl IS ■ • • • a a m 9 sgpc ■ "^ ^&l 1 eo«o 60^ CO b • co^ CO pl^ • • • • i ooo ill 5 fa H »-teo»-i Moo 1-4 1-4 r»f-<C9aoro C40bOOUId^ fl a a .g-Ssa "... © I I I fa« — p,o I CO b> CO 04 cqrooqaD eoeotHiH 2 I SB nsSia '■' • • 9 .t^ B 'CI 7 -^ fcO* CO ^dedo B m oeo w'di-icieo 9& b « S B Q '4* « ^wddop to IO »-• o o I IO IO * o l:»o 0:10 3» I a u s, ess O * • S o 3 S S CO ■gS. n a 11 ^g H Hi IQ 8 COOT ^©1 d a i ?al n §: ZcOiH*-l P^ • • • a e ooc4^eo5 t«0«i0'^ e<{cJc4rHf-4' aa B 4* o e "1 fa< •2:2 i^l III J I ^0> a p V ^SS5 00(010 OCDCOO • • • • IH.-IOO fhoO a 00 1^ h a B O ^ Is II ^•S^ 00«lO • • • o til Hi li o • •a : 5|g •g 4> I TS S 0* .0 a • fc.A ■ • ■ Aj « f • • 00 TO^CQ 0,OOC««-( fiSeoiQ SiotHCO ^^^ • • • OONTBBSION TABLES. 247 • • • • • toao 0<-iO C49aO E • • • " • I S SSI£:S d a|b a o •eoeow ^L • • • ■I'S&i 8o6o»io • • • • • BOO • • • • d • a a & Iff^ll OB e I s • a el ® V OQ tt oQ S 1^ ^ t> trf e o • • • • II 3 »H lO iH o o I (O »o o « o i o o « «s >ooco 0I« SJ M I OOO04O Zo-^oo 4 f«-t»H *" o-do^a a* fa WliQ00CO»-< flOOO'-'r-t So 248 OONVEBBION TABLES. <5 WJIJ 5 & s a «-lC«fH .a «4 S e ■ B • • ■ • 8*^ o o >^ 3 3 o bP }g ^^ • • ■ 0) Aoaoo s 7-1 lO o o I •o o BO 9 !^ a a 00 to CD lO ad TO10( 04 001 |QO«P>0 ou3aot>: C4eooo au ht *■» ^ « q» « 2^3 S-2 3SS 0) h l« h IS o*t; S 3 <* a U .0 S4 ^1 ^<D 2 CO 3 »o CD« gOi-if-iio »4C4«-4eoc4 <l\ • • • • « S>» h X k( tn » Q oqIqco^ iQiMOOtx • • ■ • o<-ioo a OQ fl fl B f) ■4i> « 43 I «*« fl fl »N O O « o B 2 0'tliS fl aa o fl fl fl>,«o .1. «00 -OOi 90)00 04 I illi „ 5! iCOtofP •HOO • • • gill •m g;fl;fl 9 ^ fa ^ •o-** s-** fl OfiOty ^^ • • • • eoo<D*o I OOITYBBBION TABLES. 249 I a I o GOOO*^ <OC*3^ CO • • • • 1-tOOO a ^ c0(O cqc4o ^o»-« ^t*CO • • ■ 7373 B ill I , Z Ih M 08 CBTS J3i.a D -o Sill u O 00 "*P NO h(0 O At) OOOO byooMJ M >»o8'C QQ C C -23 »OM»-lNi-j r4eoc4Nc<i n S e " oa fa O (N £ fa u u .0 =3 fc- 9 fa J2 O3J3 2 s r; S'^ 3 03 o P<eot^Tt*oo"3 ^ o •s 5 >o 10 o o u 10 to f-» * o 0? «;c9t« cooo Koato o« — • • • • C4rH rli^ OCO^ 00 NiiCO^ <ococo o • • • • 00>-ii-« d a <0r-(0 OiCOiO n Jill u gfa-2 2«»H d c « faSxco»6 fa o II ftOoifCO ^606 (3 « 10900) «ot>cp o»-»ciN d d d Oft 100 ooo lOOAiO • • • d d d iH«0 1HO9 d d d ft » S 9 Jj 5 O Q ^ O Ilii fa lOcO Ar-l-t-lO-^ o^ooo GO o a o 08 c s c 4> S B V ft fioO(N a •©00 CO . 000 C8 h p 7 d P o d gd ^ ft Oodd fa e8 IB 08 a tt d ■♦* 08 4> ft S OONTBRSION TABLES. s§| lip S|S^ i^l 1^1 i .-if "s IS ,s 'I B |i .11 ^1 II i im III 11 Is " t Hi si I jg|l| H H 8 '}pi I'l ill" ,tei 11 ? i !"• f i° !|rs ^fi ff PI' I 9ig^f:S SSsS ^S^!* ?:s SS g| ss ^= I i ti * g •fill Slfl 3i'i « II Jl B ISsE a|S| isfl S.sl-ji ■ sisj isJE.s iS3|.a g'"?" S ill i life |l ill S|IS|li jjiss: Sag EsB Sis III' "■4 ■ fcrtS fti^-e ^« ^a« ss« ■ ftoo >,** B**" ^'" a«— u £ I £ a OONVEBSIOK TABLES. 251 H H Soo tr- ie A £>« goo 6375 4721 •^331 coo a fl t*»-l l-4<0 ooesi eoci a a C90 too • • H p Hi o CO h- c<i»H eo'^ lOb- a>r-* to AC» CDQ Q«0 ^0» t« O0>-i «HO tOkO ^0» 3 CO o© »Hc4 ^s I o & -< too «S «ooco ^O .^coco ©O OrHi-l O © to ao >o iH o 6 I ao »0 * o I OB I .S OOSVEHBION TABLBB, ^H d rtiO t-oon .gi ^r-oa 'c^ a-« ov n^^r- lOrt 9 3 rxr. o^S°Q I i^ k£^ °^ '^^ "" S ' v^ p g,;i" "«:: if!3 |-S 25 ,2« i*^ n:;d sss SI I s-. iTiit! ;d44iiniHii|4f| go Sis § i*S«gs "s «|| «la 6S8| *|s == SSi^ ^H «§ oqs »S o|8 »S§ P"sS »S2 * § i i i" i ri g I I p Is B SS !: si fe^?^^&^^ s5 ii' s Ra at Sa £■ SB -g- ■£«* ■« ;|i|£!-!°!l5lte| 5 r r Is II r if S 3 B sw s b" u u u u u o 11 ii 11 i « IS Ss T Sg Ss l| / : Eg s«s .Si s«s ■ °d5 Bb- S^^ «b^ w. 68g Sga *l| hO ft-t^ ft'^'! — B ;is ^Si ssi :ss »■ g== i== |-= i i s OOKVEBSIOK TASLBS. 253 Si hSo IS S Q 0» 00 •-• 00 1*» nO00<Hl O no 5!l ^1 I 3 O •'•-'0000 o oDk3 9 9 S^^^ 3 go! ffi • • • • S3 CI CO <*o d e OD Is 1$ o^- O^iHOOr-l iSooci ia»oo» • * • • • eoci»-'»HfH i^i o I e • ■ " ill 5* Ml a?«S2 a^SSS® £ or* S Si Si ii K ^ r I s o £lS 5"- a ^ • • « :i o 1 9 s cii I •d e Si On eg S « GO J, p8 % I fl o OD u « A « 5- & 8 04 I •d fl o s| H : WEIOHTB AKD HEAHCBRS. TABI.E or ACRES ■XaiJIBED pme mU*, a tor dimrent wldtka. ijk" jiT ^. ^ k! iH ^fSi' E ,H ^ jk" .fiK aa 62 0^ ft JTO' w .006 .002 Mt .urn 28 8.80 Mi 80 t .486 at 8.82 MI 6« &87 .120 81 JB8 s .806 so S.M JMO 82 .138 Ihs .Olfl t2 SJg 'jaa K 0.91 .183 83^ ijl JM 8 .wo .018 IS 4.00 ssm 88'* j!o3 iB8 84 0.2 JB3 a JH S6 !24 J88 1023 J40 0.6 joo 11 IM -OM a .48 Mh 02 1.62 .142 88 0.7 .302 .028 88 4.81 03 J4G 0.8 MO M 4.TS .14T OA .(B2 W 4.81 aa 88 tIbs J40 K 11. 10 ixt .OM .W 80 8. J6l 91* LO .100 u m* !j 87 8.13 u 12 !o> 88 8.24 !l60 SS .213 h'* %1S .03S U .a .33 .» m 8.48 J68 90 lie .310 u 2.K 98 9) a.42 .048 48 :68 TS !l86 .228 .10 .08 8.66 .168 98 .226 JIO .32T 23 48 !o4 H 100 13.1 jno M 2^1 fl*6 K 0- :i4 TS'* OJW »« ■ 10 azi J74 T»bl« vr rntde" P«r mile, and per 100 fket newiaiwd hop|> sontallr, aad evrreapondlnr te dUferrat auBlea ot iaelb Ab7 bmw dl>t is — sloping dist > » alOBlnsdtot 19-hordlat i " vertltelKbt IS'hardJgt > or = sloping dlBl> A gnde of n fKt rlH per 100 f«et li uwiillf ci WeiUUXS AND HE&SCREa. H PBKT PHR 100 FT, HOROOHTAIh The trutlou of mfnnteg us eiren onlj la 34 feet In 100. A eUnonutcr gisduaud by Uie 3d column, ind numbuwd by the flnt on*, will gin U Hgbt tb* ilopH In feel per 11X1 Uei. So (inn. Origiunl. ltJ «"£ i Sc *s- ly STc' '.■a- 1 1 ML Si si «. HIE. is! 11 ii i DX. UlD. il SI II is 1 II 1 1 - UugHh A ; *ad thii )iit,^S,of Ui. 4dlx T;iprcz<fniif«^^pn>portioaHl luS; but tbje et^gpegt grad^p surmounted by traction onlT^ even on olecttlo ft horijoiiial ' B """"*■' '" ft TanicaJ • " A" '"" ^ Is the cotangent of the augle, a, with the horiiontil, or ths Ungenl of the int-le (9g°-a) wtth the >eiilcaJ. Tbus stated, a dope of 2 to 1 means t, slope of 2 OBASBS. 257 Table of nrrades per mile; or per 100 feet meaaared liorl« ■ontally. Grade Grade Grade Grade Grade Grade Grade Grade in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. per mile. per 100 ft. per mile. per 100 ft. per mile. per 100 ft. per mile per 100 ft. 1 .01894 39 .73S64 77 1.46833 115 2.17803 2 .03788 40 .76758 78 1.47727 116 2.19607 S .05682 41 .77652 79 1.49621 117 2.21601 4 .07576 42 .79546 80 1.51615 118 2.28486 5 .09470 43 .81439 81 1.53409 119 2,25379 6 -11364 44 .83333 82 1.55803 120 2.27278 7 .13258 46 .85227 88 1.67197 1-21 2.20167 8 .15152 46 .87121 84 1.59091 122 2.31061 9 .17045 47 .89016 86 1.60985 123 2.32966 10 .18939 48 .90900 86 1.62879 124 2.34848 LI .20833 49 .92803 87 1.64773 126 2.36742 12 .22727 60 .94697 88 1.66666 126 2.38686 18 .24621 61 .96591 89 1.68661 127 2.40680 14 .26515 52 .98485 90 1.70466 128 2.42424 16 J28409 63 1.00379 91 1.72848 129 2.44818 16 .80803 64 1.02278 92 1.74212 130 2.46212 17 .32197 65 1.04167 93 1.76186 131 2.48106 18 .31001 66 1.06061 94 1.78080 132 2.50000 10 .35985 57 1.07956 95 1.79924 133 2.51894 20 .87879 68 1.09848 96 1.81818 134 2.53788 21 ^773 69 1.11742 97 1.83712 135 2.56682 22 .41667 60 1.13636 96 1.85606 136 2.57676 28 .43561 61 1.15530 90 1.87500 137 2.59470 24 .45455 62 1.17424 100 1.89391 138 2.61364 25 .47348 63 1.19318 101 1.91288 139 2.63258 26 • .49242 64 1.21212 102 1.93182 140 .'T.65152 27 .51186 66 1.23106 103 1.95076 141 2.67046 28 .53030 66 1.25000 104 1.96970 142 2.68039 29 M924 67 1.26894 106 1.98864 143 2.70833 80 .56818 68 1.28788 106 2.00768 144 2.72727 81 .58712 69 1.30682 19Z 2.02662 145 2.74621 82 .60606 70 1.32576 108 2.04646 146 2.76516 83 .62500 71 1.34470 109 2.06439 147 2.78409 84 .64804 72 1.36364 110 2.08333 148 2.80308 86 .66288 73 1.38258 111 2.10227 149 2.82197 36 .68182 74 1.40152 112 2.12121 150 2.84091 87 .70076 75 1.42045 1.43939 113 2.14016 151 2.85986 88 .7mo 76 114 2.15909 152 2.87870 If the grade per mile should consist of feet and tenffuj add to tbe grade per 100 iMt in the foregoing table, that corresponding to the number of tenths taken firom the tabl« below ; thus, for a grade of 48.7 feet per mile, we have .81439 -f- .01826 « .82766 feet per 100 feet. Ft. per Mile. Per 100 Feet. Ft. per Mile. Per 100 Feet. Ft. per Mile. Per 100 Feet. .06 .00094 .4 .00768 .7 .01328 .1 .00189 .46 .00852 .75 .01420 .16 .00283 .6 .00947 .8 .01516 J .00379 .65 ,01041 .86 .01609 ?fi .00473 .6 .01136 .9 .01706 4 .00668 .66 .01230 .95 .0179i .36 .00662 258 WEIGHTS AND MEASUBE8. TABUE OF HEADS OF WATEB COBBESPONDIHO TO OIYEN PBESSVBES. Water at maximum density, 62.425 lbs. per cubic foot ^ 1 gram per cubit centimeter ; corresponding to a temperature of i° Centigrade = ^.2^ Fahrenheit. Head in feet — 2.306768 X pressure in lbs. per square inch. *• ** ^ 0.0160192 X pressure in lbs. per square foot. Heads corresponding to pressures not given in the table can be found by theae formulc. or taken from the table by simple proportion. Premare. Head. Preaanre* Head. Preaanre. Head. lbs. pei ■q. in. ' lbs. per sq. ft Feet. lbs. pel sq. in. > lbs. per sq. It Feet Ibe. per •q. in. lbs. per sq. ft. Feet 1 144 2.3068 61 7344 117.646 101 14644 282.984 2 288 • 4.6135 62 7488 119.952 102 14688 235.290 8 432 6.9203 68 7682 122.259 108 14832 287JS97 4 676 9.2271 64 7776 124.565 104 14976 289.904 5 720 11.6338 65 7920 126.872 106 16120 242.211 6 864 13.8406 66 8064 129.179 106 16264 244.617 7 1008 16.1474 67 8208 181.486 107 16408 246.824 8 1162 18.4541 68 8352 133.793 108 16652 249.181 9 3296 20.7609 69 8496 186.099 109 16696 261.488 10 1440 23.0677 60 8640 188.406 110 15840 268.744 11 1684 25.3744 61 8784 140.718 111 16984 256.061 12 1728 27.6812 62 8928 143.020 112 16128 268.868 13 1872 29.9880 68 9072 145.326 113 16272 260.666 14 2016 82.2948 64 9216 147.633 114 16416 262.972 16 2160 84.6016 65 9360 149.940 116 16560 266.278 16 2304 86.9083 66 9504 162.247 116 16704 267Jm 17 2448 39.2151 67 9648 164.568 117 16848 269.892 18 2692 41.5218 68 9792 156.860 118 16992 272.199 19 2736 43.8286 69 9936 159.167 119 17186 274jW 20 2880 46.1354 70 > 10080 161.474 120 17280 276J12 21 8024 48.4421 71 10224 163.781 121 17424 279.119 22 8168 60.7489 72 10368 166.087 122 17568 281.426 23 8312 68.0367 78 10512 168.394 128 17712 28S.7«2 24 8456 55.3624 74 10656 170.701 124 17856 286.088 26 8600 67.6692 76 10800 173.008 125 18000 288.84« 26 3744 69.9760 76 10944 175.814 126 18144 290.698 27 3888 62.2827 77 11088 177.621 127 18288 292.960 28 4032 64.5895 78 11232 179.928 128 18432 295.266 29 4176 66.8963 79 11376 182.235 129 18576 297J$7S 80 4320 69.2030 80 11520 184.541 130 18720 299.880 81 4464 71.5098 81 11664 186.848 181 18864 802.187 82 4608 73.8166 82 11808 189.166 132 19008 804.498 83 4752 76.1233 88 11952 191.462 138 19162 806.800 84 4896 78.4301 84 12096 193.769 184 19296 809.107 85 5040 80.7369 86 12240 196.075 186 19440 811.414 86 6184 83.0436 86 12384 198.382 186 19684 818.720 87 5328 85.3504 87 12528 200.689 187 19728 816.027 88 6472 87.6572 88 12672 202.996 188 19872 8184184 39 6616 89.9640 89 12816 205.302 189 20016 820J641 40 6760 92.2707 90 12960 207.609 140 20160 822.946 41 6904 94.5775 91 13104 209.916 141 20804 826.264 42 6048 96.8843 92 13248 212.223 142 20448 827.961 48 6192 99.191U 93 13392 214.529 143 20592 829.668 44 6836 101.4978 94 13536 216.836 144 20736 882.175 46 6480 103.8046 96 13680 219.143 145 20880 884.461 46 6624 106.1113 96 13824 221.450 146 21024 886.766 47 6768 108.4181 97 18968 223.756 147 21168 48 6912 110.7249 98 14112 226.063 148 21812 641.402 49 7056 113.0:U6 99 14266 228.870 149 21466 846.706 60 7200 115.3384 100 14400 280.677 160 21600 M6.016 I WEIGHTS AKD MEAStTBES. 259 TABUB OF PRESSURES COBRESPOMDINQ TO OITEH HEADS OF WATER. Water at maximum density, 62.425 lbs. per cubic foot » 1 gram per cubio •tntlmeter ; eorrespondiug to a temperature of 4° Centigrade — Z9:J9 Fahrenheit. Pressure in lbs. per square inch = 0.433507 X head in feet. Pressure in lbs. per square foot = 62.425 X head in feet. Pressures corresponding to heads not given in the table can be found by these formulK, or taken from the table by simple proportion. Head. Pressure. Head. Inches. Pressure. Inches. lbs. per sq. in. lbs. per sq. ft. lbs. per sq. in. lbs. per sq. fL 0.086126 0.072251 0.108377 0.144502 0.180628 0.216753 5.202083 10.4U4167 15.606250 20.808333 26.010417 31.212500 7 8 9 10 11 12 0.252879 0.289005 0.825130 0.861256 0.897381 0.488507 86.414583 41.616667 46.818750 52.020833 57.222917 62.425000 Prevnife. 1 2 3 4 5 6 7 8 9 tb 11 12 18 14 10 16 17 18 19 20 21 24 2S 26 rf 28 29 SO 31 88 0.4885 0.8670 1.3005 1.7340 2.1675 2.6010 ZJ0S45 3.4681 3.9016 4.3801 4.7686 5.2021 8.6806 6.0691 6JM)26 6.9361 7.3696 7.8031 a2366 8.6701 9.1036 9.0372 9.9707 10.4042 10.8377 11.2712 11.7047 12.1382 12.6717 18.0002 1&48S7 18.8722 14.8087 14.7392 16.1727 16.8008 Pressure. Ibe. per sq. in. 62.425 88 124.850 80 187.275 40 249.700 41 312.125 42 374.500 48 486.975 44 499.400 40 561.825 46 624.250 47 686.675 48 749.100 49 811.625 60 873.950 01 986.375 02 99SJBO0 08 1061.225 64 1123J650 05 1166i)76 56 1248.000 57 1310.925 58 1373.350 09 1435.775 60 1498.900 61 1560.626 62 1628.050 63 1685475 64 1747J0O 65 1810|25 66 1872.750 67 1935475 68 1997 JOO 69 2060105 70 2122J0O 2184i70 71 72 28O8J20 78 74 16.4733 16.9068 17.3403 17.7738 18.2073 18.6408 19.0743 19.0078 19.9413 20.8748 20.8088 21.24;8 21.6758 22.1089 22.5424 22.9759 23.4094 23.8429 24.2764 24.7099 25.1434 25.5769 26.0104 26.4439 26.8774 27.3109 27.7444 28.1780 28.6115 29.04.50 29.4785 29.9120 30.3455 30.7790 31.2125 31.6460 32.0795 Ibe. per sq. ft. 2372.150 2434.575 2497.000 2559.425 2621.850 2684.275 2746.700 2809.125 2871.550 2933.970 2996.400 3058.82.5 3121.250 3183.675 3246.100 3308.525 3870.960 8438.378 3495.800 3558.225 3620.650 3683.075 3745.500 3807.925 3870.350 3932.77.') 3995.200 4057.625 4120.060 4182.475 4244.900 4307.825 4369.750 4432.175 4494.600 4557.025 4619.400 Head. Feet. Pressure. 76 76 77 78 79 80 81 82 83 84 80 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 Ibe. per sq. in. 82.0130 32.9460 83.8800 83.8180 34.2471 84.6806 35.1141 35.5476 35.9811 86.4146 86.8481 37.2816 37.7151 38.1486 38.5821 39.0156 39.4491 39.8826 40.3162 40.7497 41.1832 41.6167 42.0502 42.4837 42.9172 43.3507 43.7842 44.2177 44.6512 45.0847 45.5182 45.9517 46.3852 46.8188 47.2523 47.6858 48.1193 lbs. per sq. ft. 4681.870 4744.300 4806.720 4869.150 49.^1.575 4994.000 5056.4^ 5118.850 5181.275 5243.700 5306.125 5368.550 5430.975 5493.400 5555.825 5618.250 5680.675 5743.100 5805.525 5867.950 5930.375 5992.800 6055.225 6117.650 6180.075 6242.500 6304.925 6367.350 6429.775 6492.200 6554.625 6617.050 6679.475 6741.900 6804.325 6866.750 6929.170 260 WEIGHTS AND MEASURES. TAMIiE OF PBESSVBES (€iMitinaed). Pressure. | Presirare. Presiare. Head. Feet. Hemd. Feet. Head. Feet lbs. per lbs. per lbs. per lbs. per lbs. per lbs.psr sq. ft sq. in. aq. ft. sq. in. sq.ft. sq. in. 112 48.5528 6991.600 144 62.4260 8989.200 176 76.2972 10986.800 lis 48.9868 7054.025 145 62.8686 9051.626 177 76.7807 11049.226 114 49.4198 7116.450 146 63.2920 9114.060 178 77.1642 11111.660 115 49.8533 7178.875 147 63.7266 9176.476 179 77.5978 11174.076 116 50.2868 7241.300 148 64.1690 9238.900 180 78.0313 11236.600 117 50.7203 7303.725 149 64.5926 9301.825 181 78.4648 11298.926 118 51.1538 7366.160 160 65.0260 9363.750 182 78.8988 11861.360 119 51.5878 7428.576 161 65.4596 9426.176 183 79.3318 11423.776 .120 52.0208 7491.000 162 65.8931 9488.600 184 79.7658 11486.200 121 52.4543 7653.425 163 66.3266 9651.026 186 80.1988 11648.626 122 52.8879 7615.860 164 66.7601 9613.460 186 80.6828 11611.060 123 53.3214 7678.275 165 67.1936 - 9675.876 187 81.0668 11678.475 124 53.7549 7740.700 166 67.6271 9738.900 188 81.4998 11736.900 126 54.1884 7803.125 157 68.0606 9800.726 189 81.9328 11798.826 126 54.6219 7865.530 168 68.4941 9863.150 190 82.8668 11860.760 127 55.0554 7927.975 169 68.9276 9925.675 191 82.7998 11923.176 128 55.4889 7990.400 160 69.3611 9988.000 192 83.2338 11986.600 129 55.9224 8052.825 161 69.7946 10050.425 193 83.6669 12048.025 180 56.3569 8115.260 162 70.2281 10112.850 194 84.1004 12110.460 131 66.7894 8177.675 163 70.6616 10175.276 196 84.6889 12172.876 132 67.2229 8240.100 164 71.0951 10237.700 196 84.9674 12236JI00 183 57.6664 8302.625 165 71.5287 10300.125 197 85.4009 12297.726 184 58.0899 8364.950 166 71.9622 10362.550 198 85.8344 12860.150 183 58.5234 8427.876 167 72.3957 10424.975 199 86.2679 12422JJ75 186 58.9570 8489.800 168 72.8292 10487.400 200 86.7014 12485.000 187 59.3905 8552.226 169 73.2627 10549.825 201 87.1349 12647.426 188 59.8240 8614.650 170 73.6962 10612.250 202 87.6684 12609.860 139 60.2575 8677.075 171 74.1297 10674.675 203 88.0019 12672.276 140 60.6910 8739.500 172 74.5632 10737.100 204 88.4364 12734.700 141 61.1245 8801.925 173 74.9967 10799.525 205 88.8689 12797.126 142 61.5580 8864.350 174 76.4302 10861.950 206 89.3024 12869.650 143 61.9915 8926.775 176 75.8687 10924.375 207 89.7359 12921.976 Table sbowlnar the total pressure against a Tertleal plane one foot wide, extending froip the surface of the water to tJie depth named in the first column. Water at its maximum density, 62.425 lbs per cubic foot =» 1 gram p«r cubic centimeter, correBpondins to a temperature of 4° Cent. = 39.2° Fahr. Total pressure in pounds = 31.2125 X square of depth in feet. Depth. Total pressnre. Depth. Total presrare. Depth. Totol prewire. Depth. Total pt-essare Feet Pounds. Feet Pounds. Feet Pounds. Feet Pounds. 1 31.21 17 9020 33 38990 49 74941 2 124.85 18 10113 34 36082 50 78081 3 280.9 19 11268 35 38235 51 8118« 4 499.4 20 12485 86 40461 62 84899 6 780.3 21 13765 37 42730 63 87676 6 1124 22 15107 38 45071 64 9101C 7 1529 23 16511 39 47474 65 94418 8 1998 24 17978 40 49940 60 112866 9 2528 25 19508 41 62468 . 65 181878 10 3121 26 21100 42 55069 70 162941 11 3777 27 22754 43 57712 76 176570 12 4495 28 24471 44 60427 80 199760 13 5275 29 26260 45 63205 86 225610 14 6118 30 28091 46 66046 90 2S2821 16 7023 31 29995 47 68948 96 28169S 16 7990 32 31962 48 71914 100 8121S8 WEIGHTS AND MEASUSES. 261 TABIiE OF 1»ISCHAB«1» Ilf CUBIC F£ET PCR SECOSTB coBBESPonrBiire to eiysjir DiscuABOfis in v. s. eAI.I.ONS P£R 24 HOVBS. n. S. gallon Discharge in cubic feet per second 231 cubic inches. 1.54723 X discharge in miUiwu of U. S. gal- lons per 24 hours. Millions Millions Millions Millions ofU. a Cubic feet of U.S. Cubic feet of U. 8. Cubic feet of U. S. Cubic feet gals, per per second. gals, per per second. gals, per per second. gals, per per second. 24hrs. 24hrB. 24hr8. 24hr8. .010 .0164728 18 20.1140 43 66.6808 72 111.400 .020 .0809446 14 21.6612 44 68.0781 73 112.948 .080 .0464169 16 28.2084 46 69.6258 74 114.496 .040 .0618891 16 24.75P7 46 71.1726 76 116.042 J080 .0778614 17 26.8029 47 72.7197 76 117.689 .060 .0928837 18 27.8601 48 74J»70 77 119.137 .070 .108806 19 29.3978 49 76.8142 78 120.684 .080 .128778 20 80.9446 60 77.8614 79 122.281 .000 .189261 21 82.4918 61 78.9087 80 123.778 .100 .164728 22 84.0390 52 80.4569 81 126.326 .200 .309446 28 36.6868 63 82.0081 82 126.873 .800 .464169 24 87.1886 64 83.6508 83 128.420 .400 .618891 26 38.6807 56 86.0976 84 129.967 .600 .778614 26 40.2279 66 86.6448 85 131.614 .600 .938887 27 41.7752 67 88.1920 86 133.062 .700 1.08806 28 43.8224 68 89.7398 87 134.609 .800 1.28778 29 44.8696 69 • 91.2866 88 136.156 .900 1.89261 80 46.4169 60 92.8337 89 137.703 1 1.64728 81 47.9641 61 94.3809 90 139.251 2 8.09446 82 49.6118 62 96.9282 91 140.798 3 4.64169 88 61.0586 63 97.4764 92 142.345 4 6.18891 84 62.6068 64 99.0226 93 143.892 5 7.78614 &9 64.1530 66 100.670 94 145.489 6 9.28887 86 e».7002 66 102.117 95 146.987 7 10.8806 87 67.2476 67 103.664 96 148.584 8 12.8778 38 68.7947 68 105.212 97 150.081 9 13.9261 89 60.8419 69 106.759 98 151.628 !• 16.4728 40 61.8891 70 108.306 99 153.176 n 17.0196 41 68.4364 71 109 J68 100 154.728 12 18.6667 42 64.9836 262 WEIGHTS AND MBASURBS* TABIiE OF BISCHAROlMi IN CUBIC FEfiT PEB SBOOUD CORRESPONDING TO OITEN BISCHABOES IN IM- PERIAIi GAIiliONS PER 24 HOURS. Imperial gallon «> 277.274 cubic inches. Discharge in cubic feet per second = 1.85717 X discharge in Imperial gallons per 24 hours. Millions MilUons Millions Millions of Imp. Cubic feet of Imp. Cubic feet of Imp. Cubic feet of Imp. Cubic feet gals, per per second. gals, per per second. gals, per per second. gals, per per second. 24hrs. 24hr8. 24hr8. 24hr8. .010 .0185717 13 24.1432 43 79.8583 72 133.7162 .020 .0871434 14 26.0004 44 81.7155 83.5727 73 135.5734 .030 .0557151 15 27.8576 45 74 187.4306 .040 .0742868 16 29.7147 46 86.429A 76 139.2878 .050 .0928585 17 31.5719 47 87.287^1 76 141.1449 .000 .111430 18 33.4291 48 89.1442 77 143.0021 .070 .130002 19 35.2862 49 91.0013 78 144.8593 .080 .148574 20 37.1434 50 92.8585 79 146.7164 .090 .167145 21 39.0006 51 94.7157 80 148.6736 .100 .185717 22 40.8577 52 96.5728 81 160.4308 .200 .371434 23 42.7149 53 98.4300 82 162.11879 .900 .557151 24 44.5721 54 100.2872 88 164.1451 .400 .742868 25 46.429$ 55 102.1444 84 156.0028 .500 .928585 26 48.2864 56 104.0015 86 167.8595 .600 1.11430 27 50.1436 67 105.8587 86 169.7166 .700 1.30002 28 52.0008 58 107.7159 87 161.6738 .800 1.48574 29 53.8579 69 109.5730 88 168.4310 .900 1.67145 80 55.7151 60 111.4302 89 166.2881 1 1.85717 31 57.5728 61 113.2S74 90 167.1453 2 3.71434 32 59.4294 62 115.144$ 91 169.0025 3 5.57151 33 61.2866 68 117.0017 92 170.8696 4 7.42868 34 63.1438 64 118.8589 98 172.7168 5 9.28585 35 65.0010 66 120.7160 94 174.6740 6 11.1430 36 66.8581 68 122.5732 96 176.4S12 7 13.0002 37 68.7153 67 124.4304 96 178.2883 8 14.8574 38 70.5725 68 126.287$ 97 180.1465 9 16.7145 39 72.4296 69 128.1447 98 182.0027 10 18.5717 40 74.2868 70 130.0019 99 183.8698 11 20.4289 41 76.1440 71 131.8591 100 186.7170 12 22.2860 . 42 78.0011 WEIGHTS AND MEASURES. 263 TABIiE OF DISCHAB«ES IN OAIil^OMS PER 84 HOUIIA COBKESPONDINO TO OITEST DISCHARGES IN CUBIC FEET PER SECOND. U. S. gallon = 231 cubic inches. Imperial gallon = 277.274 cubic inchea- Diaoharge in U. S. gallons per 24 hours = 646317 X discharge in cubic feet per second. Discharge in Imperial gallons per 24 hours » 538454 X discharge in cubic fe«i per second. Onb. ft. Millions of Millions of Cub. ft. Millions of Millions of U. S. gHllons Imperial gallons per sec. U. S. gallons Imperial gallons per 24 hours. per 24 hoars. per 24 hours. per 24 hours. 1 0.646317 0.538454 53 34.254795 ,28.5880U 2 1.292634 1.0769O7 54 34.901112 29.076488 8 1.938951 1.615361 55 85.547428 29.614951 4 2.685268 2.158815 56 36.193745 30.153405 i 3.281584 2.692266 CT 36.840062 30.691859 6 3.877901 3.230722 58 37.486379 81.230312 7 4.524218 8.769176 59 38.132696 31.768766 8 5.170535 4.307629 60 38.779013 32.307220 9 5.816852 4.846088 61 39.425330 32.845678 10 6.463169 5.384537 62 40.071647 33.384127 11 7.109486 5.922990 68 40.717963 33.922581 12 7.755808 6.461444 64 41.364280 34.461034 18 8.402119 6.999898 65 42.010597 34.999488 14 9.0484S6 7.538351 66 42.656914 85.537942 15 9.694753 8.076805 67 43.303231 36.076395 16 10.341070 8.615259 68 43.949548 36.614849 17 10.987387 9.153712 69 44.595865 37.153303 IB 11.633704 9.692166 70 45.242182 37.691756 19 12.280021. 10.230620 71 45.888498 38.230210 20 12.926338 10.769073 72 46.534815 38.768664 21 13.572654 11.307527 78 47.181132 89.307117 22 14.218971 11.845981 74 47.827449 39.845571 28 14.865288 12.384434 75 48.473766 40.384025 24 15.511605 12.922888 76 49.120083 40.922478 28 16.157922 13.461342 77 49.766400 41.460932 .28 16.804289 13.999795 78 50.412717 41.999385 27 17.450556 14.538249 79 51.059034 42.537838 28 18.0968(73 15.076702 80 51.705350 43.076293 .29 18.743190 15.615156 81 52.351667 43.614746 80 19.889506 16.158610 82 52.997984 44.153200 81 20.085828 16.692063 83 63.644301 44 691654 82 20.682140 17.280517 84 54.290618 45.230107 88 21.328457 17.768971 85 54.936935 45.768561 84 21.974774 18.307424 86 55.583252 46.307015 85 22.621091 18.845878 87 56.229569 46.845468 86 23.267408 19.384332 88 66.875885 47..'W3922 87 23.913725 19.922785 89 67.522202 47.922376 88 24.560041 20.461239 90 58.168519 48.460829 89 25.206a'W 20.999693 91 68.814836 48.999283 40 25.852675 21.588146 92 59.461153 49.537737 41 26.498992 22.076600 93 60.107470 50.076190 42 27.145309 22.615054 94 60.753787 50.614644 48 27.791626 23.158507 95 61.400104 51.153098 44 28.487943 23.691961 96 62.046420 51.691561 46 29.084260 24.280415 97 62.692737 52.230006 46 29.730576 24.768868 98 63.389054 52.768459 47 30.376893 25.307322 99 63.985371 53.306912 48 81.028210 25.845776 100 64.631688 53.845366 49 81.669627 26.384229 101 65.278005 54.383820 50 32.315844 26.922683 102 65.924322 54.922273 61 32.962161 27.461187 103 66.570639 55.4WJ727 62 83.608476 27.999590 104 67.216956 55.999181 264 WEIGHTS AKD MEASURES. TABI4E OF BISCHABOES (Continned). Cub ft. Millions of Minions of Cub. ft. MilUonsof Millions of per sec. U. S. galloDB Imperial gallons per sec. U. S. gallons Imperial gallons per 24 hours. per 24 hours. per 24 hours. per 24 hours. i05 67.863272 66.687684 167 107.934919 89.921761 106 68.509589 67.076088 168 108.581236 90.460215 107 69.155906 57.614542 169 109.227553 90.998669 208 69.802223 58.162995 170 109.873870 91.537122 109 70.448540 68.691449 171 110.520186 92.075576 110 71.094867 69.229903 172 111.166503 92.614030 111 71.741174 69.768356 173 111.812820 93.152488 112 72.387491 60.306810 174 112.459137 93.690937 118 73.033807 60.845264 175 113.105454 94.229891 lU 73.680124 61.383717 176 113.761771 94.767844 115 74.326441 61.922171 177 114.898088 95.806298 116 74.972768 62.460625 178 115.044406 96.844761 117 75.619075 62.999078 179 115.690722 96.388206 118 76.265392 63.537532 180 116.337038 96.921669 119 76.911709 64.075986 181 116.983355 97.460112 120 77i»8026 64.614439 182 117.629672 97.998666 121 78.204342 66.152893 188 118.275989 98.537020 122 78.850659 66.691347 184 118.922306 99.075478 123 79.496976 66.229800 185 119.568623 99.618927 124 80.143293 66.768254 186 120.214940 100.152881 125 80.789610 67.306708 187 120.861257 100.690684 126 81.435927 67.845161 188 121.507578 101.229288 127 82.082244 68.383615 189 122.153890 101.767742 128 82.728561 68.922068 190 122.800207 102.806196 129 83.874878 69.460522 191 123.446524 102.84464» 180 84.021194 69.998976 192 124.092841 108.388108 131 84.667511 70.537429 193 124.739158 103.921666 132 85.313828 71.075883 194 125.»85475 104.460010 183 85.960145 71.614337 195 126.081792 106.098464 184 86.606462 72.162790 196 126.678108 106336917 186 87.262779 72.691244 197 127.324425 106.076S71 186 87.899096 73.229698 198 127.970742 106.618825 187 88.545413 73.768151 199 128.617059 107.162278 188 89.191729 74.306605 200 129.268376 107.690782 189 89338046 74.845059 201 129.909698 106.229186 140 90.484363 76.383612 202 130.566010 108.767689 141 91.130680 76.921966 203 131.202327 109.306098 142 91.776997 76.460420 204 131.848644 • 109344647 148 92.423314 76.998873 205 , 132.494960 110.388000 144 93.069631 77.637327 206 133.141277 110.921464 146 93.715948 78.075781 207 133.787594 111.45990S 146 94.362264 78.614234 208 134.433911 111.998861 147 95.008581 79.152688 209 135.080228 112.536815 148 96.664898 79.691142 210 135.726545 113.075269 149 96.301215 80.229596 211 136.872862 118.618722 150 96.947532 80.768049 212 137.019179 114.152176 151 97.593849 81.306503 213 137.665495 114.690680 152' 98.240166 81.844956 214 138.311812 116.229088 163 98.886483 82.383410 215 138.958129 116.767887 154 99.532800 82.921864 216 139.604446 116.806891 155 100.179116 83.460317 217 140.25U768 116344444 156 100.825433 83.998771 218 140.897080 117.882898 157 101.471750 84.537225 219 141.643397 117321882 158 102.118067 85.075678 220 142.189714 118.468806 159 102.764384 85.614132 221 142.836030 118.998SB» 160 108.410701 86.152586 222 143.482347 119386n8 161 104.057018 86.691039 228 144.128664 120.07616ft 162 104.703335 87.229498 224 144.774981 120.618620 163 105.349651 87.767947 225 146.421298 121.163074 164 ia5.995968 88.306400 226 146.067615 121.600B87 165 106.642285 88.844854 227 146.713982 122.228881 166 107.288602 89.883308 228 147.860249 122.78704 TIME. 265 TABIiE OF I»lS€HAReES (Contlnae^i). Oab. ft MillioDs of Millions uf Oub tt MilUons of Millious of per sec. U. S. gallons Imperial gallons per sec. U. S. gallons Imperial gallou per 24 hours. per 24 hours. per 24 hours. per 24 hours. 229 148.006566 123.306888 240 155.116061 129.228878 230 148.652882 123.844342 241 155.762368 129.767332 231 149.299199 124.382795 242 156.408685 180.305786 232 149.945516 124.921249 243 157.065002 130.844239 233 150.591833 125.459703 244 167.701819 131.382693 234 151.238150 125.998156 245 158.847636 131.921147 235 151.884467 126.536610 246 158.993962 132.459600 236 152.680784 127.075064 247 169.640269 132.998054 237 163.177101 127.613517 248 160.286586 133.636608 238 163.828417 128.151971 249 160.932903 134.074961 239 164.469734 128.690426 250 161.579220 134.613416 TIME. 60 seconds,*! marked s, =■ 60 minutes,! *' m, = 24 hours, " h, = 7 days, " d, = Arc Time 1° = 4 minutes r s= 4 seconds V = 0.066... second 1 minute 1 hour = 1 day = 1 week = 3600 seconds 1440 minutes = 86400 seconds 168 hours = 10U80 minutes Time Arc 24 hours =360° Ihour = 15° 1 minute = 0° 15' 1 second =* 0° 0' 15" Bletbods of reekonins time. Astronomers distinguish between mean solar time, true or api)arent solar time, and sidereal time. At a standard meridian (see page 267) mean solar time is the same at ordinary clock time. At any point not on a standard meridian, standard time is the local mean solar time of the meridian adopted as standard for such point ; and local time is = time at a standard meridian phu correction for longitude from that meridian if the place is east of the meridian, and vice versa. For the amount of such correction, see second table above. A true or apparent aolnr day is the interval of time between two successive culminations of the sun, «.«., between two successive transits or passages of the sun across the meridian of the same point ou the earth ; but, since these intervals are unequal, they do not correspond with the uniform movement of clock time. A fictitious or imaginary sun, called the "mean sun," is therefore supposed to move along the equator in such a way that the interval between its culminations is con- stant. This interval is called a day, or mean solar day, and is the average of the lengths of all the apparent solar days in a vear. Apparent and mean time agree at four points in the year, viz., about the middle of April and of June, September 1 and December 24. The sun is sometimes behind and sometimes in advance of the mean sun, and is called " slow " or " fast " accordingly. The sun is " slow " in winter, the maximum being about February 11, when it passes any standard meridian, or "souths" (making of^Mrent noon), about 14m, 28s, after noon by a correct clock. The sun is " fast," or in advance of the clock, in MJty and in the £all, with a maximum, about l^ovember 2, of about 16m, 20s. The difference between apparent and mean time is called the equation of time. It can be obtained from the Nautical Almanac, or, approximately, by taking the mean between the times of sunrise and sunset, as given in ordinary almanacs. As solar time is measured by the apparent daily motion of the sun, so sidereai time is measured by that of the fixed stars, or, more strictly speaking, by the motion of the vernal equinox which is the point where the sun crosses the equator in the spring. * The second was formerly divided into 60 equal parts called thirds (marked '") ; but it is now divided decimally. f The old and confusing practice of designating minutes, seconds and thirds of time (see footnote *) as % " and ''', is no longer in vogue. Days, hours, min- utes and seconds are now designated by d, h, m, and s, respectively, thus : 2d, 20h, 48ni, 65.43 s.j and the symbols ' and " designate minutes and seconds of are. 266 <TIMB. A sidereal dAy" is the interval of time between two tueeeisiye paaaages of the vernal equinox (or. practically, of auy star) past the meridian of a ^ven point on the earth. It is, practically, the time required for one complete revo- lution of the earth on its axi£, relatively to the stars. The length of the sideral day is 23 h, 56 m, 4.U9 s, of mean solar time, or S m, 56.91 A of mean solar time less than the mean solar day of 24 hours. In other words, a star will, on any night, appear to set 3 m, 55.91 s earlier by a correct clock than it did on the preceding night. Hence, substantially, the number of sidereal days in a year is greater by 1 than the number of solar days. The sidereal day, like the solar day, is divided into 24 hours. These hours are. of oourse, shorter than those of tne solar day in the same proportion as the sidereal day is shorter than the solar day. They are counted from to 24, com- mencing with sidereal itoon, or the instant when the vernal equinox passes the ujmer meridian. Tlie etwil day (» 24 hours of clock or mean solar time) commences at mid- night ; and the astronomical solar day at noon on the civil day of the same date. Thus, on a standard meridian, Thursday, May 9, 2 a. m . civil time, is Wednesdav, May 8, 14 h, astronomical time; but Thursday, May 9, 2 p. M., eivil time, is Thursday, May 9, 2 h, astronomical time. Tbe cItII month is the ordinary and arbitrary month of the calendar, varying in length from 28 to 31 mean solar days. A sidereal montb is the time required for the moon to perform an entire revolution with reference to the stars. Its mean length, in mean solar time, is about 27 d, 7 h, 43 m, 12 s. A lunation, or synodic month is the time from new moon to new moon. Its mean length is about 29 d, 12 h, 44 m, 8 s. The tropical or natural year is the time during which the earth describes the circuit from either equinox to the same again. Its mean length, in mean solar time, is now about 365 d, 5 h, 48 m, 49 s. The sidereal year is the time during which the earth describes its orbit with reference to the stars. Its mean length, in mean solar time, is about 365 d, 6 h, 9 m, 10 s. The elwll year is that arbitrary or conventional and variable division of time comprised between the 1st of January and tbe 31st of the following Decem- ber, both inclusive. It contains ordinarily 365 mean solar days of 24 hours, bat each yenr whose number is divisible by 4 contains 366 days, and is called a leap year, except that those years whose numbers end in 00 and are not multipMB of 400 are not leap years. To regulate a watch hy the stars. The author, after having rega- ' lated his chronometer for a year by this method onlv,diffiereid but a few seconds from the actual time as deduced from careful solar observations. Select a window, facing west if possible, and commanding a view of a roof-crest or oth^ fixed horizontal line, preferably about 40^ above the horizon, in order to avoid disturbance due to refraction, and distant say 50 feet or more. Note the time when any bright fixed star (not a planet) passes the range formed between the roof, etc., and any fixed horizontal line about the window frame, as a pin fixed in <>it her Jamb. The sight in the window, and the watch, must be illumi- nated. The star will pass the range 3 m. 55.91 s. earlier on each suooeeding evening. Those stars which are nearest the equator appear to move the fastest, and are therefore best suited to the purpose. If the first observation of a given star lie made as late as midnight, that saron star will answer for about three months, until at last it will begin to pass the range in daylight. Before this happens, transfer the time to another star which sets later. By thus tabidating, throughout the year, about half a dozen stars which follow each other at nearly equal intervals of time, we may provide a standard by means of which correct clock time may be ascertained on any clear night. Experinfenting in this way with two of the best chronometers, the author found that tWr rates varied, at times, as much as from three to eight seconds per day. An average man takes two steps (one right, one left) per B c ca»d« Hence, march music usually takes one second per measure (or ** bar "). Modem watches usually tick five times, and clocks either one, two, or four tlmes^ per second. STANDARD RAILWAY TIME. 267 STANBARD RAII.WAT TIME, ADOPTED I8SS. The following amtngement of standard time was recommended by the General and Southern Time Gonyentions of the railroads of the United States and Canada, held respectiyely in St. Louis, Mo., and New York city, April, 18S3, and in Chicago, m., and New York city, in October, 1883, and went into effect on most of the rail- itMMls of the United States and Canada, NoTembar 18th, 1888. Most of the principal cities of the United States hare made their respective local times to correspond with it. This system was proposed by Mr. W. V. Allen, Secretary of the Time Gonyen- tions, and its adoption was largely due to his efforts. We are indebted to Mr. Allen for documents from which the following has been condensed, five standards of time or five ** times," have been adopted for the United States and Canada. These are, respectively, the mean times of the 60th, 76th, 90th, 106th, and 120th meridians west of Greenwich, England. As each of these meridians, in the above order, is 16<> west of its predecessor, its time is one hour slower. Thus, when it is noon on the OOch meridian, it is 1 p.m. on the 76th, and 11 a. m. on the 106th. vThe following gives the name adopted for the standard time of each meridian, and the conventional color adopted, and uniformly adhered to, by Mr. Allen, for the purpose of designat* ing it and its time, Ac, on the maps published under his anspioess Longitude west from Greenwich. Name of Standard Time. Conventional color. W 76P 9(P 106° laoo Intercolonial. Eastern. Central. Mountain. Pacific. Brown. Red. Blue. Oreen. Yellow. Theoretically, each meridian may be said to give the time for a strip of country ttP wide, running north and south, and having the meridian for its center. Thus ths meridian on which the change of time between two standard meridians is sup- p sssd to take place, lies half>way between them. But it would, of course, not be practiesble for the railroads to use an imaginary line in passing from one time standard to another. The changes are made at prominent stations forming the ter- mini of two or more lines; or, as in the case of the long Pacific roads, at the ends (tf divisions. As far as practicable, points at which changes uf time had previously basn made, were selected as the changing points under the new system. Detroit, Wch., Pittoburgh, Pa., Wheeling and Parkersburg, W. Va., and Augusta, Ga., al- though not situated upon the same meridian, are points of change between «a$tem and central standard times. A train arriving at Pittsburgh from the east at noon, and leaving for ths west 10 minutes after its arrival, leaves (by the figures shown npon its time-table, and by the watches of its train hands) not at 10 minutes afker ISjbat at 10 minntss alter 11. The necessity for making the changes of time at principal points, instead of on a true meridian line, necessitates also some "overlapping** of the times, or of their eolors on the map. Thus, most of the roads between Buffalo and Detroit, on the north side of Lake Brie, run Irf ** eastern," or **red,** time; while those on the $ouih side of the Ijske, between Buffalo and Toledo, immediately opposite to and directly south of them, run by ** central ** or " blue ** time. If the chauMs of time were made at ths meridians midway between the standard ones, it woula not be necessary for any town to change its time more than 30 min- utes. As it is, somewhat greater changes had to be made at a few points. Thus, standard time at Detroit is 32 minutes ahead, and at Savannah 86 minutes back, of mean locaf time. In most cases the necessary change was made upon the railways by simply setting docks and watches ahead or back the necessary number of minutes, and without making any change in time-tables. Raliux, and a few adjacent cities, use the time of the 60th meridian, that being the nearest one to them ; but the railroadM in the same district have adopted the T6th meridian, or eastern, time; so that, for railroad purposes, intercolonial time has never come into force. In 1878 there were 71 time standards in use on the railroads of the United States and Canada. At the time of the adoption of the present system this number had ■been reduced, by consolidation of roads, Ac, to hS, By its adoption, the number be- tame 5, or, practicslly, 4, owing to the adoption of eastern time by the intercolonial roads; as aJrcHsdy explained. 268 DIAIA DIALLING. To malKe a borlxontal San^dlal, Draw a line a h ; and at right angles to it, draw 66. From any convenient point, bb c, in a fr, draw the perp c o. Make the angle cao equal to the lat of the place ; aJfo the angle e o « equal to the same ; Join o e. Bfake e n equal to o e; and from n as a center, with the rad e n, describe a quadraat e «; and div it into 6 equal parts. Draw c y, parallel to 6, 6; and firom n, through the 5 ^ DIAL ^ points on the quadrant, ^ draw lines n t^n t, ^c, terminating in ey. From a draw lines a 6, a 4, Ac, passing through t, i, Ac. From any convenient point, as c, describe an arc r nt A, as a kind of fin- ish or border to half the dial. All the lines may now be effaced, except the hour lines a 6, a 6, a 4, Ac, to a 12, or a A; unless*, as is generally the case, the dial is to be divided to quarters of an hour at least. In this case each of the divisions on the quad- rant « «, must be subdivided into 4 equal parts; and lines drawn from n, thioaf^ the points of subdivision, terminating in ty. The quarter-hour lines must be drawn from a, as were the hour Unes. Subdivisions of 6 min may be made in the same way ; but these, as well as single min, may usually be laid off around the border, by eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi- nary dial. To draw the other half of the Fig, make a d equal to the intended thick- ness of the gnomon, or style, of the dial ; and draw d 12, parallel, and equal to a 12 ; and draw the arc x^ to, precisely similar to the arc rmh. Between x and to, on the arc ng «0, space off divisions equal to those on the arc r7nh\ and number them for the hoan, as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be triangular; its thickness must throughout be equal to a<2 or &«o; its base murt cover the space adhv)\ its point will be at ad; and its perp height Av, over A.«^ must be such that lines vd^uii, drawn from its top, down to a and d, will make the angles u a A, « d io, each equal to the lat of the place. Its thickness, if of metal, may conveniently be fh>m ^ to ^ inch ; or if of stone, an inch or two, or more, aooording to the siie of the dial. Usually, for neatness of appearance, the back A u « to of the style is hollowed inward. The opper edges, ua, v d, which cast the shadows, moat be sharp and straight. The dial must be fixed in place hor, or perfectly level ; ah and dw must be placed truly north and south ; ad being south, and A«o north. Th» dial givee only sun or solar time ; but clock time can be found by means of the ** fiurt' or slow of the sun," as given by all almanacs. If by the almanac the tun is 6 miB. Ac, fast, the dial will be the same ; and the clock or watoh, to be correct, must be f Bin slower than it ; and vice versa. To make a Vertical Snn-Dlal. Proceed as directed above, except that the angles eao and eo« on the drawing, and the angle t«a A or v dir of the style, must lie equal to the oo-latitnde (» dif- ference between the latitude and 90^) of the place, and the hours must be num- bered the opposite wav from those in the above flgare ; i e, from A to y number 12, 11, 10, 9, 8, 7 ; and from to tog number 12. 1, 2, 8, 4, 6. The dial plate muat be placed vertically, in the position shown in the figure, (kcing ezacuy south, and with a A and dw vertical. BOABD HBABCBE. BOABD HEASTTBE. ■ fMlowlnv t»Me. Tha u. BOARD HEASURB. niMe at Bo»r« Mcaanre— (ConUaud.) £i -- „.. "Kr-tf d.M Id a "loriilil.) f 35 P IM THIOUt MM SS IKOIUS. P i i 1 1 J .«™ if 1 1 1 ,i S 1 1 1 ^ .HSU i i 1 1 i 1 t I 1 ■a 1 1 1 s i 1 1 i i nuK. 1 1 1 ) J 3 1 I." „ 1 1 BOASO HEABURB. T»M« of Btmrd Heu DTC— (OonUn Md.) 1 a|- sx M( TK OKK> Ba DT nraa ■S. 'H IH 1^ 5 i i i i i s J s It" i! i 1 1 i is 1 i if i 1 1 if IS sir ;.s 1 1 i is Is is !:| i 1 1 J 1 i •- i BOARD UEASUBE. Table of Board Mcaaar* — (Contlnutd.} BOARD MEASURE. 273 Table of Board Biewiare~(Continued.) si H H 1. H 2. H H H 8. H H H 4. H 6. 7. H H 8. H H H 9. H 10. H IS. IS. H u. 16. It. 17. 18. 10. ao. 21. ». IS. M. of Board Meuure oontaiBOd In on« raaning tfnA of Softotlinga of dilftrent dimenaiODS. < Original.) THIOKNEStt IK ZKOHIBS. 10 lOii io« lOH Ft Rd.M. FtBd.M. ptBa.M: FtBd.lC .1083 .2136 .2186 JfM .4167 .4271 .4375 .4479 .6250 .6406 .6363 .671* .8333 .8642 .8750 .8956 1.042 1.068 1.094 1.120 1.250 1.821 1.318 1.344 1.458 1.495 1.631 1.568 1.667 1.708 1.750 1.793 1.875 1.922 1.969 2.016 2.0A3 2.135 2.188 2.240 2.292 2.349 2.406 2.464 2.500 2.563 2.625 2.688 2.708 2.776 2.844 2.911 2.917 2.990 3.063 3.135 3.126 3.208 3.281 3.359 .S.333 3.417 3.600 3.583 3.542 3.630 8.719 3.807 3.750 3.844 3.938 4.031 3.958 4.057 4.m 4.255 4.167 4.271 4.479 4.375 4.484 4.594 4.703 4.5a3 4.608 4.813 4.927 4.792 4.911 5.061 6.161 5.000 5.126 5.250 6.376 5.208 5.339 5.469 6.599 5.417 5.562 6.688 6.898 5.625 6.766 6.906 6.047 5.833 5.979 6.126 6.271 &04S 6.193 •M4 6.486 «.2S0 <.406 •.56B 6.719 6.468 6690 «TB1 6.943 6.667 6.833 T.00O 7.167 6.875 tMl T.219 7.391 7.083 T.960 7.438 7.616 7.292 1JIT4, 7.056 T.838 7.500 7J88 7.676 8.068 7.708 7.901 8.004 8.286 7.917 8.115 8.313 8.510 8.125 8.828 8.631 8.734 8.3S3 8.542 8.760 8.968 8.643 8.766 8.960 9.182 8.760 8.969 9.188 9.406 8.958 9.182 9.406 9.630 9.167 9.396 9.626 9.864 9.876 9w699 9.844 10 08 9.583 9.823 10.06 10.30 9.792 10.04 10.28 10.63 10.00 10.26 10.50 10.75 10.42 10.68 10.94 11.20 10.83 11.10 11.88 11.66 11.26 11.68 11.81 12.09 11.67 11.96 12.26 12.64 12.06 12.39 12.69 12.99 12.50 12.81 13.13 13.44 12.92 1324 18.66 13.89 18.83 13.67 14.00 14.33 13.76 14.09 14.44 14.78 14.17 14.58 14.62 14.88 15.23 14.95 15.81 15.77 16.00 15.88 16.76 16.13 15.88 16.23 16.63 17.02 16.67 17.08 17.60 17.92 17.60 17.94 18.38 18.81 18.33 18.79 19.26 19.71 19.17 19.06 90.13 90.60 WJOO 99.60 21.00 21.60 1 11 rtBd-M. .9893 .4683 .6875 .9167 1.146 1.376 1.604 1.833 3.063 2.292 2.621 2.750 2.979 8.308 .3.438 3.667 8.896 4.136 4.354 4.583 4.813 5.042 5.271 5.500 5.729 6J68 6.188 6.417 6.646 6.876 7.104 7.333 7.563 7.792 8.021 8.250 8.479 8.709 8.939 9.167 9.396 9.626 9.854 10.06 10.81 10.64 10.77 11.00 11.46 11.92 12.38 12.83 13.29 13.76 14.21 14.67 16.13 16.68 16.04 16.50 17.42 18.33 19.25 20.17 21.08 22.00 llji rt.Bi.lL .23U .4688 7031 .9376 1.172 1.406 1.641 1.875 2.109 2.344 2.578 2.813 3.017 3.281 3.516 8.730 3.984 4.219 4.453 4.688 4.922 5.156 6.391 5.625 5.869 6.094 6.328 6.363 6.797 7.081 7.366 7.500 7.734 7.969 8.303 8.438 8.672 S.906 9.141 9.376 9.600 ]0j08 10.31 10.66 10.78 11.02 11.26 11.72 12.19 12.66 13.13 13.59 14.06 14.63 15.00 16.47 13.94 16.41 16.88 17.81 18.75 19.69 20.63 21.56 32.50 UH FtBd.lC .9306 .4792 .7188 .9688 1.198 1.438 1.677 1.917 2.156 2.396 2.636 2.876 3.113 3354 .3.594 3.833 4.073 4.313 4.552 4.791 6.031 5.270 6.510 6.750 5.990 6.229 6..469 6.708 6.948 7.188 7.427 7.667 7.906 8.146 8.386 8.625 IIH 9.104 9.3a 9.583 9.823 10.06 10.30 10.54 10.78 11.02 11.36 11.60 11.98 12.46 12.94 13.42 13.90 14.38 14.85 15.33 13.81 16.29 18.77 17.26 18.21 19.17 20.13 21.08 22.04 23.00 FtBd.M. .2448 .4896 .7344 1.224 1.469 1.714 1.958 2.203 2.448 2.693 4.938 3.182 3.427 3.67S 8^17 4.161 4.406 4.651 4.896 5.141 5.385 ,5.680 5.875 6.120 6.366 6.609 6.854 7.090 7.344 7.589 7.833 8.078 8-32B 8.566 8.813 9.057 9.302 9.547 9.793 10.04 10.28 10.53 10.77 11.02 11.36 11.61 11.75 12.24 12.73 13.22 13.71 14.20 14.69 15.18 15.67 16.16 16.65 17.14 17.63 1840 10.58 20.56 21.54 32.52 23.60 12 FCBd.M. .8600 .5000 .7500 1.000 1.250 1.500 1.730 2.000 2.250 2.600 2.750 8.000 8.250 3.600 8.750 4.000 4.250 4.500 4.730 5.000 6.250 5.500 6.750 6.000 6.250 6.500 6.750 7.000 7.250 7.500 7.750 8.000 8.250 8.500 8.750 9.000 9.250 9.500 9.750 10.00 10.26 10.50 10.76 11.00 11.25 11.50 11.75 12.00 12.50 13.00 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 19.00 20.00 21.00 22.00 23.00 94.00 *>2 ♦"9 $ 1. H H 2. If H 3. 14 H H 4. H H H 3. H H 6. H H t • . H H H 8. H 9. H 11. 12. H 13. H 14. >i 15. H 16. H 17. H 18. 19. 20. 21. 32 IS. 24 18 274 IiAITD SUKYBZIHa. LAND SURVEYING. In surveyliie • tnet of gimiml, the sites which eoMpose its outline are deri» nated by nuraben in the order in which they ocoor. Thst end of each side which first presents itself in theooarseof the surrey, may be called its near end ; and the other its /or end. The oamber of each side is plaoed at its far end. Thus, in Figr. 1, the sarTey being supposed to comroeDce at the corner 6, and to follow the direc- tion of the arrows, toe irst side is <>, 1 : and its number is placed at its far end at 1 ; and so of the rest. Let NS be a meridian line, that is, a north and south line; and EW an east and west line. Than in any side which runs northwaidly; Flff.1. whether northeast, as side 2; or north westL as sides 8 and 1; or doe north; the distance in a due north direction between its near end and its far end, is called its lunihing; thus, a 1 is the northing of side 1; Ibthe northing of side 2 ; 4e of idde 5. In like manner, if any side runs in a southwardly direction, whether southeastwardly, as side 8; or south westwardly, as sides 4 and 6; or due south ; the corresponding distance in a due south direction between its near end and its far end, is called its southing; thus, d3 is the southing of side 8; 80 of side 4; /6 of side 6. Both northinss and southings are included in the general term jD^erence of Latitude of a side ; or, more commonly but erroneously, its kUiiude, The distance due east, or due west, between the near and the far end of any side, is in like manner called the Muting^ or westing^ of that side, as the case nuy be; thus, 6 a is the westing of side 1; 6/ of side 6; e6 of side 5; e4 of side 4; and 6 2 is the easting of side 2 ; 2 d of side 8. Both eastings and westing are included in the general term Dqaarture of a side; implying that the side d^xxrU so far from a north or south direction. We may say that a side norths, wests, sontheasta^ Ac. We shall call the northings, southings, Ac. the Ks, Ss, £b, and Ws ; the lati- tudes, lats; and the departures, d^. Perfect accuracy is unattainable in any operation inyolyinff the measur»^ meuts of angles and distances.* That work is accurate enough, which cannot be made more so without an expenditure more than commensurate with the object to be gained. There is no great difficulty in confining the uncertainty within about one-half per cent, of the content, and this probably never pre- ▼ents a transfer in farm transactions. But errors always become apparent when we come to work out the field notes; and since the map or plot of the surrby, and the calculations for ascertaining the content, should be consistent within them- selres, we do what is usually called eorreding the errors, but what in fact is simply humoring them, in, no matter how scientific the nrocess may appear. We distrib- ute them all around the survey. Two methods are used for this purpose, both based upon precisely the same principle * one by means of drawing; the other, more exact but much more trouolesome. by calculation. The graphic method, in the hands of a correct draftsman, is sufficiently exact for all ordinarv purposes. Add all the sides in feet together; and divide the sum by their number, for the average length. IMvide this average by 8 ; the quotient will be the proper scale in feet per inch. In other words, take about 8 ins. to represent an average side. We shall take it for granted that an engineer does not consider it accurate work to • A 100 ft. ehalii may Tary Its length 5 feet per mile, between winter and sammer. bj m«rc ehange of temperature; and this alone will make a differenoe of about 1 acre in 6X1. The turn- dent aboald praetiao ploitlng from perfeetUr accurate dau : aa tnoL tSa ejuunpto la table. ^ 181, ot LAXD BUBYETINQ. 275 ■Mwatv hto MiglM t9 the nearwi qoarter of a degree, wtaieh 1« tbe atnal prMtiM amonf land'torrey tn. Tbey OMi, Df idmbi of tbe engineer's tmntlt, now in aniTonal ose on our pobllo works, be readfq^ ■eMMOd within a minute or two ; and being thus nocb more accurate than the oompass oonrsee, (wtaiob eanoot be read off so eloselr, and which are moreover subject to many lonroes of error,) th«f serve to correct the Utter in the oflloe. The noting of the coarses, however, should not be confined t« the nearest quarters of a degree, btit should be read as closely as tbe observer oan guess at the minutes. The back courses also should be taken at every comer, as an additionid cheek, and for tbe deteetioa ef local attraction. It la well in taking the oom- pass bearings, to adopt as a rule, always to point the north of tbe compass* box toward tbe ohJeet whose bearing is to be taken, and to read off from tbe north end of the needle. A person who uses indUEerentiy th» M and tbe S of the box, and of the needle, will be very liable to make mistakee. n ie beet to measure the least angle (shown by dotted arcs, Tig 2.) at the sther it ; whether it be exterior, ae that at oomer ft; or interior, as all the others; because it is al- ways less than 180° ; so ^ , . •,. ^ that there is less danger >; .' Fig. 8. ef reading it off ineor- '" reetly, than if Itezeeeded 180P; tiUdBf It for grant. ed that the transit InstmnMDt Is graduated fhnn the same lero to 180° each way ; If it is gradnatai fkvm sevo to 180° tfte preeaatlon is useless. When the small angle is exterior, subtract it from SIMP for the interior one. Snppoelng the fleld work to be finished, and that we require a plot from which the oontenta may be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be measnred br scale, and thtir areas calculated one by one,} a protraction of it may be made at once from tbe field notes, either by uslQg tbe angles, or by first oorrtictiag the bearinga by means of the angles, and then nsing them. The last is tbe best, because in the first tbe protractor must be moved to each angle ; whereas In tbe last it will remain sUtionary while all the bearings are being pricked off. Kverj movement of it Inoreasea the liability to errors. The manner of oorreotlng the bewrings Is explained on tbe next page. In either case the protracted plot will oertainly not eloee precisely ; not only in oonsequence of errors in tbe field work, but also in the protracting itself. Thus the last side. No 6, Fig S, Instead of closing in at eomer 6, will end somewhere else, say, for instance, at (; the diet 1 6 being the etoting orror, which, however, as represented in Pig 3, is more than ten times as great, proportionally to the siie of the snrrey. as would be allowable in praetice. Now to hnmor-ln this error, rule through every oomer a short line parallel to ( d; and. in all eases, in the direetion from t (wherever it mav be) to tbm Btartlag point 6. Add all the sidoB together ; and measure ( fi by the scale of the plot. Then befl)i> BiBg at oomer 1, at the fsr end of side 1, say« as the Sum of aU . Total dosing . • oiii^ i • Error the sides • error «d •• ""^* • Ibrsidel. Lay eff tbia error fh>m 1 to a. Then at comer 3, say, as the Sum of all . Total olosing . , Sum of • Error the sldea • error 16 • • sides 1 and S • for side 1 Which error lay off from 2 to 6 ; and so at each of the comers; always using, a« the third term, the sum of Uie sides between the starting point and the ^ven ooAier. Finally, Join the points a, b, e, li, e, 6 ; and the plot la finished. The oerreotiec has evidently changed the length of every side ; lengthening some and shortening others. U has also changed the angles. Tbe new lengths and angles may with tolerable accuracy be fonnd by means of the scale and protractor ; and be marked on the plot Instead of the old ones. tnm those to be fbond in books on survering. This Is the only way In which be oan learn what la Mt by aecorate work. His semlolrealar protractor should be about 9 to 12 Ins in diam. and gradn- I to 10 min. His straight edge and triangle should be of metal: we prefer (vorman silver, which I not rast as steel does ; and they should be made with teniptUou* aeeuraey by a skilfUl lustra- jt-naker. A very fine needle, with a sealing-wax beiul, should be used for pricking off disU and aaglcs; it mnst be held vertically ; and the eye of tbe draftsman most be directly over it. The lead peaeU should be hard (Paber's No. 4 is good for protracting), and must be kept to a sharp point by rabMiv on a fine file, after nsing a knife for removing the wood. Tbe scale should be at least as long aa the longest side of tbe plot, and should be made at the edge of a strip of tbe same paper as the plot Is drawn on. This will obviate to a considerable extent, errors arising from contracUon and expao- ilea. Unfortunately, a sheet of paper does not contract and expand in the same proportion length* •Iss and eroaswlae, thus preventing the paper scale n-om being a perfect corrective. In plots of com- 1MB farm survi^s, iko, however, the errors rh>m this source may be neglected. For such plott as mav m pretraoted. divided, and computed within a time too short to admit of appreciable change, theordi- iarf seales of wood, ivory, or metal may be used ; but satisfHctory accuracy oannot be obtained with Asm on plots requiring several days, if tbe air be meanwhile alternately moist and dry, or subject to ssnsldarable variations in temperature. What is called parehmont paper is worae in this respect thaa fsed ordinary drawing-paper. With tba ArMoliic preoaatii«8 wa maj work tnm a drawing^ with as mnoh aoenra^ as is iwnaQf ~i in tli« Md WW*. 276 LAND BUBYETINa. When U)« plot taM nuny sldM. tula Mlonlating the error for eaob eC tfieai _ 4aoe, In a weU'performea aurrey and protraoUon, the entire error will be but a verj unall qoanti^, jjA abould not exoeed about -r^jr P^>^ of the periphery,) it may uanallj be divided among the sidee by merely placing about ^, ^, and H of it at oomera aboat ^ yi, and H way around the plot ; and at Intermediate cornera propor- tion It by eye. Or caloulatioB may be avoided Mtlrely bt drawing a line a 6 of a length Sual to the united lengtha all the aidea ; dividing it Into diatanoea a, 1 ; 1, 3 ; Sm. equal to the reapeotive aidea. Make b e equal to the entire oloaing error ; join a e ; and ilraw 1 , 1' ; 2. 2' , 4o, which will give the error at each oorner. When the plot ia thus completed, it may be divided by One pencil llnea into trianglea, whoaa baaea and heights may be measured by the aoale, in order to compute the oontenta. With care In both the anrrey and the drawing, the error ahould not exeeatf about -r-Itt V^ ot the true area. At leaat two distinot aeta of trianglea abould be drawn and computed, as a guard against miatakea ; and If the two aeta dlflbr in calculated oontenta more than about -^^ part, they have not been aa carefully frepared aa they abould have been. The doaing error due to imperfect fleld- work, may be accurately Mloulated, aa we shall ahow, and laid down on the paper before beginning the plot ; thua furnishing • perfect teat of the accuracy of the protraction work, which, if correctly done, will not cloae at the point of beginning, but at the point which indicates the error. But this calculation of the error, by a little additional trouble, furniahea data alao for dividing it by calculation among the diff aides; besides the means of drawing the plot co)-r«c(Zy at once, without the use of a protractor ; thna en»> bling uB to make the aubaequent meaaurementa and oomputationa of the triangles with more oar- tainty. We shall now describe thia proceaa, but would recommend that even when it la employed, and aapeeially in complicated surveys, a rough plot should first be made and oorreoted, by the first of the two mechanical methods already alluded to. It will prove to be of great service in using the method by oalonlation, inaamuoh aa it fumisbes an eye check to vexations mistaken which are otherwise apt to occur: for, although the principles involved are extremely simple, and easily remembered when once understood, yet the oonUnual changes in the directions of the sides will, without great ears, •auae na to uae Na inatead of Sa; Bs instead of Wa, Ac. We auppose, then, that such a rough plot has been prepared, and that the angles, bearings, and diatancea, aa taken ft'om the field book, are figured upon it in leadptneU. Add together the interior angles formed at all the cornera : call their sum a. Unit the number o* aidea by 1909 ; from the prod aubtract 360" : if the remainder la equal to the aum a, it ia a proof that the anglea have been correctly meaanred.* This, however, will rarely if ever ooeur ; there wHl always be aome discrepancy ; but if the field work has been performed with moderate eare, tliis wUl not cxcMd about two mln for each angle. In this case div it <n tqttal part* among all the anglea, adding or aubtracting, as the caae may be, unleaa It amounta to leaa than a min to each angle, when it may be entirely disregarded in common farm surveys. The corrected angles may then be marked 0n the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be aa •hown in Fig S. Next, by meana of these oorreoted angles, oorreet ths bearings alao. thua. Fig t ; Select some aide (the longv the better) trom. the two enda of which the bearing and ths reverse bearing agreed ; thns showing that that bearinc was probably not infloenesd by local attraction. Let ilds t be the one so selected ; ••» sume iM bearing, N 76° ST I, as taken on the ground, to be correct; through either end of it, as at its far end S, draw the short meridian line ; par- allel to which draw others through every ooraer. Now, having the bearing of side S, M nP 8*i' B, and reqnirfaig that of side S, it is pltfn that the reverse bearing fromoor> ner 8 is 8 75° S2' W ; and that therefore the angle 1. %, m, is 76° 32'. Therefore, if we take IfP 38' trom the entire oorreoted angle 1, 8, S, or lUP 67', the rem 68° 86' wiU bn the angle m 83 ; consequently the bearing of aideS mstaC be 8 MO 86' E. For finding the bearing of aide 4, we now hare the angle 88 a of the reveraebearing af •Ide S, alao equal to 6»o 26' : and if we add this to the entire corrected angle 234. or tofito 88*. we havs theangleaS4 = «8O23'+e»°S3' = 1380 67'; which taken f^m 180°. leaveo the angle 684= il^S'; FI9.8. • BecaoM in evenr atralght*llned figure the sum of all its Interior 1 light angles as the figure has sides, minus 4 right angles, or 300°. iglos Is eqnal to twlea a« LANS SUBYEYINa. 277 Mrtftal obMrrstion Is BMestaiy to B«e how tbe aereral angles are to be employed at eaeh oanmt, Biilea are sometimes given for this purpose, but unless frequently used, they are soon forgotten. The plot ueehanioally prepared obviates the necessity for such rules, inasmuch as the principle of proceeding thereby beoomes merely a matter of sight, and tends greatly to prarent error from asing the wrong bearings ; while the protractor will at onoe detect any serions mistakes as to the angles, and thus prevent their being carried farther along. After having obtained all the corrected bearings, Utev may be figured on the plot instead of those taken in the field. Thej will, however, require a slUi farther oorreetion after a while, since they will be affected by the adjustment of the closing error. We now prooeed to ealoalate the closing error <6 of Fig t, which is done on th« principle that in a aorreet survey the northings will be equal to the southings, and the eantings to the westings. Pre* pare a tabia of 7 columns, as below, and in the first S cols place the numbers of the sides, and their '^or. rsotedooarsee; also the diets or lengths of the Mdes, as meanured on the mugh plot, ifsnchaonQ has been prepared ; bnt if not, then as measured on the ground. Let them be as follows : Side. Bearing. Dist. Ft. Latitudes. Departures. N. 8. £. W. 1 3 8 4- 6 • N10O40'W N 750 82' X 8 69° 25' X 8 41° 3' W N 790 40' W 8 53030'W 1060 1202 1110 850 802 706 1015.5 300.3 143.9 800.2 <U1. 419.3 11fl3.9 1039.2 804. 658.2 789. 566,7 1459.7 1450.6 1460.5 Error In Lat. 2203.1 Error in Dep. 2217.9 2203.1 9.2 14.8 Kow. bj means of tne Table of Sines, etc., And the N, 8, R, W, of the several sides, and place them in the oorrespAoding four columns. Thus, for side 1, which is 1(M0 feet long, with bearing N 1|0 40' W ; cos ItP 4(K &s 0.9580 ; sin 16P 40' = 0.2868. Hare N s 1000 x 0.9580 s 1015.5; and W s 1060 X 0.2^ = 304. Prooeed tbvs with all. Add vp the foor eols ; find the dllT between the N and S ools ; and also between the B and W ones. In this instance we find that the Ns are 0.2 feet greater than the Ss ; and that the Wa are 14.8 ft greater than the Is ; in other words, there is a eleslntf error which wonld cause a mrrtct protraotion of oar first three eels, to terminate 9.2 feet too far north of Um starting point : and 14.8 feet too ter west of it. 80 that by placing this error npon the paper before beginning to protraet, We should bare a ten ftnr the aoenraoy of the protracting work ; bnt, aa before remarked, a little more IrenUe will now enable us to div the error proportionally amonc all the Ms, Ss, Sa, and Ws, and thereby give aa data for drswing the plot correctly at once, without using a protractor at all. To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaeea are farther apart : and that the addings-np ef the old N, S, B, W oolunns are omitted. The additioai here aotloed are made subseqaently. The saw table is on (ha nasi pafs. Bkm AKX. Tbe l>earinir And ibe reverse bearing from the two ends of a line will not read preciHt'ly the same argle; and the differauce varies with the latitode and with the length of the line, but not in the same proportion with either. It is, however, generally too small to be detected by the needle, bein^p, according ^o Gummare, only three quarters of a minute in a liue one mile long in lat 40°. In higher lata it is more, and in lower ones less. It is caused by the fact that meridians or north and soath lines are not truly parallel to each other; but would if extended ■eet at the poles. Heaee tbe only bearing (bat can be run in a straigbt line, eilh ttrlet aocnraey, is a true N and 8 one ; except on the very equator, where alone a due E and w one will also be straight. But a true curved E and W line may be found ■lywhere with suffioient accuracy for the survevor's purposes thus. Having first by means of the N ttMrmtUt or otherwise got a true N and 8 bearing at the starting point, lay off from it 90*, for a true land W DMtring at that point. This B and W bearing will be tangent to the true E and W curve. Baa this tangent carefully : and at intervals (say at the end of each mile) lay off ftrom it (towards the N If in N lat, or vice versa) an ofltet whose length in /Ml is equal to the proper one from the Wlowinff (able, multiplied by the sotiare of the distanee in mtlM from the star«iug point. These •bets will mark points in the tme K and W curve. 10° lao SOO liatitade IT or H. 250 80° 960 409 46° 500 550 003 «• OAieUi in ft one mile ft*oni startinfr point. 4M .118 .179 .34S .311 .885 .467 .559 .667 .795 .952 1.15 1.43 te, any offiiet in ft = .6666 X Total Dist in miles> X Nat Tane of Lat. A rtiainb line is any one that crosses a meridian obliquely, that is, ia ■•flher d«S ir ttitf 8, nor E and W. 278 LAND SURVEYING. Side. Bearing. Dist. Ft. Latitudes. Departures. N. S. K. W. 1 N 16° 40^ W N 75° 32' E S 69° 25' E a 410 3' w N790 40^ W S 53° 30' W * 1060 1202 1110 850 802 705 1015.6 1.7 3O4.0 2.7 1013.8... ... 301.3 2 300.3 1.9 390.2 1.8 1163.9 3.1 3 298.4 143.9 1.3 ... 1167.0 1039.2 2.9 4 392 ... 641.0 1.3 ... 1042.1 558.2 2.2 5 642.3... 419.3 1.1 .. 556.0 789.0 2.1 6 142.6... ... 786.9 666.7 1.8 420.4... 664.9 5729 Sum of Sides. 1454.8 Cor*d Na. 1464.7 Cor'd Ss. 2209.1 Cor'd Es. 2209.1 Cor'd Ws. Kow we have alrewlj foaAd by the old Uble that the Ns and th« W« are too long; oonaoquent^ fhey must be shortened ; while the Be, and E«, maet be lengthened ; all in the following proportieBa: ▲•the Sum of all . Any given .. Total err of . Err oflat, erdep, the Eidee * side * * lat or dep • of giren elde. Thng, oommencing with the lat of side 1, we hare, as Sum of all the aides. . Sldel. .. Total lat err. . Lat err of side L. 6729 • 1060 • • 9.2 • l.t Now as the lat of side 1 is north, It mnst be shortened ; henee tt keooma«'=:10IS.5-~l.T3dCtaj^ as Bgured oat in the new table. Again we hare for the departinv of side 1, Snm of all the aides. . Sldel. .. Total dep err. . Dep err of aide 1. 6729 • 1060 • • 14.8 • 2.7 Vow as the dep of side 1 is west, it most be shortaned; faenes it beaoiMB9M— S.T=^m;S, «a figvraa out in the new table. Prooeedlng thus with eaeh side, we obtain all the corrected lats and deps as shown in the new table : where thej are oon- nected wfth their reepeotlT* sides by dotted lines; but la praotioe it is better to oross oal the original ones when the oal" onlatlon is finished and proved. If we now add upthe 4 eols of oorrected N, S, S, W,w« And *^ %t the Ns =: the Ss ; and tha S8= the Ws; thus proving (hat the work is right. There la. It la 5fi \ / true, a dlsorepanoy of .1 of a ft I \- ^^j^ — y betweentbeNs, andtheSs; bat tbis is owing to oar oarryiBg out the oemotions to only oaa deoimat plaoet and la too small to be regarded. Diaerepmnofaa of 8 «r 4 t^thi of a foot wtn sometimes ooear f^m this cause; but may ha n^lootad. The oorrsolod late and dioM mast ovUaatty ehaiifa tha bearing aad dlstanoa or a bnt wttheut knowing either of these, we eaa aew plot the survey by means of the FUr.4. LAND SUBTEYIMQ. ir.iM. i. iM. -"'"."■ -',-■"■ 1 su no.« ino.o i |g«;^5^?;^|,s-£ Stt-J'A i. ^ «,d^. •W.i,^ KJKE USi i § 280 LAND BURTEYING. •r the •orragr.* The oomoted northings and southian we have already found ; ae alio the eaatinfi and wesUngi. The middle diata are fouDd by meau of the latter, by employing their holvM ; adkUng hair eaatinge, and lubtraeting half wectinga. Thne it ia evident that the middle dist 2' of aide a, is Snal to hair the easting of side S. To this add the other half easting of side 2, and a half easang side S ; and the sum is plainly equal to the middle dist 8' of side 8. To this add the other half easting of Ride 3, and subtract a half westing of side 4. for the middle dist 4' of side 4. From this subtract the other half westing of side 4, and a half westing of side 6, for the middle dist 6' of side 6i and se on. The actual calAulation mi^ be made thus : Half easting of side 3 = 2 = fi8lS.5 E £= mid dUt of side 1 S8S.6 I Half easting of side 8 = IMll 1167.0 E — = 521.0 E 1688.0 E = mid dist of side t. 621.0 E ■Of 556 ting of ride 4 = — 2 2209.0 E = 278.0 W 19S1.0 E = mid dist Of aide 4» 278.0 W 786.t 166S.0 E Balf vesting of side 6= = 8W.5W 2 1259.5 E = mld«iator«ide6. 88S.5W Half westing of side 6 = 564.9 866.0 E 282.4 W 688.6 EsmMdlstefiUett. 282.4 W Balf veeting of side 1 = 801.8 801.2 E lfi0.6W 160.6 Est mid dist of side 1. The work always proves Itself by the last two results being equal. Next make a table like the following, in the first 4 ools of whioh plaoe the numbers of th« sldaa, the middle dists. the northings, and loathings. Mult each middle dist by its corresponding northing or southing, and place the products in their proper col. Add up each col ; subtract the least flrom the Side. 1 2 8 4 6 6 Middle dist. 150.6 583.5 1688 1931 1259.5 583.6 Northing. 1013.8 298.4 142.6 Southing. 392 642.3 420.4 North prod. 152678 174116 179605 506390 Sonth prod. 661606 1240281 245345 2147322 506399 43560)1640923(37.67 Aont. • Proof. To lllnatrate the principle npon whioh this mle is based, let a 6, be, and c a. Fig 6, represent in order the 8 sides of the triangular plot of a survey, with a meridian line <l^ drawn through the extreme west cor* ner, a. Let lines o d and ef be drawn from eaeh oomer, perp to the meridian line ; also from the middle of eaeh side draw lines w e, m n, « o, also perp to meridian ; and representing the middle dlsts of the sides. Then sinoe the sides are regarded in the order a 6, 5 e, e a, it is plain that a d represents the northing of the side a b ; fa the northing of ea; and d/ the southing of 6e. Aow if we mult the nothing ad ot the side ab, by its mid dist ew, the prod Is the area of the triangle abd. In like manner the northing fa of the side ea, mult by its mid dist « o, gives the area of the triangle a ef. Again, the $otUhing dfot the side b e, mult by lu mlddistmn, gives the area of the entire flg dhefd. If ftom this area we subtract the areas of the two triangles at tf, and aef, the rem is evidently the area of the plot •6«. ^ith any other plot, however oomi^lflated. Fi|r.& IJLND SURVKTINQ. 281 ■natMt. Th« ran will be tbe area of the rarvey in aq ft ; which, div by 4S6M, (the namber af aq ft la an aore,) will be tbe area in aor^a ; in this iusiauoe, 37.67 ac. It now remaina enly to oaloalate the eorreeted beariugs and lengptha of the sides of the sorrey, all of which are neceaaarUy changed by the adoption of tbe eorreeted lau and deps. To And the bearing of any aide, dir lu departure (K or W) by Ita 1m (N or S) ; in the table of nat tang, find (he qnot ; HOI 3 W the angle opporite It Is (he reqd angle of bearing. Thus, for the oourae of aide 1, we hare >-— ' — — =: .3972=rnat tang ; oppoaite which in the table is the reqd angle, l(P 8S' ; the bearing, therefore. Is K 1«» M' W. Again : fer the dial or length ef any aide, from the table of nat cosines take the cos opposite to tbe angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the oos. Tons for tlie diet of side 1, we find opposite 16° S3', the coa .9686. And Lat. Cos. 1013.8 -i- .9686 » 1067.6 the reqd disk Tte MlaiwiBc table oontaias all the cMreotifOiis ef the foregoing snnr^y ; eonaeqaeatly, if the bear. Side. Bearing. Dist.7t. 1 S 8 4 6 6 N 16® 33' W N 760 Sy E S e«0 23'K S40O63' W N 78«> 44' W 8 63® 21' W 1057.6 12M.0 1118.3 849.6 800.1 704.3 .*. tags anA dlsts are correctly plotted, they will close perfictly. The yeang asatatant Is adTised ta prafBtiae doing thla, as well as dtviding the plot Into triangles, and oempottng the content. In this manner be will soon learn what degree of care is neoeseary to insiue aocarats resalis. The following hlsta may often be ef serrloe. 1st. ATold taking bearings and Aisle along a eirenitoas bound- a atyUnelikeate, Fig7;bQtma •. ......................_.._=' » .«> the etralght line a c ; and al - . -r* right anglea to It, measure ofT sets to tbe crooked line. 94. iTisblng to surrey a straight flna fMm a to e, bat being ana" ble to direct the instrument precisely toward e, on account ef iBierreainv woods, or ether ebattMlea; first nm atrialUnab as • «». as nearly in the proper direotlon aa can be guessed at. . Measure m e. and say, as a m is to in e, so ts 100 ft to T Lay off a o equal to 100 ft, and o • equal to r ; and run the final line a s e. Or. if m is quite small, calculate offsets like o s for erery 100 ft alnc a », and thus avoid the aeeesslty for running a second line. Sd. When e is Tisible from a, but dia uitervenlng ground dllBcnIt to measure along, on account of marshes, Ice, extend the side y a to good ground at t : then, making the angle ytd equal to y a o, run the line t n to that point d at wlaiA the ma^ ndel» found by trial to be equal to the angle atd. It will rarely be necessary to mmkm asore than one trial for this point d; for, suppose it to be made at x, see where it strikes a e at <; aioaeaw 4 e, and eontinoe ftxmi x, making a <( =< c 4th. In case of a very irregular piece of laad. or a lake, Fig 8, surround it by straight lines. Surrey these, and at right angles to them, ■MMaro ofbets to the crooked boundary, ftth. SurTeyiBg a straight line from w toward y, Fig ft m Ffff.ft. « d Flff.lO. n FI9.0. s o. Is net To iMMs It, lay off aright aagletptw; measure any <«; make It* OS I v; make «» v < =90°; make « < = ( i»; make •<y = 90°. Then is ti = uv; and ly la in the straight line. Or, with less trouble, at g make I g a=aOPt measure any g a; make #«s3=d0O; and«s = |r0: make a«< = 60O. Then is y • = 9 a or ••; and < s, continued toward r. Is la the etralght Hue. fth. Being between two ol^eets, m and n. and wishing to place myself ia laagi with them, I lay a straight rod s b on the ground, and point it to one ef the objects m ; then to the end e, I And that It does not point to the otaT ofejeet. By suoeessire trials, I find tbe e # te vhleh H polats to both otjects, and eoaseq. wtly is ia range with them. 282 CHAINING. CHAINUrO. Chains. EDgineers have abandoned the Gunter's chain of 6& ft, divided into 100 links of 7.92 ins each. They now use a chain of 100 ft^ with 100 links of 1 ft each, and calculate areas In sq ft, the number of which, divided by 43,560, reduces to acres and decimals, instead of to acres, roods, and perches, Giinter's chain is used on U. S. Government land surveys. Chains are commonly made of iron or steel wire. Each link is bent &i each of its ends, to form an eye, by which it is connected with the adjacent linki, either directly, as in the Grumman patent chain, or, more commonly, by from 1 to 3 small wire links. The wear of tnese links is a fruitAil source of inaccuracy, inasmuch as even a very slight wear of each link considerably increases the length of the chain. Hence, chains should be compared with some standard, sucn as a target rod, every few days while in use. For transportation, the lengths are folded on each other, making a compact and sheaf-like bundle. Tapes. With improved facilities for the manufacture of steel tape, the chain is going out of use. The tape, being much lighter, requires much less pull, and, as there are no links to wear, its length is much more nearly constant than that of the chain. It is replacing, to some extent, the base-measuring rod for accurate geodetic work. Steel tapes are made in continuous lengths up to 600, 600, and even 1000 ft, but those of 100 ft are the most commonly used. Very long tapes are liable to breakage in handling. Even the shorter lengths, unless handled carefully^ are apt to kink and breaC Breaks are difficult to mend, and the repaired joint is seldom satisfactory ; whereas a kink in a wire chain seldom involves more than a temporary change of length. Being run over by a car or wagon will often kink steel tapes very badly, if it does not break them.* How* ever, the lightness, neatness, and reliability of the tape ofiG^et these disadvan* tages, which, indeed, the surveyor soon learns to overcome. Tapes for general field work are usually narrow (from 0.10 to 0.25 in) and thick (from 0.018 to 0.025 in),t and are graduated by means of small brass and copper rivets, spaced, in general, 6 ft apart, 1 ft apart in the 10 ft at eac^ end, and 0.1 ft apart in the ft at each end. They are usually mounted on reels. Tapes for city work are wider (from 0.25 to 0.5 in) and thinner (from 0.007 to 0.010 in)t and are graduated (usually to 0.01 ft) throughout their length by means of lines and numerals etched on the steel. Pins are ordinarily of wire, pointed at the lower end, and bent to a ring at the upper end. They can be forced into almost any ground that is not exceed- ingly stony. A steel ring, like a large key rin^ is often used for carrying the pins. Each pin should have a strip of bright red flannel tied to its top, in order that it may be readily found, among the grass, etc., by the rear chainman. Corrections for Hofs and tStretcll. The following diagram ^ (seep. 283) gives the correction for a steel tape weighing 0.75 fi> per 100 ft.t *The Nichols Engineering & Contracting Ck>., Chicago, guarantees that its tapes will not be injured by beins run over by wagons. fThe sizes of tapes, as made by different manufacturers, vary greatly. In applying the corrections, therefore, the width and thickness of the tape to be used should be carefully measured, and its weight per ft computed. X Deduced from diagrams constructed by Mr. J. O. Clarke, Proceedings Engi- )ers' Club of Philadelphia, April, 1901, Vol. XVIII, No. 2. from the formuU : Stretch, in feet neers' PS EA where P = pull on tape, in fl>s. S = span of tape, in feet. E = modulus of elasticity for steel = 27,600,000 flt>s per sq in. A = area of cross-section of tape weighing 0.76 B> per 100 ft. = 0.0022 square ins, and from the equation of the parabola, according to which W> S* shortening by sag, in feet = ^ where W = weight of tape, in pounds per foot. Except for very light pulls, this last formula gives practically the same reaalts as the equation of the catenary, which is absolutely correct, but much more cumbersome. , an StHi Tape Wallihing f, TbuA, a tupBj of uj teiigthf weiohlug 1 lb iDj-giTenooiiKtioD,m pull oti-^j=lHy. J, - r le OOfTactlan on tbaata-adard tape, weighing 0.70 CoDveTselT : cItct a pull Qf 10 bs on a SO ft ipan of a tape wdfthlnd; D.fl Tb per lOOrt; requiredtheaorrectian. Ta produaelbeBameemirln tbeUpe welgbtng 0.7S lb per 100 ft "onlii require the diagram at 1Z.G Bn on tb Tble ia thfl proper AorrAcliQi li^itar tape vllh 10 *■ pull. bB of ■Undn'd ferrglh at M^»^r. For' ordinarr eteel tape, Uie t MJ nperatu re it about (10000085 ft pCT ft per degree " ' lU of y = 10 X j;^ - 12.0 lbs. Beftrrl 'or » » ft span, we flna comictinn " - □ ight, a Wben measuring oter slopliiL , . tapesbnuld beheld as Dearlf boriioutal as possible, trsnsferrlug the poaitloD of Ibe raised end to the ground bj means of a plumb line. Where the ground Is ■teep, It b^xiines necessary to use a short length of tape, as the down-hul ebain- psraliel with theslope, and the disUncecarrecledGr the (ullowlDg form 284 LOCATION OF THE MERIDIAN. IiO€ATIOIir OF THE HERIDIAHT. By means of clrcampolar stars. (1) Seen from a point O (Figs. 1 and 2) on the earth, a circumpolar star e (•tar near the pole P) ap(>ears to describe daily* and counterclockwise a small circle, euwl, about the pole. The angle P O e, P O u, etc., subtended by the radius P e, P u, etc., of this circle, or the apparent distance of the star from the pole, is called its polar distance. The polar distances of stars vary sligntly from year to year. See Table 3. They vary slightly also during each year. In the case of Polaris this latter yariation amounts to about 50 seconds of arc. (3) The altitude of the pole is the angle N O P of the pole's elevation above the horizon N E S W, and Is = the latitude of the point of obser- FiG. 1. Pig. 2. ration. Decl Inatlon = angular distance north or south from the celestial equator. Thus, declination of pole = 90°. Declination of any star = 90°— its polar distance. (3) Let Z e H be an arc of a vertical circlet passing through a circumpolar star, e, and let H be the point where this arc meets the horizon N E S W. Then the angle N Z H at the zenith Z, or N O H at the point O of observa- tion, between the plane N Z O of the meridian and the plane H Z O of th© star's vertical circle (or the arc N H), is called the azlmutlkt of the star. If this angle N O H be laid off from O H, on the ground, the line O N will be in the plane of the meridian N Z S, or will be a nortb-and-sontii llne.|| (4) When a star is on the meridian Z N of the observer, above or below the pole P, as at u or ^, it is said to be at its upper or lower culmina- tion, respectively. Its azimuth is then = 0, tne line O H coinciding with the meridian line O N. (5) When the star has reached its greatest distance east or west ftom the pole, as at e or w, it is said to be at its eastern or western eloni^A- tlon.{ « In 23 h. 56.1 m. t A great circle is that section of the surface of a sphere which is formed by a plane passing through the center of the sphere. A vertical circle is a great circle passing through the zenith Z. I Astronomers usually reckon azimuth from the south point around through the west, north, and east points, to south again ; but for our pur- pose it is evidently much more convenient to reckon it f^om the north point, and either to the east or to the west, as the case may be. II The point N, on the horizon; is called the north point, and must not be confounded with the north pole P. g As seen ttova. the equator, a star, at either elongation, is, like the pole Itself, on the horizon ; and the two lines Pe,Tw, joining it with the pole, * — I a single straight line perpendicular to the meridian, and lying in the LOCATION OF THE MERIDIAN. 285 (6) The boar anffle of any star, at any given mconent, is the time which has elapsed since it was in upper culmination.'" (7) Evidently the azimuth of a star is continually changing. In cir- cumpolar stars it varies from OP to maximum (at elongation) and back to (P twice daily, as the star appears to revolve about the pole ; but when the star is near either elongation the change in azimuth takes place so slowly that, for some minutes, it is scarcely perceptible, the star appearing to travel vertically. (8) Given the polar distance of a star and the latitude of the point of observation, the aaimutli of the star, at eloiiirAtlon, may be found by the formula.f Sine of azimuth of star = sine of polar distance of star cosine of latitude of point of observation or see (11) and Table 3. (9) The following circumpolar stars are of service in connection obeervations for determining the meridian. See Fig. 3. Constellation Letter Ursa minor (Little bear) a (alpha) Ursa major (Great bear) € (epsilon) ( " " i <(zeta) with Cassiopeia S (delta) Called Polaris Alioth Mizar Deltas Jfora»r^.^tet:» July Fig. 8. (10) Polaris^ or the nortb star, is fortunately placed for the determi- nation of the meridian, its polar distance being only about 1%^. See Table 3. Fig. 3 shows the circumpolar stars as the}r appear about midnight in July ; inverted, as in January ; with the left side uppermost, as in April ; ana, with the right side uppermost, as in October. R horizon. The azimuth of the star is then == its polar distance. But in other latitudes Pc and Pit; form acute angles with the meridian, as shown, and these angles decrease, and the azimuth of the star at elongation in- creases, as the latitude increases. * In lat. 40° N., the hour angle, ZPc = ZP«>, of Polaris, at elongation, is = 5 h. 55 m. of solar time. Caation. It will be noticed that, except for an observer at the equator, the elongations do not occur at 90° from the meridian. t In the spherical triangle Z P «, we have : sin e Z P ^ sinPe sin Z e P ^ Bin P Z But, since Z « P = 90°, sin Z « P = 1. Also, sin P Z = cos (90° — P Z), and < Z P — azimuth of e. sin Pe _ sin polar d ist ance P O e cos latitude Hence, sin azimuth of e . ^ „ sm F Z 1 « Cassiopeia is here called Delta, for brevity. I Polaris is easily fonnd by means of the two well-known stars called the *^ pointers '' in " the dipper," Fig. 3, which forms the binder 286 LOCATION OF THE MERIDIAN. (11) Table 3 ffives the polar distances of Polaris and their log sines for January 1 in each third year from 1900 to 1990 inclnsive, the log cosines of each fifth deeree of latitude from '2/iP to 50°, and the corresponding azimuths of Polaris at elongation. Intermediate values may be taken by interpolation.* (12) By olMervatlon of Polaris at elonntlon. This method has the convenience, that at and near elongation the star appears to travel vertically for some minutes, its azimuth, during that time, remaining practically constant : but during certain parts of tne year (see Table 1;, the elongations of Polaris take place in daylight; so that this method cannot then be used. | See (18), (19), (22). Nor can it be used at any time in places south of about 4° N. lat., because there Polaris is not visible. (18) The approximate times of elongation of Polaris for certain dates, in 1900, are given in Table 1, with instructions for finding the times for other dates. Or, watch Polaris in connection with any of those stars which are nearly in line with it and the pole, as Delta, Mizar, and Alioth. See Fig. 3. The time of elongation is approximated, with sufficient clofleneas for the determination of the azimuth, by the cessation of apparent hori- zontal motion duriftg the observation. (14) From fifteen to thirty minutes before the time of elongation, have the transit, see (21). set up and carefully centered over a stake previously driven and marked with a center point. The transit must be in adjust- ment, especially in regard to the second adjustment, p. 294, or that or the horizontal axis, by which the line of collimation is made to describe a ver- tical plane when the transit is leveled and the telescope is swung upwMrd or downward. (15) Means must be provided for illuminating the cross-hairs of the tran- sit. X I'h^ T^^y ^ done by means of a bull's eye, or a dark lantern, so neld as not to throw its light into the eye of the observer ; or, better, by means of a piece of tin plate, cut and per- forated as in Fig. 4, bent at an angle of 45^, as in Fig. 5, and painted white on the surface next to the telescope. The ring, formed by bending the long sirip, is placed around the object end of the telescope. A li^ht, screened from the view of the observer^ is then held, at one side of the instrument, in sucb a way Fig. 4. . Fig. 6. that its rays, falling upon the oblique and whitened surface of the tin plate, are reflected directly into the telescope. (16) Bring the vertical hair to cut Polaris, and, bv means of the tangent screw, follow the star as it appears to move, to the right if approaching eoM.- em elongation, and mce versa, keeping the hair upon the star, as nearly as may be. As elongation is approached, the star will appear to move more and more slowly. When it appears to travel vertically along the hair, it has practically reached elongation, and the vertical plane of the transit, vriih the vertical hair cutting the star, is in the plane of the star's vertical circle. Depress the telescope, and fix a point in the line of sight, preferably 300 feet or more distant from the transit.f Immediately reverse the transit, (swinging it horizontally through an arc of 19XP), sight to the star again. ^ portion of the " great bear " (Ursa major), a line drawn through these two stars passing near Polaris. .\s the stars in the handle of the dipper form the tail of the great bear, as shown on celestial maps, so Polaris and the stars near it form the tail of the little bear (Ursa minor.) Polaris is also nearly midway and in line between Delta and Mizar. Polaris forms, with three other and less brilliant stars, a quite symmetrical cross, with Polaris at the end of the right arm. In Fig. 3 this cross is inverted. Its height is about 5°, or == the distance between the pointers. * Part of a table computed by the Surveying Class of 1882-8, School of Engineering, Vanderbilt University, Nashville, Tenn., and published by Prof Clin H. Landreth. t The stake must be illuminated. This may be done bv throwina' light upon that side of the stake which faces the transit, or, better, by holding a sheet of white paper behind the stake, with a lantern behind the paper. In the latter case, the cross-hairs of the transit, as well as the stake, and the knife-blade or pencil-point with which the assistant marks it, show out dark against the illuminated surface of the paper. \ See Note, page 290. LOCATION OF THE MERIDIAN. 287 •gain depress, and» if the line of sight then coincides perfectly -with the mark first set, both are in the plane of the star's vertical circle. If not, note where the line of sight does strike, and make a third mark, midway between the two. The line of sight, when directed to this third mark, is in the required plane, from which the azimuth, found as in (8), has yet to be laid off to the meridian, to the l^ from. eaMem elongation, and vice vena, (17) To avoid driving the distant stake and marking it during the night, a fixed target at any convenient point may be used, and the horizontal angle formed between the line ox sight to the star and that to the target merely noted, for use in ascertaining and laying off the azimuth of the tarvet. (19) By otMervation of Polaris at cnlmtnaiioii. Owing to its greater difficulty, this method will generally be used only when that by elongation is impracticable. It consists in watching Polaris in connec- tion with another circumpolarstar (such asMizar *or Delta) until Polaris is seen in the same vertical ]^ane with such star, and then waiting a short and known time T, as follow8,t until Polaris reaches calminatlon, where- upon Polaris is stehted and the line of collimation is in the meridian. At their upper culniinations, Mizar and Delta are too near the zenith to be conveniently observed at latitudes north of about 25° and BOP respectively. At their lower culminations they are too near the horizon to be used to advantage at places much below about 88° of N. latitude. In general. Delta is conveniently obeexved at lower culmination ttom. February to August, and Mizar during the rest crf^kie year. Mizar Delta T= T = In 1900 2.6 mins 8.4 mins In 1910 6.5 mins 7.2 mins Mean annual increase, 1900-1910 . 0.39 min 0.38 min (19) "By obsenration of Polaris at any point In Its path* Table 1 gives the mean solar times of upper culmination of Polaris on the 1st of each month in 1900, and directions ibr ascertaining the times on other dates ; and Table 2 gives the azimuths of Polaris corresponding to different values of its hour angle in civil or mean solar time, for different latitudes fh)m 30° to 50°, and for the years 1901 and 1906. For hour angles and lati- tudes intermediate of those in the table, the azimuths may be taken by interpolation. See Caution and formula, p. 290. (SO) The local time} of observation must be accurately known, and the time of the preceding upper culmination (as obtained from Table 1) dedu<!ted from it. The difference is the hour angle. If the hour angle, thus found, is 11 h. 58 m. or less, the star is west of the meridian. If it is greater than 11 h. 58 m., the star is east of the meridian. In that case deduct the hour angle from 28 h. 56 m. and enter the table with the remaiTuier as the hour an^le. See Fig. 1. (»1) Where great accuracy is not required, Polaris may be observed by means of a plumb-line and sight. A brick, stone, or other heavy object will answer perfectly as a plumb-bob. It should hang in a pail of water. A compass sight, or any other device with an accurately straight slit about 1/16 inch wide, may be used. The sight must remain always perfectly verti- cal, but must'be adiustable horizontally for a few feet east and west. The plumb-line and sight should be at least 15 feet apart, and so placed that the star and plumb-line can be seen together through the sight, throughout the observation. The plumb-line must be illuminated. It is well to arrange all these matters on an evening preceding that of the observation. When the star reaches elongation, the sight must be fastened in range with the plumb-line and the star. From the line thus obtained, lay off the azimuth ; to the toest for ea^em elongation, and vice versa. {fSS9) Bjr any star at eqnal altitudes. This method, applicable to south as well as to north latitudes, consists in observing a star when it is at any two equal altitudes, £. and W. of the meridian, thus locating, on the horizon, two points of equal and opposite aziQiuth. The meridian will be midway between the two points. • Mizar will be recognized by the small star Alcor, close to it. t Deduced from values calculated in astronomical time (p. 266) by the U. S. Ckiast and Oeodetic Survey. X Ijocal time agrees with standard time (p. 267) on the standard meridians only. For other points add to standard time 4 minutes for each degree of longitude east of a standard meridian, and trice versa. 288 LOCATION OF THE MERIDIAN. (as) By e^aal sliadows from the sun. Piir. 6 ADDroximAtP At the solstices (about June 21 and December 21) the path a b c <J traveraed before and after noon, by the end of «*«'<'"■ tniveraea the solar shadow O o, etc., of a verti- cal object O, or by the shadow of a knot tied in a plumb-line suspended over O, will intersect a circular arc a N d, described about O, at equal dis- tances, am^ md, from the meridian O N. The observations should be made within two hours before and after noon. At the vernal equinox (March 21) the line thus located will then be west, and at the autumnal equinox (Sept. 21) east, of the merid- ian, by less than 7.}4 minutes of arc. For intermediate dates the error is nearly proportional to the time elapsed. It is well to draw several arcs of different radii, O a, O 6, etc., note two points where the path of the shadow intersects each arc, and take the mean of all the results. A small piece of tin plate, with a hole pierced through it, may be placed with the hole vertically over O ; and the bright spot, formed by the light shining through the hole, used in place of the end of the shadow. Table 1. ^^^S'V^^*'?"^** **'^" times of elongratlon and calmlnatlon muilJh hTlScX)" ■^•' ^ong. 90° W. from Greenwich, on the first of each The times given in this table are mean solar or local times. fn^ti^^^^o iS^Y.^"" 5^22i^.^i,.TJi»l^.^iL«^ i« bold-faee. In lattude 25^, W. elongations occur later and E. earlier K« , . , latitude 50°, W. " " earlier and E. later f ^^ nearly 2 mins. le correction fc%r Inno-iti-iHA amr\tt-n*a *ex ana■m.^^^■^ » «..• *. jfj.i , In 'TK^ —-w > y. cttiijcrtiiiuji. later) * -f -•"**«'• For other days of the month, deduct 8.94 min. for each succeedinp fl*v In general, the times are a little later each vear In iSith^^S?! i^^v ^: b}A minutes later, but in 1905, only about 3 mlnm^s latefthan^iJT^iJ? *S2!}* discrepancy is due to the occurrence of leS^yeaMni^'^ ^ ^^' ^^^ Inasmuch as this table serves chiefly to out the obsPrvlV «« «r.,««^ ^ he- should be at his post from 15 to S m?nmk in advance^S^^^ ""S^^ the gradual increase in the times is of little conseauence Thi^oUl'^®*; the star at.elongation is determined by observS ^ position ot At culmination, where the change in azimuth is most ranid a»i o-,^. <« At elongation, an error in time of 20 minutes 10 minutes 5 minutes 1 minute will make an error in azimuth of less than 90 seconds less than 6 " less than 2 " about 0.06 second , „, ~' — aooui 0.06 second Jan. 1. 12.31 A. July 1. 12.51 A. W. M. E. M. Jan. 1. 6.38 P. July 1. «.44 P. U. M. L. Elongratlons. (E, eastern : W, western.) 1900. AVp.V IfSSkli. .^SSk^. Km«: \Zl .»P.^«. i%-k^. .?J^kV rj.J:S: V^l Cnlmlnatlons. (U, upper ; L, lower.) 1900. E Mi w! M. Feb. 1. L. 4.38 A. M. U. M. Aug. 1 4.45 A. }^fn'}'h >P^- 1- ^- May 1. L. 2.47 A. M. 12.45 A. M. 10.48 P. M. Sept. 1. U. Oct. 1. U. Nov. 1. U. 2.43 A.M. 12.46 A.M. 10.40 P.M. Jane 1. S.«8P. Dec. 1. S42P. ^ LOCATION OP THE MERIDIAM. l«.»5°4e°4IS'' HW 410 43 47I 51 440 47 5ll Sa m'o 58 11 SI i 1 wu" uiiuillr ba 'S 290 LOCATION OF THE MERIDIAN. Table a. POLARIS. POLAR DISTANCES, AND AZIMUTH AT ELONGATION. Azimuth at Elongation, in Latitude u Polar Dist. of Polaris Log sin poldist. 1 S0<> JWO BOO 85° 40<> 400 50^ O / ft o / o t O f o / O f o / o / 1900 1 18 33 8.38027 1 18.8 1 21.1 1 24.9 1 29.8 1 36.1 1 44.1 1 64.4 1908 1 12 37 8.32 472 1 17.3 1 20.1 1 28.8 1 28.7 1 34.8 1 42.7 1 58.0 1906 1 11 41 8.31 910 1 16.3 1 19.1 1 22.8 1 27.6 1 33.6 1 41.4 1 51JS 1909 1 10 45 8.31 341 1 15.3 1 18.1 1 21.7 1 26.4 1 32.3 1 40.1 1 60.1 1912 1 9 49 8.80 765 ] 14.3 1 17.0 1 20.6 1 25.2 1 31.1 1 88.7 1 48.6 1915 1 8 53 8.30 181 1 13.3 1 16.0 1 19.6 1 24.1 1 29.9 1 37.5 1 47.2 1918 1 7 58 8.29594 1 12.3 1 15.0 1 18.6 1 28.0 1 28.7 1 36.1 1 46.7 1921 17 2 8.28 999 1 11.4 1 14.0 1 17.4 1 21.9 1 27.6 1 34.8 1 44.8 1924 16 7 8.28 401 1 10.4 1 13.0 1 16.3 1 20.7 1 26.8 1 33.5 1 42.9 1927 1 6 12 8.27 794 1 9.4 1 11.9 1 16.3 1 19.6 1 25.1 1 82.2 1 41.4 1980 1 4 16 8.27 169 1 8.4 1 10.9 1 14.2 1 18.5 1 28.9 1 30.9 1 40.0 Log 008 Ut 9.97 299 9.95 728 9.98 753 9.91 337 9.88426 9.84949 9.80807 . Owing to changes in the position of Polaris during the year, the positions given in the table may at times be in error by as much as a minute. The error is greater in the nigher latitudes. Having the north polar distance,/), of a star, and the latitude, L, of the point of observation, we have, declination of star = 6 = 90° — p ; and ^e aslmutb, a, of the star, corresponding to any hour angle, a, may be found by the following formulas : TanM = ^ = -^. Then Tan a = <^ " ' ^° * . cos h cos h cos (L— M) The declinations, fi, of Polaris are given in the U. S. Ephemeris or Nautical Almanac. From these the polar distances may be obtained more accurately than from our Table 3. Caution. When it is desired to determine the meridian within one minute of arc, it is well to use more than one method and compare the results. For example, observe Polaris both E. and W. of the meridian, aitd a star at equal altitudes south of the zenith. NoTK. — Lf Polaris be found during twilight, iu the morning or evening, obsei-- ▼atinns of it luuy be made without artificial illumiaation of the cross-haira. For times of elongation, see Table 1, CouTertiion of Arc Into Time, and vice versa. Arc Tike 1° = A minutes 1' = 4 seconds l» =1 0.066... second Time Abc 24 hours =860° Ihour = 150 1 minute = OP 16' 1 second « (PVl^ 1 TBE ENGINEBB's TRANSIT. 291 THE ENGINEER'S TRANSIT. 292 TtTB EHQINEE&B TRAITBET. Thb dtMIlB of the transit, like thme of the IstgI, are dllTerCDtlT trmtgei hf diff nukem, and to mlt pirtkuUr purpoAU. We deocribe it In iti modern Ibrm, SB uude by Heller ud Brightly, arPhlladiL without the lone bBbU«-tBke F F, Fig 1, onder the telescope, and the BrrndBstcd an p, It la theli plklB teBMalt. With tb«e sHiendage*, or nther vltta a, eradusted cirde ia fite* of the Bra It becomea Tirtiullr s COBipl«te Ttaeodiillle. B D D, Fig 1, Is the tripo<l>ke«d. The Krew-tbRwds at v loeelfe the sciew of a wooden trlpod-head-cover vhen the inetniment Ia out of use. S B A la Qu l«w«r panulel |»l«t«. After the traaiit has been set tstt dmtIt oier the center ofa sl^e, the mlilftlns-plat«, <f d e c, enables lu, bf illabClr lonealng the I«TelllBK-BCrem K, to shlA the upper paiU boriiontallT a (rifle, and ■haa bring the plumb-bob eiactlj OTer tbe center -with leaa trouUe than bf the elder method of puiblog one or tiro of the legs furibei Into the giouod. or apread- InE tbem more or leaa. Tbe acreirfl, E, are taea tightened, thereby puablDg up> ward the upper BBiwllel pl«M n « ni z i, and vitb It tbe balT-bkll t, ibni pr^alng o c llghtl; up afalnat the under lida at 8. Ths plomb-UnB paana throngb the yert hols in 6- Scraw-eaja, / g, protect the leTalUnMcrewi ttom. dual, ia The feet, i. of tbe icrewa, work In looea aocketa^^, made flat at bottom. to^presene S from being Indented, The paita thui far dTeKribed are guaiBUw left atUiched lo [he legs at all Uraea. Flj? 1 show, the method of attachmmt. To set (he upper puM up*a «m panllal l>l«te|k Plaoe tbe lowerendor UU Id 1 1, holding tbe Instrument so that the thrw bloekaonaawe (of which the one ahown at Fis morable) mar ^oter the three oorreapondtiiK THE engineer's TRANSIT. 293 rMeeses in a, thus allowing a to bear fully on m, upon which the upper pute then rest. (The inner end of the spring-catch, I, in the meantime enters agroov6 around U, Just below a, and prevents the upper parts from falling off, if the in* strument is now carried over the shoulder.) Kevoive the upper parts horizontaUj a trifle, in either direction, until thev are stopped by the striking of a small lug on a against one of the blocks F. Tne recesses in a are now clear of the blocks. Tighten g, thereby pushing inward the movable block F, which clamps the bevelled flange a between it and the two flxed blocks on m m, and confines the spindle U to the fixed parallel plates. It remains so clamped while the instrument is being used. To remoTe the upper parts ft^m theparallel plates. Loosen g, bring the recesses in a opposite the blocks F. Hold back I, and lift the upper parts, which are then held together by the broad head of the screw inserted into the foot of the spindle w. T T is the oater reTOlTlng: spindle, cast in one with the support* Ing^plate Z 2^, to which is fastenea the s^radnated limb O. The limb extends beyond the compass-box, and thus admits of larger graduations than would otherwise be obtainable, to wis the Inner revolving^ spindle. At its top it has a broad flange, to which is fastened the vernier plate P. To the latter are fastened the corapass-box C, the two bubble-tubes M M, the standards y Y, supporting the telescope, &c. Each bubble-tube is supported and adjusted by four capstan-head nuts, two at each end. The bent strip, curving over the tnbe, protects the glass from accidental blows in swinging the telescope. <k»iatrol of motions of ir>*»dnAl«d limb O O and wernler plate P. — ^The tangent-screw 6 and a spiral spring (not shown) opposite to it are fixed to the graduated limb 00, and hold between them a projection y from the loose collar t, which is thus confined to the limb and made to travel with it. The clamp-screw H passes through the collar t and presses against the small lug shown at its inner end. When H is tightened, this lug is pressed against the fixed spindle U U, to which the graduated limb is thus made fast. A slow mo- tion may, however, still be given to the limb by means of the tangent-screw G. The motion of the vernier plate P over the graduated limb O is simUarly governed by the tangent-screw 6 and its spiral spring (not shown), fixed to the ternier plate P, and the clamp-screw e, which passes tnrough the collar z, and {>re88es against the small lug shown at its inner end. In Heller and Brightly's nstraments, the screw b is provided with means for taking up its ** wear," or "lost-motion." There are two verniers. One is shown at ja. Fig 1. Both may be read, and their mean taken, when great accuracy is required. Ivory reflectors, c, facilitate their reading. Before the instrument is moved from one place to another, the eompaas-needle, ib. Fig 2, should always be pressed up against the glass cover of the compass-box by means of the upright miUed-head screw seen on the ver^ nier-plate m Fig 1, Just to the right of the nearest standard. The pivot^point is thus protected from injury. R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope in any required position. It is oest to let the eye-end. 1C, of the telescope revolve dowHiffard, as otherwise the shade on O, if in use, may fall off. The tangent-screw, il. moves a vert arm attached to R, and is thus used for slightly changing the elevation of the telescope. In the arm is a slit like that seen in the vernier-arm L Bt mesns of the screw D. the movable vernier-arm Y may be clamped at tDT desired point on the vertical limb g. When (P of the vernier is placed at 9(Pon the arc ^, and the index of the opposite arm is placed over a small notch on the horizontal brace (not seen in our figs) of the standards, the two slits will be opposite each other, and may be used for laying off offsets, oc, at right-angles to the line of sight. One end, R, of the telescope axis rests in a movable box, under which is a screw. By means of the screw, the box may be raised or lowered, and the axis thus ad- justed for very slight derangements of the standards. For E, B, O, and A, see iaulf p 306. a is a dust-guard for the object-slide. StaaiA Kalrs. Immediately behind the capstan-screw, p. Fig 1, is seen a nnaller one. This and a similar one on the opposite side of the telescope, work in a ring inside the telescope, and hold the ring in position. Across the ring are itretched two additional horizontal hairs, called stadia hairs, placed at such a distance apart, vertically, that they will subtend say 10 divisions of a graduated rod placed 100 ft from the instrument, 15 divisions at 150 ft, Ac. They are thus used for asttsuring hor and sloping distances. Tbe lonff babble-tube« F F, Fig 1, enables us to use the transit as a level. •Ithoof h it Is not so well adaotsd as the latter to this purpose. 294 THE engineer's transit. To aAinmt a plain Transit* When either a lerel or a transit is purchased, it is a good precaution (but one which the writer has never seen alluded to) to first screw the oltject-glass firmly home to its place ; and then make a short continuous scratch upon the ringt>f the glass, and upon its slide ; so as to be able to see at any time when at work, that the glass is always in the same position with regard to the slide. For if, after all the adjustments are completed, the position of the glass should become clumged, (as it is apt to be if unscrewed, and afterward not screwed up to the same precise spot,) the acyustments may thereby become materially deranged ; especially if the object-glass is eccentric, or not truly ground, which is often the case. Such scratches should be prepared by the maker. In making adjustments, as well as when using a transit or lerel, be careful that the eye-glass and object-glass are so drawn out that there shall be ne parallax. The eye-glass must first be drawn out so as to obtain perfect distinctness of the cross-hairs ; it must not be disturbed afterward; but the object-glass must be moved for different distances. First, to ascertain tliat tlie bnbble-tnbes, M Bf • are placed parallel to the vernier-plate, and that therefore when both bubbles are in the centers of their tubes the axis qf the inst is vert. By means of the four levelling- screws, K, bring both bubbles to the centers of their tubes in one position of the inst ; then turn the upper parts of the inst half-way round. If the Dubbles do not remain in the center, correct half the error by means of the two capstan-nuta rr; and the other half by the levelling-screws K. Repeat the trial until both bubbles remain in the center while the inst is being turned entirely around on its spindle. Second, to see that the standards have snfTered no deranire- ment ; that is, that they are of equal height and perpendicular to the vernier- plate, as they always are when they leave the makers hands. Level the inst perfectly ; then direct the intersection of the hairs to some point of a high object (as the top of a steeple) near by ; clamp the inst by means of screws H and e, and lower the telescope until the intersection strikes some point of a low object. (If there is none sucn drive a stake or chain-pin, Ac, in the line.) Then un- clamp either H or e, and turn the upper parts of the inst half-way round ; fix the intersection again upon the high point ; clamp ; lower the telescope to the low point. If the intersection still strixes the low point, the standards are in order. If not, correct one-ltalf of the difference by means of the adjusting-block and screw at the end, R, of the telescope axis. Fig. 1, and repeat the trial de novo, resetting the stake or chain-pin at each trial. If the inst has no adjusting-block for the axis, it should be returned to the maker for correction of any derange- ment of the standards. A transit may be used for running ^raight lines^ even if the standards become slightly bent, by the process described at the end of the fourth adjustment. Third, to see that the cross-hairs are traly vert and hor ^rhen the inst is level. When the telescope inverts, the cross-hairs are nearer the eye-end than when it shows objects erect. The maker takes care to place the cross-hairs at right-angles to each other in their ring, or diaphragm ; and gene- rally he so places the ring in the telescope, that when levelled, they shaJl be reii and hor. sometimes, however, this is neglected ; or the ring may by accident be- come turned a little. To be certain that one hair is vert, (in which case the other must, by construction, be hor,) after having adjusted the bubble-tubes, level the in« strnment carefully, and take sight with the telescope at a plumb-line, or other yert straight edge. If the vert hair coincides with this object, it is, sofar^ in adjustment ; but if not, then loosen sKghtlv only two adjacent screws of the four, pp i t. Fig 1 ; and with a knife, key, or other small Instrument, tap verj gently against the screw-heads, so as to turn the rin^ » little in the telescope; persevering until the hair be* comes truly vertical. When this icr done, tighten the screws. In the absence of a plumb-line, or vert stsulgfat edge, sight the cross-hair at a Tery small distinol point; and see if the hair still cuts that point, when the telescope is raised or lowered by revolring it on its axis. The mode of performing the foregoing will be readily understood ft'om this Fig, which represents a section across the top part of the tele> acope, and at the cross-hairs. The hair-ring, or diaphragm, a; vert hair, v; tele* scope tube, g ; ring outside of telescope tube, d; & is one of the four capstMi-headed screws which hold the hair-ring, a, in its place, and also serve to a^jnst it. The lower ends of these screws work In the thickness of the hair-ring; so that when they are loosened somewhat, they do not lose their hold on the ring. Small THE EKOIKEES'S TIUXSIT. 295 mO washers, c, are placed under the heads h of the screws. A space ^ y is left around each screw where it passes through the telescope tube, to allow the screws aud ring together to be moved a little sideways when the screws b are slightly loosened. Fourth, to see tliat the wertical hair is In the line of colU- matlon. Flant the tripod firmly upon the ground, as at a. Level the inst ; clamp it; and direct the vert hair by means of tangent-screw O ffigs. 1 and 2) upon some convenient object h\ or if there is none such, drive a thin stake, or a ennin-pin. Then revolving the telescope vert on its Hxis, ^ observe some object, as c, where the vert hair now strikes ; ^ a ^^^ or if there is none, place a second pin. Uoclamp the instru- « «^^ ment by the clamp-screw H ; and turn the whole upper • " part of it around until the ven hair again strikes b. JPig, 4, Clamp again ; and again revolve the telescope vert on its axis. If the vert h»ir now strikes e, as it did before, it shows that c is really at ; and that 6, a, e, are in IM^ same straight line ; and therefore this adjustment is in order. If not, observe where it does strike, say at m, (the dist a m being taken equal to a c,) and place a pin there also. Measure m c ; and place a pin at v, in tne line m c, making m v <— one-fourth of m c. Also put a pin at 0, half- way between m and c, or in range with a and b. By means of the two hor screws that move the ring carrying the cross-hairs, adjust the vert hair until it euts V. Now repeat the fntire operation ; and persevere until the telescope, after being directed to b, shall stVike the same object 0, both Hmes, when revolved on its axis. See whether the movement of the ring in this 4th adjustment has dis- turbed the verticality of the hair. If it has, repeat the 3d adjustment. Then re- peat the 4th, if necessary ; and so on until both adijustments are found to be right at the same time. Thus a straight line mav be run, even if the hairs are out of adjustment ; but with somewhat more trouble. For at each station, as at a, two back-sights, and two fore-sights, a c and a m, may be taken, as when making the adjustment ; and the point 0, half-way between c and m, will be in the straight line. The inst may then be moved to 0, and the two back-sights be taken to a ; and so on. Angles measured by the transit, whether vert or hor, will evidently not be tifected by the hairs being out of a4justment, provided either that the vert liair is truly Tert. or that we use the inler^oHon of the hairs when measuring. The foreproiniT ^^^^ All the a^instments needed, unless the tran- sit is reqnlrea for levelUi^, in which case the following one muse be attended to : To adjust the lontr bnbble*t1Ihe« F F, Fie. l, we first place the line of sight of the telescope hor, and then make the bubble-tube hor, so that the two are parallel. Drive two pegs, a and b Fig. 5, with their tops at precisely the same level (see Bem. p. 296) and at least about 100 ft. apart ; 800 or more will be better. Plant the inst Armly, in range with them, as at c, making^ c an aliquot part of a b, and as short as will permit focusing on a rod at 6. The inst need not be leveled. Suppose the line of sight to cut e and d. Take the readings b e and a d. Their diff is be — ad=^an — ad=*dn\ and ah-.ac: dnids'i s being the height of the target at a when the readings (a «, b 0) on the two stakes are equal. as==ad-\-ds^ad-\ r — ' If the reading on a taceeeds that on b (as when the line of sight is vfg) the diff of readings is = a ^ — bf=sag — ai^gi\ smd as = a g — g s=aaff — ^ — j- — • Sight to «, bring the babble to the cen of its tube by means of the two small nuts n n at one end of the tube. Fig. 1, and assume that the telescope and tube are parallel.* The zeros of * Thla B«0eeM s mnmll «iTor due to the oarralnre of the earth ; fDr a hor line at v ia v h, tao* flaatiml to tlM earved (or " tofwl") torfiaoe of still water at «, whereae » • Is tangential to water aarf at a point midwaj between a and h. Henoe if the telesoope at « points to a li will not be parallel te the level bobbto-tnbe. To allow for this, and for the reftvotloa bj the air, wUeh diminUhM the error, rsiae the tarfet on • to a point h above a. h* — .0000000205 x square of a in (I ; bat when • e is S30 ft, Jk a is only aboni one tenth of an inoh and barely oovers the apparent thlekness of Um -bnlrlatkn ' 296! THE ENaiNEER'S TRANSIT. the vert circle, and of its vernier, may now be aAjiitted, if they require it, by loosening the vernier screws and then moving the vernier until the two coin- cide. ^ , . - , Rem. If no level is at hand for levelling the two pegs o and &, it may be done by the transit itself, thus : Carefully level the two short bubbles, by means of the levelling-screws K. Drive a peg m, from 100 to 300 feet from the instrument o. Then placing a target-rod on m, clamp the target tight at whatever height, as sv, the hor hair happens to cut it ; it being of no im- l^ L portance whether the telescope is level or not; TV (J) although it might as well be as nearly so as can \ X conveniently be guessed at. Clamp the telescope g^ — JJ. in its position by the clamp-ring K, Fig. 1. Re- ^ volve the inst a considerable way round; say iJifiT. 0. nearly or quite half way. Place another peg n, atprecUdy the same diet from the instrument that m is; and continue to drive it un- til the hor hair cuts the target placed on it, and still kept clamped to the rod, at the same height as when it was un m. When this is done, the tops of the two pegs are on a level with each other, and are ready to be used as before directed. When a transit is intended to be used for surveying farms, Ac, or for retracing lines of old surveys, it is very useful to set the compass so as to allow for the ** va- riation" during the interval between the two surveys. For this purpose a '' TArlatton- vernier " is added to such transiCB ; and also to the oompaos. When the graduations of a transit are figured, or numbered, so as to read both H) 10 ways from aero, thus, i n ii 1 1 1 h i 1 1 1 1 1 1 1 1 1 1 1 1 I m the vernier also is mada double ; that is, it also is graduated and numbered from its sero both ways. In thia case, if the angle is measured from zero toward the right hand, the reading must be made from the right hand half of the vernier ; and vice versa. If the figuring la single, or only in one direction, from zero to 360^, then only the single vernier la necessary, as the angles are then measured only in the direction that the figuring counts. ICngineers differ in their preferences for various manners of figuring the graduations. The writer prefers from zero each way to 180^, with two double ver- niers. To replace cross-hairs in a IcTel, or transit. Take out tiie tube from the eye end of the telescope. Looking in, notice which side of the oroM- hair diaphragm is turned toward the eye end. Then loosen the four screws which hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new hairs with beeswax, varnish, glue, or gum-arabic water, Ac. This requires care. Then, to return the diaphi-agm to its place, press firmly into one of the screw-holes on the circumf of the diaphragm itself, the end of a piece, of stick, long enough to reach easily into the telescope as far as to where the diaphragm l^Iongs. By this stick, as a handle, insert the diaphragm edgewise to its place in me telescope, and hold it there until two cpposUe screws are put in place and screwed. Then draw the stick out of the hole in the diaphragm ; and with it turn the diaphragm until the same side presents itself toward the eye end as before ; then put in the other two screws. The so-called cross- hairs are actually spider-web, so fine as to be barely visible to the naked eye. Holler A Brightly use very fine platina wire, which is much better. Human hair is entirely too coarse. To replace a spirit-level, or bnbble^lass. Detach the level from the instmment; draw off its sliding ends; push out the broken glass vial, and the cement which held it ; insert the new one, with the proper side up (the upper side is always marked with a file by the maker); wrapping some paper around its ends, if it fits loosely. Finally, put a little putty, or melted beeswax over the ends of the vial, to secure it against moving in its tube. In purchasing instruments, especially when they are to be used far from a maker, it is advisable to provide extras of such parts as may be easily broken or lost ; such as glass compass-covers, and needles; atjQusting pins; level vials; magniflen, Ao, Theodolite adjustments are performed like those of the level and transit. let. That of the cross-hairs; the same as in the level. 2d. The long bubble-tube of the telescope ; also as in the level. 8d. Th^ two short bubble-tubes ; as in tne transit. 4th The vernier of the vert limb ; as in the transit with a vert circle. 5th. To see that the vert hair travels vertically ; as in the fourth adjustment of the transit. In some theodolites, no adjustment is provided for this ; but in Isrm onaa it is provided for by screws under the feet of the standards. Somttimw • second telescofKi is added ; it Is p^iic«d belov the hor limb, and to THE BOX OB POCKET SEXTANT. 297 called a toate?ur. It has its own clamp, and tangent-screw. Its use is to ascertain whether the sero of that limb has moved during the measurement of hor angles. When, previously to beginning the measurement, the zero and upper telescope are directed to^inund the first object, point the lower telescope to any small distant object, and then clamp it. During the subsequent measurement, look through i^ from time to time, to be sure that it still strikes that object ; thus proving that nt slipping has occurred. THE BOX OR POCKET SEXTANT. Ths portability of the pocket sextant, and the fact that It reads to single minutes, render it at times very useful to the engineer. By it, angles can be measured while in a boat, or on horseback ; and in many situations which preclude the use of a transit. It is useful for obtaining latitudes, by aid of an artificial horizon. When closed, it resembles a cylindrictu brass box, about 3 inches in diameter, and 1)^ inches deep. This box is in two parts ; by unscrewing which, then inverting one i>art,,and then screwing them to- gether again, the lower part becomes a handle for holding the instrument. Looking down upon its top when thus arranged, we see, as in this figure, a movable arm I C, called the index, which turns on a center at C, and car- ries the vernier Y at its other end. Q 6 is the graduated arc or limb. It actually subtends about 13P, but is di- vided into about 146^. Its zero is at one end. Its graduations are not shown in the Fig. Attached to the index is a small mov- able lens, (not shown in the figure,) likewise revolving around C, for read- ing the flue divisions of the limb. When measuring an angle, the index is moved by turning the milled-head P of a pinion, which works in a rack placed within the box. The eye is applied to a eir* cnlar hole at the side of the box, near A. A small telescope, about 3 inches long, ; accompanies the instrument; but may generally be dispensed with. When so, the eye-hole at A should be partially closed by a slide which has a very small eye-hole in it ; and which is moved by the pin A, moving in the curved slot. Another slide, at the nde of the box, carries a.dark glass for covering the eye-faole when observing the ran. When the telescope is used, it is fastened on by the milled-head screw T. The top part shown in our figure, can be separated from the cylindrical part, by removing 3 or 4 small screws around its edge ; and the interior can then be exam- ined, and cleaned if necessary. Like nautical, and other sextants, this one bm two principal glasses, both of them mirrors. One, the Index-fplass, is attached to the underside of the index, at C; its upper" edge being indicated by the two dotted lines. The other, the Moriaon-KliMiS) (because, when meas- uring the vert angles of celestial bodies, it is directed toward the horizon,) is also within the box; the position of its upper edge being shown by the dotted lines at R. The horizon-glass is silvered only half-way down ; so that one of the observed objects may.be seen directly through its lower half, while the image of the other object is seen in the upper half, reflected from the index-glass. That the instrument may be in a4justment, ready for use, these two glasses must be at right angles to the plane of the instrument ; that is, to the under side of the top of the box, to which they are attached; and must also be parallel to each other, when the zeros of .the vernier and of the Umb coincide. The index-glass is already permanently fixed by the ma^T, and requires no other a4ju8tment. But the horizon-glass has two adjust- ments, which are made by a key like that of a watch, and having a milled-head K. It is screwed into the top of the box, so as to be always at hand for use. When needad, it is unscrewed. This key fits upon two small square-heads, (like that for 298 THE COMPASS. winding a watch;) one of which is ihown at S; while the other is near it, but on the SIDE of the box. These squares are the heads of two small screws. Jf the horlEon glass H should, aa in this sketch, (where it is shown endwise,) not be at right angles to the top U HJ of the box, it is brought right by turning the square- bead S of the screw S T ; and if, after being so far rectified, it still is not parallel to thn index-glass when the zeros coincide, it is moved a little backward or forward by the mjuare head at the side. To adjust a box sextant, bring the two aeros to coincide precisely ; then look through the eye-hole, and the lower or unsilvered part of the horixon-glasB, at some distant object. If the instru- ment is in adjustment, the object thus seen directly, will coincide precisely with its reflected image, seen at the same time, at the same spot. But if it is not in ac^ustment, the two will appear separated either hor or vert, or both, thus, * • ; in which case apply the key E to the square-head S ; and by turning it slightly in whichever direc- tion may be necessary, still looking at the otjject and its ima^e^ bring the two into a hor position, or on a level with each other, thus, * •. Then apply the key to the square- head in the side of the box; and by turning it slightly, bring the two to coincide perfectly. The instrument is then ai^justed. In some instruments, the hor glass has a hinge at v, to allow it play while being adjusted by the single screw S T ; but others dispense with this hinge, and use two screws like S on top of the box, in addition to the one in the side. If a sextant is used for measuring vert angles by means of an artificial boriEon, the actual altitude wilF be but one-half of that read off on the limb ; because we then read at once both the actual and the reflected angle. The great objection to the sextant for engineering purposes, is that it does not measure angles horizontally, as the transit dues ; unless when the observer, and the two ob> jects happen to be in the same hor plane. Thus an observer with a sextant at A, if measuring the angle subtended by the mountain-peaks B and C, must hold the graduated plane of the sextant in the plane of A B C ; and must actually meas- ^^ ,' ,-' I ^ ; ure the angle BAG; whereas what he g^k^*^':'- ' wants is the hor angle nAm. This is ^f""" -'Wl greater than BAG, because the dista An A and A m, are shorter than A B and A G. The transit gives the hor angle n A m, be- <iau8e its graduated plane is first fixed hor by the levelling-screws ; and the subse- Suent measurement of the angle is not affected by his directing merely the line of [ght upward, to any extent, in order to fix it upon B and G. For more on this sub- ject ; and for a method of partially obviating this objection to the sextant, see the note to Example 2, Case 4, of " Trigonometry." Tbe nautical sextant, used on ships, is constructed on the same principle as the box sextant ; and its adjustments are very similar. In it, also, the index- glass is permanently fixed by the maker ; and the horizon-glass has the two adjust- ments of the box sextant. It also has its dark glasses for looking at the sun ; and a small eight-hole, td be used when the telescope is dispensed with. •-•- THE COMPASS. To adjust a Compass* The first adjustment is that of the bubbles. Plant firmly ; and level th« Instrument, in any position ; that is, bring the bubbles to the centers of their tubes. Then turn the instrument half-way round. If tlie bubbles then remain at the cen- ters, they are in adjustment;. but if not, correct one-half the diff" in each bubble, by means of the adjusting-screws of the tubes. Level the instrument again ; tun it half roimd ; and if the bubbles still do not remain at the center, the atiUusting- ■crews must be again moved a little, so as to rectify half the remaining diff. Gener* THE COMPASS. 299 ally several trials must be thus made, until the bubbles will remain at the oente while the compass is being turned entirely around. Seeond adjustmeiit* Level the compass, and then see that the needle it hor; and if not, make it so by means of the small piece of wire which is wrapped around it ; sliding the wire toward the high end. A needle thus horizontally ad- justed at one place, will not remain so if removed fietr north or south from that place. If carried to tiie north, the north end will dip down ; and if to the south, the soutii end will do so. The sliding wire is intended to counteract this. Tliird a^Jnatment. This is always fixed right at first by the maker; that is, the sights, or slits for sighting through, are placed at right angles to the compass plate ; so that when the latter is levelled by the bubbles, the sights are vert. To test whether they are so, hang up a plumb-line ; and having levelled the compass, take sight at the line, and see if the slits coincide with it. If one or both slits should prove to be out of plumb, as shown to an exaggerated extent in this sketch. It should be unscrewed from the compass, and a portion of its foot on the high side be filed or ground off, as per the dotted line ; or as a temporary expedient, a small wedge may be placed under the low side, so as to raise it. Foortb BdJaBtmeilt, to straighten the needle, if it should become bent. The compass being levelled, and the needle hor, and loose on its pivot, see whether its two ends continue to point to exactly opposite graduations, (that is, graduations 18€P apart ;) while the compass is turned completely around. If it does, the needle is straight ; and its pin is in the center of the graduated circle ; bat if it does not, then one or both of these require adjusting. First level the compass. Then turn It until some graduation (say 90^) comes precisely to the north end of the needle. If the south end does not then point precisely to the opposite 90° division, lift off the needle, and bend the pivot-point until it does ; remembering that every time said point is bent, the compass must be turned a hairsbreadth so as to keep the north end of the needle at its 90^ mark. Then turn the compass half-way round, or until the opposite 90° mark comes precisely to the north end of the needle. Make a fine pen* <^ mark where the touth end of the needle now points. Then take off the needle, and bend it until its south end points ha^f^ay between its 90° mark and. the pencil mark, while its north end is kept at 90° by moving the eompass round a hairsbreadth. Tlie needle will then be straight, and must not be altered in making the following, adjostment, although it will not yet cut opposite degrees. Flfih a4ius^i»eiit, of the pivot-pin. After being certain that the needle is straight, turn the compass around until a part is arri ved at where the two ends of the needle happen to cut opposite degrees. Then turn the compass quarter way around, or through 90°. If the needle then cuts opposite degrees, the pivot-point is already in adjustment ; but if the needle does not so cut, bend the pivot-point until it does. Bapeat, if necessary, until the needle cuts opposite degrees while being turned entirely •round. Oare and nicety of observation are necessary in making these adjustments properly ; because the entire enor to be rectified is, in itself^ a minute quantity; and the novice it must be held parallel to the graduated circle. Otherwise annoying errors of several minutes will be made in a single observation ; and the accumulation of two or three such errors, arising from a cause unknown to him, may compel him to abandon the ac^ustments in despair. This su^estion applies also to the reaiding of angles taken by the transit, Ac ; although the errors are not then likely to be so great as in the case of the compass. In purchasing a magnifier for a compass, see that DO part of it, as hinges, or rivets, are made of iron ; for such would change the direction of the needle. If the sight-slits of a compass are not fixed by the maker in line with the two opposite zeros, the engineer cannot remedy the defect. This can be ascertained by passing a piece of fine thread through the slits, and observing whether it stands precisely over the zeros. THE COMPAfiS. THE COUFABB. I! II' |3|3i||| III |3 111 til 1^ > ts f i i i II if! I L, ; Hi bSj 302 OOlffTOtm LIKEB. United StatflB, by Henry GtuuMtCi In 17th Annual Beport ef tf. 8. Geological Survey, 1896-^ Electrietty, either atmospheric, or excited by rubbing the glass ooy«r of the compass box, sometimes gives trouble. It may be removed by touching the glass with the moist tongue or finger. DEMAOHETIZATIOV. The needle, if of sqft metal, Bometimeo loses part of its magnetism, and consequently does not work well. It may be restored by simply drawing the north pole of a common magnet (either straight or horseshoe) about a dozen times, from the center to the end of the south half of the needle ; and the south pole, in the same way, along the north half; pressing the magnet gently upon the needle. After each stroke, remove the magnet several inches from the needle, while bringing it back to the center for making another stroke. Each half of the needle in turn, while being thus operated on, should be held flat upon a smooth hard surface. Sluggish action of the needle is, however, more generally produced by the dulling or other iujury of the point of the pivot. RemagnetiEing will throw the needle out of balance ; which must be counteracted by the sliding wire. In order to prevent mistakes by readlnn^ sometimes from one end, and sometimes from the other end of the needle, it is best to always point the N of the compass-box toward the object whose bearing is to be taken ; and to read off from the north end of the needle. This is also more accurate. OONTOUB LINES. A OOHTOUB um is a curved hor one, every point in which represents the same level ; thus each of the contour lines SSc, 91c, 94c, itc. Fig 1, indicates that every point in the ground through which it is traced is at the same level ; and that that level or height is everywhere 88, 91, or 94 ft above a certain other level or height called datum ; to which all others are referred. Frequently the level of the starting point of a survey is taken as being 0, or zero, or datum ; and if we are sure of meeting with no points lower than it, this answers every purpose. But if there is a probability of many lower points, it is better to assume the starting point to be so far above a certain supposed datum, that none of these lower points shall become minus quantities, or bdow said supposed datum or zero. The only object in this is to avoid the liability to error which arises when some of the levels are -|-» or plus ; and some — ^ or minus. Hence we may assume the level of the starting point to be 10, 100, 1000, Ac, ft above datum, according to circumstances. The vert dists between each two contour lines are supposed to be equal ; and in railroad surveys through well-known districts, where the engineer knows that his actual line of survey will not require to be much changed, the dist may be 1 or 2 ft only ; and the lines need not be laid down for widths greater than 100 or 200 ft on each side of his center-stakes. But in regions of which the topography is compara-' tively unknown ; and where consequently unexpected obstacles may occur which require the line to be materially changed for a considerable dist back, the observa- tions should extend to greater widths ; and for expedition the vertical dists apart may be increased to 3, 5, or even 10 ft, depending on the character of the country, Ac. AlsOj when a survey is made for a topographical map of a State, or of a county, vert dists of 5 or 10 ft will generally suffice. Let the line A B, Fig 1, starting from 0, represent three stations (S 1, S 2, 8 3,) of the center line of a railroad survey ; and let the numbers 100, 108, 101, 104, along that line denote the heights at the stakes above datum, as determined by levelling. Then the use of the contour lines is to show in the offlcH what would be the effect of changing the surveyed center line A B, by mrving any part of it to the right oi CONTOUB JUNES. 303 Iflft hand.* Thug, if it should be moved 100 ft to the left, the starting point wonl^ be on ground about 6 ft higher than at present ; inasmuch as its leyel would then be about 106 ft above dktum, instead of 100. Station 1 would be about 7 ft higher, or 110 ft instead of 103. Station 2 would be about 7 ft higher, or 108 ft instead of 101. If the line b<« thrown to the right, it will plainly be on lower ground. The field obeervat^'ons for contour lines are sometimes made with the spirit-level; but more frequently oy a slope-man. with a straight 12-ft graduated rod, and a slope instniment, or clinometer. At each station he lays his rod upon the ground, as FIg.l. • nearbr a^ right angles to the center line A B as he can Judge by eye ; and placing the slope instrument upon it, he takes the angle of the slope of the ground to the nearest ^ of a degree. He also observes how far beyond the rod the slope continuee the same ; and with the rod he measures the dist. Then laying down the rod at that Kint also, he takes the next slope, and measures its length ; and so on as far as may Judged necessary. His notes are entered in Ids field-book as shown in Fig 2 ; the angles of the slopes being written above the lines, and their lengths below ; and should be accompanied by such remarks as the locality suggests ; such as woods, rocks, maryih. sand, field, garden, across small run, ftc, Ac. * la thni aiing the word* right and left wc an lUppoMd to have our baeki turned to the ■tartiog point of the survey. In a river, the rliplit bniik or shore is that which IS on the right band as we descend it, that is, in speaking of its right or left huk. ve are lODpoMd to hare oar backs turned toward! Ita head, or origin ; and bo with a surrey 804- CONTOUR LINES, I- 91 ''m^i' 64- 70 It is not abeolately necessary to represent the slopes roughly in the fleld-book, aa in Fig 2; for by usin^ the sign + to signify "up;" — "4own;" and = "'leTel,*' the slopes may be vrnt- ten in a straight line, as in Fig 2^. The notes naving been taken, the preparation of the contour lines by means of them, is of course office-work ; and is usually done at the same time as the draw- ing of the map, &c. The. field observations at each station are then sepa- rately drawn by protrac- tor and scale, as shown in Fig 3 for the starting point O. The scale should not be less than about -^ inch to a ft, if anything Iik« accuracy is aimed at. Suppose that at said station the slopes to the right, taken in their order, are, as in Fig 2, U°, 4°, and '26P ; and those to the left, 20°, lO^, and IQP ; and their lengths as in the same Fig. Draw a hor line h o. Fig 3 ; and consider the center of it to be the station-stake. From this point as a center, lay off these angles with a protractor, as shown on tho arcs in Fig 3. Then beginning say on the right hand, with a parallel ruler draw the first dist a c, at its proper slope of 16^ ; and of its proper length, 45 ft, by scale. Then the same with c y and yt.Do the same with those on the left hand. We then have a cross-sectitm of the ground at 8ta 0. Then on the map, as in Fig 1, draw a line as m n, or A 10, at right angles to the line of road, and passing through tha station-stake. On this line lay down nie Jior dists a d, d «, s «, ae^eg^gk^ marking them with a small star, as is done and lettered in Fig 1, at 8ta O. When extreme accuracy is pretended to, these hor dists must be found by measure on Fig 3 ; but as a general rule it will be near enough, when the slopes do not ex- ceed 10°, to assume them to be the same as the sloping diets measured in the field. Next ascertain how high each of the points cy tint is above datum. Thus, measure by scale the vert dist ae. Suppose it is found to be 5 ft ; or in other words, that e is 5 ft below stationHBtake 0. Then since the level at stake is 100 ft above datum, that at c must be 6 ft less, or 100 — 6 = 95 ft above datum ; which may be marked in light lead-pencU figures on the map, as at d, Fig 1. N6xt for the point y, suppose we find « 2/ to be 11 ft, or y to be 11 ft below stake ; then its heiglit above datum must be 100 — 11 =s 89 ; which also write in pencil, as at s. Proceed in the same way with t. Next going to the left hand of the station-stake, we find « I to be say 2 ft ; but Z is above the level of the station-stake, therefore its height above datum is Biff. 8. 100 4- 2 » 102 ft, as figured at e on the map. Let ng be 5 ft; then is n, 100 -f- ^ 105 ft above datum, as marked at a ; and so on at eacn station. When this has been done at several stations, we may draw in the contour lines of that portion by hand thus: Suppose they are to represent vert heights of 3 ft. Beginning at Station O (of which the height above datum is 100 ft) to lay down a contour line 103 ft abova datum, we see at once that the height of 103 ft must be at ^, or at ^ the dist from « to g. Make a light lead-pencil dot at t ; and then go to the next fetation 1. Here we see that the height of 103 ft coincides with the station-stake itself; place a dot there, and go to Sta 2. The ^evel at this stake is 101 ; therefore the contour for lOP CONTOUli LDcaa. 305 ft mtut evidently be 2 ft higher, or at <, ^ of the dl^t fh>m Sta 2 to +104 ; theretiort make a dot at i. Then go to Sta 3. Here the leTel being 104 aboye datum, the con- tour of 103 must be at y, or i of the diet from Sta 3 to +99 ; put a dot at y. Finally draw by hand a curving line through ^ SI, i, and y ; and the contour line of 103 ft ii done. All the others are prepared in the same way, one by one. The level of each must be figured upon it at short intervals along the map, as at 103 c, 106 c, Ac Or, instead of first placing the + points on the map,l;o denote the slope dists actu- ally measured upon the ground, we may at once, and with lees trouble, find and show those only which represent the points ty S 1, t, y, Ac, of the contours themselves. Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec- tion c < of the ground has been prepared as before ; and that we want the hor dista from the stake, to contour linea for 94, 97, 100 ft, Ac, 3 ft apart vert. Draw a vert line t; 2, through the station-stake, and on it by scale mark levels of 94, 97, 100, dba ft. This is readily done, inasmuch as we have the level 104 of the stake already given. Through these levels draw the hor lines a. b, m, n, <&c. to the ground-slopes. Then these lines, measured by the scale, plainly give the requirea dists. When the ground is very irregular transversely, the cross-sections must be taken in the field nearer together than 100 ft. The preparation of contour lines will be greatly facilitated by the use of paper ruled into small squares of not less than about ^ inch to a side, for drawing the cross-sections upon. When the ground is very steep, it is usual to shade such portions of the map to represent hill-side. The closer together the contours come, the steeper of course is the ground between them ; and the shading should be proportionally darker at such portions. But for working maps it is best to omit the shading. In surveys of wide districts, the transit instrument with a graduated vertical circle or arc, g, p. 291, ia used for measuring the angles of slope, instead of the common slope-instrument. In many cases, notes similar to the following will serve the purpose of contour lines on railroad surveys. BUCO.. 61.. es.. 6S. ... — S.1B. +S. IL. ... + 2.2B. — 1.8L. ... = 1. E. + 4. 1 L. Wblek meaai tbat at ttotlon 40, the slope of tbe groand on the right, m nearly as he can Jadge by 0jm, •r by hi* band-lerel, is aboat S ft downward, for 1 ehain, or 100 ft ; and on the left, about 2 ft apward In 1 ehnln. At 61, 2 ft ap, in Zehatns t* the right; and 1 ft down in S chains to the left. A% tS, l«y«l for 1 ohaln to tbe right; and ascending 4 ft in 2 chains to the left. At 6S, the same as at n, Ai aoBie spots it will be well to add a sketch of a orons-seotion, like Fig 2 ; only, instead of the ■agies, use ft of rise or fall, to indicate the slopes, as J udged bj eye, or by a haod-level. By this ■ethod, the resolt at every station will be somewhat in error; bat these small errors will balance •aeh other m» nearly that the total may be regarded as sufBeiently correot for all the parpoees of a pnUmioMxy eettmate of the oost of a rood. When the final stakes for guiding the workmen are pioflod* the slopes should be sorefliUy taken, in order to ooloalato the qnontity of ezeavation aooa- ratoly for payment. 20 TH E LEV EL. Qui ptDS 1 J which coDHoe the semlclrculsr clipi 1 1, aud iheu oprnlg; the clip*. The pins should be tied l« the Ys, by pieces ot string, to preveut Ihofr being iom. (be ilide of the oMwt-gbai O, is nio>ed burkward ur rorw&rd by a rauk niid plnian, bf meBDB or the mlHeS hewl A. The slide of the lyt-gkia £. la moved Id [be same WB* br the milled head e. A cTlindrlual lube ef brass, oallal s lAmfe, is usua]]* hirnMied with «*eh kTel. It la Intended to bo slid on to the objeci-cnd O of the teleicape, to prerent Ibe ^are of the sun upua the objecl-glass, when the nun ia low. At Biaui outer rlDseiiclrcUag the telescope, and carrrlOE 4 small cspstan- beadea wrewa; tmof wfilcb.pp, are at lop and boiiom; while the other two, of whkh I la ODe, areittba^ea, and M right inelei top p. laslde a[ this outer ring la another, loaldo of the telescope, atid wblc^b bas stretched acrosa it two «, when cairjlQ^ .._ _... th'm ™r'!J'^ be Juii^ed'bf ejar^ls euablea the lereller lo'^see^ tbaTt/o i(^m»n hulda b"^ ia desired, as la KHuetimei the case, when itsking out work, ^t may ba obtained (^ IA* tiutrumrnt ij in perfiet a^ailmtnl, and UvUai) by tighllng at a plmnb-llne. or olhor »ert oltjoct, and then turning tbe tideetopo a little in Its tiw aa to bring the Uw teleKopa and Y, to aare that tron'cle In fiiiure. Heller & Brightly, howaTw. The small holes around the beadaofthe 4 small capstaii-screwsti,l,JustnrerfedU^ are for admitting the end of a small steel pin, or lerer, fbr tumlogtbem. If flnt will be lDworM^ and Iba liorltnntal hair with it. But un loiAing through the tal» I THE LEVBU 307 ■cope th«7 will appear to be mSsed. If first the lower one be looeened, and the npper one tightened, the hor hair will be Mctnally raised, but apparently lowered. This is because the glasses iu the eye-piece B reTerse the apparent position of objects intid€ cf the telescope ; which effect is obTiated, as regcurds exterior ol^Jects, by means of the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a hair appears to strike too high, it must be raised still higher ; if it appears to be already too &r to the right or left, it must be actually movcKl still more in the same direction. This remark, however, does not apply to teleacopn which make objects appear iUTerted. There is no danger of li^urlng the hairs by these motions, inasmuch as the four screws act against the ring only, and do not come in contact with the hairs them- t^lves. Under the telescope is the bubble-tube D D. One end of this tube can be raised or lowered slightly by means of the two capstan-headed nuts n n, one of which must be looeened before the other is tightened. On top of the bubblo'tube are scratches for showing when the bubble is central in the tube, frequently these scratches, or marks, are made on a strip of brass placed above the tube, as in our fig. There are several of them, to allow for the lengthening or shortening of the bubble by changes of temperatuie. At the other end of the bubble-tube are two smidl capstan-screws, placed on opposite sides horizontally. The circular head of one of them is shown near L By means of these two screws, that end of the tube can be slightly moved hor, or to right or left. Under the bul>ble-tube is the bak Y F ; at one end of which, as at y, are two large capstan-nuts to w, which operate upon a stout interior screw which forms a prolongation of the Y. The holes in these nuts are lai^r than the others, as they require a larger lever for turning them. If the lower nut is loosened and the upper one tightened, the Y above is raised ; and that end of the telescope becomes farther removed from the bar; and vice versa. Some makers place a similar screw and nuts under both Ys ; while others dispense with the nuts entirely, and substitute beneath one end of the bar a large circular milled head, to be turned by the fingers. This, however, is exposed to accidental alteration, which should be avo&ded. When the portions above m are put upon m. and fastened bv the screw Y, all the upper part may be swung round hor, in either direction, oy loosening the elamp-serew H ; or such motion may be prevented by tightening thatecrew. It frequently happens, after the telescope has been sighted very nearly upon an object, and then clamped by H, that we wish to bring the cross-hairs to coincide more precisely with the object than we can readily do by turning the telescope by kand: and in this case we uee the tanfrent-ticrew 5, by means of which a Bliffht but steady motion may be given after the instrument is clamped. For fuller remarks on the clamp and tangent-screws, see '* Transit." The parallel plates m and S are operated bv four levelllnipHierews ; three or which are seen in the figure, at K K. The screws work in sockets B; which, aa weU as the screws, extend above the upper plate. When the instrument is placed on the ground for levelling, it is well to set it so that the lower parallel plate S shall be as nearly horizontal as can be roughly judged by eye ; in order to avoid much turning of the levelling screws K ^ in making the upper plate m hor. The lower plate S, and the brass oarts below it, are together called the tripod-taead ; and, in connection with three wooden legs Q Q Q, constitute the tripod. In the figure are seen the heads of wing-nuts J which confine the legs to the tripod-head. Under the center of the tripod-head should always be placed a small ring, from which a plumb-bob may be suspended. This is not needed in ordinary levelling, but becomes useful when rangmg center-stakes, &c. To adjast a Irevel. This is a qnite simple operation, but requires a little patience. Be careful to avoid thraininff any of the screws. The large Y nuts ie w sometimes require some force to ttoH them ; but it should be applied by pressure, and not by blows. Before begin- rJDg to su^nst, attend to the o^ect-glass, as directed in the first sentence under ^^ To •i^nst a plain transit.** Three at^nstments are necessary ; and rrnist be made in the following order: First, that of tlie cross-bairs ; to secure that their intersection shall toatinue to strike the same point of a distant object, while the telescope is being tnnu'd round a complete revolution in its Ys. This is called ac^usting the line sf eolllmation, or sometimes, the line of sight; but it is not strictly the line of (tight until all the adjustments are finished; for until then, the line of coUimation vni not serve for taking levelling sights. If eross-liairs brealK* see p 296. Second* Miat of Uie bnbble-tnbe D D, to place it parallel to the Une 308 TBB LBYBL. 0f coUimatlon. preTiomly •4|asted; so that when the bahble stands at the centra o( ItD tube, indicating that it is lerel, we know that onr sight through the telescope is hor. To replace broken bubble tabe, see p 296. Tbird, tbat of tbe Ts, by which the telescope and bubble-tube a^re supported; flo that the bubble-tube, and line of sight, shall be perp to the yert axis of the instru- ment; so as to remain hor while the telescope is pointed to objects in diff directions, as when taking back and fore sights. To make tbe first adjastmenty or that of the cross-hairs, plant the tripod ^r/n2y upon the ground. In this adjustment it is not necessary to lerel the instrument. Open the clips of the Ys ; unclamp ; draw out the eye-glass E, until the cross-hairs ieure aeen perfectty cUar ; sight the telescope toward some clear dis- tant point of an object ; or still better, toward some straight line, whether yert or not. More the object-glass 0, by means of the milled head A, so that the object shsJI be clearly seen, wltbout parallax, that is, without any apparent dancing about of the cross-hairs, if the eye is moved a little up or down or sideways. To secure this, the object-glass alone is moved to suit different distances ; the eye-glass is not to be changed after it lb once properly fixed upon the cross-hairs. The neglect of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the tangent-screw d, bring either one of the cross-hairs to coincide x>reciM/y with the object. Then gently, and without jarring, revolve the telescope naif-way round in its Ys. When this is done, if the hair still coincides precisely with the object, it is in adjustment ; and we proceed to try tbe other hair. But if it does not coincide, then by means of the i screws p, t, move the ring which carries the hairs, so as to rectify, as nearly as can be judged hy eye, only one-fuUf of the error; remembering that the ring must be moved in the direction opposite to what appean to be the right one ; unless the telescope is an inverting one. Then turn the telescope back again to its former position : and again by the tangent-screw bring the cross-hair to coincide with the object. Then again turn the telescope half-way round as before. The hair will now be found to be more nearly in its right place, but, in all probabil- ity, not precisely so ; inasmuch as it is difficult to estimate one-half the error accu- rately by eye. Therefore a little more alteration of the ring must be made ; and it may be necessary to repeat the operation several times, before the adjustment is perfect. Afterward treat the other hair in precisely the same manner. When both are adjusted, their intersection will strike the same precise spot while the telescope is being turned entirely round in its Ys. This must be tried before the aci^ustment can be pronounced perfect; because at times the adjustment of the second hair, slightly deranges that of the first one ; especially if both were much out in the b» ginning. To make the second adjustment, or to place the bubble-tube paralW to the line of collimation. This consists of two dis> tinct adjustments, one vert, and one hor. The first of these is effected by means of the two nuts n n on the vert screw at one end of the tube ; and the second by tbe two hor screws at the other end,^, of the tube. Looking at the bubble-tube endwise, from t in tbe foregoing Fig, its two hor adjusting-screws 1 1 are seen as in this sketch. The larger capstan-headed nut helov), has nothing to do with the adjustments ; it merely hold^ the end of the tube in its place. . To make the vert adjustment of the bubble-tube, by means of the two nuts nn. Place the telescope over a diagonal pair of the levelling-ecreWH K. K ; and clamp it there. Open the clips of the Ys; and by means of the levelliug-screws bring the bubble to the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and put it back again in its reversed position. This being done, if tbe bubble still remains at the center of its tube, this adjustment is in order ; but if it moves toward one end, that end is too high, and must be lowered ; or else the other end must be rftised. First, correct htdf the error by means of the levelling-screws K K, and then the re- maining half by means of the two small capstan->headed nuts n». To roiM the end n, first loosen the upper nut and then tighten the lower one ; to do which, turn each nut so tiiat the near side moves toward your right. To louwr it, first loosen the lowei nut, then tighten ttie upper one, moving the lutar side of each nut toward your ^fU Having thus brought the bubble to the middle again, again lift the telescope out of its Ys ; turn it end for end, and replace it. The bubble will now settle nearer the center than it did before, but will probably require still further adjustment. If so, correct haif the remaining error by the levelling-screws, and half by the nuts, as be* fhre; and so continue to repeat tbe operation until the bubble remidns at the cental in both positions. For another method, see '* To adjust the long bubble-tube,** p 2ML Horizontal adjustment of bubble-tube ; to see that its axis is in the same plans with nhat of the telescope, as it usually is in new instruments. It is not eesily d» TEE LEVEL. 309 ranged, except by blows. Have the bubble-tube, as xxearly as may be, directly under the telescope, or over the center of the bar T F. Bring the telescope over two of the leTellingHScrews K K ; clamp it there ; center the bubble with said screws ; turn the telescope in its Ts, say about ^ inch, bringing the bubble-tube out from over the center of the bar, first on one side, then on the other. If the bubble stays centered irhile so swung out, this adjustment is correct. It it,runs towajrd opposite ends of its tabe when swung out on opposite sides of the center, move the end t of the tube by the two horizontal screws 1 1 until the bubble stays centered when the tube is swung out on either side. If the bubble runs toward the same end of its tube on both sidesy tiie tube is not truly cylindrical, but slightly conical,* so that if the telescope is tamed in its Ts the bubble will leave the center, even when the horizontal a^just- ment is correct. It is known to be correct, in such tubes, if the bubble runs the Kune diikmce from the center when swung out the same distance on each side. Having made the horizontal adjustment, turn the telescope back in its Ys until the bubble-tube is over the bar. Bepeat the vertUxU adjustment (p 308), which may have become deranged in making this horizontal one. Persevere until both adjustments are found to be correct at the same time. To mabe tibe tliird adjustment, or to a4just the heights of th« Ts, m ■s to make the line of coUimation parallel to the bar V F, or perp to the vert axis of the instrument. The other adjustments being made, fasten down the clips of the Ts. Make the instrument nearly level by means of all four of the levelling-screws K. Place the telescope over two of the levelling-screws which stand diagonally; and leave it there undamped. Then bring the bubble to the center of its tube, by the two levelling-screws. Swing the upper part of the instrument half-way around, BO that the telescope shall again stand over the same two screws; but end for end. This done, if the bubble leaves the center, bring it half-way back by the large cap- stan nuts to, 10 ; and the other half by the two levelling-screws. Remember that to raise the T, and the end of the bubble over «o, io, the lower tv must be loosened ; and the upper one tightened ; and vice versa. Now place the telescope over the pttier diagonal pair of levelling-screws; and repeat the whole operation with them, ilav- Ing completed it, again try with the first pair; and so keep on until the bubble re- mains at the center of its tube, in every position of the telescope. Correct levelling may be performed even if all the foregoing adjustments are out of order; provided each fore-sight he taken at preeiidy the tame distance from the instrument as the back-sight is. But a good leveller will keep his instrument always in acyustment; and will test the ac^ustments at least once a day when at work. As much, however, depends upon the rodman, or target-man, as upon the leveller. A rod- man who is careless about holding the rod vert, or about reading the sights correctly, ibould he discharged without mercy. The levelling-screws in many instruments become very hard to turn if dirty. Clean with water and a tooth-brush. Use no oil on field instruments. Forma for level note-books. When the distance is short, so as not to fsqnire two sets of books, the following is perhaps as good as any. I 8^olI.'S£tU^".,.| »»• |l*"l.|«»«««.| Cut. I «IL I Bat on pnblic works generally the original field-books have only the first five cols. After the grades have been determined by means of the profile drawn from these, the re«nlta are placed in another book, which has only the first col and the last four. In both cages, the right-hand page is reserved for memoranda. The writer considers it best, both witii the level and with the transit, to consider the term " Station " to apply to the whole dist between two consecutive stakes; and that its number shall be that vrrftten on the last stake. Thus, with the transit, Station 6 means the dist fin>m stake 5 to stake d; that it has a bearing or ocnirse of so and so; and its length is so and mo. And with the level, Station 6 also means the dist from stake 5 to stake 6; the back-sight for that dist being taken at stake 5, and the Ibre-sight on stake 6; and thait the level, grade, cut, or fill is that at stake 6. The starting-point of the nwej, wbether a stake, or any thing else, we call and mark simply 0. • This defect can be remedied only by removing the tube and inserting a correctly- ihaped one, and this is best done by an instrument-maker ; but correct work can be done in qpite of it, Ihus: Make all the acyustments as nearly correct as possible. Level the instrument. By turning the telescope in its Ts, make the vertical hair coincide with a plumb-line or other vertical line, and make a short continuous knife- Kiatch on the collar nearest the object-glass, and on the adjoining T. Lift the tele- Kope ont of its Ts, turn it end for end, replace it in its Ts ; again bring the upright hair vertical, and make on the other T a scratch coinciding with that on the collar. Then, in levelling or in a4justing, always see that the scratch on the collar coincides Mitt thai on the ac^oining T when the bubble-tube is under the telescope. THB HAKD-IiBVKL TOE BASD-LETEI. ffll. M arpuTged bj Prof€«ir Locke, of (;indlDll»U,l« SLmpljhuKIl .IR it in DM hind, u idlookinethroBgh 'nd!^TinVni^1 BDd Ihe oW«t . ebotlomof-hlA ™,ghtl,.top KO.'^mmrftottl^ .»1dopcniog..nd for sijurtizig tie «irs, (an be 1 loFhed hKkwird a ir poahod fom •»rdby»™al1>pri p1»c«J at so 1 iiglBo''HS=,«M^ ■h the f.>rfB.B. nlioned DpBniogB, lil'y* Jrf fa^ a^^ f. M shown I.) ' tho^nlle dotl^' linMCMdK; Mid nWUi of™h8 tnbe rTi. Throup?^' (b« wire shull Btaoir no piinllu ; bat ■pp**' tHd; BCBinM the dIi)boI irEui Ibe <J« la allghll; moTod Dp or don. At «ch and oT t)i» tube B O la ■ dmdu pl«oB of To adlaat tbe bond-level, lint fli &U fbet Id 100 J'ards ^art, 'nitB beiDg done« ] level marlij. ud take atght a the oUisr. If, then, tbe wire does not appau- aleht a the oUisr. If, then, tbe wire does not appau- to be ■ illghtly huckwiLrt or forward, M the hand-level tWelf, eieii If i[ la onUrely od( o( odjoil- ^ "^ f nhlecL u d. an that tha wire aoneui to cnl the eenter ro''"iVhVr°CMermito"ri^rk"f^"h»l1--w'ay"to\wee"c°i^^^^ Then (> und in will be Che two iBTel mirkB reijuirod. With o»re, these adjualnimti, when once msdo, will remain in ordet for ream. The Intlrumenl gsnenllyhas aBmall ring r, for hanging eiplorlng a roule. The heigh 1 of « bar* iiil I can be found bybeelnning st the ftiot. and ijgtiling aheed at anj little chance objei^t which the onm-wlre ma; Btrlka, ■• a pebble, cnlg, Ic; then going fonrard, ataud at Ibat object, and fix Che win m a height eqnal 10 thac of the eye, lay bK^ feet, or whateTar it may be, WheUier going DP or down It, If the bill la coTered with grau, bnihea, te, a target rod moR be need for the fore-aighw ; and the tonstant height of the eye may be reganlsdH IiBTXU. 311 To adiast a bailder*s plnmb- leTei, todi stand it npon any two sup- borta «» and it, and mark where the plumb- line cuts at o. Then reverse It, placing the foot t upon n, and d upon m, and mark where the line now cute at e. Half-way between o and e make the permuient mark. Whenerer the line cuts this, the fiaet t and d are on a level. To adjast a slope-lnstrament, or clinometer. As usually made, the bubble-tube is attached to the movable bar by a screw near each end^ and the head of one of the screws conceals a small slot in the bar, which allows a slight vert motion to the scr^w when loose, and with it to that end of the tube. Therefore, in order to adjust the bubble, this screw is first loosened a little, and then moved up «r down a trifle, as may be reqd. It is then tightened again. 312 ZJSVBLLING BY THB BABOMETEB. liETEIililire BT THE BAROIHETER. 1. Many drcnmstancM combine to render the results of this kind of WTellino^ no* reliable where great accuracy is required. This fact was most concluslyely proved by the observations made by Captain T. J. Cram, of the U. 8. Coast Stirvey. See Beport of U. 3. C. S., toI. for 1864. It is difficult to read oiT from an aneroid (the kind of barom generally employed for engineering purposes) to within from two to five or six ft, depending on its size. The moisture or dryness of the air aflTects the results; also winds, the ricinity of mountains, and the daily atmospheric tides, which cause incesHant and irregular fluctuations in the barom. A barom hanging quietly in a room will often vary -^jf of an inch within a few hours, corresponding to a diff of elev.ition of nearly 100 ft. No formula can posiiibly be deyised that shall •mbrace these sources of error. The variations dependent upon temperature, latir tnde, Ac, are in some measure provided for; so that with very ddicate instruments, • skilful observbr may measure the diff of altitude of two points dose together, such as the bottom and top of a steeple, with a tolerable confidence that he is within two or three feet of the truth. But if as short an interval as even a few hours elapses between his two observations, such changes may occur in the condition of the atmo- sphere that he may make the top of the steeple to be lower than its bottom ; or at least, cannot feel by any means certain that he is not ten or twenty ft in error; and this may occur without any perceptible change in the atmosphere. Whenever prac- ticable, therefore, there should be a person at each station, to observe at both points at the same time. Single observations at points many miles apart, and made on dif- ferent days, and in different states of the atmosphere, are of little value. In such cases the mean of many observations* extending over several days, weeks, or months, and made when the air is apparently undisturbeid, will give tolerable approximAtionB to the truth. In the tropics the rang^ of the atmospheric pres is much leas than in other regions, seldom exceeding ^ inch at any one spot; also more regular in time, and, therefore, less productive oferror. Still, the barometer, especially eitiier the aneroid, or Bourdon^s metallic, may be rendered highly useftil to the civil engi- neer, in cases where great accuracy is not demanded. By hurrying from point tO point, and especially by repeating, he can form a Judgment as to which of two sum- mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying party, may materially lessen their labors, especially in a rough country, by select- ing the general route for them in advance. The accounts of the agreement within a few inches, in the measurements of high mountains, by diff observers, at diff periods ; and those of ascertaining accurately the grades of a railroad, by means of an aneroid, while riding in a car, will be believed by those only who are ignonmt of the subject. Such results can happen only by chance. When possible, the observations at different places should be taken at the same time of day, as some check upon the effects of the daily atmospheric tides ; and In very important cases, a memorandum should be made of the year, month, day, and hour, as well as of the state of the weather, direction of the wind, latitude of the place, Ac, to be referred to an expert, if necessary. The effecto of latitade are not included in any of our formulas. When reqd they may be found in the table page 814. Several other corrections must be made when great accuracy is aimed at ; Dut they require extensive tables. In rapid railroad exploring, however, such refinements may be neglected, Inas- much as no approach to such accuracy is to be expected ; but on the contrary, errors 01 from 1 to 10 or more feet in 100 of he^ht, wul frequently occur. As a very roa§rli avera^r® ^^ iQ^y assume that the barometer falls -J^ inch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But in fact its rate of tall decreases continually as we rise ; so that at one mile high it fiEdls ^ inch for about 106 ft rise. Table 2 shows the true rate. JLEVSLLING BY THE BABOM£T£B. 813 To «aeert«in tlie dUT of lieiirbt belweew two points. Jlcn^E 1. Take readings of the barom and therm (Fah) in tlie siiade at both stations. Add together the two readings of the barom, and div their sum bj 2, for their mean ; which call b. Do the same with the two readings of the thermom,*and call the mean t. Subtract the least reading of the barom from the greatest ; and call the diff d. Then mult together this diff d; the number from the next Tablt: No. 1, opposite ( ; and the constant number 30. Div the prod by b. Or Height Diff (d) of ^ Tabular number opposite v, n«„„*..„* on in feet "^ barom ^ mean (f) of thermom X constant du . mean (b) of barom. ExAMPLi. Beading of the barom at lower station, 26.64 ins ; and at the upper sta 20.82 ins. Thermom at lowest sta, 70^; at upper sta, 4^. What is the diff io height of the two stations? Here, Sarom, 26.64 Therm, 70^ " 20.82 *• iOP — — Also^ — — 2)47.46 2)110 23.78 mean of bar, or b. 669 mean of therm, or t. The tabular number opposite 66°, is 917.2. Bar. Bar. Again, 26.64 — 20.82 = 5.82, diff of bar ; or d. Hence, d. Tab No. Con. Height _ 5.82 X 917 Si X 30 _ 160143.12 ^^^ ^ ^^^,^ in feet 23.73 (or 6) "*" 23.73 Then oorrect for latitude, if more aooaracy is reqd, by rule on next page. mie screw at tlie baekof an aneroid Is for adjusting the index by a stand- ard barom. After this has been done it must by no means be meddled with. In some instruments specially made to order with that intention, this screw may bo used also for turning the index back, after having risen to an elevation so great that the index has reached the extreme limit of the graduated arc. After thus turning it back, the indications of the index at greater heights must be added to that at- tained when it was turned back. TABIiB 1. For Rale 1. Mean Mmd Mean Mean • of No. of No. of No. of No. Ther. Ther. Ther. Ther. oo 801.1 80° 864.4 60O 927.7 90O 991.0 1 803.2 31 866.6 61 929.8 91 993.1 3 805.3 32 868.6 62 981.9 92 995.2 S 807.4 38 870.7 63 934.0 98 997.3 4 809.6 84 872.8 64 936.1 94 999.4 6 811.7 86 874.9 66 938.2 95 1001.6 . « 818.8 36 817.0 66 940.3 96 1003.7 7 815.9 87 879.2 67 942.4 97 1005.8 8 818.0 38 881.8 68 944.6 98 1007.9 9 820.1 80 883.4 69 946.7 96 1010.0 10 822.2 40 886.4 70 948.8 100 1012.1 11 824.3 41 887.6 71 950.9 101 1014.2 12 826.4 42 869.6 72 953.0 102 1016.3 13 828.5 48 891.7 73 955.1 103 1018.4 li 880.6 44 893.8 74 967.2 104 1020.5 16 833.8 46 896.0 76 969.3 105 1022.7 16 834.9 46 898.1 76 961.4 106 10i4.8 17 887.0 47 900.2 n 968.6 107 1026.9 18 889.1 48 902.3 78 965.6 108 1029.0 1» 8«1.3 49 904.6 79 967.7 lOB 1031.1 20 84SJI 60 906.6 80 860.9 110 ia'M.2 21 8A5.4 61 908.7 81 972.0 111 1035.3 23 847.6 63 910.8 82 974.1 112 1037.4 28 848.6 63 913.0 83 976.2 118 1039.5 3i 861.8 64 916.1 84 978.3 lU 1041.6 25 853.9 66 917.2 86 980.4 116 1043.8 96 8G6.0 66 919.3 86 982.6 116 1045.9 27 868.1 67 921.4 87 964.7 117 1048.0 28 800.2 68 923.6 88 966.8 118 1050.1 » 863.8 69 925.6 89 988.9 119 1052.2 314 LEVSLLINO BT THE BAROMETEB. RuLi 2. BelTlUe's short approx rale is the one beit adapted to rapid Aeld use, namely, add together the two readings of the barom only. Also find the diir between said two readings; then, as tbe sam of the two readlnffs is to tbelr dlff, so Is 55000 feet to the reqd altitude. <3orreetion for latitude is usually omitted where great accuracy is not required. To apply it, first find the altitude by the rule, as before. Then divide it by the number in the following table opposite the latitude of the place. (If the two places are in different latitudes, use their mean.) Add the quotient to the altitude if the latitude is leea than 45°. Subtract it if the Utitude is more than 45°. No cor- rection required for latitude 45°. Table of corrections for latitude. Lat. Lat. Lat. Lat. Lat. Lat. 0° S52 14° 890 280 630 420 8867 640 1140 680 490 a S54 16 416 80 706 44 10101 66 941 70 460 4 856 18 486 82 . 804 46 00 68 804 72 486 6 860 ao 460 U 941 46 10101 60 705 74 416 8 867 22 490 86 1140 48 8867 62 680 76 990 10 8T5 M 527 88 1468 60 9028 64 572 78 886 IS 886 26 672 40 9038 1 63 1458 66 527 80 876 lieTCllins by Barometer; or bjr the bollini^ point. Rule 3. The following table. No. 2, enables us to measure heights either by means of boiling water, or by the barom. The third column shows the approximate alti- tude above sea-level corresponding to diif heights, or readings of the barom ; and to the diif degrees of Fahrenheit's thermom,at which water boils in the open air. Thus when the barom, under undisturbed conditions of the atmosphere, stands at 24.08 inches, or when pure rain or distilled water boils at the t«mp of 201° Fah ; the place is about 5764 ft above the level of the sea, as shown by the table. It is therefore rery easy to find the diffoi altitude of two places. Thus : take out from table No 2, the altitudes opposite to the two boiling temperatures ; or to the two barom readings. Subtract the one opposite the lower reading, from that opposite tbe upper reading. The rem will be the reqd height, as a rough approximation. To correct this, add together the two therm readings ; and div the sum by 2, for their mean. From teble for temperature, p 816, take out the number opposite this mean. Mult the ap- proximate height just found, by this tabular number. Then correct for lat if reqd. Ex. The same as preceding ; namely, barom at lower sta, 26.64 ; and at npper ata, 20.82. Thermom at lower sta, 70° Fnh ; and at the npper one, 40°. What is the diff of height of the two stations ? Alt. Here the tabular altitudes are, for 20.82 9579 and for 26.64 3115 To correct this, we have 70° + 40° 110° 6464 ft, approx height. . 65° mean ; and in table p 816, opp to 55°, we find 1.048. Therefore 6464 X 1.048 = 6774 ft, the reqd height. This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft In each 100 ft. At 70° Fah, pure water will boil at 1° less of temp, for an average of about 660 ft of elevation above sea-level, up to a height of U a mile. At the height of 1 mile, V* of boiling temp will correspond to about 560 ft of elevation. In table p315 the mean of the temps at the two stations is assumed to be 32° Fah ; at which no correc- tion for temp is necessary in using the table ; hence the tabular number opposite 32°, in table p 316, is 1. This diff produced in the temp of the hailing pointy by change of elevation, most not be confounded with that of the atmotpherej due to the same cause. The air be- comes cooler as we ascend above sea-level, at the rate (very roughly) of about 1^ Fah for every 200 ft near sea-level, to 350 ft at the height of 1 mile. The followingr table, "So. 2, (so tar as it relates to the barom^ was da^ dncnd by the wnter from the standard worU on the barom 'by Lieut.-Ool. R. S. Wil- liamson, U. S. army."* • FablUbed by penaiMton of OoTernmeni In 1868 by Vao Koetraod. N. T- lAVELLINQ BT THE BABOKBTEB, ETC. 315 TABI.E 9. I.«ivellliifc by Bfkrometer ; or by the bnllliift p»liil. imed templn theebide 32° Full. JI pot S2°, mult harnni sk us per TBbIe,p 316 SOUND. Corre«il«iis f«r temperatare; to be used in eonnecUon wltb Bule 3, wlien irreater aecuracy is necessary. Also in con- nection witli TaMe 2 wlien tlie temp is not 33°. Mean • Mean Mean Mean * temp Malt temp Mult temp Mnlt temp Mult in the by in the by In the by in the by shade. shade. » ihade. shade. Zero. .933 28° .992 5«o 1.050 84° 1.108 20 .937 30 .996 68 1.064 86 1.112 4 .942 32 1.000 60 1.058 88 1.117 6 .946 34 1.004 62 1.062 90 1.121 8 .960 36 1.008 64 1.066 92 1.126 10 .954 38 1.012 66 1.071 94 1.129 12 .958 40 1.016 68 1.076 96 1.133 14 .962 42 1.020 70 1.079 98 1.138 16 .967 44 1.024 72 1.083 100 1.142 18 .971 46 1.028 74 1.087 102 1.146 20 .976 48 1.032 76 1.091 104 1.150 22 .979 60 1.036 78 1.096 1U6 1.154 U .983 62 1.041 80 1.100 108 1.168 » .987 64 1.046 82 1.104 110 1.163 SOUND. u — 20° M 1040 « — 10° u 1060 it u 1060 it 10° « 1070 U 20° u 1080 M • 32° u 1092 « 40° u 1100 M 50° u 1110 t( 60° it 1120 H 70° M 1130 U 80° U 1140 (« 90° U 1160 K 100° t( 1160 M 110° <( 1170 « 120° a 1180 (( (( t: u u it It It it t( M U M tt It tt tt U tt tt 4( <t tt it it tt tt U tt u M (« « tt «. 1 tt 6.08 .. 1 u 5.03 ■B 1 tt 4.98 *■ 1 « 4.93 ^ 1 (( 4.8S IBS X u 4.83 ■> I « 4.80 ^ 1 « 4.78 ^ 1 H 4.73 m^ 1 U 4.68 m= 1 It 4.63 *B 1 l( 4.69 ■B X u 4.65 IM \ tt 4.61 — 1 tt 4.47 « (« -reloeitjr at sound in quiet open air, haa been experimentally deter> mined to be very approximately 1090 feet per second, when the temperature is at freezing point, or 32° Fahienheit. For every degree Fahrenheit uf increase of temperature, the velocity increases by from V^ foot to 1^ feet per second, according to different authorities. Taking the iucreasu at 1 foot per second for each degree (which agrjBes closely with theoretical calculations), we have at ^ 30° Fahr 1030 feet per sec '^ 0.1951 mile per sec — 1 mile in 6.13 seconds. — 0.1970 — 0.1989 — 0.2008 » 0.2027 — 0.2045 — 0.2068 — 0.2083 — 0.2102 — 0.2121 — 0.2140 — 0.2169 — 0.2178 — 0.2197 — 0.2216 — 0.2236 If the air is calm, fog or rain does not appreciably affect the retult ; but wisds do. Very loud sounds appear to travel somewhat faster than low ones. The watchword of sentinels has been heard across still water, on a calm night, 10^ miles ; and a cannon 20 miles. Separate sounds, at intsrvals of ^ of a second, cannot be distin- guished, but appear to be connected. The distances at whieh a speaker can be understood, in front, on one side, and behiud him, are about ab 4, S, and 1. Dr. Charles M. Cresson informs the writer tliat, by repeated trials, he found that in a Philadelphia gas main 20 inches diameter and 16000 feet long, laid and covered in the earth, but empty of gas, and having one horizontal bend of 90^, and of 40 fast reuUus, the sound of a pistol-shot travelled 16000 feet in precisely 16 seconds, or 1000 feet per second. The arrival of the sound was barely audible ; but was rendered very apparent to the eye by its blowing off a diaphragm of tissue-paper placed over the end of the main. Turo bosits anchored some distance apart may serve as a base line for triangulating objects along the coast; the distance between them being first found by firing guns on board one of them. In ivater tliie velocity is about 4708 feet pef second, or about 4 times that in air. In iwroodsy it is from 10 to 16 times ; and in metalSf fh>m 4 to 10 times greater than in air, according to some authorities. w t« g!^;.'.: Eaeb 13^ M IS" of bekt prodncaln wr*t Ir^ i °* ^ I"" '° "" '^t^' Id Iki no «!• niv U(>. n IM Iha I«(Ilu i , „ ezpsnalon of HtVD* wUlfmi* TinM mcRlnS points «re qnlM lUiccrtatM. W« ^n (be miu of •atanwIborMH. iMoWlii tUiMlJi »(aiMjtor Bnool^tioiit l«l»,»t«|Mlro» "HlilHriH ■aiBnUHT wUh Uu H« olulsiuil • «'n n» will diiim lu l>ii|lk [au? of in Iniik. THEBUOIfETEBS. T« «liBnc« derreea of Fitlirenbelt 1« Ike eorrMipOBdIns de- ■re«a •rc«ntl)?«de) l&kBiir>)irBidliiK32°liivn-ihitnih« ilit|aoiis: mnlt — lD°IMiit Agali,— 190F>b = r— II— a])>cCi-t = — •&XC'i-?=— l^Oaal. ~ ~ To cli»iiKe P>h taMBOi uks & Fati rudlug 32^ Iswrthu ihs (Ina • ti^-eoBi.o. linln,— IPlVlisI— ls-ijfx'*+»='--«SXt-H'=— *1°B*M. ToeliAnce £«ntto l'nb|niiill ihe Oni nwllDg b> 0; dirlds by ! eIlAn» O 5l*i>^'=r^^?"° >. Tkkaft i^taii.—tfOmn=i--xx'-T6>+ai=—*''r^. shannlMml to Fohri- niiu:Tir'i«i™=("X9*«)+»t ,^'-4i« '^^oekanreBtenMCentiiDnltbyH; div by 4. Thna: -fB°R«u — + 8° TABI>E1> FHtarenhelteomiMredwItliCeiitlBr^deaiidK^a* THERHOUETERS. Zl^ TABLE 3. CcntlrnMie eom|»a>«d with rahrcnbclt a>« C F. K. C. F. R. C. F. I K. C F. R. TABLE S. Ktaaoiar coiapitrfld wltb FabMiBbelt luul tlTradc. K. F. C. B. F. C. R. F. C. R. F. C. M III.M KnOO 4< 4119 tl.iS I* TS Jt.n —1 1.16 — I1.1S II M!ffi K.a M ullS SI.M It 00 wiot — < O^UI — IiIm ig 303.01 (MM 11 lU.ie M.it IS lb e.11 — s —1.75 — i8.7e Ts via.Ti 0.11 H 9ij» u.m 11 u 1.U — s -4.in _».og TS IK.r, •l!«i •> MM i'lUI II M s!lM - 8 -H 60 -h'm II I "" — SLIl (U n.oi> uiloo ID M LU — » — isigo ~ia!oi> a Bi.n 91 ot!eo iiiw i ti.ui l.oo ~h ~ie'^ -si'm a i nils n mIn leM i km i.m - la ~-9i.oo — SSM « 17.50 II 01,11 a.ib I U.1S ].^ —a -u.a -x.u, 11 iZn MM 10 J7!oo »!oo -10 g'so '\-ii.sb -« -».a> -ooiot 320 Aia AIR-ATMOSPHERE. The atmospliere is known to extend to at least 4S miles abore the earth. It is a mixture of about 79 measures of nitrogen gas and 21 of oxygen gas ; or about 77 nitrogen, 23 oxygen, br weight. It generallr con- tains, however, a trace of water, and of carbonic acid and carbu retted hydrogen gaaes, and still less ammonia. Density of air. Under *' normal ** or " standard " conditions (sea level, lat 45^, barometer 760 mm => 29.922 ins, temperature O^C^ZTP F) dry air weljirhs 1.292673 kilograms per cubic meter * = 2.17888 fi>s avoir per cubic yard. For other lats and elevations — Density, in kg per cu m, =i 1.292673 X j^^^A ^ ^^ —0.002837 oos 2 lat) • where B = earth's mean radius =» 6,366,198 meters ; A >« eleTation above aea level, in meters. For other temperatures, see below. Under normal conditions, but with 0.04 parts carbonic acid (0 O,) in 100 parts of air, density = 1.293052 kg per cu m.f » 2.17952 fi» avoir per cu yd.^ The atmospherie pressure, at any given place, may yarr 2 inches or more from day to day. 'rhe averagr® pressure, at sea level ^ varies from about 745 to 770 millimeters of mercury according to the latitude and locality. 760 millimeters * is generally accepted as the mean atmospheric pressure, and called an atmosphere. The '* metrie atmosphere,** taken arbitrarily at 1 kilogram per square centimeter, is in general use in Continental Europe. The pressure diminishes as the altitude increases.f Therefore, a pump in a high region will not lift water to as great a height as in a low one. The pressure of air, like that of water, is, at any given point, equal in all directions. It is often stated that the temperature of the atmosphere lowers at the rate of 1<^ Fah for each 300 feet of ascent above the earth's snrfhees but this is liable to many exceptions, and varies much with local causes. Actual observation in balloons seems to show that, up to the first 1000 feet, 1^ in aboat 200 feet is nearer the truth ; at 2000 feet, 1° in 250 feet ; at 4000 feet, 1° in 300 feet; and, at a mile, 1° in 350 feet. In breathingr, a grown person at rest requires from 0.25^ to 0.35 of a cubic foot of air per minute : which, when breathed, vitiates from 8.5 to 5 cubic feet. When walking, or hard at work, he breathes and vitiates two or three times as much. About 5 cubic feet of fresh air per person per minute are required for the perfect ventilation of rooms in winter; 8 in summer. Hospitals M to 80. Beneath the ipeneral level of the surface of the earth, in temperate regions, a tolerably uniform temperature of about 50° to 60^ Fah exists at the depth of about 50 to 60 feet ; and inereases about 1° for each additional 50 to 60 feet ; all subject, however, to considerable deviations owing to many local causes. In the Rose Bridge Colliery, England, at the depth of 2424 feet, the temperature of the coal is 93.5° Fah ; and at the bottom of a boring 4169 feet d eep, near Berlin, the temperature is 119°. The air is a werjr slow eondnetor of heat; hence hollow walls serre to retain the heat in dwellings ; besides keeping them dry. It mahea into a waeunm near sea level with a velocity of about 1157 feet per second ; or 13.8 miles per minute ; or about as fast as sound ordinarily travels through quiet air. See Sound. ^ Iiike all other elastie fluids, air expands eoually witik e^ual increases of temperature. Every increase of o° Fah, expands the bulk of any of them slightly more than 1 per cent of that which it has at 0^ Fah ; or 500° about doubles its bulk at xero. The bulk of anv of them diminishes inversely in proportion to the total pressure to which it is subjected. This holds good with air at least up to pressures of about 750 fte per sqnare inch, or 50 times its natural pressure ; the air in this case occupying one-flxtietii of its natural bulk. In like manner the bulk will increase as the total preasuiv is diminished. Substances which follow these laws, are said to be perCeetiy * H. V. Regnault, M6moires de 1* Acaddmie Royale des Sciences de Plnstitiit de France, Tome XXI, 1847. Translation in abstract. Journal Of Franklin Insti- tute, Phila., June, 1848. fTravaux et M6moire8 du Bureau International.desPoidset Mesnres, Tomel £age A 54. Smithsonian Meteorological Tables, 1898, publiabed In Smithsooian [iscellaneous Collections, Vol. XXXV, 1897. I See Conversion Tables. f See Leveling by the Barometer. WIND. 321 1 elAstle. Under apressure of about 6^ tons persqiiaie Indi, air would become as dense as wa^er. Since the air at the surface of the earth is pressed 14^ !ba per square inch by the.atmosphere above it, and since this is equal to the we^ht of a oolumn of water 1 inch square and 34 feet high, it follows that at the depths of 84, 68, 102 feet, &4i, below water, air will be compressed into ^, 3^, 3^ Ac, 01 its bulk at the surface. In a divliiK-bell, men, after some experience, can readily work for seyeral hours at a depth of 51 feet, or under a pressure of 2^ atmospheres ; or 37^ ftis Kir square inch. But at 90 feet deep, or under 3.64 atmospheres, or nearly 55 8 per square inch, they can work for but about an hour, without serious suffer^ ing from paralvsis. or even danger of death. Still, at the St Louis bridge, work was done at a deptn of 1103>^ feet ; pressure 63.7 9>8 per square inch. The dew point is that temp (varying) at which the air deposits its vapor. Tlie gnreatest beat of tlie air in the sun probably never exeeeds 145° Fah J nor the greatest cold — 74P at u ight. About 130° above, and 40° below zero, are the extremes in the U. S. east of the Mississippi ; and 65^ below in the N. W.; all at common ground level. It is stated, however, that —81° has been observed in N. E. Siberia: and +10lo Fah in the shade in Paris; and +153° in the sun at Greenwich Observatory, both in July, 1881. It has frequently ex- 'beaded -i-l(XP Fah in the shade in Philadelphia during recent years. WIND. The relation between the weloeity of wind, and its preas* lire against an obstacle placed either at right angles to its course, or inclined to it, has not been well determined ; and still less so its pressure against curved surfaces. The pressure against a laige surface is probably proportionally greater than gainst a small one. It is generally supposed to vary nearly as the squares of the velocities; and when the obstacle is at right angles to its direction, the Sressure in lbs per square foot of exposed surface is considered to be equal to lie square jof the velocity in miles per hour, divided by 200. On this basis, which is probably quite aefective, the following table, as given by Smeaton, is prepared. YeiL in MUes Vei. m Ft. Frea. in Lbs. Remarks. per Hour. per Sec. per Sq. Ft. 1 1.467 .005 Hardlj perceptible. ^..^^ PleMsnt. ^C~J>g s 2.933 .020 8 4.400 .045 ^ 4 . 5.867 .OBO ^ 5 10 7.38 14.67 .125 .5 zJo/rt 12H 18.S3 .781 Fresh breexe. O lb n. 1.125 20 S9.33 8. ^ . Th« prei acainit 25 86.67 3.125 Brlakwind. « iiemioylindrioal so 44. 4.5 Strong wind. sarfftoe ac&nom 40 S6.67 8. High wind. ig about half that 60 73.88 12.5 Storm. against the flat 60 88. 18. Violent storm. gnrf abnni. SO 117.3 32. Hurricane. 100 146.7 60. Violent hunieane, uprooting large trees. TreddTOld reeommends to allow 40 lbs per sq ft of roof for the pras of wind against it ; but aa roob are oonstruoted with a slope, and oonsequentty do not receive <ke ftill foree or the wind, this is plainly too much.* Moreover, only one>half of a roof is usually ex- I, even thas partially, to the wind. Probably the force in suoh cases varies approximately as the of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per prodmsed a pre* of 14 lbs per sq ft againut an object perp to it: and one of 70 miles, per hour, (the Mvterect gale on reoord at that city.) 43 lbs per sq foot. These would make the ores per sq ft, More nearly equal to the ■qoAre of the vel iq miles per hour, dlv by 100 ; or nearly twice as great as glvea in Smaaton's table, we should ourselves give the preference to the Liverpool observations. A very violent gale in Scotland, registered by an excellent anemometer, or wind-gauge, 45 lbs per sq ft. It la stated that aa high as 55 lbs has been observed at Glasgow. High winds often l^ roots. The gaoge at Oirard Coliese, Fhilada, broke onder a strain of 43 lbs per sq ft ; a tornado passing St the moment, within a mils. By inrersion of SoMaton's rule, if the force in Iba per sq ft, be mult by 200, the sq rt of the prod Igive the vel in milec per hoar. Smeaton's rule is used by the U. S. Signal Service. «i/c • The writer thinks 8 lbs per sq foot of mrdinarn doubte-aloping roofi, or 10 lbs for •Ked-rooft, suffl ■ «imt allowanee for prea of wind. 21 322 RAIN AND SNOW. RAIN AND SNOW. The annaal preelpitatlon * at any giyen place varies greatly from year to year, the ratio between maximum and minimum being frequently greater than 2 : 1. Beware of averai^es. In estimating ^oo^«, take the maximum falls, and in estimating water supply, the mtnimttm, not only per annum, but for short periods. In estimating water supply, make deductions for evaporatios and leakage. Maxima and minima deduced fh>m observations covering only 4 or 5 years are apt to be misleading. Data covering even 10 or more years may just miss includ- ing a very severe flood or drought. Becords of from 15 to 20 years may usually be accepted as sufficient. Table 1. Averafre Preelpltatlon * In tbe United States, in ins. (Frmn Bulletin C of U. S. Department of Agriculture, compiled to end of 1891.) Steto. Spr. Alabama 14.9 Ariisona 1.3 Arkansas 14.8 California. 6.2 Colorado 42 Connecticut 11.1 Delaware 10.2 Dist. Columbia.11.0 Florida 10.2 Georgia 12.4 Idaho 4.4 Illinois 10.2 Indiana 11.0 Indian T'y 10.6 Iowa 8.3 Kansas 8.9 Kentucky 12.4 Louisiana 13.7 Maine 11.1 Maryland 11.4 Massachusetts. ..11.6 Michigan 7.9 Minnesota 6.5 Mississippi 14.9 Missouri 10.0 8am. Aat. Win. Atxn'l 13.8 10.0 149 53.6 43 2.2 3.1 10.9 12.5 11.0 12.8 50.6 0.3 3.5 11.9 21.9 5.5 2.8 2.3 148 12.5 11.7 11.5 46.8 11.0 10.0 9.6 40.8 12.4 9.4 9.0 41.8 21.4 14.2 9.1 549 15.6 10.7 12.7 51.4 2.1 3.6 7.0 17.1 11.2 9.0 7.7 38.1 11.7 9.7 10.3 42.7 11.0 8.9 6.7 36.2 12.4 8.1 41 32.9 11.9 6.7 3.5 31.0 12.5 9.7 11.8 46.4 15.0 10.8 144 53.9 10.5 12.3 11.1 45.0 12.4 10.7 9.5 440 11.4 11.9 11.7 46.6 9.7 9.2 7.0 83.8 10.8 5.8 8.1 26.2 12.6 10.1 15.4 53.0 12.4 9.1 6.5 38.0 SUte. Spr. Sum. Aat. Win. Annl Montana 4.2 Nebraska S.9 Nevada 2.3 N. Hampshire. 9.8 New Jersey 11.7 New Mexico..... 1.4 New York 8.5 N. Carolina 12.9 N. Dakota 46 Ohio 10.0 Oregon 9.8 Pennsylvania...l0.3 Rhode Island. ..11.9 S. Carolina 9.8 S. Dakota 7.2 Tennessee 18.6 Texas 8.1 Utah 3.4 Vermont 9.2 Virginia 10.9 Washington 8.6 W. Virginia 10.9 Wisconsin 7.8 Wyoming 4.8 United States... 9.2 49 2.6 2.8 140 10.9 49 2.2 26.9 0.8 1.3 3.2 7.6 12.2 11.4 10.7 44.1 13.3 11.2 11.1 47.8 5.8 8.5 2.0 12.7 10.4 9.7 7.9 86.5 16.6 12.0 12.2 68.7 8.0 2.8 1.7 17.1 11.9 9.0 9.1 40.0 2.7 10.5 21.0 440 12.7 10.0 ».6 42.6 10.7 11.7 12.4 46.7 16.2 9.7 9.7 46.4 9.7 8.5 2.5 22.9 12.5 10.2 145 60.7 8.6 7.6 6.0 80.3 1.5 2.2 8.5 lao 12.2 11.4 9.8 42.1 12.5 9.5 9.7 42.6 3.9 10.5 16.8 89.8 12.9 9.0 10.0 42.8 11.6 7.8 6.2 82.6 8.5 2.2 1.6 11.0 10.3 8.3 8.6 80.8 At Philadelphia, in 1869, during which occurred the greatest drought known there for at least 50 years, 43.21 inches fell ; August 13, 1873, 7.3 inches in 1 day ; August, 1867, 15.8 inches in 1 month ; July, 1842, 6 inches in 2 hours ; 9 inches per month not more than 7 or 8 times in 25 years. From 1825 to 1893, greatest in one year, 61 inches, in 1867 \ least, 30 inches, in 1826 and 1880. At Norristown, Pennsylvania, in 1865, the writer ^aw evidence that at least 9 inches fell within 5 hours. At Genoa, Italy, on one occasion, 32 inches fell in 24 hours ; at Geneva, Switzerland, 6 inches in 3 hours ; at Marseilles, France, 13 inches in 14 hoars; in Chicago, Sept., 1878, .97 inch in 7 minutes. Near iJondon, Eng^land, the mean total fall for many years is 28 inches. On one occasion, 6 inches fell in 1% hours! In the mountain districts of the English lakes, the fall is enormous: reaching in some years to 180 or 240 inches; or from 15 to 20 feet ! while, in tne adjacent neighborhood, it is but 40 to 00 inches. At Liverpool, the average is 34 inches ; at Ckiinburgh, 30 : Glasgow, 22; Ireland, 36; Madras, 47; Calcutta, 60; maximum for 16 years, 82; Delhi, Si; Gibraltar, 80 ; Adelaide, Australia, 23 ; West Indies, 36 to 96 ; Rome, 89. On the Khassya hills north of Calcutta, 500 inches, or 41 feet 8 inches, have Allien in the 6 rainy months I In other mountainous districts of India, annual falls of 10 to 20 feet are common. A moderate steady rain , continuing 24 hours, will yield a depth of about an indu As a seneral rule, more rain fhlls in warm tban in 99MA €SonntrIes; and more in elevated regions than in low ones. Local pecuUaxw * Precipitation includes snow, hail, and sleet, melted, estimated at 10 inches snow » 1 inch rain. Unmelted snow ia BAIV AND SNOW. 323 KieB, howerer, sometimeB reyerae this : and also oanse great differences in the amounts in places quite near each other ; as in the English lake districts Just alluded to. It is sometimes difficult to account for these variations. In some lagoons in New Granada, South. America, the writer has known three or four heavy raiiio to occur weekly for some months, during which not a drop fell on hills about 1000 feet high, within ten miles' distance, and within full sight. At another locality, almost a dead-level plain, fully three-quarters of the rains that fell for two years, at a spot two miles from his residence, occurred in the morn- ing ; while those which fell about three miles from it, in an opposite direction, were in the afternoon. Tlie relation between precipitation and stream»0ow is greatly ai^cted by the existence of forests or crops, by the slope and character of ground on the water-shed, especially as to rate of absorption, by the season of the year, the frost in the ground, etc. The stream-flow may ordinarily be taken as vary* ing between 0.2 and 0.8 of the rainfall. Streams in limestone regions frequently loee a very large proportion of their flow through subterranean caverns. Aasnminff a fall of 2 feet in 1 year (=3 76,379 cubic feet per square mile per day), that half the rainfall is available for water supply, and that a per capita consumption of 4 cubic feet (^t 30 gallons) per day is sufficient, one square mile will supply 19,095 persons ; or a square of 88.26 feet on a side will supply one person. Ineb of rain amonnto to 3630 enble fiBet; or 27156 U. SL EkUonB ; or 101.3 tone per acre ; or to 2323200 cubic feet ; or 17378743 U. S. gal^ ns ; or 64821 tons per squ&re mile at 62^^ fts per cubic foot. • ^ The most destructive rains are usualhr those which fall upon snow, nnder which the ground is frozen, so as not to absorb water. Table 2. Kaxlmnm intensify of rainlMl for periods of 5, 10, and 60 minutes at Weather Bureau stations equipped with self-registering gauges, compiled from all available records to the end of 1896. (From Balletin D of U. S. Department of Agriculture.) Stations. Rate per hour for— Stations. Rate per hour for— 6min. lOmins. 60 mins. 6min. 10 mins. 60 mins. Bismarck. Ins. 9.00 8.40 8.16 7.80 7.80 7.50 7.44 7.20 7.20 6.72 6.60 6.60 6.60 Inches. 6.00 6.00 4.86 4.20 6.60 6.10 7.08 6.00 4.92 4.98 6.00 3.90 4.80 Inches. 2.00 1.30 2.18 1.25 2.40 1.78 2.20 2.15 1.60 1.68 2.21 L60 1.86 Chicago Ins. 6.60 6.48 6.00 6.00 5.76 6.64 6.46 5.40 6.40 4.80 4.56 8.60 3.60 Inches. 6.92 6.58 4.80 4.20 6.46 3.66 5.46 4.80 4.02 3.84 4.20 3.30 240 Inches. 1 60 St. Paul Galveston... Omaha 2 55 Kew Orleans 1.65 Milwaukee Dodge City Norfolk 1.84 1 55 Washington Jacksonville Cleveland 'Atlanta. 1.12 1.50 Detroit. Key West Philadelphia... St Louis... Cincinnati Denver... 2.26 1.60 2.25 New York aty> Boston Savannah 1.70 1 18 Indianapolis...... Memphis.. Duluth— 1.35 The welirbt of firesbly flallen snow, as measured by the author, varies from aoont 5 to 12 lbs per cubic foot ; apparently depending chiefly upon the degree of humidity of the air through whicn it had passed. On one occasion, when minsled snow and hail had fallen to the depth of 6 inches, he found its weight to Be 81 fbs per cubic foot. It was very dry and incoherent. A cubic foot of heavy snow mav, by a gentle sprinkling of water, be converted into abont half a cubic foot of slush, weighing 20 9>s.; which will not slide or mn oflf from a shingled roof sloping 30^, if the weather is cold. A cubic block of snow •atorated with water until it weighed 45 Tba per cubic foot, Just slid on a rough board inclined at 45''; on a smoothly planed one at 30^ ; and on slate at 18° : all ipproximate. A prism of snow, saturated to 62 lbs per cubic foot, one inch square, and 4 inches high, bore a weif^t of 7 fi»s ; which at first compressed it abont one-quarter part of its length. European engineers consider 6 n>s per square foot of roof to oe snffielent allowance for the weight of snow; 324 RAIN AND 6NOW. and 8 lbs for the pressare of wind ; total. 14 lbs. The writer thinks that in the U. S. the allowance for snow should not be taken at leu than 12 fi>8 ; or the total for snow and wind, at 20 Bm. There is no danger that snow on a roof will become saturated to the extent Just alluded to ; because a rain that would supply the necessary q^uantity of water would also by its violence wash away the snow ; but we entertain no doubt whatever that the united pressures from snow and wind, in our Northern States, do actually at times reach, and even surpass. 20 fbs per square foot of root The limit of perpetnal snow at the equator is at the height of about 16000 feet, or say 3 miles above sea-level; in lat 45° north or south, It is libout half that neight; while near the poles it is about at searleveL Rain Oaoi^es. Plain cylindrical vessels are ill adapted to service as rain gauges ; because moderate rains, even though sufficient to yield a large run-off from a moderate area, are not of sufficient depth to be satisfactorily measured unless the depth be exaggerated. The inaccuracy of measurement, always con- siderable, is too great relatively to the depth. In its simplest and most usual form, the gauge (see Fig.) consists essentially of a funnel. A, which receives the rain and leads it into a measuring tube, B, of smaller cross-section. The funnel should have a verticci and fairly sharp edge, and, in order to minimize the loss through xA/ evaporation, it should fit closely over the tube, and its lower end ^ diould be of small diameter. ' The depth of water in the tube is ascertained by inserting, to the bottom of the tube, a measuring stick of some unpolished wood which will readily show to what aepth it has been wet. The stick may be permanently graduated, or it may be compared with an ordi- nary scale at each observation. The tube is usually of such diameter that the area of its cross-section, minus that of the btick, is one-tenth of the area of the funnel month. The depth of rainiaU is then one- tenth of the depth as measured by the stick. B DiiCENsiONS OP Standard U. S. Wbathbb Bubbau Bain Gauge. Ins. A. Beceiver or funnel. Diameter 8 B. Measuring tube. Height 20 ins. " 2.53 C C. Overflow attachment and snow gauge. " 9 Such gauges, with the tubes carefully made from seamless drawn brass tubing, «08t about $5.00 each ; but an intelligent and careftil tinsmith, given the dimen- sions accurately, can construct, of galvanized iron, for about ^.00 a gauge that will answer every purpose of the engineer. Tbe exposure has a very marked effect upon the results obtained. The funnel should be elevated about 3 ft, in order to prevent rain from splashing back into it from the ground or roof. If on a roof, the latter shoald be nat, and pref- erably 50 ft wide or wider, and the gause should be placed as far as possible from tbe edges ■ else the air currents, produced by the wind striking the side of the building, will carry some of the rain over the gauge. No objects much higher than the gauge should be near it, as they produce variable air currents which •may seriously affect its indications. An overflow tank, G, should be provided, for cases of overfilling the tube. Water, freezing in the gauge, may burst it, or force the bottom off, or at least <ao deform the gauge as to destroy its accuracy. To measure snow, the funnel is removed, and the snow is collected in the overflow attachment or other cylindrical vessel deep enough to prevent the snow from being blown out, and the cross-sectional area of which is accurately known. The snow is then melted, either by allowing it to stand in a warm place, or, with less loss through evaporation, by adding an accurately known quantity of luke-warm water. In the latter case, the volume of the added water must of course be deducted from tbe measurement. Rainfall equivalent of snow. Ten inches of snow are usually taken as equivalent to 1 in of rain ; but, according to various authorities, the equiva- lent may vary between 2>^ and 34; i. e., between 25 and 1.84 &». per cubic foot. Self-reeordinir g^ngr^s, of which several forms are on the market, are Jiuite expensive, and, even when purchased from regular makers, seldom per- ectly reliable. Gauges using a small tipping bucket register inaccurately la heavy rains ; those using a float are limitea as to the total depth which they c xegister ; while those which weigh,tbe rain, if exposed, are aflbcted by wind. BAIir AND 81I0V. Bulletia Cot U.S. DeparUuent of AftlcaUura, IBM.) •F0riinlaiu)^Abbmmm,UablIa,tiioata0.2Mniih,34.S mauuthBton!4,3per pt. of the dnjA ombrsc^d ivltfain tb* 30 yean, ram fall to a depth of from a tFMaiOiitobarU7Bcail;. t Fnm Juaarr 1S14 oeOj. iFiomliUy ISTi odI;. 326 WATEB. WATER. Pure water, as boiled and distilled, Is eomposed of the tiro gases, hydro- gen and oxygen ; in the proportions of 2 measures hydrogen to 1 of oxygen ; or 1 weight of hydrogen to 8 of oxvgen. Ordinarily, however, it contains sev- erid foreign ingredients, as carbonic and other acids ; and soluble mineral, or organic substances. When it contains mirch lime,- it is said to be h€a^; and will not make a good lather with soap. Tbe air in its ordinary state conlwiiis about 4 grains of water per cubic foot. The average pressure of tlie air at sea level, will balamee a colamn of water 34 feet high ; or about 30 inches of mercury. At its boil- ing point of 212° Fah, its bulk is about one twenty-third greater than at IQP. Its welg^lit per cubic foot is taken at 62^ fi>fl,or 1000 ounces avoir; but 62}^ lbs would be nearer the truth, as per table beh>w. It is about 816 times hearier than air, when both are at the temperature of 62°; and the barometer at 80 inches. With barometer at 30 inches the weight of perfectlv pure water is as follows. At about 39*^ it has its maximum density of 62.425 ros per cubic foot. Temp, Fah. Lbs per Cub Ft. 929 62.417 40° 62.423 50° 62.409 60° 62^7 Temp, Fah. Lbs per Cub Ft. 70° 62.302 80® - 62.218 90°- 62.119 212°- « 69.7 Weifflil; of sea ivater 64.00 to 64.27 B>s per cubic foot, or say 1.6 to 1.9 9>8 per cubic foot more than fresh water. See also p 328. Water has its maxlmnm density when its temperature is a littler above 89° Fah ; or about 7^ above the freezing point. By best authorities 39.2°. From about 39° it expands either by cold, or by heat. When the temperature of 320 reduces it to ice, its weight is but about 57.2 lbs. per cubic foot ; and its specific gravity about .9176, according to the investigations of L. Dufour. Hence, as ice, it has expanded one- twelfth of its original bulk as water; and the sadcleii expansive force exerted at the moment of freezing, is sufficiently great to split iron water-pipes; being probably not less than 30000 lbs per square inch. Instances have occurred of its splitting cast tubular posts of iron bridges, and of ordinary buildings, when full of rain water Arom exposure. It also loosens and throws down masses of rock, through the Joints or which rain or spring water has found its way. Retaining- walls also are sometimes overthrown, or at least bulged, by the freezing of water which has settled between their backs and the earth filling which they sustain ; and walls which are not founded at a sufficient depth, are often lifted upward by the same process. It is said that in a irlass tube ^ Incli in diameter, water will not freeze until the temperature is reduced to 23°; and in tubes of less than^ inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in rapid running streams. Ancbor lee, sometimes found at depths as great as 26 feet, consists of an aggregation of small crystals or needles of ice frosen s* the surface of rapid open water ; and probably carried below by the fbroe of ths stream. It does not form under frozen water. Since ice floats in waters and a floatinff body displaces a weight of the liquid equal to its own weight, it follows that a cubic foot of floating ice weighing 57.2 lbs, must displace 57.2 fSs of water. But 67.2 lbs of water, one foot square, is 11 inches deep: therefore, floating ice of a cubical or paralleloplpedal shape, will have \^ of its volume under water; and only ^ above: and a square foot of ice of any thickness, will require a weiffht equal to ^ of its own weight to sink it to the surface of the water. In practice, however, this must be regarded merely as a close approxima}iion, since the weight of ice is somewhat iSfocted by en- closed air-bubbles. Pure water is usually assumed to boll at 212° Fah In the open air, at the level of the sea ; the barometer being at SO inches ; and at about 1^ less for every 620 feet above sea level, for heights within 1 mile. In fsct, its boiling point Varies like its freezing point, with its purity, the density of the air, the material 4>f the vessel, dbc. In a metallio vessel, it may boil at 210°; and in a glass one, at from 212° to 220°; and it is stated that if all air be previously extracted, it requires 275°. It evaporates at all temperatures; dissolves more substances than any other agent : and has a greater capacity for heat than any other known substanosi It is eomjpressfHl at the rate of about one-21740th. (or about ^^ of an inch in 18^ feet,) by each atmosphere or pressure of 16 lbs per square Inclk When the pressure is removed, it* »>\»uHniXj restores its orisinal boUk J WATER. .327 Effeet on metals. The lime contained in many waters, forms deposits In metallic water-pipes^ and in channels of earthenware, or of masonry ; especially if the current oe slow. Some other substances do the same ; obstructing the flow of the water to such an extent, that it is always expedient to use pipes of diameters larger than would otherwise be necessary. The lime also forms very hard inemstatioiis at tbe bottoms of boilers^ very much impair- ing their efficiency ; and rendering them more liable to burst. Such water is unfit for locomotives. We have seen it stated that the Southwestern B R Ck>, England, prevent this lime deposit, along their limestone sections, by dissolving 1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms similar deposits in boilers; as uso does mud, and other impurities. Water, either when very pure, as rain water; or when it contains carbonic acid, (as most water does,) produees carbonate of lead in lead pip^ ; and as this is an active poison, such pipes should not be used for such waters. Tinned lead pipes may be substituted for them. If, however, sulphate of lime also be present, as is very frequently the case, this effect is not always produced; and several other substances usually found in spring and river water, also diminish it to a greater or less degree. Fresh uraier corrodes vrronslit Iron more rapidly tban cast; but the reverse appears to be the case with sea water; although it also affects wrought iron very quickly ; so that thick flakes may be detached from it with case. The corrosion of iron or steel by sea water increases with the carbon. Cast-iron cannons from a vessel which had been sunk in the fresh water of the Delaware River for more than 40 years, were perfectly free from rust. Gen. Pasley, who had examined the metals found in the ships Royal George, and Edgar, the first of which had remained sunk in the sea for 62 years, and the last for 133 years, "stated that the cast iron had generally become quite soft; and in some cases resembled plumbago. Some of the shot when exposed to the air became hot; and burst into many pieces. The wrought iron was not so much injured, except when in eantaet vkth copper, or brcus gun^metal. Neither of these last was much affected, except when in contact with iron. Some of the wrought iron was reworked by a blacksmith, and pronounced superior to modern iron." **Mr. Cottam stated that some of the guns had been carefully removed in their soft state, to the Tower of London : and in time (within 4 years) returned their orig^ inal hardneu. Brass cannons rrom the Mary Rose, which had been sunk in the sea for 292 years, were considerably honevcombed in spots only ; (perhaps where iron had been in contact with them.) The old cannons, of wrought-iron bars hooped together, were corroded about }^ inch deep; but had probskoly been pro- tected bv mud. The cast-iron shot became redhot on exposure to the air; and fell to pieces like dry clay I" ** Unprotected parts of cast-iron sluice-valves, on the sea gates of the Cale- donian canal, were converted into a soft plumbaginous substance, to a depth of % of an inch, within 4 years; but where they had been coated with common Swedish tar, they were entirely uninjured. This softening effect on cast iron appears to be as rapid even when the water is but slightly orackish ; and that only at intervals, it also takes place on cast iron imbedded in salt earth. Some water pipes thus laid near the Liverpool docks, at the expiration of 20 years were soft enough to be cut by a knife ; while the same kind, on higher ground beyond the influence of the sea water, were as good as new at tne end of 60 years." Observation has, however, shown that the rapidity of this action depends ntncn on the quality of the Iron ; that which is dark- colored, and contains much carbon mechanically combined with it, corrodes most rapidly : while hard white, or light-gray castings remain secure for a long time. Some cast-iron sea-piles of this character, showed no deterioration in 40 years. Contact wltli brass or copper is said to induce a galvanic action which greatly hastens decay in either fresh or salt water. Some muskets were recovered from a wreck which had been submerged in sea water for 70 years near New York. The brass parts were in perfect condition ; but the iron parts had entirely disappeared. Galwanlstng: (coating with zinc) acts as a pre* serrative to the iron, but at the expense of the sine, which soon disappears. The iren then corrodes. If iron be well heated, and then coated with toot coal-tar, it will resist the action of either salt or freshwater for many years. It is very important that the tar be perfectly purified. Sucji a coat« ing, or one of paint, will not prevent barnacles and other shells from attaching themselves to the iron. Asphaltum, if pure, answers as well aa 4M>a]->tar. Copper and bronse are very little affected by sea water. Ko galvanic action has. been detected where bnun leroles are inserted intt the water-pipea in Philadelphia. 328 TIDES. Tbe most prejudicial exposure for Iron, as well as for wood, is that to alternate wet and dry. At some dangerous spots In Long Island Sound, it has heen the practice to drive round bars of rolled iron about 4 inches diam- eter, for supporting signals. These wear away most rapidly between high and low water; at the rate of about an inch in depth in 20 years ; in which time the 4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under frenh water especially, or under ground, a thin coating of coal-pitch vamishi carefully applied, will protect iron, such as water-pipes, Ac, for a long time. See page 655. *The sulphuric acid contained in the water from coal minei corrodes iron pipes rapidly. In tbe ft'esli water of canals, iron boata have continued In service from 20 to 40 years. Wood remains sound for centuries under either fresh or salt water, if not exposed to be worn away by the action of currents : or to be destroyea by marine insects. fitea urater welgrns from 64 to 64.27 ft>s per cubic foot, or say from 1.6 to 1.9 ft)s per cubic foot more than fresh water, varying with the locality, and not appreciably with the depth. Theexcess, over the weight of fresh water, is chiefly common salt. At 64 lbs per cubic foot, 35 cubic feet weigh 2240 fi>s. Sea water freezes at about 27° Fahr. The ice is fresh ; but (especially at low tempera- tures) brine may be entrapped In the ice. A teaspoonful of powdered alum, well stirred into a bucket of dirty w^ater, will generally purify it sufficiently within a few hours to be drinkable. If « hole 3 or 4 feet deep be dug in the sand of the sea-shore, the infiltrating watei will usually be sumciently fresh for washing with soap; or even for drinking. It is also stated that water may be preserved sweet for many years by placing in the containing vessel 1 ounce of black oxide of manganese for each gallon of water. It is said that water kept in zinc tanks ; or flowing through iron tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc formed thereby; and it is recommended to coat zinc surfaces with asphalt varnish to prevent this. Yet, in the city of Hartford, Conn, service pipes of iron, galvanized inside and out, were adopted in 1855, at the recommendation of the water commissioners ; and have been in use ever since. They are like- wise used in Philadelphia and other cities to a considerable extent. In many hotels and other builaings in Boston, the *' Seamless Drawn Brass Tube" of the American Tube Works at Boston, has for many years been in use for service Eipe ; and has given great satisfaction. It is stated that the softest water may e kept in brass vessels for years without any deleterious result. Tlie action of lead upon some waters (even pure ones) is highlr poison- ous. The subject, however, is a complicated one. An injurious ingredient may be attended by another which neutralizes its action. Organic matter, whether vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm- less. Ice may be so impure that its water is dangerous to drink. Tke popular notion tbat hot water freezes more qniclLljr than cold, with air at the same temperature, is erroneous. TIDES. . The tides are those well-known rises and falls of the surface of the sea and of some rivers, caused by the attraction of the sun and moon. There are two rises, floods, or high tides ; and two falls, ebbs, or low tides, every 24 hoars and 50 minutes (a lunar day) ; making the average of S hours 12^^ minutes between high and low water. These intervals are, however, subject to fpreat variations; as are also the heights of the tides; and this not only at different places, but at the same place. These irregularities are owing to the shape of the coast line, the depth of water, winds, ana other causes. ImuMy at new and full moon, or rather a day or two after, (or twice in each lunar month, at intervals of two weeks,) the tiaes rise higher, and fall lower than at other times; and these are called spring tides. Also, one or two days after the moon is iu her quarters^ twice in a lunar month, they both rise and fall less than at other times ; and are then called neap tides. From neap to spring they rise and fall more daily ; and vice versa. The time of hifrii water at any place, is generally two or three hours after the moon has passed over either the upper or lower meridian ; and is called the establishment of that place; because, when this time is established, the time of high water on any other day may be found from it in most cases. The total height of spring tides is generally from 1}^ to 2 times as great as that of neaps. The great ii<t*I wave is merely an undulation, unattended by any current, or progressive motion of the particles of water. Each successive hijgh tide occurs STOUt 24 mlnatei later than the preceding one ; anil so with the Um tides- EVAPOBATIOK AND LEAKAGE. 329 EVAPOEATION, F UTBATIO N, AND LEAKAGE. Tbe amount of evaporation from surfaces of water exposed to tlM natural effect* of the open air, is of cooree greater in aammer than in winter ; althoagh It is quite perceptible in even, the coldest weather. It is greater in ahalloir water than in deep, inasmuch aa th* bottom also beoomes heated by the sun. It is greater in running, than in standing water ; on much the same principle that it is greater daring winds than calms. It is probable that the average dailj loss from ^ reservoir of moderate depth, m>m evaporation alone, throughout the 3 warmer months of the year, (June, Jniy, Aagust,) rarely exceeds about -^ inch, in any part of the United States. Or JL inch daring the 9 colder months ; except in the Soathem States. These two averages would give adaily one of .16 inch ; or a total annual loss of $6 ins, or 4 ft 7 ins. It probably is S.5 to 4 ft. By some trials by the writer. In the tropics, ponds of pure water 8 ft deep, in a stiff retentive day, and ftally exposed to a very hot san all day, lost during the dry sea- son, preoijiely 2 ins in 16 days ; or H ^^oh per day ; while the evaporation from a glass tumbler was V inch per day. The air in that region is highly charged with moisture ; and the dews are heavy. Every day during the trial the thermometer reached ftt>m 115° to 126° in the sun. The total annual evaporation in several parts of England and Scotland is stated to average fhmi 22 to 38 ins ; at Paris, 84; Boston, Mass, 32 ; many places in the U. 8.,' SO to 36 ins. This last would give a dailj average of -aA^ ineh for the whole year. Such statements, ho.wever, are of very little value, nnless accompanied by memoranda of the circumstances of the case ; such as the depth, exposure, sixe and nature of the vessel, pond. Ac, which contains the water, Ac. Sometimes the total annua) evaporation from a district of country exceeds the rain fall ; and vice versa. On canals, reservoirs, Ac, it is usual to combine the lofis bj eyaporation* with that by filtration. The last is that which soaks into the earth ; and of which some portion passes entirely through the banks, (when in embankt;) and if in very small quantity, may be dried up by the son and air as fast as it reaches the outside ; so as not to exhibit itself as water ; but if is greater quantity, it becomes apparent, as leakage. E. H. Gill, € E, stat^ the average evaporation and filtra- tion on tlie Sandy and Beaver canal, Oliio, (38 ft wide at ^ater snr- Cmo; 26 ft at bottom ; and 4 ft deep.) to be but IS cub ft per mile per minute, in a dry secuon. Here the exposed water surf in one mile is 200640 sq ft; and in order, with this surf, to lose 13 cub ft per mln, or 18720 cub ft per day of 34 hours, the quantity lost must be innjWV ~ '^'^^ f^> — ^H loch fa depth per day. Moreover, one mile of the canal contains 675840 cab ft ; therefore, the number of days teqd for the combined evaporation and filtration to amount to as mach as all the water in the canal, is ^-I^ ^J^ = 36 days. Observations in warm weather on. a 22'mile reaeh of the Chenango canal, N 18720 York, (40; 28 ; and 4 ft,) gave 9SH cub ft per mile per min ; or 6 times as much aa in the preceding ease. This rate would empty the canal in about 8 days. Besides this there was an excessive leakage at the gates of a look, (of only bH ft lift,) of 479 cub ft per min, 22 cub ft per mile per min ; and at aqnedneta, and waste-weirs, others amounting to 19 cub ft per mileper min. The leakage at other locks with lifts of 8 ft, or. less, did not excMa about 350 cub ft per min, at each. On other canals, it has been found to be fhom 60, to 500 ft per min. On the Chesapeake and Ohio canal, (where 60, 82, and 6 ft.) Mr. Fisk, C E, estimated the loss by evap and filtration in 2 weeks of warm weather, to be Moai 10 all the water in the canal. Professor Baublue assumes 2 Ins per day, for leafcaffe of canal bed, and evaporation, on Eni^llsb canals* i. B. Jervls, B, estimated the loss trom evap, filtration, and leakage through lock' gates, on the original Erie canal, (40, 28, and 4 ft.) at 100 eub ft per mile per min; or 144000 cub It per day. The water surf in a mile Is 211200 sq ft ; therefore, the daily loss would be equal to a dsjpth of <Hi tbe Belaware division of tbe Pennsylvania canals, when the sapply is temporarily shut off f^m any long reach, tbe water falls from 4 to 8 ins per day. The filtration will of course be muoh greater on embankta, than in eota. In some of our canals, the depth at high embankta beoomes quite considerable ; the earth, from motives of economy, not being filled in level under the bottom of the canal ; but merely left to form its own natural slopes. At one spot at least, on tbe Ches and Ohio canal, where one side Is a natural face of vertical rock, this depth is 46 ft. Sooh depths increase the leakage very greatly ; especially when, as is frequently the case, the em- baakta are not paddled; and the practice Is not to be commended, for other reasons also. Tbe total averaire loss from reservoli^ of moderate deptbs. In ease tbe earthen dams be constmeted with proper oare, and well settled bv time, will not exoeed ahont f^om ^ to 1 inch per day ; Imt in new ones, it will usually be oonsiderabiy greater. Tbe loss flrom dltcbes, or cbannels of small area, is much greater than that from navigable canals ; so that long canal feeders usually deliver but a small pre* psrtion of the water which enters them at their heads. 330 FORGE IN RIGID BODIES. MECHANICS. FORCE IN BIQID BODIES. In the following pages we endeavor to make clear a few elementary prinoiples of Mechanics. The opening articles are devoted chiefly to the subject of matter m motion; for, while an acquaintance with this is perhaps not absolutely required in obtaining a loorking Itnowledge of those principles of Statics which enter so largely into the computations of the civil engineer, yet it must be an Important aid to their intelligent appreciation. Art. 1 (a). Meolianlcs may b« dellned as that branch of science which treats of the effects of force upon matter. This broad definition of the word *' Mechanics" includes hvdroetatics, hydraulics, pneumatics, etc., if not also electricity, optics, acoustics, and indeed all branches of physics ; but we f^hall here confine ourselves chiefly to the consideration of the action of extraneous forces upon bodies supposed to be rigid, or incapable of change of shape. S) Mechanics is divided into two branches, namely : Inematlos $ or the study of the moliona of bodies, without reference to the causei of motion ; and Dynamlesy or the study of force and its efiiects. The latter is sob-divided into Kinetics; which treats of the relations between force and motion; and Statics t which considers those special, but very numerous, cases, where etpui and opporite forces counteract each other and thus destroy each other's motions. Art. $8 (a). Matter, or substaitoey may be defined as whatever occupies spao^ as metal, stone, wood, water, air, steam, gas, etc. (b) A iMKly is any portion of matter which is either more or less completely separated in fact from all other matter, or which we take into consideration by itself and as if it were so separated. Thus, a stone is a body, whethsr it be falling thronngh the air or lying detached upon the ground, or built up into a wall. Alao^ the wall is a body ; or, if we wish, we may consider any portion of the wall, as any particulsr cubic foot or inch in it, as a body. The earth and the other planets are bodies, and their smallest atoms are bodies. A train of cars may be regarded as a body; as may also each car, each wheel or axle or other part of the car, each passenger, etc., etc Similarly, the ocean is a body, or we may take as a body any portion of it at plsss- nre, such as a cubic foot, a certain bay, a drop, etc. (c) But in what follows we shall (as already stated) consider chiefly rigid bodies: i. «., bodies which undergo no change in shape^ such as by being crushed or str^chea or pulled apart, or penetrated by another body. AH actual bodies are of course more or less subject to some such changes of shape ; t. «., no body i* in fact absolutely rigid; but we may properly, for convenience, suppose such bodies to exist, because many bodies are so nearly rigid that under ordinary circumstances they undergo little or no change of shape, and because such change as does occur may be con- sidered under the distinct head of Strength of Materials. (d) But while bodies are thns to be regarded as incapable of change at form, it Is squally important that we regard them as smeeplihle to change of p^ititm as wholm. Thus, they may be upset or turned around horizontally or in any other direction, or moved along in any straight or curved line, with or without turning around a point within themselves. In short they are capable of moHon, as wholes. FORCE IN RIGID BODIES. 331 A.ictm 3 (a). Motion of a body is change of its poeitton fn relation to another body or to some real or imaginary point, which (for conyenieiice) we regard as fixed, or at rest. Thns, while a stone &11b from a roof to the ground, its position, relatively to the roof, is constantly changing, as is also that relatively to the ground and that relatively to any given point in the wall ; and we say that the stone is in motion relor tively to either of tkote bodies, or to any point in them. But if two stones, A and B, flail from the roof at the same instant and reach the jironnd at the same (subsequent) instant, we say that although each moves, relatively to roof and ground, yet they have no fi^otum rebxtivdy to each other; or, they are at rest relatively 1o each other; for their position in regard to each other does not change ; i. e., in whatever direction and at whatever distance stone A may be from stone B at the time of starting, it remains in that same direction, and at that same distance from B during the whole time of the fall. Similarly, the roof, the wall and the ground are at rest relatively to each other, yet they are in motion relatively to a falling stone. They are also in motion relatively to the sun, owing to the earth's daily rotation about its axis, and iti annual movement around the sun. (b) If a train-man walks toward the rear along the top of a freight train Just as flwt as the train moves forward, he is in motion relatively to the train; but, as a whole, be is at real relatively to ImUdingSf etc. near by ; for a spectator, standing at a little distance from the track, sees him continually opposite the same part of such building, etc. If the man on the train now stops walking, he comes to rest relatively to the irotn, but at the same time comes into motion relatively to the surrounding bnHdinffSt etc., for the spectator sees him begin to move along with the train. (c) Since we know of no absolutely fixed point in space, we cannot say, of any body, what its absoltUe motion is. Consequently, we do not know of such a thing as absolute re«^ and are si^e in saying that all bodies are in motion. Art* 4 (u). The ▼eloetty of a moving body is its rate of motion. A body (as a railroad train) is said to move with uniform -velocttFy or constant velooit^y when the distancee moved over in equal times are equal to each other^ no matter how tmall those times may be taken. (b) The -velocity la cxprcsacd by stating the dittance passed over during some giv0n feme, or which tBovid be passed over during that time if the uniform motion continued so long Thus, if a railroad train, moving with constant velocity, passes over 10 miles in half an hour, we may say that its velocity, during that time, is (». «., that it moves at (he rate of) 20 miles per hour, or 105,600 feet per hour, or 1780 feet per minute, or 2Si^ feet per second. Or, we may, if desirable, say that it moves at the rate of 10 miles in half an hour, or 8R feet in three seconds, etc. ; but it is generally more convenient to Htate the distance passed over in a unit of time, as in one day. one hour, one second, etc. (c) I^ of two trains, A and B. moving ^ith constant velocity, A moves 10 miles in half an hour, B moves 10 miles in quarter of an hour, then the veloeitieB are, A, SX) miles per hour, B, 40 miles per hour. In other words, the velocity of a body (which may be defined as the distance passed over in a given time) is inversely as the time required to pafis over a given distance. (d) By nnlt velocity is meant that velocity whieh, by common consent, is taken as equal to unity or one. Where English measures are used, the unit velocity gen- erally adopted in the study of Mechanics is 1 foot per second. (e) When we say that a body has a velocity of 20 miles per hour, or 10 feet per second, etc.. we do not imply that it will necessarily travel 20 miles, or 10 feet, etc. ; for it may nc^ have snfBcient time for tbat. We mean merely that it is traveling at the rate of 20 miles per hour, or 10 feet per second, etc. ; so that if it coniimied to move at that same rate for an hour, or a second, etc., it would travel 20 miles, or 10 feet. etc. (t) When velocity inereaget. it is said to be accelerated. When it decreases. It is said to be retarded. If the acceleration or retardation is in exact proportion to the time ; that is, when during any and every equal interval of time, the same degree of change takes place, it is uniformly accelerated, or retarded. When otherwise, the words vcuriahle and variaMy are used. (s) A body may have, at the same time, tivro qr more Independent veloel- requlring to be considered. For instance, a ball fired vertically upward from a J 332 FOBOE IK RIGID BODIES. Sn, and then falling again to the earth, has, daring the whole time of its rise and 1, (iBt) the tmiform vptoard Telocity with which it leaves the muzzle, and (2nd) the continually acceUrated dovmward Telocity given to it by gravity, which acts upon it daring the whole time. Its remUant (or apparent) velocity at any moment is the d^ertnoe between these two. Thus, immediately after learlng the gun, the downward velocity given by gravity is very small, and the resultant velocity is therrfore npwanl and Teiy nearly equal to the whole upward velocity due to the powder. But after awhila the downward velocity (by constantly increasing) beoomes equal to the upward velocity ; i. «., their difference, or the resultant velocity, becomes nothing ; the ball at that instant stands still ; but its downward velocity continues to increase, and immediately becomes a little greater than the upward velocity ; then greater and greater, until the ball strikes the ground. At that instant its resultant velocity is rthe downward Telocity which it would ) , ( the uniform upward *» •€ have acquired by falling dwring the V — < velocity given by the (, vahoU tivM of its rite and faU. ) ( powder. We have here neglected the resistance of the air, which of course retards botb flie ascent and the descent of the ball. (li) As a further illustration, regard a b n c as a raft drifting in the direction ca ox nh. A man on the rait walks with uniform velocity from comer n ta corner c while the raft drifts (with a uniform velocity a little greater than that of the man) through the distance n b. /^a\ Therefore, when the man reaches corner c, that comer has v'H^vVs^ moved to the point which, when he started, was occupied by xTff-"-^^ a. The man's resultant motion, relatively to the bed of the / ; / river or to a point on shore, has therefore been » a. His / j. / motion at right angles to n a, due to his walking, is t c, but ^<" — -fi / that due to the drifting of the raft is o 6. These two are ***--.i'''* equal and opposite. Hence his resultant motion <U right il angles to n a is nothing ; he does not move from the line n a. His walking moves him through a distance equal to n i, in the direction n a; and the drifting through a distance equal to t a, and the sum of these two is n a. (i) All the motions which we see given to bodies are but €hang«a in their unknown absolute motions. For convenience, we may conflne our attention to some one or more of these changes, neglecting others. Thus, in the case of the ball fired upward from a gun (see. (9) above) we may neglect Its uniform upward motion and consider only its constantly accelerated downward motion under the action of gravity ; or, as is more usual, we may oonaldar only the retuUard or appatrmi motion, which is first upward and then downvrard. In both cases we neglect the motions of the ball caused by the several motions of the earth in spaed. Art. 5 (•)• Forcoy <be «Miiu« of change of motion. Suppose • perfectly smooth ball resting upon a perfectly hard, frictionless and level surfiMS^ and suppose the resistance of the air to be removed. In erder to merely move the ball horizontally (i. e., to set it in motion — ^to change its state of motion) some /orc« must act upon it. Or, if such a ball were already in motion, we could not retard or hasten it, or turn it from its path without exerting force upon it. For, as stated in Neiirton's flrat \wk-v¥ of motion, ''•-rerjr body continues In its •tstte of rest or of motion in a straight line, except in so far as it may be com- pelled by impressed forces to change that state." On the other hand, if a force act* upon a body, the motion of the body must undergo change. (b) Force Is an action betifreen t-wo bodies, fending eitber i» separate them or to bring them closer togeU&er. For Instance, when a stone falls to the ground, we explain the Csct by saying that a force (the attnction of gravitation) tends to draw the earth and the stone together. Magnetic and electric attraction, and the cohesive force between the particles of a body, are other instances of ottmcttee force. (c) Force applied by contsMst. In practice we apply force to a body (B) by causing contact between it and another body (A) which has a tendency to motwti toward B. A repulsive force is tljus called into action between the two bodies (io ■omo way which we cannot understand), and this force pushes B forward (or in the FOBCE IN RIGID BODIEBL 333 direction of A's tendency to move) and pushes A backward, thus diminishii^ its for- ward tendency * If, for instance, a stone be laid npon the ground, it tends to moTe downward, bat does not do so, because a repulsive force pushes it and the earth apart Just as hard as the force of gravity tends to draw them together. Similarly, when we attempt to lift a moderate weight with our hand, we do so by giving the hand a tendency to move upward. If the hand slips from the weight this tendency moves the hand rapidly upward before our will force can dieck it. But otherwise, the repulsive force, generated by contact between the hand (tending upward) and the weight, moves the latter upward in spite of the force of gravity, and pushes the hand downward, depriving it of much of the upward velocity which it would otherwise have. It is perhaps chiefly fh>m the eftortf of Vhich we are conscious in such cases, that we derive our notions of <Yorce." When a moving billiard ball. A, strikes another one, B, at rest, the tendency of A to continue moving forward is resisted by a repulsive force acting between it and B. This force pushes B forward, and A backward, retarding its former velocity. As explained in Art. 23 (a), ' the repulsive force does not exist in either body ontil the two meet (d) The repulsive force thus generated by contact between two bodies, continues to act only so long as they remain in contact, and only so long as they tend (from * ■ome extraneous cause) to come closer together. But it is genenJly or always accompanied by an additional repulsive force, due to the compreuion of the particles of the bodies and their tendency to return to their original positions. This eUutic repulsive force may continue to act after the tendency to compression has ceased. (e) Force acts either sui a P^I or sui a puatai. Thus, when a weight Is susiMnded by a hook at the end of a rope, gravity jmU« the weight downward, the weifrht ptuhn the hook, and the hook puUi the rope, each of these actions being accompanied, of course, by its corresponding and opposite "reaction.** When two bodies collide, each pushps the other, generally for a very short time. (1) EjqiiaUtjr of actloni aad reaction. A force always exerts itself equally upon the two bodies between which it acts. Thus, the force (or attraction) of gravitation, acting between the earth and a stone, draws the earth upward just as hard as it draws the stone downward ; and the repulsive force, acting between a table and a stone resting upon it, pushes the table and the earth downward just as bard as it pushes the stone upward.' This is the fact expressed by Ne'vrton's tl&lrd lainr ot motloiiy that **to every action there is always an equal and contrary reaction.*' For measures of force, see Arts. 11, 12, 13. If a cannonbidl in its flight cuts a leaf from a tree, we say that the leaf has reacted against the batt with precisely the same force with which the ball acted against the leaf. That degree of force was sufficient to cut off a leaf, but not to arrest the ball. A ship of war, in running against a canoe, or the fist of a pugilist strikint; his opponent in the foce, receives as violent a blow as it gives ; but the same blow that will upset or sink a canoe, will not opprecto&Iy affect the motion of a ship, and the blow which may seriously damage a nose, mouth, or eyes, may have no such effect upon hard knuckles. The resistance which an abutment opposes to the pressure of an arch ; or a retain- ing^wall to the pressure of the earth behind it, is no greater than those pressures themselves ; but the abutment and the wall are, for the sake of safety, made capable of sustaining much greater pressures, in case accidental circumstances should pro- duce such. (§p) In most practical cases 'we liaT-e to consider only one of the two bodies between which a force acts. Hence, for convenience, we commonly speak as if the force were divided into two equal and opposite forces, one for each of the two bodies, and confine our attention to one of the bodies and the force acting upon it, neglect- ing the other. Thus we may speak of the force of steam in an engine as acting upon the pitton, and neglect its equal and opposite pressure against the head of the founder. (h) That point of a body to which, theoretically, a force is applied, is called the pplnt ot application. In practice we cannot apply force to a point, according to the seientlflo meaning of that word ; but have to apply it distributed over an ap- preciable area (sometimes very large) of the surface of the body. * We ordinarily express all this by saying simply that A pushes B forward, and this is sufficiently exact for practical purpoees ; but it is well to recognize that it iH merely a convenient expression and does not fully state the facts, and that every force neees- aarUff consists of two equal^and opposite pulls or pushes exerted between two bodiai. 334 FORCE IN RIGID BODIBS. For the present we shall aasume that the line of action of the force passes through toe center of gravitf of the body and forms a right angle with the sur- face at the point of application. Art* 7 (a). Acoeleratlon. When an unresisted force, acting upon a body, sets it in motion (i. «., gi^es it Telocity) in the direction of the force, this velocity increases as the force continues to act; each equal interval of time (if the force remains constant) bringing its own equal increase of velocity. Thus, if a stone bu let full, the furce of gravity gives to it, in the first in- conceivably short interval of time, a small velocity downward. In the next equal interval of time, it adds a second equal velocity, and so on, so that at the end of the second interval the velocity of the stone is twice as great, at the end of the third interval three times as great, as at the end of the first one, and so on. We may divide the time into as small equal intervals as we please. In each such interval the constant* force of gravity gives to the stone an equal increase of velocity. Such increase of velocity is called accelerstion-f When a body is thrown verticaUy uptoardy the downward acceleration of gravity appears as a retardation of the upward motion. When a force thus acu offaitut the motion under consideration, its acceleivp tion is called negatim. Art* 8 (a). Tbe rate of aujoeleratloiii is the acceleration which takes place in a given Hmsj as one second. rb) The unit rate of acceleration is that which adds unit of velocity in a unit of time ; or, where Bnglish measures are used, one foot per second, per geeond, (o) For a given rate of acceleratioo, the total accelerations are of course propor- tional to the HmsM during which the velocity increases at that r&te. Art. (tt), Iia^rs of acceleration* Suppose two blocks of iron, one fwhich we will call A^ twice as large as the other (a), placed each upon a perfectly fricnonless and horizontal plane, so that in moving them horizontally we are opposed by no force tending to hold them still. Now apply to each block, through a spring balance, a pull such as will keep the pointer of each balance always at the same mark, as, for instance, constantly at 2 in both balances. We thus have equal forces acting upon unequ^ masses.^ Here the rate of acceleration of a Is double that of A ; for nrlien the forces are equal tbe rates of auseelera- ration are Inversely as tike masses* In other words, in one second (or in any other given time) the small block of iron, a, will acquire twice the increase of velocity that A (twice as lai^e) vdll acquire ; so that if both blocks start at the same time from a state of rest, the smaller one, a, will have, at the end of any given time, twice the velooitff of A, which has twice its mass. (b) Again, let the two masses, A and a, be equal, but let the foree exerted upon a be twice that exerted upon A. Then the rate of acceleration of a will (as before) be twice that of A ; for, 'vrl&en tl&e masses are cqnatly tbe rates of aoeelera* ration are alrectljr as the forces* (e) We thus arrive at the principle that, in any case, the rate of acceleration Is dlreotljr proportlonsil to the force and Inwerselw proportional to the n&ass* * We here speak of the force of gravity, exerted in a given place, as constant, because it is so for all practical purposes. Strictly speaking, it increases a very little as the stone approaches the earth. t Since the rtUe of acceleration is generally of frreater conseq-qence. in Meohanies, than the total acceleration, or the "acceleration" proper, srienttfic writers (for the sake of brevity) use the term "acceleration" to denote that rate, and the term "total acceleration" to denote the total increase or decrease of velocity occnrrinK during any given time. Thus, the rate of acceleration of gravity (about 32.2 ft. per second per second) is called, simply, the "acceleration of gravity.'* As we shall not have to use either expression very frequently, we shall, generally, to avoid misappre' hension, give to each idea its full name ; thus, <* total acceleration " for the whoU change of velocUy in a given case, and " rate of acceleration " for the rate of that change. t The mass of a body Is the quantity of matter that it contains. FOBCB IN BIOID BODIES. 335 * (d) Htticat if we make the two forces propmrtloiial to the two maases, tbe rases of aoceleratioQ will be equal ; or, t»r m fpiwea vmtm of acosleimtloii^ tbe forces most be dlrectljr as the masses. (e) Hence, also, a greater force is required to Impart a g^Ten Teloci^ to a girea body in a short time than to impart the same Telocity in a longer time. For instance, the forward coupling links of a long train of cam wonld snap instantly nnder a pull safBoIent to give to the train in two seconds a Telocity of twenty miles per hour, sup- Ciing a suflBoiently powerftil looomotiTe to exist In many such cases, therefore) we Te to be contented with a slow, instead of a rapid acceleration. A string may safely sustain a^ weight of one pound suspended from our hand. If we wish to impart a great upward Telocity to the weight in a very sJiort timey we eTi- dently can do so only by exerting upon it a great force; in other words, by Jerking the sMng Tiolently upward. But if the string has not tensile strength sufficient to transmit this force from our hand to the weight, it will break. We might safely giTe to the weight the desired Telocity by applying a le$» Jbre^ during a longer time. {t) When a stone falls, the fi>rce pulling the earth upward is (as remarked aboye) equal to that which pulls the stone downward, but the tncun of tne earth is so Tastly greater than that of the stone that its motion is totally imperctptible to us, and would still be so. eTen if it were not counteracted by motions in other directions in other parts of the earth. Hence we are pracHcaUy^ though not abtolutelif, right when we say that the earth remains at rest while the stone fiJls. (§;) Bat in the case of the two billiard balls (Art 5e. p. 388), we can dearly see the result of die action of the force upon each of the two bodies; for tbe second ball, B» which was at rest, now moTes forward, while the forward Telocity of tbs lint OB», A, is dimiidshed or destroyed, its backward mention thus appearing as a . ntenlaMBa of ita forward motion. And, (since the same force acts upon both balls) mass . mass . rate of acceleration . rate of negatlTe acceleration ofA'ofB'* ofB ofA or (siaee the ibrce acts Ibr tbe saate time upon both balls) miuMy mass forward Telodty . loss of forward Telocity oTA * ofB *' OfB OfA ' Ok) "RgwAng. A man oaamot 1^ a weight of 20 tons; but if it be placed upon prv^r friction rollers^ he can move it horisontally, as we sea in some drawbridges, tumtablea, Ac. ; and if friction and the resistance of the air could be entirely remoTed, he could BOTC it by a ringle breath ; and it would continue to uoto forerer after the foiee of the brecrth had ceased to act upon it. It would, howsTer, moTe Tory slowly, because the force of tbe single breath would hsTe to diffuse itself among 20 tons of matter. He can more it, if it be placed in a suitable Teasel in water, or if snqiended from a long rope. A powerful locomotlTe that may moTe 2000 tons, cannot lift 10 tons Terticaltar. If we imagine two bodies, each as large and heaTy as the earth, to be precisely balanced in a pair of scales without friction, a single grain of sand added to either icale'paa, would giTe motion to both bodies. ' Art. lO (a). The constant force of gravity is a uniformly accelerating force when it acts upon a body falling freely ; for it then Increases the Telocity at uie uni- form rate of .322 of a foot per second during every hundredth part of a second, or 32,2 feet per second in eTery second. Also when it acts upon a body moving down an in- clined plane; although in this case the increase is not so rapid, becatise it is caused l^ only a part of the graTity, while another pert preeses the body to the plane, and a third part OTercomes the friction. It is a uniformly retarding force, upon a body thrown Tertically upward; for no matter what may be the Telocity of the body when projected upward, it will be diminished .322 of a foot per second in each hundredth part of a second during its rise, or 82.2 feet per second during each entire second. At least, such would be the case were it not for the varying resistance of the air at difforent Telocities. It is a uniformly straining force when it causes a body at rest, to press ux)on another body ; or to pull upon a strinfi; by which it is suspended. The foregoing expressions, like those of momentum, strain, push, pull, lift, work, &c., do not indicate different hinde of force ; but merely different kinds of ^eets producM by the one grand principle, force. (b) The aboTe 82.2 feet per second is called the aeceleratton otgrm.'vltr f and by scientiile writers is conTcntlonally denoted by a small g % or, more correctly qieak- 336 FOBOB IN BIGTD BODIEa. tag, ifnce the aoc«l«ratloii li not precisely the eame at ftll parti of Che Mrtb, g denoteethe aooeleratloii ptf aeoond, whateTer it may be, at Any particular idaoe. . Art. 11 (a). Ralatton b«tw«eit force and nuuM* The mass of a body is the quantity of matter which It contains. Ons cubic foot of water has ttotei AS great a mass as /la^ a cubic foot of water, but a lesi mass than one cubic foot of iron. Thus, the n'Mof a body is a measure of mass between bodies of the tame material, but not between bodies of different materials. (b) When bodies are allowed to fall freely in a racuum at a given place, 4hey are found to acquire equal velocities in any eiven time, of whatever different materials they may be Qomposed. From ibis we know (Art. 9 (dV, p. 335). that the forcee moving them downward, viz. : their respective tMighu at that place, must be proportional to their maeaee. Thus, in any given placet the weight of a bodv is a perfect measure of its mam. But the weight of a given bodv changes when the body is moved from one level above the sea to another, or from one latitude to another; while the mass of the body of course remains t/ie same in all places. ThuSga piece of iron which weighs a pound at the level of the sea, will weigh leee than a pound by a spring balance, upon the top of a mountain close by. because the attraction between the earth and a eiven mass diminishes when tne latter recedes from the earth's center. Or if tne piece of iron weighs one pound near the North or South Pole, it will, for the same reason, weigh leet toan a pound by a spring balaliioe if weighed nearer to the equator and at the same level above the sea. The difference in the weight of a body in different localities is so slight as io be of no account in questions of ordinary practical Mechanics ;• bat scientific exactness requires a measure of mass which will give the same expression for the quantity of matter in a given body, wherever it may be; and, since weighing Is a verv convenient way of arriving at the quantity of matter in a body, it is desirable that we should still be able to express tiie mass in terms of the weight. Now, when a given body is carried to a hieher level, or to a lower latitude, its loss of weight is simply a decrease in the Jores with which gravity draws it downward, and this same decrease also causes a decrease of the velocitu which the body acquires in falling during any given time. The change m velocity, by Art. (6), p. 884, is necessarily propor* - nonal to the change in weight Therefore, if the weight of a body at any place be divided by the velocity which gravity imparts in one second at the same place (and called sr^ or the aeceUrcttion of gravity for that place), the quotient will be tne same at aU plaoei^ and therefore serves as an invariable mei^ure of the mass. (c) By common consent, the m&it ot mass, in scientific Mechanics, is said io be that quantity of matter to which a unit of force can give unit rate of acceleration. This unit rate, in countries where English measures are used, is one foot per second, per second. It remains then to adjust the units offeree and of maee. Two methods (an old and a new one) are in use for doing this. We shall refer to them here as methods A and B respectively. fd) In metl&od A, still generally used in questions of etatics^ the untt ox n»roe is fixed as that force which is equal to the weiaht of one pound in a certain place; i.e.. the force with which the earth at that place attracts a certain standard piece of platinum called a pound: and the unit of maee is not this standard piece of metal, but, as stated in (c)^ that mass to which this unit force of one pound gives, in one second, a velocity of one foot per second. Now the one pound attraction of the earth upon a mass of one pound will (Art. 1, p. 330) in one second give to that mass a velocity — (/ or about 32 feet per second; and (Art 9 (a), p. 834), for a given force the masses are inversely as the velocities imparted in a given time. Therefore, to give in one second a velocity of only one foot per second (instead of g or about 32) the one pound unit of force would have to act upon a mass g times (or about 82 times) that which weighs one pound. This could be accomplished, with an Attwood*s machine. Art 16 (e), p. 889, by making the two equsU weights each «- 15 ^ lbs. and the third weight *■ 1 Ibw *The greatest discrepancy that can occur at various heights and latitadeS| by adopting weight as the measure of quantity, would not oe likely to s x ess a 1 in 300; or. under ordinary circumstances, 1 in 1000. FOBOE IN BiaiD BODIES. 337 By method A, therefore, the unit of masn is g times (or about 33 times) the mass of the standard piece of metal called a pound; i. e., a body containing one such unit of mass wei^s g lbs. or about 32 lbs.; or, tijr method A, the weight of any given body ^ ^ y the mass of the body, in lt«. — Sf A jjj oaitg Qf mass. file moss of a body, in units of mass - l^.^ ^^^g^^ ^^ ^^^ ^Q^y> ^^ POP°<^ g For instance: in a body weighing the mass is about y^ pound ^ unit of mass 1 •* 2 « 82 " i 64 « 2 *• •• It has been suggested to call this unit of mass a *' Matt/* (•) In naetlMPd By the moM of the standard pound piece of platinum is taken as toe unit of nuum and is called a pound} and the force which will give to it in one second a velocity of one foot per second is taken as the unit of force. This small unit of force is called a ponndal* In order that it may in one second give to the mass of one pouna a velocity of only one foot per second, it must (by Art f b), be -i. f or about Jjj of the weight of said pound mass. Hence, "by m«tl&od By the ma8$ of any given body, in jxrunds - the weight of the body in poundaU and the weiifht of a body, in potmdale — gr X the maea of the body in pound§. Forinstar'jce: in a body weighing the mass of the body is about ^ pouitdal — > JL pound JL pound 82 " — 1 " 1 •• 64 " — 2 « 2 " tty VoT coi&'renlenoey we sometimes disregard the scientific require- ment that the unit of force must be that which will give unit rate of accele- ration to nnit mass, and take a pound of matter as our unit of maes^ and a pound weight as our unit of force. Our unit of force will then in one second give ft velocity of g (or about 32.2 feet per second) to our unit of mass. In Sicties^ we are not concerned with the masses of bodies, but only with the fijrees acting upon them, including their weights. Art* 13 (a). Impnlse. By taking, as the unit of force, that force which, in one second, will give to unit mass a velocity of one foot per second, we have (by Art. 9, p. 334), in any case of unbalanced /orc« acting upon a mass during a iSvon timei Velocity - force X time ^^ mass Force - volocity X mass ^gj time Mass - . force X time ^3^ velocity Time - niass Xvelocity ^^ force Force X time — mass X velocity. . . . (5> * 25 338 FORCE IN RIQID BODIES. Tb the prodnoft^ force X time, in equation (5), writers now give the name Impolee^ which was formerly given to eoUUion (now called liiip««t}* See Art 24 (a). The term impuUe, as now used^ conveys merely the idea of force acting through a certain length of time. Equation (5) tells us that an impulse (the product of a force by the time of its action) is numerically equal to the momentum* which it produces. Eqilation (2) tells us that any force is numerically equal to the momentum which it can produce in one second. In other words, the monftentmn of a body moving with a given veloci^ is numerically equal to the force which in one second can produce or destroy that velocity in that body; or, a force is numerically equal to the rate pw second at which it can produce momentum. Thus, forces are proportional to the momentums which they can produce in a given time; or, in a given time^ equal forces produce equal momentums. Therefore a force must always give equal and opposite momentums to the two bodies between which it acts. Art* 13 (»)• Tlk* luiiial -wajr of meaanrln^ a fbrce is by ascertaining the amount of some other force which it can counteract. Thus we may meas- ure the weight of a body by hanging it to a spring balance. The scale of the balance then indicates the amount of tension m the spring: and we know thai the weight of the body is equal to the tension, because the weight just pre vents the tension firom drawing the hook upward. Thus, fbremm are conveniently expressed In -vrelfpltt|i9 as in pounds, tons, &c., and they are generally so measured in Statics, and in our following articles. (b) A fbroe mav' be aonstant or Tarlable* When a stone rests upon the ground, the pull of gravity upon it (i. e., its weight) remains constanlh neither increasing nor decreasing. But when a stone is thrown upward its weight decreases very slightly as it recedes from the earth, and again increases as it approaches it during its fall. In this case, the force of gravity, acting upon the stone, decreases or increases eteadUy, But a force may change euddenlyf or irregtUariyf or may be intermittent $ as when a series of uneqiul blows are struck by a nammer. In what follows we shall have to do only with forces supposed to be conatarU, Art. 14b («)• "DonuUjr* The deneitiea of materials are proportional to the mauee contained in a given volume, as a cubio inch j or inoerselff as the volume required to contain a given mass. Or^ since the weights at a given place are proportional to the masses, the densities are proportional to the weights per unit of volume (or ** specific gravities **) of the materials. Thus, a body weigh* inff 100 lbs. per cubio foot is twice as dense as one weighing only 60 Iba. per cuoic foot ait the same place. Art. 15 (a). Inertia. The inability of matter to set itself in motion^ or ta change the rato or direction of its motion, is called its inertia, or inertneeaL \Blien we say that a certain body has twice the inertia (inertness) of a smaller one, we mean that twice the /or<:0 is required to give it an equal ratoof acoete* ration ; and that, since all force (Art. 5f)t acts equally in both direo* tions, we experience twice as great a reaction (or so-called ** resistance*^ from the larger body as from the smaller one. The ** inertia** of a body is therefora a measure of the/ore« required to produce in it a given rato of acceleration; Ob which is the same thing, it is a measure of the mass of the body. We mi^ therefore consider ** inertia'* and **mass** as identical. (b) What is called the ** resistance of inertia** of a body, ia simply reaction, (i s., one of . the two equal and opposite actions) of whatever force we apply to the body. Hence, its amount depends not only upon tiia mass of the Dody, but also upon the rato of acceleration which we choose to *The momentam of a body (sometimes called its ** quantity of motion") is equal to the product obtained by multiplying its ma»9 by its velocity. If «• adopt the pound as the unit of mass, as in '* method B/* Art. 11 («), tha proauct, voeight in pownds X velocity, is numerically either exactly or neurly the same as the product, m(M8 in pounds X velocity, depending upon whether or not the body is in that latitude and at that level where a ma8$ of one poufltdl is said to weigh one pound. But the product, weight in poundats X velocity; It exactly a times (about 82.2 times) the product, mass in pounds X velocity; afeo^ k^ ** menod A,** iMi^M in |K>uncte X velocity — y X «MM in ** matte '* X FORCE IN RIGID BODIES. 339 giTe to it. Therefore we cannot tell, from the mass or weight of a bodj alone, what its " reeistance of inertia " in any given case will be. Art. 16 (a). Forees In opposite directions. When two equal and opposite forces act upon a body at the same time, and in the same straight line, we say that they destroy each other's tendencies to more the body, and it remains at rest. If two unequal forces thus act in opposition, the smaller force and an equal portion of the greater one are said to counteract each other in the same way, but the remainder of the greater force, acting as an unbalanced or unresisted force, moves the body in its own direction, as it would do if it were the only force acting upon it. Thus, when we move bodies, in practice, we encounter not only the " resist- ance of inertia" (i. e., we not only have to exert force in order to move inert matter), but we are also opposed by other /otom, acting against us, as friction, the resistance of the air. and, often, all or a part of the wHght of the body. By '' resistances," in the following, we mean such resisting /oroM, and do not include in the term the " resistance of xHertia," (b) If separated, the two bodies, A and B, of 8 &m and 2 lbs respectively, would fidl with equal accelerations = g ; each unit, — , of mass being acted upon by its own weight, W, Bat. connected as they are, A will move downward, and B upward, with an acceler- T^2A ation =» only f ; for now an unbalanced force of 5 only 8 — 2 = lfb must give acceleration to a mass (J'*^0 T*2j4 ation =» only | ; for now an unbalanced force of 5 ,8 + 2 6 « " 2 T-2.4 of = -. But, to give to a mass, B, of -, an aoeelof |, requires a force of -. I »^]b=»a4 lb. 3|liji EjI^ This, plus 2 lbs (required to balance the weight of Al B) is the tension, 2.4 lbs existing throughout the cord. Exerted at A, this tension balances 2.4 of the 3 lbs weight of A. The remainder (8 — 2.4 = 0.6 !b) of the weight, acting downward upon the mass, -, of A, gives to it the required acceleration of ^; , - force .- 8 0.6 g .. g for here = 0.6 -«- - = -r-2 = 0.2 g = f . mass g 8 * 6 ^ Or we may regard the total tension, 2.4 lbs, in the cord at A, as acting upon A O and giving to it a negative or upward acceleration of 2.4 -t- - = 0.8 g, which, g dedacted from g (the acceleration which A would otherwise have) leaves Acceleration = g — 0.8 g = 0.2 g = |. Let W = weight of A w ss weight of B F a> net force available for acceleration » W — w mr -4- w M =3 combined mass of both bodies = — — — g m => mass of B » - g a => acceleration T a tension in cord. Then: a = ^ = (W - w) ^ :5^^ = «-^£i:^ M ^ g W + w m . . ^ , ^ g(W — W) / W — W\ T = w + ma = w +— a==wH ^-vi?— ; = w 1 1 + ^5,— — |. g 'gW + w \Wh-w/ (e) An ** Ati¥00«l*s Machine'* consists essentially of a pulley, a flexible cord passing over the pulley, two equal weights (one suspended at each end of the cord), and a third weigfnt, generally much lighter than either of the other two. The two equal weights balance each other by means of the pulley and cord. The third weight is laid upon one of the other two weights. The force of gravity, acting upon the third weight, then sets the masses of the three weights in motion at a small but constantly increasing velocity. In order to do this it mast also overcome the friction of the pulley and cord, and the rigidity 340 FORCE IN RIGID BODIES. of the latter ; bat, as these are made as slight as possible, they are, fbr ooo- venience, neglected. The machine is used for Illustrating the acceleration given to inert matter by unbalanced force, and forma an excel^nt example of the two distinct duties which a moving force generally has to perform, vis: (1st) the balancing of resistance, and (2nd) acceleration. (d) In the case of a lo«oiiiotlire. drawfngr a train on a leTel, fHc- tioa and the resistance of the air are the only resistances to be balanced ; for Uie weight of the train here opposes no resistance. Unless the force of the steam is more than sufficient to balance the resistances, it cannot mote the train. If it exceeds the resistances, the excess, however slight, gives motion to the inert matter of the train. If, at any moment while the train is moving, the force of the steam becomes jtut equal to the resUtcmces (whether by an increase of the latter or by diminishing the force) the train will move on at a uniform velocity equal to that which it had at the moment when the force and resistance were equalized ; and, if these could always be kept equal, it would so move on forever. But so lone as the excess of steam pressure over the resistances continues to act, the velocity Is increased at each instant ; for during eaeh such instant liie excess of force gives a small velocity in addition to that already existing. On a level railroad, let P »- the total tractive force of the locomotive = say 13 tons <• W sa weight of locomotive = 50 tons w sa weight of train = 336 tons R *= resistance of locomotive (including internal fHction, etc.) «> 8 tona r a resistance of train =■ 1 ton F » net force available for acceleration — P — R — r-s9 toms M « mass of engine and train =■ — — ,. _ -* 12 * g 82.2 - ^ , w 8S6 ^^ , . m «■ mass of train = - — —- = 10.44 g 32.2 a = acceleration T = tension on draw-bar. F 9 Then : Acceleration at a » ^ — r^ = 0.75 ft per second per second. The tension T on the draw-bar « resistance of train + force causing accel- eration a, orT=r + ma — 1 + 10.44 X 0.76 = 1 + 7.83 = 8.83 tons. This tension, T, pulling backward against the locomotive, causes there a T a aa «r retardatim, or negative acceleration, of masa of tocomoUve = -go- = »» « per sec per sec, and thus reduces, by that amount, the acceleration which the (P ■■'' r) s 10 X 8S.8 locomotive would otherwise have, and which would be — ^ — ka ■■ ,. — oO 60 ~ 6.44. This, less 6.69, » 0.75 ft per sec per sec — acceleration of train. (e) If the tractive force of a locomotive exceeds the resistances, due to friction, grades, and air, the velocity will be accelerated ; but it then heoomeB more dilB- cult to maintain the excess of force, for the pistons must travel fast«r through the cylinders, and the boiler can no longer supply steam fast enough to maintain the original cylinder pressure Besides, some of the resistances increase with increase of velocity. We thus reach a speed at which the engine, alUiough exerting its utmost force, can do no more than balance the resistances. T^e train then moves with a uniform velocity equal to that which it had when thia condition was reached. When it becomes necessary to stop at a station some distance ahead, steam It shut off, so that the steam force of the engine shall no longer counterbalance or destroy the resisting forces; and the number of the resistances themselves is in- creased by adding to them the friction of the brakes. The reBistanoea, thus incneased, are now the only forces acting npon the train, and their acoeleration is negative, or a retardation. Hence, the train moves more and more slowly, and must eventually stop. (f) Caution. When two opposite forces are in equilibrium, an addition to one of the forces does not always form an unbalanced force ; for in many cases the other force increases eguallyy up to a certain point. For Instance, when we attempt to lift a weight, W, its downward resistance^ R, remains constantly Just equal to our upward pull, P, however P may vary, until P exceeds W. Thas, R can never exceed W, but may be much less than It. Indeed, when we atop pull- ing, R ceases, although W (the attraction between the eartn and the weight) of FORCE IN RIGID BODIES. 841 eoarse remnins unchanged throaghout. Such Tariation of resisting force, to meet varying demands, occurs in all those innumerable cases where structures sustain varying loads within their ultimate strength. Art. 17 (a). Work. Force, when it moves a body,* is said to do " work " upon it. The whole work done by the force in moving the body through any dis- tance is measured by multiplying the force by thedutance; or: Work = Force X distance. If the force is taken in pounds, and the distance in feet, the product ([or the work done) will be in foot-pounde ; if the force is in tons and the distance in inches, the product will be in inch-tons ; and so on.f Thus, if a force of moves a body through we have work = 1 pound 10,000 feet 10,000 foot-pounds 100 pounds 100 '* 10,000 " 10,000 " 1 foot 10,000 " or, in any case, if the fiprce be F pounds, the whole work done by it in moving a body through s feet, is F « foot-pounds. (I») The foot-pound, the foot-ton, the inch-pound, the inch-ton, etc., etc., are called unlto oi wwrfc.f For practical purposes, in this country, forces are most frequently stated in pounds, and the distances (through which they act) in feet. Hence tbe ordi- nary anii of work, is the foot-pound. The metric nnit of work is the klloflrram-meter, i e. l Kilogram raised 1 meter = 2.2046 pounds raiaed 3.2800 feet, = 7.23S1 foot-pounds. 1 foot-pound = 0.13825 kilogram-meter. («) In most cases, a portion at least of the work done by a force is ex- pended in owereomlnv reflistiunees. Thus, when a locomotive begins to move a train, a portion of its force works against, and balances, the resist- anoM of friction or of an up-grade, while the remainder, acting as unbalanced toroe upon the inert mass of the train, increases its velocity. An upward pull of exactly one pound will not raise a one pound weight, but will merely biuanoe the downward force of gravity. If we increase the upward pail from one pound (=» 16 ounces) to 17 ounces, the ounce so added, being unbalanced foroe, will give motion to the mass, and will acceleirate its upward velocity as long as it continues to act. If we now reduce the upward pull to 1 pound, thus miking it just equal to the downward pull of gravity, the body will move on upward with a uniform velocity : but if we reduce the upward force to 15 ounces (= || pound), then there will be anjunbalanced dovmward force of 1 ounce acting upon the body, and this downward force will generate in the body a downward or negative acceleration or retardation, and will destroy the upward velocity in the same time aa the upward excess of 1 ounce required to produce it. Daring any time, while the 17 ounces upward ** force" were acting against the 16 ounces downward " resistance," the product of total upward force X distance mast be gre<Uer than that of resistance X distance. The excess is the work done in accelerating the velocity, by virtue of which the body has acquired kinetic energy or capacity for doing work in coming to rest. On the other hand, while the npward velocity was being retarded, the product of total upward force X dist was less than that of resistance X dist, the difference being the work done by the kinetic energy against the resistance of gravity. In practice, the term " work" is usually restricted to that j9or<ion of the work which a force performs in balancing the resistances which act against it ; in other words, to the work done by so much of the force as is equal to the resistance. With this restriction, we have work ^ force X dist, = resistance X dist. Thus, if the resistance be a friction of 4 ft>s., overcome at every point along a distance of 8 feet; or if it be a weight of 4 S>s., lifted 3 feet high, then the work done amounts to 4 X 8 » 12 foot-9>8, provided the initial and the final velocities are equal. (d) In cases wbere tbe weloeity Is nnlform, as in a steadily running macbine, tbe force is necessarily equal to the resistance ; and where the velocities at the beginning and end of any work are equal (as where the machine starts from rest and conies to rest again) the mean force is equal to the mean resistance. In such cases, therefore, the two products, mean force X distance, and mean resistance X distance, are equal, and we have, as before, Work =^ force X dist = resistance X dist. ♦ A man who Is standing still is not considered to be working, any more than is a post or a rope when sustaining a heavy load ; although he may be support- ing an oppressive burden, or holding a car-brake with all his strength ; for his force moves nothing in either case. t These products must not be confounded with momerUs, — force X leverage. 342 FOKCE IN RIGID BODIES. (f ) In calculating the work done by machinery, etc., allowance must be made for this expenditure of a portion of the work in overcoming resistances. Thus, in pump- ing water, part of the applied force is required to balance the friction of the different parts of the pump; so that a steam or water "power,** exerting a force of 1(H) &8., and moving 6 feet per second, cannot raise 100 fi>8. of water to a height of 6 feet per second. Therefore machines, so far from gaining power ^ according to the popular idea, actually lose it in one sense of the word. In Uarting a piece of machinery, the forces employed have (1st) to balance, react a^rainst, or destroy the resisting force of friction and the cohesive forces of the material which is to be operated on ; and (2d) to give motion to the unresisting matter of the machine and of the material operated on, after the resisting forces which had acted upon them have thus been rendered ineffective. But after the desired velocity has been established, the forces have merely to bcUance the resistances in order that the velocity may continue uniform. (g) That portion of the work of a machine, etc., which is expended against fric- tion is sometimes called <* lost -work " or ** prejudicial ^rorky" M'hile only that portion is called " useful -vrork " which renders visible and tangible service in the shape of output, etc. Thus, in pumping water, the work done in overcoming the friction of the inimp and of the water is said to be lost or prejudicial, while the useful work would be represented by the product, weight of water deliverwl X height to which it is lifted. The distinction, although artiflcial| and somewhat arbitrary, is often a very con- venient one ; but the work is of course not actually ** lost," and still less is it ** pre- judicial ;" for the water could not be delirered without first overcoming the resist- ances. A merchant might as well call that portion of bis money lost which he expends for clerk-hire, etc. (it) For a given force and distance^ tlie i^ork done is independent of the time $ for the product, force X distance, then remains the same, whatever the time may be. But the distance through which a given force will work at a given velocity is of course proportional to the time during which it is allowed to work. Thus, in order to lift 50 pounds 100 feet, a man must do the same work, (= 6000 foot-pounds) whether he do it in one hour or in ten ; but, if he exerts constantly the scrnie foroey he will lift 50 &>s. ten times as high in ten hours as in one, and thus will do ten times the work. Thus, for a given force, the vrork is proportional to the tinte* Art. 18 (a), Poorer. The quantity of any work may evidently be considered without regard to the time required to perform it ; but we often require to know the rate at which work can be done ; that is, how much can be done within a certain time. The rate at which a machine, etc. can work is called its -power. Thus, in selecting a steam-engine, it is important to know how much it can do per minute, hour, or dag. We therefore stipulate that it shall be of so many horse-powers; which means nothing more than that it shall be capable of overcoming resisting forces at the rate of so many times 33,000 foot-pounds per minute when running at a uniform velocity, i. e., when force X distance = resistance X distance. (b) The liorse-poiver, 33,000 foot-pounds per minute, or 550 foot-pounds per second, is the unit of ponrer, or of rate of ivork, commonly used in connec- tion with engines. The metric horse-poorer, called "force dt cheval," " cheval-vapeur," or (German) " Pferdekraft," is 75 kilogram-meters pel second = 542.48 ft-ibs. per sec. = 32,549 ft.-ft>s. per minute = 0.9863 horse-power. 1 horse-power = 1.0138 " force de cheval." In theoretical Mechanics the foot-ponud per second is used in English measure ; and the lUlo§;ram-meter per ceo- ond in metric measure, 1 foot-pound per second =» 0.13826 kilogram-meter per second. 1 kilogram-meter per second = 7.2331 foot-pounds per second. (c) Up to the time when the velocity becomes uniform, the po-wer, or rate 9t vrork, of the train, in Art. 16 (d), is variable, being gradually axelerated. For in each second it overcomes its resistances (and moves its point of application) through a greater distance than during the preceding second. Also, after the steam is shut off, the rate of work is variable, being gradually retarded. When the force of the steam just balances the resistances, the rate of work is uniform. (d) Po-«rer = force X velocity. Since the rate of work is equal to the work done in a given Hrne, as so m&xxy foot-pounds per second, we may find it by dividing the work in foot-pounds done during any given time by the number of seconds in tkst time. Thus _ ^ * , force in pounds X distance In feet Power =■ rate of work = \. , ; • time in seconds FOBGE IN RIGIB BODIES. 343 Bat this is eqaivalent to - . * . J V ^ distance in feet Power -rate of work -force in pounds X time in seconds — -orce in lbs. X velocity in feet per second. Or if we treat only of the work of that force which overcomes resUtancea: or i« eawes where the velocity is either uniform throughout or the same at the beginning and end of the work; Power rate of work _ resistance, w velocity, in ft-lbs. per sec " in ft-lbs. per sec in lbs. ^ in ft per sec. Thus if the resistance is 3300 lbs. and is overcome thrpugh a distance of 10 feet in every minute; or if the resistance is 33 lbs. and is overcome through ?di8tonce of 1000 f4et per minute, the rate. of the work i^J^J^^^J'^ the same, namely, 33,000 foot-pounds per mmute, or one horso-power; Sat lbs. vel. lbs. vel. . 8300 X 10 — 33 X 1000 — 33,000 foot-pounds per mmute. M The same "power" which will overcome a given resistance through* riven distance, in agiven time, will also overcome any other resistance through Wiy other distance, in that same time, provided the «:<»w**^°®**°^.*^^®.5S!? when multiplied together give the same amount as m the first case. Thus. the power that will lift 60 pounds through 10 feet in asecond, will m a second Hft 600 pounds, 1 foot; or 25 pounds. 20 feet; or 6000 pounds ^ oi a foot. El practice, the adjustment of the speed to suit different resistances, is usually effected by the medium of cog-wheels, belts^or lever.. By "^eans of these the engine, watei>wheel, horse, or other motive power, exerting a given force and ruhning at a given velocity, may be made to overcome small resist* ances rapidly, or great ones slowly, as desired. Art. 19 (a). The 'vrork 'vrhldi a bodjr ean do hy -rlrtiie ot its motion j or (which is the same thing) the 'vrorh reonircd to brins the body to rest. Kinetic energy* -vim -viTa^ or "living ttorce.'* As already remarked, a force equal to the weight of any body, at any place, will, in one second, give to the mass or matter of the body a velocity — g, or (on the earth's surface) about 32.2 feet per second. Or if a body be thrown \Lpward with a velocity — ■ g, its weight will stop it in one second. Since, in the latter case, the velocity at the beeinning and at the end of the ■econd are, respectively,— g feet per second, ana — 0, the mean velocity of the iody is -£- feet per second. Therefore, during the second it will rise _^ feeC^ 2 2 or about 16 feet. In other words, the work which any body can do, by virtue of being thrown vertically upward with an initial velocity (velocity at the gtart) otg feet per second, is equal to the product of its weight multiplied of -J- feet Or, work in foot-pounds — weight X -^ Ifotioe that in this ease (since the initial velocity v Is equal to jy), JL. — 1. ^ 9 Smppose now that the same body be thrown upward with double the former velocity; i. e., with an initisd velocity equal to.2 g (or about 64 feet per seconds dince gravity requires (Art 8 c), two seconds to impart or destroy this velocity, the body will now move upward during two seconds, or twice as long a Urns as before. But its mean velocity now is p. or twice as great as before. Therefore, moving for double the time and with double the velocity, it will teavel /our times as far, overcoming the same resistance as before (viz. : its own weight) through /our times the distance. Thus, by making its initial velocity v — 2 p, {. «., by doubling its -L-. making g it — 2, we have enabled the body to do four times the work which it could io when its — !L was 1; so that the work in the second case is equal to the 9 344 FOBOE IN RIGID BODIE& product of that in the first case multiplied by the 8quar$ of -2L( Qg^ - weight X -2- X ^ — weight X — And it is plain that this would be ithe case for any other velocity. Now the total amount of the work which the body can do, is independent of the amount of the resistance against which it is done; for if we increase the resistance we diminish the distance in the same proportion, so that their product, or the amount of work, remains the same. The above formula^ therefore, applies to all cases ; i. 6., the total amoiuit ot 'vrorfc, in fo^ pounds, whicn any body will do, f^ainst any resistance, by virtue of its motioii Alone, in coming to rest, is Work - weight of moving body, in lbs. X square of its velocity in ft per sec^d f/ — weight of moving body, in lbs. X fall in ft required to give the velocity _ weight of moving body, in lbs, y square of its velocity in ft per second g 2 In these equations, the weight is that which the body has in any given plaoe^ and g is the acceleration of gravity at that same place. (b) Since the weight of a body j^ j^^ ^^^ ^^^^ 1^^ ^ 336), the last formula becomes, by "method A,^* Art. 11 (d). mass of moving body w square of its velocity in ft per second in foot^ot^mb " in "matU^' '^ 2 and by "method B," Art. U (e), mass of moving body v> square of its velocity in ft per eeobad infoo^poundato" in potmdij ^ 2 (c) In the above equations the left hand side represents the work (or resis- tance overcome through a. distaiice) in any given case, while the right hand side represents the Unetlo energy of the body, by which it is enabled to do that work. Some writers call this energy "via ▼!▼»,»» or " living force" a name formerly given (for convenience) to a quantity just double the energy, or — mass X velocity*. (d) As an illustration of the foregoing, take a train weighing 1,120,008 pounds, and moving at the rato of 22 feet per second. The kinetic energy ef such a train is energy - weight X I5!2^; or. 1,120,000 lbs. X — — 8,400,000 ft.-lbs. 64.4 That is, if steam be shut off, the train will perform a work of 8,400^000 fL-lba. in coming to rest. Thus, if the sum of all the resistances (of friction, air, grades, curves, ete.) remained constantly — 6000 lbs.,* the train would travel 8,400,000 ft.-lb8. _ lesott, 5000 lbs. (e) We thus see that the total quantity of work which a body can do by virtua of its motion alone, and without assistance ft-om extraneous forces, is in pi^ portion to the weight of the body and to the square of its velocity when it begins to do the work. For example, suppose that a train, at the momaDft when steam is shut off, has a velocity of 10 miles an hour and that the kinetio energy, which that velocity gives it, will by itself carry the tram against th» •In practice, this would not be the case. 9OB0B IK RIGID BODIES. 345 CMistances of Che road, etc^ for it distance of ons quarter of a mile before it stops. Then, if steam be shut off while the train is moTing at 5, 20, 30 or 40 miles per hour (t. e^ with ^^ 2, 8 or 4 times 10 miles per hour) the train will tiavel JL, 1, 2 ^ or 4 miles (or ^ 4, 9 or 10 times ^ mile) before coming to rest* Bat the rate of work done is proportional simply to the resistance and the ntoeity (Art* IBd, p. 842). Therefore, the locomotive whose steam is shat oft at 20, 80 or 40 miles per honr, will require, for running its 4. 9 or 16 quarters tf a mile, but 2, 3 or 4 times as many seconds ae it required at 10 miles per hour. The same principle applies to all cases of acceleration or of retardation.f For instance, in the case of a falling body, the distance through which it mnst fall in order to acquire any giren velocity is as the square of that Telocity, but the time required is simply as the velocity. Also, if a body is ttirown Terticanlly upward with any given velocity, the height to which it will rise bvh the time gravitv destroys that velocity, will be as the square of the Yelooity,but the time wiU be simply as the velocify. Art. SO (a). The momentnin of a moving body (or the product of its mass by its velocity) is the rate, in foot-pounds per second, at which it works against a resisting force equal to its own weighty as in the case of a body thrown vertically upward. At the instant when it comes to rest, its momentum, or rate of work, is of course = nothing. Therefore its mean rate of work, or mean momentum, is one-half of that which it has at the moment of startiug. Thus, suppose such a body to weigh 5 lbs. Then, whatever its velocity may be, 6 pounds is the resisting force, against which it must work while coming to resL Let the initial velocity be 96 feet per second. Then its momentum ■• mass X velocity «— 6 X 96 — 480 foot-pounds per second? Mid, while ooming to rest, its •Moa momentum -» mass X T . ^r^ ■« 240 foot-pounds per second. Now, in falling, the weight of the body (5 lbs.), would ^ve it a velocity of 96 foet per second in about three seconds. Consequently, in rising, it will destroy im lelooity in the tame time. In other words, the time — ,. velocity ^ velocity •^ acceleration g M £| 1. 3. Three seconds, therefore, is the time during which it can work. How, if the mean rate of work in foot-pounds per seeond (at which a body ean work against a resistance) be multiplied by the time during which it can ooBtinue so to work, the product must be the total work done. Or, in this case^ work mean rate of work v^ time, oji* v <» ion *r^* »wvn»^. to IWbe, - in flrlbs. per sec. X or No. of sees. - 240 X 3 - 720 footrpounds. -weight X 12}2£ife X ^^l^^ifc 2 g .weight X y^'?^ ,asinAjt.l9(o),-6 X ^ - 720 ft.ponnda (b) We may notice also that since, in the case of a falling body, or of one ihixywn upward, . ^"^^ is the time during which it must fall in order to acquire a given velocify, or during which it must rise in order to lose it^ therefore, Telocity ^ reloaiij ^ ^ ^^^ velocily X time — distance traversed; so that weight X 1212215? - weight X H^SpLx I2!22!5 ^ weight X dislanee traversed -« the work. — - ^' ' ™" ■^-- l■^■■^■ — ■-■I --■-■■■■■■ ■-■ — ■ . ■■ 1^ ■■ ■■■III, ■■■■■■■■■ _^ ■ ■ I I ■^■^■^M— — i— ^M^ • This sappofes, for oonvenience, that the resistances remain uniform through* out, and are the same in all the cases, which, however, would not hold good in praotioe. t Retardation is merely acceleration in a direction opposite to that of the motion which we happen to be coasidering. 346 fOBCE IN BIOIB BODIES. Art. 91 (a)* Bnawrf to toJ — irucU blc. Energy, expended In wortt, to not destroyed. It is either transterred to other bodies, or eue stored ap in the body itself; or part may be ithua transferred, and the re^t thus stored. Bnt^ althoagh ener^ cannot be destroyed, it may be rendered useless to us. Thn^ amoTing train, in coming to rest on » level track, transfers its kinetic enei into other kinetio energy: namely, the useless heat due tofidctioo at the r brakesand Journals ; and this heat, although none of itiadeatrayed, is disai] Jed the earth and air so as to be practicallyoeyond our recovery. Alt. sa (a). Potential •nergy* or possible energy, may be defined as •toted-np energy. We lift a one-pound body one-foot oy expending upon it one foot-pound of energy. But this foot-pound is stored up in the **sy8tem ** (composed of the earth and the body) as an addition to its stock of potential energy. For, while the stone falls through one foot, the system wilt acquire a kinetic energy of one foot-pound, and will part with one foot-pound of its potential energy. • (b) The potentiai energy of a ''system*' of bodies (such as the earth and a weight raised above it, or the atoms of a mass of powder, or those of a bent spring) depends upon the relative poaitiona of those bodies, and upon their tendencies to change those positions. The kinetie energy of a system (such as the earth and a moving train of cars) depends upon the tnaM«6 m its bodies and upon their motion relatively to each other. Familiar instances of potential energy are— the weight or spring of a clock When fully or partly wound up, and whether moving or not; the pent-up water In a reservoir; the steam pressure in a boiler; and the explosive energy of powder. We have mechanical energy in the case of the weight or springs or water; heat energy in the case of the steam, and obemica! energy in that df the powder. (o) In many oases we ma3r conveniently estimate the total potential enei^ of a systenu Thus (neglecting the resistance of Uie air) the explosive energy of a pound of powder is » the weight of any given cannon ball X the height to which the force of that powder could throw it. •» the weight of the ball X (the square of the initial velocity given to it by the explosion) -i- 20. But in other cases we care to find only a certain definite portion of the total potential energy. Thus, the toM potential energy of a olock-weight* would not be exhausted until the weight reached the center of the earth: but we generally deal only with that portion which was stored In it by winding-up. and which tt will give out again as kinetio energy in running down. This portion is -• th^ weight X the height which it has to run down -• the weight X (the square of the velocity which it would acquire in fallin^/V>oe{y through that height) -i- 2if. (d) There are many cases of energy in which we may hesitate as to whether the term "kinetic" or "potential** Is the more appropriate. Thus, the pres- sure of steam in a boiler is believed to be due to tne violent motion of the particles of steam, which bombard the inner surface of the boiler-shell; so that, from this point of view, we should call the energy of steam kinetie. But, on tne other hand, the shell itself remains stationary; and, until the steam is permitted to escape from the boiler, there fs no outward evidence of energy in the shape of work. The energy remains stored up in the boiler ready kt nse. From this point of view, we may call th e energy of steam potential energy. (e) It seems reasonable to suppose that further knowledge as to the nature of other forms of energy, apparently potential (as is that of steam), might reveal the fact that all energy is ultimatiely kinetio. Art. 23 (a). There is much confusion of ideas in regard to those actions to which, in Mechanics, we give the names, *' force," *• enerfry«'* ** power," etc. This arises from i he fact that in every-day language these terms are used indiscriminately to express the sime ideas. Thus, we commonly speak of the " force " of a cannon-ball flying through the air, meaning, however, the repulsive force which would be exerted between the ball and a building, etc. with which it might come into contact. This force would tend to move a part of the building along in the direction of the flight of the ball, and would move the ball backward ; (i. e., would retard Its forward motion). But this great repulsive "force" does not exist until the ball strikes the building. Indeed, we cannot even tell, from the velocity and weight of the ball, what tne amount of the force will be, for this depends upon the strength, etc., of the building. If the building is of glass, the foroe mav be so slight as scarcely to retard the motion of the ball perceptibly, while,'if the building is an * For convenience we may thus speak of the energy of a mdem of bodies (the earth and the clock-weight) as resiaing in only one of the bodies. FORCE IN RIOIB BODIES. 347 earth embankment, the force will be much greater, and may retard the motion oX the ball so rapidly as to entirely stop it before it has gone a foot farther. The moving ball has great (kinetic) energy; but the only force that it exerts during its flij^ht is the comparatively very slight one required to push aside the particles of air. The energy of the ball, and therefore the total work which it can do, are inde^ pendent of the nature or the obstruction which it meets ; but since the work is the product of the resistance oifered and the distance throu^^h which it can be overcome, the distance must be inversely as the resistance offered ; or (which is the same thing) inversely as the force required of, and exerted by, the ball in balancing that resistance. Since work, in ft.-lb8. => force, in &>s., X distance traversed, in feet, we have force in lbs. = work, in ft.-lbs. _ rate of work, ' distance traversed, in feet in ft.-lbs. per fool. Art. S4 (a). An impact, blow, stroke or collision takes place when a moving body encounters another body. The peculiarity of such cases is that the time of adion of the repulsive force due to the collision Is so short that een- erally it is impossible to measure it, and we therefore cannot calculate the force ttovsx the momentum produced by it in either of the two bodies : but since both bodies undergo a great change of velocity (i.e., a great acceleration) during this Short time, we know that the repulsive force acting between them must be very great. We shall consider only cases of direet Impact, or impact where the centers of gravity of the two bodies approach each other in one straieht line, and where the nature of the surfaces of contact is such that the repulsive force caused by the impact also acts through those centers and in their line of approach. (b) This forcCj acting equally upon the two bodies (Art. fi/), for the same length of time (namely, tne time during which they are in contact), neces- sarily produces equal and opposite changes in their momentums (Art. 12, p. 888). Hence, the total momentum (or product, mass X velocity) of the ttoo bodies is always the same after impact as it was before. (c) But the relative behavior of the two bodies, after collision, depends upon their elasticity. If they could be perfectly inelastic, their velocities, after im- pact, would be equal. In other words, they would move on together. If they could be perfectly elastic, they would separate from each other, after collision, with the same velocity with which they approached each other before collision. (d) Between these two extremes, neither of which is ever perfectly realized in Enictice, there are all possible degrees of elasticity , with corresponding differences 1 the behavior of the bodies. The subject, especially that of indirect impact, is a very complex one, but seldom comes up in practical civil engineering. (e) " In some careful experiments made at Portsmouth dock-yard, England, a man of medium strength, and striking with a maul weighing 18 lbs., the handle •f which was 44 inches long, barely started a bolt about '% of an inch at each blow ; and it required a quiet pressure of 107 tons to press the bolt down the same quantity ; but a smsQl additional weight pressed it completely home." 348 GRA.VITY — ^PALLING BODI£S. «RATITT. FAIililire BOBIK8. Bodies flAlllngr Tertleally. A body, falling freely in racuo from a state of rest, acquires, by the end of tbe first second, a Telocity of about 32.2 feet per second ; and, in each succeeding second, an cuidition of velocity, or aoceleratiod, of about 82.2 feet per second. In other worda^ tbe Telocity receivM in each second an acceleration of about '62.2 feet per second, or is accelerated at tbe raU of about 32.2 feet per second, per B^cond. This rate ie generally called (fbr brerity, see foot-Bote,t p. 334), simply the sM)oeleratloia of gravity (bat see * below), and is denoted by |p« It increases ftx>m about 82.1 f«et per second, par second, at the equator, to about 32.5 at the poles. In the latitude of London it if 82.19. These are its values at sea-level ; but at a height of 6 miles above that level it is diminished by only about 1 part in iOO. For most practical purpoeee it may be taken at 32.2. Caution. Owlnar to tbe resistance of the air none of the follow- ing rules give perfectly accurate results in practice, especially at great vela. The greater the specific gravity of the body the better will oe the rMnlt. The air ffeelets botn rislnir and fklllnir bodies. If a body be tbrown vertically upwards with a given vel, it will rise to the same height from wiiich it must have fallen in order to acquire said vel; and its vel will be retarded in each second 32.2 It per lec* Its average ascend' ing velocity will be half of that with which it startled ; as in all other cases of uniformly retarded vel. In falling it will acquire the same vel that it started up with, and in the same time. See above Caution. Acceleration acquired* in a given time = ff X time in a given fall from rest = \^ 2 g X fall. in a given fall from rest ) __ twice the fall and given time j *~ time Time required - , , x» acceleration for a giyen acceleration >- — 9 for a given fall flrom rest fall fall 3^ final velocity fall for a given fall from rest i _^ or otherwise / ™ mean vel ~~ J^ (initial vel + final vel) FaU In a given time (starting from rest) — time X H ^^"^ ^^ ^ timeS X ^iff in . giren time (.t«:ttagi _ inltl.1 t«1 + ftn.l r«| from rest or otherwise) J 2 reqd for a given acceleration "i __ acceleration^ (starting from rest) ) 2g during any one given second (counting from rest) ■» ^ X (number of the second (Ist, 2d, Ac) — \\ during any equal consecutive times (starting from rest) « 1, 3, 5, 7, 9, Ae. wfti^e f ^^*- 2d. 3d. 4th. 6th. 6th. 7th. 8th. 9th. 10th. ' seconds Velocity; ft per sec. Dist fallen since end of preceding sec ; ft. Total diet fallen; ft. 32.2 16.1 64.4 48.3 96.6 80.5 16.1 1 64.4 144.9 128.8 112.7 257.6 161.0 144.9 402.6 193.2 177.1 679.6 225.4 209.3 788.9 267.6 241.6 1090.4 289.8 278.7 1904.1 822.0 805.9 1610.0 * By " acceleration,** in thi» article, we mean the total aooelerstion ; C «., tbe whole change of velocity occarring in the givwi tins or fUl. For the raft oC *rflHtwrtn> we use simnly the letter g. DESCENT ON INCLINED PLANES. 349 I^escent on Inclined plirnes. When a body, U. is placed upon an inclined plane, AC, its whole weight W is not employed m giviug it ▼elocity (as in the case of bodies falling vertically) but a portion, P, of it (= W X cosine of o = W X cosine of a*) is expended in perpendicular pressure against the plane; while only S, (= W X sine of o — W X sine of a*) acts upon U in a direction parallel to the surface AC of the plane, and tends to slide it down that surf. The acceleration, generated in a given body in a given time, is proportional to the force acting upon the body in the direction of the acceleration Hence If we make W to represent bv scale tbe ttccfeleration g (say 32.2 ft per l*c) which gray would give to U in a sec if falling freely, then S will give, by the same scale, the acceleration in ft per sec which the actual sliding force 8 would give to U in one sec if there were no friction between U and the plane. We have therefore theontio^ acceleration down the plane = gr x siae of a. Therefore we have only to substitute "^. sin a" in place of "flr;" and the </omn^ distance or "slide" AC in place of the corresponding vertical distance or " fall " A £ in the equations, in order to obtain the acceleratioos etc as follows : on an inclined plane witbont friction. In the foUowingr* tbe slides A € are in feet, tbe times in seconds, and tbe velocities and accelerations in feet per second.t Accelerationfof sliding velocity i« - -j««« n^^ "^^rt accel acquired in falling) w i^ _ in a given time = ^^^^ ^^^j^^ the same time / X sin a B g. sin a X time in agiven slide, as AC,> slide from rest i 14 ti°>e f vert accel acquired in falling) =< freely thro the corresponding >•■ { verthtAE J » y' 2 ^. sin a X slide V^7^'^ for a given sliding acceleration Time required sliding acceleration ff, sin a for a given slide, as A C, lirom _^ slide wst "" y^ final sliding veloc « /__8lid iity "" V H flf. si sl ide sin a time reqd to fall freely thro the correspond- ing verthtAE sin a slide slide for a given slide, from > ^ ^ rest or otherwise J "* mean sliding vel "" H (initial + final sliding vels) Cosine a Sine a horizontal stretch, as E C» base EC of any length, aa A C ^ l/AC» — Al? length A C ^ that length A C height A E _ fall, A E. in any given length, A C ^ T/AC2 — te<> length AC"" "^ that length AC * Because o and a are equal. tHr acceleration,*! flW» cartielet we mean the total acceleration, t. «., the whok eha&flle in telodty occurring in the given time or slide, for ttie rate of acceleration ire nse tiaiolT the letter a. 350 GRAVITY — PENDULUMS. Slide, u A C in a glyen time, starting from rest = time X }4 final sliding Tel = time *X}iff. sin a. in a given time, s<«rting from rest ., ,, or otherwise — ""*® X mean sliding Tel - time X H (initial + final, sliding rels) required for a ffiren sliding accel- „ sliding acceleration* oration (starting from rest) *" 2 p. sin a But in praetlce the sUdlmr on the plane ts always on- £!?;^ ^X ™««»- To Inclnde the emJt of Metionrwe hJ^ only to substitute sin a - (cos a. ooeff fric)] " in place of « g. sin a " in the abore eqoatlona. Lse Friction = Perpendicnlar pressure P X coefficient of friction = weight W X cosine a X coefficient of friction and retardation of firletlon '^gX cosine a X coefficient of friction. Besnitant slidinir acceleration « theoretical sliding accel (due to the sliding force, S) — retardation of fHo =- iff. sin a) — (g. cosine a. coeff fric) = ffX fsin a — (cosine a. coelTfrlc) j If the retardation of friction (•= y. cos a X coeff fVic) is not leu than the total •r theoretical accel ("^. sin a") the body cannot slide down the plane. "PX Because • ^ PENDULUMS. Tex numbers of ribrations which diff pendulums will make in any ^Ten place la a giren time, are inversely as the square roots of their lengths : thus, if one of them Is 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; but its number of vibrations will be but ^ /^i or i^ as great. The times in which diff pendulums will make a yibration, are directly as the sq rts of their lengths. Thna, if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, S, or 4 times aa great ; and so also will be the time occupied in one of its vibrations. The length of a pendulum vibrating seconds at the level of the sea, in a Taonmn. in the lat of London (51^ North) is 39.1393 ins ; and in the lat of N. York (409^ North) 39.1013 ins. At the equator about ^ inch shorter ; and at the poles, about -ffg Inch longer. Approximately enough for experiments which occupy but a few sec, we may at any place call the length of a seconds pendulum in the open air, 89 ins ; half sec, fl^^ ins ; and may assume that long and short ribrations of the same pen- dulum are made in the same time ; which they actually are, very nearly. For mea»- nring depths, or dists by sound, a sufficiently good sec pendulum may be* made of a pebble (a small piece of metal is better) and a piecfi of thread, suspended fh>m a common pin. The length of 39 ins should be measured from the centre of the pebble. PXBJSTDULUMS, ETC. 351 In Btartliig tlie Tibratlons, the pebble, or boby must not be thrown into motion, but meroly lei drop^ after extending the string at the proper height.. To find the lenKrtb of a pendalam read to make a given number of vibrations in a min, divide 375 by said reqd number. The square of the quot will bo the length in ins, near enough for such temporary purposes as the foregoing. Thus, for a pendulum to make 100 vibrations per min, we have |^^ =» 3.75 ; and the square of 3.75 = 14.06 ins, the reqd length. To find (lie namber of ▼ibrationti per min for a pendulum of given length, in ins, take the sq rt of said length, and div 375 by said sq rt. Thus, for a pendulum 14.06 ins long, the sq rt is 3.75 ; and z-=i » 100, the reqd number. Rkk. 1. By practising before the sec pendulum of a dock, or one prepared as Just stated, a person will soon learn to ooant 5 in a sec, for a few sec in succession ; and will thus be able to divide a sec into 5 equal parts ; and this may at times be oseftil for ▼ery rough estimating when he has no pendalam. Oentre of Oscillation and Pereusslon* Bsv. 2. When a pendulum, or any other suspended body, is vibrating or oscillating backward and forward, it is plain that those particles of it which are far front the point of suspension move faster than those which are near it. But there is always a certain point in the body, such that if all the particles were concentrated at it, so that all should move with the same actual vel, neither the number of oscillations, nor their angular vel, would be changed. This point is called the center of oKiUa- Hon. It is not the same as the cen of grav, and is always farther than it fh)m the point of suspension. It is also the cerUre of percussion of the suspended vibrating body. The dist of this point fh>m the point of snap is found thus : Suppose the body to be divided into many (the more the better) small parts ; the smaller the better. Find the wc^gnt of each part. Also find the cen of grav of each part ; also the dist firom each such con of gray to the point of susp. Square each of these diets, and mult each square by the wt of the corresponding small part of the body. Add the products together, and call their sum p. Next mult the weight of the entire body by the dist of its cen of grav from the point of susp. Gall the prod p. Divide p hyg* Thinp is the moment of inertia of the body, and if divided by the wt of the body the sq rt of the quotient will be the Radius of Gyration. Angrnlar Telocity. When a body revolves around any axis, the parts which are farther from that axis move faster than those nearer to it. Therefore we cannot assign a stated linear velocity in feet per second, or miles per hour etc, that shall apply to every patriot it. But every part of the body revolves around an entire circle, or through an angle of 860P, in the same time. Hence, all the part« have the same ▼elocity in deare^i per second, or in revolutions per seoond. This is called the angular velocity. Scientific writers measure it by the length of the arc de- scribed by any point In the body in a given time, as a second, the length of the arc being measured by the number of times the length of Us ottn radius la con- tained in it. When so measured, Angular velocity __ liaear velocity (in feet etc) per sec in radU per second - length of radius (in feet etc) Here, as before, the angular velocity is the same for all the points in the body, because the velocities of the several points are directly as their radii or dis- tances from the axis of revolution. In each revolution, each point describes the circumference of the circle in which ft revolves =» 2 v r (ir = 3.1416 etc ; r = radius of said circle). 0>nse- qaently, if the body makes n revolutions per second, the length of the arc de- scribed by each point in one second is 2irrn; and the angular velocity of the body, or linear velocity of any point measured in its own radii, is . 2irr» 2 w » «= say 6.2832 X revs per second = say .1047 X revs pe» f^inute. Moment of Inertia. Sappose a body revolving around an axis, as a grindstone; or oscillating, like apeodnlum. Suppose that the distance from the axis of revolution (which, in the pendulum, Is the point of suspension) to each individual particle of the body, has been measured; and that the square of each such distance has been multiplied by the weight of that particle to which said distance was measured. 352 MOMENT OF INERTIA. The sum of all these products is the moment of inertia of the bodf . Or Moment of Inertia -{ the sum, lor all the particles }•' r weight square of dist -< of X of particle from (.particle axis of revolution or, I = 'S,<fiw. Scientific writers frequently use the mass of each particle ; ie, its weight instead of its weight, in calcnlatiug acceleration (g) of gravity, or about 32.2 the moiueiit of inertia. Ill practice we may suppose the body to be divided into portions measuring a cubic inch (or some other small size) each : and use these insteaO of the theo- retical infioitely small particles. The smaller these portions are taken, the more nearly correct will be the result. When the moment of inertia of a mere surface is wanted (instead of that of a body), we suppose the surface to be divided into a numl)er of small areaSf and use them instead of the weights of the small portions of the body. weight of body, Muare of Table of Radii of Clyratlon. Body Any body or fig^nre Solid cylin- der ditto ditto, infinitely short (circular surface) Hollow cyl* inder ditto, infinitely thin ditto, of any thickness ditto, infinitely thin ditto, infinitely thin and infinitely short (circumfer- ence of a circle) Solid spbere Rewolwinff around any given axis Its longitudinal axis adiam, midway between Its enas a diameter its longitudinal axis ditto a diam midway between its ends ditto a diameter a diameter V Badlas of Gyratioii ' moment of inertia around the given axis weight of body, or area of surface radius of cylinder X 'Xj-k- * radius of cylinder X aboat .7071 V ' length' radiu8> of cylinder 12 "*" 4 V radius of oylinder i nner rad» + outer radi 2 nidlUB of eylfnder V din ner rad' + outer rad* ^ length^ 4 ■*" 12 V radius^ of cylinder length* 12 radias of cylinder X ■at radius of cylinder X about .7071 V radius* of sphere 2.5 = radfus of sphere X V'Ti" = rndins of sphere X about .68246 BADU OP GYRATION. 353 Table of Radii of Oyration,— CoimiruBD. Hollow •plioro of any thickness ditto, thin ditto, inflnitelT thin (spherical surface) 8traiirl>t line, ab Solid eone Circular plate, of rect- angular cross seo- tion Circular ring^, of rectan- $oIftr cross section Square, rect- angrle and otlier snr* RevolviniT aroand a diameter ditto ditto any point, £, In its length either end, a or 6 Its center, e its axis S«e Solid cylin- der See Hollow cylin- der Badins of Oyration V 2 (outer rad* — inner rad») 5 (outer rad* — inner rad') approz (outer rad + inner rad) x ^065 radius of sphere ■■ radios of sphere X al>but .8166 Sab length aft X "\-^ — length abX about JB776 •- length abX about .2887 radius of base of cone X '\/~S' M radius of base of cone x .5477 For the thidcnest of plate or ring, measured perpendicularly to the plane of the circumference, take the length of the cylinder. For Uasl radius of gyration, or that around the longe$t aafs, see p 496 and 497. 2a 364 CENTRIFUGAL FORCE. GEBTTRIFVOAI. FORCE. When a body a, Fig. 1 , moves in a circular path abd^ it tends, at each point, as a or 6^ to move in a tangent at or bif to the circle at that point. But at each point, as a, etc., in the path, it is <ie;fiected from the tangent by a force acting toward the center, c, of tlie circle. This force may be the tension of a string, ca, or the attraction between a planet at e and its moon a^ or the inward pressure of the rails, ah,OB & curve, etc., etc. Like all force, it is an action between two bodies, tending either to separate them or to draw them closer together, and act- ing equally upon both. (See Art. 5 (6), p. 882). In the case of the string, itpnlU the body a, Un/xird the center, r, and the nail or hand, etc., at c, toward the body at a or 6, etc. ; i. e.^from the center. In the case of a oar on a curve it pushes the car toward the center, and the rails from the center. The pull or push on the revolving body toward the center is-called the eentripetttl forc«; while the pull or push tending to move the defecting body from the center is called the cenArira^al force. These two *^ forces," being merely the two ** sides " ^as it were) of the same stress, are necessarily equal and opposite, and can onlv exist toffether. The moment the stress or tension exceeds ihe strength (or inherent conesive force) of the string, etc., the latter breaks. The centripetal and centrif- ugal forces therefore instantly cease ; and the body, no longer disturbed by a deflecting force, moves on, at a uniform velocity,* in a tangent, at or M', etc., to its circular path*; «. «., at right angles to the direction whloh the centrifugal force had at the moment it ceased. 4 (a). A singrle revoliringr body, a, Fig. 1. Let = the centrifugal or centripetal force, in pounds. = the weight of the body a, in pounds, = the radius ca of the path of the center of arc V = the uniform velocity of the body a in ftt circular path dbd,'iu feet per = the radius ca of the path of the center ofgraviiy of the body a, in feet. second, ft — the number of revolutions per minute, ^ a the acceleration of gravity = say 32.2 feet per second per second, 900 ^ = about 28980. «■ = circumference -^ diameter » say 3.1416. ir* « about 9.869ft. Then, for the centriAigal force, /: If we have the velocity v in feet per second : / = W ^ t • • • (1) If we have the number n of revolutions per minute : / = W ' t • • • (2) 9U0 g /» about .0008406 WB»* 2 ... (8). * Neglecting friction, gravity, the resistance of the air, etc. t For let a/. Fig. 1, represent the amount and direction of the velocity • of the body at a in feet per second. Then at the end of one second the body will have reached the point b (the arc ab being made = a/), and the amount and direction of its velocity at b will then be represented by the line bt' = a< in lengtli, but differing; in direction. Drawing cu and cm' at the center, equal and parallel respectively t<i at and bt'y we find that the change in the direction of the motion (».«., the acceleration towaid the center) during the second is represented by the arc mm' ; and, since angle aeb = angle ttcu', we have the proportion, radius H or m : ab or at :: cu or at: arc Mt\ In other words, the acceleration tnt' in one second, or rate of acceleration, is ^ aC V* ■o '^ ^* ^"^> ^^^ ^^® f<°^^ causing that acceleration, we have / = mass of body X rate of acceleration =» mass of body X v "■ ^ ^S~' JByformula(l),/ = W . But»=— — — :andv* ^g 60 * 3600 900 ' It X „, »r*R'n« _.ir«Rn» $ Formula (3) is obtained from (2) by substituting the values 9.8696 and 2898U for v" and 900 g respectively. CENTRIFUGAL POftCE. 355 (b) Wbecls and dines. Suppose the rim of a wheel to be cut into verj short dices, as shown (much exaggerated) at a, Fig. 2. Then for each slice, as a, by formula (1): /= weight W of slice X ^ ;* and if each slice were connected ti m o with tb« eenter by a separate string, the mini of the titresses in all the strings (taeglectlhg friction between adjacent slices) would be: F — sum of centrifugal forces of all the slices f = weight of riin X 'Big' (4). But the stress with which we ure usually concerned in such cases (viz. : the tension f n tbe rim Itself in the direction of a taiugent to its own cir- cumference) is much Uss than the theoretical quantity F obtained from formula (4), being in fact only T^j^n ^^ ^^* ^^^ suppose first that the same thin rim is cut only at two opposite points m and n, Fig. 3, and that its two halves are held together only by toe string S. * If the rim is very thin in proportion to itB diameter mn^ we may take the center •f gravity of each slice as bein^ io a circle mn midway between the inner and outer M * A*. _» ^ Ai..^-.. inner radius 4- outer radius - - - . ^, edges of the rim, so thAt K = ^ . In a rim of appreciable thidnMSS, this is not the case, because each slice is a little thicker at its outer than at its inner end. See Fig. 6. Hence its center of grHvity is a little outside of the curved line AMI, Fig. 2. t In a perfectly balanced rim (». «., a rim whose center of gravity coincides with its eenter of rotation, as in Fif;. 3) the centrifugal forces of tbe particles on one side of c counterbalance those on the opposite side. Here, too, K = 0. Hence, as a lehole^ nich a rim hss lu> centrifugal force ; i. «., no tendency to leave the center in any one Abrection by rirtue of its rotation. But if the two centers do not coincide (Fig. 4), then the rim is a single revolving body, and its centrifugal force is : / = weight •f entire rim X ^~ ; where B is the distance between the two centers, and v the &g velocity of the center of grarity a. The force / acts in the line Joining the two wnters. 356 CENTRIFUGAL FORCE. Then : • F semi-circumference mzn : diameter tnn : '• 'tt ' pull on the string S ; 80 that pull on half weight ^ i^ ^ _2 weight t>* F F ■trtny S "^ of rim '^ R^ ^ ir "" of rim '^ R ^ir"" ir"" 3.1416 ' ' * ^ '' and if the rim is now made complete bv joining the ends at m and n, and if the string S is removed, then the pull on the string by formula (5) will be equallv iLivided between m and n. Hence each cross-section, as m orn, of the rim, will sustain a tensile stress equal to half the pull on the string; or «^».«^« *- «««. il ?- weight of rim Xt>' .-. tension m rim »- - ^28^ ^ 6.2882 Ry ' <®>' The centripetal force,/, Fig. 2, holding any part o of the rim to its circular path, is the resultant of the two equal tensions at the ends of that part. For the streu per square inch of cross-section of rim, we have : _ ten sion in rim ~ area A of cross-eection of rim, in square inches F _ weight of rim X v* ,-. ""6.2832A~ 6.2882 A R^ ^ '* We shall arrive at the same result if we reflect that the pull in the string S or the sum of the two tensions at m and n, is equal to the centrifugal force /of either half of the rim, revolving, as a sinsle body, about the center e. Find the center of gravity G of the half rim, and then, in formula (1), use the velocity of that point, and the radius cG instead of velocity at g and radius cz reepectively ; thus: «.,ii *« »*^«« — / ^ centrifugal force_ „^i„ut «<• u„i/ ^„ >• (velocity at G)« . pull in string == / = ^^ haff-rim = '^^'^^^ ^f half-rim X ^^^ ; and half of this Is the tension in each cross-section of the rim.t If the rim were Infinitely thin, cG, Fig. 3, would be 0.6366 ex. If Its thickness must be taken Into consideration, and If it is of rectangular crofls-section, find the centers of gravity g and jK, Fig. 6, of the whole semiolrcular segment cz and of the small segment c6 respectively (eg *» 0.4244 oe, and eg' = 0.4244 eb. Then . , area of entir e segment cz g'^ = gg'X area~of halTrim * For rims of other than rectangular cross-secticHi, use formulae (4), (5) and r6). In a disc, sncli as a irrlndstone, the tension In each full cross-section mn. Fig. 7, is equal to the centrifugal force / of ha^ the disc. Let W » weight of half disc. The distance cG from the center c to the center of gravity G of the half disc, Is cG = 0.4244 cz ; and the * In Fig. 2, let the centrifugal force of any slice, o, be represented by the diagonal, /, of a rectangle, whose sides. H and Y, are respectively parallel and perpendicular to the given diameter mn. Tnen H and V represent the components of / in those two directions. The equal and opposite horizontal components H, of o and of th<* corresponding slice o', being parallel to mn, have no tendency to pull the rim ^^art at m or n. Hence, the pull on a string S, Fig. 3, perpendicular to mn, is the sum of the components Y of all the slices. For each very thin slice. Fig. 6 (greatly exaggerated) we have (since angle A = angle A') : Length I . its horizontal . . centrifugal force , its vertical of slice • projection, p ' ' /, of slice * component V. Hence, for the entire half-rim mn^ Fig. 3 (made up of such slices), we have: «rh.lf.rim • prelection «, ' ' sT^or^. llfiim' ' f*^ J,,^'. •>' which is identical with the proportion at top of page. t The rim* of revolving wheels are usually made strong enough to resist the tension due to the centrifugal force, without aid from the apohe$^ which thus have merely to support the weight of the wheel. But if the rim breaks, the centrifugal forces of its fragments come entirely upon the spokes; and, since the breakage is always irregn- lar, some of the spokes will always receive more than their share. CENTRIFUGAL FORCE. 357 rad. cGXfl' 0.4244 czx^ (8). (»). = W = w 0.4244 (vel. at g)« czXg 0.4244 g« n» cz 900^ The stress per square inch in any full section mn is tension in mn unit stress => area of cross-section in square inches 0.4244 (velocity at g)' = W -W diam. mm, ins. X thickness, ins. XczXff 0.4244 ir« w« cz diam. mn, ins. X thickness, ins. X 900 ^ . .(10)1 . . (11). Fig. 5 n m c n Fig. 7 f= the centripetal force, in pounds, acting uvon a single revolving body, a, Figs. 1, 2, 4 and 5, or upon the halt-rim or half-disc, Figs. 8, 6 and 7 = the centrifugal force exerted by such body.' TP = the sura of the centrifugal forces f, of all the particles of a rim, Fig. 3. W = the weight of the body, in pounds. R = the radius c<iy Figs. 1, 4 aud 5, of the path of the center of gravity of the body. V = the uniform velocity of the body in its circular path, in feet per second. n = tlie number of revolutions per minute. g = the acceleration of gravity = say 32.2 feet per second. 900 g = about 28980. oircumfereuce w = = say 3.1416. ir* = about 9.8696. diameter In m rolling wlieel, each point in the rim, during the moment when it touches the ground, is stationary tpith respect to the earth; but each particle has the same velocity abont the center as if the latter were stationary, and hence the •entrifugal force has no effect upon the weight. 368 8T4TICS. STATICS. FORCES. !• Statics Defined. The science of 3tatics, or of equilibrium of forces; takes account of those very numerous cases where the forces under con- sideration are in equilibrium, or balanced. It embraces, therefore, all cases of bodies which are said to be "at rest."* 2. In the problems usually presented in civil engineering, a certain given force, or certain given forces, applied to a stationary* body (as a bridge or building) tend to produce motion, either in the structure as a whole or in one or more of its members; and it is required to find and to apply another force or other forces which will balance the tendency to motion, and thus permit the structure and its members to remain at rest. See If 33, below. 3. Equilibrium* Suppose a body to be acted upon by certain forces. Then those forces are said to be in equilibrium, when, as a whole, they pro- duce no change in the body's state of rest or of motion, either as regards its motion as a whole along any particular line (motion of translation), or as regards its rotation about any point, either within or without the body. In such cases the body also is said to be in equilibrium. See % 84, below. 4* A body may be in equilibrium as regards the forces imder consideration, even -though not in equilibrium as regards other forces. Thus, a. stone, held between the thumb and finger, is in equilibrium as regards their two equal pressures, even though it may be lifted upward by the excess of the muscular force of the arm over the attraction between the earth and the stone. Simi- larly, on a level railroad, a car is in equilibrium as regards gravity and the upward resistance of the rails, although the horizontal pull of the locomotive may exceed the resistance to traction. 5. molecular Action. Any force, applied to a body, is in fact made up of a system of forces, often parallel or nearly so, applied to the several particles of the body. Thus, the attraction exerted by the earth upon a grain of sand or upon the moon is, strictly speaking, a cluster of nearly par- allel forces exerted upon the several particles of those bodies ; but, for con- venience, and so far only as concerns their tendency to move the body as a whole, we conceive of such forces as replaced by a single force, equal to their sum and acting in one line. In thus considering the forces, we as- ^me that the bodies are absolutely rigid, so that each of them acts as a angle " material particle" or " material point." 6. Transmission of Force. The upward pressure of the ground, upon a stone resting upon it, acts directly only upon those particles which are nearest to the ground. These, in turn, exert a (practically) equal upward force upon those immediately above them, and so on; and the i<^rce is thus transmitted throughout the stone. 7. Rigid Bodies. In treating of bodies as rigid, we assume that the intermolecular forces hold the several particles absolutely in their original relative positions. It is not the material that resists being broken, but the forces which hold its particles in their places. Thus, a cake of ice may sustain a great pressure; but its particles yield readily when its cohesive forces are destroyed by a melting temperature. 8. Force Units. The force units generally used in statics are those of weight, as the pound and the kilogram. See Conversion Tables, p. 235. In statics we have no occasion to consider the masses of bodies (except * Strictly speaking, absolute rest is scarcely conceivable, since all bodies are actually in motion (see Art. 3, p. 331). so that unbalanced forces produce merely changes in the states of motion oi bodies. Yet, for a body to be at rest, relative to other bodies, is a very common condition, and, in practical statics, we usually regard the body under consideration as being at rest relatively to the earth or to some other large body, so that the oaange of state of motion, due to the action of unbalanced force upon it, consists in a change from relative rest to relative motion. See % 33, below. FORCES. 359 in so far as these determine their weights, or the force of gravity exerted upon them), bodies being regarded merely as the media upon and through which the forces under consideration are exerted. Hence we require, in statics, no units of mass; and, as the bodies are regarded as being "at rest," no upits of time, velocity, acceleration, momentum, or energy. 0. Forces, how Petermlned. A force is fully determined when we know (1) its amount (as in pounds, or in some other weight unit), (2) its direction, (3) its sense (see % 10), and (4) its position or its point of applica- tion. 10. When a force is represented by a line, the length of the line mav be made to represent by scale the amount of the force, and its direction and position may often be made to indicate those of the force, while the sense of the force may be shown by arrows or letters affixed to the lines, or by the signs, + and — . Thus, the directuma of the forces represented by lines a and 6, Fig. 1, are vertical, and those of e and d are horizontal. The sense of a is upward, of b downward, of c right-handed, of d left-handed. Thus, a and b are of like direction, but of opposite sense; and so with c and d. In treating of vertical or horieontal forces, we usually call upward or right-handed forces posi- tive, and downward or left-handed forces nef^rative, as indicated by the signs^ 4- and — , in Fig. 1.^ When a force is designated by two letters, at- tached to the line representing it, one at each end of the line, the sense of the force may be indicated by the order in which the letters are taken. Thus, in Fig. 1, having regard to the directions of the arrows, we have forces, ef, ha* k Cand n m, 11. Hfine of Action, etc. The point (see ^ 6) at which a force P, Fig. 2, is supposed to be applied, as a, is called its point -of application. But the force is transmitted, by the particles, throughout the body (see ^ 6), and :t:i k I *n n — y ■< — g ri». 1. the e€fect of the force, as regards the body as a whole, is not changed if it be re^rded as acting at any other ix>int, as 6, in its line of action. We may therefore regard any point in that line as a point of application of the force. For instance, the tendency to move the stone, Fig. 2, as a whole, will not bo changed if, instead of pushing it, at a, we apply a puU (in the same direction and in the same sense) at b; and if a weight, P, be laid upon the top of the hook, at b. Fig. 3, it will have the same tendency, to move the hook as a whole, as it has when suspended from the hook as in the Fig. A force cannot actually be applied to a body at a point outside of the sub- stance of the body, as between the upper and lower portions of the hook in Fig. 3, yet this portion also of the line a 6 is a part of the line of action of the force. The vertical force, exerted by the weight, P, is transmitted to b by means c^ bending moments in the bent portion of the hook. 12. Stress. (See Art. 1, Strength of Materials, p. 454.) Opposing forces, applied to a body by contact (see Art. 5 c, p. 332), cause stress, or the exertion of intermolecular force, within it, or between its particles, tending to pull them apart (tension) or to press them closer together (compression). The stress, due to two equal opposing forces, is equal to one of them. Tension and Compression. Ties, Struts, etc. If the action of the forces tends to pull farther apart the particles of the body upon which they act, the stress is called a tension or pull, or a tensile stress. If it tends to press them claser together, the stress is called a pressure, com- tvession or push, or a compressive stress. A long slender piece sustaining tension is called a tie. One sustaining compression is called a strut or |X)8t. One capable of sustaining either tension or compression is called a tie-strut or strut-tie. 360 STATICS. MOMENTS. 13. Moments. If, from any point, o, or </, Fig. 4, a line, o c or o' «, be drawn normally to the line of action, n m, of a force. Pi, whether the point, o or o\ be within or outside of the body upon which the force, Pi, is acting, said line, ocot </ «, is called the arm or leverage of the force about such point; and if the amount of the force, in lbs., eto., be multiplied by the length of the arm, in ft., etc., then the product, in ft.-lbs., etc., is called the moment of the force about that point.* The moment represents the total tendency of the force to produce rotation about the given point. A force has evidently no moment about any point in its line of action. 14. Sense of Moments. Since the moment of Pi about o. Fig. 4, tends to cause rotation (about that point) in the direction of the motion of the hands of a clock, as we look at the clock and at the figure, or from left to right, as indicated by the arrow on the circle around o, it is called a clock- wise or right-hand moment ; but the moment of the same force about </ tends to produce rotation from right to left. Hence it is called a counter- clockwise or left-hand moment, as is also that of P« about o. Right- hand or clockwise moments are conventionally considered as positive, or +t and left-hand or counter-clockwise moments as negativet or — ; 15. The pl£ine of a moment is that plane in which lie both the line of action and the arm of the force. 16. The resultant or combined tendencv of two or more moments in the same plane is equal to the algebraic sum ox the several moments. Thus, Fig. 4, if the forces, Pi, P2, and Pa, are respectively 6, 5, and 3 lbs., and if the arms, oc, oy, and o 0, of their moments about o are respectively 7* 6, and 8 ft., we have Pi . c — Pi .0 y 4- Ps . o « -6X7—6X6 + 3X3 - 42 — 30 + 9 =21 ft.-lbs. ^Zy'm. I K — n — • ©i^^o 5*— IF— ^ k- — f ^ FIgr. S. Figr. 6. 17. If the algebraic sum of the moments is zero, they are in equilibrium and tend to cause no rotation of the body about the given point. Thus, in Fig. 6, where W is the weight, and G the center of ^pavity of the body, and R the upward reaction of the left support, a, taking moments about the right support, b, we have R / — W a; — zero ; or R i — W «. Hence, W X having W, x and Z, to find R, we have R — - .- . Similarly, in Fig. 6, where W — weight of beam alone, and g^ the center of gravity of W, is at the center of the span /, so that the leverage b g of the weight of the beam about h, is -■ - -, we take moments about &, thus: R Z i- O o — W- - — Mm — N n — zero; or Mm + Nn + W — — Oo R- 2 . I '*'Note that a very small force may have a great moment about a point, while a much greater force, passing nearer to the same point, may have a smaller moment about it ; or, passing through the jwint, no moment at all. MOMENTS. 861 In Fig. 7, where W is the weight of the beam itself, and w its leverage, tak- ing moments about b, we have + RZ + O0 — Nn — Ww-|-Mm = 0; Wi£> + Nn — Mm — Oo Hence, Reaction at a R I In any case, if W be the combined weight and G the common center of gravity, of the beam and its several loads, and x the horizontal distance of that center from the right support, h\ and if I be the span, R the reaction of the left support, a, and R' that of the right support, 6, we have R - Wx I W Ifx-|-, Ria-^ -R'. I and R' - W — R. Flff. 7. Note that the moments^ of two or more forces, about a given point, may be in equilibrium, while the forces themselves are not in equilibrium. See 1 84, below. 18. Center of Moments* So far as concerns equilibrium of moments, it is immaterial what point is selected as a center of moments ; but it is gen- erally convenient to take the .center of moments in the line of action of one (or more, if there be concurrent forces, see ^ 19) of the unknown forces, for we thus eliminate that force or those forces from the equation. CLASSIFICATION OF FORCES. 19. Classification of Forces, and Parallel Forces. Concurrent, Colin ear, Coplanar. Forces are called concurrent when their lines of Figr* s. Figr. 9. action meet at one point, as a, b, c, d, e and /, or / and g. Fig. 8 ; non-concur- rent when they do not so meet, as c and g; colinear when their lines of action coincide, as a and b. or c and d; non-colinear when they do not coincide, as b and /; coplanar when their lines of action lie in one plane,* as a, b, c, d and c, or b, f and (7, etc. ; non-coplanar, as c and g, or 6, / and d, when they do not he in one plane; parallel wnen their lines of action are parallel, as and g\ non-parallel when those lines are not parallel, as b and /. *Acting wpon a plane, as in Fig. 9, must not be confounded with acting in that plane, as in Figs. 70, etc. 862 STATICS. Any two parallel forces must be coplanar. Three or more parallel forces may or may not be coplanar. Any two concurrent forces must be coplanar. Three or more concurrent forces mav or may not be coplanar. Any two ooplanar forces must be either parallel or concurrent. COMPOSITION AND BESOLUTION OF FORCES. SO. Kesultant. A single force, which can produce, upon a body con« sidered as a whole, the same effect as two or more given forces combined, is called the resultant of those forces. Thus, in Fig. 10 (b), a downward pres- sure, G, ■= to + W, is the resultant of the downward pressures w and W; and, in Fig. 11 (6), a downward pressure, =■ W — tr, is the resultant of the downward pressure W and the upward pull w of the leit-hand string.* 31. Component.- Any two or more forces which, together, produce, upon a body considered as a whole, the same effect as one given force, are called the components of that force, which thus' becomes their resultant. Thus, in Fig. 10 (6), w and W are the components of the total force, G, = «; + W. In Fig. 1 1 (6), + W ( = 5) and m) ( - — 3) are the components of G.* 22. If we take into account the resultant of any given forces, those forces (components) themselves must of course be left out of account, as regards their action upon the body as a whole; although we may still have to con- eider their effect upon its particles. Vice versa, if the forces (components) are considered, their resultant must be neglected. Fflff. 10. 6 (C) 3 S^ » FI9. 11. 23. Anti-resultant. The anti-resultant of one or more forces is a sinsle force which, acting upon any body or system of bodies considered as a wh(ue, produces an effect eoual, but opposite, to that of their resultant. In other words, the anti-resultant is the force reouired to hold the given force or forces in equilibrium. Thus, in Fig. 10 (o), the upward reaction, G, of the sround, is the anti-resultant of the two downward forces, w and W ; and the downward resultant, W 4- to, of W and to, is the anti-resultant of G. In Fig. 11 (6), G (upward) is the anti-resultant of W (downward) and to (acting upward through the left-hand string). Similarly, this upward pull of tff is the anti-resultant of W and G. 24. In any system of balanced forces (forces in equilibrium), any one of the forces is the anti-resultant of all the rest ; and any two or more of them have, for their resultaht, the anti-resultant of all the rest. In such a system, the resultant (and the anti-resultant) of all the (balanced) forces is zero. 25. Anti-component. The anti-components of a given force, or of a given system of forces, are any two or more forces whose resultant is the anti- resultant of the given force or of the given system of forces. 26. Composition and Resolution of Forces. The operation of finding the resultant of any given system of forces is called the composition of forces; while that of finding any desired components of a given force is called the resolution of the force. ♦ For convenience, we here reverse the convention of H 10. COLINEAR FORCES. 363 Colinear Forces. 27* Let the vertical line, w. Fig. 10 (6), represent, by any oonyenient scale, the weight of the upper stone in Fi^. 10 (a), and W that of the lower stone. Then, w + W, ■". G, ~ the combined length of the two lines, gives, by the same scale, the combined weight of the two stones, and a verticu line G, coincident with them, equal to tneir sum, and pointing upward, would represent their anti-resultuit, or the reaotioii of the ground. (a) \ \ V / / a \ \ A / / 6 . W^ w* lb) (C) z=io< JUve to to to to to to Tauat» t9 t9 t» t9 t» ^ r^) ^ J ::i B'-'Sei 1>« >t» 94 \ J Fi«. 13. !88. Similarly, if, at each panel point of the lower chord in the bridgo truss in Fig. 12 (a), we have 2 tons dead load (weight of bridge and floor, etc.*) axul 10 tons live load (train, vehicles, cattle, passengers, etc.), the com* bined length of the two lines in Fig. 12 (b), L - 10, and D - 2, gives the tota*. panel load of 12 tons. 29. In Fig. 11 the prenure, 5 lbs., of W upon the ground, is diminished by the 3 lbs. upward pull of the cord, transmitted from the smaller weight i9, leaving 2 lbs. upwara pressure to be exerted bv the ground in order to main- tun equilibrium. The upward reaction, R, of the pulley is — w + W — G -■8 + 6 — 2 -* 6. This is represented graphically in Fig. 11 (c). 30. In the truss shown in Fig. 12 (a), the total dead and live load is — 6 X 12—72 tons, and half this total load, or 36 tons, rests upon each abut- ment. Hence, to preserve equilibrium, each abutment must exert an up- ward reaction of 36 tons; but, in order to ascertain how much of these 36 tons is iranamiUed through the end-pott, a e, we must deduct from it the 12 tons which we assume to be originally concentrated, as dead and live load* at the panel jpoint a; for this portion is evidently not transmitted through a e. Accordingly, in Fig. 12 (c), we draw R upward, and equal by scale to 36 tons: and, from its upper end, draw p downward and — 12 tons. The remainder of R, — R — p -• 36 ^ 12 — 24 tons, is then the pressure trans- mitted through a e. 31* Golinear forces are called similar when they are of like sense, and opposite when of opposite sense. The tame distinction applies to result* ants. b h a ■^— f o c Figr. 13. d • 33* For equilibrium, under the action of colinear forces, itia, ci oo^irse, necessary that toe sum of the forces acting in one sense be equal to the sum of those acting in the oppomte sense, or, in other words, that the algebraic sum of ail the forces be zero. Thus, in Fig. 13, if the forces are in equilibrium, the sum, b a ■{• a o, ot the two right-handed forces must be equal to the sum, ed + dc + co, of the three left-handed forces. Or, con- sidering the right-handed forces, b a and a o, as positive, and the left-handed forces, e dj d c and c o, as negative, as in ^ 10, we have, as the condition of equilibrium of colinear forces : ba •{- ao — oc—'cd — de — O. *The dead load is, of course, never actually concentrated upon one chord, as here indicated ; but It is often assumed, for convenience, that it is so concentrated. 364 STATICS. In other words, the algebraic sum of all the forces must be zero; or, more briefly, 2 forces — 0, where the Greek letter S (sigma), or sign of summation, is to be read "The sum of — ." 33* Two equal and opposite forces, acting upon a body, are com- monly said to keep it at rest ; but, strictly speaking, they merely prevent each other from moving the body, and thus permit it to remain at rest, so far as they are concerned ; for they cannot keep it at rest against the action of any third force, however slight and in whatever direction it may act; and the body itself has no tendency to move. 34. Unequal Opposite Forces. If two opposite forces, acting upon a body, are unequal, the smaller one, and an equal portion of the greater one, act against each other, producing no effect Uf^n the body as a whole; while the remainder, the resultant, moves the body in its own direction. Concurrent Coplanar Forces. The Force Parallelogram. 35. Composition. Let the two lines, ao,bo, in any of the diagrams of Fig. 14, represent, in magnitude, direction and sense, concurrent forces whose lines of action meet at the point o. Then, in the parallelogram, acbo, formed upon the lines a o^ b o, the resultant of those two forces is repre- sented, in magnitude and in direction, by that diagonal, R, which passes through the point, o, ci concurrence. The parallelogram, a c & o, is called a force parallelogram. a' (a) o "^^^^V* ligr. 14. 36. Resolution. Conversely, to find the components of a given force, o c, Fig. 14, when it is resolved in any two ^ven directions, o a, o 6, draw the lines, o a\ o b\ in those directions and of mdefinite length, and upon these lines, with the diagonal R » o c, construct the force parallelogram a ch o» The sides, o a, ob, of the parallelogram then represent the required compo- nents in amotmt and in direction. ^ 37. Caution. The two forces, a o and b o. Fig. 14, may act either toward or from the point o; or, in other words^ they may act either as pulls or as pushes ; but the lines representing them m the parallelogram, and meeting at the point, o, must be drawn, either both as pushes or both as pulls; and the resultant, R, as represented by the diagonal of the pandlelogram, will be a pull or a push, according as the two forces are represented as pulls or as pushes. 38. Thus, in Fig. 15 (^a), the inclined end-post of the truss pushes obliouely downward toward o, with a force represented by a' o, while the lower chord pulls away from o, toward the ri^ht, with a force represented by o V, If, now, we were to construct, in Fig. 15 (a), the parallelogram o a' cf V^ we should obtain the diagonal o cf or c' o, which does not represent the true re- sultant. In fact, as one of the two forces acts toward, and the other from, the point, o, we could not tell (even if R' were the direction of the resultant) in which sense its arrow should point. We must first either suppose the push, a' o, in the end-post, toward o,^ to be carried on beyond o, so as to act as a pull, o a. Fig. 16 (o) (of course, in the same direction and sense as before), thus treating both forces as pulls; or FOBCE PABALIiELOOKAH. 366 rise we must similarly suppose the pull, o V, in the chord, to be transformed into the push, 6 o, of Fig. 15 (c), thus treating both forces as pushes. In either case we obtain the true resultant, R ( » a' 5', Fig. 15a), which, in this ease, represents the vertical downward pressure of the end of the truss upon the abutment. FtfT- IS- Caution. The tensile force, exerted at the end of a flexible tie, neces- sarily acts in the line of the tie; but, in general, the pressure, exerted at the end of a strut, acts in the line of the axis of the strut only when all the forces producing it are applied at the other end of the strut. Thus, in Fig. 15 id), the components, R and H, of the weight, W, do not coin- cide with the axis of the beam which supports the Toad; but in Fig. 15 (e), where the weight acts at the intersection of the two struts, its com- ponents, R and H, do coincide with the axes of the struts. See idso Figs. 143 and 145 (b). 39. Demonstration. The rational demonstration of the principle of the force parallelogram is given in treatises on Mechanics. (See Bioliog- raphy.) It may be established experimentally as indicated in Fig. 16, where c o represents by scale the pull shown by the spring balance C, while o a and o h represent those shown by A and B respectively. 40.- Equations for Components and Resultant. Given the amounts of the forces, a and c, or of the resultant, R, and the angles formed between them. Fig. 17 (a), we have'*': ♦ See dotted lines, Fig. 17 (a), noting that c* ^ c; c. sin (x + i/) -» R. sin X, and a. sin (x + y) >« K. sin y. 366 STATICS. rt = c sin (x 4- y) R^ Bin X sin X — a si n (x + y) sin y „ = R »>» » sin (x + y)* sin (X -i- 1/)* , . ^, If the angle between the two forces is 90**, Fig. 17 (b), these formulas be< me: come: cos y cos X c — R cos y; a =» R cos x. FiiT- 17. 41. Position and Sense of Resultant. Figs. 18. If the lines representing the components be drawn in accordance with Iff 37 and 38, and if a straight line, m n or m' n', be drawn through the point, o. of concur- rence, in such a way that both forces are on one side of that line, then the line representing the resultant will be found upon the same side of that line with the components, and between them ; and it will act toward the line, m n or tn* n'r ii the components act toward it, and vice versa. The resultant is necessarily in the same plane with its two components. tm /^^MiosS^ nC^T^^^' ^r -s/' Fiir. 18. Fis. 19. 42* If one of the components is colinear with the force, it is the force itself, and the other component is zero. In other words, a force cannot be resolved into two non-colinear components, one of which is^n the line of action of the force. Thus the rope, o e. Fig. 19, may receive assistance from tu}o ad- ditional ropes, pulling in the directions a c, and c b; for the resultant of their pulls may coincide with o c; but, so long as o c remains vertical, no aingU force, as c a or c b, can relieve it, imless acting in its own direction c o. 43. In Fig. 20, the load, P, placed at C, ia suspended entirely by the verti- cal member B C, and exerts directly no pull along the horizontal member, C £. Neither does a puU in the latter exert any eneot upon the force acting in B C, so long as B C remains vertical. But the tension in B C, acting at B, does exert a thrust o a along B D, although that member is at right angles to B C; for B G meets there also the inclined member A B; and the tension o d \3 thus resolved into o a and o 6, along B D and B A respectively. The horizontal thrust, o a, in B D, is really the anti-resultant of the horizontal comp>onent, db, of the oblique thrust in the end-poet B A, at its head, B, which thrust is — the pull in A £, due to P. FORCE TRIAKGLE. 367 44. In Fig. 21, the tension, o e, in the inclined tie, D G, is resolved, at D, into o a and o b, acting at right angles to each other along D F and D £ re- spectively. 45. A resultant may be either greater or less than either one of its two oblique components, but it is always less than their sum. If the components are equal, and if the angle between them » 120^, the resultant is eaueii to one of them. Therefore the same weight which would break a single vertical rope or post, would break two such ropes or posts, each inclined 60° to the vertical. Fly. 91. The Force Triangle. 46. The Force Triangle. Inasmuch as the two triangles, into which a paralldogram is divided by its diagonal, are similar and equal. It is suffi- cient to cu^w either one of these triangles, aoc or h oc. Figs. 14, 16, 18, in- stead of the entire parallelogram. 47. If three concurrent coplanar forces are in equilibrium, the lines repH resenting them form a triangle; and the arrows, indicating their senses, foUow each other around the triangle. Thus, in Fi^. 22 (a), we have, acting at o and balancing each other there, three forces: vu., (1) the vertical down- ward force o c of the weight, acting as a pull through the rope o c, (2) the horizontal thrust a o through the oeam a o, and (3) the upward inclined thrust 6 of the strut o b, all acting in the senses (o c,ao,b o) in which the letters are taken, and as indicated by the arrows. 48. Each of the forces in Fig. 22 (&) and (c) is th6 anti-resultant of the other two in the same triangle ; and, if its sense be reversed, it becomes their resultant. Thus, o c, Fisj. 22 (b), is the anti-resultant, and c o the resultant, ofea and a o; and o c. Fig. 22 (c), is the anti-resultant, and c o the resultant of e & and bo,cb being parallel to a o. Fig (b), and representing the thruflt exerted by the horizontal beam against the joint o, Fig. (a).* ib) (c) id) (e) ^^ • c e^t, Flff. 33. *Fig. 22 (tO and (e), representing the same two forces, a o, b o, of Fig. 22 (a), show the erroneous resultant (a b) obtained if the lines are drawn with their arrows pointing both toward or both from the meeting-point of the lines. See ^1f 37, 38. A comparison of any force parallelogram, as that in Fig. 18, with either of the two force triangles composing it, will show that this, while apparently contradicting Ht 37 and 38, is merely another statement of the same fact. The apparent contradiction is due to the fact that, in the force triangle, the lines representing the forces do not meet at the point, o, of concurrence of the forces.' 368 STATICS. 40. Converselsr, if the three sides of a trian^e be taken as representing, in direction and in amount, three concurrent forces whose senses are such that arrows, representing them and affixed to their respective sides in the triangle, follow each other around it, then those forces are in equilibrium. 50. The three forcest Fig. 23, are proportional, respectively, to the sines of their opposite angles. Thus: Force a : force b : force e — Sin A : sin B : sin C. Fly. 2S. 51* Example. In Fig. 24, the half arch and its spandrel, acting as a nngle rigid bodv, are assumed to be held in equilibrium by their combined weight, W, the horizontal pressure h at the crown, and the reaction R of the skewback, which is assumed to act through the center of the skewback. In the force triangle c « t, e «, acting through the center of gravity of the half arch and spandrel, represents the known weight W, and 8 t ia drawn hori- sontal, or parallel to h . From c, where h, produced, meets the line of ac- tion of W, draw c t through the center of the skewback. Then • t and e I give us the amounts of h and R respectively. Fig. 24. Figr- 9Xi, 52. Example. Let Fig. 25 represent a roof truss, resting upon its abut- ments and carrying three loads, as shown by the arrows. Draw a R ver- tically, to represent the proportion of the loads carried by the left abut- ment, a, or, which is the same thing, the vertical upward reaction of that abutment. Then, drawing R c, parallel to the chord member, a <2, to inter- sect a 6 in c, we have, for the stresses in a e and a d, due to the three loads: Stress in a « s a e " od = Re It %4 ^'^bA 63. While any two or more given forces, as o 6 and h c. Fig. 26 (a) (arrows reversed), or o b' and b' c,oroa and a c, or o a' and o' c, can nave but one re- sultant c; a sinffle force, as o c. may be resolved into two or more concur- rent components in any desired directions. In other words, there is an infinite number of possible systems of concurrent forces which have o c for their resultant. SECTANOULAB COMPONENTS. 869 Bectangular Gomponenti. 54. ResoluteSt or Rectangular Components. A very common case of resolution of forces is that where a force, as the pressure, c n, of the post, fig. 27, is to be resolved into components at rieht angles to each other, as are the vertical and horizontal components c t and tn in Fig. 27 (a). Two such components, taken together, are called the resolutes or rectangular compo- nents of ibjb force. The joint, o d, in Fig. 27 (a), is properly placed at right angles to e n; but the joint c ib. Fig. 27 (5), provides also against accidental changes in the direction of c n. In Fig. 27 (6), the surfaces, c i and i b, are preferably {proportioned as the components, c i and t ih Fig. 27 (a), respec- tively, by simUarity of triangles, ctb, ctn^ Tig. 27. Fl«r. 28. 55* Example. In bridge and roof trusses it is often required to find the vertical and horizontal resolutes of the stress in an inclined member, or to find the stress brought ui>on an inclined member by a given vertical or hori- zontal stress applied at one of its ends, in conjunction with another stress (whose amount may or may not be given) at right angles to it. Thus, in Fig. 28, the tension C p in the diagonal C d is resolved into a com- pression e p along the upper chord member CD* and a compression C e in the • post Cc.*^ Addmg to C c the load at c, and representing their sum by / c, we nave tension f g in chord member e d, and tension c g in the diagonal B c. Making B A = c g,-we have i A, compression in B C, and B j, compression in the end-post or batter poet B A. But the load at b also sends to B, through the hip vertical B 6, a load (tension) equal to itself. Representing this by B ;fc, we have ( A; as its component along the chord member B O, and B I as its oom]M)nent along the end-post B A. Now, making A *» = the sum of B; and B /, we find the vertical resolute A » = so much of the vertical reaction of the abutment as is due to the three loads only, and the horizontal resolute mn '^ the corresponding stress in the chord member, A c. '\ >a Flff. 80. 56. Example. Inclined Plane. Again, in Fig. 29, let it be required to find the two resolutes of P (the weight of the ball) respectively parallel and perpendicular to the inclined plane. The former is the tendency of the ball to move down the plane, and is called the tangential component. The ^'Ilie stress^ thus found is not necessarily the total stress in the member. The compression in C c (neglecting its own weight and that of the top chord) fe due entirely to the tension C p in C <i, acting at its top, and hence C e rep- tmenta the total compression in C c; but e p ia only a portion of the com- pression sustained by C D ; for B C also contributes its share toward this. 24 btl«rut ofthebBUaCUDBt h« plui«, aod ia (wlled the noimal compon enl. Herew to draw the triftogl of for. OEOC-Pto JdiiMtkma. S£ Iho weight of the b •11. Uld » undo s^/F- ely the do mid wid the taaeeatia 87. If the ineli ed plane g m, Fig 29, to be fri«t o»le«, and if the body 018 to be prevented t rom sliding down 'o're^J^^'g."- umotafone ■.ppliedm&direotiDTi parsJIel to the plwi the pluu e. that thua, n Fig. 30, B ihe stoi be friotionlees, mhave a e ™ agamsl SS. Table, of a tordiffenot T>Ft. H«. i: i' 1: t Id Il.t \>»« Dl-K • Or * c. It both triangles are drawn, we have the foro trhs line a « (or c a) is called the prolecllon of o c ui BTBE8S CX>HFONENT8. 371 59. Equations. In Fig. 29. o a »- P . cos e o a a c ■" P . sin e o a and, since the angle eoa between the vertical o e and the normal component o a is equal to the angle A of inclination between the plane g m and the hori- sontal a n, we have : Normal component, o a » P . cos A. Tangential component, a c » P . sin A. 60. When a force is resolved into rectangular components, as in Figs. 29 and 30, each of these components represents the total effort or tendency which that force alone ean exert in that direction. FI9. 81. Thus, in Fig. 31, the utmost force which the weight o e alone can exert perpendicularly againat tke plane is that represented by the component o a. iVue, if, in order to prevent the bo<ly from sliding down the plane, we apply a force in some other direction, such as the horisontal one, h o, instead of the tangential one h o, and find the components of o c in the directions h o and o a, weuiall find the normal component o d greater than before; but the increase a d is due entirely to the normal component, h fr, of the horisontal force h o. Thus, the only effect upon the body o, and upon the plane, of substituting h o for b o, is to add the normal component, h 2>, of the former, to that (o a) doe. Stress Components. 61. Stress Components. In Fig. 32, let a o and & o be any two forces, and c o their resultant. From a and 6 draw a a' and h 6' at right angles to the diagonal o c of the force parallelogram a o b c^ and construct the sub-> parallelograms (rectangles), oa' a a" and oVh If'. Each of the original com- ponents, o a, o- h, is thus resolved into two sub-components, perpendicular to each other, one of which is perpendicular also to the resultant, o c, while the other coincides with o c in position and in sense. Now, perpendiculars, let fall from the opposite angles of a paralldogram upon its diagonal, are equaL .// (a) .// ,, (6) ^ / 4 < / f< "-, / 6 -X Flff. 32. Bence the two colinear forces, o a'\ and o 6", acting upon the body at o, are equal and opposite (although the lines, a' a and h' 6, representing them, are not opposite). Hence also they are in equilibrium, and their only effect upon the body is a stress of compression in Fig. 32 (a), and of tension in Fig. 32 (6). They may therefore be called the stress components. The other two sub-components (o a' of o a, and o 5' of o h) combine to form the resultant o e, which is equal to their sum, and which tends to move the body in its own dir&ction. 372 STATICS. 62. The two great forces, o a, ob, in Fig. 33 (6) have the same reeultant, oc, = o c', as the two small forces, o a' o b\ in Fig. 33 (a), although their stress components, a" a, = V b, are much greater. 63. It often happens that one of the components is itself normal to the resultant. Thus, in Fig. 22, where o c is vertical, its component, o a, is hori- zontal, and the perpendicular, let fall from a upon o c, represents its hori- zontal anti-component, a o. Here the horizontal and the inclined beam sustain equal horizontal pressures; but the vertical pressure, o c, "^ the weight, W, is borne entirely by the inclined beam. Flip. 33. Flip. 34. 64. When, as in Fig. 34, the resultant, o c, forms, with one of the original components, o a and o b, an angle, aoc, greater than 90^, the perpendicularB, a a', b 6', from a and &, must be let fall upon the line of the resiiltant produced. Here, however, as before, the two equal and opposite sub-components, o a" and o b"j are in equilibrium at o, while the other two sub-components, o b* and o a', go to make up the resultant o c; which, however (since o 6' and o a* here act in oppottte senses) is equal to their difference, and not to their sum, as in Fig. 32. Fig. 34 shows that a dowrvward force, o e, may be so resolved that one of its components is an upward force, o a, greater than the original downward force, and that the pressure, o 6, has a component, o b* or V &, parallel to o c, and greater than o c itself; for b" b — o 6' '^ o c -\- cV. Applied and Imparted Forces. 65. Applied and Imparted Forces* In Fig. 29, the ball is free to roll down the inclined plane. Hence, although the entire weight P of the ball is applied to the body g mn, only the normal component o a is imparted to it or exerts any pressure upon it, and this pressure is in tlie direction o a. But in Fig. 30, the body g mn ceeeives and resists not only the normal component o a, but also (by means of the stop «) the tanaential component o b; and the entire force P, or o c, is thus imparted to the body g mn, pres»- ing it in the direction o c. Comiposition and Kesolution of Concurrent Forces by Means of Co-ordinates. 66. In Fig. 35 (a) let the three coplanar forces E, F and G act through the point x. Draw two lines, H H, and V V, Fig. 35 (b), crossing each other at right angles, as at o.* These lines are called rectangular co-ordin- ates. From o, draw lines E o, F o, G o, parallel to E a:, F x, (jrx, Fig. 35 (o), and equal respectively to the forces E, F, and G by any convenient scale. Re- solve each of these forces, Fig. 35 (6), into two components, parallel to H H and V V respectively. Thus, E o is resolved into t o and n o, F o into u o and e o, G o into i o and m o. Then, summing up the resolutes, we have: Sum of horizontal resolutes = u o — io — to — — so, and Sum of vertical resolutes == no + e o — m,o — ao, ao; *It is only for convenience that the co-ordinates are usually drawn (as in Fig. 35) at right angles. They may be drawn at any other angle (see Fig. 36) ; but. in any case, the forces must of course be resolved into components EaraUel to the co-ordinaUa, whatever the directions of those co-ordinatee may e. COMPOSITION AND RESOLUTION. 373 and — 9 and a o are the resolutes of the resultant, R, of the three forces, E, F and G. 67. When a system of (concurrent) forces is in equilibrium, the algebraic* sum of the components of all the forces, along either of the two co-ordinates, is zero. Thus, in Fig. 35 (6) or 36, if the sense of R be such that it shall act as the anti-resultant of the other three forces E, F and G, its component, o « or o a, along either co-ordinate, will be found to balance those of the other forces along the same co-ordinate. Flff. 35. Henoe we have the very important proposition that : When a system of ooncurrent coplanar forces is in equilibrium, the algebraic sums of their com- ponents, in any two directions, are each equal to zero. Fig:. S6. 68. Conversely, in a system of concurrent forces, if the algebraic sums of the components in any two directions are each jequal to zero, the forces are in equilibrium. If the sum of the components in one of anv two directions is not equal to zero, the forces cannot be in equilibrium. Thus, in Fig. 35 (6) or 36 (b), the sum of the components, along either one (as VV) of the two co-ordinates, may be zero; and yet, if the sum of those along the other co-ordinate is not zero, their resultant, or algebraic sum, will move the body, on which they act, in the direction of that resultant. ♦The components being taken as + or — , according to the sense of each. 374 STATICS. 69. With Tertical and horizontal co-ordinates, the condition of equilibrium* becomes: . The sum of the horizontal resolutes must be equal to zero ; The sum of the vertical resolutes must be equal to sero; or, more briefly: 2 horizontal resolutes ■- 2 vertical resolutes ■» Conversely, if these conditions are fulfilled, the forces are in equilibrium. Tig. 37. Tig. 3S. Flip. 39. 70. Resultant of More than Two Coplanar Forces. Where it is required to find the resultant of more than two concurrent and coplanar forces, as in Fig. 37, we may first find the resultant Ri of any two of them, as of P] and Ps; then the resultant, R^, of Ri and a third force, as Pa; and so on, until we finally obtain the resultant R of all the forces. This resultant is evidently concurrent and coplanar with the given forces. 71. It is quite immaterial in what order the forces are taken. Thus, we may, as in Fig. 38, first combine Pi and Ps; then their resultant Ri with Ps, obtaining R2; and, finally, R^ with P4, obtaining R;or, as in Fig. 39, we may first combine any two of the forces, as Pi and Ps, obtaining their resultant Ri ; then proceed to any other two forces, as Ps and P4, and obtain their resultant R^; and finally combine the two resultants, Ri and R^, ob- taining the resultant R. The Force Polygon. 73. The Force Polygon. Comparing Figs. 37 and 38 with Figs. 40 and 41, respectively, we see that we may arrive at the same resultant R by simply drawing, as in Fig. 41, lines representing the several forces in any order, but following each other according to their senses. It will be noticed that this is merely an abbreviation of the process of drawing the several force parallelograms. 73. Resultant and Anti-resultant. The line, — R, required to com- plete the polygon, represents the an<i-resultant of the other forces if its sense IS such that it follows them around the polygon, as in Fig. 40. If its sense is opposed to theirs, as in Fig. 41, it is their reavUant, R. 74. In other words, if any number of concurrent forces, as Pj, Pj, Pj, P* and R, Figs. 37 and 38, f are in equilibrium, the lines representing them, if drawn in any order, but so that tneir senses follow each other, will form a closed F>olygon, as in Fig. 40 (or in Fig. 41 if the sense of R be reversed). 75. Conversely, if the lines representing any system of concurrent coplanar forces, when drawn with their senses following each other, form a closed polygon, as in Fig. 40, those forces are in equilibrium. *With non-concurrent forces, another condition must be satisfied. See ^ 83. tR is here regarded as tending upward, so as to form the anft-resultant of the other forces. FORCE POLYGON. 375 It will be noticed that the force triangle, and the straight line representing a system of colinear forces, Figs. 10 and 11. Hlf 20, etc., or a system of parallel forces, Figs. 55, etc., tf 111, etc., are merely special cases of the force polygon. 76. In a force polygon. Fig. 42, any one of the forces is the anti-resultant of all the rest. Any two or more of the forces balance all the rest ; or, their resultant is the anti-resultant of all the rest. If a line a c or 6 d, Fig. 42, be drawn, connecting any two comers of a force rig. 40. Fiff. 49. polygon, that line represents the resultant, or the anti-resultant (according i its arrow is drawn) of all the forces on either side of it. Thus : a c is the resultant of Pi Ps and the anti-resultant of P3 P4 Pg <5 a " " " Ps P4 Ps * " '* Pi Fa 6 rf " " " Ps Ps " " " P4 P6 Pi d b " •* " P4 Pft Pi " " " Pi P3 77* Knowing the directions of all the forces of a system, as Pi P5, Fig. 42, and the am&unta of all but two of them, as Ps and P3, we may find the amounts of those two by first drawing the others, P4, Ps and Pi, as in the figure. Then two lines b c and c d, drawn in the directions of the other two and dosing the polygon, will necessarily give their amounts. Tig. 48. Tig. 44. 78. If any two points, as o and c. Fig. 43. be taken, then the force or forces represented by any line or system of lines joining those two points will be equivalent to o c. Thus :oe''oabc'^ode^onpc''ohkmc = on mc '^ o fc " o gc, etc., etc. Similarly, in Fig. 42, the force polygon abe deais equivalent to the force polygon ab fdea, and to the force triangle, abca, eacn being = zero. • Non-concuirent Coplanar Forces. 79. Non-concurrent Coplanar Forces. Fig. 44. The process of finding the resultant of three or more coplanar but non-concurrent forces is the same as if they were concurrent. Thus, let Pi, Ps and Ps represent three sueh forces.* We may first find the resultant Ri of any two of them, as Ps *Apy two coplanar non-parallel forces, as P; and P2, or P^ and Ps are necessarily concurrent (see % 19); but there is no single pomt in which the three forces meet. 376 STATICS. and P3; and then, by combining Ri with the remaining force Pi, we find the resultant R of the three forces. Here the line R represents the resultant, not only in amount and in direction, but also in position. That ls, the line of action of the resultant coincides with R. 80. The resultant R is the same, in amount and in direction, as if the forces were concurrent, and its position is the same as it would have been if their point of concurrence were m the line of R. If there are more than three forces, we proceed in the same waj'. 81. Conversely, the resultant R, or any other force, may be resolved into a system of any number of concurrent or nonconcurrent coplanar forces, in any direction^, at pleasure. Thus, we may first resolve R into Pi and Ri; then either of these into two other forces, as Ri into P2 and P3, and so on. 83. If a system of non-concurrent coplanar forces is in equilibrium, the forces will still be in equilibrium if they are so placed as to be concurrent; provided, of course, that their directions, senses and amounts remain un- changed ; but it does not follow that a system of forces, whicl> is in equilib- rium when concurrent, will remain in equilibrium when so placed as to be non-concurrent. Thus, the five forces, Pt Pr„ Fig. 45 (a), may be so placed, as in Fig. 45 (6), that the resultant a c, of Pi and Pa, does not coincide with the re- sultant c a of P3, P4 and Ps. but is panUlel to it. These two resultants -then form a couple. (See tlf 155, etc.) Fig. 45. 83. Third Condition of Equilibrium. Hence, equilibrium for concurrent forces, stated in \ 69, the oondHiona of 2 2 vertical horizontal components ■■ components -= do not suffice for non-concurrent forces, and a third condition must be added, viz. : — 2 moments « 0; t. e., the moments of the forces, taken about any point, must be in equilib- rium. A system of forces in equilibrium has no resultant ; hence it has no moment about any point. In other words, the moments of the forces, as well as the forces themselves, are in equilibrium. 84. The resultant of a system of unbalanced non-concuireiit forces, acting upon a body, may be either (1) a single force, acting through the center of gravity of the body; or (2) a couple; t. e., two equal and parallel forces of opposite sense (see m 155, etc.) ; or (3) either (a) a single force, acting through the center of gravity of the body, and a couple ; or (b) a single forca acting elsewhera th»r throu^k ( he center of gravity of the body. ^ In Case (3), the two alternative resultants are interchangeable; t. e.. a single force, acting elsewhere than through the center of gravity of the body, may always be replaced by an equivalent combination consisting of an eqijuu CORD POLYGON. 377 parallel force*, acting through the center of gravity of the body, and a couple, and vice versa. See HI 161', etc. The resultant gives to the body, in Case (1), motion of translation in a straight line, without rotation; in Case (2), rotation without translation; and m Case (3), both translation and rotation. See foot-note (*), t !• 85. The force polygon, ^ 72, Figs. 40, etc., and the method by co- ordinates. H 66, Fig. 35, therefore, give us only the amount, direction and sense of the resultant of non-ooncurrent forces, and not its position. To find the position of the resultant of non-concurrent forces, we may have recourse to a figure, like Fig. 44. where the forces are represented ib their actual posi- tions, or to the cord polygon, H1[ 86, etc., Fig. 46. The Cord Polygon. 86. In the force triangle any two of the three lines may be regarded as representing, by their directions, the positions of two members (two struts or two ties, or one strut and one tie) of indefinite length, resisting the third force ; while their lengths give the amounts of the forces which those mem- bers must exert in oTaer to maintain equilibrium. FlfT* 20 (repeatefl). 87. Thus, in Fig. 26 (6), are shown four different systems, of two mem- bers each, inclined respectively like the forces c h and b o in Fig. 26 (a) and balancing the third force o c. The stresses in these two members are given by the lengths of the lines e b and b o in Fig. 26 (a). Tlie members acting as struts are represented, in Fig. 20 (b), as abutting against flat surfaces, while those acting as ties are represented as attached to hooks, against which they pull. In Fig. 26 (c) and (d) are indicated systems of members, inclined like the forces c a' and a' o, ca and a o, respectively, of Fig. 26 (a), by which the third force o c might be supported. 88. In the force polygon abed ea. Fig. 46 (6), representing the four forces, Pi, Ps, P3, Pj, of Fig. 46 (a), if we select, at pleasiue, any point o (called the pole) and draw from it a series of straight lines oa,ob, etc. (called mys), radiatinff to the ends, a, b, c, etc., of the lines Pi, Ps, etc., representing the forces, we snatl form a series of force triangles, aobthoc, etc. Thus, in the triangle d b o we have the force Pi, or a b, balanced by the two forces o a and b o; m the triangle b c o, the force P2, or b c, balanced by the two forces o b and c o; and so on. 89. The Cord Polygon. If, now, in Fig. 46 (a), we draw the lines a and b, parallel respectively to the rays o a and o b of Fig. 46 (b) and meeting in the une representing the force Pi, they will represent the positions of two tension members of indefinite length, which will balance the force Pi by ex- erting forces represented, in amount as well as in direction, by the rayS a and b o, Fi^. 46 (b). Again, taking pol? o'. Fig. 46 (b), instead of o, we have a' and b*. Fig. 46 (aO, parallel respectively to the rays, o' a and 0' b, and rep- resenting a pair of struts performing the same duty. 90. Similarly, the lines b and e. Fig. 46 (a), parallel respectively to rays o h and o c, represent two tension members, which, with stresses equal respec- tively to o b and c 0, Fig. 46 (b), balance the force Pg. 378 STATICS. 01. We thus obtain, finally, a system of five tension members, ab e de. Fig. 46 (a), which, if properly fastened at the ends a and e respectively, will^ by exerting forces represented respectively by the rays, o a, ob, oc, etc.. Fig. 46 (6), balance the four given forces Pi, P{, Ps and P4. 92. The figure abode. Fig. 46 (a), is called a cord polygon, funicular polygon, or equilibrium polygon. 03. Resultant, Anti-resultant. Amount and Direction. In the force polygon, Fig. 46 (6) or (d), the line e a, joining the end of the last force- line d e with the beginning of the first one a b, represents the anti-resultant of the given system of four forces, and a e their resultant. Evidently, there- fore, the rays^ a o and o e, which represent two components of a e, represent also, in direction and in amount, two forces which would balance e a, or which would be equivalent to the given system of (four) forces. Flffs. 46 (a), (a') and (fr). 04. Position of Resultant. Hence, in the cord |>olygon. Fig. 46 (a)^ the intersection, i, of the cords a and e, parallel respectiv^y to the rays o a and e o, is a point in the line of action of the resultant R; and. if we imasine a i and e i to be rigid rods, and apply, at t, a force, — R, equal and parallel to a e, but of opposite sense, that force will be the anti-resultant of the (four) given forces, and we shall have a frame-work be di of cords and rods, kept in equilibrium by the action of the five forces, Pi, Pg, Pg, P4 and — R. 06. By choosing other positions of the pole, as o\ Fig. 46 (fi), or by differ* ently arranging the given forces, as in Fig. 46 (c), we merely change the shape of the cord polygon, and (in some cases) reverse the sense of the stresses in the members. Thus, in Fig. 46 (a), all the stresses are tensions, or pulls : while in Fig. 46 (c) a, b, d and e are tensions or pulls, and c is a com- pression or push. 06. In constructing the cord polygon, Fig. 46 (a), (aO. (c), and (e), car* must be taken to draw the cords m their proper places ; and for this it is neo- essary to remember, simply, that the two rays pertaining to any particular force line in the force polygon. Fig. 46 (6), represent those members which, in the cord polygon. Fig. 46 (a), take the components of that force. CORD POLYGON. 379 Thus, o a and h o» Fig. 46 (6), pertain to the force Pi ; o b and e o to the force Pj. Hence, in Fig. 46 (a") or {c) we draw a and h (parallel respectively to o a and 6 o) meeting in the line of action of Pi : h and c (parallel respect- ively to o 6 and c o) meeting in the line of action of Ps, etc., etc. 97. Each ray in the force polygon. Fig. 46 (6), including the outside ones, is thus seen to pertain to two force?, and each force has two rays. The two oords,^ parallel respectively to the two rays of any force, must be drawn to meet in the line of^aetion of that force; and each cord must join the lines of action of the two forces to which its parallel ray pertains. The lines, a, &, c. etc., in the cord polygon. Fig. 46 (a) and (c), give merely the incLinatwM oi members which, as there arran^d, would sustain the given forces. ' The lengths of these lines have nothing to do with the amounts of the stretaes. These are given by the lengths of the corresponding raya in the force polygon, Fig. 46 (6). Flffs. 46 (o), {d) and («). 08« If the anti-resultant force, — R, is not applied, the cords a and e may be supposed fastened to firm supports, against which they exert stresses rep- resented, in amount and in direction, by the rays a o and o e respectively. But the resistances of those two supports are plainly equal and opposite to those stresses, or equal to o a and e o respectively. Hence, their resultant is the anti-resultant, — R, of the foiu> origmal forces. 99* If, Fig. 46 («), the two end members a and e were attached merely to two ties, V and V, parallel to the anti-resultant, — R, they would evidently draw the ends of those ties inward toward each other. To prevent this, let the strut k be inserted, making it of such length that the ties V and V may remain parallel to — R, and draw o k, Fig. 46 (6), parallel to k. Then a k and k e give the stresses in V and V respectively. ipO. If the anti-resultant, — R, found by means of the force pply^n, be applied in a line passing through the intersection of the outer (initial and final) members in the cord polvgon, all the forces, includinff of course the aati-resuliant, will be in equilibrium. In other words, coplanar forces are in equilibrium if they may be so drawn as to form a dosed force polygon, and if a closed cord polygon may be drawn between them. But if the anti-re- soltant be applied elsewhere, we shall have a couple, composed of the anti- rwnltant, — K, and the resultant R of the forces. 380 STATICS. Concurrent Xon-eoplanar Forces. . 101. Any two of the concurrent forces, as o o and o c. Fig. 47 (a) or (6), are necessarily coplanar. Find their resultant, o r, which must be coplanar with them and witn a third force o h. Then the resultant, R, of o r and o 6 is the resultant of the three forces. If there are other forces, proceed in the same way. 102. No three non-coplanar forces, whether concurrent or not, can be in equilibrium. 103. Force Parallelopiped. The resultant of any three concurrent non-4ioplanar forces, o a, o\ o c. Figs. 47, will be represented by the diagonal a R, of a parallelopiped, of which three converging edges represent the three forces. 104. Methods by Models, (a) For three forces. , Construct a box, Kg. 47 (a) or (6), with three conver^nt edges representing the three- forces in position and amount. Then a stryig o R, joining the proper corners, will represent the resultant. Fig. 47. Or, let ao,ho, c o. Fig. 48 (o), be three forces, meeting at o, "DnM on pasteboard the three forces a o, b o, e o, as in Fig. 48 (6), with their actual angles aob, boc, coa, and find the resultant wooi the middle pair, b o and c o. Cut out neatly the whole figure, a o a c w b a. Make deep knife- scratches along o 6, o c, so that the two outer triangles may be more readily turned at angles to the middle one. Turn them until the two edges o ci^oa meet, and then paste a piece of thin paper along the meeting joint to keep \ \ / w («) (ft) Fig:. 48. («) them in place. Stand the model upon its side o & tp c as a base, and we aball have the slipper shape a ob w. Fig. 48 ic)\o w being the sole, and aob the hollow foot. In the model, the force a o and the resultant to o of the other two forces, are now in their actual relative positions. To find their resultant, cut out a separate piece of pasteboard, R a o to, with R a and R w parallel respectively to w o and a o. Draw upon each side of it the diagonal R o. Paste this piece inside the model, with its lower edge tt; o on the line to o. Fig. 48 (6), and its edge a o in the comer a o. This done, R o represents the re« sultant oia o,b o, c o, Fig. 48 (a), in its actual position relative to them. 105. (b) For four forces, aaa o,bo,co,d o, in Fig. 49. Draw them as in Fig. 40 (a), with their angles aob, boc, etc. Draw also the resultants « o, of c o and b o; and wo,oico and d o. Then out out the entire figiire, as before, and paste together the two edges a o, a o. Hold the model in such a way that two of its jylanea (as a o 6 and boc) form the same angle with each other NON-COPLAXAR FOBCES. 881 as do the two corresponding; planes between the forces. Then we have the two resultants vo^wo, Fig. 49 (6), in their ctctiMl relative poeitiona. Cut out a separate piece of pasteboard R v o w, Fig. 40 (&), draw the diagonal R o on each side of it, and paste it inside the model, with o v and o to on the oorre* sponding lixteB of the model. Then R o will represent the resultant of the four forces, ao^bo,cOtdo, in its actiial position relative to them. The model may be made ol wood, the triangles aobth oc, etc., being cut out separately, the joining edges bevelled, and then glued to«^ther. («) FUr. 49. (*) Non-concurrent Non-coplanar Forces. lOG. Non-concurrent Non-coplanar Forces. Fig. 50 (a). (For par- allel non-coplanar forces, see ^'^ 110, etc.) Resolve each force mto two rec- tangular components, one normal to an assumed plane, the other coin- ciding with the plane.* Find the resultant of the (coplanar) components coinciding with the plane, by methods already given, and that of the normal (parallel) components, by 1ft 110, etc. If these two resultants are coplanar, tney are also concurrent, and their resultant (which is the resultant of the system) is readily found. 107. If not, let V, Fig. 50 (6), be the resultant normal to the plane, and H the resultant lying in tl^e plane. By If 162, substitute, for H, the eqtial and parallel force H', meeting V at O, and the couple H . O a, and find the result- ant, R', of V and H'. The system of forces is thus reduced to the single force R' and the couple H . O a. For Couples, see If 155. 108. Moments of Non-coplanar Forces. Th<% action of the weight W of the wall. Fig. 51 (a), and of the non-coplanar forces Pi and Pe, may be represented as in Fig. 51 (&), where the axle a* cf represents the edge a c about which the wall tends to turn, while the bars or levers represent the leverages of the forces. So far as regards the overturning stability of the wall, regarded as a rigid body and as capable of turning only about the edge a e, it is immaterial whether an extraneous force, as Pi, is applied at p or at g; but it is plainly not immaterial as regards a tendency to swing the wall around horizontally, or to fracture it; or as regards pressures (and conse- quent friction) between the axle a' <f and its bearings. For equilibrium. Pi vik ■- Pc A + W. — . Here a torsional or twisting stress is exerted in the axle. *Wires, stuck in a board representing the plane, will facilitate this. 382 STATICS. and the presBures of its ends in the bearings are more or less modified ; bui, so far as merely the equilibrium of the moments is oonoerned, we may sup- pose all of the forces and their moments to be shifted into one and the same plane, as in Fig. 51 (c). 109* In oases like that represented in Fig. 51, it is usual, for convenience, to restrict ourselves to a supposed vertical alice, «, 1 foot thick, and to the forces acting upon such slice ; supposing the weight of the slice to be concen- trated at its center of cavity, and the extraneous forces to be applied in the same vertical plane with gravity. In eflfect, we are then dealing with a slice indefinitely thin, but luiving the weight of the 1-ft. slice. Flff. 51. PARALLEL FORCES. 110. The resultant of any number of parallel forces, whether they are in the same plane or not, and whether in the same direction or not, is parallel to them and — their algebraic sum. Coplanar Parallel Forces. 111. The resultant of any number of coplanar parallel forces is in the same plane with them, whether the forces are of the same or of opposite sense; and the leverages, or arms, of such forces, and of their resultant, about any given point in the same plane, are in one straight line. Thus, in Fig. 56 (a), where the five forces, a, b, c, d and e are in one plane, their resultant, R, is in that same plane; and tne levera^ of the forces, and of R, about any point, as 6 or v, in the same plane, are in the straight line R v. Fig. 02. 113. The resultant, R, or anti-resultant, Q, Fig. 52, of two parallel forces, a and b, intersects any straight line, u v, joining the directions of the two forces. Hence, if three parallel forces are in equilibrium, they ara m the same plane. In Fig. 62 (a), the two forces, o and 6, are of like sense. R is then between a and b, and R = 6 -H a. In Fig. 52 (6), a and b are of opposite sense. R is then not between a and 6, and R — fr — a. PABALL£1< FORCES. 383 113* To find the position of the resultant, draw and measure any straight line* u V, joininjs the lines of action of the forces. It is immaterieil whether u « is perpendicular to said directions, or not. The line representing the resultant cuts u v, and its position is found thus: M i — tt « X -p- ; and v i = u v X -^. FliT* 93. *-i 114. This may be conveniently done by making u v equal, by any conve- nient scale, to the sum of the forces, as in Fig. 53, where uv^ 42. Then make u i equal, by the same scale, to the force at v,oxvi equal to the force at u. Then a line, R, Fig. 52 (a), drawn through t parallel to a and h, gives the position and direction of their resultant ; and its amount is equal to the sum of a and h; or R =- a + 6. In other words, if a force, Q, parallel to a and 6, and equal to their sum, but of opposite sense, be applied to the body any- where in a line passing through i, it will balance a and 6, or will be their anti- resultant. \~ — .-.^1^ /^ I x^l y Ftgr. 55. (ft) 115. The position of the resultant, so found, satisfies the condition of equilibrium of moments : thus, h.vi — a.ui « zero. If the two forces are equa